문제 정의: 데이터 집약적 애플리케이션의 증가로 인메모리 컴퓨팅에 대한 관심이 증대되었으며, 전통적인 2D 크로스바 설계는 저항 및 커패시턴스 기생 요소로 인해 성능 한계에 직면하고 있다.
목표: Boolean 함수를 3D 나노 크로스바 설계로 자동 합성하는 첫 번째 프레임워크인 FLOW-3D를 제안하여, 반둘레(semiperimeter)를 최소화하고, 면적, 에너지 소비, 지연 시간 등의 측면에서 기존 2D 도구보다 우수한 성능을 달성하는 것이 목적이다.
연구 방법
기본 아이디어 및 문제 정의
Boolean 함수의 합성을 위해 BDD(Binary Decision Diagram)와 3D 크로스바 사이의 유사성을 활용.
BDD의 노드와 에지에 해당하는 3D 크로스바의 금속 와이어와 멤리스터를 적절히 매핑하는 문제를 “L-labeling 문제”로 정의하고, 이를 ILP(정수 선형 계획법)로 최적 해결한다.
L-labeling 단계: 각 노드에 대해 할당 가능한 금속 층의 범위를 결정하고, 인접 층 간의 연결 제약(에지 제약 및 노드 제약)을 만족하도록 레이블링 수행.
크로스바 할당: 레이블링 결과를 바탕으로 실제 3D 크로스바 구조를 구성하여 Boolean 함수를 구현하는 하드웨어 디자인을 도출.
성능 평가
제안된 FLOW-3D 프레임워크는 2D 크로스바 기반의 기존 합성 도구와 비교하여, 반둘레, 면적, 에너지 소비, 지연 시간에서 각각 최대 61%, 84%, 37%, 41%의 개선 효과를 보임.
RevLib 벤치마크를 통해 실험적으로 평가되었으며, 3D 크로스바 설계의 효율성과 성능 향상을 입증하였다.
주요 결과
자동 합성 도구 제안: Boolean 함수를 3D 크로스바 설계로 자동 합성하는 최초의 프레임워크를 제안.
최적화 성능: FLOW-3D는 ILP 기반 L-labeling 문제 해결을 통해 3D 크로스바의 반둘레를 최소화하고, 면적 및 전력 소비를 현저히 감소시킴.
비교 평가: 기존 2D 기반 합성 도구 대비, 제안된 프레임워크는 에너지 효율과 응답 속도 면에서 우수한 성능을 나타냄.
결론 및 향후 연구
제안된 FLOW-3D 프레임워크는 3D 나노 크로스바를 이용한 흐름 기반 컴퓨팅에서 Boolean 함수 합성을 효율적으로 수행할 수 있음을 입증.
향후 연구에서는 더 복잡한 회로 및 대규모 데이터셋에 대한 확장성과, 다양한 하드웨어 제약 조건을 고려한 추가 최적화 기법이 연구될 필요가 있다.
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Culvert Outlet Scouring의 영향 매개변수 예측 및 최적화: FLOW-3D와 서로게이트 모델링을 활용한 연구
연구 배경
문제 정의: 박스형 수로(culvert) 출구에서 발생하는 침식(scouring)은 구조물 설계에 중요한 영향을 미친다.
목표: 침식 깊이와 위치를 예측하여 구조적 실패를 방지하고, 설계를 최적화하는 새로운 방법론을 제안한다.
접근법: 수치 모델링(FLOW-3D)과 Box-Behnken 설계 기법을 이용한 서로게이트 모델링을 결합.
연구 방법
FLOW-3D:
Reynolds 평균 Navier-Stokes 방정식을 기반으로 유체 흐름 시뮬레이션을 수행.
침식 예측을 위해 RNG 난류 모델을 사용.
Box-Behnken 설계:
세 가지 주요 변수: 유량(Flow Discharge, QQQ), 수로 기울기(Slope, SSS), 토양 입자 크기(d50d_{50}d50).
총 15개 모델을 통해 변수와 침식 깊이 및 위치 간 상호작용 분석.
민감도 분석:
각 변수의 변화가 결과(침식 깊이와 위치)에 미치는 영향을 정량화.
최적화:
침식 깊이 및 위치를 최소화하거나 최대화하기 위한 설계 변수의 조합 도출.
주요 결과
모델 성능:
침식 깊이 예측 정확도: R2=0.931R^2 = 0.931R2=0.931
침식 위치 예측 정확도: R2=0.969R^2 = 0.969R2=0.969
민감도 분석:
유량 증가: 침식 깊이와 위치에 선형적(또는 비선형적) 영향을 미침.
기울기 증가: 일정한 비선형 패턴 관찰.
토양 입자 크기 증가: 복잡하고 비선형적인 패턴 확인.
최적 설계:
침식 깊이 최소화: 유량과 토양 입자 크기를 낮게, 기울기를 높게 설정.
침식 위치 최대화: 유량, 토양 입자 크기, 기울기의 조합을 조절.
결론
FLOW-3D와 서로게이트 모델링: 침식 예측과 최적화에 효과적인 도구로 확인.
설계 최적화 가능성: 구조적 침식 문제를 예방하기 위해 설계 단계에서 주요 변수의 영향을 정밀히 평가.
향후 연구 제안: 추가적인 변수 도입 및 데이터를 통한 모델 개선.
이 논문은 수치 해석과 통계적 설계 접근법을 결합하여 수로 설계 문제를 해결하는 새로운 방법론을 제시하며, 향후 관련 연구에 중요한 기초 자료를 제공할 수 있습니다.
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날카로운 정상부를 가진 삼각형 허들(Sharp-Crested Triangular Hump) 위의 유동 특성 수치 모델링
연구 배경
문제 정의: 수리 구조물의 성능 및 수면 프로파일을 정확히 예측하는 것은 실험적으로 어렵고 비용이 많이 듦.
목표: CFD(Computational Fluid Dynamics)를 활용하여 삼각형 허들 위의 유동 특성을 보다 효율적이고 정확하게 분석.
접근법: FLOW-3D 기반 시뮬레이션을 수행하여 실험 데이터와 비교 검증.
연구 방법
삼각형 허들(Weir) 개요
위어(Weir)는 개수로에서 유량 조절과 방류 역할을 수행하는 중요한 수리 구조물.
본 연구에서는 크기가 50 cm × 30 cm × 7 cm인 Sharp-Crested Triangular Hump 모델을 사용.
수치 모델링
FLOW-3D를 사용하여 RANS(Reynolds-Averaged Navier-Stokes) 방정식과 VOF(Volume of Fluid) 방법을 적용.
FAVOR(Fractional Area-Volume Obstacle Representation) 기법을 사용하여 메쉬 내 장애물 영향을 반영.
총 1,920,000개의 격자 셀을 사용하여 시뮬레이션 수행.
실험 설정
Universiti Teknologi PETRONAS(UTP)의 수리 실험실에서 실험 수행.
30cm 폭, 60cm 높이, 10m 길이의 플룸(flume)에서 실험 진행.
4가지 유량 조건(30, 51.3, 75.3, 31 m³/h) 및 경사 조건(0, 0.006, 0.01)으로 실험 설계.
주요 결과
수치 시뮬레이션 vs 실험 데이터 비교
수치 시뮬레이션과 실험 결과 간의 차이는 4~5% 이내로 매우 높은 정확도를 보임.
수면 프로파일, 평균 유속, 프로우드 수(Froude Number) 등이 실험과 잘 일치.
유동 특성 분석
프라우드 수(Froude Number) 변화:
상류(Upstream)에서는 Froude Number < 1.0 → 서브크리티컬(Subcritical) 흐름.
하류(Downstream)에서는 Froude Number > 1.0 → 슈퍼크리티컬(Supercritical) 흐름.
유속(Flow Velocity) 변화:
하류로 갈수록 유속 증가, 삼각형 허들이 흐름을 방해하면서 압력 변화를 유발.
수심(Flow Depth) 변화:
상류에서는 높은 수심 유지, 하류에서는 급격한 감소 확인.
수치 시뮬레이션의 유용성
FLOW-3D가 삼각형 허들 및 수리 구조물의 유동 해석에 효과적임을 확인.
기존의 실험적 접근보다 비용이 낮고 신속한 설계 검토 가능.
결론 및 향후 연구
FLOW-3D 기반 CFD 시뮬레이션이 삼각형 허들의 유동 해석 및 설계 최적화에 효과적임을 검증.
실험 데이터와 비교했을 때 높은 정확도(오차 4~5%)를 나타내며, 초기 설계 검토에 유용함.
향후 연구에서는 다양한 난류 모델(k-ε, RNG, LES) 적용 및 추가적인 수리 구조물 연구가 필요.
연구의 의의
이 연구는 수리 구조물의 유동 해석을 위해 CFD 시뮬레이션을 실험적으로 검증하여, 위어 및 삼각형 허들 설계의 최적화 및 성능 예측을 위한 신뢰성 높은 방법론을 제시했다는 점에서 큰 의미가 있다.
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레이저 빔 형상이 용접 품질과 금속 혼합에 중요한 역할을 하며, 적절한 코어-링 직경 비율 설정이 필요.
향후 연구에서는 더 다양한 빔 형상과 용접 속도 조건을 고려하여 최적 설계 도출.
연구의 의의
이 연구는 CFD 기반 다중 물리 모델링을 활용하여 레이저 빔 형상의 금속 혼합 및 용접 품질에 미치는 영향을 체계적으로 분석하였으며, EV 배터리 제조에서 신뢰성 높은 용접 기술 개발을 위한 기초 데이터를 제공한다.
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지표 산사태로 발생한 파랑의 3차원 시뮬레이션: OpenFOAM과 FLOW-3D HYDRO 모델 비교
Ramtin Sabeti, Mohammad Heidarzadeh, Alessandro Romano, Gabriel Barajas Ojeda & Javier L. Lara
Abstract
The recent destructive landslide tsunamis, such as the 2018 Anak Krakatau event, were fresh reminders for developing validated three-dimensional numerical tools to accurately model landslide tsunamis and to predict their hazards. In this study, we perform Three-dimensional physical modelling of waves generated by subaerial solid-block landslides, and use the data to validate two numerical models: the commercial software FLOW-3D HYDRO and the open-source OpenFOAM package. These models are key representatives of the primary types of modelling tools—commercial and open-source—utilized by scientists and engineers in the field. This research is among a few studies on 3D physical and numerical models for landslide-generated waves, and it is the first time that the aforementioned two models are systematically compared. We show that the two models accurately reproduce the physical experiments and give similar performances in modelling landslide-generated waves. However, they apply different approaches, mechanisms and calibrations to deliver the tasks. It is found that the results of the two models are deviated by approximately 10% from one another. This guide helps engineers and scientists implement, calibrate, and validate these models for landslide-generated waves. The validity of this research is confined to solid-block subaerial landslides and their impact in the near-field zone.
1 Introduction and Literature Review
Subaerial landslide-generated waves represent major threats to coastal areas and have resulted in destruction and casualties in several locations worldwide (Heller et al., 2016; Paris et al., 2021). Interest in landslide-generated tsunamis has risen in the last decade due to a number of devastating events, especially after the December 2018 Anak Krakatau tsunami which left a death toll of more than 450 people (Grilli et al., 2021; Heidarzadeh et al., 2020a). Another significant subaerial landslide tsunami occurred on 16 October 1963 in Vajont dam reservoir (Northern Italy), when an impulsive landslide-generated wave overtopped the dam, killing more than 2000 people (Heller & Spinneken, 2013; Panizzo et al., 2005). The largest tsunami run-up (524 m) was recorded in Lituya Bay landslide tsunami event in 1958 where it killed five people (Fritz et al., 2009).
To achieve a better understanding of subaerial landslide tsunamis, laboratory experiments have been performed using two- and three-dimensional (2D, 3D) set-ups (Bellotti & Romano, 2017; Di Risio et al., 2009; Fritz et al., 2004; Romano et al., 2013; Sabeti & Heidarzadeh, 2022a). Results of physical models are essential to shed light on the nonlinear physical phenomena involved. Furthermore, they can be used to validate numerical models (Fritz et al., 2009; Grilli & Watts, 2005; Liu et al., 2005; Takabatake et al., 2022). However, the complementary development of numerical tools for modelling of landslide-generated waves is inevitable, as these models could be employed to accelerate understanding the nature of the processes involved and predict the detailed outcomes in specific areas (Cremonesi et al. 2011). Due to the high flexibility of numerical models and their low costs in comparison to physical models, validated numerical models can be used to replicate actual events at a fair cost and time (e.g., Cecioni et al., 2011; Grilli et al., 2017; Heidarzadeh et al., 2020b, 2022; Horrillo et al., 2013; Liu et al., 2005; Løvholt et al., 2005; Lynett & Liu, 2005).
Table 1 lists some of the existing numerical models for landslide tsunamis although the list is not exhaustive. Traditionally, Boussinesq-type models, and Shallow water equations have been used to simulate landslide tsunamis, among which are TWO-LAYER (Imamura and Imteaz,1995), LS3D (Ataie-Ashtiani & Najafi Jilani, 2007), GLOBOUSS (Løvholt et al., 2017), and BOUSSCLAW (Kim et al., 2017). Numerical models that solve Navier–Stokes equations showed good capability and reliability to simulate subaerial landslide-generated waves (Biscarini, 2010). Considering the high computational cost of solving the full version of Navier–Stokes equations, a set of methods such as RANS (Reynolds-averaged Navier–Stokes equations) are employed by some existing numerical models (Table 1), which provide an approximate averaged solution to the Navier–Stokes equations in combination with turbulent models (e.g., k–ε, k–ω). Multiphase flow models were used to simulate the complex dynamics of landslide-generated waves, including scenarios where the landslide mass is treated as granular material, as in the work by Lee and Huang (2021), or as a solid block (Abadie et al., 2010). Among the models listed in Table 1, FLOW-3D HYDRO and OpenFOAM solve Navier–Stokes equations with different approaches (e.g., solving the RANS by IHFOAM) (Paris et al., 2021; Rauter et al., 2022). They both offer a wide range of turbulent models (e.g., Large Eddy Simulation—LES, k–ε, k–ω model with Renormalization Group—RNG), and they both use the VOF (Volume of Fluid) method to track the water surface elevation. These similarities are one of the motivations of this study to compare the performance of these two models. Details of governing equations and numerical schemes are discussed in the following.
Numerical models
Approach
Developer
FLOW-3D HYDRO
This CFD package solves Navier–Stokes equations using finite-difference and finite volume approximations, along with Volume of Fluid (VOF) method for tracking the free surface
Flow Science, Inc. (https://www.flow3d.com/)
MIKE 21
This model is based on the numerical solution of 2D and 3D incompressible RANS equations subject to the assumptions of Boussinesq and hydrostatic pressure
Danish Hydraulic Institute (DHI) (https://www.mikepoweredbydhi.com/products/mike-21-3)
OpenFOAM (IHFOAM solver)
IHFOAM is a newly developed 3D numerical two-phase flow solver. Its core is based on OpenFOAM®. IHFOAM can also solve two-phase flow within porous media using RANS/VARANS equations
IHCantabria research institute (https://ihfoam.ihcantabria.com/)
NHWAVE
NHWAVE is a 3D shock-capturing non-Hydrostatic model which solves the incompressible Navier–Stokes equations in terrain and surface-following sigma coordinates
Kirby et al. (2022) (https://sites.google.com/site/gangfma/nhwave, https://github.com/JimKirby/NHWAVE)
GLOBOUSS
GloBouss is a depth-averaged model based on the standard Boussinesq equations including higher order dispersion terms, Coriolis terms, and numerical hydrostatic correction terms
Løvholt et al. (2022) (https://www.duo.uio.no/handle/10852/10184)
BOUSSCLAW
BoussClaw is a new hybrid Boussinesq type model which is an extension of the GeoClaw model. It employs a hybrid of finite volume and finite difference methods to solve Boussinesq equations
Clawpack Development Team (http://www.clawpack.org/)Kim et al. (2017)
THETIS-MUI
THETIS is a multi-fluid Navier–Stokes solver which can be considered a one-fluid model as only one velocity is defined at each point of the mesh and there is no mixing between the three considered fluids (water, air, and slide). It applies VOF method
TREFLE department of the I2M Laboratory at Bordeaux, France (https://www.i2m.u-bordeaux.fr/en)
LS3D
A 2D depth-integrated numerical model which applies a fourth-order Boussinesq approximation for an arbitrary time-variable bottom boundary
Ataie-Ashtiani and Najafi Jilani (2007)
LYNETT- Mild-Slope Equation (MSE)
MSE is a depth-integrated version of the Laplace equation operating under the assumption of inviscid flow and mildly varying bottom slopes
A simplified 3D Navier–Stokes model for two fluids (water and landslide material) using VOF for tracking of water surface
Horrillo et al. (2013)Kim et al. (2020)
(Cornell Multi-grid Coupled Tsunami Mode (COMCOT)
COMCOT adopts explicit staggered leap-frog finite difference schemes to solve Shallow Water Equations in both Spherical and Cartesian Coordinates
Liu et al. (1998); Wang and Liu (2006)
TWO-LAYER
A mathematical model for a two-layer flow along a non-horizontal bottom. Conservation of mass and momentum equations are depth integrated in each layer, and nonlinear kinematic and dynamic conditions are specified at the free surface and at the interface between fluids
Imamura and Imteaz (1995)
Table 1 Some of the existing numerical models for simulating landslide-generated waves
In this work, we apply two Computational Fluid Dynamic (CFD) frameworks, FLOW-3D HYDRO, and OpenFOAM to simulate waves generated by solid-block subaerial landslides in a 3D set-up. We calibrate and validate both numerical models using our physical experiments in a 3D wave tank and compare the performances of these models systematically. These two numerical models are selected among the existing CFD solvers because they have been reported to provide valuable insights into landslide-generated waves (Kim et al., 2020; Romano et al., 2020a, b ; Sabeti & Heidarzadeh, 2022a). As there is no study to compare the performances of these two models (FLOW-3D HYDRO and OpenFOAM) with each other in reproducing landslide-generated waves, this study is conducted to offer such a comparison, which can be helpful for model selection in future research studies or industrial projects. In the realm of tsunami generation by subaerial landslides, the solid-block approach serves as an effective representative for scenarios where the landslide mass is more cohesive and rigid, rather than granular. This methodology is particularly relevant in cases such as the 2018 Anak Krakatau or 1963 Vajont landslides, where the landslide’s nature aligns closely with the characteristics simulated by a solid-block model (Zaniboni & Tinti, 2014; Heidarzadeh et al., 2020a, 2020b).
The objectives of this research are: (i) To provide a detailed implementation and calibration for simulating solid-block subaerial landslide-generated waves using FLOW-3D HYDRO and OpenFOAM, and (ii) To compare the performance of these two numerical models based on three criteria: free surface elevation of the landslide-generated waves, capabilities of the models in simulating 3D features of the waves in the near-field, velocity fields, and velocity variations at different locations. The innovations of this study are twofold: firstly, it is a 3D study involving physical and numerical modelling and thus the data can be useful for other studies, and secondly, it compares the performance of two popular CFD models in modelling landslide-generated waves for the first time. The validated models such as those reported in this study and comparison of their performances can be useful for engineers and scientists addressing landslide tsunami hazards worldwide.
2 Data and Methods
2.1 Physical Modelling
To validate our numerical models, a series of three-dimensional physical experiments were carried out at the Hydraulic Laboratory of the Brunel University London (UK) in a 3D wave tank 2.40 m long, 2.60 m wide, and 0.60 m high (Figs. 1 and 2). To mitigate experimental errors and enhance the reliability of our results, each physical experiment was conducted three times. The reported data in the manuscript reflects the average of these three trials, assuming no anomalous outliers, thus ensuring an accurate reflection of the experimental tests. One experiment was used for validation of our numerical models. The slope angle (α) and water depth (h) were 45° and 0.246 m, respectively for this experiment. The movement of the sliding mass was recorded by a digital camera with a sampling frequency of 120 frames per second, which was used to calculate the slide impact velocity (vs). The travel distance (D), defined as the distance from the toe of the sliding mass to the water surface, was D=0.045 m. The material of the solid block used in our study was concrete with a density of 2600 kg/m3. Table 2 provides detailed information on the dimensions and kinematics of this solid block used in our physical experiments.
Figure 1. The geometrical and kinematic parameters of a subaerial landslide tsunami. Parameters are: h, water depth; aM, maximum wave amplitude; α, slope angle;vs, slide velocity; ls, length of landslide; bs, width of landslide; s, thickness of landslide; SWL, still water level; D, travel distance (the distance from the toe of the sliding mass to the water surface); L, length of the wave tank; and W, width of the wave tank and H, is the hight of the wave tank
Figure 2. a Wave tank setup of the physical experiments of this study. b Numerical simulation setup for the FLOW-3D HYDRO Model. c The numerical set-up for the OpenFOAM model. The location of the physical wave gauge (represented by numerical gauge WG-3 in the numerical simulations) is at X = 1.03 m, Y = 1.21 m, and Z = 0.046 m. d Top view showing the locations of numerical wave gauges (WG-1, WG-2, WG-3, WG-4, WG-5)
Parameter, unit
Value/type
Slide width (bs), m
0.26
Slide length (ls), m
0.20
Slide thickness (s), m
0.10
Slide volume (V), m3
2.60 × 10–3
Specific gravity, (γs)
2.60
Slide weight (ms), kg
6.86
Slide impact velocity (vs), m/s
1.84
Slide Froude number (Fr)
1.18
Material
Concrete
Table 2 Geometrical and kinematic information of the sliding mass used for physical experiments in this study
We took scale effects into account during physical experiments by considering the study by Heller et al. (2008) who proposed a criterion for avoiding scale effects. Heller et al. (2008) stated that the scale effects can be negligible as long as the Weber number (W=ρgh2/σ; where σ is surface tension coefficient) is greater than 5.0 × 103 and the Reynolds number (R=g0.5h1.5/ν; where ν is kinematic viscosity) is greater than 3.0 × 105 or water depth (h) is approximately above 0.20 m. Considering the water temperature of approximately 20 °C during our experiments, the kinematic viscosity (ν) and surface tension coefficient (σ) of water become 1.01 × 10–6 m2/s and 0.073 N/m, respectively. Therefore, the Reynolds and Weber numbers were as R= 3.8 × 105 and W= 8.1 × 105, indicating that the scale effect can be insignificant in our experiments. To record the waves, we used a twin wire wave gauge provided by HR Wallingford (https://equipit.hrwallingford.com). This wave gauge was placed at X = 1.03 m, Y = 1.21 m based on the coordinate system shown in Fig. 2a.
2.2 Numerical Simulations
The numerical simulations in this work were performed employing two CFD packages FLOW-3D HYDRO, and OpenFOAM which have been widely used in industry and academia (e.g., Bayon et al., 2016; Jasak, 2009; Rauter et al., 2021; Romano et al., 2020a, b; Yin et al., 2015).
2.2.1 Governing Equations and Turbulent Models
2.2.1.1 FLOW-3D HYDRO
The FLOW-3D HYDRO solver is based on the fundamental law of mass, momentum and energy conservation. To estimate the influence of turbulent fluctuations on the flow quantities, it is expressed by adding the diffusion terms in the following mass continuity and momentum transport equations:
quation (1) is the general mass continuity equation, where u is fluid velocity in the Cartesian coordinate directions (x), Ax is the fractional area open to flow in the x direction, VF is the fractional volume open to flow, ρ is the fluid density, R and ξ are coefficients that depend on the choice of the coordinate system. When Cartesian coordinates are used, R is set to unity and ξ is set to zero. RDIF and RSOR are the turbulent diffusion and density source terms, respectively. Uρ=Scμ∗/ρ, in which Sc is the turbulent Schmidt number, μ∗ is the dynamic viscosity, and ρ is fluid density. RSOR is applied to model mass injection through porous obstacle surfaces.
The 3D equations of motion are solved with the following Navier–Stokes equations with some additional terms:
where t is time, Gx is accelerations due to gravity, fx is viscous accelerations, and bx is the flow losses in porous media.
According to Flow Science (2022), FLOW-3D HYDRO’s turbulence models differ slightly from other formulations by generalizing the turbulence production with buoyancy forces at non-inertial accelerations and by including the influence of fractional areas/volumes of the FAVOR method (Fractional Area-Volume Obstacle Representation) method. Here we use k–ω model for turbulence modelling. The k–ω model demonstrates enhanced performance over the k-ε and Renormalization-Group (RNG) methods in simulating flows near wall boundaries. Also, for scenarios involving pressure changes that align with the flow direction, the k–ω model provides more accurate simulations, effectively capturing the effects of these pressure variations on the flow (Flow Science, 2022). The equations for turbulence kinetic energy are formulated as below based on Wilcox’s k–ω model (Flow Science, 2022):
where kT is turbulent kinetic energy, PT is the turbulent kinetic energy production, DiffKT is diffusion of turbulent kinetic energy, GT is buoyancy production, β∗=0.09 is closure coefficient, and ω is turbulent frequency.
2.2.1.2 OpenFOAM
For the simulations conducted in this study, OpenFOAM utilizes the Volume-Averaged RANS equations (VARANS) to enable the representation of flow within porous material, treated as a continuous medium. The momentum equation incorporates supplementary terms to accommodate frictional forces from the porous media. The mass and momentum conservation equations are linked to the VOF equation (Jesus et al., 2012) and are expressed as follows:
where the gravitational acceleration components are denoted bygj. The term u¯i=1Vf∫Vf0ujdV represents the volume averaged ensemble averaged velocity (or Darcy velocity) component, Vf is the fluid volume contained in the average volumeV,τ is the surface tension constant (assumed to be 1 for the water phase and 0 for the air phase), and fσi is surface tension, defined as fσi=σκ∂α∂xi, where σ (N/m) is the surface tension constant and κ (1/m) is the curvature (Brackbill et al., 1992). μeff is the effective dynamic viscosity that is defined as μeff=μ+ρνt and takes into account the dynamic molecular (μ) and the turbulent viscosity effects (ρνt). νt is eddy viscosity, which is provided by the turbulence closure model. n is the porosity, defined as the volume of voids over total volume, and P∗=1Vf∫∂Vf0P∗dS is the ensemble averaged pressure in excess of hydrostatic pressure. The coefficient A accounts for the frictional force induced by laminar Darcy-type flow, B considers the frictional force under turbulent flow conditions, and c accounts for the added mass. These coefficients (A,B, and c) are defined based on the work of Engelund (1953) and later modified by Van Gent (1995) as given below:
where D50 is the mean nominal diameter of the porous material, KC is the Keulegan–Carpenter number, a and b are empirical nondimensional coefficients (see Lara et al., 2011; Losada et al., 2016) and γ = 0.34 is a nondimensional parameter as proposed by Van Gent (1995). The k-ω Shear Stress Transport (SST) turbulence is employed to capture the effect of turbulent flow conditions (Zhang & Zhang, 2023) with the enhancement proposed by Larsen and Fuhrman (2018) for the over-production of turbulence beneath surface waves. Boundary layers are modelled with wall functions. The reader is referred to Larsen and Fuhrman (2018) for descriptions, validations, and discussions of the stabilized turbulence models.
2.2.2 FLOW-3D HYDRO Simulation Procedure
In our specific case in this study, FLOW-3D HYDRO utilizes the finite-volume method to numerically solve the equations described in the previous Sect. 2.2.1.1, ensuring a high level of accuracy in the computational modelling. The use of structured rectangular grids in FLOW-3D HYDRO offers the advantages of easier development of numerical methods, greater transparency in their relation to physical problems, and enhanced accuracy and stability of numerical solutions. (Flow Science, 2022). Curved obstacles, wall boundaries, or other geometric features are embedded in the mesh by defining the fractional face areas and fractional volumes of the cells that are open to flow (the FAVOR method). The VOF method is employed in FLOW-3D HYDRO for accurate capturing of the free-surface dynamics (Hirt and Nichols 1981). This approach then is upgraded to method of the TruVOF which is a split Lagrangian method that typically produces lower cumulative volume error than the alternative methods (Flow Science, 2022).
For numerical simulation using FLOW-3D HYDRO, the entire flow domain was 2.60 m wide, 0.60 m deep and 2.50 m long (Fig. 2b). The specific gravity (γs) for solid blocks was set to 2.60 in our model, aligning closely with the density of the actual sliding mass, which was approximately determined in our physical experiments. The fluid medium was modelled as water with a density of 1000 kg/m3 at 20 °C. A uniform grid comprising of one single mesh plane was applied with a grid size of 0.005 m. The top, front and back of the mesh areas were defined as symmetry, and the other surfaces were of wall type with no-slip conditions around the walls.
To simulate turbulent flows, k-ω model was used because of its accuracy in modelling turbulent flows (Menter 1992). Landslide movement was replicated in simulations using coupled motion objects, which implies that the movement of landslides is based on gravity and the friction between surfaces rather than a specified motion in which the model should be provided by force and torques. The time intervals of the numerical model outputs were set to 0.02 s to be consistent with the actual sampling rates of our wave gauges in the laboratory. In order to calibrate the FLOW-3D HYDRO model, the friction coefficient is set to 0.45, which is consistent with the Coulombic friction measurements in the laboratory. The Courant Number (C=UΔtΔx) is considered as the criterion for the stability of numerical simulations which gives the maximum time step (Δt) for a prespecified mesh size (Δx) and flow speed (U). The Courant number was always kept below one.
2.2.3 OpenFOAM Simulation Procedure
OpenFOAM is an open-source platform containing several C++ libraries which solves both 3D Reynolds-Averaged Navier–Stokes equations (RANS) and Volume-Averaged RANS equations (VARANS) for two-phase flows (https://www.openfoam.com/documentation/user-guide). Its implementation is based on a tensorial approach using object-oriented programming techniques and the Finite Volume Method (McDonald 1971). In order to simulate the subaerial landslide-generated waves, the IHFOAM solver based on interFoam (Higuera et al., 2013a, 2013b), and the overset mesh framework method are employed. The implementation of the overset mesh method for porous mediums in OpenFOAM is described in Romano et al. (2020a, b) for submerged rigid and impermeable landslides.
The overset mesh technique, as outlined by Romano et al. (2020a, b), uses two distinct domains: a moving domain that captures the dynamics of the rigid landslide and a static background domain to characterize the numerical wave tank. The overlapping of these domains results in a composite mesh that accurately depicts complex geometrical transformations while preserving mesh quality. A porous media with a very low permeability (n = 0.001) was used to simulate the impermeable sliding surfaces. RANS equations were solved within the porous media. The Multidimensional Universal Limiter with Explicit Solution (MULES) algorithm is employed for solving the (VOF) equation, ensuring precision in tracking fluid interfaces. Simultaneously, the PIMPLE algorithm is employed for the effective resolution of velocity–pressure coupling in the Eqs. 7 and 8. A background domain was created to reproduce the subaerial landslide waves with dimensions 2.50 m (x-direction) × 2.60 m (y-direction) × 0.6 m (z-direction) (Fig. 2c). The grid size is set to 0.005 m for the background mesh. A moving domain was applied in an area of 0.35 m (x-direction) × 0.46 m (y-direction) × 0.32 m (z-direction) with a grid spacing of 0.005 m and applying a body-fitted mesh approach, which contains the rigid and impermeable wedges. Wall condition with No-slip is defined as the boundary for the four side walls (left, right, front and back, in Fig. 1). Also, a non-slip boundary condition is specified to the bottom, whereas the top boundary is defined as open. The experimental slide movement time series is used to model the landslide motion in OpenFOAM. The applied equation is based on the analytical solution by Pelinovsky and Poplavsky (1996) which was later elaborated by Watts (1998). The motion of a sliding rigid body is governed by the following equation:
where, m represents the mass of the landslide, s is the displacement of the landslide down the slope, t is time elapsed, g stands for the acceleration due to gravity, θ is the slope angle, Cf is the Coulomb friction coefficient, Cm is the added mass coefficient, m0 denotes the mass of the water displaced by the moving landslide, A is the cross-sectional area of the landslide perpendicular to the direction of motion, ρ is the water density, and Cd is the drag coefficient.
2.2.4 Mesh Sensitivity Analysis
In order to find the most efficient mesh size, mesh sensitivity analyses were conducted for both numerical models (Fig. 3). We considered the influence of mesh density on simulated waveforms by considering three mesh sizes (Δx) of 0.0025 m, 0.005 m and 0.010 m. The results of FLOW-3D HYDRO revealed that the largest mesh deviates 9% (Fig. 3a, Δx = 0.0100 m) from two other finer meshes. Since the simulations by FLOW-3D HYDRO for the finest mesh (Δx = 0.0025 m) do not show any improvements in comparison with the 0.005 m mesh, therefore the mesh with the size of Δx = 0.0050 m is used for simulations (Fig. 3a). A similar approach was followed for mesh sensitivity of OpenFOAM mesh grids. The mesh with the grid spacing of Δx = 0.0050 m was selected for further simulations since a satisfactory independence was observed in comparison with the half size mesh (Δx = 0.0025 m). However, results showed that the mesh size with the double size of the selected mesh (Δx = 0.0100 m) was not sufficiently fine to minimize the errors (Fig. 3b).
Figure 3. a, b Sensitivity of numerical simulations to the sizes of the mesh (Δx) for FLOW-3D HYDRO, and OpenFOAM, respectively. The location of the wave gauge 3 (WG-3) is at X = 1.03 m, Y = 1.21 m, and Z = -0.55 m (see Fig. 2d)
In terms of computational cost, the time required for 2 s simulations by FLOW-3D HYDRO is approximately 4.0 h on a PC Intel® Core™ i7-8700 CPU with a frequency of 3.20 GHz equipped with a 32 GB RAM. OpenFOAM requires 20 h to run 2 s of numerical simulation on 2 processors on a PC Intel® Core™ i9-9900KF CPU with a frequency of 3.60 GHz equipped with a 364 GB RAM. Differences in computational time for simulations run with FLOW-3D HYDRO and OpenFOAM reflect the distinct characteristics of each numerical methods, and the specific hardware setups.
2.2.5 Validation
We validated both numerical models based on our laboratory experimental data (Fig. 4). The following criterion was used to assess the level of agreement between numerical simulations and laboratory observations:
where ε is the mismatch error, Obsi is the laboratory observation values, Simi is the simulation values, and the mathematical expression |X| represents the absolute value of X. The slope angle (α), water depth (h) and travel distance (D) were: α = 45°, h = 0.246 m and D = 0.045 m in both numerical models, consistent with the physical model. We find the percentage error between each simulated data point and its corresponding observed value, and subsequently average these errors to assess the overall accuracy of the simulation against the observed time series. Our results revealed that the mismatch errors between physical experiments and numerical models for the FLOW-3D HYDRO and OpenFOAM are 8% and 18%, respectively, indicating that our models reproduce the measured waveforms satisfactorily (Fig. 4). The simulated waveform by OpenFOAM shows a minor mismatch at t = 0.76 s which resulted from a droplet immediately after the slide hits the water surface in the splash zone. In term of the maximum negative amplitude, the simulated waves by OpenFOAM indicates a relatively better performance than FLOW-3D HYDRO, whereas the maximum positive amplitude (aM) simulated by FLOW-3D HYDRO is closer to the experimental value. The recorded maximum positive amplitude in physical experiment is 0.022 m, whereas it is 0.020 m for FLOW-3D HYDRO and 0.017 m for OpenFOAM simulations. In acknowledging the deviations observed, it is pertinent to highlight that while numerical models offer robust insights, the difference in meshing techniques and the distinct computational methods to resolve the governing equations in FLOW-3D HYDRO and OpenFOAM have contributed to the variance. Moreover, the intrinsic uncertainties associated with the physical experimentation process, including the precision of wave gauges and laboratory conditions, are non-negligible factors influencing the results.
Figure 4. Validation of the simulated waves (brown line for FLOW-3D HYDRO and green line for OpenFOAM) using the laboratory-measured waves (black solid diamonds). This physical experiment was conducted for wave gauge 3 (WG-3) located at X = 1.03 m, Y = 1.21 m, and Z = -0.55 m (see Fig. 2d). Here, ε shows the errors between simulations and actual physical measurements using Eq. (13)
3 Results
Following the validations of the two numerical models (FLOW-3D HYDRO and OpenFOAM), a series of simulations were performed to compare the performances of these two CFD solvers. The generation process of landslide waves, waveforms, and velocity fields are considered as the basis for comparing the performance of the two models (Figs. 5, 6, 7 and 8).
Figure 5.Comparison between the simulated waveforms by FLOW-3D HYDRO (black) and OpenFOAM (red) at four different locations in the near-field zone (WG-1,2,4 and 5). WG is the abbreviation for wave gauge. The mismatch (Δ) between the two models at each wave gauge is calculated using Eq. (14)
Figure 6. Comparison of water surface elevations produced by solid-block subaerial landslides for the two numerical models FLOW-3D-HYDRO (a–c) and OpenFOAM (e–g) at different times
Figure 7. Snapshots of the simulations at different times for FLOW-3D HYDRO (a–c) and OpenFOAM (e–g) showing velocity fields (colour maps and arrows). The colormaps indicate water particle velocity in m/s, and the lines indicate the velocities of water particles
Figure 8. Comparison of velocity variations at (WG-3) for FLOW-3D HYDRO (light blue) and OpenFOAM (brown)
3.1 Comparison of Waveforms
Five numerical wave gauges were placed in our numerical models to measure water surface oscillations in the near-field zone (Fig. 5). These gauges offer an azimuthal coverage of 60° (Fig. 2d). Figure 5 reveals that the simulated waveforms from two models (FLOW-3D HYDRO and OpenFOAM) are similar. The highest wave amplitude (aM) is recorded at WG-3 for both models, whereas the lowest amplitude is recorded at WG-5 and WG-1 which can be attributed to the longer distances of these gauges from the source region as well as their lateral offsets, resulting in higher wave energy dissipation at these gauges. The sharp peaks observed in the simulated waveforms, such as the red peak between 0.8–1.0 s in Fig. 5a from OpenFOAM, the red peak between 0.6–0.8 s in Fig. 5b also from OpenFOAM, and the black peak between 1.4–1.6 s in Fig. 5d from FLOW-3D HYDRO, are due to the models’ spatial and temporal discretization. They reflect the sensitivity of the models to capturing transient phenomena, where the chosen mesh and time-stepping intervals are key factors in the models’ ability to track rapid changes in the flow field. To quantify the deviations of the two models from one another, we apply the following equation for mismatch calculation:
where Δ is the mismatch error, Sim1 is the simulation values from FLOW-3D HYDRO, Sim2 is the simulation values from OpenFOAM, and the mathematical expression |X| implies the absolute value of X. We calculate the percentage difference for each corresponding pair of simulation results, then take the mean of these percentage differences to determine the average deviation between the two simulation time series. Using Eq. (14), we found a deviation range from 9 to 11% between the two models at various numerical gauges (Fig. 5), further confirming that the two models give similar simulation results.
3.2 Three-Dimensional Vision of Landslide Generation Process by Numerical Models
A sequence of four water surface elevation snapshots at different times is shown in Fig. 6 for both numerical modes. In both simulations, the sliding mass travels a constant distance of 0.045 m before hitting the water surface at t = 0.270 s which induces an initial change in water surface elevation (Figs. 6a and e). At t = 0.420 s, the mass is fully immersed for both simulations and an initial dipole wave is generated (Figs. 6b and f). Based on both numerical models, the maximum positive amplitude (0.020 m for FLOW-3D HYDRO, and 0.017 m for OpenFOAM) is observed at this stage (Fig. 6). The maximum propagation of landslide-simulated waves along with more droplets in the splash zone could be seen at t = 0.670 s for both models (Fig. 6c and g). The observed distinctions in water surface elevation simulations as illustrated in Fig. 6 are rooted in the unique computational methodologies intrinsic to each model. In the OpenFOAM simulations, a more diffused water surface elevation profile is evident. Such diffusion is an outcome of the simulation’s intrinsic treatment of turbulent kinetic energy dissipation, aligning with the solver’s numerical dissipation characteristics. These traits are influenced by the selected turbulence models and the numerical advection schemes, which prioritize computational stability, possibly at the expense of interface sharpness. The diffusion in the wave pattern as rendered by OpenFOAM reflects the application of a turbulence model with higher dissipative qualities, which serves to moderate the energy retained during wave propagation. This approach can provide insights into the potential overestimation of energy loss under specific simulation conditions. In contrast, the simulations from FLOW-3D HYDRO depict a more localized wave pattern, indicative of a different approach to turbulent dissipation. This coherence in wave fronts is a function of the model’s specific handling of the air–water interface and its targeted representation of the energy dynamics resulting from the landslide’s interaction with the water body. They each have specific attributes that cater to different aspects of wave simulation fidelity, thereby contributing to a more comprehensive understanding of the phenomena under study.
3.3 Wave Velocity Analysis
We show four velocity fields at different times during landslide motion in Fig. 7 and one time series of velocity (Fig. 8) for both numerical models. The velocity varies in the range of 0–1.9 m/s for both models, and the spatial distribution of water particle velocity appears to be similar in both. The models successfully reproduce the complex wavefield around the landslide generation area, which is responsible for splashing water and mixing with air around the source zone (Fig. 7). The first snapshot at t = 0.270 s (Fig. 7a and e) shows the initial contact of the sliding mass with water surface for both numerical models which generates a small elevation wave in front of the mass exhibiting a water velocity of approximately 1.2 m/s. The slide fully immerses for the first time at t = 0.420 s producing a water velocity of approximately 1.5 m/s at this time (Fig. 7b and f). The last snapshot (t = 0.670 s) shows 1.20 s after the slide hits the bottom of the wave tank. Both models show similar patterns for the propagation of the waves towards the right side of the wave tank. The differences in water surface profiles close to the slope and solid block at t = 0.67 s, observed in the FLOW-3D HYDRO and OpenFOAM simulations (Figs. 6 and 7), are due to the distinct turbulence models employed by each (RNG and k-ω SST, respectively) which handle the complex interactions of the landslide-induced waves with the structures differently. Additionally, the methods of simulating landslide movement further contribute to this discrepancy, with FLOW-3D HYDRO’s coupled motion objects possibly affecting the waves’ initiation and propagation unlike OpenFOAM’s prescribed motion from experimental data. In addition to the turbulence models, the variations in VOF methodologies between the two models also contribute to the observed discrepancies.
For the simulated time series of velocity, both models give similar patterns and close maximum velocities (Fig. 8). For both models the WG-3 located at X = 1.03 m, Y = 1.21 m, and Z = − 0.55 m (Fig. 2d) were used to record the time series. WG-3 is positioned 5 mm above the wave tank bottom, ensuring that the measurements taken reflect velocities very close to the bottom of the wave tank. The maximum velocity calculated by FLOW-3D HYDRO is 0.162 m/s while it is 0.132 m/s for OpenFOAM, implying a deviation of approximately 19% from one another. Some oscillations in velocity records are observed for both models, but these oscillations are clearer and sharper for OpenFOAM. Although it is hard to see velocity oscillations in the FLOW-3D HYDRO record, a close look may reveal some small oscillations (around t = 0.55 s and 0.9 s in Fig. 8). In fact, velocity oscillations are expected due to the variations in velocity of the sliding mass during the travel as well as due to the interferences of the initial waves with the reflected wave from the beach. In general, it appears that the velocity time series of the two models show similar patterns and similar maximum values although they have some differences in the amplitudes of the velocity oscillations. The differences between the two curves are attributed to factors such as difference in meshing between the two models, turbulence models, as well as the way that two models record the outputs.
4 Discussions
An important step for CFD modelling in academic or industrial projects is the selection of an appropriate numerical model that can deliver the task with satisfactory performance and at a reasonable computational cost. Obviously, the major drivers when choosing a CFD model are cost, capability, flexibility, and accessibility. In this sense, the existing options are of two types as follows:
Commercial models, such as FLOW-3D HYDRO, which are optimised to solve free-surface flow problems, with customer support and an intuitive Graphical User Interface (GUI) that significantly facilitates meshing, setup, simulation monitoring, visualization, and post-processing. They usually offer high-quality customer support. Although these models show high capabilities and flexibilities for numerical modelling, they are costly, and thus less accessible.
Open-source models, such as OpenFOAM, which come without a GUI but with coded tools for meshing, setup, parallel running, monitoring, post-processing, and visualization. Although these models offer no customer support, they have a big community support and online resources. Open-source models are free and widely accessible, but they may not be necessarily always flexible and capable.
OpenFOAM provides freedom for experimenting and diving through the code and formulating the problem for a user whereas FLOW-3D HYDRO comes with high-level customer supports, tutorial videos and access to an extensive set of example simulations (https://www.flow3d.com/). While FLOW-3D-HYDRO applies a semi-automatic meshing process where users only need to input the 3D model of the structure, OpenFOAM provides meshing options for simple cases, and in many advanced cases, users need to create the mesh in other software (e.g., ANSYS) (Ariza et al., 2018) and then convert it to OpenFOAM format. Auspiciously, there are numerous online resources (https://www.openfoam.com/trainings/about-trainings), and published examples for OpenFOAM (Rauter et al., 2021; Romano et al., 2020a, b; Zhang & Zhang, 2023).
The capabilities of both FLOW-3D HYDRO and OpenFOAM to simulate actual, complex landslide-generated wave events have been showcased in significant case studies. The study by Ersoy et al. (2022) applied FLOW-3D HYDRO to simulate impulse waves originating from landslides near an active fault at the Çetin Dam Reservoir, highlighting the model’s capacity for detailed, site-specific modelling. Concurrently, the work by Alexandre Paris (2021) applied OpenFOAM to model the 2017 Karrat Fjord landslide tsunami events, providing a robust validation of OpenFOAM’s utility in capturing the dynamics of real-world geophysical phenomena. Both instances exemplify the sophisticated computational approaches of these models in aiding the prediction and analysis of natural hazards from landslides.
As for limitations of this study, we acknowledge that our numerical models are validated by one real-world measured wave time series. However, it is believed that this one actual measurement was sufficient for validation of this study because it was out of the scope of this research to fully validate the FLOW-3D HYDRO and OpenFOAM models. These two models have been fully validated by more actual measurements by other researchers in the past (e.g., Sabeti & Heidarzadeh, 2022b). It is also noted that some of the comparisons made in this research were qualitative, such as the 3D wave propagation snapshots, as it was challenging to develop quantitative comparisons for snapshots. Another limitation of this study concerns the number of tests conducted here. We fixed properties such as water depth, slope angle, and travel distance throughout this study because it was out of the scope of this research to perform sensitivity analyses.
