Effects of ramp slope and discharge on hydraulic performance of submerged hump weirs

Effects of ramp slope and discharge on hydraulic performance of submerged hump weirs

Arash Ahmadi a, Amir H. Azimi b

Abstract

험프 웨어는 수위 제어 및 배출 측정을 위한 기존의 수력 구조물 중 하나입니다. 상류 및 하류 경사로의 경사는 자유 및 침수 흐름 조건 모두에서 험프 웨어의 성능에 영향을 미치는 설계 매개변수입니다.

침수된 험프보의 유출 특성 및 수위 변화에 대한 램프 경사 및 유출의 영향을 조사하기 위해 일련의 수치 시뮬레이션이 수행되었습니다. 1V:1H에서 1V:5H까지의 5개 램프 경사를 다양한 업스트림 방전에서 테스트했습니다.

수치모델의 검증을 위해 수치결과를 실험실 데이터와 비교하였다. 수면수위 예측과 유출계수의 시뮬레이션 불일치는 각각 전체 범위의 ±10%와 ±5% 이내였습니다.

모듈 한계 및 방전 감소 계수의 변화에 대한 램프 경사의 영향을 연구했습니다. 험프보의 경사로 경사가 증가함에 따라 상대적으로 높은 침수율에서 모듈러 한계가 발생함을 알 수 있었다.

침수 시작은 방류 수위를 작은 증분으로 조심스럽게 증가시켜 모델링되었으며 그 결과는 모듈 한계의 고전적인 정의와 비교되었습니다. 램프 경사와 방전이 증가함에 따라 모듈러 한계가 증가하는 것으로 밝혀졌지만, 모듈러 한계의 고전적인 정의는 모듈러 한계가 방전과 무관하다는 것을 나타냅니다.

Hump weir 하류의 속도와 와류장은 램프 경사에 의해 제어되는 와류 구조 형성을 나타냅니다. 에너지 손실은 수치 출력으로부터 계산되었으며 정규화된 에너지 손실은 침수에 따라 선형적으로 감소하는 것으로 나타났습니다.

Hump weirs are amongst conventional hydraulic structures for water level control and discharge measurement. The slope in the upstream and downstream ramps is a design parameter that affects the performance of Hump weirs in both free and submerged flow conditions. A series of numerical simulations was performed to investigate the effects of ramp slope and discharge on discharge characteristics and water level variations of submerged Hump weirs. Five ramp slopes ranging from 1V:1H to 1V:5H were tested at different upstream discharges. The numerical results were compared with the laboratory data for verifications of the numerical model. The simulation discrepancies in prediction of water surface level and discharge coefficient were within ±10 % and ±5 % of the full range, respectively. The effects of ramp slope on variations of modular limit and discharge reduction factor were studied. It was found that the modular limit occurred at relatively higher submergence ratios as the ramp slope in Hump weirs increased. The onset of submergence was modeled by carefully increasing tailwater level with small increments and the results were compared with the classic definition of modular limit. It was found that the modular limit increases with increasing the ramp slope and discharge while the classic definition of modular limit indicated that the modular limit is independent of the discharge. The velocity and vortex fields in the downstream of Hump weirs indicated the formation vortex structure, which is controlled by the ramp slope. The energy losses were calculated from the numerical outputs, and it was found that the normalized energy losses decreased linearly with submergence.

Introduction

Weirs have been utilized predominantly for discharge measurement, flow diversion, and water level control in open channels, irrigation canal, and natural streams due to their simplicity of operation and accuracy. Several research studies have been conducted to determine the head-discharge relationship in weirs as one of the most common hydraulic structures for flow measurement (Rajaratnam and Muralidhar, 1969 [[1], [2], [3]]; Vatankhah, 2010, [[4], [5], [6]]; b [[7], [8], [9]]; Azimi and Seyed Hakim, 2019; Salehi et al., 2019; Salehi and Azimi, 2019, [10]. Weirs in general are classified into two major categories named as sharp-crested weirs and weirs of finite-crest length (Rajaratnam and Muralidhar, 1969; [11]. Sharp-crested weirs are typically used for flow measurement in small irrigation canals and laboratory flumes. In contrast, weirs of finite crest length are more suitable for water level control and flow diversion in rivers and natural streams [7,[12], [13], [14]].

The head-discharge relationship in sharp-crested weirs is developed by employing energy equation between two sections in the upstream and downstream of the weir and integration of the velocity profile at the crest of the weir as:

where Qf is the free flow discharge, B is the channel width, g is the acceleration due to gravity, ho is the water head in free-flow condition, and Cd is the discharge coefficient. Rehbock [15] proposed a linear correlation between discharge coefficient and the ratio of water head, ho, and the weir height, P as Cd = 0.605 + 0.08 (ho/P).

Upstream and/or downstream ramp(s) can be added to sharp-crested weirs to enhance the structural stability of the weir. A sharp-crested weir with upstream and/or downstream ramp(s) are known as triangular weirs in the literature. Triangular weirs with both upstream and downstream ramps are also known as Hump weirs and are first introduced in the experimental study of Bazin [16]. The ramps are constructed upstream and downstream of sharp-crested weirs to enhance the weir’s structural integrity and improve the hydraulic performance of the weir. In free-flow condition, the discharge coefficient of Hump weirs increases with increasing downstream ramp slope but decreases as upstream ramp slope increases (Azimi et al., 2013).

The hydraulic performance of weirs is evaluated in both free and submerged flow conditions. In free flow condition, water freely flows over weirs since the downstream water level is lower than that of the crest level of the weir. Channel blockage or flood in the downstream of weirs can raise the tailwater level, t. As tailwater passes the crest elevation in sharp-crested weirs, the upstream flow decelerates due to the excess pressure force in the downstream and the upstream water level increases. The onset of water level raise due to tailwater raise is called the modular limit. Once the tailwater level passes the modular limit, the weir is submerged. In sharp-crested weirs, the submerged flow regime may occur even before the tailwater reaches the crest elevation [8,14], whereas, in weirs of finite crest length, the upstream water level remains unchanged even if the tailwater raises above the crest elevation and it normally causes submergence once the tailwater level passes the critical depth at the crest of the weir [7,17]. The degree of submergence can be estimated by careful observation of the water surface profile. Observations of water surface at different submergence levels indicated two distinct flow patterns in submerged sharp-crested weirs that was initially classified as impinging jet and surface flow regimes [14]. [8] analyzed the variations of water surface profiles over submerged sharp-crested weirs with different submergence ratios and defined four distinct regimes of impinging jet, surface jump, surface wave, and surface jet.