5 Conclusions
We configured, calibrated, validated and compared two numerical models, FLOW-3D HYDRO, and OpenFOAM, using physical experiments in a 3D wave tank. These validated models were used to simulate subaerial solid-block landslides in the near-field zone. Our results showed that both models are fully compatible with investigating waves generated by subaerial landslides, although they use different approaches to simulate the phenomenon. The properties of solid-block, water depth, slope angle, and travel distance were kept constant in this study as we focused on comparing the performance of the two models rather than conducting a full sensitivity analysis. The findings are as follows:
Different settings were used in the two models for modelling landslide-generated waves. In terms of turbulent flow modelling, we used the Renormalization Group (RNG) turbulence model in FLOW-3D HYDRO, and k-ω (SST) turbulence model in OpenFOAM. Regarding meshing techniques, the overset mesh method was used in OpenFOAM, whereas the structured cartesian mesh was applied in FLOW-3D HYDRO. As for simulation of landslide movement, the coupled motion objects method was used in FLOW-3D HYDRO, and the experimental slide movement time series were prescribed in OpenFOAM.
Our modelling revealed that both models successfully reproduced the physical experiments. The two models deviated 8% (FLOW-3D HYDRO) and 18% (OpenFOAM) from the physical experiments, indicating satisfactory performances. The maximum water particle velocity was approximately 1.9 m/s for both numerical models. When the simulated waveforms from the two numerical models are compared with each other, a deviation of 10% was achieved indicating that the two models perform approximately equally. Comparing the 3D snapshots of the two models showed that there are some minor differences in reproducing the details of the water splash in the near field.
Regarding computational costs, FLOW-3D HYDRO was able to complete the same simulations in 4 h as compared to nearly 20 h by OpenFOAM. However, the hardware that were used for modelling were not the same; the computer used for the OpenFOAM model was stronger than the one used for running FLOW-3D HYDRO. Therefore, it is challenging to provide a fair comparison for computational time costs.
Overall, we conclude that the two models give approximately similar performances, and they are both capable of accurately modelling landslide-generated waves. The choice of a model for research or industrial projects may depend on several factors such as availability of local knowledge of the models, computational costs, accessibility and flexibilities of the model, and the affordability of the cost of a license (either a commercial or an open-source model).
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In this study, experimental investigations were conducted on rectangular side weirs with different widths and heights. Corresponding simulations were also performed to analyze hydraulic characteristics including the water surface profile, flow velocity, and pressure. The relationship between the discharge coefficient and the Froude number, as well as the ratios of the side weir height and width to upstream water depth, was determined. A discharge formula was derived based on a dimensional analysis. The results demonstrated good agreement between simulated and experimental data, indicating the reliability of numerical simulations using FLOW-3D software (version 11.1). Notably, significant fluctuations in water surface profiles near the side weir were observed compared to those along the center line or away from the side weir in the main channel, suggesting that the entrance effect of the side weir did not propagate towards the center line of the main channel. The proposed discharge formula exhibited relative errors within 10%, thereby satisfying the flow measurement requirements for small channels and field inlets.
1. Introduction
Sharp crested weirs are used to obtain discharge in open channels by solely measuring the water head upstream of the water. Side weirs, as a kind of sharp-crested weir, are extensively used for flow measurement, flow diversion, and flow regulation in open channels. Side weirs can be placed directly in the channel direction or field inlet, without changing the original structure of the channel. Thus, side weirs have certain advantages in the promotion and application of flow measurement facilities in small channels and field inlets. The rectangular sharp-crested weir is the most commonly available, and many scholars have conducted research on it. Research on side weirs started in 1934. De Marchi studied the side weir in the rectangular channel and derived the theoretical formula based on the assumption that the specific energy of the main flow section of the rectangular channel in the side weir section was constant [1]. Ackers discussed the existing formulas for the prediction of the side weir discharge coefficient [2]. Chen concluded that the momentum theorem was more suitable for the analytical calculation of the side weir based on the experimental data [3]. Based on previous theoretical research, more and more scholars began to carry out experimental research on side weirs. Uyumaz and Muslu conducted experiments under subcritical and supercritical flow regimes and derived expressions for the side weir discharge and water surface profiles for these regimes by comparing them with experimental results [4]. Borghei et al. developed a discharge coefficient equation for rectangular side weirs in subcritical flow [5]. Ghodsian [6] and Durga and Pillai [7] developed a discharge coefficient equation of rectangular side weirs in supercritical flow. Mohamed proposed a new approach based on the video monitoring concept to measure the free surface of flow over rectangular side weirs [8]. Durga conducted experiments on rectangular side weirs of different lengths and sill heights and discussed the application of momentum and energy principles to the analysis of spatially varied flow under supercritical conditions. The results showed that the momentum principle was fitting better [7]. Omer et al. obtained sharp-crested rectangular side weirs discharge coefficients in the straight channel by using an artificial neural network model for a total of 843 experiments [9]. Emiroglu et al. studied water surface profile and surface velocity streamlines, and developed a discharge coefficient formula of the upstream Froude number, the ratios of weir length to channel width, weir length to flow depth, and weir height to flow depth [10]. Other investigators [11,12,13,14] have conducted experiments to study flow over rectangular side weirs in different flow conditions. Numerous studies have been conducted in laboratories to this day. Compared to experimental methods, the numerical simulation method has many attractive advantages. We can easily obtain a wide range of hydraulic parameters of side weirs using numerical simulation methods, without investing a lot of manpower and resources. In addition, we can conduct small changes in inlet condition, outlet condition, and geometric parameters, and study their impact on the flow characteristics of side weirs. Therefore, with the development and improvement of computational fluid dynamics, the numerical simulation method has begun to be widely applied on side weirs. Salimi et al. studied the free surface changes and the velocity field along a side weir located on a circular channel in the supercritical regime by numerical simulation [15]. Samadi et al. conducted a three-dimensional simulation on rectangular sharp-crested weirs with side contraction and without side contraction and verified the accuracy of numerical simulation compared with the experimental results [16]. Aydin investigated the effect of the sill on rectangular side weir flow by using a three-dimensional computational fluid dynamics model [17]. Azimi et al. studied the discharge coefficient of rectangular side weirs on circular channels in a supercritical flow regime using numerical simulation and experiments [18]. The discharge coefficient over the two compound side weirs (Rectangular and Semi-Circle) was modeled by using the FLOW-3D software to describe the flow characteristics in subcritical flow conditions [19]. Safarzadeh and Noroozi compared the hydraulics and 3D flow features of the ordinary rectangular and trapezoidal plan view piano key weirs (PKWs) using two-phase RANS numerical simulations [20]. Tarek et al. investigated the discharge performance, flow characteristics, and energy dissipation over PK and TL weirs under free-flow conditions using the FLOW-3D software [21]. As evident from the aforementioned, the majority of studies have primarily focused on determining the discharge coefficient, while comparatively less attention has been devoted to investigating the hydraulic characteristics of rectangular side weirs. Numerical simulations were conducted on different types of side weirs, including compound side weirs and piano key weirs, in different cross-section channels under different flow regimes. It is imperative to derive the discharge formula and investigate other crucial flow parameters such as depth, velocity, and pressure near side weirs for their effective implementation in water measurement. In this study, a combination of experimental and numerical simulation methods was employed to examine the relationship between the discharge coefficient and its influencing factors; furthermore, a dimensionless analysis was utilized to derive the discharge formula. Additionally, water surface profiles near side weirs and pressure distribution at the bottom of the side channel were analyzed to assess safety operation issues associated with installing side weirs.
2. Principle of Flow Measurement
Flow discharge over side weirs is a function of different dominant physical and geometrical quantities, which is defined as
where Q is flow discharge over the side weir, b is the side weir width, B is the channel width, P is the side weir height, v is the mean velocity, h1 is water depth upstream the side weir in the main channel, g is the gravitational acceleration, μ is the dynamic viscosity of fluid, ρ is fluid density, and i is the channel slope (Figure 1).
Figure 1. Definition sketch of parameters of rectangular side weir under subcritical flow. Note: h1 and h2 represent water depth upstream and downstream of the side weir in the main channel, respectively; y1 and y2 represent weir head upstream and downstream of the side weir in the main channel, respectively.
In experiments when the upstream weir head was over 30 mm, the effects of surface tension on discharge were found to be minor [22]. The viscosity effect was far less than the gravity effect in a turbulent flow. Hence μ and σ were excluded from the analysis [23,24]. In addition, the channel width, the channel slope, and the fluid density were all constant, so the discharge formula can be simplified as:
According to the Buckingham π theorem, the following relationship among the dimensionless parameters is established:
Selected h1 and g as basic fundamental quantities, and the remaining physical quantities were represented in terms of these fundamental quantities as follows:
In which
Based on dimensional analysis, the following equations were derived.
Namely
So the discharge formula can be simplified as:
In a sharp-crested weir, discharge over the weir is proportional to 𝐻1.51H11.5 (H1 is the upstream total head above the crest, namely H1 = y1 + v2/2 g), so Equation (6) can be transformed as follows:
Consequently, the discharge formula over rectangular side weirs is defined as follows, in which 𝑚=𝑓(𝑏ℎ1m=f(bh1,𝑃ℎ1,𝐹𝑟1)Ph1,Fr1). Parameter m represents the dimensionless discharge coefficient. Parameter Fr1 represents the Froude number at the upstream end of the side weir in the main channel.
3. Experiment Setup
The experimental setup contained a storage reservoir, a pumping station, an electromagnetic flow meter, a control valve, a stabilization pond, rectangular channels, a side weir, and a sluice gate. The layout of the experimental setup is shown in Figure 2. Water was supplied from the storage reservoir using a pump. The flow discharge was measured with an electromagnetic flow meter with precision of ±3‰. Water depth was measured with a point gauge with an accuracy of ±0.1 mm. The flow velocity was measured with a 3D Acoustic Doppler Velocimeter (Nortek Vectrino, manufactured by Nortek AS in Rud, Norway). In order to eliminate accidental and human error, multiple measurements of the water depth and flow velocity at the same point were performed and the average values were used as the actual water depth and flow velocity of the point. The main and side channels were both rectangular open channels measuring 47 cm in width and 60 cm in height. The geometrical parameters of rectangular side weirs are shown in Table 1.
Figure 2. Layout of the test system.
Table 1. The geometrical parameters of rectangular side weirs.
When water passes through a side weir, its quality point is affected not only by gravity but also by centrifugal inertia force, leading to an inclined water surface within that particular cross-section before reaching the weir. In order to examine water profiles adjacent to side weirs, cross-sectional measurements were conducted at regular intervals of 12 cm both upstream and downstream of each side weir, denoted as sections ① to ⑩, respectively. Measuring points were positioned near the side weir (referred to as “Side I”), along the center line of the main channel (referred to as “Side II”), and far away from the side weir (referred to as “Side III”) for each cross-section. The schematic diagram illustrating these measuring points is presented in Figure 3.
Figure 3. Schematic diagram of measurement points.
4. Numerical Simulation Settings
4.1. Mathematical Model
4.1.1. Governing Equations
Establishing the controlling equations is a prerequisite for solving any problem. For the flow analysis problem of water flowing over a side weir in a rectangular channel, assuming that no heat exchange occurs, the continuity equation (Equation (9)) and momentum equation (Equation (10)) can be used as the controlling equations as follows:
The continuity equation:
Momentum equation:
where: ρ is the fluid density, kg/m3; t is time, s; ui, uj are average flow velocities, u1, u2, u3 represent average flow velocity components in Cartesian coordinates x, y, and z, respectively, m/s; μ is dynamic viscosity of fluid, N·s/m2; p is the pressure, pa; Si is the body force, S1 = 0, S2 = 0, S3 = −ρg, N [24].
4.1.2. RNG k-ε Model
The water flow in the main channel is subcritical flow. When the water flows through the side weir, the flow line deviates sharply, the cross section suddenly decreases, and due to the blocking effect of the side weir, the water reflects and diffracts, resulting in strong changes in the water surface and obvious three-dimensional characteristics of the water flow [25]. Therefore the RNG k–ε model is selected. The model can better handle flows with greater streamline curvature, and its corresponding k and ε equation is, respectively, as follows:
where: k is the turbulent kinetic energy, m2/s2; μeff is the effective hydrodynamic viscous coefficient; Gk is the generation item of turbulent kinetic energy k due to gradient of the average flow velocity; C∗1εC1ε*, C2ε are empirical constants of 1.42 and 1.68, respectively; ε is turbulence dissipation rate, kg·m2/s2.
4.1.3. TruVOF Model
Because the shape of the free surface is very complex and the overall position is constantly changing, the fluid flow phenomenon with a free surface is a typical flow phenomenon that is difficult to simulate. The current methods used to simulate free surfaces mainly include elevation function method, the MAC method [26] and the VOF (Volume of Fluid) method [27]. The VOF method is a method proposed by Hirt and Nichols to deal with the complex motion of the free surface of a fluid, which can describe all the complexities of the free surface with only one function. The basic idea of the method is to define functions αw and αa, which represent the volume percentage of the calculation area occupied by water and air, respectively. In each unit cell, the sum of the volume fractions of water and air is equal to 1, i.e.,
The TruVOF calculation method can accurately track the change of free liquid level and accurately simulate the flow problems with free interface. Its equation is:
where: u_¯m is the average velocity of the mixture; t is the time; F is the volume fraction of the required fluid.
4.2. Parameter Setting and Boundary Conditions
To streamline the iterative calculation and minimize simulation time, we selected a main channel measuring 7.5 m in length and a side channel measuring 2.5 m in length for simulation. Three-dimensional geometrical models were developed using the software AutoCAD (version 2016-Simplified Chinese). The spatial domain was meshed using a constructed rectangular hexahedral mesh and each cell size was 2 cm. A volume flow rate was set in the channel inlet with an auto-adjusted fluid height. An outflow–outlet condition was positioned at the end of the side channel. A symmetry boundary condition was set in the air inlet at the top of the model, which represented that no fluid flows through the boundary. The lower Z (Zmin) and both of the side boundaries were treated as a rigid wall (W). No-slip conditions were applied at the wall boundaries. Figure 4 illustrates these boundary conditions.
Figure 4. Diagram of boundary conditions.
5. Results
5.1. Water Surface Profiles
Water surface profiles were crucial parameters for selecting water-measuring devices. Upon analyzing the consistent patterns observed in different conditions, one specific condition was chosen for further analysis. To validate the reliability of numerical simulation, measured and simulated water depths of rectangular side weirs with different widths and heights at a discharge rate of 25 L/s were extracted for comparison (Table 2 and Figure 5). The results in Table 2 and Figure 5 indicate a maximum absolute relative error value of 9.97% and all absolute relative error values within 10%, demonstrating satisfactory agreement between experimental and simulated results.
Figure 5. Comparison between measured and simulated flow depth.
P/cm
Section Position
b = 20 cm
b = 30 cm
b = 40 cm
b = 47 cm
hm/cm
hs/cm
R/%
hm/cm
hs/cm
R/%
hm/cm
hs/cm
R/%
hm/cm
hs/cm
R/%
7
④
21.49
19.4
9.73
17.74
16.9
4.74
16.07
14.51
9.71
13.79
12.50
9.35
④′
20.48
19.05
6.98
17.78
16.14
9.22
15.69
14.31
8.80
⑥
20.71
19.02
8.16
17.82
16.31
8.47
15.92
14.53
8.73
15.23
13.80
9.39
⑧′
22.00
20.22
8.09
18.27
16.74
8.37
16.59
14.96
9.83
⑧
22.37
20.17
9.83
17.73
16.80
5.25
16.27
15.08
7.31
15.36
14.36
6.51
10
④
24.15
22.6
6.42
19.96
18.84
5.61
19.03
18.58
2.36
16.83
15.85
5.82
④′
24.21
22.05
8.92
19.49
18.19
6.67
18.75
18.35
2.13
⑥
24.01
21.78
9.29
19.65
18.34
6.67
18.95
18.63
1.69
17.52
16.09
8.16
⑧′
24.88
22.4
9.97
20.65
19.21
6.97
20.12
19.29
4.13
⑧
24.03
22.96
4.45
21.16
19.34
8.60
19.71
19.43
1.42
18.39
17.36
5.60
15
④
28.85
27.56
4.47
25.86
24.09
6.84
24.05
21.89
8.98
22.73
20.80
8.49
④′
28.49
26.97
5.34
25.19
23.84
5.36
23.42
21.46
8.37
⑥
28.85
26.98
6.48
25.72
23.99
6.73
23.23
21.82
6.07
23.10
21.05
8.87
⑧′
28.96
27.30
5.73
26.38
24.19
8.30
24.18
22.27
7.90
⑧
29.18
27.96
4.18
26.57
24.54
7.64
24.57
22.33
9.12
23.20
21.10
9.05
20
④
33.29
32.34
2.85
30.63
29.02
5.26
28.49
26.87
5.69
26.99
25.81
4.37
④′
33.14
31.95
3.59
29.75
28.62
3.80
28.11
26.79
4.70
⑥
33.32
31.79
4.59
30.04
28.45
5.29
28.99
26.86
7.35
27.42
26.72
2.55
⑧′
34.02
32.39
4.79
30.69
28.95
5.67
29.59
27.25
7.91
⑧
34.62
32.84
5.14
31.44
29.29
6.84
29.51
27.31
7.46
28.21
27.00
4.29
Table 2. Comparison of measured and simulated water depths on Side I of each side weir at a discharge of 25 L/s
Due to the diversion caused by the side weir, there was a rapid variation in flow near the side weir in the main channel. In order to investigate the impact of the side weir on water flow in the main channel, water surface profiles on Side I, Side II, and Side III were plotted with a side weir width and height both set at 20 cm at a discharge rate of 25 L/s (Figure 6). As depicted in Figure 6, within a certain range of the upstream end of the main channel, water depths on Side I, Side II, and Side III were nearly equal with almost horizontal profiles. As the distance between the location of water flow and the upstream end of the weir crest decreased gradually, there was a gradual decrease in water depth on Side I along with an inclined trend in its corresponding profile; however, both Side II and Side III still maintained almost horizontal profiles. When approaching closer to the side weir area with flowing water, there was an evident reduction in water depth on Side I accompanied by a significant downward trend visible across an expanded decline range. The minimum point occurred near the upstream end of the weir crest before gradually increasing again towards downstream sections. At the crest section of the side weir, there is an upward trend observed in the water surface. The water surface tended to stabilize downstream of the main channel within a certain range from the downstream end of the weir crest. There was no significant change in the water surface profiles of Side Ⅱ and Side Ⅲ in the crest section. It can be inferred that the side weir entrance effect occurred only between Side Ⅰ and Side Ⅱ. M. Emin reported the same pattern [10].
Figure 6. Water surface profiles on Side I, Side II, and Side III with a side weir width of 20 cm and height of 15 cm at a discharge of 25 L/s.
For a more accurate study on the entrance effect of the side weir on the Water Surface Profile (WSP) for Side I; a comparative analysis conducted using different widths but the same height (15 cm) at a discharge rate of 25 L/s is presented through Figure 7, Figure 8, Figure 9 and Figure 10.
Figure 7. Water surface profile on Side Ⅰ with a side weir width of 20 cm and height of 15 cm at a discharge of 25 L/s.
Figure 8. Water surface profile on Side Ⅰ with a side weir width of 30 cm and height of 15 cm at a discharge of 25 L/s.
Figure 9. Water surface profile on Side Ⅰ with a side weir width of 40 cm and height of 15 cm at a discharge of 25 L/s.
Figure 10. Water surface profile on Side Ⅰ with a side weir width of 47 cm and height of 15 cm at a discharge of 25 L/s.
According to Figure 7, Figure 8, Figure 9 and Figure 10, the water depth upstream of the main channel started to decrease as it approached the upstream end of the weir crest and then gradually increased at the weir crest section. In other words, the water surface profile exhibited a backwater curve along the length of the weir crest. The water depth remained relatively stable downstream of the main channel within a certain range from the downstream end of the weir crest. Additionally, there was a higher water depth downstream of the main channel compared to that upstream of the main channel. Furthermore, an increase in the width of the side weir led to a gradual reduction in fluctuations on its water surface.
5.2. Velocity Distribution
The law of flow velocity distribution near the side weir is the focus of research and analysis, so the simulated and measured values of flow velocity near the side weir were compared and analyzed. Take the discharge of 25 L/s, the height of 15 cm, and the width of 30 cm of the side weir as an example to illustrate. Figure 11 shows the measured and simulated velocity distribution in the x-direction of cross-section ④. As can be seen from Figure 11, the diagrams of the measured and simulated velocity distribution were relatively consistent, and the maximum absolute relative error between the measured and simulated values at the same measurement point was 9.37%, and the average absolute relative error was 3.97%, which indicated a satisfactory agreement between the experimental and simulated results.
Figure 11. Velocity distribution in the x-direction of section ④: when the discharge is 25 L/s, the height of the side weir is 15 cm and the width of the side weir is 30 cm. (a) Measured velocity distribution; (b) Simulated velocity distribution.
From Figure 11, it can be seen that the flow velocity gradually increased from the bottom of the channel towards the water surface in the Z-direction, and the flow velocity gradually increased from Side Ⅲ to Side Ⅰ in the Y-direction. The maximum flow velocity occurred near the weir crest.
Figure 12 shows the distribution of flow velocity at different depths (z/P = 0.3, z/P = 0.8, z/P = 1.6) with a side weir width of 30 cm and height of 15 cm at a discharge of 25 L/s. The water flow line began to bend at a certain point upstream of the main channel, and the closer it was to the upstream end of the weir crest, the greater the curvature. The maximum curvature occurred at the downstream end of the weir crest. The flow patterns at the bottom, near the side weir crest, and above the side weir crest were significantly different. There was a reverse flow at the bottom of the main channel, where the forward and reverse flows intersect, resulting in a detention zone. The maximum flow velocity at the bottom layer occurred at the upstream end of the side weir crest. When the location of water flow approached the weir crest, the maximum flow velocity occurred at the upstream end of the weir crest. The maximum flow velocity on the water surface occurred at the downstream end of the weir crest. As the water depth decreased, the position of the maximum flow velocity gradually moved from the upstream end of the side weir to the downstream end of the side weir.
Figure 12. Distribution of flow velocity at different depths with a side weir width of 30 cm and height of 15 cm at a discharge of 25 L/s. (a) z/P = 0.3; (b) z/P = 0.8; (c) z/P = 1.6.
5.3. Side Channel Pressure Distribution
When water flowed through the side weir, an upstream water level was formed, resulting in a pressure zone at the junction with the side channel. This pressure zone led to increased water pressure on the floor of the side channel, which affected its stability and durability. In small channels or fields where erosion resistance is weak, excessive pressure can cause scour holes. Therefore, analyzing the pressure distribution in the side channel is necessary to select an appropriate height and width for the side weir that effectively reduces its impact on the bottom plate.
To investigate the impact of side weir width on hydraulic characteristics, pressure data was collected at a discharge rate of 25 L/s for side weirs with heights of 20 cm and widths ranging from 20 cm to 47 cm. The pressure distribution map was drawn, as shown in Figure 13.
Figure 13. Comparison of pressure distribution on the bottom plate of the side channel with different widths of side weirs when the discharge is 25 L/s and the height of side weirs is 20 cm. (a) P = 20 cm, b = 20 cm; (b) P = 20 cm, b = 30 cm; (c) P = 20 cm, b = 40 cm; (d) P = 20 cm, b = 47 cm.
As can be seen from Figure 13, the pressure at the bottom of the side channel decreased as the width of the side weir increased. This uneven distribution of water flow on the weir was caused by the sharp bending of water flow lines and the influence of centrifugal inertia force over a short period. After passing through the side weir, the water flow became symmetrically distributed with respect to the axis of the side channel, leaning towards the right bank at a certain distance. As we increased the width of the side weir, we noticed that its position gradually approached the side weir and maximum pressure decreased at this location where the water tongue formed due to flowing through it (Figure 13). For a constant height (20 cm) but varying widths (20 cm, 30 cm, 40 cm, and 47 cm), we measured maximum pressures at these positions as follows: 103,713 Pa, 103,558 Pa, 103,324 Pa, and 103,280 Pa, respectively. Consequently, increasing width reduced the impact on the side channel from water flowing through it while changing pressure distribution from concentration to dispersion in a vertical direction. In the practical application of side weirs, appropriate height should be selected based on the bottom plate’s capacity to withstand the pressure exerted by flowing water within channels.
To investigate how height affects the hydraulic characteristics of rectangular side weirs further (Figure 14), we extracted pressures on bottom plates when discharge was fixed at 25 L/s while varying heights were set as follows: 7 cm, 10 cm, 15 cm, and 20 cm, respectively.
Figure 14. Comparison of pressure distribution on the bottom plate of the side channel with different heights of side weirs when discharge is 25 L/s and the width of side weirs is 20 cm. (a) P = 7 cm, b = 20 cm; (b) P = 10 cm, b = 20 cm; (c) P = 15 cm, b = 20 cm; (d) P = 20 cm, b = 20 cm.
As shown in Figure 14, when the width of the side weir was constant, the pressure at the bottom of the side channel increased with the height of the side weir. As the height of the side weir increased, the water tongue formed by flow through the side weir gradually moved away from it in a downstream direction. In terms of vertical water flow, as the height of the side weir increased, the position of maximum pressure at which the water tongue falls shifted closer to the axis of the side channel from its right bank. Moreover, an increase in height resulted in higher maximum pressure at this falling point. For a constant width (20 cm) and varying heights (7 cm, 10 cm, 15 cm, and 20 cm), corresponding maximum pressures at this landing point were measured as 102,422 Pa, 102,700 Pa, 103,375 Pa, and 103,766 Pa, respectively. Consequently, increasing width led to a greater impact on both flow through and pressure distribution within the side channel; transforming it from scattered to concentrated along its lengthwise direction. Therefore, when applying such weirs practically one should select an appropriate width based on what pressure can be sustained by their respective channel bottom plates.
5.4. Discharge Coefficient
Based on dimensionless analysis, the influencing parameters of the discharge coefficient were obtained. To study the effect of parameters Fr1, b/h1, and P/h1, discharge coefficient values were plotted against Fr1, b/h1, and P/h1, shown in Figure 15, Figure 16 and Figure 17. The discharge coefficient decreased as parameters Fr1 and b/h1 increased. The discharge coefficient increased as parameter P/h1 increased. As Uyumaz and Muslu reported in a previous study, the variation of the discharge coefficient with respect to the Froude number showed a second-degree curve for a subcritical regime [4].
Figure 15. Variation of discharge coefficient values against Froude number.
Figure 16. Variation of discharge coefficient values against the percentage of the side weir width to the upstream flow depth over the side weir.
Figure 17. Variation of discharge coefficient values against the percentage of the side weir height to the upstream flow depth over the side weir.
Quantitative analysis between discharge coefficient values and parameters Fr1, b/h1, and P/h1 was conducted using data analysis software (IBM SPSS Statistics 19). The various coefficients obtained are shown in Table 3.
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig
B
Std. Error
Beta (β)
Constant
−1.294
0.155
−8.369
0.000
Fr1
3.430
0.286
3.401
12.013
0.000
b/h1
−0.004
0.004
−0.045
−0.944
0.348
P/h1
2.401
0.167
4.064
14.394
0.000
Table 3. Coefficient.
The value of t and Sig are the significance results of the independent variable, and the value of Sig corresponding to the value of t is less than 0.05, indicating that the independent variable has a significant impact on the dependent variable. Therefore, the values of Sig corresponding to the parameters Fr1 and P/h1 were less than 0.05, indicating that the parameters Fr1 and P/h1 have a significant impact on the discharge coefficient. On the contrary, the parameter b/h1 has less impact on the discharge coefficient. Therefore, quantitative analysis between discharge coefficient values and parameters Fr1, and P/h1 was conducted using data analysis software by removing factor b/h1. The model summary, ANOVA, and coefficient obtained are shown respectively in Table 4, Table 5 and Table 6. R and adjusted R square in Table 4 were approaching 1, which indicated the goodness of fit of the regression model was high. The value of Sig corresponding to the value of F in Table 5 was less than 0.05, which indicated that the regression equation was useful. The values of Sig corresponding to the parameters Fr1 and P/h1 in Table 6 were less than 0.05, indicating that the parameters Fr1 and P/h1 have a significant impact on the discharge coefficient.
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
1
0.913 a
0.833
0.829
0.03232
Table 4. Model Summary b. Note: a. Predictors:(Constant), Fr1, P/h1; b. Discharge coefficient.
Model
Sum of Squares
df
Mean Square
F
Sig
1
Regression
0.402
2
0.201
192.545
0.000 a
Residual
0.080
77
0.001
Total
0.483
79
Table 5. ANOVA b. Note: a. Predictors:(Constant), Fr1, P/h1; b. Discharge coefficient.
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig
B
Std. Error
Beta (β)
Constant
−1.326
0.151
−8.796
0.000
Fr1
3.479
0.281
3.449
12.396
0.000
P/h1
2.427
0.164
4.108
14.765
0.000
Table 6. Coefficient a. Note: a. Predictors:(Constant), Fr1, P/h1.
Based on the above analysis, the flow coefficient formula has been obtained, shown as follows:
Discharge formula were obtained by substituting Equation (15) into Equation (12), as shown in Equation (16).
where Q ∈ [0.006, 0.030], m3/s; b ∈ [0.20, 0.47], m; P ∈ [0.07, 0.20], m.
Figure 18 showed the measured discharge coefficient values with those calculated from discharge formulas in Table 3. The scatter of the data with respect to perfect line was limited to ±10%.
Figure 18. Comparison of the measured discharge coefficient values with those calculated from discharge formulas in Table 3.
6. Discussions
Determining water surface profile near the side weir in the main channel is one of the tasks of hydraulic calculation for side weirs. As the water flows through the side weir, discharge in the main channel is gradually decreasing, namely dQ/ds<0. According to the Equation (17) derived from Qimo Chen [3], it can be inferred that the value of 𝑑ℎ/𝑑𝑠 is greater than zero in subcritical flow (Fr < 1), that is, the water surface profile near the side weir in the main channel is a backwater curve. Due to the side weir entrance effect at the upstream end, water surface profiles drop slightly at the upstream end of the side weir crest, as EI-Khashab [28] and Emiroglu et al. [29] reported in previous experimental studies.
In this study, the water surface profile exhibited a backwater curve along the length of the weir crest. Therefore, during side weir application, it is crucial to ensure that downstream water levels do not exceed the highest water level of the channel.
The head on the weir is one of the important factors that flow over side weirs depends on. At the same time, the head depends on the water surface profile near the side weir in the main channel. Therefore, further research on the quantitative analysis of water surface profile needs to be conducted. Mohamed Khorchani proposed a new approach based on the video monitoring concept to measure the free surface of flow over side weirs. It points out a new direction for future research [8].
The maximum flow velocity, a key parameter in assessing the efficiency of a weir, occurs at the upstream end of the weir crest, typically near the crest. This is attributed to the convergence of the flow as it approaches the crest, resulting in a significant increase in velocity. It was found that in this study the minimum flow velocity occurred at the bottom of the main channel away from the side weir. Under such conditions, the accumulation of sediments could lead to siltation, which in turn can affect the accuracy of flow measurement through side weirs. This is because the presence of sediments can alter the flow patterns and cause errors in the measurement. Therefore, it becomes crucial to explore methods to optimize the selection of side weirs in order to minimize or eliminate the effects of sedimentation on flow measurement.
Pressure distribution plays a crucial role in ensuring structural safety for side weirs since small channels and field inlets have relatively limited pressure-bearing capacities. Therefore, it is important to select an appropriate geometrical parameter of rectangular side weirs based on their ability to withstand the pressure exerted on their bottom combined with pressure distribution data at the bottom of the side channel we have obtained in this study.
The discharge coefficient formula (Equation (15)), which incorporates Fr1 and P/h1, was derived based on dimensional analysis. However, it is worth noting that previous research has contradicted this formula by suggesting that the discharge coefficient solely depends on the Froude number. This conclusion can be observed in this study such as in Equations (18)–(23) in Table 7 of the manuscript [30,31,32,33,34,35], which clearly demonstrate the dependency of the discharge coefficient on the Froude number. In contrast, our derived discharge coefficient formula (Equation (15)) offers a more streamlined and simplified approach compared to Equation (25) [36] and Equation (29) [10]—making it easier to comprehend and apply—an advantageous feature particularly valuable in fluid dynamics where intricate calculations can be time-consuming. Furthermore, our derived discharge coefficient formula (Equation (15)) exhibits a broader application scope than that of Equation (24) [37] as shown in Table 8. Equation (26) [38] and Equation (27) [5] are specifically applicable under high flow discharge conditions. Conversely, our derived discharge coefficient formula (Equation (15)) is better suited for low-flow discharge conditions.
Table 7. Discharge coefficient formulas of rectangular side weirs presented in previous studies.
Discharge/(L·s−1)
Width of Side Weir/cm
Height of Side Weir/cm
Number of Formula
10~14
10~20
6~12
(24)
35–100
20~75
1~19
(26), (27)
6~30
20~47
7~20
(15)
Table 8. Application scope of discharge coefficient formulas.
In addition to the factors studied in the paper, factors such as the sediment content in the flow, the bottom slope, and the cross-section shape of the channel also have a certain impact on the hydraulic characteristics of the side weir. Further numerical simulation methods can be used to study the hydraulic characteristics and the influencing factors of the side weir. Water measurement facilities generally require high accuracy of water measurement, the flow of sharp-crested side weirs is complex, and the water surface fluctuates greatly. While conducting numerical simulations, experimental research on prototype channels is necessary to ensure the reliability of the results and provide reference for the body design and optimization of side weirs in small channels and field inlets.
7. Conclusions
This paper presents a comprehensive study that encompasses both experimental and numerical simulation research on rectangular side weirs of varying heights and widths within rectangular channels. A thorough analysis of the experimental and numerical simulation results has been conducted, leading to the derivation of several notable conclusions:
A comparative analysis was conducted on the measured and simulated values of water depth and flow velocity. Both of the maximum absolute relative errors were within 10%, which indicated that the numerical simulation of the side weir was feasible and effective.
The water surface profile exhibited a backwater curve along the length of the weir crest. The side weir entrance effect occurred only between Side Ⅰ and Side Ⅱ. This indicates that flow patterns and associated hydraulic forces at the weir entrance play a crucial role in determining water level distribution along the weir crest.
The maximum flow velocity of the cross-section at the upstream end of the weir crest occurred near the weir crest, while the minimum flow velocity occurred at the bottom of the main channel away from the side weir. As the water depth decreased, the position of the maximum flow velocity gradually moved from the upstream end of the side weir to the downstream end of the side weir.
When the height of the side weir remains constant, an increase in the width of the side weir leads to a decrease in pressure at the bottom of the side channel. Conversely, when the width of the side weir is kept constant, an increase in its height results in an increase in pressure at the bottom of the side channel. Therefore, during practical applications involving side weirs, it is crucial to select an appropriate weir width based on the maximum pressure that can be sustained by the channel’s bottom plate.
The discharge coefficient was found to depend on the upstream Froude number Fr1 and the percentage of the side weir height to the upstream flow depth over the side weir P/h1. The relationship between the discharge coefficient and parameters Fr1 and P/h1 was obtained using multiple regression analysis, which was of linear form and provided an easy means to estimate the discharge coefficient. The discharge formula is of high accuracy with relative errors within 10%, which met the water measurement accuracy requirements of small channels in irrigation areas.
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Enhanced understanding of flow structure in braided rivers is essential for river regulation, flood control, and infrastructure safety across the river. It has been revealed that the basic morphological element of braided rivers is confluence-bifurcation units. However, flow structure in these units has so far remained poorly understood with previous studies having focused mainly on single confluences/bifurcations. Here, the flow structure in a laboratory-scale confluence-bifurcation unit is numerically investigated based on the FLOW–3D® software platform. Two discharges are considered, with the central bars submerged or exposed respectively when the discharge is high or low. The results show that flow convergence and divergence in the confluence-bifurcation unit are relatively weak when the central bars are submerged. Based on comparisons with a single confluence/bifurcation, it is found that the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit reign over those of the downstream central bar. Concurrently, the high-velocity zone in the confluence-bifurcation unit is less concentrated than that in a single confluence while being more concentrated than that observed in a single bifurcation. The present work unravels the flow structure in a confluence-bifurcation unit and provides a unique basis for further investigating morphodynamics in braided rivers.
1 Introduction
Confluences and bifurcations commonly exist in alluvial rivers and usually are important nodes of riverbed planform (Szupiany et al., 2012; Hackney et al., 2018). Flow convergence and divergence in these junctions result in highly three-dimensional (3D) flow characteristics, which greatly influence sediment transport, and hence riverbed evolution and channel formation (Le et al., 2019; Xie et al., 2020). Braided rivers, characterized by unstable networks of channels separated by central bars (Ashmore, 2013), have confluence-bifurcation units as their basic morphological elements (Ashmore, 1982; 1991; 2013; Federici & Paola, 2003; Jang & Shimizu, 2005). In particular, confluence-bifurcation units exhibit a distinct morphology from single confluences/bifurcations and bifurcation-confluence regions because two adjacent central bars are included. Within a confluence-bifurcation unit, two tributaries converge at the upstream bar tail and soon diverge to two anabranches again at the downstream bar head. Therefore, the flow structure in the unit may be significantly influenced by both the two central bars, and thus considerably different from that in single confluences, single bifurcations, and bifurcation-confluence regions, where the flow is affected by only one central bar. Enhanced understanding of flow structure in confluence-bifurcation units is urgently needed, which is essential for water resources management, river regulation, flood control, protection of river ecosystems and the safety of infrastructures across the rivers such as bridges, oil pipelines and communication cables (Redolfi et al., 2019; Ragno et al., 2021).
The flow dynamics, turbulent coherent structures, and turbulent characteristics in single confluences have been widely studied since the 1980s (Yuan et al., 2022). Flow dynamics at river channel confluences have been systematically and completely analyzed, which can be characterized by six major regions of flow stagnation, flow deflection, flow separation, maximum velocity, flow recovery and distinct shear layers (Best, 1987). For example, the field observation of Roy et al. (1988) and Roy and Bergeron (1990) highlighted the flow separation zones and recirculation at downstream natural confluence corners. Ashmore et al. (1992) measured the flow field in a natural confluence and found flow accelerates suddenly at the confluence junction with two separated high-velocity cores merging into one single core at the channel centre. De Serres et al. (1999) investigated the three-dimensional flow structure at a river confluence and identified the existence of the mixing layer, stagnation zones, separation zones and recovery zones. Sharifipour et al. (2015) numerically studied the flow structure in a 90° single confluence and found that the size of the separation zone decreases with the width ratio between the tributary and the main channel. Recently, three main classes of large-scale turbulent coherent structures (Duguay et al., 2022) have been presented, i.e. vertical-orientated vortices or Kelvin-Helmholtz instabilities (Rhoads & Sukhodolov, 2001; Constantinescu et al., 2011; 2016; Biron et al., 2019), channel-scale ‘back-to-back’ helical cells, (Mosley, 1976; Ashmore, 1982; Ashmore et al., 1992; Ashworth, 1996; Best, 1987; Rhoads & Kenworthy, 1995; Bradbrook et al., 1998; Lane et al., 2000), and smaller, strongly coherent streamwise-orientated vortices (Constantinescu et al., 2011; Sukhodolov & Sukhodolova, 2019; Duguay et al., 2022). However, no consensus on a universal turbulent coherent structure mode has been reached so far (Duguay et al., 2022). In addition, some studies (Ashworth, 1996; Constantinescu et al., 2011; Sukhodolov et al., 2017; Le et al., 2019; Yuan et al., 2023) have focused on turbulent characteristics, e.g. turbulent kinetic energy, turbulent dissipation rate and Reynolds stress, which can be critical parameters to further explaining the diversity of these turbulent coherent structure modes.
Investigations on the flow structure in single bifurcations have mainly focused on hydrodynamics in anabranches (Hua et al., 2009; van der Mark & Mosselman, 2013; Iwantoro et al., 2022) and around bifurcation bars (McLelland et al., 1999; Bertoldi & Tubino, 2005; 2007; Marra et al., 2014), whereas few studies have considered the effects of bifurcations on the upstream flow structure. Thomas et al. (2011) found that the velocity core upstream of the bifurcation is located near the water surface and towards the channel center in experimental investigations of a Y-shaped bifurcation. Miori et al. (2012) simulated flow in a Y-shaped bifurcation and found two circulation cells upstream of the bifurcation with flow converging at the water surface and diverging near the bed. Szupiany et al. (2012) reported velocity decreasing and back-to-back circulation cells upstream of the bifurcation junction in the field observation of a bifurcation of the Rio Parana River. These investigations provide insight into how bifurcations affect the flow patterns upstream, yet there is a need for further research on the dynamics of flow occurring immediately before the bifurcation junction.
Generally, the findings of studies on bifurcation-confluence regions are similar to those concerning single confluences and bifurcations. Hackney et al. (2018) measured the hydrodynamic characteristics in a bifurcation-confluence of the Mekong River and found the velocity cores located at the channel centre and strong secondary current occurring under low discharges. Le et al. (2019) reported a high-turbulent-kinetic-energy (high-TKE) zone located near the bed in their numerical simulation of flow in a natural bifurcation-confluence region. Moreover, a stagnation zone was found upstream of the confluence and back-to-back secondary current cells were detected at the confluence according to Xie et al. (2020) and Xu et al. (2022). Overall, these studies have further unraveled the flow patterns in river confluences and bifurcations.
Unfortunately, limited attention has been paid to the flow structure in confluence-bifurcation units. Parsons et al. (2007) investigated a large confluence-bifurcation unit in Rio Parana, Argentina, and no classical back-to-back secondary current cells were observed under a discharge of 12000 m3·s−1. To date, the differences in flow structure between confluence-bifurcation units and single confluences/bifurcations have remained far from clear. In addition, although the effects of discharge on flow structure have been investigated in several studies on single confluences/bifurcations, (Hua et al., 2009; Le et al., 2019; Luz et al., 2020; Xie et al., 2020; Xu et al., 2022), cases with fully submerged central bars were not considered, which is typical in braided rivers during floods. In-depth studies concerning these issues are urgently needed to gain better insight into the flow structure in confluence-bifurcation units of braided rivers.
This paper aims to (1) reveal the 3D flow structure in a confluence-bifurcation unit under different discharges and (2) elucidate the differences in the flow structure between confluence-bifurcation units and single confluence/bifurcation cases. Using the commercial computational fluid dynamics software FLOW-3D® (Version 11.2; https://www.flow3d.com; Flow Science, Inc.), fixed-bed simulations of a laboratory-scale confluence-bifurcation unit are conducted, and cases of a single confluence/bifurcation are also included for comparison. Two discharges are considered, with the central bars fully submerged or exposed respectively when the discharge is high or low. Based on the computational results, the 3D flow structure in the confluence-bifurcation unit conditions is analyzed from various aspects including free surface elevation, time-averaged flow velocity distribution, recirculation vortex structure, secondary current, and turbulent kinetic energy and dissipation rate. In particular, the flow structure in the confluence-bifurcation unit is compared with that in the single confluence/bifurcation cases to unravel the differences.h
2. Conceptual flume and computational cases
2.1. Conceptual flume
In this paper, a laboratory-scale conceptual flume is designed and used in numerical simulations. Figure 1(a–d) shows the morphological characteristics of the flume. To ensure that the conceptual flume reflects morphology features of natural braided channels, key parameters governing the flume morphology, e.g. unit length, width, and channel width-depth ratio, are determined according to studies on morphological characteristics of natural confluence-bifurcation units (Hundey & Ashmore, 2009; Ashworth, 1996; Orfeo et al., 2006; Parsons et al., 2007; Sambrook Smith et al., 2005; Kelly, 2006; Ashmore, 2013; Egozi & Ashmore, 2009; Redolfi et al., 2016; Ettema & Armstrong, 2019).
Figure 1. The sketch of the conceptual flume: (a) the original flume, (b) the central bar: (c) the sketch of cross-section C-C, (d) the sketch of cross-section D-D, (e) the modified part for the single confluence, (f) the modified part for the single bifurcation, (g) the position of different cross-sections. The red dashed boxes denote the regions of primary concern.
Figure 1. The sketch of the conceptual flume: (a) the original flume, (b) the central bar: (c) the sketch of cross-section C-C, (d) the sketch of cross-section D-D, (e) the modified part for the single confluence, (f) the modified part for the single bifurcation, (g) the position of different cross-sections. The red dashed boxes denote the regions of primary concern.
2.1.1. Length and width scales of the confluence-bifurcation unit
The length and width scales of the flume are first determined. The inner relation among the length LCB and average width B of a confluence-bifurcation unit and the average width Bi of a single branch was statistically studied by Hundey and Ashmore (2009), which indicates the following relations: 𝐿CB =(4∼5)𝐵 (1) 𝐵 =1.41𝐵𝑖 (2) In addition, Ashworth (1996) gave B = 2Bi in his experimental research on mid-bar formation downstream of a confluence, while the confluence-bifurcation unit of Rio Parana, Argentina has a relation of B≈1.71Bi (Orfeo et al., 2006; Parsons et al., 2007). Accordingly, the following relations are used in the present paper: 𝐿CB =4𝐵 (3) 𝐵 =1.88𝐵𝑖 (4) where LCB = 6 m, B = 1.5 m and Bi = 0.8 m.
2.1.2. Central bar morphology
The idealized plane pattern of central bars in braided rivers is a slightly fusiform leaf shape with a short upstream side and a long downstream side (Ashworth, 1996; Sambrook Smith et al., 2005; Kelly, 2006; Ashmore, 2013). To simplify the design, the bar is approximated as a combination of two different semi-ellipses (Figure 1(b)). The major axis Lb is two to ten times longer than the minor axis Bb according to the statistical data in Kelly’s study, and the regression equation is given as (Kelly, 2006): 𝐿𝑏=4.62𝐵0.96𝑏 (5) In this study, the bar width Bb is set as 0.8 m, whilst the lengths of downstream (LT1) and upstream sides (LT2) are 2 and 1.5 m, respectively (Figure 1(b)). Thus, the relation of Lb and Bb is given as: 𝐿𝑏=(𝐿𝑇1+𝐿𝑇2)=4.375𝐵𝑏 (6) The lengths of the inlet and outlet parts are determined as Lin = Lout = 8 m, which ensures negligible effects of boundary conditions without exceptional computational cost.
2.1.3. Width-depth ratio
Channel flow capacity can be significantly affected by cross-section shapes. For natural rivers, cross-section shapes can be generalized into three sorts based on the following width-depth curve (Redolfi et al., 2016): 𝐵=𝜓𝐻𝜑(7) Braided rivers usually have ψ = 5∼50 and φ>1, which indicates a rather wide and shallow cross-section. The central bar form should also be taken into account, so a parabolic cross-section shape is used here with ψ = 8 and φ>1 (Figure 1(c,d)).
2.1.4. Bed slope
In addition, natural braided rivers are usually located in mountainous areas and thus have a relatively large bed slope. According to flume experiments and field observations, the bed slope Sb is mostly in the range of 0.01∼0.02, and a few are below 0.01 (Ashworth, 1996; Egozi & Ashmore, 2009; Ashmore, 2013; Redolfi et al., 2016; Ettema & Armstrong, 2019). In this study, Sb takes 0.005.
2.1.5. Complete sketch of the conceptual flume
In summary, the flume is 29 m long, 2.4 m wide, and 0.6 m high. The plane coordinates (x-direction and y-direction) used in the calculation process are shown in Figure 1 (a). Note that the inlet corresponds to x = 0 m, and the centreline of the flume is located at y = 1.3 m. Besides, the thalweg elevation of the outlet is set as z = 0 m.
2.2. Computational cases
As stated before, the first aim of this paper is to reveal the flow structure in the confluence-bifurcation unit under different discharges. Therefore, two basic cases are set first: (1) case 1a under a low discharge (0.05 m3·s−1) with exposed central bars and (2) case 2a under a high discharge (0.30 m3·s−1) with fully submerged central bars. A total of 22 cross-sections are identified to examine the results (Figure 1(g)).
Further, cases of a single confluence/bifurcation are generated by splitting the original confluence-bifurcation unit into two parts. Part 1 only includes the upstream central bar and focuses on the flow convergence downstream of CS04 (Figure 1(e)), while Part 2 only includes the downstream central bar and focuses on the flow divergence upstream of CS19 (Figure 1(f)). Notably, the numbers of corresponding cross-sections in the original flume are reserved to facilitate comparison. The outlet section of the single confluence as well as the inlet section of the single bifurcation is extended to make the total length equivalent to the original flume (29 m). Also, two discharge conditions (0.05 and 0.30 m3·s−1), which correspond to exposed and fully submerged central bars, are considered for the single confluence/bifurcation. In total, six computational cases are conducted, as listed in Table 1. As the conceptual flume is designed to be symmetrical about the centreline, the momentum flux ratio (Mr) of the two branches should be 1 in all six cases. This is confirmed by further examining the computational results.
Case
Configuration
Qin (m3·s−1)
Zout (m)
Mr
Condition of bars
1a
CBU
0.05
0.15
1
Exposed
1b
SC
0.05
0.15
1
Exposed
1c
SB
0.05
0.15
1
Exposed
2a
CBU
0.30
0.34
1
Submerged
2b
SC
0.30
0.34
1
Submerged
2c
SB
0.30
0.34
1
Submerged
Table 1. Computational cases with inlet and outlet boundary conditions.
3. Numerical method
In this section, the 3D Large Eddy Simulation (LES) model integrated in the FLOW-3D® (Version 11.2; https://www.flow3d.com; Flow Science, Inc.) software platform is introduced, including governing equations and boundary conditions. Information on computational meshes with mesh independence test can be found in the Supplementary material.
3.1. Governing equations
The LES model was applied in the present paper to simulate flow in the laboratory-scale confluence-bifurcation unit. The LES model has been proven to be effective in simulating turbulent flow in river confluences and bifurcations (Constantinescu et al., 2011; Le et al., 2019). The basic idea of the LES model is that one should directly compute all turbulent flow structures that can be resolved by the computational meshes and only approximate those features that are too small to be resolved (Smagorinsky, 1963). Therefore, a filtering operation is applied to the original Navier-Stokes (NS) equations for incompressible fluids to distinguish the large-scale eddies and small-scale eddies (Liu et al., 2018). The filtered NS equations are then generated, which can be expressed in the form of a Cartesian tensor as (Liu, 2012):
(10) where ¯𝑢𝑖 is the resolved velocity component in the i – direction (i goes from 1 to 3, denoting the x-, y – and z-directions, respectively); t is the flow time; ρ is the density of the fluid; ¯𝑝 is the pressure; ν is the kinematic viscosity; τij is the sub-grid scale (SGS) stress; ¯𝐺𝑖 is the body acceleration. In FLOW–3D®, the full NS equations are discretized and solved using the finite-volume/finite-difference method (Bombardelli et al., 2011; Lu et al., 2023).
Due to the filtering process, the velocity can be divided into a resolved part (¯𝑢(𝑥,𝑡)) and an approximate part (𝑢′(𝑥,𝑡)) which is also known as the SGS part (Liu, 2012). To achieve model closure, the standard Smagorinsky SGS stress model is introduced here (Smagorinsky, 1963): 𝜏ij−13𝜏kk𝛿ij=−2𝜈SGS¯𝑆ij(11) where νSGS is the SGS turbulent viscosity, and ¯𝑆ij is the resolved rate-of-strain tensor for the resolved scale defined by (Smagorinsky, 1963): ¯𝑆ij=12(∂¯𝑢𝑖∂𝑥𝑗+∂¯𝑢𝑗∂𝑥𝑖)(12) In the standard Smagorinsky SGS stress model, the eddy viscosity is modelled by (Smagorinsky, 1963): 𝜈SGS=(𝐶𝑠¯𝛥)2∣¯𝑆∣,∣¯𝑆∣=√2¯𝑆ij¯𝑆ij(13) ¯𝛥=(ΔxΔyΔz)1/3(14) where Cs is the Smagorinsky constant, Δx, Δy, and Δz are mesh scales. In FLOW–3D®, Cs is between 0.1 to 0.2 (Smagorinsky, 1963). One of the key problems in simulating 3D open channel flow is the calculation of free surface. FLOW–3D® uses the Volume of Fluid (VOF) method (Hirt & Nichols, 1981) to track the change of free surface. The VOF method introduces a fluid phase fraction function f to characterize the proportion of a certain fluid in each mesh cell. In that case, the surface position can be precisely located if the mesh cell is fine enough. To monitor the change of f with time and space, the following convection equation is added:
For open channel flow, only two kinds of fluids are involved: water and air. If f is the fraction of water, the state of the fluid in each mesh cell can be defined as:
In FLOW–3D®, the interface between water and air is assumed to be shear-free, which means that the drag force on the water from the air is negligible. Moreover, in most cases, the details of the gas motion are not crucial for the heavier water motion so the computational processes will be more efficient.
3.2. Boundary conditions
Six boundary conditions need to be preset in the 3D numerical simulation process. Discharge boundary conditions are used for the inlet of the flume, where the free surface elevation is automatically calculated based on the free surface elevation boundary conditions set for the outlet. The specific information on the inlet and outlet boundary conditions for all computational cases is shown in Table 1. Moreover, because the free surface moves temporally, the free surface boundary conditions are just set as no shear stress and having a normal pressure, and the position of the free surface will be automatically adjusted over time by the VOF method in FLOW–3D®. Furthermore, the bed and two side walls are all set to be no-slip for fixed bed conditions, and a standard wall function is employed at the wall boundaries for wall treatment.
The inlet turbulent boundary conditions also need to be considered. They are set by default here. The turbulent velocity fluctuations V’ are assumed to be 10% of the mean flow velocity with the turbulent kinetic energy (TKE) (per unit mass) equaling 0.5V’2. The maximum turbulent mixing length is assumed to be 7% of the minimum computational domain scale, and the turbulent dissipation rate is evaluated automatically from the TKE.
4. Results and discussion
4.1. Flow structure in the confluence-bifurcation unit
4.1.1. Free surface elevation
Figure 2 shows the free surface elevation at five different longitudinal profiles (i.e. α = 0.2, 0.4, 0.5, 0.6, 0.8) for cases 1a and 2a. The parameter α was defined as follows:𝛼=𝑠𝐵(17) where s is the transverse distance between a certain profile and the left boundary of the flume. In general, the longitudinal change of free surface in the two cases is very similar despite different discharge levels. The free surface elevation decreases as the channel narrows from the upstream bifurcation to the front of the confluence-bifurcation unit. On the contrary, when the flow diverges again at the end of the confluence-bifurcation unit, the free surface elevation increases with channel widening. However, whether the fall or rise of free surface elevation in case 1a is much sharper than that in case 2a, especially at profiles with α = 0.2 and 0.8 (Figure 2(a)), which indicates there may be distinct flow states between the two cases. To further illustrate this finding, the Froude number Fr at different cross-sections (CS08∼CS15) is examined. In case 2a, the flow remains subcritical within the confluence-bifurcation unit. By contrast, in case 1a, a local supercritical flow is observed near the side banks of CS09 (i.e. α = 0.2 and 0.8), with Fr being about 1.2. This local supercritical flow can lead to a hydraulic drop followed by a hydraulic jump, which accounts for the sharp change of the free surface. The foregoing reveals that when central bars are exposed under relatively low discharge, supercritical flow is more likely to occur near the side banks of the confluence junction due to flow convergence.
Figure 2. Five time-averaged free surface elevation profiles in the confluence-bifurcation unit, in which α denotes the lateral position of the certain profile. Note that the black dashed line denotes the position of CS09, where Fr is about 1.2 near the side banks (α = 0.2 and 0.8) in case 1a. Z’ = z/h2, X’ = x/B, h2 is the maximum flow depth at the outlet boundary of cases 2a, 2b and 2c, h2 = 0.34 m.
Moreover, in both cases 1a and 2a, the free surface is higher at the channel centre than near the side banks, whether at the front or the end of the confluence-bifurcation unit. Thus, lateral free surface slopes from the centre to the side banks are generated. For example, the lateral free surface slopes at CS09 are 0.022 and 0.016 respectively for cases 1a and 2a. These lateral slopes can lead to lateral pressure gradient force whose direction is from the channel centreline to the side banks. Notably, the lateral surface slope in case 1a is steeper than that in case 2a, which may also result from the effect of the supercritical flow.
4.1.2. Time-averaged streamwise flow velocity
Figure 3. Time-averaged flow velocity distribution at three different slices over z-direction in the confluence-bifurcation unit: (a)∼(c) case 1a, (d)∼(f) case 2a. The flow direction is from the left to the right. StZ = Stagnation Zones, MiL = Mixing Layer. X’ = x/B, Y’ = y/B, Ui’ = Ui/Uti, Ui denotes the time-averaged streamwise flow velocity in case series i (i = 1,2), Uti denotes the cross-section-averaged streamwise flow velocity in case series i, Ut1 = 0.385 m/s, for case 2a Ut2 = 0.714 m/s.Figure 4. Time-averaged flow velocity contours at eight different cross-sections in the confluence-bifurcation unit: (a) case 1a, (b) case 2a.
Besides the shared features described above, some differences between the two cases are also identified. First, flow stagnation zones at the upstream bar tail are found exclusively in case 1a as the central bars are exposed (Figure 3 (a–c)). Second, in case 1a the mixing layer is obvious in both the lower or upper flows (Figure 3 (a–c)), while in case 2a the mixing layer can be inconspicuous in the upper flow (Figure 3 (f)). Third, in case 1a, two high-velocity cores gradually transform into one single core downstream of the confluence [Figure 4 (a), CS08∼CS11] and are divided into two cores again at the downstream bar head [Figure 4 (a), CS15]. By contrast, in case 2a, the two cores merge much more rapidly [Figure 4 (a), CS08∼CS09], and no obvious reseparation of the merged core is found at the downstream bar head (Figure 3 (d–f)). The latter two differences between cases 1a and 2a indicate that the flow convergence and divergence are relatively weak when the central bars are fully submerged. It is noticed that when the central bars are exposed, the flow in branches needs to steer around the central bar, which can cause a large angle between the two flow directions at the confluence, and thus relatively strong flow convergence and divergence may occur. By contrast, when the central bars are fully submerged, the flow behavior resembles that of a straight channel, with flow predominantly moving straight along the main axis of the central bars. Therefore, a small angle between two tributary flow forms, and thus flow convergence and divergence are relatively mild.
4.1.3. Recirculation vortex
A recirculation vortex with a vertical axis is a typical structure usually found where flow steers sharply, and is generated from flow separation (Lu et al., 2023). This vortex structure is found in the confluence-bifurcation unit in the present study, marking several significant flow separation zones. Figure 5 shows the recirculation vortex structure at the bifurcation junction of the confluence-bifurcation unit. In both cases 1a and 2a, two recirculation vortices BV1 and BV2 are found at the bifurcation junction corner. Moreover, BV1 and BV2 seem well-established near the bed but tend to transform into premature ones in the upper flow, and there is also a tendency for the cores of BV1 and BV2 to shift downstream as they transition from the lower to the upper flow (Figure 5(a–c,d–f)). This finding indicates that flow separation zones exist at the bifurcation junction corner, and the vortex structure is similar in the separation zones under low and high discharges. These flow separation zones are generated due to the inertia effect as flow suddenly diverges and steers towards the curved side banks of the channel (Xie et al., 2020). Notably, two additional vortices BV3 and BV4 are found at both sides of the downstream bar in case 1a (Figure 5(a–c)), but no such vortices exist in case 2a. This difference shows that flow separation zones at both sides of the downstream bar are hard to form when the bars are completely submerged under the high discharge.
Figure 5. Recirculation vortices at the bifurcation junction (streamline view at three different slices over z-direction): (a)∼(c) case 1a, (d)∼(f) case 2a. The red solid line marked out the position of these vortices (BV1∼BV4).
Similarly, Figure 6 shows the recirculation vortex structure at the confluence junction of the confluence-bifurcation unit. No noteworthy similarities but a key difference between the two cases are observed at this site. Two vortices CV1 and CV2 are found downstream of the confluence junction corner in case 1a (Figure 6(c)), which mark two separation zones. Conversely, no such separation zones are found in case 2a. In fact, separation zones were reported at similar sites under relatively low discharges in some previous studies (Ashmore et al., 1992, Luz et al., 2020, Sukhodolov & Sukhodolova, 2019; Xie et al., 2020). Nevertheless, the flow separation zones at the confluence corner are very restricted in the present study (Figure 6(c)). Ashmore et al. (1992) also reported that no, or very restricted flow separation zones occur downstream of natural river confluence corners, primarily because of the relatively slow change in bank orientation compared with the sharp corners of laboratory confluences where separation is pronounced (Best & Reid, 1984; Best, 1988). In the present study, the bank orientation also changes slowly, which may explain why flow separation zones are inconspicuous at the confluence corner.
Figure 6. Recirculation vortices at the confluence junction (streamline view at three different slices over z-direction): (a)∼(c) case 1a, (d)∼(f) case 2a. The red solid line marked out the position of these vortices (CV1 & CV2).
The differences in the distribution of recirculation vortices discussed above may be mainly attributed to the difference in the angle between the tributary flows under different discharges. Some previous studies have reported that the confluence/bifurcation angle can significantly influence the flow structure at confluences/bifurcations (Best & Roy, 1991; Ashmore et al., 1992; Miori et al., 2012). Although the confluence/bifurcation angle is fixed due to the determined central bar shape in the present study, the angle between two tributary flows is affected by the varying discharge. When the central bars are exposed under the low discharge, the flow is characterized by a more pronounced curvature of the streamlines, and a large angle between the two tributary flows is noted (Figure 6(b)), causing strong flow convergence and divergence. By contrast, a small angle forms as the central bars are submerged, thereby leading to relatively weak flow convergence/divergence (Figure 6(e)). Overall, the differences mentioned above can be attributed to the differences in the intensity of flow convergence and divergence under different discharges.
It should be noted that some previous studies (Constantinescu et al., 2011; Sukhodolov & Sukhodolova, 2019) presented that there is a wake mode in the mixing layer of two streams at the confluence junction. The wake mode means that in the mixing layer, multiple streamwise coherent vortices moving downstream will form, which is similar to the flow structure around a bluffing body (Constantinescu et al., 2011). However, no such structure has been found within the confluence-bifurcation unit in this study. According to the numerical simulations of Constantinescu et al. (2011), a wake mode was found at a river confluence with a concordant bed and a momentum flux ratio of about 1. The confluence has a much larger angle (∼60°) between the two streams when compared to the confluence junction of the confluence-bifurcation unit in the present study where the angle is about 25°. As flow mechanics at river confluences may include several dominant mechanisms depending on confluence morphology, momentum ratio, the angle between the tributaries and the main channel, and other factors (Constantinescu et al., 2011), the relatively small confluence angle in the present study may explain why the wake mode is absent. The possible effects of the confluence/bifurcation angle are reserved for future study. Additionally, flow separation can lead to reduced local sediment transport capacity, thus causing considerable sediment deposition under natural conditions. Hence, the bank may migrate towards the inner side of the channel at the positions of CV1, CV2, BV1, and BV2, while the bar may expand laterally at the positions of BV3 and BV4.
4.1.4. Secondary current
Secondary current is the flow perpendicular to the mainstream axis (Thorne et al., 1985) and can be categorized into two primary types based on its origin: (1) Secondary current generated by the interaction between centrifugal force and pressure gradient force; (2) Secondary current resulting from turbulence heterogeneity and anisotropy (Lane et al., 2000). There are some widely recognized definitions of secondary current strength (SCS) (Lane et al., 2000). In this paper, the secondary current cells are identified by visible vortex with a streamwise axis, and the definition of SCS proposed by Shukry (1950) is used:
where ux, uy, and uz are flow velocities in x, y, and z directions and ux represents the mainstream flow velocity.
Figure 7 presents contour plots of SCS and the secondary current structure at key cross-sections of the study area. When the central bars are exposed, at the upstream bar tail (CS08), intense transverse flow occurs with flow converging to the centreline, but no secondary current cell is formed (Figure 7(a)). This is consistent with the findings of Hackney et al. (2018). At the confluence junction (CS09), transverse flow still plays a major role in the secondary current structure, with flow converging to the centreline at the surface and diverging to side banks near the bed (Figure 7(b)). Moreover, ‘back-to-back’ helical cells, which are two vortices rotating reversely, tend to generate at CS09 with their cores located near the side banks (Figure 7(b)) (Mosley, 1976; Ashmore, 1982; Ashmore et al., 1992), yet their forms are rather premature. As the flow goes downstream, the cores of the helical cells gradually rise to the upper flow and approach towards the centreline, and the helical cells become well-established (Figure 7(c–e)). When the flow diverges again at the downstream bar head (CS15), the helical cells attenuate rapidly, and the secondary current structure is once again characterized predominantly by transverse flow (Figure 7(f)).
Figure 7. Distribution of secondary current strength and secondary current cells at six different cross-sections: (a)∼(f) case 1a, (g)∼(l) case 2a. The secondary current cells are identified by visible lateral vortices (streamline view). The zero distance of each cross-section is located on the right bank.
When the central bars are fully submerged under the high discharge, the secondary current structure at the upstream bar tail and the confluence junction exhibits a resemblance to that under the low discharge (Figure 7(g,h)). However, at CS09, two pairs of cells with different scales tend to form under the high discharge (Figure 7(h)). The large and premature helical cells are similar to those under the low discharge, whereas the small helical cells are located near side banks possibly due to wall effects. As the flow moves downstream, the large helical cells tend to diminish rapidly and merge with the small ones near both side walls (Figure 7(i–k)). Moreover, the secondary current structure is once again characterized predominantly by transverse flow at CS14 under the high discharge, which occurs earlier than that under the low discharge (Figure 7(k)). At the downstream bar head, transverse flow still takes a dominant place, while the helical cells seem to become premature with increased scale (Figure 7(l)).
In general, in both cases 1a and 2a, the lateral distribution of SCS at all cross-sections is symmetrical about the channel centreline, where SCS is relatively small. A relatively high SCS is detected at both the upstream bar tail and the downstream bar head due to the effects of centrifugal force caused by flow steering. SCS decreases rapidly from the upstream bar tail (CS08) to the entrance of the downstream bifurcation junction (CS14), followed by a sudden increase at the downstream bar head (CS15) (Figure 7 (a–e, g–k)). However, the distribution of high-SCS zones is different between the two discharges. Under the low discharge, high-SCS zones appear along the bottom near the centerline and at the free surface on both sides of the centreline. Although similar high-SCS zones are found along the bottom near the centerline under the high discharge, the high-SCS zones are not found at the free surface. Furthermore, it is noticed that more obvious high-SCS zones appear under the low discharge compared with the high discharge, especially at CS09. This may be attributed to the differences in the intensity of flow convergence and divergence under different submerging conditions of the central bars. When the central bars are exposed, flow convergence and divergence are strong and sharp flow steering occurs, thereby causing large SCS. By contrast, when the central bars are fully submerged, flow convergence and divergence are relatively weak, and thus small SCS is observed.
4.1.5. Turbulent characteristics
Turbulent characteristics reflect the performance of energy and momentum transfer activities in flow (Sukhodolov et al., 2017). Comprehensive analysis of turbulent characteristics is crucial as they greatly impact the incipient motion, settling behavior, diffusion pattern, and transport process of sediment. Here, the TKE and turbulent dissipation rate (TDR) of flow in the confluence-bifurcation unit are analyzed.
Figure 8 shows the distribution of TKE on various cross-sections in cases 1a and 2a. In the same way, Figure 10 shows the distribution of TDR. The values of TKE and TDR are nondimensionalized with mid-values of TKE = 0.005 m2·s−2 and TDR = 0.007 m3·s−2. In both cases 1a and 2a, the distributions of TKE and TDR show symmetrical patterns concerning the channel centreline. High-TKE and high-TDR zones exhibit a belt distribution near the channel bottom (McLelland et al., 1999; Ashworth, 1996; Constantinescu et al., 2011), indicating that turbulence primarily originates at the channel bottom due to the influence of bed shear stress. A sudden increase of TKE (Weber et al., 2001) and TDR occurs near the channel bottom at the confluence junction [Figure 8 and 9, CS08∼CS09] and from the entrance of the bifurcation junction (CS14) to the downstream bar head (CS15) (Figures 8 and 9).
Figure 8. Turbulent kinetic energy contours at eight different cross-sections in the confluence-bifurcation unit: (a) case 1a, (b) case 2a. TKE = turbulent kinetic energy. TKE’ = dimensionless value of TKE, with regard to a mid-value of TKE = 0.005 m2·s−2.Figure 9. Turbulent dissipation rate contours at eight different cross-sections in the confluence-bifurcation unit: (a) case 1a, (b) case 2a. TDR = turbulent dissipation rate. TDR’ = dimensionless value of TDR, with regard to a mid-value of TDR = 0.007 m3·s−2.Figure 10. Comparison of the distribution of time-averaged streamwise flow velocity along the flow depth at different cross-sections between the confluence-bifurcation unit and the single confluence. (a)∼(f) 1a vs. 1b, (g)∼(l) 2a vs. 2b.
Despite the common turbulent characteristics between cases 1a and 2a, additional high-TKE zones are found in the upper flow at the upstream bar tail (CS08), the confluence junction (CS09) and the downstream bar head (CS15) (Figure 8) when the central bars are fully submerged. The formation mechanism of these high-TKE zones near the water surface is more complicated, which may result from interactions of velocity gradient, secondary current structure and wall shear stress (Engel & Rhoads, 2017; Lu et al., 2023).
4.2. Comparison with single confluence/bifurcation cases
In this section, the results of a single confluence (cases 1b and 2b) and a single bifurcation (cases 1c and 2c) are compared with those of the confluence-bifurcation unit (cases 1a and 2a) under two discharges. Flow structure at CS08∼CS15 is mainly concerned below.
4.2.1. Comparison with single confluence cases
First, the patterns of time-averaged streamwise velocity, TKE and TDR within the single confluence (presented by contour plots in the supplementary materials) are assessed and then compared with those within the confluence-bifurcation unit (Figures 4, 8, and 9). It is found that distributions of these parameters are similar in the confluence-bifurcation unit and the single confluence from the upstream bar tail (CS08) to the entrance of the bifurcation junction (CS14), despite varying discharges. As the existence of the downstream central bar is the main difference between the single confluence and the confluence-bifurcation unit, this finding indicates that the downstream bar may have limited influence on the flow structure in the confluence-bifurcation unit. In other words, the flow structure in the confluence-bifurcation unit appears to be mainly shaped by the presence of the upstream bar, with its impact potentially reaching as far as the entrance of the bifurcation (CS14). Moreover, under the low discharge, the two high-velocity cores seem to merge later (at CS11) in the single confluence than in the confluence-bifurcation unit (at CS10), which indicates the convergence of two tributary flows may achieve a steady state faster in the confluence-bifurcation unit. To further elucidate the differences, results on the distribution of time-averaged streamwise velocity and TKE along the flow depth are discussed below.
4.2.1.1. Time-averaged streamwise velocity
Figure 10 shows the distribution of time-averaged streamwise flow velocity along the flow depth at different cross-sections. Note that α = 0.5 denotes the channel centreline and α = 0.7 denotes a position near the side banks. As only marginal differences are found at α = 0.3 and 0.7, only profiles at α = 0.7 are displayed for clarity.
Under the low discharge, no obvious difference in the distribution of time-averaged streamwise flow velocity is observed at the upstream bar tail (Figure 10(a)). At the confluence junction (Figure 10(b)), the velocities near the side banks (α = 0.7) are larger than those at the centre (α = 0.5) in both the confluence-bifurcation unit and the single confluence, which suggests that the two tributary flows have not sufficiently merged. The two tributary flows achieve convergence at CS11 in both the confluence-bifurcation unit and the single confluence (Figure 10(c)), with the velocity at the centre (α = 0.5) is larger than that near the side banks. Nevertheless, the velocities at the centre (α = 0.5) and near the side banks (α = 0.7) are closer to each other in the confluence-bifurcation unit than those in the single confluence, which represents less sufficient flow convergence in the confluence-bifurcation unit than in the single confluence. Therefore, it can be inferred that the convergence of two tributary flows may achieve a steady state faster in the confluence-bifurcation unit. After reaching the steady state, the velocity near the side banks (α = 0.7) is smaller in the single confluence than in the confluence-bifurcation unit despite close values at the centre (α = 0.5) (Figure 10(d,e)). This leads to a more pronounced disparity between velocities at the centre and near the side banks in the single confluence than that observed in the confluence-bifurcation unit. In other words, the high-velocity zone is more concentrated on the channel centreline in the single confluence, while the lateral distribution of flow velocity tends to be more uniform in the confluence-bifurcation unit. This may be attributed to the influence of the downstream central bar, which is further proved by comparing the velocity profiles at CS15 (Figure 10(e)).
As for the high discharge condition, from CS08 to CS14, the quantitative differences in velocity distribution between the confluence-bifurcation unit and the single confluence seem small. This indicates that the effect of morphology appears to be subdued when the central bars are fully submerged under the high discharge. It should be also noted that under both the low and high discharge, velocity profiles at the corresponding location exhibit the same shapes in the confluence-bifurcation unit and the single confluence, which indicates that the upstream confluence may dominate the flow structure in the confluence-bifurcation unit.
4.2.1.2. Secondary current
Figure 11 shows contour plots of SCS and the secondary current structure for single confluence cases. Compared with Figure 7, under both low and high discharge conditions, the distribution of SCS and the structure of helical cells in the confluence-bifurcation unit and the single confluence are very similar from CS08 to CS12 (Figure 7(a–d, g–j) and Figure 11(a–d, g–j)]. This indicates that the secondary current structure in the confluence-bifurcation unit exhibits certain consistent features when compared to those in the single confluence, thus proving that the effects of the upstream central bar may dominate the flow structure in the confluence-bifurcation unit. However, the secondary current structure at CS14 and CS15 is different between the confluence-bifurcation unit and the single confluence (Figure 7 and 11(e, f, k,l)). Under the low discharge, transverse flow is from the side banks to the centre and relatively high SCS is found near the side banks at CS14 in the single confluence, while the transverse flow is always from the centre to the side banks and SCS is relatively low at the corresponding sites in the confluence-bifurcation unit (Figure 11(e)). Under the high discharge, the helical cells near the side walls almost diminish in the single confluence, while they still exist in the confluence-bifurcation unit at CS14 (Figure 11(k)). Under both low and high discharges, the secondary current pattern at CS15 is similar to that at CS14 in the single confluence, while they are different in the confluence-bifurcation unit due to the existence of the downstream central bar. This comparison indicates that the existence of the downstream central bar can influence the upstream secondary current structure, nevertheless, the effects are fairly limited.
Figure 11. Secondary current at different cross-sections in the single confluence condition: (a)∼(f) case 1b, (g)∼(l) case 2b. The zero distance of each cross-section is located on the right bank.
4.2.1.3. Turbulent kinetic energy
Figure 12 shows TKE distribution along the flow depth at different cross-sections. Under the low discharge, in general, the maximum TKE tends to appear near the channel bottom in both the confluence-bifurcation unit and the single confluence. No obvious difference is observed at the upstream bar tail (CS08) (Figure 12(a)). Downstream this site (at CS09), the maximum TKE near the side banks (α = 0.7) is larger than that at the channel centre in the single confluence, while they are close to each other in the confluence-bifurcation unit (Figure 12(b)). This can also be attributed to the insufficient convergence of the two tributary flows. At CS11, flow convergence achieves a steady state in the confluence-bifurcation unit, while it remains insufficient in the single confluence. As flow convergence reaches a steady state at CS12, the maximum TKE in the single confluence exhibits a more concentrated distribution on the channel centre than that in the confluence-bifurcation unit (Figure 12(d)). This effect becomes more obvious downstream at CS14 (Figure 12(e)). The less-concentrated distribution of the maximum TKE in the confluence-bifurcation unit can be owing to the effects of the downstream central bar as well, which appears analogous to that mentioned in 4.2.1.1.
Figure 12. Comparison of the distribution of TKE along the flow depth at different cross-sections between the confluence-bifurcation unit and the single confluence. (a)∼(f) 1a vs. 1b, (g)∼(l) 2a vs. 2b.
Under the high discharge condition, two peaks of TKE appear in both the confluence-bifurcation unit and the single confluence (Figure 12(g–l)). Moreover, in both the confluence-bifurcation unit and the single confluence, from the upstream bar tail to the downstream bar head, the peak of TKE in the upper flow is larger at the channel centre (α = 0.5), while the peak of TKE in the lower flow is larger near the side banks (α = 0.7). However, the disparity between the TKE near the side banks and at the channel centre seems to be larger in the single confluence, while the TKE in the confluence-bifurcation unit takes a more uniform distribution. Even though, TKE profiles at the corresponding location exhibit highly similar shapes in the confluence-bifurcation unit and the single confluence, suggesting that the effects of channel morphology seem to be inhibited when the central bars are submerged under the high discharge.
4.2.2. Comparison with single bifurcation cases
Distributions of time-averaged streamwise velocity, TKE and TDR at corresponding cross-sections are also compared between the single bifurcation (see the Supplementary material) and the confluence-bifurcation unit (Figures 4, 8 and 9). Unlike the high similarity in flow characteristics exhibited between the confluence-bifurcation unit and the single confluence, significant differences are found between the confluence-bifurcation unit and the single bifurcation, especially at CS08∼CS14. On the one hand, the high-velocity zones are broader and asymmetrical concerning the channel centreline in the single bifurcation, with a belt-like and an approximately elliptic-like distribution respectively under the low and high discharges. By contrast, the high-velocity zone is a core that concentrates on the channel centre in the confluence-bifurcation unit. Moreover, the maximum velocity seems smaller in the single bifurcation than that in the confluence-bifurcation unit. On the other hand, the high-TKE belt near the channel bottom appears to be narrower in the single bifurcation than in the confluence-bifurcation unit, especially at CS08∼CS14 under the low discharge. Furthermore, additional high-TKE zones are found near the side walls at CS08∼CS11 in the single bifurcation, of which the scale is obviously smaller than those in the confluence-bifurcation unit. In addition, TKE at the channel centre is smaller near the free surface in the single bifurcation than that in the confluence-bifurcation unit. Nevertheless, the distributions of velocity, TKE and TDR seem to be similar in the confluence-bifurcation unit and the single bifurcation at CS15. As the existence of the upstream central bar is the main difference between the single confluence and the confluence-bifurcation unit, all the above findings indicate that the upstream central bar greatly influences the flow structure in the confluence-bifurcation unit. On the other hand, the downstream central bar may have a restricted influence on the flow structure in the confluence-bifurcation unit, whose impact may be limited to a range between the entrance of the bifurcation (CS14) and the downstream bar head (CS15). To further elucidate the differences, results on the distribution of time-averaged streamwise velocity and TKE along the flow depth are discussed below.
4.2.2.1. Time-averaged streamwise velocity
Figure 13 shows the distribution of time-averaged streamwise velocity along the flow depth at different cross-sections. Under the low discharge, distinct distribution patterns of flow velocity between the confluence-bifurcation unit and the single bifurcation are found at CS08, CS09 and CS11, which can be attributed to the effects of upstream flow convergence (Figure 13(a–c)). However, when the flow convergence reaches a steady state in the confluence-bifurcation unit (Figure 13(d–f)), the high-velocity zone is more concentrated in the confluence-bifurcation unit than in the single bifurcation due to to the significant influence of the upstream central bar on the flow structure. The velocity profiles at the downstream bar head can be a shred of evidence as well, with the maximum velocity larger at the channel centre but smaller near the side banks in the confluence-bifurcation unit than in the single bifurcation.
Figure 13. Comparison of the distribution of time-averaged streamwise flow velocity along the flow depth at different cross-sections between the confluence-bifurcation unit and the single bifurcation. (a)∼(f) 1a vs. 1c, (g)∼(l) 2a vs. 2c.
Under the high discharge, the distribution of velocity seems to exhibit limited differences between the two kinds of morphology, which indicates that the effects of channel morphology may be less noticeable when the central bars are fully submerged under the high discharge. Nevertheless, the velocity in the lower flow (below a relative depth of 0.45) shows a uniform lateral distribution in the single bifurcation, as the velocity profile at the channel centreline (α = 0.5) is in line with that near the side banks (α = 0.7) (Figure 13(g–l)). However, in the confluence-bifurcation unit, different velocity distributions in the lower flow can be observed at the channel centreline (α = 0.5) and near the side banks (α = 0.7). The foregoing results indicate that when the central bars are fully submerged, the high-velocity zones are more concentrated on the channel centreline in the confluence-bifurcation unit, while the lateral distribution of flow velocity within the single bifurcation tends to be more uniform, especially near the bifurcation junction (Figure 13(j,k)). This can also be attributed to the dominant influence of the upstream central bar over the downstream central bar.
It is also noted that the flow velocity distribution along the flow depth in the confluence-bifurcation unit is of a similar pattern despite varying discharges. As a critical point, the maximum velocity appears in the upper flow. The distribution above the critical point is approximately linear whereas it appears logarithmic below. By contrast, despite the similarity observed under the low discharge, the flow velocity distribution along the flow depth within the single bifurcation exhibits a distinct pattern under the high discharge, especially near the side banks (Figure 13(e–h)). On the one hand, the critical point in the upper flow no longer corresponds to the maximum velocity. On the other hand, the velocity distribution deviates from logarithmic below the critical point, with the maximum velocity appearing at a relative depth of 0.45. Succinctly, the distribution of streamwise velocity along the flow depth may retain the same pattern regardless of discharge levels in the confluence-bifurcation unit, while it may exhibit distinct patterns under different discharge levels in the single bifurcation.
4.2.2.2. Secondary current
Figure 14 shows contour plots of SCS and the distribution of secondary current for single bifurcation cases. In general, the value of SCS near the side banks at CS08∼CS14 (Figure 14(a–d, g–j)) in the single bifurcation seems smaller than that in the confluence-bifurcation unit (Figure 7(a–d, g–j)), especially under the low discharge. SCS distribution at CS14 is similar in the confluence-bifurcation unit and the single bifurcation under both low and high discharges. This difference in SCS distribution between the confluence-bifurcation unit and the single bifurcation indicates that the downstream bifurcation may have a restricted influence on the flow structure in the confluence-bifurcation unit. This influence is limited to a range between the entrance of the bifurcation (CS14) and the downstream bar head (CS15).
Figure 14. Secondary current at different cross-sections in the single bifurcation condition: (a)∼(f) case 1c, (g)∼(l) case 2c. The zero distance of each cross-section is located on the right bank.
In addition, the secondary current structure may also present different patterns in response to varying channel morphologies and discharge conditions. Under the low discharge condition, multiple unstable helical cells with asymmetrical distribution are formed from CS08 to CS12 in the single bifurcation (Figure 14(a–d)), while no obvious helical cells are found at CS14 and CS15 (Figure 14(d,e)). These findings are quite different from the stable and symmetrical helical cells at all cross-sections shown in the confluence-bifurcation unit (Figure 7). This difference may be attributed to the significant influence of the upstream central bar and the limited influence of the downstream central bar. Under the high discharge condition, only one pair of premature helical cells are found from CS08 to CS12 in the single bifurcation with their cores located near the side banks (Figure 14(e,f)). As the flow moves downstream, the helical cells gradually develop and become well-established (Figure 14(g,h)). These helical cells in the single bifurcation show symmetric cross-sectional distribution and a similar longitudinal development as in the confluence-bifurcation unit. However, in the confluence-bifurcation unit, two pairs of helical cells appear upstream of CS12 and CS14 and gradually fuse to one pair under the high discharge. As the ‘two-pairs’ structure in the confluence-bifurcation unit origins from the upstream confluence, the differences in the secondary current structure between the single bifurcation and the confluence-bifurcation unit under the high discharge can also be owing to the effects of the upstream central bar in excess of those of the downstream central bar.
4.2.2.3. Turbulent kinetic energy
Figure 15 shows the TKE distribution along the flow depth at different cross-sections. Under the low discharge, when the two tributary flows have not achieved sufficient convergence in the confluence-bifurcation unit, the maximum TKE is more concentrated in the single bifurcation (Figure 15(a–c)). As flow convergence achieves a steady state, more concentrated high-TKE zones appear at the channel centre within the confluence-bifurcation unit, confirming the finding that the effects of the upstream central bar reign over those of the downstream central bar in the confluence-bifurcation unit. However, things can be very complicated under the high discharge. For TKE distribution at the channel centreline, two peaks appear in the confluence-bifurcation unit with one close to the free surface and the other near the bed (Figure 15(g–l)). By contrast, only one peak near the bed is present in the single bifurcation. Therefore, a larger TKE can be found in the upper flow of the channel centreline in the confluence-bifurcation unit. For TKE distribution near the side banks, two peaks appear in both the confluence-bifurcation unit and the single bifurcation at CS09∼CS14 (Figure 15(h–l)). The upper peak is larger but the lower peak is smaller within the single bifurcation than those within the confluence-bifurcation unit. These significant discordances in TKE distribution under the high discharge further prove that the effects of the upstream bar on the flow structure in the confluence-bifurcation unit are more prominent than those of the downstream central bar.
Figure 15. Comparison of the distribution of TKE along the flow depth at different cross-sections between the confluence-bifurcation unit and the single bifurcation. (a)∼(f) 1a vs. 1c, (g)∼(l) 2a vs. 2c.
4.2.3. Further discussion of the comparisons
The above subsections have revealed significant differences in flow structure within the confluence-bifurcation unit and the single confluence and bifurcation, which directly result from the distinct channel morphologies and vary with the discharge conditions as well. These differences are summarized and further discussed below.
The distinctive morphology of a confluence-bifurcation unit plays a pivotal role in governing streamwise flow velocity distribution, secondary current structure, and turbulent kinetic energy distribution within the channel. Generally, from the upstream bar tail (CS08) to the entrance of the bifurcation (CS14), the flow structure in the confluence-bifurcation unit is highly similar to that in the single confluence, while it exhibits great differences (as shown in 4.2.2) between the confluence-bifurcation unit and the single bifurcation. This indicates that the upstream central bar greatly influences the flow structure in the confluence-bifurcation unit, with the effects spreading to the entrance of the bifurcation. At the downstream bar head (CS15), the flow structure (e.g. the transverse flow patterns) in the confluence-bifurcation unit exhibits high similarity to that in the single bifurcation. However, these similarities do not spread to upstream cross-sections, suggesting that the influence of the downstream central bar is limited at the bifurcation junction. In a word, the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit are in excess of those of the downstream central bar.
However, despite the influence of channel morphology, discharge may also have some important effects on the streamwise flow velocity distribution. On the one hand, when the central bars are exposed under the low discharge, the high-velocity zone is less concentrated in the confluence-bifurcation unit than in the single confluence, while it is more concentrated in the confluence-bifurcation unit than in the single bifurcation. On the other hand, it is noticed that when the central bars are fully submerged under the high discharge, reduced differences in flow structure between the confluence-bifurcation unit and the single confluence/bifurcation are witnessed, and thus the morphology effect seems to be subdued.
4.3. Implications
The present work unravels the flow structure in a laboratory-scale confluence-bifurcation unit and takes the first step to further investigating morphodynamics in such channel morphology. Based on the comparison with a single confluence/bifurcation, the findings provide insight into the complex 3D interactions between water flow and channel morphology. The distinct flow structure in the laboratory-scale confluence-bifurcation unit may appreciably alter sediment transport and morphological evolution, of which research is underway. As the basic morphological element of braided river planform is confluence-bifurcation units, the present work should have direct implications for flow structure in natural braided rivers. This is pivotal for the sustainable management of braided rivers which deals with water and land resources planning, eco-hydrological well-being, and infrastructure safety such as cross-river bridges and oil pipelines (Redolfi et al., 2019; Ragno et al., 2021).
Notably, braided rivers worldwide (e.g. in the Himalayas, North America, and New Zealand) have undergone increased pressures and will continue to evolve due to forces of global climate change and intensified anthropogenic activities (Caruso et al., 2017; Hicks et al., 2021; Lu et al., 2022). In particular, channel aggradation caused by increased sediment supply as well as exploitation of braidplain compromise space for flood conveyance, making the rivers prone to flooding. In this sense, an enhanced understanding of the flow structure under high discharge when central bars are fully submerged is essential for mitigating flooding hazards.
5. Conclusions
This study has numerically investigated the 3D flow structure in a laboratory-scale confluence-bifurcation unit based on the LES model integrated in the FLOW–3D® software platform. Two different discharges are considered with the central bars fully submerged or exposed respectively when the discharge is high or low. Cases of a single confluence/bifurcation are included for comparison. The key findings of this paper are as follows:
Several differences are highlighted in the comparison of the flow structure in the confluence-bifurcation unit between the two discharges. When the central bars are fully submerged under the high discharge, the mixing layer of two tributary flows is less obvious, and two high-velocity cores merge more rapidly as compared with those under the low discharge. Besides, flow separation zones are found neither at the confluence corner nor on both sides of the downstream bar when the central bars are fully submerged. Moreover, SCS seems to be smaller near the side banks under the high discharge than under the low discharge. Therefore, it is suggested that flow convergence/divergence is relatively weak in the confluence-bifurcation unit when central bars are fully submerged under the high discharge.
From the upstream bar tail to the entrance of the bifurcation, the flow structure in the confluence-bifurcation unit is highly similar to that in the single confluence, while it exhibits great differences from that in the single bifurcation. Only at the downstream bar head does the flow structure in the confluence-bifurcation unit exhibit high similarity to that in the single bifurcation. Consequently, the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit reign over those of the downstream central bar.
Despite the influence of channel morphology, discharge may also have significant effects on the distribution of streamwise flow velocity. On the one hand, when the central bars are exposed under the low discharge, the high-velocity zone is less concentrated in the confluence-bifurcation unit than in the single confluence, while it is more concentrated in the confluence-bifurcation unit than in the single bifurcation. On the other hand, when the central bars are fully submerged under the high discharge, reduced differences in flow structure between the confluence-bifurcation unit and the single confluence/bifurcation are witnessed, and thus the morphology effect seems to be subdued.
It is noticed that the effects of other factors (e.g. confluence and bifurcation angles, bed discordance) on the flow structure in the confluence-bifurcation unit are not discussed here. Studies on these issues are warranted and reserved for future work.
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자유 표면 흐름은 가정과 사무실 환경 모두에서 사용되는 소비자 제품의 설계 및 제조에서 일반적입니다. 예를 들어, 병 채우기는 매일 대규모로 이루어지는 프로세스입니다. 생산 속도를 극대화하면서 낭비를 최소화하도록 이러한 프로세스를 설계하면 시간이 지남에 따라 상당한 비용 절감으로 이어질 수 있습니다. FLOW-3D는 또한 스프레이 노즐을 설계하고 다공성 재료 및 기타 소비재 구성 요소의 흡수 기능을 모델링하는 데 사용할 수 있습니다. FLOW-3D 의 공기 유입, 다공성 매체 및 표면 장력을 포함한 고급 다중 물리 모델을 사용하면 소비자 제품 설계를 정확하게 시뮬레이션하고 최적화하는 것이 쉽습니다.
충전재
유입된 공기는 생산 라인에서 용기를 채울 때 액체의 부피를 늘릴 수 있습니다. 아래 왼쪽 이미지는 높이가 약 20cm인 병을 1.2초 동안 채우는 것을 보여줍니다. 색상 음영은 액체에 있는 공기의 부피 분율을 나타냅니다. 병에서 혼합 시간이 짧고 혼합 정도가 높기 때문에 공기가 표면으로 올라가 빠져나갈 시간이 없었습니다. 그러나 오른쪽 이미지에서 볼 수 있듯이 약 1.7초의 추가 시간이 지나면 공기가 표면으로 올라가면서 발생하는 액체 부피 감소가 명확하게 보입니다. FLOW-3D 의 드리프트 플럭스 모델을 사용하면 액체에 있는 기포와 같은 구성 요소를 분리하여 분리할 수 있습니다.
이 기사에서는 FLOW-3D를 사용하여 새로운 타이드 병 디자인의 충전을 모델링하는 방법을 설명하며, Procter and Gamble Company의 기술 섹션 책임자인 John McKibben이 기고했습니다 .
지금 오전 9시인데 긴급 이메일을 받았다고 상상해보세요.
방금 새로운 Tide® 병 디자인 중 하나가 손잡이에 채워지고 충전 장비에 문제가 생길 수 있다는 것을 깨달았습니다. 우리는 프로토타입 병이 없으며 몇 주 동안 없을 것입니다. 디자이너와 소비자는 디자인의 모습을 좋아하지만, 채우는 방식이 생산 시설에 쇼스토퍼가 될 수 있습니다.
이런 상황이 제게 주어졌을 때, 저는 3D 지오메트리(그림 1)의 스테레오 리소그래피(.stl) 파일을 요청하여 응답을 시작했고, 제가 무엇을 할 수 있는지 알아보고자 했습니다. 저는 FLOW-3D가 .stl 파일을 사용하여 지오메트리를 입력하고 충전을 위한 자유 표면 문제를 해결할 수 있을 것이라는 것을 알고 있었습니다. 저는 이것이 잠재적인 문제에 대한 좋은 정성적 이해를 제공할 것으로 기대했지만, 이 애플리케이션에 얼마나 정확할지에 대해 약간 불확실했습니다.