[18] characterized the onset of submergence by defining the modular limit as a stage when the free flow head increases by +1 mm due to tailwater rise. The definition of modular limit is somewhat arbitrary, and it is difficult to identify for large discharges because the upstream water surface begins to fluctuate. This definition did not consider the effects of channel and weir geometries. The experimental data in triangular weirs and weirs finite-crest length with upstream and downstream ramp(s) revealed that the modular limit varied with the ratio of the free-flow head to the total streamwise length of the weir [17]. Weirs of finite crest length with upstream and downstream ramps are known as embankment weirs in literature [1,19,20] and Azimi et al., 2013) [19]. conducted two series of laboratory experiments to study the hydraulics of submerged embankment weirs with the upstream and downstream ramps of 1V:1H and 1V:2H. Empirical correlations were proposed to directly estimate the flow discharge in submerged embankment weirs for t/h > 0.7 where h is the water head in submerged flow condition. He found that the free flow discharge is a function of upstream water head, but the submerged discharge is a function of submergence level, t/h [21]. studied the hydraulics of four embankment weirs with different weir heights ranging from 0.09 m to 0.36 m. It was found that submerged embankments with a higher ho/P, where P is the height of the weir, have a smaller discharge reduction due to submergence. Effects of crest length in embankment weirs with both upstream and downstream ramps of 1V:2H was studied in both free and submerged flow conditions [1]. It was found that the modular limit in submerged embankment weirs decreased linearly with the relative crest length, Ho/(Ho + L), where Ho is the total head and L is the crest length.

In submerged flow condition, the performance of weirs is quantified by the discharge reduction factor, ψ, which is a ratio of the submerged discharge, Qs, to the corresponding free-flow discharge, Qf, based on the upstream head, h [12]. In submerged-flow conditions, flow discharge can be estimated as:��=���

[1] proposed a formula to predict ψ that could be used for embankment weirs with different crest lengths ranging from 0 to 0.3 m as:�=(1−��)�where n is an exponent varying from 4 to 7 and Yt is the normalized submergence defined as:��=�ℎ−[0.85−(0.5��+�)]1−[0.85−(0.5��+�)]where H is the total upstream head in submerged-flow conditions [7]. proposed a simpler formula to predict ψ for weirs of finite-crest length as:�=[1−(�ℎ)�]�where m and n are exponents varying for different types of weirs. Hakim and Azimi (2017) employed regression analysis to propose values of n = 0.25 and m = 0.28 (ho/L)−2.425 for triangular weirs.

The discharge capacity of weirs decreases in submerged flow condition and the onset of submergence occurs at the modular limit. Therefore, the determination of modular limit in weirs with different geometries is critical to understanding the sensitivity of a particular weir model with tailwater level variations. The available definition of modular limit as when head water raises by +1 mm due to tailwater rise does not consider the effects of channel and weir geometries. Therefore, a new and more accurate definition of modular limit is proposed in this study to consider the effect of other geometry and approaching flow parameters. The second objective of this study is to evaluate the effects of upstream and downstream ramps and ramps slopes on the hydraulic performance of submerged Hump weirs. The flow patterns, velocity distributions, and energy dissipation rates were extracted from validated numerical data to better understand the discharge reduction mechanism in Hump weirs in both free and submerged flow conditions.

Section snippets

Governing equations

Numerical simulation has been employed as an efficient and effective method to analyze free surface flow problems and in particular investigating on the hydraulics of flow over weirs [22]. The weir models were developed in numerical domain and the water pressure and velocity field were simulated by employing the FLOW-3D solver (Flow Science, Inc., Santa Fe, USA). The numerical results were validated with the laboratory measurements and the effects of ramps slopes on the performance of Hump

Verification of numerical model

The experimental observations of Bazin [16,17] were used for model validation in free and submerged flow conditions, respectively. The weir height in the study of Bazin was P = 0.5 m and two ramp slopes of 1V:1H and 1V:2H were tested. The bed and sides of the channel were made of glass, and the roughness distribution of the bed and walls were uniform. The Hump weir models in the study of Seyed Hakim and Azimi (2017) had a weir height of 0.076 m and ramp slopes of 1V:2H in both upstream and

Conclusions

A series of numerical simulations was performed to study the hydraulics and velocity pattern downstream of a Hump weir with symmetrical ramp slopes. Effects of ramp slope and discharge on formation of modular limit and in submerged flow condition were tested by conducting a series of numerical simulations on Hump weirs with ramp slopes varying from 1V:1H to 1V:5H. A comparison between numerical results and experimental data indicated that the proposed numerical model is accurate with a mean

Author contributions

Arash Ahmadi: Software, Validation, Visualization, Writing – original draft. Amir Azimi: Conceptualization, Funding acquisition, Investigation, Project administration, Supervision, Writing – review & editing

Uncited References

[30]; [31]; [32]; [33].

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.

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Fig. 9 From: An Investigation on Hydraulic Aspects of Rectangular Labyrinth Pool and Weir Fishway Using FLOW-3D

An Investigation on Hydraulic Aspects of Rectangular Labyrinth Pool and Weir Fishway Using FLOW-3D

Abstract

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 [12]. 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].

figure 1
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 [79]. 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.

figure 2
Fig. 2

Table 1 Experimental conditions considered for calibration

Full size table

2.2 Numerical Models

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.

figure 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 (xyzt). 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) [413]. 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 (xyz) 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 (uvw) are the velocity components, (AxAyAz) are the flow area components, (Gx, Gy, Gz) are the mass accelerations, and (fxfyfz) are the viscous accelerations in the directions (xyz), 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 kε 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 kε 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]:

∂(��)∂�+∂(����)∂��=∂∂��[������∂�∂��]+��−�ε

(5)

∂(�ε)∂�+∂(�ε��)∂��=∂∂��[�ε�eff∂ε∂��]+�1εε��k−�2ε�ε2�

(6)

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

Full size table

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. 4x1 = 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.

figure 4
Fig. 4

The apparent index of convergence (p) in the GCI method is calculated as follows:

�=ln⁡(�3−�2)(�2−�1)/ln⁡(�)

(7)

f1f2, 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 variables is shown in Table 3. Since the GCI values for the smaller grid (GCI12) are lower compared to coarse grid (GCI23), it can be concluded that the independence of the grid is almost achieved. No further change in the grid size of the solution domain is required. The calculated values (GCI23/rpGCI12) are close to 1, which shows that the numerical results obtained are within the convergence range. As a result, the meshing of the solution domain consisting of a block mesh with a mesh size of 0.012 m and a block mesh within a larger block mesh with a mesh size of 0.009 m was selected as the optimal mesh (Fig. 5).Table 3 GCI calculation

Full size table

figure 5
Fig. 5

The boundary conditions applied to the area are shown in Fig. 6. The boundary condition of specific flow rate (volume flow rate-Q) was used for the inlet of the flow. For the downstream boundary, the flow output (outflow-O) condition did not affect the flow in the solution area. For the Zmax boundary, the specified pressure boundary condition was used along with the fluid fraction = 0 (P). This type of boundary condition considers free surface or atmospheric pressure conditions (Ghaderi et al. [19]). The wall boundary condition is defined for the bottom of the channel, which acts like a virtual wall without friction (W). The boundary between mesh blocks and walls were considered a symmetrical condition (S).

figure 6
Fig. 6

The convergence of the steady-state solutions was controlled during the simulations by monitoring the changes in discharge at the inlet boundary conditions. Figure 7 shows the time series plots of the discharge obtained from the Model A for the three main discharges from the numerical results. The 8 s to reach the flow equilibrium is suitable for the case of the fish ladder with pool and weir. Almost all discharge fluctuations in the models are insignificant in time, and the flow has reached relative stability. The computation time for the simulations was between 6 and 8 h using a personal computer with eight cores of a CPU (Intel Core i7-7700K @ 4.20 GHz and 16 GB RAM).

figure 7
Fig. 7

3 Results

3.1 Verification of Numerical Results

Quantitative outcomes, including free surface and normalized velocity profiles obtained using FLOW-3D software, were reviewed and compared with the results of Ead et al. [6]. The fourth pool was selected to present the results and compare the experiment and simulation. For each quantity, the percentage of mean absolute error (MAPE (%)) and root-mean-square error (RMSE) are calculated. Equations (10) and (11) show the method used to calculate the errors.

MAPE(%)100×1�∑1�|�exp−�num�exp|

(10)

RMSE(−)1�∑1�(�exp−�num)2

(11)

Here, Xexp is the value of the laboratory data, Xnum is the numerical data value, and n is the amount of data. As shown in Fig. 8, let x1 = distance from a given weir in the x-direction and Y1 = water depth in the y-direction from the bottom. The trend of the surface profiles for each of the numerical results is the same as that of the laboratory results. The surface profiles of the plunging flows drop after the flow enters and then rises to approach the next weir. The RMSE and MAPE error values for Model A are 0.014 m and 3.11%, respectively, indicating acceptable agreement between numerical and laboratory results. Figure 9 shows the velocity vectors and plunging flow from the numerical results, where x and y are horizontal and vertical to the flow direction, respectively. It can be seen that the jet in the fish ladder pool has a relatively high velocity. The two vortices, i.e., the enclosed vortex rotating clockwise behind the weir and the surface vortex rotating counterclockwise above the jet, are observed for the regime of incident flow. The point where the jet meets the fish passage bed is shown in the figure. The normalized velocity profiles upstream and downstream of the impact points are shown in Fig. 10. The figure shows that the numerical results agree well with the experimental data of Ead et al. [6].

figure 8
Fig. 8
figure 9
Fig. 9
figure 10
Fig. 10

3.2 Flow Regime and Discharge-Depth Relationship

Depending on the geometric shape of the fishway, including the distance of the weir, the slope of the bottom, the height of the weir, and the flow conditions, the flow regime in the fishway is divided into three categories: dipping, transitional, and flow regimes [4]. In the plunging flow regime, the flow enters the pool through the weir, impacts the bottom of the fishway, and forms a hydraulic jump causing two eddies [220]. In the streamwise flow regime, the surface of the flow passing over the weir is almost parallel to the bottom of the channel. The transitional regime has intermediate flow characteristics between the submerged and flow regimes. To predict the flow regime created in the fishway, Ead et al. [6] proposed two dimensionless parameters, Qt* and L/w, where Qt* is the dimensionless discharge, L is the distance between weirs, and w is the height of the weir:

��∗=���0���

(12)

Q is the total discharge, B is the width of the channel, S0 is the slope of the bed, and g is the gravity acceleration. Figure 11 shows different ranges for each flow regime based on the slope of the bed and the distance between the pools in this study. The results of Baki et al. [21], Ead et al. [6] and Dizabadi et al. [22] were used for this comparison. The distance between the pools affects the changes in the regime of the fish ladder. So, if you decrease the distance between weirs, the flow regime more likely becomes. This study determined all three flow regimes in a fish ladder. When the corresponding range of Qt* is less than 0.6, the flow regime can dip at values of L/B = 1.83. If the corresponding range of Qt* is greater than 0.5, transitional flow may occur at L/B = 1.22. On the other hand, when Qt* is greater than 1, streamwise flow can occur at values of L/B = 0.61. These observations agree well with the results of Baki et al. [21], Ead et al. [6] and Dizabadi et al. [22].

figure 11
Fig. 11

For plunging flows, another dimensionless discharge (Q+) versus h/w given by Ead et al. [6] was used for further evaluation:

�+=��ℎ�ℎ=23�d�

(13)

where h is the water depth above the weir, and Cd is the discharge coefficient. Figure 12a compares the numerical and experimental results of Ead et al. [6]. In this figure, Rehbock’s empirical equation is used to estimate the discharge coefficient of Ead et al. [6].

�d=0.57+0.075ℎ�

(14)

figure 12
Fig. 12

The numerical results for the conventional weir (Model A) and the rectangular labyrinth weir (Model B) of this study agree well with the laboratory results of Ead et al. [6]. When comparing models A and B, it is also found that a rectangular labyrinth weir has larger Q + values than the conventional weir as the length of the weir crest increases for a given channel width and fixed headwater elevation. In Fig. 12b, Models A and B’s flow depth plot shows the plunging flow regime. The power trend lines drawn through the data are the best-fit lines. The data shown in Fig. 12b are for different bed slopes and weir geometries. For the conventional weir and the rectangular labyrinth weir at submerged flow, Q can be assumed to be proportional to 1.56 and 1.47h, respectively. In the results of Ead et al. [6], Q is proportional to 1.5h. If we assume that the flow through the orifice is Qo and the total outflow is Q, the change in the ratio of Qo/Q to total outflow for models A and B can be shown in Fig. 13. For both models, the flow through the orifice decreases as the total flow increases. A logarithmic trend line was also found between the total outflow and the dimensionless ratio Qo/Q.

figure 13
Fig. 13

3.3 Depth-Averaged Velocity Distributions

To ensure that the target fish species can pass the fish ladder with maximum efficiency, the average velocity in the fish ladder should be low enough [4]. Therefore, the average velocity in depth should be as much as possible below the critical swimming velocities of the target fishes at a constant flow depth in the pool [20]. The contour plot of depth-averaged velocity was used instead of another direction, such as longitudinal velocity because fish are more sensitive to depth-averaged flow velocity than to its direction under different hydraulic conditions. Figure 14 shows the distribution of depth-averaged velocity in the pool for Models A and B in two cases with and without orifice plates. Model A’s velocity within the pool differs slightly in the spanwise direction. However, no significant variation in velocity was observed. The flow is gradually directed to the sides as it passes through the rectangular labyrinth weir. This increases the velocity at the sides of the channel. Therefore, the high-velocity zone is located at the sides. The low velocity is in the downstream apex of the weir. This area may be suitable for swimming target fish. The presence of an opening in the weir increases the flow velocity at the opening and in the pool’s center, especially in Model A. The flow velocity increase caused by the models’ opening varied from 7.7 to 12.48%. Figure 15 illustrates the effect of the inverted slope on the averaged depth velocity distribution in the pool at low and high discharge. At constant discharge, flow velocity increases with increasing bed slope. In general, high flow velocity was found in the weir toe sidewall and the weir and channel sidewalls.

figure 14
Fig. 14
figure 15
Fig. 15

On the other hand, for a constant bed slope, the high-velocity area of the pool increases due to the increase in runoff. For both bed slopes and different discharges, the most appropriate path for fish to travel from upstream to downstream is through the middle of the cross section and along the top of the rectangular labyrinth weirs. The maximum dominant velocities for Model B at S0 = 5% were 0.83 and 1.01 m/s; at S0 = 10%, they were 1.12 and 1.61 m/s at low and high flows, respectively. The low mean velocities for the same distance and S0 = 5 and 10% were 0.17 and 0.26 m/s, respectively.