시뮬레이션 설정 및 실행
오후 1시경에 저는 지오메트리 파일, 유량, 유체 특성을 받았습니다. 몇 시간 이내에 시뮬레이션이 실행되어 예비 결과가 나왔습니다. 저는 제 고객을 초대하여 결과를 잠깐 살펴보게 했고 그는 “사장의 상사”를 데려와서 살펴보게 했습니다. 그래서 저녁 5시경에 예비 결과를 살펴보고 원래 우려했던 것이 문제가 아니라는 것을 확인했습니다.
하지만 결과는 몇 가지 다른 의문을 제기했습니다. 손잡이에 채우면 유입 유체 제트가 많이 깨졌습니다. 이렇게 하면 유입 공기와 거품의 양이 늘어날 것이라는 걸 알았습니다(결국 세탁 세제를 채우고 있으니까요). FLOW-3D 공기 유입 모델을 테스트하기로 했습니다. 이 모델은 원래 난류 제트용으로 개발되었고, 이 층류 문제를 살펴보면 얼마나 잘 수행될지 확신할 수 없었습니다.
그림 2: 채워진 결과그림 3: 실험 비교
그림 2는 공기 유입 모델이 있는 경우와 없는 경우 병 충전 모델의 결과를 보여줍니다. 유입 공기가 포함되면 충전 레벨이 상당히 증가한다는 점에 유의하십시오. 유입 공기가 병 상단에서 유체를 강제로 밀어내지는 않지만 공기 유입 정확도를 확인해야 할 만큼 충분히 가깝습니다. 그림 3은 공기 유입 레벨을 몇 주 후에 실행한 실험 이미지와 비교합니다(시제품 병이 출시된 후). 제트 분리 및 충전 레벨의 질적 일치는 우수하며 시뮬레이션이 병 설계를 선별하기에 충분히 정확하다는 것을 확인했습니다.
홍조
변기가 어떻게 작동하는지 궁금한 적이 있나요? 사실 꽤 복잡합니다. 손잡이를 밀면 물이 변기 그릇을 채우기 시작합니다. 변기 그릇의 유체 수위가 트랩 상단(변기 그릇 뒤) 위로 올라가면 웨어 유형의 흐름이 시작됩니다. 흐름이 충분히 빠르면 변기 그릇에 거품이 형성되어 사이펀이 생성됩니다. 그 지점에서 사이펀이 변기 그릇에서 물을 끌어내고 변기가 물을 흘립니다. 많은 지역에서 물 절약은 중요한 문제이며, 저유량 변기는 가정과 상업용 모두에 필요합니다. 하지만 변기가 첫 번째 시도에서 제 역할을 하지 못하면 물 절약 목표는 달성되지 않습니다. FLOW-3D를 사용하면 다양한 설계를 모델링하여 최적의 결과를 얻을 수 있습니다.
식품 가공
식품 가공 산업은 복잡한 유체, 일반적으로 비뉴턴 유체, 슬러리, 고체와 유체의 혼합물을 관리하여 분배 장비를 최적으로 설계하고 제조하기 위한 다양한 요구 사항이 있습니다. 이는 상업용 장비의 일관성과 내구성 및 품질에 필수적입니다. 또한 포장 디자인의 혁신을 통해 한 제품을 다른 제품과 명확히 구별할 수 있습니다. 예를 들어, 꿀, 케첩 또는 크리머를 깨끗하고 정확하게 분배하는 것은 소비자가 매장에서 내리는 선택일 수 있습니다. 운송 및 보관 요구 사항에는 더 나은 모양 엔지니어링과 더 많은 용기 재료 선택이 필요합니다. 1.5리터 물병이나 세탁 세제를 움직이거나 떨어뜨리는 동안의 유체 하중은 상류 설계의 중요한 부분이 될 수 있습니다.
꿀, 옥수수 시럽, 치약과 같은 점성 유체는 일반적으로 고체 표면에 닿으면 코일을 형성하는 경향이 있습니다. 이 효과는 관찰하기에 흥미롭고 재미있지만, 공기가 제품에 끌려들어 포장이 어려워질 수 있는 포장 공정에서는 환영받지 못할 수 있습니다. 코일링이 발생하는 조건은 유체의 점도, 유체가 떨어지는 거리, 유체의 속도에 따라 달라집니다. FLOW-3D는 다양한 물리적 공정 매개변수를 연구하여 효율적인 공정을 설계하는 데 도움이 되는 정확한 도구를 제공합니다.
혼입
지난 수십 년 동안 컴퓨터화된 측정 및 시뮬레이션 기술의 발전으로 인해 혼합에 대한 이해가 크게 진전되었습니다. 유동 모델링 기술의 지속적인 발전 덕분에 혼합 장비의 유동 의존적 프로세스에 대한 자세한 통찰력을 CFD 소프트웨어를 사용하여 쉽게 시뮬레이션하고 이해할 수 있습니다. 오늘날 블렌딩에서 고체 현탁액, 재킷 반응기의 열 전달에서 발효에 이르기까지 광범위한 응용 분야가 FLOW-3D 의 혼합 기술을 사용하여 모델링됩니다. FLOW-3D 시뮬레이션은 임펠러의 모든 구성과 모든 용기 형상의 혼합 조건에서 블렌딩 시간, 순환 및 전력 수와 같은 주요 혼합 매개변수를 평가하는 데 도움이 될 수 있습니다. 이러한 시뮬레이션은 실험적 방법을 사용하여 보완합니다. 이러한 장비의 유동 의존적 프로세스를 예측하고 이해하기 위해 CFD 소프트웨어를 사용하면 제품 품질을 향상시키고 많은 제품의 비용과 출시 시간을 모두 줄일 수 있습니다.
비뉴턴 유체
혈액, 케첩, 치약, 샴푸, 페인트, 로션과 같은 비뉴턴 유체는 다양한 점도를 가진 복잡한 유동학을 가지고 있습니다. FLOW-3D 는 변형 및/또는 온도에 따라 달라지는 비뉴턴 점도를 가진 이러한 유체를 모델링합니다. 전단 및 온도에 따른 점도는 Carreau, 거듭제곱 법칙 함수 또는 단순히 표 형식의 입력을 통해 설명됩니다. 일부 폴리머, 세라믹 및 반고체 금속의 특징인 시간 종속 또는 틱소트로피 거동도 시뮬레이션할 수 있습니다.
핸드 로션 펌프는 종종 여러 가지 설계 문제와 관련이 있습니다. 펌프가 공기 공극을 가두지 않고 효과적으로 작동하고 로션의 연속적인 흐름을 생성하는 것이 중요합니다. 좋은 설계는 노력이 덜 필요하고 이상적으로는 로션을 원하는 곳으로 향하게 합니다. FLOW-3D 의 이동 객체 모델은 노즐이 아래로 눌리는 것을 시뮬레이션하여 저장소의 로션을 가압하는 데 사용됩니다. 로션의 압력과 로션을 추출하는 데 필요한 힘을 연구할 수 있습니다. 여러 설계 변수는 동일한 고정 구조 메시 내에서 쉽게 분석할 수 있습니다.
다공성 재료
다공성 매체에서 유체의 이동에 대한 수치 모델링은 어려울 수 있지만 FLOW-3D 에는 다공성 재료와 관련된 문제를 해결하는 데 유용한 기능이 많이 포함되어 있습니다. FAVOR™ 기술에는 사용자가 연속적인 다공성 매체를 표현할 수 있도록 하는 데 필요한 다공성 변수가 포함되어 있습니다. FLOW-3D를 사용하면 사용자가 포화 및 불포화 흐름 조건을 모두 시뮬레이션할 수 있습니다. 거듭제곱 법칙 관계를 사용하면 불포화 흐름 조건에서 모세관 압력 과 포화 사이의 비선형 관계를 모델링 할 수 있습니다. 별도의 충전 및 배수 곡선을 사용하여 히스테리시스 현상을 모델링할 수 있습니다. 서로 직접 접촉하는 경우에도 서로 다른 다공성, 투과성 및 습윤성 속성을 서로 다른 장애물에 할당할 수 있습니다. 투과성은 흐름 방향에 따라 지정할 수 있으므로 사용자가 다공성 매체의 이방성 동작을 모델링할 수 있습니다. 유체와 다공성 매체 간의 열 전달을 고려할 수 있습니다.
분무
소용돌이 분무 노즐은 화학 세정제, 의약품 및 연료에서 액체를 분사하는 일반적인 방법입니다. 액체를 성공적으로 분무하려면 일반적으로 노즐로 침투하는 공기 코어를 형성해야 합니다. CFD는 최적의 분무 콘에 대한 기하학, 소용돌이 속도 및 유체 특성의 영향을 탐색하는 효과적인 방법입니다.
이 예에서 2차원 축대칭 소용돌이 흐름이 시뮬레이션되었습니다. 대칭 축을 따라 공기 코어가 노즐의 전체 길이를 거의 관통했습니다. 왼쪽 플롯은 평면에서 속도 분포를 나타내는 벡터가 있는 압력 분포입니다. 오른쪽 플롯은 속도의 소용돌이 구성 요소로 채색되어 있으며 빨간색은 더 높은 값을 나타냅니다.
분무 콘의 규모와 입자 크기가 너무 광범위하기 때문에 분무의 완전한 분무를 직접 계산하는 것은 불가능합니다. 또한 분무는 외부 교란, 노즐의 미세한 결함 및 기타 영향과 밀접하게 관련된 혼란스러운 프로세스입니다. 그러나 노즐을 떠날 때 분무 콘의 특성(예: 벽 두께, 콘 각도, 축 및 방위 속도)을 예측할 수 있다면 이러한 유형의 흐름 장치를 최적화하는 데 큰 도움이 됩니다.
소용돌이 분무 노즐의 FLOW-3D 시뮬레이션
Products
자유 표면 흐름은 가정과 사무실 환경 모두에서 사용되는 소비자 제품의 설계 및 제조에서 일반적입니다.
예를 들어, 병 채우기는 매일 대규모로 진행되는 프로세스입니다. 생산 속도를 최대화하면서 낭비를 최소화하도록 이러한 프로세스를 설계하면 시간이 지남에 따라 상당한 비용 절감으로 이어질 수 있습니다. FLOW-3D는 또한 스프레이 노즐을 설계하고 다공성 재료 및 기타 소비재 구성 요소의 흡수 기능을 모델링하는 데 사용할 수 있습니다.
공기 혼입, 다공성 매질 및 표면 장력을 포함한 FLOW-3D의 고급 다중 물리 모델을 사용하면 소비자 제품 설계를 정확하게 시뮬레이션하고 최적화 할 수 있습니다.
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PCI-Express(또는 PCI-E) 표준을 사용하는 최근 출시된 AMD 비디오 카드(예: AMD RX 6950 XT)와 nVidia 그래픽 카드(예: nVidia GeForce RTX 3090)는 하이엔드 비디오 카드 차트에서 흔히 볼 수 있습니다.
PassMark – G3D Mark High End Videocards / Price
FLOW-3D POST 성능과 밀접한 그래픽카드의 이해
FLOW Science, inc의 최첨단 POST Processor인 FLOW-3D POST를 최대한 활용하려면 좋은 하드웨어가 있어야 합니다. 이 블로그에서 소프트웨어 엔지니어링의 GUI 개발자/관리자인 Stephen Sanchez는 이러한 하드웨어 권장 사항에 따라 최적의 FLOW-3D POST 경험을 얻을 수 있는 방법에 대해 정보를 제공 합니다.
고품질 그래픽 하드웨어
최소 3GB의 VRAM 이 있는 그래픽 카드로 시작하는 것이 좋습니다 . 이것은 많은 볼륨 렌더링을 수행할 경우 특히 중요합니다. 볼륨 렌더링은 FLOW-3D POST 의 고급 기능으로 iso-surface가 아닌 유체 도메인 전체에서 변수의 세부 사항을 시각화합니다. 이 기능은 매우 통찰력 있지만 후 처리 중에 효과적으로 사용하려면 좋은 하드웨어가 필요합니다.
다음으로 Intel 통합 그래픽을 기본 그래픽 하드웨어로 사용해서는 안됩니다. 인텔 통합 그래픽은 전용 그래픽 하드웨어가 있는 랩톱에서도 대부분의 랩톱에서 일반적입니다(자세한 내용은 아래 참조).
대부분의 FLOW-3D POST 기능은 이 구성에서 작동하지 않으므로 Intel 통합 그래픽을 지원하지 않습니다.
FLOW-3D POST 는 NVIDIA 그래픽 카드 와 함께 사용할 때 가장 잘 수행됩니다. FLOW-3D POST 가 잘 작동하는 것으로 확인되었으므로 Maxwell 아키텍처 제품군 이상의 NVIDIA 그래픽 하드웨어를 적극 권장 합니다.
NVIDIA Quadro 카드는 가장 안정적인 것으로 입증되었습니다. 고급 AMD 카드도 작동해야 하지만 NVIDIA 하드웨어 및 드라이버만큼 안정적이지 않다는 사실을 발견 했으므로 항상 AMD보다 NVIDIA를 권장합니다.
노트북의 듀얼 그래픽 카드 – 간단하지만 숨겨진 솔루션
이제 많은 노트북에 NVIDIA 그래픽 카드와 Intel 통합 그래픽 간에 전환 할 수 있는 기능이 있습니다. NVIDIA 카드로 FLOW-3D POST 가 실행되고 있는지 확인하는 것이 중요합니다 . NVIDIA 제어판을 통해 NVIDIA 카드로 노트북을 강제로 실행할 수 있습니다.
비디오 드라이버 업데이트
비디오 드라이버가 업데이트 되었는지 확인하는 것이 좋습니다. FLOW-3D POST 에서 비디오 드라이버를 업데이트하여 쉽게 해결할 수 있는 아티팩트 및 디스플레이 문제에 대한 보고가 있었습니다 . 비디오 드라이버를 최신 상태로 유지하는 것은 이러한 문제를 방지하는 좋은 방법입니다.
RAM, RAM, RAM!
메모리가 충분하지 않으면 시뮬레이션 후 처리가 불가능할뿐만 아니라 메모리 요구 사항을 인식하는 것이 중요합니다. 최대 10 배의 성능 저하로 이어질 수 있습니다! FLOW-3D POST 에 필요한 RAM 양은 여러 요소, 특히 시뮬레이션 크기에 따라 다릅니다. 사용자에게 최대한의 유연성을 제공하기 위해 메시의 셀 수에 따라 다음과 같은 RAM 권장 사항이 있습니다.
초대형 (2 억 개 이상의 셀) : 최소 128GB
대용량 (6 천 ~ 1 억 5 천만 셀) : 64-128GB
중간 (3 천만 ~ 6 천만 셀) : 32-64GB
소형 (3,000 만 셀 이하) : 최소 32GB
FLOW-3D POST 는 메모리 집약적 일 수 있습니다. 실행할 시뮬레이션 크기에 대한 대략적인 아이디어가 있는 경우, 이 지침을 가능한 한 잘 따르는 것이 좋습니다. 즉, 유연성을 극대화하고 가장 원활한 FLOW-3D POST 경험을 보장하기 위해 문제 크기에 관계없이 가능한 한 많은 RAM을 확보하는 것이 좋습니다.
그래픽 카드를 업그레이드 교체 설치하는 방법
그래픽 카드를 업그레이드하는 것은 성능 향상을 위한 좋은 방법이다. 그래픽 카드 업그레이드를 통해 시각적으로 고사양을 요구하는 POST 작업을 쉽게 소화할 수 있는 컴퓨터로 진화할 수 있다.
업그레이드를 위한 그래픽 카드 구매시 고려 사항, 기존 PC에 적합한가?
원하는 그래픽 카드를 결정하는 것은 복잡하고 미묘한 문제다. AMD와 엔비디아는 200달러 미만에서부터 최대 1,500달러에 이르는 지포스(GeForce) RTX 3090에 이르기까지 거의 모든 예산에 대한 선택지를 제공하기 때문이다.
카드의 소음, 발열, 전력 소비 등과 같은 사항을 고려할 수 있겠지만, 일반적으로는 비용 대비 가장 큰 효과를 제공하는 그래픽 카드를 원한다.
컴퓨터가 새 그래픽 카드를 지원하는 적절한 하드웨어인지 확인한다.
사용자가 겪는 가장 일반적인 문제는 부적절한 파워 서플라이(power supply)다. 충분한 전력을 공급할 수 없거나 사용 가능한 PCI-E 전원 커넥터가 충분하지 않을 수 있다. 필자의 경험상 파워 서플라이는 적어도 제조업체에서 권장하는 파워 서플라이의 요구 사항을 충족해야 한다. 예를 들어, 350W를 소비하는 지포스 GTX 3090을 구입했다면 8핀 전원 커넥터 한 쌍과 함께 엔비디아에서 제안한 최소 750W의 전력 공급 장치를 갖춰야 한다.
현재 파워 서플라이가 얼마나 많은 전력을 제공하는지 알아보려면 PC 본체를 열고 모든 파워 서플라이에 기본 정보가 나열된 표준 식별 스티커를 확인하면 된다. 또한 사용 가능한 6핀 및 8핀 PCI-E 커넥터의 수를 확인할 수 있다.
ⓒ Thomas Ryan 파워서플라이
마지막으로 본체 내부에 새 그래픽 카드를 넣을 충분한 공간이 있는지 확인한다. 일부 고급 그래픽 카드는 길이가 상당히 길어 30Cm 이상일 수 있으며, 확장 슬롯이 2개 또는 3개가 될 수 있다. 해당 그래픽 카드의 실제 크기는 제조업체 웹사이트에서 찾을 수 있다.
여기까지 해결했다면 이제 본격적으로 설치 작업에 착수한다.
생각보다 간단한 그래픽 카드 설치 작업
그래픽 카드 설치에는 새 그래픽 카드, 컴퓨터, 그리고 십자 드라이버 3가지만 있으면 된다. 설치하기 전 PC를 끄고 전원 플러그를 뽑는다.
기존 GPU를 제거해야 하는 경우가 아니면, 먼저 프로세서의 방열판에 가장 가까운 긴 PCI-E x16 슬롯을 찾아야 한다. 이 슬롯은 메인보드의 첫 번째 또는 두 번째 확장 슬롯이다.
이 슬롯에 접근을 차단하는 느슨한 전선이 없는지 확인한다. 기존 그래픽 카드를 교체하는 경우, 연결된 케이블을 모두 분리하고, PC 본체 후면 내부에 고정 브래킷에서 나사를 제거한 다음, 카드를 제거한다. 대부분의 메인보드에는 그래픽 카드를 제자리에 고정하는 PCI-E 슬롯 끝에 작은 플라스틱 걸쇠(latch)가 있다. 이 걸쇠를 눌러 이전 그래픽 카드의 잠금을 해제하고 분리한다.
ⓒ Thomas Ryan PCI-E x16 슬롯에 설치
이제 새 그래픽 카드를 개방형 PCI-E x16 슬롯에 설치할 수 있다. 카드를 슬롯에 완전히 삽입한 다음, PCI-E 슬롯 끝에 있는 플라스틱 걸쇠를 눌러 제자리에 고정한다. 그런 다음 나사를 사용해 그래픽 카드의 금속 고정 브래킷을 PC 본체에 고정한다. 덮개 브래킷 또는 이전 그래픽 카드를 고정했던 나사를 재사용할 수 있다.
ⓒ Thomas Ryan 그래픽 카드에는 추가 전원 커넥터 연결
대부분의 게임용 그래픽 카드에는 추가 전원 커넥터가 필요하다. 추가 전원이 필요한 경우, 해당 PCI-E 전원 케이블을 연결했는지 확인한다. 전원이 제대로 공급되지 않으면 그래픽 카드가 제대로 작동하지 않는다. 이 PCI-E 전원 케이블을 연결하지 않으면 PC 자체가 부팅되지 않을 수 있다.
그래픽 카드를 고정하고 난 후, 전원을 켠 상태에서 본체 측면 패널을 제자리로 밀어넣고 디스플레이 케이블을 새 그래픽 카드에 연결해 작업을 완료한다. 이제 컴퓨터를 켠다.
이제 그래픽 카드의 소프트웨어를 업그레이드할 단계가 왔다.
새 그래픽 카드가 이전 카드와 동일한 브랜드일 경우에는 절차가 간단하다. 제조업체의 웹사이트로 이동해 운영체제에 맞는 최신 드라이버 패키지를 다운로드한다. 그래픽 드라이버는 일반적으로 약 500MB로, 상당히 크다. 인터넷 연결 속도에 따라 다운로드하는 데 시간이 걸릴 수도 있다. 드라이버를 설치하고 컴퓨터를 다시 시작하면 이제 새 그래픽 카드가 제공하는 부드럽고 매끄러운 프레임 속도를 즐길 수 있다.
그래픽 카드 제조업체가 바뀐 경우(인털에서 AMD로, 혹은 AMD에서 인텔로), 새 그래픽 카드용 드라이버를 설치하기 전에 이전 그래픽 드라이버를 제거하고 컴퓨터를 다시 시작해야 한다. 이전 드라이버를 제거하지 않으면 새 드라이버와 충돌할 수 있다.
editor@itworld.co.kr 기사 일부 발췌 인용
그래픽 카드 GPU 온도 확인하는 방법
그래픽 카드 온도 확인은 아주 쉽다. 윈도우에서 바로 온도를 확인할 수 있는 내장 도구도 추가됐다. 또한, 무료 GPU 모니터링 도구가 많이 있고 그중 대다수가 온도를 측정해준다. 조금 더 자세히 알아보자.
ⓒ MARK HACHMAN / IDG 그래픽카드 온도 확인
마이크로소프트가 윈도우 10 2020년 5월 업데이트에서 GPU 온도 모니터링 툴을 작업 관리자에 추가했다. 무려 24년이나 걸렸다.
Ctrl+Shift+Esc를 열어 작업 관리자 대화창을 열거나 Ctrl+Alt+Delete에서 ‘작업 관리자’를 선택하거나 윈도우 시작 메뉴 아이콘을 오른쪽 클릭해서 ‘작업 관리자’를 선택한다. 여기에서 ‘성능’ 탭으로 들어가면 왼쪽에 GPU를 확인할 수 있을 것이다. 윈도우 10 2020년 5월 업데이트 혹은 그 이후 버전의 윈도우가 설치되어 있을 때만 사용할 수 있는 기능이다.
하지만 이 기능은 매우 단순하다. 시간 흐름에 따른 온도 변화를 추적하지 않고, 현재의 온도만을 보여준다. 그리고 업무를 하거나 오버클럭 조정 중에 작업 관리자를 여는 것도 귀찮을 수 있다. 마침내 윈도우에 GPU 온도를 확인할 수 있는 기능이 들어간 것은 환영하지만, 뒤이어 설명할 서드파티 도구가 훨씬 더 나은 GPU 온도 확인 옵션을 제공한다.
AMD 라데온 그래픽 카드 사용자가 라데온 세팅(Radeon Setting) 앱을 최신 버전으로 유지하고 있다면 방법은 쉽다. 2017년 AMD는 시각 설정을 변경할 수 있는 라데온 오버레이(Radeon Overlay)를 출시했다. 여기에도 GPU 온도와 다른 중요한 정보를 확인할 수 있는 성능 모니터 기능이 있다.
프로그램을 활성화하려면 Alt+R 키를 눌러 라데온 오버레이를 불러온다. 성능 모니터링 섹션에서 원하는 탭을 선택한다. Ctrl+Shift + 0을 눌러서 성능 모니터링 도구 설정을 단독으로 불러올 수 있다.
라데온 세팅 앱에서 오버클럭 도구인 와트맨(Wattman)으로 이동해 GPU 온도를 확인할 수 있다. 윈도우 바탕 화면을 우클릭하고, 라데온 설정을 선택한 후 게이밍(Gaming) > 글로벌 세팅(Global Setting) > 글로벌 와트맨(Global Wattman) 항목으로 이동한다. 도구를 사용해 지나친 오버클럭으로 그래픽 카드를 날려버리지 않겠다고 서약한 후에는 와트맨에 액세스하고 GPU 온도, 그리고 그래프 형태로 된 핵심적 통계 수치를 볼 수 있다. 여기까지가 전부다.
라데온 사용자가 아닌 사람도 많을 것이다. 스팀의 하드웨어 설문 조사는 전체 응답자 PC 중 75%가 엔비디아 지포스 그래픽 카드를 탑재했다는 결과를 발표했다. 그리고 지포스 익스피리언스 소프트웨어는 GPU 온도 확인 기능을 제공하지 않아서 서드파티 소프트웨어의 손을 빌려야 한다.
그래픽 카드 제조 업체는 보통 GPU 오버 클럭을 위한 특수한 소프트웨어를 제공한다. 이 도구에는 라데온 오버레이처럼 가장 중요한 측정을 실행할 때 OSD(On-Screen Display)를 지속하는 옵션 등이 있다. 여러 종류 중에서 가장 추천하는 것은 다재다능함을 갖춘 MSI의 애프터버너(Afterburner) 도구다. 이 제품은 오랫동안 인기를 얻었는데 엔비디아 지포스, AMD 라데온 그래픽 카드 두 제품 모두에서 잘 작동하고, 반길 만한 다른 기능도 더했다.
이제 그래픽 카드를 모니터링하는 소프트웨어를 갖췄다. 하지만 화면을 채우는 숫자는 맥락이 없이는 아무것도 아니다. 그래픽 카드 온도는 어디까지 괜찮은 것일까?
쉬운 대답은 없다. 제품마다 다르다. 이럴 때는 구글이 친구가 된다. 대다수 칩은 섭씨 90도 중반에도 작동하고, 게이밍 노트북에서도 90도까지 온도가 올라가는 경우가 흔히 있다. 그러나 일반 데스크톱 PC 온도가 90도 이상으로 올라간다면 구조 신호나 다름없다. 공기 흐름이 원활한 GPU 1대 시스템에서는 80도 이상 올라가면 위험하다. 팬이 여러 개 달린 커스텀 그래픽 카드는 무거운 워크로드 하에서도 60~70도가 적당하고, 수냉쿨러가 달린 GPU라면 온도가 더 낮아야 할 것이다.
그래픽 카드가 최근 5년 안에 생산된 제품이고 90도 이상으로 뜨거워진다면, 또는 최근 몇 주간 온도가 급격히 상승했다면 다음의 냉각 방법을 고려해보자.
그래픽 카드 온도 낮추는 법
그래픽 카드 온도가 높아졌을 때 하드웨어 업그레이드에 돈을 들이지 않고 개선하지 않기란 어렵다. 그러나 돈을 쏟아붓기 전에 정말 그래야 하는지 필요성을 점검해 보자. 다시 한번 강조하지만 그래픽 카드는 뜨거운 온도를 버틸 수 있도록 설계되어 있다. PC가 무거운 게임이나 영상 편집 중에 강제 종료되는 경우가 아니라면 아마도 걱정할 필요가 없을 것이다.
우선, 시스템의 케이블을 깨끗하게 정리해 GPU 주변의 공기가 원활하게 순환되는지 확인하라. 케이블이 깔끔하게 정리됐다면 케이스에 팬을 추가하는 것도 고려한다. 모든 PC는 최적의 성능을 위해 공기를 빨아들이고 내보내는 팬이 여럿 달려 있는데, POST PC라면 팬은 더 많아야 한다. 저렴한 팬은 10달러부터 구입할 수 있고, RGB 조명이 붙은 화려한 제품은 조금 더 가격이 높다.
마지막으로, GPU와 히트싱크의 써멀 페이스트가 오래되어 말라 있다면 효율이 떨어질 수 있다. 특히 오래된 그래픽 카드라면 더더욱 그렇다. 그리고 아주 드문 경우지만 품질이 좋지 않은 써멀 페이스트가 발라져서 출시되는 경우도 있다. 다른 방법이 모두 효과가 없다면 써멀 페이스트를 다시 바르는 것을 시도해보자. 그러나 과정이 매우 어려울 수 있고 카드마다 조금씩 다르고, 잘못 손댈 경우 사용자 보증 기한의 보호를 받을 수 없게 된다.
온도를 확실하게 낮추려면 수랭 쿨러를 위한 쿨링 시스템을 고려한다. 대다수 사용자에게는 지나친 모험이지만 대부분 수냉쿨러는 발열과 노이즈 감소 효과가 확실하고 공기 냉각에 있어 병목 현상도 없다.
“업무 효율 향상의 기본” 멀티 모니터 구축 가이드
듀얼 모니터를 사용하면 업무 생산성이 높아진다는 연구 결과가 있지만, 모니터가 많을수록 생산성이 높아지는지 여부에 대해서는 아직 이렇다 할 근거는 없다. 그러나 업무 생산성을 생각하지 않더라도 모니터를 여러 대(3대~6대까지) 사용하는 것은 멋진 일이며, 많은 화면을 봐야 하는 엔지니어는 정말 필요할지도 모른다.
모니터를 세로로 세워두면 긴 문서를 볼 때 스크롤을 적게 해도 된다는 장점이 있다. 멀티 디스플레이 환경을 구축하기 위해 고려해야 할 모든 것들을 살펴보겠다.
멀티 모니터 구축 가이드(www.itworld.co.kr)
1단계 : 그래픽 카드 확인하기
보조 모니터를 구입하기 전에 컴퓨터가 물리적으로 이 모든 모니터들을 감당할 수 있을지 점검해 봐야 한다. 가장 쉬운 방법은 PC의 뒷면을 보고, 그래픽 포트(DVI, HDMI, 디스플레이포트, VGA 등)가 몇 개나 있는지 확인하는 것이다.
별도의 그래픽 카드가 없다면 포트를 2개밖에 발견하지 못할 것이다. 그래픽이 통합된 대부분의 마더보드는 모니터 2개 밖에 설치하지 못한다. 별도의 그래픽 카드가 있다면, 마더보드의 포트를 제외하고 최소 3개의 포트를 발견할 수 있을 것이다.
팁 : 마더보드와 별도 그래픽 카드의 포트를 모두 이용해서 멀티 모니터를 설치할 수도 있지만, 이 경우 성능 저하와 모니터끼리의 속도 차이가 발생할 것이다. 그래도 이렇게 하고 싶다면, PC의 BIOS에서 Configuration > Video > Integrated graphics 로 진입한 다음, ‘always enable’로 설정한다.
그러나 별도의 그래픽 카드에 3개 이상의 포트가 있다고 해서 이것을 모두 동시에 사용할 수 있다는 의미는 아니다. 예를 들어서 구형 엔비디아 카드는 포트가 2개 이상이어도 하나의 카드에 모니터를 2개 이상 연결할 수 없다. 자신의 그래픽 카드가 멀티 모니터를 지원하는지 판단하는 가장 좋은 방법은 그래픽 카드 모델명을 찾아서 원하는 모니터 개수와 함께 검색하는 것이다. 예를 들어, ‘엔비디아 GTX 1660 모니터 4대’라고 검색하면 된다.
EVGA 지포스 RTX 2060 KO 같은 현대적인 그래픽카드는 여러 디스플레이를 동시에 연결할 수 있다. ⓒ BRAD CHACOS/IDG
그래픽 카드가 원하는 만큼 충분히 모니터를 지원할 수 있으면 좋지만, 그렇지 않다면 추가 그래픽 카드를 구입해야 한다. 그래픽 카드를 추가로 구입하기 전 타워 안에 충분한 공간(PCI 슬롯)이 있는지, 전원 공급은 충분한지 확인해야 한다.
멀티 모니터용으로만 그래픽 카드를 구입한다면 최신 그래픽 카드 중에서도 저렴한 옵션을 선택하는 것이 좋다.
아니면 멀티 스트리밍이 지원되는 디스플레이포트를 탑재한 신형 모니터를 사용하는 방법도 있다. 그래픽 카드의 디스플레이포트 1.2에 연결하고, 디스플레이포트 케이블을 사용해 다음 모니터로 연결하는 것이다. 모니터의 크기나 해상도가 같지 않아도 된다. 뷰소닉(ViewSonic)의 VP2468이 이런 제품 중 하나다. 아마존에서 약 210달러에 판매되는 이 24인치 모니터는 디스플레이포트 아웃 외에도 프리미엄 IPS 스크린, 아주 얇은 베젤 등 멀티 모니터 설정에 이상적인 특징을 제공한다.
2단계 : 모니터 선택하기
그래픽 카드에 대해서 파악했다면 이제 추가 모니터를 구입할 차례다. 사용자에 따라서 기존에 사용하고 있는 모니터, 책상 크기, 추가 모니터 용도 등에 따라서 완벽한 모니터가 달라질 것이다.
필자의 경우, 이미 24인치 모니터 2대를 가지고 있었기 때문에 중앙에 설치할 더 큰 모니터가 필요해서 27인치 모니터를 선택했다. 게임을 하지 않기 때문에 모니터 크기 차이는 상관없었다. 하지만 사용자에 따라서 멀티 모니터로 POST를 하거나 동영상을 보기 위해서는 이러한 구성보다 같은 모니터를 연결하는 것이 더 좋을 것이다.
모니터를 구입하기 전에 PC와 모니터의 포트 호환성을 설펴야 한다. DVI-HDMI 혹은 디스플레이포트-DVI 등 전환해주는 케이블을 이용할 수도 있지만 다소 귀찮다. 그러나 PC나 모니터에 VGA 포트가 있다면, 교체를 권한다. VGA는 아날로그 커넥터이기 때문에 선명도가 떨어진다.
3단계 : PC설정
모니터를 구입하고 나면 PC에 연결하고 PC의 전원을 켠다. 이것으로 모니터 설치가 끝났다. 하지만 완전히 끝난 것은 아니다.
윈도우가 멀티 모니터 환경에서 잘 동작하게 만들어야 하는데, 윈도우 7이나 윈도우 8 사용자라면 바탕화면에서 오른쪽 클릭하고 ‘화면 해상도’를 선택한다. 윈도우 10 사용자라면 ‘디스플레이 설정’을 클릭한다. 그러면 디스플레이를 정렬할 수 있는 창이 나타난다.
ⓒ ITWorld 디스플레이 설정
여기서 모니터들이 모두 탐지되는지 확인할 수 있다. ‘식별’을 클릭하면 각 디스플레이에 큰 숫자가 나타난다. 주 모니터(작업 표시줄과 시작 버튼이 나타나는 모니터)로 사용할 모니터에 1번이 나타나야 하는데, 원하는 것을 선택한 다음 아래 여러 디스플레이 설정에서 ‘이 디스플레이를 주 모니터로 만들기’를 클릭한다. 그 다음 ‘다중 디스플레이’ 드롭다운 메뉴에서 복제할 것인지 확장할 것인지를 선택하면 되는데, 대부분의 경우 ‘디스플레이 확장’이 적합하다.
GPU 제어판에서도 다중 모니터를 설정할 수 있다. 바탕화면에서 오른쪽 클릭을 하고 엔비디아, AMD, 인텔 등 그래픽 제조사의 제어판 메뉴를 열어 윈도우와 유사한 방식으로 디스플레이를 설정할 수 있다.
멀티 디스플레이를 구축할 경우에는 같은 모델을 이용하는 것이 해상도나 선명도, 색보정 등의 문제가 발생하지 않아 ‘끊김 없는’ 경험을 할 수 있다.
하부 테일워터를 위한 USBR 및 쐐기형 배플 블록 분지를 사용한 유압 점프의 수치적 조사
Muhammad Waqas Zaffar; Ishtiaq Hassan; Zulfiqar Ali; Kaleem Sarwar; Muhammad Hassan; Muhammad Taimoor Mustafa; Faizan Ahmed Waris
Abstract
The stilling basin of the Taunsa barrage is a modified form of the United States Bureau of Reclamation (USBR) Type-III basin, which consists of baffle and friction blocks. Studies revealed uprooting of baffle blocks due to their vertical face. Additionally, the literature highlighted issues of rectangular face baffle blocks: less drag, smaller wake area, and flow reattachment. In contrast, the use of wedge-shaped baffle blocks (WSBBs) is limited downstream of open-channel flows. Therefore, this study developed numerical models to investigate the effects of USBR and WSBB basins on the hydraulic jump (HJ) downstream of the Taunsa barrage under lower tailwater conditions. Surface profiles in WSBB and modified USBR basins showed agreement with previous studies, for which the coefficient of determination (R2) reached 0.980 and 0.970, respectively. The HJ efficiencies reached 57.9 and 58.6% in WSBB and modified USBR basins, respectively. The results of sequent depths, roller length, and velocity profiles in the WSBB basin were found more promising than the modified USBR basin, which further confirmed the suitability of the WSBB basin for barrages. Furthermore, WSBB improved flow behaviors in the basin, which showed no fluid reattachment on the sides of WSBB, increased wake regions, and decreased turbulent kinetic energies.
Barrages in Pakistan were built about 50–100 years ago and play an important role in the economy. However, as time passed, the stability of these barrages was compromised due to hydraulic and structural deficiencies (Zaidi et al. 2011). Similarly, Taunsa Barrage Punjab, on the mighty river Indus, is one of the major hydraulic structures, which was constructed about 65 years ago. The barrage was designed for a design discharge capacity of 28,313 m3/s, and its stilling basin was a modified form of the United States Bureau of Reclamation (USBR) Type-III basin that consisted of USBR impact friction and baffle blocks. These arrangements dissipate excessive kinetic energy, enhance turbulence, kill rollers, and stabilize the hydraulic jump (HJ) even in case of less tailwater depth. The barrage consists of 64 bays, and the total width of the barrage between the abatements is 1,324.60 m. Of total width, 1,176.5 m is a clear waterway (Zaffar & Hassan 2023a). Figure 1 shows the typical cross-section of the Taunsa barrage.
Figure 1
Typical cross-section of the Taunsa barrage.
Soon after the barrage operation in 1958, multiple problems occurred on the barrage downstream, such as uprooting of the impact baffle blocks due to their vertical face, damage to the basin’s floor, lowering of tailwater levels, and bed retrogression (Zulfiqar & Kaleem 2015). During 1959–1962, repair works were carried out to cater to these issues, but the problems remained persistent. To resolve these issues, the Punjab Government constituted a committee of experts in 1966 and 1973, but no specific measures were taken, and the issues continued to aggravate (Zaidi et al. (2004). Additionally, these traditional impact blocks also face flow reattachment on the sides that decreases the drag force (Frizell & Svoboda 2012). On the contrary, after investigating the wedge-shaped baffle blocks (WSBBs) downstream of pipe outlets, research scholars (Pillai et al. 1989; Verma & Goel 2003; Verma et al. 2004; Goel 2007; Goel 2008; Tiwari et al. 2010) reported that these blocks increased the energy dissipation and created more eddies and wake regions on either side. These studies further mentioned that upon the use of WSBBs, the overall length of stilling basins was also reduced from 15 to 25%.
Energy dissipation is the most common issue faced in the design of hydraulic structures. The kinetic energy typically comes from the upstream of the dams (El Baradei et al. 2022), spillways (Sutopo et al. 2022), chute, sluice gates, and weirs, which are further induced by the HJ and its turbulent structure (Elsaeed et al. 2016). In the HJ, flow suddenly changes from supercritical to subcritical conditions which dissipate the energy of the upstream flow, thereby saving the hydraulic structures from damage. The HJ occurs when Froude Number (Fr) falls below unity, which is the ratio of inertia to gravitational forces that can be calculated by the following equation (Bayon-Barrachina & Lopez-Jimenez 2015; Bayon-Barrachina et al. 2015).
(1)
where v, h, and g are stream-wise velocities, flow depth, and acceleration due to gravity, respectively. Hager & Sinniger (1985) investigated characteristics of the HJ for abrupt changes in horizontal bed and proposed the following equation to compute the efficiency of HJs.
(2)
where Fr1 is the Froude number in the supercritical flow before the HJ.
Bakhmeteff & Matzke (1936) developed HJ similarity models and proposed dimensionless Equation (3) for the free surface profile of HJs.
(3)
where is the water depth at x (hi) and the variable X is the dimensionless longitudinal coordinates (x), as shown in the following dimensionless equations, respectively.
(4)
(5)
where D is the gate opening and h1 and h2 are the water depths in supercritical and subcritical regions, respectively. X1 and X2 are the functions of variable X, and their values can be calculated at the toe of the HJ and the end of the roller region, respectively. The components of Equations (4) and (5) are shown in Figure 2.
Figure 2
Schematic diagram for the dimensionless free surface profiles of HJs.
Habibzadeh et al. (2012) conducted experiments to investigate the role of baffle blocks for submerged HJs and energy dissipation downstream of low-head hydraulic structures. Chachereau & Chanson (2011) and Wang & Chanson (2015) investigated free surface profiles and turbulent fluctuation within the HJ for a wide range of initial Froude number (Fr1). The velocity profiles showed a wall jet-like profile, and turbulence intensities were high due to the fluctuations at the free surface. Maleki & Fiorotto (2021) developed a semi-empirical method to investigate HJs on a rough bed. The results showed that the characteristic length scale was linearly changing with Fr1. Macián-Pérez et al. (2020b) carried out experiments on the USBR-II stilling basin to investigate the characteristics of HJs. The results of sequent depths and HJ efficiency agreed with the experimental studies, which reached 99.2 and 97%, respectively. The results of the dimensionless free surface profile also agreed with the previous studies for which the value of the coefficient of determination (R2) reached 0.979. Murzyn & Chanson (2009) conducted experiments for a wide range of Fr1 up to 8.3 to investigate bubbly and turbulence structure within the HJ. The results showed that void fraction (Cmax) and bubble frequency (Fmax) were found in the developing region, and vertical interfacial velocity agreed with the wall jet-like profile. Qasim et al. (2022) conducted experiments on bed discordance downstream of different weirs. The results indicated that as the bed discordance increased, the dimensionless flow depth decreased downstream of the discordance, which increased the Froude number. The results further showed that as the configuration of bed discordance was changed, the free surface profiles were also changed, which affected the flow depths and velocity profiles. Bhosekar et al. (2014) conducted experiments to investigate the characteristics of discharge downstream of orifice spillways. The results showed that free surface profiles were not elliptical due to the flat curve near the gate opening. The results further indicated that the flat curve developed a negative pressure on roof profiles, which reduced the discharge capacity of the spillway.
To stabilize the HJ in the stilling basins, different shapes of baffle blocks are employed, i.e., baffle blocks (Habibzadeh et al. 2012), friction blocks (Chaudary & Sarwar 2014), end sill (Mansour et al. 2004) and vertical sill (Alikhani et al. 2010), splitter blocks (Verma & Goel 2003), curved (Eloubaidy et al. 1999), T-shaped and triangular (Tiwari & Goel 2016), and WSBB (Pillai et al. 1989; Goel 2007; Goel 2008). These arrangements control the HJs in case of fewer tailwater depths (Peterka 1984) and minimize the erosion downstream of structures (Zaffar et al. 2023). Sayyadi et al. (2022) investigated HJ characteristics for negative steps in the stilling basin. The results showed that the negative step increased the energy dissipation up to 11%. Pillai et al. (1989) compared three different stilling basins for the Fr1 up to 4.5. The results showed that the stilling basin with the WSBB reduced the scour and overall length of the basin. Goel (2008), Goel (2007), and Tiwari et al. (2010) conducted experiments to investigate HJ characteristics downstream of square and circular pipe outlets using WSBBs. The results showed that as compared to the impact USBR-VI basin, the WSBB basin spread the fluid efficiently in the lateral direction and reduced the basin length up to 50%.
In the former section, the experimental studies on HJs, velocity distribution, free surface profiles, and turbulent kinetic energy (TKE) are discussed, which could be assisted by Computational Fluid Dynamics (CFD) models (Ghaderi et al. 2020). Furthermore, over these hydraulic structures, the flow is very complex and associated with secondary currents, which characterized it as highly turbulent in all directions. Hence, using laboratory and field experiments, it is hard to accurately measure the free surface profile, velocities, secondary currents, and TKE over these hydraulic structures (Jothiprakash et al. 2015). Furthermore, physical experiments and on-site measurements are usually expensive and time-consuming. In contrast, the improvements in computational speed, storage, and turbulence modeling have made CFD a viable complementary investigation tool for hydraulic modeling (Ghaderi et al. 2021). Consequently, the use of numerical modeling tools such as Open Foam (Bayon-Barrachina & Lopez-Jimenez 2015), ANSYS Fluent (Aydogdu et al. 2022), and FLOW-3D (Hirt & Sicilian 1985) has become prevalent to get hydraulic characteristics of grade-control structures. Such modeling tools are helpful, especially when the basic fundamental equations are unable to provide desired outputs, like in the case of multifaceted geometries (Herrera-Granados & Kostecki 2016). So far, many researchers have employed numerical models in the hydraulic investigations of HJs and energy dissipation, but only a few of the latest studies are highlighted here. FLOW-3D numerical models were employed to investigate the HJ (Zaffar & Hassan 2023a) and baffle blocks (Zaffar & Hassan 2023b) for different stilling basins of the Taunsa barrage. These studies focused on velocity distribution, TKE, free surface profiles, energy loss energy, and the effects of baffle blocks on the HJ characteristics. Macián-Pérez et al. (2020a) carried out a numerical investigation on a high Reynolds of 210,000 to study the HJ characteristics. Upon comparison, the FLOW-3D model showed 93% accuracy in the roller length of HJs. The results also indicated 94.2 and 94.3% accuracy for sequent depths and HJ efficiency, respectively. Nikmehr & Aminpour (2020) examined the HJ characteristics on rough beds using FLOW-3D, and compared results with the experiments. The results indicated that roughness height and its distance affected the HJ length. Gadge et al. (2018) conducted a numerical study to investigate the impact of roof profiles on the discharge capacity of orifice spillways and validated the models with experimental results. The study revealed that in addition to the pond level and height of orifice (d), the bottom and roof profiles also affected the discharge coefficient (Cd).