Figure 16 shows the contour of the averaged depth velocity for various distances from the weir at low and high discharge. The contour plot shows a large variation in velocity within short distances from the weir. At L/B = 0.61, velocities are low upstream and downstream of the top of the weir. The high velocities occur in the side walls of the weir and the channel. At L/B = 1.22, the low-velocity zone displaces the higher velocity in most of the pool. Higher velocities were found only on the sides of the channel. As the discharge increases, the velocity zone in the pool becomes wider. At L/B = 1.83, there is an area of higher velocities only upstream of the crest and on the sides of the weir. At high discharge, the prevailing maximum velocities for L/B = 0.61, 1.22, and 1.83 were 1.46, 1.65, and 1.84 m/s, respectively. As the distance between weirs increases, the range of maximum velocity increases.

figure 16
Fig. 16

On the other hand, the low mean velocity for these distances was 0.27, 0.44, and 0.72 m/s, respectively. Thus, the low-velocity zone decreases with increasing distance between weirs. Figure 17 shows the pattern distribution of streamlines along with the velocity contour at various distances from the weir for Q = 0.05 m3/s. A stream-like flow is generally formed in the pool at a small distance between weirs (L/B = 0.61). The rotation cell under the jet forms clockwise between the two weirs. At the distances between the spillways (L/B = 1.22), the transition regime of the flow is formed. The transition regime occurs when or shortly after the weir is flooded. The rotation cell under the jet is clockwise smaller than the flow regime and larger than the submergence regime. At a distance L/B = 1.83, a plunging flow is formed so that the plunging jet dips into the pool and extends downstream to the center of the pool. The clockwise rotation of the cell is bounded by the dipping jet of the weir and is located between the bottom and the side walls of the weir and the channel.

figure 17
Fig. 17

Figure 18 shows the average depth velocity bar graph for each weir at different bed slopes and with and without orifice plates. As the distance between weirs increases, all models’ average depth velocity increases. As the slope of the bottom increases and an orifice plate is present, the average depth velocity in the pool increases. In addition, the average pool depth velocity increases as the discharge increases. Among the models, Model A’s average depth velocity is higher than Model B’s. The variation in velocity ranged from 8.11 to 12.24% for the models without an orifice plate and from 10.26 to 16.87% for the models with an orifice plate.

figure 18
Fig. 18

3.4 Turbulence Characteristics

The turbulent kinetic energy is one of the important parameters reflecting the turbulent properties of the flow field [23]. When the k value is high, more energy and a longer transit time are required to migrate the target species. The turbulent kinetic energy is defined as follows:

�=12(�x′2+�y′2+�z′2)

(15)

where uxuy, and uz are fluctuating velocities in the xy, and z directions, respectively. An illustration of the TKE and the effects of the geometric arrangement of the weir and the presence of an opening in the weir is shown in Fig. 19. For a given bed slope, in Model A, the highest TKE values are uniformly distributed in the weir’s upstream portion in the channel’s cross section. In contrast, for the rectangular labyrinth weir (Model B), the highest TKE values are concentrated on the sides of the pool between the crest of the weir and the channel wall. The highest TKE value in Models A and B is 0.224 and 0.278 J/kg, respectively, at the highest bottom slope (S0 = 10%). In the downstream portion of the conventional weir and within the crest of the weir and the walls of the rectangular labyrinth, there was a much lower TKE value that provided the best conditions for fish to recover in the pool between the weirs. The average of the lowest TKE for bottom slopes of 5 and 10% in Model A is 0.041 and 0.056 J/kg, and for Model B, is 0.047 and 0.064 J/kg. The presence of an opening in the weirs reduces the area of the highest TKE within the pool. It also increases the resting areas for fish (lower TKE). The highest TKE at the highest bottom slope in Models A and B with an orifice is 0.208 and 0.191 J/kg, respectively.

figure 19
Fig. 19

Figure 20 shows the effect of slope on the longitudinal distribution of TKE in the pools. TKE values significantly increase for a given discharge with an increasing bottom slope. Thus, for a low bed slope (S0 = 5%), a large pool area has expanded with average values of 0.131 and 0.168 J/kg for low and high discharge, respectively. For a bed slope of S0 = 10%, the average TKE values are 0.176 and 0.234 J/kg. Furthermore, as the discharge increases, the area with high TKE values within the pool increases. Lower TKE values are observed at the apex 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 for both bottom slopes. The effect of distance between weirs on TKE is shown in Fig. 21. Low TKE values were observed at low discharge and short distances between weirs. Low TKE values are located at the top of the rectangular labyrinth weir and the downstream corner of the weir wall. There is a maximum value of TKE at the large distances between weirs, L/B = 1.83, along the center line of the pool, where the dip jet meets the bottom of the bed. At high discharge, the maximum TKE value for the distance L/B = 0.61, 1.22, and 1.83 was 0.246, 0.322, and 0.417 J/kg, respectively. In addition, the maximum TKE range increases with the distance between weirs.

figure 20
Fig. 20
figure 21
Fig. 21

For TKE size, the average value (TKEave) is plotted against q in Fig. 22. For all models, the TKE values increase with increasing q. For example, in models A and B with L/B = 0.61 and a slope of 10%, the TKE value increases by 41.66 and 86.95%, respectively, as q increases from 0.1 to 0.27 m2/s. The TKE values in Model B are higher than Model A for a given discharge, bed slope, and weir distance. The TKEave in Model B is higher compared to Model A, ranging from 31.46 to 57.94%. The presence of an orifice in the weir reduces the TKE values in both weirs. The intensity of the reduction is greater in Model B. For example, in Models A and B with L/B = 0.61 and q = 0.1 m2/s, an orifice reduces TKEave values by 60.35 and 19.04%, respectively. For each model, increasing the bed slope increases the TKEave values in the pool. For example, for Model B with q = 0.18 m2/s, increasing the bed slope from 5 to 10% increases the TKEave value by 14.34%. Increasing the distance between weirs increases the TKEave values in the pool. For example, in Model B with S0 = 10% and q = 0.3 m2/s, the TKEave in the pool increases by 34.22% if you increase the distance between weirs from L/B = 0.61 to L/B = 0.183.

figure 22
Fig. 22

Cotel et al. [24] suggested that turbulence intensity (TI) is a suitable parameter for studying fish swimming performance. Figure 23 shows the plot of TI and the effects of the geometric arrangement of the weir and the presence of an orifice. In Model A, the highest TI values are found upstream of the weirs and are evenly distributed across the cross section of the channel. The TI values increase as you move upstream to downstream in the pool. For the rectangular labyrinth weir, the highest TI values were concentrated on the sides of the pool, between the top of the weir and the side wall of the channel, and along the top of the weir. Downstream of the conventional weir, within the apex of the weir, and at the corners of the walls of the rectangular labyrinth weir, the percentage of TI was low. At the highest discharge, the average range of TI in Models A and B was 24–45% and 15–62%, respectively. The diversity of TI is greater in the rectangular labyrinth weir than the conventional weir. Fish swimming performance is reduced due to higher turbulence intensity. However, fish species may prefer different disturbance intensities depending on their swimming abilities; for example, Salmo trutta prefers a disturbance intensity of 18–53% [25]. Kupferschmidt and Zhu [26] found a higher range of TI for fishways, such as natural rock weirs, of 40–60%. The presence of an orifice in the weir increases TI values within the pool, especially along the middle portion of the cross section of the fishway. With an orifice in the weir, the average range of TI in Models A and B was 28–59% and 22–73%, respectively.