From the literature review, it is found that only a few studies are conducted on the flow characteristics downstream of the Taunsa barrage (Zaidi et al. 2004, 2011; Chaudhry 2010). These studies were carried out in laboratory flume and investigated the effects of tailwater on the location of HJs. However, the studies were lacking in providing the data for other essential hydraulic parameters, i.e., velocity distribution, free surface profiles, TKE, and relative energy loss in the stilling basin. On the contrary, the literature has revealed many experimental and numerical studies on different shapes of baffle blocks downstream of open-channel flow, but the use of WSBB downstream of river diversion barrage is found limited. In the previous studies (Pillai et al. 1989; Verma & Goel 2003; Verma et al. 2004; Goel 2008, 2007; Tiwari et al. 2010), these blocks have only been tested downstream of pipe outlet basins for the initial Froude number of 4.5. Therefore, in the present study, FLOW-3D numerical models are developed to investigate the effects of presently available USBR baffle blocks in the stilling basin of the Tuansa barrage. Due to the uprooting problems of these blocks, the study also investigates the suitability of WSBBs downstream of the studied barrage and draws a comparison between the results of modified USBR and WSBB basins. In this study, based on results from the literature, WSBB with a vertex angle of 150° and cutback angle of 90° is applied for Fr1 up to 6.64. The main objective of this study is to investigate HJs and flow behavior with USBR baffle blocks and WSBB downstream of an investigated barrage at 44 m3/s discharge. At 44 m3/s discharge, the numerical models are operated at the minimum tailwater level of 129.10 m, and investigated free surface profiles, sequent depths, roller lengths, HJ efficiency, velocity profile, and TKE in the two different stilling basins.
MATERIALS AND METHODS
Existing and proposed stilling basins, appurtenances
The present numerical models are developed downstream of the Taunsa barrage, Pakistan. The stilling basin of the barrage includes USBR impact friction and baffle blocks (Zaffar & Hassan 2023b; Zaffar et al. 2023). In the basin, floor level and weir crest are fixed at 126.79 and 130.44 m, respectively. The slopes of upstream and downstream weir glacis are maintained at 1:3 and 1:4 (H:V), respectively. In both the studied basins, the blocks are installed 14.63 m away from the centerline of the crest and are placed in a staggered position. The overall length and height of the USBR blocks is 1.37 m as shown in Figures 3(a) and 3(b). Additionally, between the two staggered rows of baffle blocks, a 1.37-m distance is maintained, while the top width of all the USBR blocks is 0.46 m, which is angled at 45° from the rear side. On the other hand, in the WSBB basin, WSBB is placed at the locations of impact USBR baffle blocks. Furthermore, in both the studied basins, two staggered rows of friction blocks are also placed at the basin’s end about 28.95 m away from the weir’s crest. These friction blocks are 1.37 m long, 1.22 m wide, and 1.37 m high. The top surface of these blocks is identical to their bottom. The overall length, width, and height of the WSBB are kept at 1.37 m, and a detailed geometry of the investigated WSBB can be seen in Figures 3(c) and 3(d). Currently, for the investigated WSBB, a vertex angle of 150° and a cutback angle of 90° are employed.
Figure 3
Baffle block geometry for the basins: (a) top view of USBR baffle blocks, (b) isometric view representing front of USBR baffle blocks, (c) top view of WSBBs and (d) isometric view representing front of WSBBs.
Numerical model implementation
Environmental flows are governed by the laws of physics and represented by Navier–Stokes Equations (NSEs), which are inherently nonlinear, time-dependent, and contain three-dimensional partial deferential schemes (Viti et al. 2018). These partial differential equations explain the procedures of continuity, momentum, heat, and mass transfer. For one- and two-dimensional models, these equations can be solved analytically, while for the solution of three-dimensional models, CFD models are employed to discretize the NSEs. In these models, flow equations, i.e., NSEs and continuity equations, are discretized in each cell. Generally, these models start with a mesh, which further contains multiple interconnected cells in the employed mesh blocks. These meshes subdivide the physical space into small volumes, which are associated with several nodes. The values of unknown parameters are stored on these nodes, such as velocity, temperature, and pressure. Different numerical techniques are available to discretize the NSEs, i.e., Direct Numerical Simulation (DNS) (Jothiprakash et al. 2015), Large Eddy Simulation (LES) (Ghosal & Moin 1995), and Reynold Averaged Navier–Stokes (RANS) Equation (Kamath et al. 2019). However, as compared to DNS and LES models, due to less computation cost and simulation time, the RANS model is frequently used in river and hydraulic investigations. Using the RANS model, two additional variables are generated, for which turbulence closure models are usually employed (Carvalho et al. 2008). These models find closure by averaging the Reynolds stress terms in NSEs and append additional variables for turbulent viscosity and transport equations.
Presently, FLOW-3D models are developed to investigate the effects of different shapes of baffle blocks on HJ downstream of the river diversion barrage. The models employ RANS equations to solve algorithms and equations of incompressible fluid in each computational cell. To further address the additional terms, i.e., Reynolds stresses and turbulent viscosity, the Renormalization group (RNG K–ɛ) method is applied. For the discretization of RANS and other algorithms, at present, the Volume of Fluid (VOF) method (finite volume method (FVM)) is employed, while the equations of the controlled volume are formulated with area and volume porosity functions. This formulation is called the ‘Fractional Area/Volume Obstacle Representation’ (FAVOR) method (Hirt & Sicilian 1985). The proceeding section describes the equations used for the present models.
Assuming the flow is steady and neglecting the fluctuation of specific weight ( = 0), Equations (6) and (7) are used for the turbulent flow (Carvalho et al. 2008).
(6)
(7)
Following the above equations, it is apparent that these relationships consist of three momentum and one continuity equation, in which there are 10 unknowns (p, u, v, w, and six Reynolds stress components). In the present study, the flow is considered incompressible, which implies the following equation to solve the flow domain (Viti et al. 2018).
(8)
In Equations (6)–(8) u, v, and w, are velocity components in x, y, and z directions, respectively. and p are total pressure and fluid density while the terms are known as the Reynolds stresses. Ax, Ay, and Az are flow areas while R,, and RSOR are the model’s coefficient, flow generic property, and mass source term, respectively.
Turbulence modeling and free surface tracking
Six turbulence models are available in FLOW-3D, which employs numerous equations to solve the closure problems. Among various models, the two-equation turbulence models such as standard K–ɛ (Bradshaw 1997, RNG K–ɛ (Yakhot et al. 1991), and K–ω (Wilcox 2008) are widely used in hydraulic investigations.
The standard K–ɛ and RNG K–ɛ models solve transport equations for TKE and its dissipation. The formulation of both models is identical, but the former derives model coefficients empirically. However, RNG K–ɛ applies a statistical approach to derive the transport equations explicitly, which has shown better capability in low-turbulence and high-shear regions (Macián-Pérez et al. 2020a, 2020b) than the standard K–ɛ model. In contrast, the K–ω model was implemented for stream-wise pressure gradient and near-wall boundaries (Wilcox 2008). The model replaces turbulent dissipation rate with turbulent frequency. In the K–ω model, the values of TKE and turbulent frequency are specified at the inlet boundaries. The above-mentioned turbulence models were investigated by Macián-Pérez et al. (2020b) for flow behaviors in the USBR-II stilling basin and the study indicated that the RNG K–ɛ model showed better accuracy. Also, RNG K–ɛ (Macián-Pérez et al. 2020a) showed more promising results for a grid convergence index (GCI), free surface, roller lengths, and HJ efficiency. Additionally, the RNG K–ɛ model predicted good results in flow rate, free surface profiles, and velocity profiles. Based on bibliographical results, this study also employed the RNG K–ɛ model, for which the following transport equations (Equations (9) and (10)) are utilized for TKE (K) and its dissipation (ɛ), respectively (Macián-Pérez et al. 2020a).
(9)
(10)
where is the coordinate in the x direction; is the dynamic viscosity; is the turbulent dynamic viscosity; K is the turbulent kinetic energy; is the turbulent dissipation; is the fluid density, and is the production of TKE. Finally, the terms, , , and are model parameters whose values are given in Yakhot et al. (1991).
For free surface modeling, the VOF method is employed. VOF applies an additional variable, which is called fraction of fluid (F), in which F represents the proportion of fluid. To compute F in the domain, the following equation (Hirt & Sicilian 1985) is used:
(11)
In FLOW-3D, the fluid fraction (F) in each cell is usually presented by three possibilities:
(A)F = 0, cell is empty.
(B)F = 1, a cell is fully occupied by fluid.
(C)0 < F < 1, cell represents the surface between the two fluids.
One fluid (water) with a free surface is considered in the present models, for which FLOW-3D automatically selects the free surface method from the availableVOF advection scheme. For the free surface tracking, 0.5 value is assigned in each computation cell.
Pressure velocity coupling
One of the major issues in solving the NSEs is pressure–velocity coupling, and for that, a network of algorithms (SIMPLE (Patankar & Spalding 1972) and PISO (ISSA 1985)) has been developed. These above-mentioned algorithms use under- and over-relaxing factors for pressure correction in the continuity and momentum equations, which contain large memory. Additionally, due to the relaxation factors, sometimes the solution becomes unstable and does not find convergence. On the contrary, FLOW-3D employs the Generalized Minimum Residual Method (GMRES) (Joubert 1994) because it possesses good convergence, high speed, and uses less memory. Additionally, GMRES does not apply any relaxation factor and possesses an additional algorithm, ‘Generalized Minimum Residual Solver (GCG),’ to treat the viscous terms.
The solid geometry of the barrage bay was designed in AutoCAD and converted into a stereo lithography file. Before importing the stereo lithography file into FLOW-3D, it was tested in Netfabb-basic software to remove any holes, facets, and boundary edges. Figures 4(a) and 4(b) show stereo lithography files used for the studied models.
Figure 4
Geometry details of the studied stilling basins: (a) modified USBR baffle block basin and (b) WSBB basin.
Meshing and boundary condition
The structured rectangular hexahedral mesh was employed to resolve the geometry and flow domain. To resolve the flow domain, a coarse mesh block was initiated upstream of the barrage (Xmin = 10 m) which ended at the upstream side of the gate (Xmax = 32.90 m). However, the fine mesh block was started from Xmin = 32.90 m which was extended up to the basin’s end (Xmax = 71 m). In total, 60 m of the model domain was simulated, out of which 22.90 m comprised upstream while the remaining included downstream side. For discharge measurement, mesh sensitivity analysis was performed. The blocks with coarse meshes were initially employed to calculate the volume flow rate (Q). The total number of mesh cells used in the coarse mesh block was 1,108,705. Out of 1,108,705 mesh cells, 481,000 cells were employed for the upstream block, while 627,705 mesh cells were utilized for the downstream mesh block. On the contrary, upon the use of fine mesh blocks, a total of 2,991,820 mesh cells were employed, out of which 481,000 cells were contained in the upstream mesh block while 2,510,820 cells were employed on the downstream side. Figure 5 shows mesh grids employed for flow and solid domains.
Figure 5
Meshing setup of the modeling domain.
It is essential to mention that in both meshing scenarios, fine mesh blocks were used on the downstream side of the bay because the focus of the present investigation was made around the baffle blocks and in the HJ regions. The details of mesh cell size and mesh quality indicators for the various mesh blocks are provided in Tables 1 and 2, respectively. Notably, except for discharge analysis, the results of other hydraulic parameters are produced from the fine meshing.
Table 1
Details of mesh blocks and cell sizes
Mesh block
Number of cells
Maximum adjacent ratio
Maximum aspect ratio
Block-1
X = 196; Y = 65; Z = 37
X
Y
Z
X–Y
Y–Z
Z–X
1.0
1.0
1.0
1.999
1.0
1.993
Block-2
X = 261; Y = 130; Z = 74
1.0
1.0
1.0
1.0
1.0
1.0
Table 2
Meshing quality indicators for various mesh blocks
Scenarios
Mesh block-1 (cell characteristics)
Mesh block-2 (cell characteristics)
Coarse meshing
Δx (m)
Δy (m)
Δz (m)
Δx (m)
Δy (m)
Δz (m)
0.142
0.284
0.284
0.142
0.284
0.284
Fine meshing
Δx (m)
Δy (m)
Δz (m)
Δx (m)
Δy (m)
Δz (m)
0.142
0.284
0.284
0.142
0.142
0.142
A vertical gate of 18.5-m width, 0.53-m length, and 6.10-m height was mounted upstream of the weir crest. Pond levels of 135.93 and 136.24 m were maintained for free and orifice flows, respectively. Table 3 shows the conditions used for models’ operation.
Table 3
Free and gated flow conditions for operation of the simulations (Zaffar & Hassan 2023b)
Discharge through barrage (m3/s)
Single bay discharge (m3/s)
Pond level (m)
Tailwater levels for jump formation (m)
Gate opening (m)
Turbulence model
28,313
444
135.93
133.80
Free flow
RNG K–ɛ
2,831
44
136.24
129.10
0.280
RNG K–ɛ
For the first mesh block, the upstream and downstream boundaries were set as pressure (P), while for the second block upstream boundary was set as symmetry (S). The lateral sides were set as rigid boundaries (W), and no-slip conditions were expressed as zero tangential and normal velocity (u=v=w = 0), where u, v, and w are the velocities in x, y, and z directions, respectively. These boundaries indicate a wall law velocity profile, which further expresses that the average velocity of turbulent flows is proportional to the logarithm of the distance from that point to the fluid boundary. For all variables (except pressure (P) (which was set to zero), upper boundaries (Zmax) were set as atmospheric pressure to allow water to null von Neumann. For both mesh blocks, the lower boundaries (Zmin) were set as walls.
For the present models, the stability and convergence at each iteration were checked by Courant number (Ghaderi et al. 2020), which affected the time steps from 0.06 to 0.0023 and 0.015 to 0.0025 for free and gated flow, respectively. It is worth mentioning here that for the free flow analysis of higher discharge such as 444 m3/s, the steady state solution can only be achieved by mass-averaged fluid kinetic energy (MAFKE) and volume flow rate (VFR) at the inlet and outlet boundaries. Therefore, the time at which the MAFKE and VFR reach the steady state is assigned as the simulation time (Ts) of models. Presently, VFRs at the inlet and outlet boundaries are considered as the stability and convergence indicators. Based on the criterion mentioned above, the present free and gated models achieved hydraulic stability at Ts = 60 s while the actual time (Ta) of models ranged between 30 and 48 h. However, to accommodate free surface fluctuations, the models were run for Ts = 80 s.
Models’ verification and validation
Analysis of design discharge
For performance assessment of the numerical models, He/Hd = 0.998 (Johnson & Savage 2006; Gadge et al. 2019; Zaffar & Hassan 2023a, 2023b) was implemented for free flow analysis, where He and Hd are effective and designed heads, respectively. This was the design discharge of the Taunsa barrage, for which the models were operated on the pond and tailwater levels of 135.93 and 133.8 m, respectively, as provided in Table 3.
Gated flow modeling
Computational discharge is of paramount importance in hydraulic modeling, and the following discharge formula is used for gated flow operations (Gadge et al. 2018). To model 44 m3/s of discharge, gate opening and designed head for orifice are set at D = 0.280 m and Hd = 136.24 m, respectively. Figure 6 illustrates the typical cross-section for gated flow operations.
(12)
where Q (m3/s) is discharged through the orifice opening, A (m2) is the area of the orifice, g (m/s2) is the acceleration due to gravity and hc is the centerline head (hc = Hd–D/2). The values of coefficients of discharge (Cd) used and simulated were 0.816 and 0.819, respectively, and were found well within the range of Cd values calculated by Bhosekar et al. (2014).
Figure 6
Cross-section showing gated flow through the orifice.
RESULTS AND DISCUSSION
Discharges and flow evolution
Figure 7 shows the time instant of flow evolution for designed discharge. At the start of the simulation, due to the inlet velocity, the free surface was found to be changed. However, when it reached a steady state, a fully developed flow on the downstream glacis was achieved, as shown in Figure 7. A stable free jump was observed at Ts = 60 s for free and gated flow, and the free surface on the downstream side was found to be stable with little fluctuation. The accuracy of FLOW-3D simulations was checked by comparing those discharges with designed values. At Ts = 80 s, the modified USBR baffle block basin underestimated the discharge, which reached 440.48 m3/s and displayed a 0.80% error. Similarly, at Ts = 80 s, for the WSBB basin, the model produced 440.17 m3/s discharge, in which the maximum error reached 0.893%. However, upon the use of coarse meshing, the errors in the computed discharge were increased, which reached −5 and −4% in modified USBR and WSBB basins, respectively. The free flow analysis of modified USBR and WSBB models showed acceptable validation with the designed flow, which allowed us to run the models for orifice discharge.
Figure 7
Flow evolution at the designed flow: USBR basin (a–c) and WSBB basin (d–f).
For the gated flow, on using fine meshing with similar meshing and boundary conditions, the modified USBR basin produced 44.14 m3/s of discharge, for which the maximum error reached 0.32%. However, in the WSBB basin, the numerical model underestimated the flow, which displayed only a 1.14% error. The evolutionary process of gated flows is shown in Figure 8. Based on the validation results of free and gated flows, further analysis was performed on the characteristics of HJs in the two different basins, and their results were compared with the relevant literature.
Figure 8
Evolutionary process of orifice discharge: USBR basin (a–c) and WSBB basin (d–f).
Free surface profiles
Due to the limited results of investigated hydraulic parameters on the studied barrage, the models’ results are compared with the previous relevant experimental and numerical studies. For such comparison, the models require some similarity in boundary and initial conditions, as obtained from Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015). Similar to the studies by Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015), in the present gated models, the upstream and downstream initial conditions are set to fluid elevation, i.e., pond and tailwater level, with hydrostatic pressure boundaries, while upstream boundaries are set to atmospheric pressure. Additionally, the sides and bottom are set to wall boundaries as described in Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015). However, the present models differ from the basin appurtenances as the compared studies have investigated HJ and other parameters on the horizontal flat beds. Furthermore, Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015) have investigated hydraulic parameters such as sequent depths, roller lengths, free surface profiles, energy dissipation, and TKE for Fr1 of 6.10 and 3.8 < Fr1 < 8.5, respectively. Similar to the above-mentioned study, the present modified USBR and WSBB basins are investigated for the Fr1 of 6.5 and 6.64, respectively. Hence, to confirm the results of free surface profiles, the studies of Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015) are utilized, for which the relevant discussion is made in the proceeding paragraphs.
Using Equation (3) of Bakhmeteff & Matzke (1936), the free surface profiles of HJs were obtained by the VOF method. Figure 9 compares the results of free surface profiles with Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015). The results of the present model agreed well with Bayon-Barrachina & Lopez-Jimenez (2015) (coefficient of determination (R2) = 0.992), for which the value of R2 in WSBB and USBR basins reached 0.980 and 0.970, respectively. Similarly, after comparing free surface profiles with Wang & Chanson (2015), the results of the present model were found to be more promising. However, as compared to the USBR basin, the results of free surface profiles in the WSBB basin showed more agreement with the compared studies, as can be seen in Figure 9.
Figure 9
Comparison of dimensionless free surface profiles of HJs with the literature.
To further assess the performance and gain deeper insight into the models’ efficiency, residual plots are drawn for the investigated hydraulic parameters. These errors referred to the difference between the observed (literature) and predicted data, which monitored the regression quality (Hassanpour et al. 2021). At a 5% level of significance, a homo-scedasticity analysis measured the residual errors, and the results of predicted residual errors were compared with the previous study.
Figure 10 compares the residual errors of free surface profiles of present models with Wang & Chanson (2015) and Bayon-Barrachina & Lopez-Jimenez (2015). Notably, the solid horizontal line in Figure 10 is the agreement line. From Figure 8(b), the maximum residual errors of free surface profiles in the experimental study of Wang and Chanson ranged between −0.2 and 0.13, while from the agreement line, the maximum negative and positive residual errors in the numerical study of Bayon-Barrachina & Lopez-Jimenez (2015) reached −0.12 to 0.20. In comparison to the previous studies, at the start of the regression line, the residual errors in the present models indicated a random scattered pattern below the agreement line, which further showed that the residual errors in the present and compared models were not normally distributed. After X = 0.3, above the agreement line, the distribution pattern of residual errors was also not normal. The stilling basin with USBR baffle blocks indicated the maximum negative and positive residual values of −0.160 to 0.10, respectively, while −0.184 to 0.124 values of residuals were noticed in the WSBB basin. Furthermore, from Figure 10, it is evident from residual analysis that the free profiles of HJs within modified USBR and WSBB basins have followed the trend of previous studies and the residuals of the present model are found less, which indicates reasonable accuracy of the models. Overall, the regression analysis of free surface profiles both for the present and compared studies revealed a curvilinear pattern that showed a heteroscedasticity residual.
Figure 10
Residual error diagram of free surface profiles of HJs with the literature.
Sequent depth ratio
In 1840, Belanger developed a famous equation for the sequent depth of HJs in smooth rectangular channels, which is widely used by numerous researchers to validate the results. Similarly, in the laboratory experimentation for different upstream and downstream tailwater water levels, Hager & Bremen (1989) developed a relationship of sequent depth (y2/y1) against a wide range of initial Froude numbers (Fr1). For the present models, 6.5 and 6.64 values of Fr1 are obtained in WSBB and USBR basins, respectively, and their relationship with sequent depths is developed as shown in Figure 11(a). The results of y2/y1 against Fr1 are compared with the experiments of other authors (Belanger 1841; Hager & Bremen 1989; Kucukali & Chanson 2008) and with the numerical study of Bayon-Barrachina & Lopez-Jimenez (2015).
Figure 11
(a) Comparison of sequent depth ratio with previous studies and (b) comparison of residual errors of sequent depths with the literature.
The sequent depths obtained from WSBB and modified USBR basins were 8.96 and 8.68, respectively, which were found to agree with the experimental results of other studies (Hager & Bremen 1989 and Belanger 1841), as shown in Figure 11(a). The results were also compared with the experiments of Kucukali & Chanson (2008), in which the value of Fr1 was 6.9. However, the Fr1 values obtained from present numerical models were 6.5 and 6.64 within WSBB and USBR basins, respectively. The comparison indicated that the present models overestimated the sequent depths, for which the errors reached 8.6 and 5.7% in WSBB and modified USBR basins, respectively. Furthermore, upon comparison with Bayon-Barrachina & Lopez-Jimenez (2015), results showed that present models underestimated the sequent depths, for which the maximum errors reached −13.2 and −9.8% errors in WSBB and USBR basins, respectively.
Figure 11(b) shows the comparison of residual errors of sequent depths with the previous studies. The maximum positive and negative residual values of 0.250 and −0.222 were found in WSBB and USBR baffle block stilling basins, respectively. The pattern of residual errors indicated an equal variance along the agreement line, which showed normal destitution of residual errors, thereby a homoscedastic pattern was noticed. The residual errors of sequent depths in both the tested basins were found within the ranges of Bayon-Barrachina & Lopez-Jimenez (2015) and Kucukali & Chanson (2008).
Roller length
Figure 12 compares the roller length (Lr/d1) of two different stilling basins with the previous studies, where Lr is the roller length of HJs and d1 is the initial flow depth before the HJ. Following Figure 10, the results showed that both the stilling basins produced almost similar roller lengths. Furthermore, in comparison to the previous studies, the relationship between roller lengths (Lr/d1) and Fr1 was found close to Kucukali and Chanson’s experiments (2008). However, the comparison with other studies indicated that the present models underestimated the roller lengths. The reason for reduced roller lengths was the effects of the basin’s appurtenance, which controlled the HJ lengths, as shown in Figures 13(a) and 13(b) (encircled regions). However, as compared to the modified USBR basin, the roller length in the WSBB basin was found to be less.
Figure 12
Comparison of roller lengths of HJ and initial Froude number with previous studies.
Figure 13
Roller lengths and energy dissipators: (a) WSBB basin and (b) modified USBR basin.
Figure 14 compares the residual errors for the roller lengths of present basins with the previous studies. The analysis showed a random distribution of residual errors and indicated homo-scedasticity of residuals. The results of residual errors for both the tested basins showed a good agreement with the compared studies and remained within the range of their residual errors. However, as compared to the USBR basin, the residual errors in the WSBB basin were found less, which reached −1.38.
Figure 14
Comparison of residual errors for the roller lengths with the previous studies.
HJ efficiency
The efficiency of the HJ is the ratio of energy loss to the upstream hydraulic head. Flow depth (hi), velocity (vi), and acceleration due to gravity (g) are the variables of HJ efficiency. The following equation was used to measure the efficiency of the HJ (Bayon-Barrachina & Lopez-Jimenez 2015).
(13)
where H1 and H2 are the specific energy heads upstream and downstream of HJs, respectively.
The results of numerical models showed 57.9 and 58.6% efficiencies in WSBB and modified USBR basins, respectively. The efficiencies for both the basins were also computed by Equation (2) and the results indicated 61.2 and 61.9% efficiencies for WSBB and modified USBR basins, respectively. The comparison further revealed that the present model underestimated the efficiencies, which reached the maximum errors of 5.41 and 5.45% in WSBB and modified USBR basins, respectively.
Figure 15(a) compares the efficiencies of present models with the previous studies. Upon comparing with Wu & Rajaratnam (1996), the results showed that the present model underestimated efficiencies for which the errors reached 6.6 and 5.54% in WSBB and modified USBR basins, respectively. Similarly, after comparing with Kucukali & Chanson (2008), the present model also showed a reduction in the HJ efficiencies for which the maximum errors reached 5.07 and 3.99% in WSBB and modified USBR basins, respectively. However, after comparing with Bayon-Barrachina & Lopez-Jimenez (2015), the results showed good agreement and indicated only 0.34 and 1.37% errors in WSBB and modified USBR basins, respectively. Overall, based on the bibliographic comparison, the overall accuracy of the present models for energy dissipation reached 93%.
Figure 15
(a) Comparison of HJ efficiency and (b) comparison of residual errors for the hydraulic jump efficiency.
Figure 15(b) indicates the residual errors of for modified USBR and WSBB basins and compares the errors with the previous experimental and numerical studies. Upon comparison with the modified USBR basin and with the literature, the HJ efficiency in the WSBB basin showed a close agreement with the zero residual line, for which the maximum error reached 0.001. On the other hand, the residual error in the modified USBR basin reached 0.003. Figure 14 also showed that the maximum residual error in the HJ efficiencies was found to be less than that was observed in previous studies, which remained within the limits of the compared studies (Wu & Rajaratnam 1996; Kucukali & Chanson 2008; Bayon-Barrachina & Lopez-Jimenez 2015).
Velocity distribution
Velocity distribution was measured at different flow depths to obtain vertical velocity profiles in two different stilling basins. Figure 16 shows the typical profiles in HJs, where (δ) is the y value at which the maximum velocity (Umax) occurs, while (b) is the length scale where u = 0.5Umax and ∂u/∂y < 0 (Ead & Rajaratnam 2002; Nasrabadi et al. 2012). The results indicated that in both basins, the velocity profiles showed a wall jet-like structure (Ead & Rajaratnam 2002). The results further showed that as the distance from the HJ-initiating locations was increased the maximum velocity decreased, thereby boundary growth layers were also decreased.
Figures 17 and 18 show that due to the supercritical velocity, a contracted jet was impinging near the beds of basins, and velocity decreased in upper fluid regions. The sections A-A and B-B in the basins indicated reverse flow and eddies in the HJs. The results of the upper fluid region of the HJ indicated typical backward velocity profiles as described by other authors (Ead & Rajaratnam 2002; Nasrabadi et al. 2012). Over time, these reverse fluid circulations were found to be stabilized and showed stagnation zones (Yamini et al. 2022). The analysis further showed a recirculation region within the HJ, and the maximum backward velocity profiles were found in the developed regions. The results also showed that after the jump termination, the negative velocity profiles converted into forward velocity profiles, as can be seen in sections C-C of Figures 17 and 18. Additionally, the results showed that after the WSBB and USBR baffle blocks, the velocity near the bed decreased and became positive at the free surface.
Figure 17
2D illustration of vertical velocity profiles in the USBR basin.
Figure 18
2D illustration of vertical velocity profiles in the WSBB basin.
Figure 19 shows the vertical velocity profile in the HJs at five different horizontal sections. The dimensionless plots between (y/b) and (U/Umax) illustrated that the velocity profiles followed the wall jet-like structure and were found to be agreed with Ead & Rajaratnam (2002), where y was the flow depth, b was the length scale, and Umax was the maximum velocity in the vertical section. The results also showed that in the HJ regions, both stilling basins produced identical structures of forward velocity profiles as can be seen in Figure 19(a) and 19(b).
From Figure 19, results showed that as the distance from the HJ toe increased, the vertical distance of Umax and inner layer thickness also increased. The analysis further indicated that as the distance from the initial location of HJs increased, the position of Umax was increased, which leveled off after the HJ, as can be seen in sections (C-C) of Figures 17 and 18. In both stilling basins, at X = 2 m, from the HJ initial location, the forward velocity profiles were found well agreed with the profile of Ead & Rajaratnam (2002) and the values of R2 reached 0.937 and 0.887 for WSBB and modified USBR basins, respectively, as shown in Figures 18(a) and 18(b), respectively. However, at X = 5.4 m, as compared to velocity (U/Umax = 0.36) in the modified USBR basin, the results showed less forward velocity (U/Umax = 0.21) in the WSBB basin at the upper fluid region.
Figure 20(a) shows the residual errors of velocity profiles in the WSBB basin. At x = 2 m, x = 3.2 m, and x = 4.3 m, the residual errors were found close to the agreement line. At the above-mentioned locations in the WSBB basin, the residual errors were also found less than Ead & Rajaratnam (2002). At x = 5.4 m in the WSBB basin, the maximum positive and negative residual errors of velocity profiles were 0.179 and −0.371, respectively, which were less than those that were found in the modified USBR basin. Figure 20(b) compares residual errors of velocity profiles in the modified USBR basin with the literature. It was evident from the residual diagrams that as the stream-wise distance from the jump-initiating location was increased, the maximum positive and negative errors also increased. The maximum positive and negative residual errors in the USBR basin were found at x = 5.4 m, which reached 1.139 and −1.352, respectively, and showed deviation from Ead & Rajaratnam (2002).
Figure 20
Comparison of residual errors with previous studies: (a) WSBB and (b) modified USBR basins.
Turbulent kinetic energy
TKE is the averaged velocity value in x, y, and z directions and describes energy dissipation at two different flow sections. The root mean square values of the velocity fluctuations are used to compute TKE. By considering the successive velocity values, the root mean square velocity (Urms) can be computed by the following equation (Gray et al. 2005).
(14)
where u1, u2, and u3 are successive velocities in the flow direction. Now, TKE can be calculated by the following equation.
(15)
where urms, vrms, and wrms are the root mean square velocities in x, y and z directions, respectively. Figure 21 shows the depth-wise (X–Y) TKE in a modified USBR basin. The results showed that the maximum TKE was found within and foreside of HJs while after the HJ, TKE continued to decline up to the end of the basin. Near the foreside of basins, flows were found to be strongly turbulent, dissipating most of the TKEs. Figure 19 shows the distribution of TKEs at seven vertical sections (X–Y), i.e., at Z = 0 m, 0.47 m, 0.93 m, 1.39 m, 1.85 m, 2.51 m, and free surface. At Z = 0 m, the maximum TKE in the USBR basin was found in the HJ region, which reached 0.30 J/kg, as shown in Figure 21(a). It is observed that behind the baffle blocks, TKEs were reduced due to eddies and fluid circulations, which dissipated the TKEs. Due to the impact of supercritical flows, a small number of eddies and fluid circulations were also noticed in front of the baffle blocks. The results showed that at the basin’s floor, TKEs traveled up to X = 21 m from the HJ-initiating location. Figures 21(b)–21(d) show that as the vertical distance from the basin’s floor was increased, the TKEs also increased, while their magnitude in the longitudinal direction was found to be reduced. The maximum TKEs at Z = 0.93 m, 1.39 m, and 1.85 m were 4.2, 4.5, and 3.6 J/kg, respectively. The results further indicated that as compared to the floor level, the TKEs from the central fluid depth to the free surface were found to be increased and their distribution in horizontal and lateral directions also increased. In the modified USBR basin, the maximum TKEs were noted at the toe of the HJ, which gradually reduced as the flow moved downstream, reaching 0.1 J/kg at the basin end, as shown in Figure 21(g).
Figure 21
Depth-wise distribution of TKEs in the USBR stilling basin at (a) Z = 0 m (floor level), (b) Z = 0.47 m, (c) Z = 0.93 m, (d) Z= 1.39 m, (e) 1.85 m, (f) Z= 2.51 m, and (g) free surface.
In the WSBB basin, at Z = 0 m (floor level), maximum TKEs reached 0.20 J/kg as shown in Figure 22(a). The results showed that as compared to USBR baffle blocks, the WSBBs were spreading the flow more efficiently in the lateral direction. Due to the spreading of fluid in the lateral direction, the results indicated that in the WSBB basin, the TKEs declined earlier in the basin and less energy was reached at the basin’s end. In the WSBB basin, only the TKEs in central fluid depths traveled downstream, which ended at X = 13 m from the toe of HJs. It is worth mentioning here that as compared to the modified USBR basin, the TKEs in the WSBB basin declined earlier, which indicated 8 m less distance than the USBR basin.
Figure 22
Depth-wise distribution of TKEs in the WSBB stilling basin at (a) Z= 0 m (floor level), (b) Z= 0.47 m, (c) Z= 0.93 m, (d) Z= 1.39 m, (e) 1.85 m, (f) Z= 2.51 m, and (g) free surface.
The results further showed that upon the use of WSBBs, no flow reattachment was witnessed on either side of the baffle blocks. Due to reduced reattachment, more wake areas were generated on the side of the WSBB basin, and the results showed agreement with the statement of other authors (Verma & Goel 2003; Verma et al. 2004; Goel & Verma 2006). At Z = 0.47 m, 0.93 m, and 1.39 m, the maximum TKEs were noticed in the HJ region, which reached 4.3, 4.6, and 3.5 J/kg, respectively, as shown in Figure 22(b)–22(d), respectively. In the WSBB basin, after the HJ, the baffle blocks declined the TKEs due to the development of sharp discontinuities in the flow. After Z = 1.39 m, the value of TKEs up to the free surface gradually reduced, as shown in Figure 22(e) and 22(f). Figure 22(g) shows 2D illustrations of TKEs on the free surface, and the results indicate that as compared to the modified USBR basin, the magnitude of TKEs was lower and traveled less distance in the WSBB basin.
CONCLUSIONS
This study developed numerical models on the rigid bed to investigate the effects of USBR and WSBB baffle blocks on the HJ downstream of the river diversion barrage using FLOW-3D. VOF and RNG K–ɛ models were employed to track the free surface and turbulence, respectively. For the proposed new basin (WSBB basin), WSBB with a vertex angle of 150° and cutback of 90° is employed in the baffle block region, while the friction block region remained unchanged. The performance of the two different basins is assessed by HJs and other hydraulic parameters such as free surface profile, sequent depths, roller lengths, HJ efficiency, velocity profile, and TKE. Furthermore, the results of the present modified USBR Type-III and WSBB basins are compared with the relevant literature, for which regression analysis is performed and residual error diagrams are plotted. However, the present models are limited to the single discharge of 44 m3/s and employ only one turbulence model, i.e., RNG K–ɛ. Additionally, the present models were designed for a single bay of the barrage.
Upon use of fine meshing, in comparison to the designed discharge, the present models showed 0.80 and 0.90% of errors in modified USBR Type-III and WSBB basins, respectively. Similarly, for the gated flow, the results indicated 0.32 and 1.14% errors in the modified USBR Type-III and WSBB basins, respectively.
After employing regression analysis, the results of free surface profiles showed agreement with the previous studies for which R2 reached 0.980 and 0.970 in WSBB and modified USBR basins, respectively. From the results, it can be believed that as compared to the modified USBR Type-III, the newly proposed WSBB basin produced a better free surface profile of HJs.
Due to the inclusion of the baffle blocks in the studied basins, the roller lengths of HJs were contained efficiently, and thereby, as compared to the literature, lesser roller lengths were observed in the modified USBR Type-III and WSBB basins.
The overall efficiency of HJs in modified USBR and WSBB basins reached 58.60 and 57.90%, respectively, which showed good agreement with the literature. Based on the results of the efficiency of HJs, the accuracy of the present models reached 93%.
In the hydraulic regions, the results of dimensionless velocity profiles indicated a wall jet-like structure, which agreed well with the literature. In addition, as compared to the modified USBR Type-III basin, the velocity profiles in the WSBB basin were found to be more promising, for which R2 reached 0.937. Additionally, after the HJ, as compared to the USBR Type-III basin, the forward velocity (U/Umax) in the WSBB basin was found to be less. Conclusively, it can be said that in comparison to the modified USBR Type-III basin, at the lower discharges, the WSBB basin decays the velocities more efficiently.
The results of TKEs indicated that the flow was strongly turbulent near the foreside of the HJs, and the maximum TKEs were noted in the central fluid depths. In the WSBB basin, no fluid reattachment was observed on either side of the baffle blocks, and the results further indicated that as compared to the modified USBR Type-III basin, fewer TKEs were found at the end of the WSBB basin.
Based on the models’ results, the study confirms the suitability of WSBB downstream of the barrage for lower tailwater conditions. From the results, it is believed that FLOW-3D is a very effective and efficient tool for the hydraulic investigation of flow behavior downstream of the barrage. However, in Pakistan, the use of such modeling tools is found very limited, therefore, the study results will help hydraulic and civil engineers to assess different energy dissipation arrangements within the stilling basins and will provide suitable alternative solutions. The present study was limited to the fixed geometry of the WSBB, therefore, it is suggested to investigate HJ and flow characteristics with other vertex and cutback angles. In addition, it is also recommended to study the hydraulics of WSBB downstream of barrages by employing multiple bays of barrage and other turbulence models.
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Mary Kathryn Walker Florida Institute of Technology, mwalker2022@my.fit.edu
Robert J. Weaver, Ph.D. Associate Professor Ocean Engineering and Marine Sciences Major Advisor
Chungkuk Jin, Ph.D. Assistant Professor Ocean Engineering and Marine Sciences
Kelli Z. Hunsucker, Ph.D. Assistant Professor Ocean Engineering and Marine Sciences
Richard B. Aronson, Ph.D. Professor and Department Head Ocean Engineering and Marine Sciences
Abstract
모노파일은 해상 풍력 터빈 건설에 사용되며 일반적으로 설계 수명은 25~50년입니다. 모노파일은 수명 주기 동안 부식성 염수 환경에 노출되어 구조물을 빠르게 분해하는 전기화학적 산화 공정을 용이하게 합니다. 이 공정은 모노파일을 보호 장벽으로 코팅하고 음극 보호 기술을 구현하여 완화할 수 있습니다.
역사적으로 모노파일 설계자는 파일 내부가 완전히 밀봉되고 전기화학적 부식 공정이 결국 사용 가능한 모든 산소를 소모하여 반응을 중단시킬 것이라고 가정했습니다. 그러나 도관을 위해 파일 벽에 만든 관통부는 종종 누출되어 신선하고 산소화된 물이 내부 공간으로 유입되었습니다.
표준 부식 방지 기술을 보다 효과적으로 적용할 수 있는 산소화된 환경으로 내부 공간을 재고하는 새로운 모노파일 설계가 연구되고 있습니다. 이러한 새로운 모노파일은 간조대 또는 조간대 수준에서 벽에 천공이 있어 신선하고 산소화된 물이 구조물을 통해 흐를 수 있습니다.
이러한 천공은 또한 구조물의 파도 하중을 줄일 수 있습니다. 유체 역학적 하중 감소의 크기는 천공의 크기와 방향에 따라 달라집니다. 이 연구에서는 천공의 크기에 따른 모노파일의 힘 감소 분석에서 전산 유체 역학(CFD)의 적용 가능성을 연구하고 주어진 파도의 접근 각도 변화의 효과를 분석했습니다.
모노파일의 힘 감소를 결정하기 위해 이론적 3D 모델을 제작하여 FLOW-3D® HYDRO를 사용하여 테스트했으며, 천공되지 않은 모노파일을 제어로 사용했습니다. 이론적 데이터를 수집한 후, 동일한 종류의 천공이 있는 물리적 스케일 모델을 파도 탱크를 사용하여 테스트하여 이론적 모델의 타당성을 확인했습니다.
CFD 시뮬레이션은 물리적 모델의 10% 이내, 이전 연구의 5% 이내에 있는 것으로 나타났습니다. 물리적 모델과 시뮬레이션 모델을 검증한 후, 천공의 크기가 파도 하중 감소에 뚜렷한 영향을 미치고 주어진 파도의 접근 각도에 대한 테스트를 수행할 수 있음을 발견했습니다.
접근 각도의 변화는 모노파일을 15°씩 회전하여 시뮬레이션했습니다. 이 논문에 제시된 데이터는 모노파일의 방향이 통계적으로 유의하지 않으며 천공 모노파일의 설계 고려 사항이 되어서는 안 된다는 것을 시사합니다.
또한 파도 하중 감소와 구조적 안정성 사이의 균형을 찾기 위해 천공의 크기와 모양에 대한 연구를 계속하는 것이 좋습니다.
Monopiles are used in the construction of offshore wind turbines and typically have a design life of 25 to 50 years. Over their lifecycle, monopiles are exposed to a corrosive saltwater environment, facilitating a galvanic oxidation process that quickly degrades the structure. This process can be mitigated by coating the monopile in a protective barrier and implementing cathodic protection techniques. Historically, monopile designers assumed the interior of the pile would be completely sealed and the galvanic corrosion process would eventually consume all the available oxygen, halting the reaction. However, penetrations made in the pile wall for conduit often leaked and allowed fresh, oxygenated water to enter the interior space. New monopile designs are being researched that reconsider the interior space as an oxygenated environment where standard corrosion protection techniques can be more effectively applied. These new monopiles have perforations through the wall at intertidal or subtidal levels to allow fresh, oxygenated water to flow through the structure. These perforations can also reduce wave loads on the structure. The magnitude of the hydrodynamic load reduction depends on the size and orientation of the perforations. This research studied the applicability of computational fluid dynamics (CFD) in analysis of force reduction on monopiles in relation to size of a perforation and to analyze the effect of variation in approach angle of a given wave. To determine the force reduction on the monopile, theoretical 3D models were produced and tested using FLOW-3D® HYDRO with an unperforated monopile used as the control. After the theoretical data was collected, physical scale models with the same variety of perforations were tested using a wave tank to determine the validity of the theoretical models. The CFD simulations were found to be within 10% of the physical models and within 5% of previous research. After the physical and simulated models were validated, it was found that the size of the perforations has a distinct impact on the wave load reduction and testing for differing approach angles of a given wave could be conducted. The variation in approach angle was simulated by rotating the monopile in 15° increments. The data presented in this paper suggests that the orientation of the monopile is not statistically significant and should not be a design consideration for perforated monopiles. It is also suggested to continue the study on the size and shape of the perforations to find the balance between wave load reduction and structural stability.
Figure 1: Overview sketch of typical monopile (MP) foundation and transition piece (TP) design with an internal j-tube (Hilbert et al., 2011)
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M.T. Mansouri Kia1,2, H.R. Sheibani 3, A. Hoback 4 1 Manager of Dam and Power Plant Construction, Khuzestan Water and Power Authority (KWPA), Ahwaz, Iran. 2 Ph.D., Department of Civil Engineering, Payame Noor University, Tehran, Iran. 3 Associate Professor of PNU University, Tehran, Iran. 4 Professor of Civil, Architectural & Environmental Engineering, University of Detroit Mercy Civil, Rome, Italy.
Abstract
Mared Dam in northern Abadan is under construction on the Karun River and it is the first ship lock in Iran. In this study, the ship’s lock was examined. Every vessel must pass through this lock in order to transport water from Arvand River to Karun and vice versa. The interior dimensions of the Mared Shipping Lock are 160 meters long, 25 meters wide and 8 meters deep. Several important times are calculated for lock operation. 𝑇is the first time the gates open, 𝑇15 the time the initial gates remain open until the height difference between the two sides reaches 150 mm, 𝑇filled is the duration between the start of the opening the gates till the difference between the two ends becomes zero after 𝑇15. Finally, T is the total time required for opening or closing the gates completely. The rotational speeds of the gates range from 5 to 35 radians per minute. Numerical modeling has been used to study fluid behavior and interaction between fluid and gates in flow 3D software. Different lock maintenance scenarios have been analyzed. Important parameters such as inlet and outlet flow rate changes from gates, water depth changes at different times, stress and strain fields, hydrodynamic forces acting on different points of the lock have been calculated. Based on this, the forces acting on hydraulic jacks and gates have been calculated. The minimum time required for the safe passage of the ship through the lock is calculated.
북부 아바단의 마레드 댐은 카룬 강에 건설 중이며 이란 최초의 선박 잠금 장치입니다. 본 연구에서는 선박의 자물쇠를 조사하였습니다. Arvand 강에서 Karun으로 또는 그 반대로 물을 운송하려면 모든 선박이 이 수문을 통과해야 합니다.
Mared Shipping Lock의 내부 치수는 길이 160m, 너비 25m, 깊이 8m입니다. 잠금 작동을 위해 몇 가지 중요한 시간이 계산됩니다. 𝑇은 게이트가 처음 열릴 때, 𝑇15는 양쪽의 높이 차이가 150mm에 도달할 때까지 초기 게이트가 열린 상태로 유지되는 시간, 𝑇filled는 게이트가 열리는 시작부터 이후 두 끝의 차이가 0이 될 때까지의 시간입니다.
𝑇15. 마지막으로 T는 게이트를 완전히 열거나 닫는 데 필요한 총 시간입니다. 게이트의 회전 속도는 분당 5~35라디안입니다. 수치 모델링은 유동 3D 소프트웨어에서 유체 거동과 유체와 게이트 사이의 상호 작용을 연구하는 데 사용되었습니다. 다양한 잠금 유지 관리 시나리오가 분석되었습니다.
게이트의 입구 및 출구 유속 변화, 다양한 시간에 따른 수심 변화, 응력 및 변형 필드, 수문의 다양한 지점에 작용하는 유체역학적 힘과 같은 중요한 매개변수가 계산되었습니다.
이를 바탕으로 유압잭과 게이트에 작용하는 힘을 계산하였습니다. 선박이 자물쇠를 안전하게 통과하는 데 필요한 최소 시간이 계산됩니다.
Fig 1. (a) The Location of the Bahman Shir dam (upstream), (b) Bahman Shir dam (downstream dam) and (c) Mared Dam. Note: The borders of the countries are not exact.
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The coupled dynamics of interfacial fluid phases and unconstrained solid particles during the binder jet 3D printing process govern the final quality and performance of the resulting components. The present work proposes a computational fluid dynamics (CFD) and discrete element method (DEM) framework capable of simulating the complex interfacial fluid–particle interaction that occurs when binder microdroplets are deposited into a powder bed. The CFD solver uses a volume-of-fluid (VOF) method for capturing liquid–gas multifluid flows and relies on block-structured adaptive mesh refinement (AMR) to localize grid refinement around evolving fluid–fluid interfaces. The DEM module resolves six degrees of freedom particle motion and accounts for particle contact, cohesion, and rolling resistance. Fully-resolved CFD-DEM coupling is achieved through a fictitious domain immersed boundary (IB) approach. An improved method for enforcing three-phase contact lines with a VOF-IB extension technique is introduced. We present several simulations of binder jet primitive formation using realistic process parameters and material properties. The DEM particle systems are experimentally calibrated to reproduce the cohesion behavior of physical nickel alloy powder feedstocks. We demonstrate the proposed model’s ability to resolve the interdependent fluid and particle dynamics underlying the process by directly comparing simulated primitive granules with one-to-one experimental counterparts obtained from an in-house validation apparatus. This computational framework provides unprecedented insight into the fundamental mechanisms of binder jet 3D printing and presents a versatile new approach for process parameter optimization and defect mitigation that avoids the inherent challenges of experiments.
바인더 젯 3D 프린팅 공정 중 계면 유체 상과 구속되지 않은 고체 입자의 결합 역학이 결과 구성 요소의 최종 품질과 성능을 좌우합니다. 본 연구는 바인더 미세액적이 분말층에 증착될 때 발생하는 복잡한 계면 유체-입자 상호작용을 시뮬레이션할 수 있는 전산유체역학(CFD) 및 이산요소법(DEM) 프레임워크를 제안합니다.
CFD 솔버는 액체-가스 다중유체 흐름을 포착하기 위해 VOF(유체량) 방법을 사용하고 블록 구조 적응형 메쉬 세분화(AMR)를 사용하여 진화하는 유체-유체 인터페이스 주위의 그리드 세분화를 국지화합니다. DEM 모듈은 6개의 자유도 입자 운동을 해결하고 입자 접촉, 응집력 및 구름 저항을 설명합니다.
완전 분해된 CFD-DEM 결합은 가상 도메인 침지 경계(IB) 접근 방식을 통해 달성됩니다. VOF-IB 확장 기술을 사용하여 3상 접촉 라인을 강화하는 향상된 방법이 도입되었습니다. 현실적인 공정 매개변수와 재료 특성을 사용하여 바인더 제트 기본 형성에 대한 여러 시뮬레이션을 제시합니다.
DEM 입자 시스템은 물리적 니켈 합금 분말 공급원료의 응집 거동을 재현하기 위해 실험적으로 보정되었습니다. 우리는 시뮬레이션된 기본 과립과 내부 검증 장치에서 얻은 일대일 실험 대응물을 직접 비교하여 프로세스의 기본이 되는 상호 의존적인 유체 및 입자 역학을 해결하는 제안된 모델의 능력을 보여줍니다.
이 계산 프레임워크는 바인더 제트 3D 프린팅의 기본 메커니즘에 대한 전례 없는 통찰력을 제공하고 실험에 내재된 문제를 피하는 공정 매개변수 최적화 및 결함 완화를 위한 다용도의 새로운 접근 방식을 제시합니다.
Introduction
Binder jet 3D printing (BJ3DP) is a powder bed additive manufacturing (AM) technology capable of fabricating geometrically complex components from advanced engineering materials, such as metallic superalloys and ultra-high temperature ceramics [1], [2]. As illustrated in Fig. 1(a), the process is comprised of many repetitive print cycles, each contributing a new cross-sectional layer on top of a preceding one to form a 3D CAD-specified geometry. The feedstock material is first delivered from a hopper to a build plate and then spread into a thin layer by a counter-rotating roller. After powder spreading, a print head containing many individual inkjet nozzles traverses over the powder bed while precisely jetting binder microdroplets onto select regions of the spread layer. Following binder deposition, the build plate lowers by a specified layer thickness, leaving a thin void space at the top of the job box that the subsequent powder layer will occupy. This cycle repeats until the full geometries are formed layer by layer. Powder bed fusion (PBF) methods follow a similar procedure, except they instead use a laser or electron beam to selectively melt and fuse the powder material. Compared to PBF, binder jetting offers several distinct advantages, including faster build rates, enhanced scalability for large production volumes, reduced machine and operational costs, and a wider selection of suitable feedstock materials [2]. However, binder jetted parts generally possess inferior mechanical properties and reduced dimensional accuracy [3]. As a result, widescale adoption of BJ3DP to fabricate high-performance, mission-critical components, such as those common to the aerospace and defense sectors, is contingent on novel process improvements and innovations [4].
A major obstacle hindering the advancement of BJ3DP is our limited understanding of how various printing parameters and material properties collectively influence the underlying physical mechanisms of the process and their effect on the resulting components. To date, the vast majority of research efforts to uncover these relationships have relied mainly on experimental approaches [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], which are often expensive and time-consuming and have inherent physical restrictions on what can be measured and observed. For these reasons, there is a rapidly growing interest in using computational models to circumvent the challenges of experimental investigations and facilitate a deeper understanding of the process’s fundamental phenomena. While significant progress has been made in developing and deploying numerical frameworks aimed at powder spreading [20], [21], [22], [23], [24], [25], [26], [27] and sintering [28], [29], [30], [31], [32], simulating the interfacial fluid–particle interaction (IFPI) in the binder deposition stage is still in its infancy. In their exhaustive review, Mostafaei et al. [2] point out the lack of computational models capable of resolving the coupled fluid and particle dynamics associated with binder jetting and suggest that the development of such tools is critical to further improving the process and enhancing the quality of its end-use components.
We define IFPI as a multiphase flow regime characterized by immiscible fluid phases separated by dynamic interfaces that intersect the surfaces of moving solid particles. As illustrated in Fig. 1(b), an elaborate IFPI occurs when a binder droplet impacts the powder bed in BJ3DP. The momentum transferred from the impacting droplet may cause powder compaction, cratering, and particle ejection. These ballistic disturbances can have deleterious effects on surface texture and lead to the formation of large void spaces inside the part [5], [13]. After impact, the droplet spreads laterally on the bed surface and vertically into the pore network, driven initially by inertial impact forces and then solely by capillary action [33]. Attractive capillary forces exerted on mutually wetted particles tend to draw them inward towards each other, forming a packed cluster of bound particles referred to as a primitive [34]. A single-drop primitive is the most fundamental building element of a BJ3DP part, and the interaction leading to its formation has important implications on the final part characteristics, such as its mechanical properties, resolution, and dimensional accuracy. Generally, binder droplets are deposited successively as the print head traverses over the powder bed. The traversal speed and jetting frequency are set such that consecutive droplets coalesce in the bed, creating a multi-drop primitive line instead of a single-drop primitive granule. The binder must be jetted with sufficient velocity to penetrate the powder bed deep enough to provide adequate interlayer binding; however, a higher impact velocity leads to more pronounced ballistic effects.
A computational framework equipped to simulate the interdependent fluid and particle dynamics in BJ3DP would allow for unprecedented observational and measurement capability at temporal and spatial resolutions not currently achievable by state-of-the-art imaging technology, namely synchrotron X-ray imaging [13], [14], [18], [19]. Unfortunately, BJ3DP presents significant numerical challenges that have slowed the development of suitable modeling frameworks; the most significant of which are as follows:
1.Incorporating dynamic fluid–fluid interfaces with complex topological features remains a nontrivial task for standard mesh-based CFD codes. There are two broad categories encompassing the methods used to handle interfacial flows: interface tracking and interface capturing [35]. Interface capturing techniques, such as the popular volume-of-fluid (VOF) [36] and level-set methods [37], [38], are better suited for problems with interfaces that become heavily distorted or when coalescence and fragmentation occur frequently; however, they are less accurate in resolving surface tension and boundary layer effects compared to interface tracking methods like front-tracking [39], arbitrary Lagrangian–Eulerian [40], and space–time finite element formulations [41]. Since interfacial forces become increasingly dominant at decreasing length scales, inaccurate surface tension calculations can significantly deteriorate the fidelity of IFPI simulations involving <100 μm droplets and particles.
2.Dynamic powder systems are often modeled using the discrete element method (DEM) introduced by Cundall and Strack [42]. For IFPI problems, a CFD-DEM coupling scheme is required to exchange information between the fluid and particle solvers. Fully-resolved CFD-DEM coupling suggests that the flow field around individual particle surfaces is resolved on the CFD mesh [43], [44]. In contrast, unresolved coupling volume averages the effect of the dispersed solid phase on the continuous fluid phases [45], [46], [47], [48]. Comparatively, the former is computationally expensive but provides detailed information about the IFPI in question and is more appropriate when contact line dynamics are significant. However, since the pore structure of a powder bed is convoluted and evolves with time, resolving such solid–fluid interfaces on a computational mesh presents similar challenges as fluid–fluid interfaces discussed in the previous point. Although various algorithms have been developed to deform unstructured meshes to accommodate moving solid surfaces (see Bazilevs et al. [49] for an overview of such methods), they can be prohibitively expensive when frequent topology changes require mesh regeneration rather than just modification through nodal displacement. The pore network in a powder bed undergoes many topology changes as particles come in and out of contact with each other, constantly closing and opening new flow channels. Non-body-conforming structured grid approaches that rely on immersed boundary (IB) methods to embed the particles in the flow field can be better suited for such cases [50]. Nevertheless, accurately representing these complex pore geometries on Cartesian grids requires extremely high mesh resolutions, which can impose significant computational costs.
3.Capillary effects depend on the contact angle at solid–liquid–gas intersections. Since mesh nodes do not coincide with a particle surface when using an IB method on structured grids, imposing contact angle boundary conditions at three-phase contact lines is not straightforward.
While these issues also pertain to PBF process modeling, resolving particle motion is generally less crucial for analyzing melt pool dynamics compared to primitive formation in BJ3DP. Therefore, at present, the vast majority of computational process models of PBF assume static powder beds and avoid many of the complications described above, see, e.g., [51], [52], [53], [54], [55], [56], [57], [58], [59]. Li et al. [60] presented the first 2D fully-resolved CFD-DEM simulations of the interaction between the melt pool, powder particles, surrounding gas, and metal vapor in PBF. Following this work, Yu and Zhao [61], [62] published similar melt pool IFPI simulations in 3D; however, contact line dynamics and capillary forces were not considered. Compared to PBF, relatively little work has been published regarding the computational modeling of binder deposition in BJ3DP. Employing the open-source VOF code Gerris [63], Tan [33] first simulated droplet impact on a powder bed with appropriate binder jet parameters, namely droplet size and impact velocity. However, similar to most PBF melt pool simulations described in the current literature, the powder bed was fixed in place and not allowed to respond to the interacting fluid phases. Furthermore, a simple face-centered cubic packing of non-contacting, monosized particles was considered, which does not provide a realistic pore structure for AM powder beds. Building upon this approach, we presented a framework to simulate droplet impact on static powder beds with more practical particle size distributions and packing arrangements [64]. In a study similar to [33], [64], Deng et al. [65] used the VOF capability in Ansys Fluent to examine the lateral and vertical spreading of a binder droplet impacting a fixed bimodal powder bed with body-centered packing. Li et al. [66] also adopted Fluent to conduct 2D simulations of a 100 μm diameter droplet impacting substrates with spherical roughness patterns meant to represent the surface of a simplified powder bed with monosized particles. The commercial VOF-based software FLOW-3D offers an AM module centered on process modeling of various AM technologies, including BJ3DP. However, like the above studies, particle motion is still not considered in this codebase. Ur Rehman et al. [67] employed FLOW-3D to examine microdroplet impact on a fixed stainless steel powder bed. Using OpenFOAM, Erhard et al. [68] presented simulations of different droplet impact spacings and patterns on static sand particles.
Recently, Fuchs et al. [69] introduced an impressive multipurpose smoothed particle hydrodynamics (SPH) framework capable of resolving IFPI in various AM methods, including both PBF and BJ3DP. In contrast to a combined CFD-DEM approach, this model relies entirely on SPH meshfree discretization of both the fluid and solid governing equations. The authors performed several prototype simulations demonstrating an 80 μm diameter droplet impacting an unconstrained powder bed at different speeds. While the powder bed responds to the hydrodynamic forces imparted by the impacting droplet, the particle motion is inconsistent with experimental time-resolved observations of the process [13]. Specifically, the ballistic effects, such as particle ejection and bed deformation, were drastically subdued, even in simulations using a droplet velocity ∼ 5× that of typical jetting conditions. This behavior could be caused by excessive damping in the inter-particle contact force computations within their SPH framework. Moreover, the wetted particles did not appear to be significantly influenced by the strong capillary forces exerted by the binder as no primitive agglomeration occurred. The authors mention that the objective of these simulations was to demonstrate their codebase’s broad capabilities and that some unrealistic process parameters were used to improve computational efficiency and stability, which could explain the deviations from experimental observations.
In the present paper, we develop a novel 3D CFD-DEM numerical framework for simulating fully-resolved IFPI during binder jetting with realistic material properties and process parameters. The CFD module is based on the VOF method for capturing binder–air interfaces. Surface tension effects are realized through the continuum surface force (CSF) method with height function calculations of interface curvature. Central to our fluid solver is a proprietary block-structured AMR library with hierarchical octree grid nesting to focus enhanced grid resolution near fluid–fluid interfaces. The GPU-accelerated DEM module considers six degrees of freedom particle motion and includes models based on Hertz-Mindlin contact, van der Waals cohesion, and viscoelastic rolling resistance. The CFD and DEM modules are coupled to achieve fully-resolved IFPI using an IB approach in which Lagrangian solid particles are mapped to the underlying Eulerian fluid mesh through a solid volume fraction field. An improved VOF-IB extension algorithm is introduced to enforce the contact angle at three-phase intersections. This provides robust capillary flow behavior and accurate computations of the fluid-induced forces and torques acting on individual wetted particles in densely packed powder beds.
We deploy our integrated codebase for direct numerical simulations of single-drop primitive formation with powder beds whose particle size distributions are generated from corresponding laboratory samples. These simulations use jetting parameters similar to those employed in current BJ3DP machines, fluid properties that match commonly used aqueous polymeric binders, and powder properties specific to nickel alloy feedstocks. The cohesion behavior of the DEM powder is calibrated based on the angle of repose of the laboratory powder systems. The resulting primitive granules are compared with those obtained from one-to-one experiments conducted using a dedicated in-house test apparatus. Finally, we demonstrate how the proposed framework can simulate more complex and realistic printing operations involving multi-drop primitive lines.
Section snippets
Mathematical description of interfacial fluid–particle interaction
This section briefly describes the governing equations of fluid and particle dynamics underlying the CFD and DEM solvers. Our unified framework follows an Eulerian–Lagrangian approach, wherein the Navier–Stokes equations of incompressible flow are discretized on an Eulerian grid to describe the motion of the binder liquid and surrounding gas, and the Newton–Euler equations account for the positions and orientations of the Lagrangian powder particles. The mathematical foundation for
CFD solver for incompressible flow with multifluid interfaces
This section details the numerical methodology used in our CFD module to solve the Navier–Stokes equations of incompressible flow. First, we introduce the VOF method for capturing the interfaces between the binder and air phases. This approach allows us to solve the fluid dynamics equations considering only a single continuum field with spatial and temporal variations in fluid properties. Next, we describe the time integration procedure using a fractional-step projection algorithm for
DEM solver for solid particle dynamics
This section covers the numerical procedure for tracking the motion of individual powder particles with DEM. The Newton–Euler equations (Eqs. (10), (11)) are ordinary differential equations (ODEs) for which many established numerical integrators are available. In general, the most challenging aspects of DEM involve processing particle collisions in a computationally efficient manner and dealing with small time step constraints that result from stiff materials, such as metallic AM powders. The
Unified CFD-DEM solver
The preceding sections have introduced the CFD and DEM solution algorithms separately. Here, we discuss the integrated CFD-DEM solution algorithm and related details.
Binder jet process modeling and validation experiments
In this section, we deploy our CFD-DEM framework to simulate the IFPI occurring during the binder droplet deposition stage of the BJ3DP process. The first simulations attempt to reproduce experimental single-drop primitive granules extracted from four nickel alloy powder samples with varying particle size distributions. The experiments are conducted with a dedicated in-house test apparatus that allows for the precision deposition of individual binder microdroplets into a powder bed sample. The
Conclusions
This paper introduces a coupled CFD-DEM framework capable of fully-resolved simulation of the interfacial fluid–particle interaction occurring in the binder jet 3D printing process. The interfacial flow of binder and surrounding air is captured with the VOF method and surface tension effects are incorporated using the CSF technique augmented by height function curvature calculations. Block-structured AMR is employed to provide localized grid refinement around the evolving liquid–gas interface.
CRediT authorship contribution statement
Joshua J. Wagner: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing. C. Fred Higgs III: Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Writing – original draft, Writing – review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by a NASA Space Technology Research Fellowship, United States of America, Grant No. 80NSSC19K1171. Partial support was also provided through an AIAA Foundation Orville, USA and Wilbur Wright Graduate Award, USA . The authors would like to gratefully acknowledge Dr. Craig Smith of NASA Glenn Research Center for the valuable input he provided on this project.
Este documento está relacionado con un proyecto en curso para el cual se está desarrollando e implementando un gemelo digital estructural del puente de Kalix en Suecia. 이 문서는 스웨덴 Kalix 교량의 구조적 디지털 트윈이 개발 및 구현되고 있는 진행 중인 프로젝트와 관련이 있습니다.
RESUMEN Las cargas ambientales, como el viento y el caudal de los ríos, juegan un papel esencial en el diseño y evaluación estructural de puentes de grandes luces. El cambio climático y los eventos climáticos extremos son amenazas para la confiabilidad y seguridad de la red de transporte.
Esto ha llevado a una creciente demanda de modelos de gemelos digitales para investigar la resistencia de los puentes en condiciones climáticas extremas. El puente de Kalix, construido sobre el río Kalix en Suecia en 1956, se utiliza como banco de pruebas en este contexto.
La estructura del puente, realizada en hormigón postensado, consta de cinco vanos, siendo el más largo de 94 m. En este estudio, las características aerodinámicas y los valores extremos de la simulación numérica del viento, como la presión en la superficie, se obtienen utilizando la simulación de remolinos desprendidos retardados (DDES) de Spalart-Allmaras como un enfoque de turbulencia RANS-LES híbrido que es práctico y computacionalmente eficiente para cerca de la pared densidad de malla impuesta por el método LES.
La presión del viento en la superficie se obtiene para tres escenarios climáticos extremos, que incluyen un clima con mucho viento, un clima extremadamente frío y el valor de cálculo para un período de retorno de 3000 años. El resultado indica diferencias significativas en la presión del viento en la superficie debido a las capas de tiempo que provienen de la simulación del flujo de viento transitorio. Para evaluar el comportamiento estructural en el escenario de viento crítico, se considera el valor más alto de presión en la superficie para cada escenario.
Además, se realiza un estudio hidrodinámico en los pilares del puente, en el que se simula el flujo del río por el método VOF, y se examina el proceso de movimiento del agua alrededor de los pilares de forma transitoria y en diferentes momentos. En cada una de las superficies del pilar se calcula la presión superficial aplicada por el caudal del río con el caudal volumétrico más alto registrado.
Para simular el flujo del río, se ha utilizado la información y las condiciones meteorológicas registradas en períodos anteriores. Los resultados muestran que la presión en la superficie en el momento en que el flujo del río golpea los pilares es mucho mayor que en los momentos posteriores. Esta cantidad de presión se puede usar como carga crítica en los cálculos de interacción fluido-estructura (FSI).
Finalmente, para ambas secciones, la presión en la superficie del viento, el campo de velocidades con respecto a las líneas de sondas auxiliares, los contornos del movimiento circunferencial del agua alrededor de los pilares y el diagrama de presión en ellos se informan en diferentes intervalos de tiempo.
요약 바람, 강의 흐름과 같은 환경 하중은 장대 교량의 설계 및 구조 평가에 필수적인 역할을 합니다. 기후 변화와 기상 이변은 교통 네트워크의 신뢰성과 보안에 위협이 됩니다.
이로 인해 극한 기상 조건에서 교량의 복원력을 조사하기 위한 디지털 트윈 모델에 대한 수요가 증가했습니다. 1956년 스웨덴 칼릭스 강 위에 건설된 칼릭스 다리는 이러한 맥락에서 테스트베드로 사용됩니다.
포스트텐션 콘크리트로 만들어진 교량 구조는 5개 경간으로 구성되며 가장 긴 길이는 94m입니다. 본 연구에서는 하이브리드 RANS-LES 난류 접근 방식인 Spalart-Allmaras 지연 분리 와류 시뮬레이션(DDES)을 사용하여 수치적 바람 시뮬레이션의 공기역학적 특성과 표면압 등 극한값을 얻습니다. LES 방법으로 부과된 벽 근처 메쉬 밀도.
바람이 많이 부는 기후, 극도로 추운 기후, 그리고 3000년의 반환 기간에 대해 계산된 값을 포함한 세 가지 극한 기후 시나리오에 대해 표면 풍압을 얻습니다. 결과는 과도 풍류 시뮬레이션에서 나오는 시간 레이어로 인해 표면 풍압에 상당한 차이가 있음을 나타냅니다. 임계 바람 시나리오에서 구조적 거동을 평가하기 위해 각 시나리오에 대해 가장 높은 표면 압력 값이 고려됩니다.
또한 교량 기둥에 대한 유체 역학 연구를 수행하여 하천의 흐름을 VOF 방법으로 시뮬레이션하고 기둥 주변의 물 이동 과정을 일시적이고 다른 시간에 조사합니다. 각 기둥 표면에서 기록된 체적 유량이 가장 높은 강의 흐름에 의해 적용되는 표면 압력이 계산됩니다.
강의 흐름을 시뮬레이션하기 위해 이전 기간에 기록된 정보와 기상 조건이 사용되었습니다. 결과는 강의 흐름이 기둥에 닿는 순간의 표면 압력이 나중에 순간보다 훨씬 높다는 것을 보여줍니다. 이 압력의 양은 유체-구조 상호작용(FSI) 계산에서 임계 하중으로 사용될 수 있습니다.
마지막으로 두 섹션 모두 바람 표면의 압력, 보조 프로브 라인에 대한 속도장, 기둥 주위 물의 원주 운동 윤곽 및 압력 다이어그램이 서로 다른 시간 간격으로 보고됩니다.
키워드: 디지털 트윈 , 풍력 공학, 콘크리트 교량, 유체역학, CFD 시뮬레이션, DDES 난류 모델, Kalix 교량
Palabras clave: Gemelo digital , Ingeniería eólica, Puente de hormigón, Hidrodinámica, Simulación CFD, Modelo de turbulencia DDES, Puente Kalix
1. Introducción
Las infraestructuras de transporte son la columna vertebral de nuestra sociedad y los puentes son el cuello de botella de la red de transporte [1]. Además, el cambio climático que da como resultado tasas de deterioro más altas y los eventos climáticos extremos son amenazas importantes para la confiabilidad y seguridad de las redes de transporte. Durante la última década, muchos puentes se han dañado o fallado por condiciones climáticas extremas como tifones e inundaciones.
Wang et al. analizó los impactos del cambio climático y mostró que se espera que el deterioro de los puentes de hormigón sea aún peor que en la actualidad, y se prevé que los eventos climáticos extremos sean más frecuentes y con mayor gravedad [2].
Además, la demanda de capacidad de carga a menudo aumenta con el tiempo, por ejemplo, debido al uso de camiones más pesados para el transporte de madera en el norte de Europa y América del Norte. Por lo tanto, existe una necesidad creciente de métodos confiables para evaluar la resistencia estructural de la red de transporte en condiciones climáticas extremas que tengan en cuenta los escenarios futuros de cambio climático.
Los activos de transporte por carretera se diseñan, construyen y explotan basándose en numerosas fuentes de datos y varios modelos. Por lo tanto, los ingenieros de diseño usan modelos establecidos proporcionados por las normas; ingenieros de construccion documentar los datos en el material real y proporcionar planos según lo construido; los operadores recopilan datos sobre el tráfico, realizan inspecciones y planifican el mantenimiento; los científicos del clima combinan datos y modelos climáticos para predecir eventos climáticos futuros, y los ingenieros de evaluación calculan el impacto de la carga climática extrema en la estructura.
Dadas las fuentes abrumadoras y la complejidad de los datos y modelos, es posible que la información y los cálculos actualizados no estén disponibles para decisiones cruciales, por ejemplo, con respecto a la seguridad estructural y la operabilidad de la infraestructura durante episodios de eventos extremos. La falta de una integración perfecta entre los datos de la infraestructura, los modelos estructurales y la toma de decisiones a nivel del sistema es una limitación importante de las soluciones actuales, lo que conduce a la inadaptación e incertidumbre y crea costos e ineficiencias.
El gemelo digital estructural de la infraestructura es una simulación estructural viva que reúne todos los datos y modelos y se actualiza desde múltiples fuentes para representar su contraparte física. El Digital Twin estructural, mantenido durante todo el ciclo de vida de un activo y fácilmente accesible en cualquier momento, proporciona al propietario/usuarios de la infraestructura una idea temprana de los riesgos potenciales para la movilidad inducidos por eventos climáticos, cargas de vehículos pesados e incluso el envejecimiento de un infraestructura de transporte.
En un proyecto en curso, estamos desarrollando e implementando un gemelo digital estructural para el puente de Kalix en Suecia. El objetivo general del presente artículo es presentar un método y estudiar los resultados de la cuantificación de las cargas estructurales resultantes de eventos climáticos extremos basados en escenarios climáticos futuros para el puente de Kalix. El puente de Kalix, construido sobre el río Kalix en Suecia en 1956, está hecho de una viga cajón de hormigón postensado. El puente se utiliza como banco de pruebas para la demostración de métodos de evaluación y control de la salud estructural (SHM) de última generación.
El objetivo específico de la investigación actual es dar cuenta de parámetros climáticos como el viento y el flujo de agua, que imponen cargas estáticas y dinámicas en las estructuras. Nuestro método, en el primer paso, consiste en simulaciones de flujo de viento y simulaciones de flujo de agua utilizando un modelado CFD transitorio basado en el modelo de turbulencia LES/DES para cuantificar las cargas de viento e hidráulicas; esto constituye el punto focal principal de este artículo.
En el siguiente paso, se estudiará la respuesta estructural del puente mediante la transformación de los perfiles de carga eólica e hidráulica en cargas estructurales en el análisis de EF estructural no lineal. Por último, el modelo estructural se actualizará incorporando sin problemas los datos del SHM y, por lo tanto, creando un gemelo digital estructural que refleje la verdadera respuesta de la estructura. Los dos primeros enfoques de investigación permanecen fuera del alcance inmediato del presente artículo.
2. Descripción del puente de Kalix
El puente de Kalix consta de 5 vanos largos de los cuales el más largo tiene unos 94 metros y el más corto 43,85 m. El puente es de hormigón postensado, el cual se cuela in situ de forma segmentaria y una viga cajón no prismática como se muestra en la Fig. 1. El puente es simétrico en geometría y hay una bisagra en el punto medio. El ancho del tablero del puente en la losa superior e inferior es de aproximadamente 13 my 7,5 m, respectivamente. El espesor del muro es de 45 cm y el espesor de la losa inferior varía de 20 cm a 50 cm.
Las pruebas en túnel de viento solían ser la única forma de examinar la reacción de los puentes a las cargas de viento Consulte [3]; sin embargo, estos experimentos requieren mucho tiempo y son costosos. Se requieren cerca de 6 a 8 semanas para realizar una prueba típica en un túnel de viento Consulte [4]. Los últimos logros en la capacidad computacional de las computadoras brindan oportunidades para la simulación práctica del viento alrededor de puentes utilizando la dinámica de fluidos computacional (CFD).
Es beneficioso investigar la presión del viento en los componentes del puente utilizando una simulación por computadora. Es necesario determinar los parámetros de simulación del puente y el campo de viento a su alrededor; por lo tanto, se pueden evaluar con precisión sus impactos en las fuerzas aplicadas en el puente.
Las demandas de diseño de las estructuras de puentes requieren una investigación rigurosa de la acción del viento, especialmente en condiciones climáticas extremas. Garantizar la estabilidad de los puentes de grandes luces, ya que sus características y formaciones son más propensas a la carga de viento, se encuentra entre las principales consideraciones de diseño [3].
3.1. Parámetros de simulación
La velocidad básica del viento se elige 22 m/s según el mapa de viento de Suecia y la ubicación del puente de Kalix según EN 1991-1-4 [5] y el código sueco BFS 2019: 1 EKS 11; ver figura 1. La superficie libre sobre el agua se considera un área expuesta a la carga de viento. La dirección del ataque del viento dominante se considera perpendicular al tablero del puente.
Las simulaciones actuales se basan en tres escenarios que incluyen: viento extremo, frío extremo y valor de diseño para un período de retorno de 3000 años. Cada condición tiene diferentes valores de temperatura, viento básico velocidad, viscosidad cinemática y densidad del aire, como se muestra en la Tabla 1. Los conjuntos de datos meteorológicos se sintetizaron para dos semanas meteorológicas extremas durante el período de 30 años de 2040-2069, considerando 13 escenarios climáticos futuros diferentes con diferentes modelos climáticos globales (GCM) y rutas de concentración representativas (RCP).
Se seleccionaron una semana de frío extremo y una semana de viento extremo utilizando el enfoque desarrollado de Nik [7]. El planteamiento se adaptó a las necesidades de este trabajo, considerando el horario semanal en lugar de mensual. Se ha verificado la aplicación del enfoque para simulaciones complejas, incluidos los sistemas de energía Consulte [7]Consulte [8], hidrotermal Consulte [ 9] y simulaciones de microclimas Consulte [10].
Para considerar las condiciones climáticas extremas de una infraestructura muy importante, el valor de la velocidad básica del viento debe transferirse del período de retorno de 50 años a 3000 años como se indica en la ecuación 1 [6]. El perfil de velocidad y turbulencia se crea en base a EN 1991-1-4 [5] para la categoría de terreno 0 (Z0 = 0,003 my Zmín = 1 m), donde Z0 y Zmín son la longitud de rugosidad y la altura mínima, respectivamente. La variación de la velocidad del viento con la altura se define en la ecuación 2, donde co (z) es el factor de orografía tomado como 1, vm (z) es la velocidad media del viento a la altura z, kr es el factor del terreno que depende de la longitud de la rugosidad , e Iv (z) es la intensidad de la turbulencia; ver ecuación 3.���50=[0.36+0.1ln12�] 1�����=��·ln��0·��� [2]���=�����=�1�0�·ln�/�0 ��� ����≤�≤���� [3]���=������ ��� �<���� [4]
Se calcula que el valor de la velocidad del viento para T = período de retorno de 3000 años es de 31 m/s; por lo tanto, los diagramas de velocidad del viento e intensidad de turbulencia se obtienen como se muestra en la figura 2.
Para que las investigaciones sean precisas en el flujo alrededor de estructuras importantes como puentes, se aplica un enfoque híbrido que incluye simulaciones de remolinos desprendidos retardados (DDES) y es computacionalmente eficiente [11][12]. Este modelo de turbulencia usa un método RANS cerca de las capas límite y el método LES lejos de las capas límite y en el área del flujo de la región separada ‘.
En el primer paso, el enfoque de simulación de remolinos separados se ha ampliado para adquirir predicciones de fuerza fiables en los modelos con un gran impacto del flujo separado. Hay varios ejemplos en la parte de revisión de Spalart Consulte [11] para varios casos que usan la aplicación del modelo de turbulencia de simulación de remolino separado (DES).
La formulación DES inicial [13] se desarrolla utilizando el enfoque de Spalart-Allmaras. Con respecto a la transición del enfoque RANS al LES, se revisa el término de destrucción en la ecuación de transporte de viscosidad modificada: la distancia entre un punto en el dominio y la superficie sólida más cercana (d) se sustituye por el factor introducido por:�~=���(�.����·∆)
Se ha empleado un enfoque modificado de DES, conocido como simulación de remolinos desprendidos retardados (DDES), para dominar el probable problema de la “separación inducida por la rejilla” (GIS) que está relacionado con la geometría de la rejilla. El objetivo de este nuevo enfoque es confirmar que el modelado de turbulencia se mantiene en modo RANS en todas las capas de contorno [14]. Por lo tanto, la definición del parámetro se modifica como se define:�~=�-�����(0. �-����·�) 6
donde fd es una función de filtro que considera un valor de 0 en las capas límite cercanas al muro (zona RANS) y un valor de 1 en las áreas donde se realizó la separación del flujo (zona LES).
3.3. Rejilla computacional y resultados
RWIND 2.01 Pro se emplea para la simulación de viento CFD, que usa el código CFD externo OpenFOAM® versión 17.10. La simulación CFD tridimensional se realiza como una simulación de viento transitorio para flujo turbulento incompresible utilizando el algoritmo SIMPLE (Método semi-implícito para ecuaciones vinculadas a presión).
En la simulación actual, el solucionador de estado estacionario se considera como la condición inicial, lo que significa que cuando se está calculando el flujo transitorio, el cálculo del estado estacionario de la condición inicial comienza en la primera parte de la simulación y tan pronto como se calcula. completado, el cálculo de transitorios se iniciará automáticamente.
La cuadrícula computacional se realiza mediante 8.057.279 celdas tridimensionales y 8.820.901 nudos, también se consideran las dimensiones del dominio del túnel de viento 2000 m * 1000 m * 100 m (largo, ancho, alto) como se muestra en la figura 3. El volumen mínimo de la celda es de 6,34 * 10-5 m3, el volumen máximo es de 812,30 m3 y la desviación máxima es de 1,80.
La presión residual final se considera 5 * 10-5. El proceso de generación de mallas e independencia de la rejilla se ha realizado utilizando los cuatro tamaños de malla que se muestran en la figura 4 para la malla de referencia, y finalmente se ha conseguido la independencia de la rejilla.
Se han realizado tres simulaciones para obtener el valor de la presión del viento para condiciones climáticas extremas y el valor de cálculo del viento que se muestra en la Fig. 5. Para cada escenario, el resultado de la presión del viento se obtiene utilizando el modelo de turbulencia transitoria DDES con respecto a 30 (s) de duración que incluye 60 capas de tiempo (Δt = 0,5 s).
Se puede observar que el área frontal del puente está expuesta a la presión del viento positiva y la cantidad de presión aumenta en la altura cerca del borde del tablero para todos los escenarios. Además, la Fig. 5. ilustra los valores negativos de la presión del viento en su totalidad en la superficie de la cubierta. El valor de pertenencia para el período de 3000 años es mucho más alto que los otros escenarios.
Es importante tener en cuenta que el intervalo de la velocidad del viento de entrada tiene un gran impacto en el valor de la presión en la superficie más que en los otros parámetros. Además, para cada escenario, el intervalo más alto de presión del viento y succión durante el tiempo total debe considerarse como una carga de viento crítica impuesta a la estructura. El valor más bajo de la presión en la superficie se obtiene en el escenario de condiciones de frío extremo, mientras que en condiciones de mucho viento, el valor de la presión se vuelve un orden de magnitud más alto.
Además, es importante tener en cuenta que el comportamiento del puente sería completamente diferente debido a las diferentes temperaturas del aire, y puede ocurrir un posible caso crítico en el escenario que experimente una presión menor. Con respecto al valor de entrada de cada escenario, el rango más alto de presión del viento pertenece al nivel de diseño debido al período de retorno de 3000 años, que ha recibido la velocidad del viento más alta como velocidad de entrada.
4. Simulación hidráulica
Los pilares de los puentes a través del río pueden bloquear el flujo al reducir la sección transversal del río, crear corrientes parásitas locales y cambiar la velocidad del flujo, lo que puede ejercer presión en las superficies de los pilares. Cuando el río fluye hacia los pilares del puente, el proceso del flujo de agua alrededor de la base se puede dividir en dos partes: aplicando presión en el momento en que el agua golpea el pilar del puente y después de la presión inicial cuando el agua fluye alrededor de los pilares [15].
Cuando el agua alcanza los pilares del puente a una cierta velocidad, el efecto de la presión sobre los pilares es mucho mayor que la presión del fluido que queda a su alrededor. Debido a los desarrollos de la ciencia de la computación, así como al desarrollo cada vez mayor de los códigos dinámicos de fluidos computacionales, se han utilizado ampliamente varias simulaciones numéricas y se ha demostrado que los resultados de muchas simulaciones son consistentes con los resultados experimentales [16].
Por ello, en esta investigación se ha utilizado el método de la dinámica de fluidos computacional para simular los fenómenos que gobiernan el comportamiento del flujo de los ríos. Para este estudio se ha seleccionado una solución tridimensional basada en cálculos numéricos utilizando el modelo de turbulencia LES. La simulación tridimensional del flujo del río en diferentes direcciones y velocidades nos permite calcular y analizar todas las presiones en la superficie de los pilares del puente en diferentes intervalos de tiempo.
4.1. Parámetros de simulación
El flujo del río se puede definir como un flujo de dos fases, que incluye agua y aire, en un canal abierto. El flujo de canal abierto es un flujo de fluido con una superficie libre en la que la presión atmosférica se distribuye uniformemente y se crea por el peso del fluido. Para simular este tipo de flujo se utiliza el método multifase VOF.
El programa Flow3D, disponible en el mercado, utiliza los métodos de fracciones volumétricas VOF y FAVOF. En el método VOF, el dominio de modelado se divide primero en celdas de elementos o volúmenes de controles más pequeños. Para los elementos que contienen fluidos, se mantienen valores numéricos para cada una de las variables de flujo dentro de ellos.
Estos valores representan la media volumétrica de los valores en cada elemento. En las corrientes superficiales libres, no todas las celdas están llenas de líquido; algunas celdas en la superficie de flujo están medio llenas. En este caso, se define una cantidad llamada volumen de fluido, F, que representa la parte de la celda que se llena con el fluido.
Después de determinar la posición y el ángulo de la superficie del flujo, será posible aplicar las condiciones de contorno apropiadas en la superficie del flujo para calcular el movimiento del fluido. A medida que se mueve el fluido, el valor de F también cambia con él. Las superficies libres son monitoreadas automáticamente por el movimiento de fluido dentro de una red fija. El método FAVOR se usa para determinar la geometría.
También se puede usar otra cantidad de fracción volumétrica para determinar el nivel de un cuerpo rígido desocupado ( Vf ). Cuando se conoce el volumen que ocupa el cuerpo rígido en cada celda, el límite del fluido dentro de la red fija se puede determinar como VOF. Este límite se usa para determinar las condiciones de contorno del muro que sigue el arroyo. En general, la ecuación de continuidad de masa es la siguiente:��𝜕�𝜕�+𝜕𝜕�(����)+�𝜕𝜕�(����)+𝜕𝜕�(����)+������=���� 10
Las ecuaciones de movimiento para los componentes de la velocidad de un fluido en coordenadas 3D, o en otras palabras, las ecuaciones de Navier-Stokes, son las siguientes:𝜕�𝜕�+1�����𝜕�𝜕�+���𝜕�𝜕�+���𝜕�𝜕�+��2�����=-1�𝜕�𝜕�+��+��-��-��������-��-��� 11𝜕�𝜕�+1�����𝜕�𝜕�+���𝜕�𝜕�+���𝜕�𝜕�+��������=-�1�𝜕�𝜕�+��+��-��-��������-��-��� 12𝜕�𝜕�+1�����𝜕�𝜕�+���𝜕�𝜕�+���𝜕�𝜕�=-1�𝜕�𝜕�+��+��-��-��������-��-��� 13
Donde VF es la relación del volumen abierto al flujo, ρ es la densidad del fluido, (u, v, w) son las componentes de la velocidad en las direcciones x, y y z, respectivamente, R SOR es la función de la fuente, (Ax, Ay, Az ) son las áreas fraccionales, (Gx, Gy, Gz ) son las fuerzas gravitacionales, (fx, fy, fz ) son las aceleraciones de la viscosidad y (bx, by, bz ) son las pérdidas de flujo en medios porosos en las direcciones x, y, z, respectivamente [17].
La zona de captación del río Kalix es grande y amplia, por lo que tiene un clima subpolar con inviernos fríos y largos y veranos suaves y cortos. Aproximadamente el 50% de las precipitaciones en esta zona es nieve. En mayo, por lo general, el deshielo provoca un aumento significativo en el caudal del río. Las condiciones climáticas del río se resumen en la Tabla 2, [18].
Contrariamente a la tendencia general de este estudio, la previsión de las condiciones meteorológicas mencionadas está utilizando la información meteorológica registrada en los períodos pasados. En función de la información meteorológica disponible, definimos las condiciones de contorno al realizar los cálculos.
Primero, según las dimensiones de los pilares en tres direcciones X, Y, Z, y según la dimensión longitudinal de los pilares (D = 8,5 m; véase la figura 7), el dominio se extiende 10D aguas arriba y 20D aguas abajo. Se ha utilizado el método de mallado estructurado (cartesiano) y el software Flow3D para resolver este problema. Para una cuadrícula correcta, el dominio se debe dividir en diferentes secciones.
Esta división se basa en lugares con fuertes pendientes. Usando la creación de una nueva superficie, el dominio se puede dividir en varias secciones para crear una malla regular con las dimensiones correctas y apropiadas, se puede especificar el número de celdas en cada superficie.
Esto aumenta el volumen final de las células. Por esta razón, hemos dividido este dominio en tres niveles: Grueso, medio y fino. Los resultados de los estudios de independencia de la red se muestran en la figura 6. Para comprobar los resultados calculados, primero debemos asegurarnos de que la corriente de entrada sea la correcta. Para hacer esto, el caudal de entrada se mide en el dominio de la solución y se compara con el valor base. Las dimensiones del dominio de la solución se especifican en la figura 7. Esta figura también contribuye al reconocimiento de los pilares del puente y su denominación de superficies.
Como se muestra en la Fig. 8, el caudal del río se encuentra dentro del intervalo admisible durante el 90% del tiempo de simulación y el caudal de entrada se ha simulado correctamente. Además, en la Fig. 9, la velocidad media del río se calcula en función del caudal y del área de la sección transversal del río.
Para extraer la cantidad de presión aplicada a los diferentes lados de las columnas, hemos seleccionado el intervalo de tiempo de simulación de 10 a 25 segundos (tiempo de estabilización de descarga en la cantidad de 1800 metros cúbicos por segundo). Los resultados calculados para cada lado se muestran en la Fig. 10 y 11. Los contornos de velocidad también se muestran en las Figuras 12 y 13. Estos contornos se ajustan en función de la velocidad del fluido en un momento dado.
Debido a las dimensiones del dominio de la solución y al caudal del río, el flujo de agua llega a los pilares del puente en el décimo segundo y la presión inicial del flujo del río afecta las superficies de los pilares del puente. Esta presión inicial decrece con el tiempo y se estabiliza en un rango determinado para cada lado según el área y el porcentaje de interacción con el flujo. Para los cálculos de interacción fluido-estructura (FSI), se puede usar la presión crítica calculada en el momento en que la corriente golpea los pilares.
Los efectos de las condiciones meteorológicas extremas, incluido el viento dinámico y el flujo de agua, se investigaron numéricamente para el puente de Kalix. Se definieron tres escenarios para las simulaciones dinámicas de viento, incluido el clima con mucho viento, el clima extremadamente frío y el valor de diseño para un período de retorno de 3.000 años. Aprovechando las simulaciones CFD, se determinaron las presiones del viento en pasos de 60 tiempos (30 segundos) utilizando el modelo de turbulencia transitoria DDES.
Los resultados indican diferencias significativas entre los escenarios, lo que implica la importancia de los datos de entrada, especialmente el diagrama de velocidades del viento. Se observó que el valor de diseño para el período de devolución de 3000 años tiene un impacto mucho mayor que los otros escenarios. Además, se mostró la importancia de considerar el rango más alto de presión del viento en la superficie a través de los pasos de tiempo para evaluar el comportamiento estructural del puente en la condición más crítica.
Además, se consideró el caudal máximo del río para una simulación transitoria según las condiciones meteorológicas registradas, y los pilares del puente se sometieron al caudal máximo del río durante 30 segundos. Por lo tanto, además de las condiciones físicas del flujo del río y cómo cambia la dirección del flujo aguas abajo, se cuantificaron las presiones máximas del agua en el momento en que el flujo golpea los pilares.
En el trabajo futuro, el rendimiento estructural del puente de Kalix será evaluado por imposición de la carga del viento, la presión del agua y la carga del tráfico, creando así un gemelo digital estructural que refleja la verdadera respuesta de la estructura.
6. Reconocimiento
Los autores agradecen enormemente el apoyo de Dlubal Software por proporcionar la licencia de RWIND Simulation, así como de Flow Sciences Inc. por proporcionar la licencia de FLOW-3D.
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번역된 기고 제목: 해류의 영향에 따른 어뢰 앵커 설치의 유체 역학 특성에 대한 수치 분석
Translated title of the contribution: NUMERICAL ANALYSIS OF THE HYDRODYNAMICS CHARACTERISTICS OF TORPEDO ANCHOR INSTALLATION UNDER THE INFLUENCE OF OCEAN CURRENTS
The gravity-installed anchor (GIA) is a type of the anchor foundation that is installed by penetrating the seabed using the weight of the anchor body. It has the advantages of high installation efficiency, low cost, and no requirement of additional installation facilities. The GIA type used in this study is the torpedo anchor, which has been ap-plied in practical cases widely. The purpose of this study is to investigate the numerical analysis of the anchor trans-porting during the installation of the torpedo anchor under the action of ocean currents. Therefore, this article con-siders external environmental conditions and the different forms of torpedo anchors by using computational fluid dynamics (CFD) software, FLOW-3D, to simulate the fluid-solid interaction effect on the torpedo anchor. The falling time, impact velocity, displaced angle, and horizontal displacement of the torpedo anchor were observed at an installation height (i.e., the distance between the seabed and the anchor release height) of 85 meters. The obtained results show that when the current velocity is greater, the torpedo anchor will have a larger displaced angle, which will affect the impact velocity of the anchor on the seabed and may cause insufficient penetration depth, leading to installation failure.
중력설치형 앵커(GIA)는 앵커 본체의 무게를 이용하여 해저를 관통하여 설치하는 앵커 기초의 일종이다. 설치 효율성이 높고, 비용이 저렴하며, 추가 설치 시설이 필요하지 않다는 장점이 있습니다. 본 연구에서 사용된 GIA 유형은 어뢰앵커로 실제 사례에 널리 적용되어 왔다.
본 연구의 목적은 해류의 작용에 따라 어뢰앵커 설치 시 앵커 이송에 대한 수치해석을 연구하는 것이다. 따라서 이 기사에서는 어뢰 앵커에 대한 유체-고체 상호 작용 효과를 시뮬레이션하기 위해 전산유체역학(CFD) 소프트웨어인 FLOW-3D를 사용하여 외부 환경 조건과 다양한 형태의 어뢰 앵커를 고려합니다.