figure 23
Fig. 23

The effect of bed slope on TI variation is shown in Fig. 24. TI increases in different pool areas as the bed slope increases for a given discharge. For a low bed slope (S0 = 5%), a large pool area has increased from 38 to 63% and from 56 to 71% for low and high discharge, respectively. For a bed slope of S0 = 10%, the average values of TI are 45–67% and 61–73% for low and high discharge, respectively. Therefore, as runoff increases, the area with high TI values within the pool increases. A lower TI is observed for both bottom slopes in the corner of the wall, downstream of the crest walls, and between the side walls in the weir and channel. Figure 25 compares weir spacing with the distribution of TI values within the pool. The TI values are low at low flows and short distances between weirs. A maximum value of TI occurs at long spacing and where the plunging stream impinges on the bed and the area around the bed. TI ranges from 36 to 57%, 58–72%, and 47–76% for the highest flow in a wide pool area for L/B = 0.61, 1.22, and 1.83, respectively.

figure 24
Fig. 24
figure 25
Fig. 25

The average value of turbulence intensity (TIave) is plotted against q in Fig. 26. The increase in TI values with the increase in q values is seen in all models. For example, the average values of TI for Models A and B at L/B = 0.61 and slope of 10% increased from 23.9 to 33.5% and from 42 to 51.8%, respectively, with the increase in q from 0.1 to 0.27 m2/s. For a given discharge, a given gradient, and a given spacing of weirs, the TIave is higher in Model B than Model A. The presence of an orifice in the weirs increases the TI values in both types. For example, in Models A and B with L/B = 0.61 and q = 0.1 m2/s, the presence of an orifice increases TIave from 23.9 to 37.1% and from 42 to 48.8%, respectively. For each model, TIave in the pool increases with increasing bed slope. For Model B with q = 0.18 m2/s, TIave increases from 37.5 to 45.8% when you increase the invert slope from 5 to 10%. Increasing the distance between weirs increases the TIave in the pool. In Model B with S0 = 10% and q = 0.3 m2/s, the TIave in the pool increases from 51.8 to 63.7% as the distance between weirs increases from L/B = 0.61 to L/B = 0.183.

figure 26
Fig. 26

3.5 Energy Dissipation

To facilitate the passage of various target species through the pool of fishways, it is necessary to pay attention to the energy dissipation of the flow and to keep the flow velocity in the pool slow. The average volumetric energy dissipation (k) in the pool is calculated using the following basic formula:

�=����0��

(16)

where ρ is the water density, and H is the average water depth of the pool. The change in k versus Q for all models at two bottom slopes, S0 = 5%, and S0 = 10%, is shown in Fig. 27. Like the results of Yagci [8] and Kupferschmidt and Zhu [26], at a constant bottom slope, the energy dissipation in the pool increases with increasing discharge. The trend of change in k as a function of Q from the present study at a bottom gradient of S0 = 5% is also consistent with the results of Kupferschmidt and Zhu [26] for the fishway with rock weir. The only difference between the results is the geometry of the fishway and the combination of boulders instead of a solid wall. Comparison of the models shows that the conventional model has lower energy dissipation than the rectangular labyrinth for a given discharge. Also, increasing the distance between weirs decreases the volumetric energy dissipation for each model with the same bed slope. Increasing the slope of the bottom leads to an increase in volumetric energy dissipation, and an opening in the weir leads to a decrease in volumetric energy dissipation for both models. Therefore, as a guideline for volumetric energy dissipation, if the value within the pool is too high, the increased distance of the weir, the decreased slope of the bed, or the creation of an opening in the weir would decrease the volumetric dissipation rate.

figure 27
Fig. 27

To evaluate the energy dissipation inside the pool, the general method of energy difference in two sections can use:

ε=�1−�2�1

(17)

where ε is the energy dissipation rate, and E1 and E2 are the specific energies in Sects. 1 and 2, respectively. The distance between Sects. 1 and 2 is the same. (L is the distance between two upstream and downstream weirs.) Figure 28 shows the changes in ε relative to q (flow per unit width). The rectangular labyrinth weir (Model B) has a higher energy dissipation rate than the conventional weir (Model A) at a constant bottom gradient. For example, at S0 = 5%, L/B = 0.61, and q = 0.08 m3/s.m, the energy dissipation rate in Model A (conventional weir) was 0.261. In Model B (rectangular labyrinth weir), however, it was 0.338 (22.75% increase). For each model, the energy dissipation rate within the pool increases as the slope of the bottom increases. For Model B with L/B = 1.83 and q = 0.178 m3/s.m, the energy dissipation rate at S0 = 5% and 10% is 0.305 and 0.358, respectively (14.8% increase). Figure 29 shows an orifice’s effect on the pools’ energy dissipation rate. With an orifice in the weir, both models’ energy dissipation rates decreased. Thus, the reduction in energy dissipation rate varied from 7.32 to 9.48% for Model A and from 8.46 to 10.57 for Model B.

figure 28
Fig. 28
figure 29
Fig. 29

4 Discussion

This study consisted of entirely of numerical analysis. Although this study was limited to two weirs, the hydraulic performance and flow characteristics in a pooled fishway are highlighted by the rectangular labyrinth weir and its comparison with the conventional straight weir. The study compared the numerical simulations with laboratory experiments in terms of surface profiles, velocity vectors, and flow characteristics in a fish ladder pool. The results indicate agreement between the numerical and laboratory data, supporting the reliability of the numerical model in capturing the observed phenomena.

When the configuration of the weir changes to a rectangular labyrinth weir, the flow characteristics, the maximum and minimum area, and even the location of each hydraulic parameter change compared to a conventional weir. In the rectangular labyrinth weir, the flow is gradually directed to the sides as it passes the weir. This increases the velocity at the sides of the channel [21]. Therefore, the high-velocity area is located on the sides. In the downstream apex of the weir, the flow velocity is low, and this area may be suitable for swimming target fish. However, no significant change in velocity was observed at the conventional weir within the fish ladder. This resulted in an average increase in TKE of 32% and an average increase in TI of about 17% compared to conventional weirs.

In addition, there is a slight difference in the flow regime for both weir configurations. In addition, the rectangular labyrinth weir has a higher energy dissipation rate for a given discharge and constant bottom slope than the conventional weir. By reducing the distance between the weirs, this becomes even more intense. Finally, the presence of an orifice in both configurations of the weir increased the flow velocity at the orifice and in the middle of the pool, reducing the highest TKE value and increasing the values of TI within the pool of the fish ladder. This resulted in a reduction in volumetric energy dissipation for both weir configurations.

The results of this study will help the reader understand the direct effects of the governing geometric parameters on the hydraulic characteristics of a fishway with a pool and weir. However, due to the limited configurations of the study, further investigation is needed to evaluate the position of the weir’s crest on the flow direction and the difference in flow characteristics when combining boulders instead of a solid wall for this type of labyrinth weir [26]. In addition, hydraulic engineers and biologists must work together to design an effective fishway with rectangular labyrinth configurations. The migration habits of the target species should be considered when designing the most appropriate design [27]. Parametric studies and field observations are recommended to determine the perfect design criteria.