어뢰앵커의 낙하시간, 충격속도, 변위각, 수평변위 등은 설치높이(즉, 해저와 앵커 해제 높이 사이의 거리) 85m에서 관찰되었다. 얻은 결과는 현재 속도가 더 높을 때 어뢰 앵커의 변위 각도가 더 커져 해저에 대한 앵커의 충격 속도에 영향을 미치고 침투 깊이가 부족하여 설치 실패로 이어질 수 있음을 보여줍니다.
Alireza Khoshkonesh1, Blaise Nsom2, Saeid Okhravi3*, Fariba Ahmadi Dehrashid4, Payam Heidarian5, Silvia DiFrancesco6 1 Department of Geography, School of Social Sciences, History, and Philosophy, Birkbeck University of London, London, UK. 2 Université de Bretagne Occidentale. IRDL/UBO UMR CNRS 6027. Rue de Kergoat, 29285 Brest, France. 3 Institute of Hydrology, Slovak Academy of Sciences, Dúbravská cesta 9, 84104, Bratislava, Slovak Republic. 4Department of Water Science and Engineering, Faculty of Agriculture, Bu-Ali Sina University, 65178-38695, Hamedan, Iran. 5 Department of Civil, Environmental, Architectural Engineering and Mathematics, University of Brescia, 25123 Brescia, Italy. 6Niccol`o Cusano University, via Don C. Gnocchi 3, 00166 Rome, Italy. * Corresponding author. Tel.: +421-944624921. E-mail: saeid.okhravi@savba.sk
Abstract
This study aimed to comprehensively investigate the influence of substrate level difference and material composition on dam break wave evolution over two different erodible beds. Utilizing the Volume of Fluid (VOF) method, we tracked free surface advection and reproduced wave evolution using experimental data from the literature. For model validation, a comprehensive sensitivity analysis encompassed mesh resolution, turbulence simulation methods, and bed load transport equations. The implementation of Large Eddy Simulation (LES), non-equilibrium sediment flux, and van Rijn’s (1984) bed load formula yielded higher accuracy compared to alternative approaches. The findings emphasize the significant effect of substrate level difference and material composition on dam break morphodynamic characteristics. Decreasing substrate level disparity led to reduced flow velocity, wavefront progression, free surface height, substrate erosion, and other pertinent parameters. Initial air entrapment proved substantial at the wavefront, illustrating pronounced air-water interaction along the bottom interface. The Shields parameter experienced a one-third reduction as substrate level difference quadrupled, with the highest near-bed concentration observed at the wavefront. This research provides fresh insights into the complex interplay of factors governing dam break wave propagation and morphological changes, advancing our comprehension of this intricate phenomenon.
이 연구는 두 개의 서로 다른 침식층에 대한 댐 파괴파 진화에 대한 기질 수준 차이와 재료 구성의 영향을 종합적으로 조사하는 것을 목표로 했습니다. VOF(유체량) 방법을 활용하여 자유 표면 이류를 추적하고 문헌의 실험 데이터를 사용하여 파동 진화를 재현했습니다.
모델 검증을 위해 메쉬 해상도, 난류 시뮬레이션 방법 및 침대 하중 전달 방정식을 포함하는 포괄적인 민감도 분석을 수행했습니다. LES(Large Eddy Simulation), 비평형 퇴적물 플럭스 및 van Rijn(1984)의 하상 부하 공식의 구현은 대체 접근 방식에 비해 더 높은 정확도를 산출했습니다.
연구 결과는 댐 붕괴 형태역학적 특성에 대한 기질 수준 차이와 재료 구성의 중요한 영향을 강조합니다. 기판 수준 차이가 감소하면 유속, 파면 진행, 자유 표면 높이, 기판 침식 및 기타 관련 매개변수가 감소했습니다.
초기 공기 포집은 파면에서 상당한 것으로 입증되었으며, 이는 바닥 경계면을 따라 뚜렷한 공기-물 상호 작용을 보여줍니다. 기판 레벨 차이가 4배로 증가함에 따라 Shields 매개변수는 1/3로 감소했으며, 파면에서 가장 높은 베드 근처 농도가 관찰되었습니다.
이 연구는 댐 파괴파 전파와 형태학적 변화를 지배하는 요인들의 복잡한 상호 작용에 대한 새로운 통찰력을 제공하여 이 복잡한 현상에 대한 이해를 향상시킵니다.
Fig. 3. Free surface and substrate profiles in all Sp and Ls cases at t = 1 s, t = 3 s, and t = 5 s, arranged left to right (note: the colour contours
correspond to the horizontal component of the flow velocity (u), expressed in m/s).
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Ali Poorkarimi1 Khaled Mafakheri2 Shahrzad Maleki2
Journal of Hydraulic Structures J. Hydraul. Struct., 2023; 9(4): 76-87 DOI: 10.22055/jhs.2024.44817.1265
Abstract
중력에 의한 침전은 부유 물질을 제거하기 위해 물과 폐수 처리 공정에 널리 적용됩니다. 이 연구에서는 침전조의 제거 효율에 대한 입구 및 배플 위치의 영향을 간략하게 설명합니다. 실험은 CCD(중심복합설계) 방법론을 기반으로 수행되었습니다. 전산유체역학(CFD)은 유압 설계, 미래 발전소에 대한 계획 연구, 토목 유지 관리 및 공급 효율성과 관련된 복잡한 문제를 모델링하고 분석하는 데 광범위하게 사용됩니다. 본 연구에서는 입구 높이, 입구로부터 배플까지의 거리, 배플 높이의 다양한 조건에 따른 영향을 조사하였다. CCD 접근 방식을 사용하여 얻은 데이터를 분석하면 축소된 2차 모델이 R2 = 0.77의 결정 계수로 부유 물질 제거를 예측할 수 있음이 나타났습니다. 연구 결과, 유입구와 배플의 부적절한 위치는 침전조의 효율에 부정적인 영향을 미칠 수 있음을 보여주었습니다. 입구 높이, 배플 거리, 배플 높이의 최적 값은 각각 0.87m, 0.77m, 0.56m였으며 제거 효율은 80.6%였습니다.
Sedimentation due to gravitation is applied widely in water and wastewater treatment processes to remove suspended solids. This study outlines the effect of the inlet and baffle position on the removal efficiency of sedimentation tanks. Experiments were carried out based on the central composite design (CCD) methodology. Computational fluid dynamics (CFD) is used extensively to model and analyze complex issues related to hydraulic design, planning studies for future generating stations, civil maintenance, and supply efficiency. In this study, the effect of different conditions of inlet elevation, baffle’s distance from the inlet, and baffle height were investigated. Analysis of the obtained data with a CCD approach illustrated that the reduced quadratic model can predict the suspended solids removal with a coefficient of determination of R2 = 0.77. The results showed that the inappropriate position of the inlet and the baffle can have a negative effect on the efficiency of the sedimentation tank. The optimal values of inlet elevation, baffle distance, and baffle height were 0.87 m, 0.77 m, and 0.56 m respectively with 80.6% removal efficiency.
Figure 3. Computed contour of velocity magnitude (m/s) for Run 1 to Run 15.
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Review on Blood Flow Dynamics in Lab-on-a-Chip Systems: An Engineering Perspective
Bin-Jie Lai
,
Li-Tao Zhu
,
Zhe Chen*
,
Bo Ouyang*
, and
Zheng-Hong Luo*
Abstract
다양한 수송 메커니즘 하에서, “LOC(lab-on-a-chip)” 시스템에서 유동 전단 속도 조건과 밀접한 관련이 있는 혈류 역학은 다양한 수송 현상을 초래하는 것으로 밝혀졌습니다.
본 연구는 적혈구의 동적 혈액 점도 및 탄성 거동과 같은 점탄성 특성의 역할을 통해 LOC 시스템의 혈류 패턴을 조사합니다. 모세관 및 전기삼투압의 주요 매개변수를 통해 LOC 시스템의 혈액 수송 현상에 대한 연구는 실험적, 이론적 및 수많은 수치적 접근 방식을 통해 제공됩니다.
전기 삼투압 점탄성 흐름에 의해 유발되는 교란은 특히 향후 연구 기회를 위해 혈액 및 기타 점탄성 유체를 취급하는 LOC 장치의 혼합 및 분리 기능 향상에 논의되고 적용됩니다. 또한, 본 연구는 보다 정확하고 단순화된 혈류 모델에 대한 요구와 전기역학 효과 하에서 점탄성 유체 흐름에 대한 수치 연구에 대한 강조와 같은 LOC 시스템 하에서 혈류 역학의 수치 모델링의 문제를 식별합니다.
전기역학 현상을 연구하는 동안 제타 전위 조건에 대한 보다 실용적인 가정도 강조됩니다. 본 연구는 모세관 및 전기삼투압에 의해 구동되는 미세유체 시스템의 혈류 역학에 대한 포괄적이고 학제적인 관점을 제공하는 것을 목표로 한다.
1.1. Microfluidic Flow in Lab-on-a-Chip (LOC) Systems
Over the past several decades, the ability to control and utilize fluid flow patterns at microscales has gained considerable interest across a myriad of scientific and engineering disciplines, leading to growing interest in scientific research of microfluidics.
(1) Microfluidics, an interdisciplinary field that straddles physics, engineering, and biotechnology, is dedicated to the behavior, precise control, and manipulation of fluids geometrically constrained to a small, typically submillimeter, scale.
(2) The engineering community has increasingly focused on microfluidics, exploring different driving forces to enhance working fluid transport, with the aim of accurately and efficiently describing, controlling, designing, and applying microfluidic flow principles and transport phenomena, particularly for miniaturized applications.
(3) This attention has chiefly been fueled by the potential to revolutionize diagnostic and therapeutic techniques in the biomedical and pharmaceutical sectorsUnder various driving forces in microfluidic flows, intriguing transport phenomena have bolstered confidence in sustainable and efficient applications in fields such as pharmaceutical, biochemical, and environmental science. The “lab-on-a-chip” (LOC) system harnesses microfluidic flow to enable fluid processing and the execution of laboratory tasks on a chip-sized scale. LOC systems have played a vital role in the miniaturization of laboratory operations such as mixing, chemical reaction, separation, flow control, and detection on small devices, where a wide variety of fluids is adapted. Biological fluid flow like blood and other viscoelastic fluids are notably studied among the many working fluids commonly utilized by LOC systems, owing to the optimization in small fluid sample volumed, rapid response times, precise control, and easy manipulation of flow patterns offered by the system under various driving forces.
(4)The driving forces in blood flow can be categorized as passive or active transport mechanisms and, in some cases, both. Under various transport mechanisms, the unique design of microchannels enables different functionalities in driving, mixing, separating, and diagnosing blood and drug delivery in the blood.
(5) Understanding and manipulating these driving forces are crucial for optimizing the performance of a LOC system. Such knowledge presents the opportunity to achieve higher efficiency and reliability in addressing cellular level challenges in medical diagnostics, forensic studies, cancer detection, and other fundamental research areas, for applications of point-of-care (POC) devices.
1.2. Engineering Approach of Microfluidic Transport Phenomena in LOC Systems
Different transport mechanisms exhibit unique properties at submillimeter length scales in microfluidic devices, leading to significant transport phenomena that differ from those of macroscale flows. An in-depth understanding of these unique transport phenomena under microfluidic systems is often required in fluidic mechanics to fully harness the potential functionality of a LOC system to obtain systematically designed and precisely controlled transport of microfluids under their respective driving force. Fluid mechanics is considered a vital component in chemical engineering, enabling the analysis of fluid behaviors in various unit designs, ranging from large-scale reactors to separation units. Transport phenomena in fluid mechanics provide a conceptual framework for analytically and descriptively explaining why and how experimental results and physiological phenomena occur. The Navier–Stokes (N–S) equation, along with other governing equations, is often adapted to accurately describe fluid dynamics by accounting for pressure, surface properties, velocity, and temperature variations over space and time. In addition, limiting factors and nonidealities for these governing equations should be considered to impose corrections for empirical consistency before physical models are assembled for more accurate controls and efficiency. Microfluidic flow systems often deviate from ideal conditions, requiring adjustments to the standard governing equations. These deviations could arise from factors such as viscous effects, surface interactions, and non-Newtonian fluid properties from different microfluid types and geometrical layouts of microchannels. Addressing these nonidealities supports the refining of theoretical models and prediction accuracy for microfluidic flow behaviors.
The analytical calculation of coupled nonlinear governing equations, which describes the material and energy balances of systems under ideal conditions, often requires considerable computational efforts. However, advancements in computation capabilities, cost reduction, and improved accuracy have made numerical simulations using different numerical and modeling methods a powerful tool for effectively solving these complex coupled equations and modeling various transport phenomena. Computational fluid dynamics (CFD) is a numerical technique used to investigate the spatial and temporal distribution of various flow parameters. It serves as a critical approach to provide insights and reasoning for decision-making regarding the optimal designs involving fluid dynamics, even prior to complex physical model prototyping and experimental procedures. The integration of experimental data, theoretical analysis, and reliable numerical simulations from CFD enables systematic variation of analytical parameters through quantitative analysis, where adjustment to delivery of blood flow and other working fluids in LOC systems can be achieved.
Numerical methods such as the Finite-Difference Method (FDM), Finite-Element-Method (FEM), and Finite-Volume Method (FVM) are heavily employed in CFD and offer diverse approaches to achieve discretization of Eulerian flow equations through filling a mesh of the flow domain. A more in-depth review of numerical methods in CFD and its application for blood flow simulation is provided in Section 2.2.2.
1.3. Scope of the Review
In this Review, we explore and characterize the blood flow phenomena within the LOC systems, utilizing both physiological and engineering modeling approaches. Similar approaches will be taken to discuss capillary-driven flow and electric-osmotic flow (EOF) under electrokinetic phenomena as a passive and active transport scheme, respectively, for blood transport in LOC systems. Such an analysis aims to bridge the gap between physical (experimental) and engineering (analytical) perspectives in studying and manipulating blood flow delivery by different driving forces in LOC systems. Moreover, the Review hopes to benefit the interests of not only blood flow control in LOC devices but also the transport of viscoelastic fluids, which are less studied in the literature compared to that of Newtonian fluids, in LOC systems.
Section 2 examines the complex interplay between viscoelastic properties of blood and blood flow patterns under shear flow in LOC systems, while engineering numerical modeling approaches for blood flow are presented for assistance. Sections 3 and 4 look into the theoretical principles, numerical governing equations, and modeling methodologies for capillary driven flow and EOF in LOC systems as well as their impact on blood flow dynamics through the quantification of key parameters of the two driving forces. Section 5 concludes the characterized blood flow transport processes in LOC systems under these two forces. Additionally, prospective areas of research in improving the functionality of LOC devices employing blood and other viscoelastic fluids and potentially justifying mechanisms underlying microfluidic flow patterns outside of LOC systems are presented. Finally, the challenges encountered in the numerical studies of blood flow under LOC systems are acknowledged, paving the way for further research.
Blood, an essential physiological fluid in the human body, serves the vital role of transporting oxygen and nutrients throughout the body. Additionally, blood is responsible for suspending various blood cells including erythrocytes (red blood cells or RBCs), leukocytes (white blood cells), and thrombocytes (blood platelets) in a plasma medium.Among the cells mentioned above, red blood cells (RBCs) comprise approximately 40–45% of the volume of healthy blood.
(7) An RBC possesses an inherent elastic property with a biconcave shape of an average diameter of 8 μm and a thickness of 2 μm. This biconcave shape maximizes the surface-to-volume ratio, allowing RBCs to endure significant distortion while maintaining their functionality.
(8,9) Additionally, the biconcave shape optimizes gas exchange, facilitating efficient uptake of oxygen due to the increased surface area. The inherent elasticity of RBCs allows them to undergo substantial distortion from their original biconcave shape and exhibits high flexibility, particularly in narrow channels.RBC deformability enables the cell to deform from a biconcave shape to a parachute-like configuration, despite minor differences in RBC shape dynamics under shear flow between initial cell locations. As shown in Figure 1(a), RBCs initiating with different resting shapes and orientations displaying display a similar deformation pattern
(10) in terms of its shape. Shear flow induces an inward bending of the cell at the rear position of the rim to the final bending position,
(11) resulting in an alignment toward the same position of the flow direction.
Figure 1. Images of varying deformation of RBCs and different dynamic blood flow behaviors. (a) The deforming shape behavior of RBCs at four different initiating positions under the same experimental conditions of a flow from left to right, (10) (b) RBC aggregation, (13) (c) CFL region. (18) Reproduced with permission from ref (10). Copyright 2011 Elsevier. Reproduced with permission from ref (13). Copyright 2022 The Authors, under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/. Reproduced with permission from ref (18). Copyright 2019 Elsevier.
The flexible property of RBCs enables them to navigate through narrow capillaries and traverse a complex network of blood vessels. The deformability of RBCs depends on various factors, including the channel geometry, RBC concentration, and the elastic properties of the RBC membrane.
(12) Both flexibility and deformability are vital in the process of oxygen exchange among blood and tissues throughout the body, allowing cells to flow in vessels even smaller than the original cell size prior to deforming.As RBCs serve as major components in blood, their collective dynamics also hugely affect blood rheology. RBCs exhibit an aggregation phenomenon due to cell to cell interactions, such as adhesion forces, among populated cells, inducing unique blood flow patterns and rheological behaviors in microfluidic systems. For blood flow in large vessels between a diameter of 1 and 3 cm, where shear rates are not high, a constant viscosity and Newtonian behavior for blood can be assumed. However, under low shear rate conditions (0.1 s
–1) in smaller vessels such as the arteries and venules, which are within a diameter of 0.2 mm to 1 cm, blood exhibits non-Newtonian properties, such as shear-thinning viscosity and viscoelasticity due to RBC aggregation and deformability. The nonlinear viscoelastic property of blood gives rise to a complex relationship between viscosity and shear rate, primarily influenced by the highly elastic behavior of RBCs. A wide range of research on the transient behavior of the RBC shape and aggregation characteristics under varied flow circumstances has been conducted, aiming to obtain a better understanding of the interaction between blood flow shear forces from confined flows.
For a better understanding of the unique blood flow structures and rheological behaviors in microfluidic systems, some blood flow patterns are introduced in the following section.
2.1.1. RBC Aggregation
RBC aggregation is a vital phenomenon to be considered when designing LOC devices due to its impact on the viscosity of the bulk flow. Under conditions of low shear rate, such as in stagnant or low flow rate regions, RBCs tend to aggregate, forming structures known as rouleaux, resembling stacks of coins as shown in Figure 1(b).
(13) The aggregation of RBCs increases the viscosity at the aggregated region,
(14) hence slowing down the overall blood flow. However, when exposed to high shear rates, RBC aggregates disaggregate. As shear rates continue to increase, RBCs tend to deform, elongating and aligning themselves with the direction of the flow.
(15) Such a dynamic shift in behavior from the cells in response to the shear rate forms the basis of the viscoelastic properties observed in whole blood. In essence, the viscosity of the blood varies according to the shear rate conditions, which are related to the velocity gradient of the system. It is significant to take the intricate relationship between shear rate conditions and the change of blood viscosity due to RBC aggregation into account since various flow driving conditions may induce varied effects on the degree of aggregation.
2.1.2. Fåhræus-Lindqvist Effect
The Fåhræus–Lindqvist (FL) effect describes the gradual decrease in the apparent viscosity of blood as the channel diameter decreases.
(16) This effect is attributed to the migration of RBCs toward the central region in the microchannel, where the flow rate is higher, due to the presence of higher pressure and asymmetric distribution of shear forces. This migration of RBCs, typically observed at blood vessels less than 0.3 mm, toward the higher flow rate region contributes to the change in blood viscosity, which becomes dependent on the channel size. Simultaneously, the increase of the RBC concentration in the central region of the microchannel results in the formation of a less viscous region close to the microchannel wall. This region called the Cell-Free Layer (CFL), is primarily composed of plasma.
(17) The combination of the FL effect and the following CFL formation provides a unique phenomenon that is often utilized in passive and active plasma separation mechanisms, involving branched and constriction channels for various applications in plasma separation using microfluidic systems.
2.1.3. Cell-Free Layer Formation
In microfluidic blood flow, RBCs form aggregates at the microchannel core and result in a region that is mostly devoid of RBCs near the microchannel walls, as shown in Figure 1(c).
(18) The region is known as the cell-free layer (CFL). The CFL region is often known to possess a lower viscosity compared to other regions within the blood flow due to the lower viscosity value of plasma when compared to that of the aggregated RBCs. Therefore, a thicker CFL region composed of plasma correlates to a reduced apparent whole blood viscosity.
(19) A thicker CFL region is often established following the RBC aggregation at the microchannel core under conditions of decreasing the tube diameter. Apart from the dependence on the RBC concentration in the microchannel core, the CFL thickness is also affected by the volume concentration of RBCs, or hematocrit, in whole blood, as well as the deformability of RBCs. Given the influence CFL thickness has on blood flow rheological parameters such as blood flow rate, which is strongly dependent on whole blood viscosity, investigating CFL thickness under shear flow is crucial for LOC systems accounting for blood flow.
2.1.4. Plasma Skimming in Bifurcation Networks
The uneven arrangement of RBCs in bifurcating microchannels, commonly termed skimming bifurcation, arises from the axial migration of RBCs within flowing streams. This uneven distribution contributes to variations in viscosity across differing sizes of bifurcating channels but offers a stabilizing effect. Notably, higher flow rates in microchannels are associated with increased hematocrit levels, resulting in higher viscosity compared with those with lower flow rates. Parametric investigations on bifurcation angle,
(21) and RBC dynamics, including aggregation and deformation,
(22) may alter the varying viscosity of blood and its flow behavior within microchannels.
2.2. Modeling on Blood Flow Dynamics
2.2.1. Blood Properties and Mathematical Models of Blood Rheology
Under different shear rate conditions in blood flow, the elastic characteristics and dynamic changes of the RBC induce a complex velocity and stress relationship, resulting in the incompatibility of blood flow characterization through standard presumptions of constant viscosity used for Newtonian fluid flow. Blood flow is categorized as a viscoelastic non-Newtonian fluid flow where constitutive equations governing this type of flow take into consideration the nonlinear viscometric properties of blood. To mathematically characterize the evolving blood viscosity and the relationship between the elasticity of RBC and the shear blood flow, respectively, across space and time of the system, a stress tensor (τ) defined by constitutive models is often coupled in the Navier–Stokes equation to account for the collective impact of the constant dynamic viscosity (η) and the elasticity from RBCs on blood flow.The dynamic viscosity of blood is heavily dependent on the shear stress applied to the cell and various parameters from the blood such as hematocrit value, plasma viscosity, mechanical properties of the RBC membrane, and red blood cell aggregation rate. The apparent blood viscosity is considered convenient for the characterization of the relationship between the evolving blood viscosity and shear rate, which can be defined by Casson’s law, as shown in eq 1.
𝜇=𝜏0𝛾˙+2𝜂𝜏0𝛾˙⎯⎯⎯⎯⎯⎯⎯√+𝜂�=�0�˙+2��0�˙+�
(1)where τ
0 is the yield stress–stress required to initiate blood flow motion, η is the Casson rheological constant, and γ̇ is the shear rate. The value of Casson’s law parameters under blood with normal hematocrit level can be defined as τ
0 = 0.0056 Pa and η = 0.0035 Pa·s.
(23) With the known property of blood and Casson’s law parameters, an approximation can be made to the dynamic viscosity under various flow condition domains. The Power Law model is often employed to characterize the dynamic viscosity in relation to the shear rate, since precise solutions exist for specific geometries and flow circumstances, acting as a fundamental standard for definition. The Carreau and Carreau–Yasuda models can be advantageous over the Power Law model due to their ability to evaluate the dynamic viscosity at low to zero shear rate conditions. However, none of the above-mentioned models consider the memory or other elastic behavior of blood and its RBCs. Some other commonly used mathematical models and their constants for the non-Newtonian viscosity property characterization of blood are listed in Table 1 below.
(24−26)Table 1. Comparison of Various Non-Newtonian Models for Blood Viscosity
The blood rheology is commonly known to be influenced by two key physiological factors, namely, the hematocrit value (H
t) and the fibrinogen concentration (c
f), with an average value of 42% and 0.252 gd·L
–1, respectively. Particularly in low shear conditions, the presence of varying fibrinogen concentrations affects the tendency for aggregation and rouleaux formation, while the occurrence of aggregation is contingent upon specific levels of hematocrit.
(28) modifies the Casson model through emphasizing its reliance on hematocrit and fibrinogen concentration parameter values, owing to the extensive knowledge of the two physiological blood parameters.The viscoelastic response of blood is heavily dependent on the elasticity of the RBC, which is defined by the relationship between the deformation and stress relaxation from RBCs under a specific location of shear flow as a function of the velocity field. The stress tensor is usually characterized by constitutive equations such as the Upper-Convected Maxwell Model
(30) to track the molecule effects under shear from different driving forces. The prominent non-Newtonian features, such as shear thinning and yield stress, have played a vital role in the characterization of blood rheology, particularly with respect to the evaluation of yield stress under low shear conditions. The nature of stress measurement in blood, typically on the order of 1 mPa, is challenging due to its low magnitude. The occurrence of the CFL complicates the measurement further due to the significant decrease in apparent viscosity near the wall over time and a consequential disparity in viscosity compared to the bulk region.In addition to shear thinning viscosity and yield stress, the formation of aggregation (rouleaux) from RBCs under low shear rates also contributes to the viscoelasticity under transient flow
(32) of whole blood. Given the difficulty in evaluating viscoelastic behavior of blood under low strain magnitudes and limitations in generalized Newtonian models, the utilization of viscoelastic models is advocated to encompass elasticity and delineate non-shear components within the stress tensor. Extending from the Oldroyd-B model, Anand et al.
(33) developed a viscoelastic model framework for adapting elasticity within blood samples and predicting non-shear stress components. However, to also address the thixotropic effects, the model developed by Horner et al.
(34) serves as a more comprehensive approach than the viscoelastic model from Anand et al. Thixotropy
(32) typically occurs from the structural change of the rouleaux, where low shear rate conditions induce rouleaux formation. Correspondingly, elasticity increases, while elasticity is more representative of the isolated RBCs, under high shear rate conditions. The model of Horner et al.
(34) considers the contribution of rouleaux to shear stress, taking into account factors such as the characteristic time for Brownian aggregation, shear-induced aggregation, and shear-induced breakage. Subsequent advancements in the model from Horner et al. often revolve around refining the three aforementioned key terms for a more substantial characterization of rouleaux dynamics. Notably, this has led to the recently developed mHAWB model
(35) and other model iterations to enhance the accuracy of elastic and viscoelastic contributions to blood rheology, including the recently improved model suggested by Armstrong et al.
Numerical simulation has become increasingly more significant in analyzing the geometry, boundary layers of flow, and nonlinearity of hyperbolic viscoelastic flow constitutive equations. CFD is a powerful and efficient tool utilizing numerical methods to solve the governing hydrodynamic equations, such as the Navier–Stokes (N–S) equation, continuity equation, and energy conservation equation, for qualitative evaluation of fluid motion dynamics under different parameters. CFD overcomes the challenge of analytically solving nonlinear forms of differential equations by employing numerical methods such as the Finite-Difference Method (FDM), Finite-Element Method (FEM), and Finite-Volume Method (FVM) to discretize and solve the partial differential equations (PDEs), allowing for qualitative reproduction of transport phenomena and experimental observations. Different numerical methods are chosen to cope with various transport systems for optimization of the accuracy of the result and control of error during the discretization process.FDM is a straightforward approach to discretizing PDEs, replacing the continuum representation of equations with a set of finite-difference equations, which is typically applied to structured grids for efficient implementation in CFD programs.
(37) However, FDM is often limited to simple geometries such as rectangular or block-shaped geometries and struggles with curved boundaries. In contrast, FEM divides the fluid domain into small finite grids or elements, approximating PDEs through a local description of physics.
(38) All elements contribute to a large, sparse matrix solver. However, FEM may not always provide accurate results for systems involving significant deformation and aggregation of particles like RBCs due to large distortion of grids.
(39) FVM evaluates PDEs following the conservation laws and discretizes the selected flow domain into small but finite size control volumes, with each grid at the center of a finite volume.
(40) The divergence theorem allows the conversion of volume integrals of PDEs with divergence terms into surface integrals of surface fluxes across cell boundaries. Due to its conservation property, FVM offers efficient outcomes when dealing with PDEs that embody mass, momentum, and energy conservation principles. Furthermore, widely accessible software packages like the OpenFOAM toolbox
(41) include a viscoelastic solver, making it an attractive option for viscoelastic fluid flow modeling.
The complexity in the blood flow simulation arises from deformability and aggregation that RBCs exhibit during their interaction with neighboring cells under different shear rate conditions induced by blood flow. Numerical models coupled with simulation programs have been applied as a groundbreaking method to predict such unique rheological behavior exhibited by RBCs and whole blood. The conventional approach of a single-phase flow simulation is often applied to blood flow simulations within large vessels possessing a moderate shear rate. However, such a method assumes the properties of plasma, RBCs and other cellular components to be evenly distributed as average density and viscosity in blood, resulting in the inability to simulate the mechanical dynamics, such as RBC aggregation under high-shear flow field, inherent in RBCs. To accurately describe the asymmetric distribution of RBC and blood flow, multiphase flow simulation, where numerical simulations of blood flows are often modeled as two immiscible phases, RBCs and blood plasma, is proposed. A common assumption is that RBCs exhibit non-Newtonian behavior while the plasma is treated as a continuous Newtonian phase.Numerous multiphase numerical models have been proposed to simulate the influence of RBCs on blood flow dynamics by different assumptions. In large-scale simulations (above the millimeter range), continuum-based methods are wildly used due to their lower computational demands.
(43) Eulerian multiphase flow simulations offer the solution of a set of conservation equations for each separate phase and couple the phases through common pressure and interphase exchange coefficients. Xu et al.
(44) utilized the combined finite-discrete element method (FDEM) to replicate the dynamic behavior and distortion of RBCs subjected to fluidic forces, utilizing the Johnson–Kendall–Roberts model
(45) to define the adhesive forces of cell-to-cell interactions. The iterative direct-forcing immersed boundary method (IBM) is commonly employed in simulations of the fluid–cell interface of blood. This method effectively captures the intricacies of the thin and flexible RBC membranes within various external flow fields.
(44) also adopts this approach to bridge the fluid dynamics and RBC deformation through IBM. Yoon and You utilized the Maxwell model to define the viscosity of the RBC membrane.
(47) It was discovered that the Maxwell model could represent the stress relaxation and unloading processes of the cell. Furthermore, the reduced flexibility of an RBC under particular situations such as infection is specified, which was unattainable by the Kelvin–Voigt model
(48) when compared to the Maxwell model in the literature. The Yeoh hyperplastic material model was also adapted to predict the nonlinear elasticity property of RBCs with FEM employed to discretize the RBC membrane using shell-type elements. Gracka et al.
(49) developed a numerical CFD model with a finite-volume parallel solver for multiphase blood flow simulation, where an updated Maxwell viscoelasticity model and a Discrete Phase Model are adopted. In the study, the adapted IBM, based on unstructured grids, simulates the flow behavior and shape change of the RBCs through fluid-structure coupling. It was found that the hybrid Euler–Lagrange (E–L) approach
(50) for the development of the multiphase model offered better results in the simulated CFL region in the microchannels.To study the dynamics of individual behaviors of RBCs and the consequent non-Newtonian blood flow, cell-shape-resolved computational models are often adapted. The use of the boundary integral method has become prevalent in minimizing computational expenses, particularly in the exclusive determination of fluid velocity on the surfaces of RBCs, incorporating the option of employing IBM or particle-based techniques. The cell-shaped-resolved method has enabled an examination of cell to cell interactions within complex ambient or pulsatile flow conditions
(51) surrounding RBC membranes. Recently, Rydquist et al.
(52) have looked to integrate statistical information from macroscale simulations to obtain a comprehensive overview of RBC behavior within the immediate proximity of the flow through introduction of respective models characterizing membrane shape definition, tension, bending stresses of RBC membranes.At a macroscopic scale, continuum models have conventionally been adapted for assessing blood flow dynamics through the application of elasticity theory and fluid dynamics. However, particle-based methods are known for their simplicity and adaptability in modeling complex multiscale fluid structures. Meshless methods, such as the boundary element method (BEM), smoothed particle hydrodynamics (SPH), and dissipative particle dynamics (DPD), are often used in particle-based characterization of RBCs and the surrounding fluid. By representing the fluid as discrete particles, meshless methods provide insights into the status and movement of the multiphase fluid. These methods allow for the investigation of cellular structures and microscopic interactions that affect blood rheology. Non-confronting mesh methods like IBM can also be used to couple a fluid solver such as FEM, FVM, or the Lattice Boltzmann Method (LBM) through membrane representation of RBCs. In comparison to conventional CFD methods, LBM has been viewed as a favorable numerical approach for solving the N–S equations and the simulation of multiphase flows. LBM exhibits the notable advantage of being amenable to high-performance parallel computing environments due to its inherently local dynamics. In contrast to DPD and SPH where RBC membranes are modeled as physically interconnected particles, LBM employs the IBM to account for the deformation dynamics of RBCs
(53,54) under shear flows in complex channel geometries.
(54,55) However, it is essential to acknowledge that the utilization of LBM in simulating RBC flows often entails a significant computational overhead, being a primary challenge in this context. Krüger et al.
(56) proposed utilizing LBM as a fluid solver, IBM to couple the fluid and FEM to compute the response of membranes to deformation under immersed fluids. This approach decouples the fluid and membranes but necessitates significant computational effort due to the requirements of both meshes and particles.Despite the accuracy of current blood flow models, simulating complex conditions remains challenging because of the high computational load and cost. Balachandran Nair et al.
(57) suggested a reduced order model of RBC under the framework of DEM, where the RBC is represented by overlapping constituent rigid spheres. The Morse potential force is adapted to account for the RBC aggregation exhibited by cell to cell interactions among RBCs at different distances. Based upon the IBM, the reduced-order RBC model is adapted to simulate blood flow transport for validation under both single and multiple RBCs with a resolved CFD-DEM solver.
(58) In the resolved CFD-DEM model, particle sizes are larger than the grid size for a more accurate computation of the surrounding flow field. A continuous forcing approach is taken to describe the momentum source of the governing equation prior to discretization, which is different from a Direct Forcing Method (DFM).
(59) As no body-conforming moving mesh is required, the continuous forcing approach offers lower complexity and reduced cost when compared to the DFM. Piquet et al.
(60) highlighted the high complexity of the DFM due to its reliance on calculating an additional immersed boundary flux for the velocity field to ensure its divergence-free condition.The fluid–structure interaction (FSI) method has been advocated to connect the dynamic interplay of RBC membranes and fluid plasma within blood flow such as the coupling of continuum–particle interactions. However, such methodology is generally adapted for anatomical configurations such as arteries
(63) where both the structural components and the fluid domain undergo substantial deformation due to the moving boundaries. Due to the scope of the Review being blood flow simulation within microchannels of LOC devices without deformable boundaries, the Review of the FSI method will not be further carried out.In general, three numerical methods are broadly used: mesh-based, particle-based, and hybrid mesh–particle techniques, based on the spatial scale and the fundamental numerical approach, mesh-based methods tend to neglect the effects of individual particles, assuming a continuum and being efficient in terms of time and cost. However, the particle-based approach highlights more of the microscopic and mesoscopic level, where the influence of individual RBCs is considered. A review from Freund et al.
(64) addressed the three numerical methodologies and their respective modeling approaches of RBC dynamics. Given the complex mechanics and the diverse levels of study concerning numerical simulations of blood and cellular flow, a broad spectrum of numerical methods for blood has been subjected to extensive review.
(65) offered an extensive review of the application of the DPD, SPH, and LBM for numerical simulations of RBC, while Rathnayaka et al.
(67) conducted a review of the particle-based numerical modeling for liquid marbles through drawing parallels to the transport of RBCs in microchannels. A comparative analysis between conventional CFD methods and particle-based approaches for cellular and blood flow dynamic simulation can be found under the review by Arabghahestani et al.
(69) offer an overview of both continuum-based models at micro/macroscales and multiscale particle-based models encompassing various length and temporal dimensions. Furthermore, these reviews deliberate upon the potential of coupling continuum-particle methods for blood plasma and RBC modeling. Arciero et al.
(70) investigated various modeling approaches encompassing cellular interactions, such as cell to cell or plasma interactions and the individual cellular phases. A concise overview of the reviews is provided in Table 2 for reference.
Table 2. List of Reviews for Numerical Approaches Employed in Blood Flow Simulation
Capillary driven (CD) flow is a pivotal mechanism in passive microfluidic flow systems
(9) such as the blood circulation system and LOC systems.
(71) CD flow is essentially the movement of a liquid to flow against drag forces, where the capillary effect exerts a force on the liquid at the borders, causing a liquid–air meniscus to flow despite gravity or other drag forces. A capillary pressure drops across the liquid–air interface with surface tension in the capillary radius and contact angle. The capillary effect depends heavily on the interaction between the different properties of surface materials. Different values of contact angles can be manipulated and obtained under varying levels of surface wettability treatments to manipulate the surface properties, resulting in different CD blood delivery rates for medical diagnostic device microchannels. CD flow techniques are appealing for many LOC devices, because they require no external energy. However, due to the passive property of liquid propulsion by capillary forces and the long-term instability of surface treatments on channel walls, the adaptability of CD flow in geometrically complex LOC devices may be limited.
3.2. Theoretical and Numerical Modeling of Capillary Driven Blood Flow
3.2.1. Theoretical Basis and Assumptions of Microfluidic Flow
The study of transport phenomena regarding either blood flow driven by capillary forces or externally applied forces under microfluid systems all demands a comprehensive recognition of the significant differences in flow dynamics between microscale and macroscale. The fundamental assumptions and principles behind fluid transport at the microscale are discussed in this section. Such a comprehension will lay the groundwork for the following analysis of the theoretical basis of capillary forces and their role in blood transport in LOC systems.
At the macroscale, fluid dynamics are often strongly influenced by gravity due to considerable fluid mass. However, the high surface to volume ratio at the microscale shifts the balance toward surface forces (e.g., surface tension and viscous forces), much larger than the inertial force. This difference gives rise to transport phenomena unique to microscale fluid transport, such as the prevalence of laminar flow due to a very low Reynolds number (generally lower than 1). Moreover, the fluid in a microfluidic system is often assumed to be incompressible due to the small flow velocity, indicating constant fluid density in both space and time.Microfluidic flow behaviors are governed by the fundamental principles of mass and momentum conservation, which are encapsulated in the continuity equation and the Navier–Stokes (N–S) equation. The continuity equation describes the conservation of mass, while the N–S equation captures the spatial and temporal variations in velocity, pressure, and other physical parameters. Under the assumption of the negligible influence of gravity in microfluidic systems, the continuity equation and the Eulerian representation of the incompressible N–S equation can be expressed as follows:
∇·𝐮⇀=0∇·�⇀=0
(7)
−∇𝑝+𝜇∇2𝐮⇀+∇·𝝉⇀−𝐅⇀=0−∇�+�∇2�⇀+∇·�⇀−�⇀=0
(8)Here, p is the pressure, u is the fluid viscosity,
𝝉⇀�⇀ represents the stress tensor, and F is the body force exerted by external forces if present.
3.2.2. Theoretical Basis and Modeling of Capillary Force in LOC Systems
The capillary force is often the major driving force to manipulate and transport blood without an externally applied force in LOC systems. Forces induced by the capillary effect impact the free surface of fluids and are represented not directly in the Navier–Stokes equations but through the pressure boundary conditions of the pressure term p. For hydrophilic surfaces, the liquid generally induces a contact angle between 0° and 30°, encouraging the spread and attraction of fluid under a positive cos θ condition. For this condition, the pressure drop becomes positive and generates a spontaneous flow forward. A hydrophobic solid surface repels the fluid, inducing minimal contact. Generally, hydrophobic solids exhibit a contact angle larger than 90°, inducing a negative value of cos θ. Such a value will result in a negative pressure drop and a flow in the opposite direction. The induced contact angle is often utilized to measure the wall exposure of various surface treatments on channel walls where different wettability gradients and surface tension effects for CD flows are established. Contact angles between different interfaces are obtainable through standard values or experimental methods for reference.
(72)For the characterization of the induced force by the capillary effect, the Young–Laplace (Y–L) equation
(73) is widely employed. In the equation, the capillary is considered a pressure boundary condition between the two interphases. Through the Y–L equation, the capillary pressure force can be determined, and subsequently, the continuity and momentum balance equations can be solved to obtain the blood filling rate. Kim et al.
(74) studied the effects of concentration and exposure time of a nonionic surfactant, Silwet L-77, on the performance of a polydimethylsiloxane (PDMS) microchannel in terms of plasma and blood self-separation. The study characterized the capillary pressure force by incorporating the Y–L equation and further evaluated the effects of the changing contact angle due to different levels of applied channel wall surface treatments. The expression of the Y–L equation utilized by Kim et al.
(9)where σ is the surface tension of the liquid and θ
b, θ
t, θ
l, and θ
r are the contact angle values between the liquid and the bottom, top, left, and right walls, respectively. A numerical simulation through Coventor software is performed to evaluate the dynamic changes in the filling rate within the microchannel. The simulation results for the blood filling rate in the microchannel are expressed at a specific time stamp, shown in Figure 2. The results portray an increasing instantaneous filling rate of blood in the microchannel following the decrease in contact angle induced by a higher concentration of the nonionic surfactant treated to the microchannel wall.
Figure 2. Numerical simulation of filling rate of capillary driven blood flow under various contact angle conditions at a specific timestamp. (74) Reproduced with permission from ref (74). Copyright 2010 Elsevier.
When in contact with hydrophilic or hydrophobic surfaces, blood forms a meniscus with a contact angle due to surface tension. The Lucas–Washburn (L–W) equation
(75) is one of the pioneering theoretical definitions for the position of the meniscus over time. In addition, the L–W equation provides the possibility for research to obtain the velocity of the blood formed meniscus through the derivation of the meniscus position. The L–W equation
(10)Here L(t) represents the distance of the liquid driven by the capillary forces. However, the generalized L–W equation solely assumes the constant physical properties from a Newtonian fluid rather than considering the non-Newtonian fluid behavior of blood. Cito et al.
(76) constructed an enhanced version of the L–W equation incorporating the power law to consider the RBC aggregation and the FL effect. The non-Newtonian fluid apparent viscosity under the Power Law model is defined as
𝜇=𝑘·(𝛾˙)𝑛−1�=�·(�˙)�−1
(11)where γ̇ is the strain rate tensor defined as
𝛾˙=12𝛾˙𝑖𝑗𝛾˙𝑗𝑖⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√�˙=12�˙���˙��. The stress tensor term τ is computed as τ = μγ̇
(12)where k is the flow consistency index and n is the power law index, respectively. The power law index, from the Power Law model, characterizes the extent of the non-Newtonian behavior of blood. Both the consistency and power law index rely on blood properties such as hematocrit, the appearance of the FL effect, the formation of RBC aggregates, etc. The updated L–W equation computes the location and velocity of blood flow caused by capillary forces at specified time points within the LOC devices, taking into account the effects of blood flow characteristics such as RBC aggregation and the FL effect on dynamic blood viscosity.Apart from the blood flow behaviors triggered by inherent blood properties, unique flow conditions driven by capillary forces that are portrayed under different microchannel geometries also hold crucial implications for CD blood delivery. Berthier et al.
(77) studied the spontaneous Concus–Finn condition, the condition to initiate the spontaneous capillary flow within a V-groove microchannel, as shown in Figure 3(a) both experimentally and numerically. Through experimental studies, the spontaneous Concus–Finn filament development of capillary driven blood flow is observed, as shown in Figure 3(b), while the dynamic development of blood flow is numerically simulated through CFD simulation.
Figure 3. (a) Sketch of the cross-section of Berthier’s V-groove microchannel, (b) experimental view of blood in the V-groove microchannel, (78) (c) illustration of the dynamic change of the extension of filament from FLOW 3D under capillary flow at three increasing time intervals. (78) Reproduced with permission from ref (78). Copyright 2014 Elsevier.
Berthier et al.
(77) characterized the contact angle needed for the initiation of the capillary driving force at a zero-inlet pressure, through the half-angle (α) of the V-groove geometry layout, and its relation to the Concus–Finn filament as shown below:
(13)Three possible regimes were concluded based on the contact angle value for the initiation of flow and development of Concus–Finn filament:
𝜃>𝜃1𝜃1>𝜃>𝜃0𝜃0no SCFSCF without a Concus−Finn filamentSCF without a Concus−Finn filament{�>�1no SCF�1>�>�0SCF without a Concus−Finn filament�0SCF without a Concus−Finn filament
(14)Under Newton’s Law, the force balance with low Reynolds and Capillary numbers results in the neglect of inertial terms. The force balance between the capillary forces and the viscous force induced by the channel wall is proposed to derive the analytical fluid velocity. This relation between the two forces offers insights into the average flow velocity and the penetration distance function dependent on time. The apparent blood viscosity is defined by Berthier et al.
(23) given in eq 1. The research used the FLOW-3D program from Flow Science Inc. software, which solves transient, free-surface problems using the FDM in multiple dimensions. The Volume of Fluid (VOF) method
(79) is utilized to locate and track the dynamic extension of filament throughout the advancing interface within the channel ahead of the main flow at three progressing time stamps, as depicted in Figure 3(c).
The utilization of external forces, such as electric fields, has significantly broadened the possibility of manipulating microfluidic flow in LOC systems.
(80) Externally applied electric field forces induce a fluid flow from the movement of ions in fluid terms as the “electro-osmotic flow” (EOF).Unique transport phenomena, such as enhanced flow velocity and flow instability, induced by non-Newtonian fluids, particularly viscoelastic fluids, under EOF, have sparked considerable interest in microfluidic devices with simple or complicated geometries within channels.
(81) However, compared to the study of Newtonian fluids and even other electro-osmotic viscoelastic fluid flows, the literature focusing on the theoretical and numerical modeling of electro-osmotic blood flow is limited due to the complexity of blood properties. Consequently, to obtain a more comprehensive understanding of the complex blood flow behavior under EOF, theoretical and numerical studies of the transport phenomena in the EOF section will be based on the studies of different viscoelastic fluids under EOF rather than that of blood specifically. Despite this limitation, we believe these studies offer valuable insights that can help understand the complex behavior of blood flow under EOF.