The current study focused on comparing a rectangular labyrinth weir with a conventional straight weir. Further research can explore other weir configurations, such as variations in crest position, different shapes of labyrinth weirs, or the use of boulders instead of solid walls. This would help understand the influence of different geometric parameters on hydraulic characteristics.

5 Conclusions

A new layout of the weir was evaluated, namely a rectangular labyrinth weir compared to a straight weir in a pool and weir system. The differences between the weirs were highlighted, particularly how variations in the geometry of the structures, such as the shape of the weir, the spacing of the weir, the presence of an opening at the weir, and the slope of the bottom, affect the hydraulics within the structures. The main findings of this study are as follows:

  • The calculated dimensionless discharge (Qt*) confirmed three different flow regimes: when the corresponding range of Qt* is smaller than 0.6, the regime of plunging flow occurs for values of L/B = 1.83. (L: distance of the weir; B: channel width). When the corresponding range of Qt* is greater than 0.5, transitional flow occurs at L/B = 1.22. On the other hand, if Qt* is greater than 1, the streaming flow is at values of L/B = 0.61.
  • For the conventional weir and the rectangular labyrinth weir with the plunging flow, it can be assumed that the discharge (Q) is proportional to 1.56 and 1.47h, respectively (h: water depth above the weir). This information is useful for estimating the discharge based on water depth in practical applications.
  • In the rectangular labyrinth weir, the high-velocity zone is located on the side walls between the top of the weir and the channel wall. A high-velocity variation within short distances of the weir. Low velocity occurs within the downstream apex of the weir. This area may be suitable for swimming target fish.
  • As the distance between weirs increased, the zone of maximum velocity increased. However, the zone of low speed decreased. The prevailing maximum velocity for a rectangular labyrinth weir at L/B = 0.61, 1.22, and 1.83 was 1.46, 1.65, and 1.84 m/s, respectively. The low mean velocities for these distances were 0.27, 0.44, and 0.72 m/s, respectively. This finding highlights the importance of weir spacing in determining the flow characteristics within the fishway.
  • The presence of an orifice in the weir increased the flow velocity at the orifice and in the middle of the pool, especially in a conventional weir. The increase ranged from 7.7 to 12.48%.
  • For a given bottom slope, in a conventional weir, the highest values of turbulent kinetic energy (TKE) are uniformly distributed in the upstream part of the weir in the cross section of the channel. In contrast, for the rectangular labyrinth weir, the highest TKE values were concentrated on the sides of the pool between the crest of the weir and the channel wall. The highest TKE value for the conventional and the rectangular labyrinth weir was 0.224 and 0.278 J/kg, respectively, at the highest bottom slope (S0 = 10%).
  • For a given discharge, bottom slope, and weir spacing, the average values of TI are higher for the rectangular labyrinth weir than for the conventional weir. At the highest discharge, the average range of turbulence intensity (TI) for the conventional and rectangular labyrinth weirs was between 24 and 45% and 15% and 62%, respectively. This reveals that the rectangular labyrinth weir may generate more turbulent flow conditions within the fishway.
  • For a given discharge and constant bottom slope, the rectangular labyrinth weir has a higher energy dissipation rate than the conventional weir (22.75 and 34.86%).
  • Increasing the distance between weirs decreased volumetric energy dissipation. However, increasing the gradient increased volumetric energy dissipation. The presence of an opening in the weir resulted in a decrease in volumetric energy dissipation for both model types.

Availability of data and materials

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비선형 파력의 영향에 따른 잔해 언덕 방파제 형상의 효과에 대한 수치 분석

비선형 파력의 영향에 따른 잔해 언덕 방파제 형상의 효과에 대한 수치 분석

Numerical Analysis of the Effects of Rubble Mound Breakwater Geometry Under the Effect of Nonlinear Wave Force

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Abstract

Assessing the interaction of waves and porous offshore structures such as rubble mound breakwaters plays a critical role in designing such structures optimally. This study focused on the effect of the geometric parameters of a sloped rubble mound breakwater, including the shape of the armour, method of its arrangement, and the breakwater slope. Thus, three main design criteria, including the wave reflection coefficient (Kr), transmission coefficient (Kt), and depreciation wave energy coefficient (Kd), are discussed. Based on the results, a decrease in wavelength reduced the Kr and increased the Kt and Kd. The rubble mound breakwater with the Coreloc armour layer could exhibit the lowest Kr compared to other armour geometries. In addition, a decrease in the breakwater slope reduced the Kr and Kd by 3.4 and 1.25%, respectively. In addition, a decrease in the breakwater slope from 33 to 25° increased the wave breaking height by 6.1% on average. Further, a decrease in the breakwater slope reduced the intensity of turbulence depreciation. Finally, the armour geometry and arrangement of armour layers on the breakwater with its different slopes affect the wave behaviour and interaction between the wave and breakwater. Thus, layering on the breakwater and the correct use of the geometric shapes of the armour should be considered when designing such structures.

파도와 잔해 더미 방파제와 같은 다공성 해양 구조물의 상호 작용을 평가하는 것은 이러한 구조물을 최적으로 설계하는 데 중요한 역할을 합니다. 본 연구는 경사진 잔해 둔덕 방파제의 기하학적 매개변수의 효과에 초점을 맞추었는데, 여기에는 갑옷의 형태, 배치 방법, 방파제 경사 등이 포함된다. 따라서 파동 반사 계수(Kr), 투과 계수(Kt) 및 감가상각파 에너지 계수(Kd)에 대해 논의합니다. 결과에 따르면 파장이 감소하면 K가 감소합니다.r그리고 K를 증가시켰습니다t 및 Kd. Coreloc 장갑 층이 있는 잔해 언덕 방파제는 가장 낮은 K를 나타낼 수 있습니다.r 다른 갑옷 형상과 비교했습니다. 또한 방파제 경사가 감소하여 K가 감소했습니다.r 및 Kd 각각 3.4%, 1.25% 증가했다. 또한 방파제 경사가 33°에서 25°로 감소하여 파도 파쇄 높이가 평균 6.1% 증가했습니다. 또한, 방파제 경사의 감소는 난류 감가상각의 강도를 감소시켰다. 마지막으로, 경사가 다른 방파제의 장갑 형상과 장갑 층의 배열은 파도 거동과 파도와 방파제 사이의 상호 작용에 영향을 미칩니다. 따라서 이러한 구조를 설계 할 때 방파제에 층을 쌓고 갑옷의 기하학적 모양을 올바르게 사용하는 것을 고려해야합니다.

Keywords

  • Rubble mound breakwater
  • Computational fluid dynamics
  • Armour layer
  • Wave reflection coefficient
  • Wave transmission coefficient
  • Wave energy dissipation coefficient

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Figure 4. Rectangular stepped spillway with (a) three baffle arrangement (b) five baffle arrangement

Prediction of Energy Dissipation over Stepped Spillwaywith Baffles Using Machine Learning Techniques

Saurabh Pujari*
, Vijay Kaushik, S. Anbu Kumar
Department of Civil Engineering, Delhi Technological University, India
Received February 23, 2023; Revised April 25, 2023; Accepted June 11, 2023
Cite This Paper in the Following Citation Styles
(a): [1] Saurabh Pujari, Vijay Kaushik, S. Anbu Kumar , “Prediction of Energy Dissipation over Stepped Spillway with
Baffles Using Machine Learning Techniques,” Civil Engineering and Architecture, Vol. 11, No. 5, pp. 2377 – 2391, 2023.
DOI: 10.13189/cea.2023.110510.
(b): Saurabh Pujari, Vijay Kaushik, S. Anbu Kumar (2023). Prediction of Energy Dissipation over Stepped Spillway with
Baffles Using Machine Learning Techniques. Civil Engineering and Architecture, 11(5), 2377 – 2391. DOI:
10.13189/cea.2023.110510.
Copyright©2023 by authors, all rights reserved. Authors agree that this article remains permanently open access under
the terms of the Creative Commons Attribution License 4.0 International License

Abstract

In river engineering, the stepped spillway of a dam is an important component that may be used in various ways. It is necessary to conduct research dealing with flood control in order to investigate the method, in which energy is lost along the tiered spillways. In the past, several research projects on stepped spillways without baffles have been carried out utilizing a range of research approaches. In the present study, machine learning techniques such as Support Vector Machine (SVM) and Regression Tree (RT) are used to analyze the energy dissipation on rectangular stepped spillways that make use of baffles in a variety of configurations and at a range of channel slopes. The results of many experiments indicate that the amount of energy that is lost increases with the number of baffles that are present in flat channels with slopes and rises. In order to evaluate the efficiency and usefulness of the suggested model, the statistical indices that were developed for the experimental research are used to validate the models that were created for the study. The findings indicate that the suggested SVM model properly predicted the amount of energy that was dissipated when contrasted with RT and the method that had been developed in the past. This study verifies the use of machine learning techniques in this industry, and it is unique in that it anticipates energy dissipation along stepped spillways utilizing baffle designs. In addition, this work validates the use of machine learning methods in this field.

Keywords

Rectangular Stepped Spillways, Baffle Arrangements, Channel Slope, Support Vector Machine (SVM), Regression Tree (RT)

Introduction

To regulate water flows downstream of a dam, a spillway structure is employed, with stepped spillways preventing water from overflowing and causing damage to the dam. These spillways consist of a channel with built-in steps or drops. Flow patterns observed include nappe flow, transition flow, and skimming flow [1]. Numerous scholars have looked at the energy dissipation in stepped spillways [2-4]. Boes and Hager [5] looked at the benefits of stepped spillways, such as their simplicity of construction, less danger of cavitation, and smaller stilling basins at downstream dam toes owing to considerable energy loss along the chute. Hazzab and Chafic [7] conducted an experimental study on energy dissipation in stepped spillways and reported on flow configurations. Additionally, the Manksvill dam spillway was examined using a 1:25 scale physical wooden model [6]. For moderately inclined stepped channels, Stefan and Chanson [8] explored air-water flow measurements. Daniel [9] discussed how the existence of steps and step heights affect stepped spillways’ ability to dissipate energy. A comparison of the smooth invert chute flow with the self aerated stepped spillway. The energy dissipation in stepped spillways was investigated using various methods. Katourany [10] compared experimental findings to conventional USBR outcomes to examine the effects of different baffle widths, spacing between baffle rows, and step heights of baffled aprons. Salmasi et al. [11] assessed the energy dissipation of through-flow and over-flow in gabion stepped spillways, discovering that gabion spillways with pervious surfaces dissipated energy more efficiently than those with concrete walls. Other forms of stepped spillways, such as inclined steps and steps with end sills, were also quantitatively studied for energy dissipation [12]. Saedi and Asareh [13] examined how the number of drop stairs affected energy dissipation in stepped drops and suggested using stepped drops to increase energy dissipation by providing flow path roughness. Al-Husseini [14] found that decreasing the number of steps and downstream slopes led to an increase in flow energy dissipation, and that the use of cascade spillways reduced energy dissipation compared to the original step spillway. MARS and ANN methods were used to estimate energy dissipation in flow across stepped spillways under skimming flow conditions, with both models proving reliable [15]. Frederic et al. [16] evaluated the energy dissipation effectiveness and stability of the Mekin Dam spillway by confirming that flow did not result in transitional flow and by calculating safety factors at various intervals. A numerical model was developed to validate a physical model examining the impact of geometrical parameters on the dissipation rate in flows through stepped spillways [17]. The regulation of the rates of dissipation is studied using a particular kind of fuzzy inference system (FIS). The findings are compared with a predefined numerical database to determine the predicted energy dissipation under various circumstances. The findings show that the suggested FIS may be a useful tool for the operational management of dissipator structures while taking various geometric characteristics into account. Nasralla [18] studied the four phases of the spillway and conducted eighteen runs to enhance energy dissipation through the contraction-stepped spillway. The study considered alternative baffle placements, heights, and widths. The results showed that downstream baffles on the stepped spillway of the stilling basin improve energy dissipation. Using the Flow 3D software, Ikinciogullari [19] quantitatively analyzed the energy dissipation capabilities of trapezoidal stepped spillways using four distinct models and three different discharges. The findings showed that trapezoidal stepped spillways are up to 30% more efficient in dissipating energy than traditional stepped spillways. In previous works, only a few machine learning algorithms were used to forecast energy dissipation across a rectangular stepped spillway without baffles. Therefore, this study used machine learning approaches such as Support Vector Machine (SVM) and Regression Tree (RT) to predict energy dissipation across a rectangular stepped spillway with varied rectangular-shaped baffle configurations at different channel slopes. The study compared these models using statistical analysis to assess their efficiency in predicting energy dissipation over rectangular stepped spillways with baffles. 2. Materials and Methods 2.1. Experimental Setup The experiments were carried out at the Hydraulics laboratory of Delhi Technological University. The tests were performed in a rectangular tilting flume of 8m long, 0.30m wide and 0.40m deep which has a facility to make it horizontal and sloping as well (shown in Figure 1). The flume consists of an inlet section, an outlet section, and a collecting tank at the downstream end which is used to measure the discharge. Figure 2 depicts the model of a rectangular stepped spillway prepared using an acrylic sheet having a width of 0.30m, a height of 0.20m and a base length of 0.40m. A total of four steps were designed with a step height of 0.05m, the step length is 0.10m and rectangular-shaped baffles of length 0.10m and height of 0.05m were arranged in different manner. Figure 3 represents the different baffle arrangements used in the experimental work. At first, the experiment was conducted for no baffle condition. Thereafter the experiment was conducted for the first arrangement of three baffles, in which two baffles were placed at a distance of 0.10m from the toe of the spillway and a distance of 0.10m was maintained between the first two baffles and the third baffle was placed between the first two baffles at a distance of 0.20m from the toe of the spillway (figure 4a). After that, the experiment was conducted for the third arrangement of baffles which consists of five baffles, two more baffles were introduced at a distance of 0.30m from the toe of the spillway and a distance of 0.10m was maintained between them (figure 4b). The baffles used in the experiment were rectangular shaped which had a height of 0.05m and length of 0.10m. The experiments were conducted for five different discharges 2 l/s, 4 l/s, 6 l/s, 8 l/s and 10 l/s. For the purpose of determining the head values both upstream and downstream of the spillway model, a point gauge with a precision of 0.1mm was used. In order to determine the average velocities of the upstream and downstream portions, respectively, a pitot static tube was used in conjunction with a digital manometer.