4.1. EOF Phenomena
Electro-osmotic flow occurs at the interface between the microchannel wall and bulk phase solution. When in contact with the bulk phase, solution ions are absorbed or dissociated at the solid–liquid interface, resulting in the formation of a charge layer, as shown in Figure 4. This charged channel surface wall interacts with both negative and positive ions in the bulk sample, causing repulsion and attraction forces to create a thin layer of immobilized counterions, known as the Stern layer. The induced electric potential from the wall gradually decreases with an increase in the distance from the wall. The Stern layer potential, commonly termed the zeta potential, controls the intensity of the electrostatic interactions between mobile counterions and, consequently, the drag force from the applied electric field. Next to the Stern layer is the diffuse mobile layer, mainly composed of a mobile counterion. These two layers constitute the “electrical double layer” (EDL), the thickness of which is directly proportional to the ionic strength (concentration) of the bulk fluid. The relationship between the two parameters is characterized by a Debye length (λ
D), expressed as
𝜆𝐷=𝜖𝑘B𝑇2(𝑍𝑒)2𝑐0⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√��=��B�2(��)2�0
(15)where ϵ is the permittivity of the electrolyte solution, k
B is the Boltzmann constant, T is the electron temperature, Z is the integer valence number, e is the elementary charge, and c
0 is the ionic density.
Figure 4. Schematic diagram of an electro-osmotic flow in a microchannel with negative surface charge. (82) Reproduced with permission from ref (82). Copyright 2012 Woodhead Publishing.
When an electric field is applied perpendicular to the EDL, viscous drag is generated due to the movement of excess ions in the EDL. Electro-osmotic forces can be attributed to the externally applied electric potential (ϕ) and the zeta potential, the system wall induced potential by charged walls (ψ). As illustrated in Figure 4, the majority of ions in the bulk phase have a uniform velocity profile, except for a shear rate condition confined within an extremely thin Stern layer. Therefore, EOF displays a unique characteristic of a “near flat” or plug flow velocity profile, different from the parabolic flow typically induced by pressure-driven microfluidic flow (Hagen–Poiseuille flow). The plug-shaped velocity profile of the EOF possesses a high shear rate above the Stern layer.Overall, the EOF velocity magnitude is typically proportional to the Debye Length (λ
D), zeta potential, and magnitude of the externally applied electric field, while a more viscous liquid reduces the EOF velocity.
4.2. Modeling on Electro-osmotic Viscoelastic Fluid Flow
4.2.1. Theoretical Basis of EOF Mechanisms
The EOF of an incompressible viscoelastic fluid is commonly governed by the continuity and incompressible N–S equations, as shown in eqs 7 and 8, where the stress tensor and the electrostatic force term are coupled. The electro-osmotic body force term F, representing the body force exerted by the externally applied electric force, is defined as
𝐹⇀=𝑝𝐸𝐸⇀�⇀=���⇀, where ρ
E and
𝐸⇀�⇀ are the net electric charge density and the applied external electric field, respectively.Numerous models are established to theoretically study the externally applied electric potential and the system wall induced potential by charged walls. The following Laplace equation, expressed as eq 16, is generally adapted and solved to calculate the externally applied potential (ϕ).
∇2𝜙=0∇2�=0
(16)Ion diffusion under applied electric fields, together with mass transport resulting from convection and diffusion, transports ionic solutions in bulk flow under electrokinetic processes. The Nernst–Planck equation can describe these transport methods, including convection, diffusion, and electro-diffusion. Therefore, the Nernst–Planck equation is used to determine the distribution of the ions within the electrolyte. The electric potential induced by the charged channel walls follows the Poisson–Nernst–Plank (PNP) equation, which can be written as eq 17.
i are the diffusion coefficient, ionic concentration, and ionic valence of the ionic species I, respectively. However, due to the high nonlinearity and numerical stiffness introduced by different lengths and time scales from the PNP equations, the Poisson–Boltzmann (PB) model is often considered the major simplified method of the PNP equation to characterize the potential distribution of the EDL region in microchannels. In the PB model, it is assumed that the ionic species in the fluid follow the Boltzmann distribution. This model is typically valid for steady-state problems where charge transport can be considered negligible, the EDLs do not overlap with each other, and the intrinsic potentials are low. It provides a simplified representation of the potential distribution in the EDL region. The PB equation governing the EDL electric potential distribution is described as
0 is the ion bulk concentration, z is the ionic valence, and ε
0 is the electric permittivity in the vacuum. Under low electric potential conditions, an even further simplified model to illustrate the EOF phenomena is the Debye–Hückel (DH) model. The DH model is derived by obtaining a charge density term by expanding the exponential term of the Boltzmann equation in a Taylor series.
4.2.2. EOF Modeling for Viscoelastic Fluids
Many studies through numerical modeling were performed to obtain a deeper understanding of the effect exhibited by externally applied electric fields on viscoelastic flow in microchannels under various geometrical designs. Bello et al.
(83) found that methylcellulose solution, a non-Newtonian polymer solution, resulted in stronger electro-osmotic mobility in experiments when compared to the predictions by the Helmholtz–Smoluchowski equation, which is commonly used to define the velocity of EOF of a Newtonian fluid. Being one of the pioneers to identify the discrepancies between the EOF of Newtonian and non-Newtonian fluids, Bello et al. attributed such discrepancies to the presence of a very high shear rate in the EDL, resulting in a change in the orientation of the polymer molecules. Park and Lee
(84) utilized the FVM to solve the PB equation for the characterization of the electric field induced force. In the study, the concept of fractional calculus for the Oldroyd-B model was adapted to illustrate the elastic and memory effects of viscoelastic fluids in a straight microchannel They observed that fluid elasticity and increased ratio of viscoelastic fluid contribution to overall fluid viscosity had a significant impact on the volumetric flow rate and sensitivity of velocity to electric field strength compared to Newtonian fluids. Afonso et al.
(85) derived an analytical expression for EOF of viscoelastic fluid between parallel plates using the DH model to account for a zeta potential condition below 25 mV. The study established the understanding of the electro-osmotic viscoelastic fluid flow under low zeta potential conditions. Apart from the electrokinetic forces, pressure forces can also be coupled with EOF to generate a unique fluid flow behavior within the microchannel. Sousa et al.
(86) analytically studied the flow of a standard viscoelastic solution by combining the pressure gradient force with an externally applied electric force. It was found that, at a near wall skimming layer and the outer layer away from the wall, macromolecules migrating away from surface walls in viscoelastic fluids are observed. In the study, the Phan-Thien Tanner (PTT) constitutive model is utilized to characterize the viscoelastic properties of the solution. The approach is found to be valid when the EDL is much thinner than the skimming layer under an enhanced flow rate. Zhao and Yang
(87) solved the PB equation and Carreau model for the characterization of the EOF mechanism and non-Newtonian fluid respectively through the FEM. The numerical results depict that, different from the EOF of Newtonian fluids, non-Newtonian fluids led to an increase of electro-osmotic mobility for shear thinning fluids but the opposite for shear thickening fluids.Like other fluid transport driving forces, EOF within unique geometrical layouts also portrays unique transport phenomena. Pimenta and Alves
(88) utilized the FVM to perform numerical simulations of the EOF of viscoelastic fluids considering the PB equation and the Oldroyd-B model, in a cross-slot and flow-focusing microdevices. It was found that electroelastic instabilities are formed due to the development of large stresses inside the EDL with streamlined curvature at geometry corners. Bezerra et al.
(89) used the FDM to numerically analyze the vortex formation and flow instability from an electro-osmotic non-Newtonian fluid flow in a microchannel with a nozzle geometry and parallel wall geometry setting. The PNP equation is utilized to characterize the charge motion in the EOF and the PTT model for non-Newtonian flow characterization. A constriction geometry is commonly utilized in blood flow adapted in LOC systems due to the change in blood flow behavior under narrow dimensions in a microchannel. Ji et al.
(90) recently studied the EOF of viscoelastic fluid in a constriction microchannel connected by two relatively big reservoirs on both ends (as seen in Figure 5) filled with the polyacrylamide polymer solution, a viscoelastic fluid, and an incompressible monovalent binary electrolyte solution KCl.
Figure 5. Schematic diagram of a negatively charged constriction microchannel connected to two reservoirs at both ends. An electro-osmotic flow is induced in the system by the induced potential difference between the anode and cathode. (90) Reproduced with permission from ref (90). Copyright 2021 The Authors, under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.
In studying the EOF of viscoelastic fluids, the Oldroyd-B model is often utilized to characterize the polymeric stress tensor and the deformation rate of the fluid. The Oldroyd-B model is expressed as follows:
𝜏=𝜂p𝜆(𝐜−𝐈)�=�p�(�−�)
(19)where η
p, λ, c, and I represent the polymer dynamic viscosity, polymer relaxation time, symmetric conformation tensor of the polymer molecules, and the identity matrix, respectively.A log-conformation tensor approach is taken to prevent convergence difficulty induced by the viscoelastic properties. The conformation tensor (c) in the polymeric stress tensor term is redefined by a new tensor (Θ) based on the natural logarithm of the c. The new tensor is defined as
Θ=ln(𝐜)=𝐑ln(𝚲)𝐑Θ=ln(�)=�ln(�)�
(20)in which Λ is the diagonal matrix and R is the orthogonal matrix.Under the new conformation tensor, the induced EOF of a viscoelastic fluid is governed by the continuity and N–S equations adapting the Oldroyd-B model, which is expressed as
(21)where Ω and B represent the anti-symmetric matrix and the symmetric traceless matrix of the decomposition of the velocity gradient tensor ∇u, respectively. The conformation tensor can be recovered by c = exp(Θ). The PB model and Laplace equation are utilized to characterize the charged channel wall induced potential and the externally applied potential.The governing equations are numerically solved through the FVM by RheoTool,
(42) an open-source viscoelastic EOF solver on the OpenFOAM platform. A SIMPLEC (Semi-Implicit Method for Pressure Linked Equations-Consistent) algorithm was applied to solve the velocity-pressure coupling. The pressure field and velocity field were computed by the PCG (Preconditioned Conjugate Gradient) solver and the PBiCG (Preconditioned Biconjugate Gradient) solver, respectively.Ranging magnitudes of an applied electric field or fluid concentration induce both different streamlines and velocity magnitudes at various locations and times of the microchannel. In the study performed by Ji et al.,
(90) notable fluctuation of streamlines and vortex formation is formed at the upper stream entrance of the constriction as shown in Figure 6(a) and (b), respectively, due to the increase of electrokinetic effect, which is seen as a result of the increase in polymeric stress (τ
xx).
(90) The contraction geometry enhances the EOF velocity within the constriction channel under high E
app condition (600 V/cm). Such phenomena can be attributed to the dependence of electro-osmotic viscoelastic fluid flow on the system wall surface and bulk fluid properties.
Figure 6. Schematic diagram of vortex formation and streamlines of EOF depicting flow instability at (a) 1.71 s and (b) 1.75 s. Spatial distribution of the elastic normal stress at (c) high Eapp condition. Streamline of an electro-osmotic flow under Eapp of 600 V/cm (90) for (d) non-Newtonian and (e) Newtonian fluid through a constriction geometry. Reproduced with permission from ref (90). Copyright 2021 The Authors, under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.
As elastic normal stress exceeds the local shear stress, flow instability and vortex formation occur. The induced elastic stress under EOF not only enhances the instability of the flow but often generates an irregular secondary flow leading to strong disturbance.
(92) It is also vital to consider the effect of the constriction layout of microchannels on the alteration of the field strength within the system. The contraction geometry enhances a larger electric field strength compared with other locations of the channel outside the constriction region, resulting in a higher velocity gradient and stronger extension on the polymer within the viscoelastic solution. Following the high shear flow condition, a higher magnitude of stretch for polymer molecules in viscoelastic fluids exhibits larger elastic stresses and enhancement of vortex formation at the region.
(93)As shown in Figure 6(c), significant elastic normal stress occurs at the inlet of the constriction microchannel. Such occurrence of a polymeric flow can be attributed to the dominating elongational flow, giving rise to high deformation of the polymers within the viscoelastic fluid flow, resulting in higher elastic stress from the polymers. Such phenomena at the entrance result in the difference in velocity streamline as circled in Figure 6(d) compared to that of the Newtonian fluid at the constriction entrance in Figure 6(e).
(90) The difference between the Newtonian and polymer solution at the exit, as circled in Figure 6(d) and (e), can be attributed to the extrudate swell effect of polymers
(94) within the viscoelastic fluid flow. The extrudate swell effect illustrates that, as polymers emerge from the constriction exit, they tend to contract in the flow direction and grow in the normal direction, resulting in an extrudate diameter greater than the channel size. The deformation of polymers within the polymeric flow at both the entrance and exit of the contraction channel facilitates the change in shear stress conditions of the flow, leading to the alteration in streamlines of flows for each region.
4.3. EOF Applications in LOC Systems
4.3.1. Mixing in LOC Systems
Rather than relying on the micromixing controlled by molecular diffusion under low Reynolds number conditions, active mixers actively leverage convective instability and vortex formation induced by electro-osmotic flows from alternating current (AC) or direct current (DC) electric fields. Such adaptation is recognized as significant breakthroughs for promotion of fluid mixing in chemical and biological applications such as drug delivery, medical diagnostics, chemical synthesis, and so on.
(95)Many researchers proposed novel designs of electro-osmosis micromixers coupled with numerical simulations in conjunction with experimental findings to increase their understanding of the role of flow instability and vortex formation in the mixing process under electrokinetic phenomena. Matsubara and Narumi
(96) numerically modeled the mixing process in a microchannel with four electrodes on each side of the microchannel wall, which generated a disruption through unstable electro-osmotic vortices. It was found that particle mixing was sensitive to both the convection effect induced by the main and secondary vortex within the micromixer and the change in oscillation frequency caused by the supplied AC voltage when the Reynolds number was varied. Qaderi et al.
(97) adapted the PNP equation to numerically study the effect of the geometry and zeta potential configuration of the microchannel on the mixing process with a combined electro-osmotic pressure driven flow. It was reported that the application of heterogeneous zeta potential configuration enhances the mixing efficiency by around 23% while the height of the hurdles increases the mixing efficiency at most 48.1%. Cho et al.
(98) utilized the PB model and Laplace equation to numerically simulate the electro-osmotic non-Newtonian fluid mixing process within a wavy and block layout of microchannel walls. The Power Law model is adapted to describe the fluid rheological characteristic. It was found that shear-thinning fluids possess a higher volumetric flow rate, which could result in poorer mixing efficiency compared to that of Newtonian fluids. Numerous studies have revealed that flow instability and vortex generation, in particular secondary vortices produced by barriers or greater magnitudes of heterogeneous zeta potential distribution, enhance mixing by increasing bulk flow velocity and reducing flow distance.To better understand the mechanism of disturbance formed in the system due to externally applied forces, known as electrokinetic instability, literature often utilize the Rayleigh (Ra) number,
(22)where γ is the conductivity ratio of the two streams and can be written as
𝛾=𝜎el,H𝜎el,L�=�el,H�el,L. The Ra number characterizes the ratio between electroviscous and electro-osmotic flow. A high Ra
v value often results in good mixing. It is evident that fluid properties such as the conductivity (σ) of the two streams play a key role in the formation of disturbances to enhance mixing in microsystems. At the same time, electrokinetic parameters like the zeta potential (ζ) in the Ra number is critical in the characterization of electro-osmotic velocity and a slip boundary condition at the microchannel wall.To understand the mixing result along the channel, the concentration field can be defined and simulated under the assumption of steady state conditions and constant diffusion coefficient for each of the working fluid within the system through the convection–diffusion equation as below:
∂𝑐𝒊∂𝑡+∇⇀(𝑐𝑖𝑢⇀−𝐷𝑖∇⇀𝑐𝒊)=0∂��∂�+∇⇀(���⇀−��∇⇀��)=0
(23)where c
i is the species concentration of species i and D
i is the diffusion coefficient of the corresponding species.The standard deviation of concentration (σ
sd) can be adapted to evaluate the mixing quality of the system.
(97) The standard deviation for concentration at a specific portion of the channel may be calculated using the equation below:
m are the non-dimensional concentration profile and the mean concentration at the portion, respectively. C* is the non-dimensional concentration and can be calculated as
𝐶∗=𝐶𝐶ref�*=��ref, where C
ref is the reference concentration defined as the bulk solution concentration. The mean concentration profile can be calculated as
𝐶m=∫10(𝐶∗(𝑦∗)d𝑦∗∫10d𝑦∗�m=∫01(�*(�*)d�*∫01d�*. With the standard deviation of concentration, the mixing efficiency
sd,0 is the standard derivation of the case of no mixing. The value of the mixing efficiency is typically utilized in conjunction with the simulated flow field and concentration field to explore the effect of geometrical and electrokinetic parameters on the optimization of the mixing results.
Viscoelastic fluids such as blood flow in LOC systems are an essential topic to proceed with diagnostic analysis and research through microdevices in the biomedical and pharmaceutical industries. The complex blood flow behavior is tightly controlled by the viscoelastic characteristics of blood such as the dynamic viscosity and the elastic property of RBCs under various shear rate conditions. Furthermore, the flow behaviors under varied driving forces promote an array of microfluidic transport phenomena that are critical to the management of blood flow and other adapted viscoelastic fluids in LOC systems. This review addressed the blood flow phenomena, the complicated interplay between shear rate and blood flow behaviors, and their numerical modeling under LOC systems through the lens of the viscoelasticity characteristic. Furthermore, a theoretical understanding of capillary forces and externally applied electric forces leads to an in-depth investigation of the relationship between blood flow patterns and the key parameters of the two driving forces, the latter of which is introduced through the lens of viscoelastic fluids, coupling numerical modeling to improve the knowledge of blood flow manipulation in LOC systems. The flow disturbances triggered by the EOF of viscoelastic fluids and their impact on blood flow patterns have been deeply investigated due to their important role and applications in LOC devices. Continuous advancements of various numerical modeling methods with experimental findings through more efficient and less computationally heavy methods have served as an encouraging sign of establishing more accurate illustrations of the mechanisms for multiphase blood and other viscoelastic fluid flow transport phenomena driven by various forces. Such progress is fundamental for the manipulation of unique transport phenomena, such as the generated disturbances, to optimize functionalities offered by microdevices in LOC systems.
The following section will provide further insights into the employment of studied blood transport phenomena to improve the functionality of micro devices adapting LOC technology. A discussion of the novel roles that external driving forces play in microfluidic flow behaviors is also provided. Limitations in the computational modeling of blood flow and electrokinetic phenomena in LOC systems will also be emphasized, which may provide valuable insights for future research endeavors. These discussions aim to provide guidance and opportunities for new paths in the ongoing development of LOC devices that adapt blood flow.
5.2. Future Directions
5.2.1. Electro-osmosis Mixing in LOC Systems
Despite substantial research, mixing results through flow instability and vortex formation phenomena induced by electro-osmotic mixing still deviate from the effective mixing results offered by chaotic mixing results such as those seen in turbulent flows. However, recent discoveries of a mixing phenomenon that is generally observed under turbulent flows are found within electro-osmosis micromixers under low Reynolds number conditions. Zhao
(99) experimentally discovered a rapid mixing process in an AC applied micromixer, where the power spectrum of concentration under an applied voltage of 20 V
p-p induces a −5/3 slope within a frequency range. This value of the slope is considered as the O–C spectrum in macroflows, which is often visible under relatively high Re conditions, such as the Taylor microscale Reynolds number Re > 500 in turbulent flows.
(100) However, the Re value in the studied system is less than 1 at the specific location and applied voltage. A secondary flow is also suggested to occur close to microchannel walls, being attributed to the increase of convective instability within the system.Despite the experimental phenomenon proposed by Zhao et al.,
(99) the range of effects induced by vital parameters of an EOF mixing system on the enhanced mixing results and mechanisms of disturbance generated by the turbulent-like flow instability is not further characterized. Such a gap in knowledge may hinder the adaptability and commercialization of the discovery of micromixers. One of the parameters for further evaluation is the conductivity gradient of the fluid flow. A relatively strong conductivity gradient (5000:1) was adopted in the system due to the conductive properties of the two fluids. The high conductivity gradients may contribute to the relatively large Rayleigh number and differences in EDL layer thickness, resulting in an unusual disturbance in laminar flow conditions and enhanced mixing results. However, high conductivity gradients are not always achievable by the working fluids due to diverse fluid properties. The reliance on turbulent-like phenomena and rapid mixing results in a large conductivity gradient should be established to prevent the limited application of fluids for the mixing system. In addition, the proposed system utilizes distinct zeta potential distributions at the top and bottom walls due to their difference in material choices, which may be attributed to the flow instability phenomena. Further studies should be made on varying zeta potential magnitude and distribution to evaluate their effect on the slip boundary conditions of the flow and the large shear rate condition close to the channel wall of EOF. Such a study can potentially offer an optimized condition in zeta potential magnitude through material choices and geometrical layout of the zeta potential for better mixing results and manipulation of mixing fluid dynamics. The two vital parameters mentioned above can be varied with the aid of numerical simulation to understand the effect of parameters on the interaction between electro-osmotic forces and electroviscous forces. At the same time, the relationship of developed streamlines of the simulated velocity and concentration field, following their relationship with the mixing results, under the impact of these key parameters can foster more insight into the range of impact that the two parameters have on the proposed phenomena and the microfluidic dynamic principles of disturbances.
In addition, many of the current investigations of electrokinetic mixers commonly emphasize the fluid dynamics of mixing for Newtonian fluids, while the utilization of biofluids, primarily viscoelastic fluids such as blood, and their distinctive response under shear forces in these novel mixing processes of LOC systems are significantly less studied. To develop more compatible microdevice designs and efficient mixing outcomes for the biomedical industry, it is necessary to fill the knowledge gaps in the literature on electro-osmotic mixing for biofluids, where properties of elasticity, dynamic viscosity, and intricate relationship with shear flow from the fluid are further considered.
5.2.2. Electro-osmosis Separation in LOC Systems
Particle separation in LOC devices, particularly in biological research and diagnostics, is another area where disturbances may play a significant role in optimization.
(101) Plasma analysis in LOC systems under precise control of blood flow phenomena and blood/plasma separation procedures can detect vital information about infectious diseases from particular antibodies and foreign nucleic acids for medical treatments, diagnostics, and research,
(102) offering more efficient results and simple operating procedures compared to that of the traditional centrifugation method for blood and plasma separation. However, the adaptability of LOC devices for blood and plasma separation is often hindered by microchannel clogging, where flow velocity and plasma yield from LOC devices is reduced due to occasional RBC migration and aggregation at the filtration entrance of microdevices.
(103)It is important to note that the EOF induces flow instability close to microchannel walls, which may provide further solutions to clogging for the separation process of the LOC systems. Mohammadi et al.
(104) offered an anti-clogging effect of RBCs at the blood and plasma separating device filtration entry, adjacent to the surface wall, through RBC disaggregation under high shear rate conditions generated by a forward and reverse EOF direction.
Further theoretical and numerical research can be conducted to characterize the effect of high shear rate conditions near microchannel walls toward the detachment of binding blood cells on surfaces and the reversibility of aggregation. Through numerical modeling with varying electrokinetic parameters to induce different degrees of disturbances or shear conditions at channel walls, it may be possible to optimize and better understand the process of disrupting the forces that bind cells to surface walls and aggregated cells at filtration pores. RBCs that migrate close to microchannel walls are often attracted by the adhesion force between the RBC and the solid surface originating from the van der Waals forces. Following RBC migration and attachment by adhesive forces adjacent to the microchannel walls as shown in Figure 7, the increase in viscosity at the region causes a lower shear condition and encourages RBC aggregation (cell–cell interaction), which clogs filtering pores or microchannels and reduces flow velocity at filtration region. Both the impact that shear forces and disturbances may induce on cell binding forces with surface walls and other cells leading to aggregation may suggest further characterization. Kinetic parameters such as activation energy and the rate-determining step for cell binding composition attachment and detachment should be considered for modeling the dynamics of RBCs and blood flows under external forces in LOC separation devices.
Figure 7. Schematic representations of clogging at a microchannel pore following the sequence of RBC migration, cell attachment to channel walls, and aggregation. (105) Reproduced with permission from ref (105). Copyright 2018 The Authors under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.
5.2.3. Relationship between External Forces and Microfluidic Systems
In blood flow, a thicker CFL suggests a lower blood viscosity, suggesting a complex relationship between shear stress and shear rate, affecting the blood viscosity and blood flow. Despite some experimental and numerical studies on electro-osmotic non-Newtonian fluid flow, limited literature has performed an in-depth investigation of the role that applied electric forces and other external forces could play in the process of CFL formation. Additional studies on how shear rates from external forces affect CFL formation and microfluidic flow dynamics can shed light on the mechanism of the contribution induced by external driving forces to the development of a separate phase of layer, similar to CFL, close to the microchannel walls and distinct from the surrounding fluid within the system, then influencing microfluidic flow dynamics.One of the mechanisms of phenomena to be explored is the formation of the Exclusion Zone (EZ) region following a “Self-Induced Flow” (SIF) phenomenon discovered by Li and Pollack,
(106) as shown in Figure 8(a) and (b), respectively. A spontaneous sustained axial flow is observed when hydrophilic materials are immersed in water, resulting in the buildup of a negative layer of charges, defined as the EZ, after water molecules absorb infrared radiation (IR) energy and break down into H and OH
+–.
Figure 8. Schematic representations of (a) the Exclusion Zone region and (b) the Self Induced Flow through visualization of microsphere movement within a microchannel. (106) Reproduced with permission from ref (106). Copyright 2020 The Authors under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.
Despite the finding of such a phenomenon, the specific mechanism and role of IR energy have yet to be defined for the process of EZ development. To further develop an understanding of the role of IR energy in such phenomena, a feasible study may be seen through the lens of the relationships between external forces and microfluidic flow. In the phenomena, the increase of SIF velocity under a rise of IR radiation resonant characteristics is shown in the participation of the external electric field near the microchannel walls under electro-osmotic viscoelastic fluid flow systems. The buildup of negative charges at the hydrophilic surfaces in EZ is analogous to the mechanism of electrical double layer formation. Indeed, research has initiated the exploration of the core mechanisms for EZ formation through the lens of the electrokinetic phenomena.
(107) Such a similarity of the role of IR energy and the transport phenomena of SIF with electrokinetic phenomena paves the way for the definition of the unknown SIF phenomena and EZ formation. Furthermore, Li and Pollack
(106) suggest whether CFL formation might contribute to a SIF of blood using solely IR radiation, a commonly available source of energy in nature, as an external driving force. The proposition may be proven feasible with the presence of the CFL region next to the negatively charged hydrophilic endothelial glycocalyx layer, coating the luminal side of blood vessels.
(108) Further research can dive into the resonating characteristics between the formation of the CFL region next to the hydrophilic endothelial glycocalyx layer and that of the EZ formation close to hydrophilic microchannel walls. Indeed, an increase in IR energy is known to rapidly accelerate EZ formation and SIF velocity, depicting similarity to the increase in the magnitude of electric field forces and greater shear rates at microchannel walls affecting CFL formation and EOF velocity. Such correlation depicts a future direction in whether SIF blood flow can be observed and characterized theoretically further through the lens of the relationship between blood flow and shear forces exhibited by external energy.
The intricate link between the CFL and external forces, more specifically the externally applied electric field, can receive further attention to provide a more complete framework for the mechanisms between IR radiation and EZ formation. Such characterization may also contribute to a greater comprehension of the role IR can play in CFL formation next to the endothelial glycocalyx layer as well as its role as a driving force to propel blood flow, similar to the SIF, but without the commonly assumed pressure force from heart contraction as a source of driving force.
5.3. Challenges
Although there have been significant improvements in blood flow modeling under LOC systems over the past decade, there are still notable constraints that may require special attention for numerical simulation applications to benefit the adaptability of the designs and functionalities of LOC devices. Several points that require special attention are mentioned below:
1.
The majority of CFD models operate under the relationship between the viscoelasticity of blood and the shear rate conditions of flow. The relative effect exhibited by the presence of highly populated RBCs in whole blood and their forces amongst the cells themselves under complex flows often remains unclearly defined. Furthermore, the full range of cell populations in whole blood requires a much more computational load for numerical modeling. Therefore, a vital goal for future research is to evaluate a reduced modeling method where the impact of cell–cell interaction on the viscoelastic property of blood is considered.
2.
Current computational methods on hemodynamics rely on continuum models based upon non-Newtonian rheology at the macroscale rather than at molecular and cellular levels. Careful considerations should be made for the development of a constructive framework for the physical and temporal scales of micro/nanoscale systems to evaluate the intricate relationship between fluid driving forces, dynamic viscosity, and elasticity.
3.
Viscoelastic fluids under the impact of externally applied electric forces often deviate from the assumptions of no-slip boundary conditions due to the unique flow conditions induced by externally applied forces. Furthermore, the mechanism of vortex formation and viscoelastic flow instability at laminar flow conditions should be better defined through the lens of the microfluidic flow phenomenon to optimize the prediction of viscoelastic flow across different geometrical layouts. Mathematical models and numerical methods are needed to better predict such disturbance caused by external forces and the viscoelasticity of fluids at such a small scale.
4.
Under practical situations, zeta potential distribution at channel walls frequently deviates from the common assumption of a constant distribution because of manufacturing faults or inherent surface charges prior to the introduction of electrokinetic influence. These discrepancies frequently lead to inconsistent surface potential distribution, such as excess positive ions at relatively more negatively charged walls. Accordingly, unpredicted vortex formation and flow instability may occur. Therefore, careful consideration should be given to these discrepancies and how they could trigger the transport process and unexpected results of a microdevice.
Zhe Chen – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; Email: zaccooky@sjtu.edu.cn
Bo Ouyang – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; Email: bouy93@sjtu.edu.cn
Zheng-Hong Luo – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; https://orcid.org/0000-0001-9011-6020; Email: luozh@sjtu.edu.cn
Authors
Bin-Jie Lai – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; https://orcid.org/0009-0002-8133-5381
Li-Tao Zhu – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; https://orcid.org/0000-0001-6514-8864
NotesThe authors declare no competing financial interest.
This work was supported by the National Natural Science Foundation of China (No. 22238005) and the Postdoctoral Research Foundation of China (No. GZC20231576).
the field of technological and scientific study that investigates fluid flow in channels with dimensions between 1 and 1000 μm
Lab-on-a-Chip Technology
the field of research and technological development aimed at integrating the micro/nanofluidic characteristics to conduct laboratory processes on handheld devices
Computational Fluid Dynamics (CFD)
the method utilizing computational abilities to predict physical fluid flow behaviors mathematically through solving the governing equations of corresponding fluid flows
Shear Rate
the rate of change in velocity where one layer of fluid moves past the adjacent layer
Viscoelasticity
the property holding both elasticity and viscosity characteristics relying on the magnitude of applied shear stress and time-dependent strain
Electro-osmosis
the flow of fluid under an applied electric field when charged solid surface is in contact with the bulk fluid
Vortex
the rotating motion of a fluid revolving an axis line
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웨어의 두 가지 서로 다른 배열(즉, 직선형 웨어와 직사각형 미로 웨어)을 사용하여 웨어 모양, 웨어 간격, 웨어의 오리피스 존재, 흐름 영역에 대한 바닥 경사와 같은 기하학적 매개변수의 영향을 평가했습니다.
유량과 수심의 관계, 수심 평균 속도의 변화와 분포, 난류 특성, 어도에서의 에너지 소산. 흐름 조건에 미치는 영향을 조사하기 위해 FLOW-3D® 소프트웨어를 사용하여 전산 유체 역학 시뮬레이션을 수행했습니다.
수치 모델은 계산된 표면 프로파일과 속도를 문헌의 실험적으로 측정된 값과 비교하여 검증되었습니다. 수치 모델과 실험 데이터의 결과, 급락유동의 표면 프로파일과 표준화된 속도 프로파일에 대한 평균 제곱근 오차와 평균 절대 백분율 오차가 각각 0.014m와 3.11%로 나타나 수치 모델의 능력을 확인했습니다.
수영장과 둑의 흐름 특성을 예측합니다. 각 모델에 대해 L/B = 1.83(L: 웨어 거리, B: 수로 폭) 값에서 급락 흐름이 발생할 수 있고 L/B = 0.61에서 스트리밍 흐름이 발생할 수 있습니다. 직사각형 미로보 모델은 기존 모델보다 무차원 방류량(Q+)이 더 큽니다.
수중 흐름의 기존 보와 직사각형 미로 보의 경우 Q는 각각 1.56과 1.47h에 비례합니다(h: 보 위 수심). 기존 웨어의 풀 내 평균 깊이 속도는 직사각형 미로 웨어의 평균 깊이 속도보다 높습니다.
그러나 주어진 방류량, 바닥 경사 및 웨어 간격에 대해 난류 운동 에너지(TKE) 및 난류 강도(TI) 값은 기존 웨어에 비해 직사각형 미로 웨어에서 더 높습니다. 기존의 웨어는 직사각형 미로 웨어보다 에너지 소산이 더 낮습니다.
더 낮은 TKE 및 TI 값은 미로 웨어 상단, 웨어 하류 벽 모서리, 웨어 측벽과 채널 벽 사이에서 관찰되었습니다. 보와 바닥 경사면 사이의 거리가 증가함에 따라 평균 깊이 속도, 난류 운동 에너지의 평균값 및 난류 강도가 증가하고 수영장의 체적 에너지 소산이 감소했습니다.
둑에 개구부가 있으면 평균 깊이 속도와 TI 값이 증가하고 풀 내에서 가장 높은 TKE 범위가 감소하여 두 모델 모두에서 물고기를 위한 휴식 공간이 더 넓어지고(TKE가 낮아짐) 에너지 소산율이 감소했습니다.
Two different arrangements of the weir (i.e., straight weir and rectangular labyrinth weir) were used to evaluate the effects of geometric parameters such as weir shape, weir spacing, presence of an orifice at the weir, and bed slope on the flow regime and the relationship between discharge and depth, variation and distribution of depth-averaged velocity, turbulence characteristics, and energy dissipation at the fishway. Computational fluid dynamics simulations were performed using FLOW-3D® software to examine the effects on flow conditions. The numerical model was validated by comparing the calculated surface profiles and velocities with experimentally measured values from the literature. The results of the numerical model and experimental data showed that the root-mean-square error and mean absolute percentage error for the surface profiles and normalized velocity profiles of plunging flows were 0.014 m and 3.11%, respectively, confirming the ability of the numerical model to predict the flow characteristics of the pool and weir. A plunging flow can occur at values of L/B = 1.83 (L: distance of the weir, B: width of the channel) and streaming flow at L/B = 0.61 for each model. The rectangular labyrinth weir model has larger dimensionless discharge values (Q+) than the conventional model. For the conventional weir and the rectangular labyrinth weir at submerged flow, Q is proportional to 1.56 and 1.47h, respectively (h: the water depth above the weir). The average depth velocity in the pool of a conventional weir is higher than that of a rectangular labyrinth weir. However, for a given discharge, bed slope, and weir spacing, the turbulent kinetic energy (TKE) and turbulence intensity (TI) values are higher for a rectangular labyrinth weir compared to conventional weir. The conventional weir has lower energy dissipation than the rectangular labyrinth weir. Lower TKE and TI values were observed at the top of the labyrinth weir, at the corner of the wall downstream of the weir, and between the side walls of the weir and the channel wall. As the distance between the weirs and the bottom slope increased, the average depth velocity, the average value of turbulent kinetic energy and the turbulence intensity increased, and the volumetric energy dissipation in the pool decreased. The presence of an opening in the weir increased the average depth velocity and TI values and decreased the range of highest TKE within the pool, resulted in larger resting areas for fish (lower TKE), and decreased the energy dissipation rates in both models.
1 Introduction
Artificial barriers such as detour dams, weirs, and culverts in lakes and rivers prevent fish from migrating and completing the upstream and downstream movement cycle. This chain is related to the life stage of the fish, its location, and the type of migration. Several riverine fish species instinctively migrate upstream for spawning and other needs. Conversely, downstream migration is a characteristic of early life stages [1]. A fish ladder is a waterway that allows one or more fish species to cross a specific obstacle. These structures are constructed near detour dams and other transverse structures that have prevented such migration by allowing fish to overcome obstacles [2]. The flow pattern in fish ladders influences safe and comfortable passage for ascending fish. The flow’s strong turbulence can reduce the fish’s speed, injure them, and delay or prevent them from exiting the fish ladder. In adult fish, spawning migrations are usually complex, and delays are critical to reproductive success [3].
Various fish ladders/fishways include vertical slots, denil, rock ramps, and pool weirs [1]. The choice of fish ladder usually depends on many factors, including water elevation, space available for construction, and fish species. Pool and weir structures are among the most important fish ladders that help fish overcome obstacles in streams or rivers and swim upstream [1]. Because they are easy to construct and maintain, this type of fish ladder has received considerable attention from researchers and practitioners. Such a fish ladder consists of a sloping-floor channel with series of pools directly separated by a series of weirs [4]. These fish ladders, with or without underwater openings, are generally well-suited for slopes of 10% or less [1, 2]. Within these pools, flow velocities are low and provide resting areas for fish after they enter the fish ladder. After resting in the pools, fish overcome these weirs by blasting or jumping over them [2]. There may also be an opening in the flooded portion of the weir through which the fish can swim instead of jumping over the weir. Design parameters such as the length of the pool, the height of the weir, the slope of the bottom, and the water discharge are the most important factors in determining the hydraulic structure of this type of fish ladder [3]. The flow over the weir depends on the flow depth at a given slope S0 and the pool length, either “plunging” or “streaming.” In plunging flow, the water column h over each weir creates a water jet that releases energy through turbulent mixing and diffusion mechanisms [5]. The dimensionless discharges for plunging (Q+) and streaming (Q*) flows are shown in Fig. 1, where Q is the total discharge, B is the width of the channel, w is the weir height, S0 is the slope of the bottom, h is the water depth above the weir, d is the flow depth, and g is the acceleration due to gravity. The maximum velocity occurs near the top of the weir for plunging flow. At the water’s surface, it drops to about half [6].
Fig. 1
Extensive experimental studies have been conducted to investigate flow patterns for various physical geometries (i.e., bed slope, pool length, and weir height) [2]. Guiny et al. [7] modified the standard design by adding vertical slots, orifices, and weirs in fishways. The efficiency of the orifices and vertical slots was related to the velocities at their entrances. In the laboratory experiments of Yagci [8], the three-dimensional (3D) mean flow and turbulence structure of a pool weir fishway combined with an orifice and a slot is investigated. It is shown that the energy dissipation per unit volume and the discharge have a linear relationship.
Considering the beneficial characteristics reported in the limited studies of researchers on the labyrinth weir in the pool-weir-type fishway, and knowing that the characteristics of flow in pool-weir-type fishways are highly dependent on the geometry of the weir, an alternative design of the rectangular labyrinth weir instead of the straight weirs in the pool-weir-type fishway is investigated in this study [7, 9]. Kim [10] conducted experiments to compare the hydraulic characteristics of three different weir types in a pool-weir-type fishway. The results show that a straight, rectangular weir with a notch is preferable to a zigzag or trapezoidal weir. Studies on natural fish passes show that pass ability can be improved by lengthening the weir’s crest [7]. Zhong et al. [11] investigated the semi-rigid weir’s hydraulic performance in the fishway’s flow field with a pool weir. The results showed that this type of fishway performed better with a lower invert slope and a smaller radius ratio but with a larger pool spacing.
Considering that an alternative method to study the flow characteristics in a fishway with a pool weir is based on numerical methods and modeling from computational fluid dynamics (CFD), which can easily change the geometry of the fishway for different flow fields, this study uses the powerful package CFD and the software FLOW-3D to evaluate the proposed weir design and compare it with the conventional one to extend the application of the fishway. The main objective of this study was to evaluate the hydraulic performance of the rectangular labyrinth pool and the weir with submerged openings in different hydraulic configurations. The primary objective of creating a new weir configuration for suitable flow patterns is evaluated based on the swimming capabilities of different fish species. Specifically, the following questions will be answered: (a) How do the various hydraulic and geometric parameters relate to the effects of water velocity and turbulence, expressed as turbulent kinetic energy (TKE) and turbulence intensity (TI) within the fishway, i.e., are conventional weirs more affected by hydraulics than rectangular labyrinth weirs? (b) Which weir configurations have the greatest effect on fish performance in the fishway? (c) In the presence of an orifice plate, does the performance of each weir configuration differ with different weir spacing, bed gradients, and flow regimes from that without an orifice plate?
2 Materials and Methods
2.1 Physical Model Configuration
This paper focuses on Ead et al. [6]’s laboratory experiments as a reference, testing ten pool weirs (Fig. 2). The experimental flume was 6 m long, 0.56 m wide, and 0.6 m high, with a bottom slope of 10%. Field measurements were made at steady flow with a maximum flow rate of 0.165 m3/s. Discharge was measured with magnetic flow meters in the inlets and water level with point meters (see Ead et al. [6]. for more details). Table 1 summarizes the experimental conditions considered for model calibration in this study.
Fig. 2
Table 1 Experimental conditions considered for calibration
Computational fluid dynamics (CFD) simulations were performed using FLOW-3D® v11.2 to validate a series of experimental liner pool weirs by Ead et al. [6] and to investigate the effects of the rectangular labyrinth pool weir with an orifice. The dimensions of the channel and data collection areas in the numerical models are the same as those of the laboratory model. Two types of pool weirs were considered: conventional and labyrinth. The proposed rectangular labyrinth pool weirs have a symmetrical cross section and are sized to fit within the experimental channel. The conventional pool weir model had a pool length of l = 0.685 and 0.342 m, a weir height of w = 0.141 m, a weir width of B = 0.56 m, and a channel slope of S0 = 5 and 10%. The rectangular labyrinth weirs have the same front width as the offset, i.e., a = b = c = 0.186 m. A square underwater opening with a width of 0.05 m and a depth of 0.05 m was created in the middle of the weir. The weir configuration considered in the present study is shown in Fig. 3.
Fig. 3
2.3 Governing Equations
FLOW-3D® software solves the Navier–Stokes–Reynolds equations for three-dimensional analysis of incompressible flows using the fluid-volume method on a gridded domain. FLOW -3D® uses an advanced free surface flow tracking algorithm (TruVOF) developed by Hirt and Nichols [12], where fluid configurations are defined in terms of a VOF function F (x, y, z, t). In this case, F (fluid fraction) represents the volume fraction occupied by the fluid: F = 1 in cells filled with fluid and F = 0 in cells without fluid (empty areas) [4, 13]. The free surface area is at an intermediate value of F. (Typically, F = 0.5, but the user can specify a different intermediate value.) The equations in Cartesian coordinates (x, y, z) applicable to the model are as follows:
�f∂�∂�+∂(���x)∂�+∂(���y)∂�+∂(���z)∂�=�SOR
(1)
∂�∂�+1�f(��x∂�∂�+��y∂�∂�+��z∂�∂�)=−1�∂�∂�+�x+�x
(2)
∂�∂�+1�f(��x∂�∂�+��y∂�∂�+��z∂�∂�)=−1�∂�∂�+�y+�y
(3)
∂�∂�+1�f(��x∂�∂�+��y∂�∂�+��z∂�∂�)=−1�∂�∂�+�z+�z
(4)
where (u, v, w) are the velocity components, (Ax, Ay, Az) are the flow area components, (Gx, Gy, Gz) are the mass accelerations, and (fx, fy, fz) are the viscous accelerations in the directions (x, y, z), ρ is the fluid density, RSOR is the spring term, Vf is the volume fraction associated with the flow, and P is the pressure. The k–ε turbulence model (RNG) was used in this study to solve the turbulence of the flow field. This model is a modified version of the standard k–ε model that improves performance. The model is a two-equation model; the first equation (Eq. 5) expresses the turbulence’s energy, called turbulent kinetic energy (k) [14]. The second equation (Eq. 6) is the turbulent dissipation rate (ε), which determines the rate of dissipation of kinetic energy [15]. These equations are expressed as follows Dasineh et al. [4]:
In these equations, k is the turbulent kinetic energy, ε is the turbulent energy consumption rate, Gk is the generation of turbulent kinetic energy by the average velocity gradient, with empirical constants αε = αk = 1.39, C1ε = 1.42, and C2ε = 1.68, eff is the effective viscosity, μeff = μ + μt [15]. Here, μ is the hydrodynamic density coefficient, and μt is the turbulent density of the fluid.
2.4 Meshing and the Boundary Conditions in the Model Setup
The numerical area is divided into three mesh blocks in the X-direction. The meshes are divided into different sizes, a containing mesh block for the entire spatial domain and a nested block with refined cells for the domain of interest. Three different sizes were selected for each of the grid blocks. By comparing the accuracy of their results based on the experimental data, the reasonable mesh for the solution domain was finally selected. The convergence index method (GCI) evaluated the mesh sensitivity analysis. Based on this method, many researchers, such as Ahmadi et al. [16] and Ahmadi et al. [15], have studied the independence of numerical results from mesh size. Three different mesh sizes with a refinement ratio (r) of 1.33 were used to perform the convergence index method. The refinement ratio is the ratio between the larger and smaller mesh sizes (r = Gcoarse/Gfine). According to the recommendation of Celik et al. [17], the recommended number for the refinement ratio is 1.3, which gives acceptable results. Table 2 shows the characteristics of the three mesh sizes selected for mesh sensitivity analysis.Table 2 Characteristics of the meshes tested in the convergence analysis
The results of u1 = umax (u1 = velocity component along the x1 axis and umax = maximum velocity of u1 in a section perpendicular to the invert of the fishway) at Q = 0.035 m3/s, × 1/l = 0.66, and Y1/b = 0 in the pool of conventional weir No. 4, obtained from the output results of the software, were used to evaluate the accuracy of the calculation range. As shown in Fig. 4, x1 = the distance from a given weir in the x-direction, Y1 = the water depth measured in the y-direction, Y0 = the vertical distance in the Cartesian coordinate system, h = the water column at the crest, b = the distance between the two points of maximum velocity umax and zero velocity, and l = the pool length.
Fig. 4
The apparent index of convergence (p) in the GCI method is calculated as follows:
�=ln(�3−�2)(�2−�1)/ln(�)
(7)
f1, f2, and f3 are the hydraulic parameters obtained from the numerical simulation (f1 corresponds to the small mesh), and r is the refinement ratio. The following equation defines the convergence index of the fine mesh:
GCIfine=1.25|ε|��−1
(8)
Here, ε = (f2 − f1)/f1 is the relative error, and f2 and f3 are the values of hydraulic parameters considered for medium and small grids, respectively. GCI12 and GCI23 dimensionless indices can be calculated as:
GCI12=1.25|�2−�1�1|��−1
(9)
Then, the independence of the network is preserved. The convergence index of the network parameters obtained by Eqs. (7)–(9) for all three network va