Figure 1. Rectangular tilting flume
Figure 2. Dimensions of classical stepped spillway
Figure 3. Arrangements of baffles in classical stepped spillway
Figure 4. Rectangular stepped spillway with (a) three baffle arrangement (b) five baffle arrangement
The distribution of the computed maximum current speed during the entire duration of the NAMI DANCE and FLOW-3D simulations. The resolution of computational domain is 10 m

Performance Comparison of NAMI DANCE and FLOW-3D® Models in Tsunami Propagation, Inundation and Currents using NTHMP Benchmark Problems

NTHMP 벤치마크 문제를 사용하여 쓰나미 전파, 침수 및 해류에서 NAMI DANCE 및 FLOW-3D® 모델의 성능 비교

Pure and Applied Geophysics volume 176, pages3115–3153 (2019)Cite this article

Abstract

Field observations provide valuable data regarding nearshore tsunami impact, yet only in inundation areas where tsunami waves have already flooded. Therefore, tsunami modeling is essential to understand tsunami behavior and prepare for tsunami inundation. It is necessary that all numerical models used in tsunami emergency planning be subject to benchmark tests for validation and verification. This study focuses on two numerical codes, NAMI DANCE and FLOW-3D®, for validation and performance comparison. NAMI DANCE is an in-house tsunami numerical model developed by the Ocean Engineering Research Center of Middle East Technical University, Turkey and Laboratory of Special Research Bureau for Automation of Marine Research, Russia. FLOW-3D® is a general purpose computational fluid dynamics software, which was developed by scientists who pioneered in the design of the Volume-of-Fluid technique. The codes are validated and their performances are compared via analytical, experimental and field benchmark problems, which are documented in the ‘‘Proceedings and Results of the 2011 National Tsunami Hazard Mitigation Program (NTHMP) Model Benchmarking Workshop’’ and the ‘‘Proceedings and Results of the NTHMP 2015 Tsunami Current Modeling Workshop”. The variations between the numerical solutions of these two models are evaluated through statistical error analysis.

현장 관찰은 연안 쓰나미 영향에 관한 귀중한 데이터를 제공하지만 쓰나미 파도가 이미 범람한 침수 지역에서만 가능합니다. 따라서 쓰나미 모델링은 쓰나미 행동을 이해하고 쓰나미 범람에 대비하는 데 필수적입니다.

쓰나미 비상 계획에 사용되는 모든 수치 모델은 검증 및 검증을 위한 벤치마크 테스트를 받아야 합니다. 이 연구는 검증 및 성능 비교를 위해 NAMI DANCE 및 FLOW-3D®의 두 가지 숫자 코드에 중점을 둡니다.

NAMI DANCE는 터키 중동 기술 대학의 해양 공학 연구 센터와 러시아 해양 연구 자동화를 위한 특별 조사국 연구소에서 개발한 사내 쓰나미 수치 모델입니다. FLOW-3D®는 Volume-of-Fluid 기술의 설계를 개척한 과학자들이 개발한 범용 전산 유체 역학 소프트웨어입니다.

코드의 유효성이 검증되고 분석, 실험 및 현장 벤치마크 문제를 통해 코드의 성능이 비교되며, 이는 ‘2011년 NTHMP(National Tsunami Hazard Mitigation Program) 모델 벤치마킹 워크숍의 절차 및 결과’와 ”절차 및 NTHMP 2015 쓰나미 현재 모델링 워크숍 결과”. 이 두 모델의 수치 해 사이의 변동은 통계적 오류 분석을 통해 평가됩니다.

The distribution of the computed maximum current speed during the entire duration of the NAMI DANCE and FLOW-3D simulations. The resolution of computational domain is 10 m
The distribution of the computed maximum current speed during the entire duration of the NAMI DANCE and FLOW-3D simulations. The resolution of computational domain is 10 m

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Acknowledgements

The authors wish to thank Dr. Andrey Zaytsev due to his undeniable contributions to the development of in-house numerical model, NAMI DANCE. The Turkish branch of Flow Science, Inc. is also acknowledged. Finally, the National Tsunami Hazard Mitigation Program (NTHMP), who provided most of the benchmark data, is appreciated. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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  1. Deniz Velioglu SogutPresent address: 1212 Computer Science, Department of Civil Engineering, Stony Brook University, Stony Brook, NY, 11794, USA

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  1. Middle East Technical University, 06800, Ankara, TurkeyDeniz Velioglu Sogut & Ahmet Cevdet Yalciner

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Correspondence to Deniz Velioglu Sogut.

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Velioglu Sogut, D., Yalciner, A.C. Performance Comparison of NAMI DANCE and FLOW-3D® Models in Tsunami Propagation, Inundation and Currents using NTHMP Benchmark Problems. Pure Appl. Geophys. 176, 3115–3153 (2019). https://doi.org/10.1007/s00024-018-1907-9

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  • Received22 December 2017
  • Revised16 May 2018
  • Accepted24 May 2018
  • Published07 June 2018
  • Issue Date01 July 2019
  • DOIhttps://doi.org/10.1007/s00024-018-1907-9

Keywords

  • Tsunami
  • depth-averaged shallow water
  • Reynolds-averaged Navier–Stokes
  • benchmarking
  • NAMI DANCE
  • FLOW-3D®
Effects of surface roughness on overflow discharge of embankment weirs

표면 거칠기가 제방 둑의 오버플로 배출에 미치는 영향

Effects of surface roughness on overflow discharge of embankment weirs

Abstract

A numerical study was performed on the embankment weir overflows with various surface roughness and tailwater submergence, to better understand the effects of weir roughness on discharge performances under the free and submerged conditions. The variation of flow regime is captured, from the free overflow, submerged hydraulic jump, to surface flow with increasing tailwater depth. A roughness factor is introduced to reflect the reduction in discharge caused by weir roughness. The roughness factor decreases with the roughness height, and it also depends on the tailwater depth, highlighting various relations of the roughness factor with the roughness height between different flow regimes, which is linear for the free overflow and submerged hydraulic jump while exponential for the surface flow. Accordingly, the effects of weir roughness on overflow discharge appear nonnegligible for the significant roughness height and the surface flow regime occurring under considerable tailwater submergence. The established empirical expressions of discharge coefficient and submergence and roughness factors make it possible to predict the discharge over embankment weirs considering both tailwater submergence and surface roughness.

자유 및 침수 조건에서 방류 성능에 대한 둑 거칠기의 영향을 더 잘 이해하기 위해 다양한 표면 거칠기와 테일워터 침수를 갖는 제방 둑 범람에 대한 수치 연구가 수행되었습니다.

자유 범람, 수중 수압 점프, 테일워터 깊이가 증가하는 표면 유동에 이르기까지 유동 체제의 변화가 캡처됩니다. 위어 거칠기로 인한 배출 감소를 반영하기 위해 거칠기 계수가 도입되었습니다.

조도 계수는 조도 높이와 함께 감소하고, 또한 테일워터 깊이에 따라 달라지며, 서로 다른 흐름 영역 사이의 조도 높이와 조도 계수의 다양한 관계를 강조합니다.

이는 자유 범람 및 수중 수압 점프에 대해 선형인 반면 표면에 대해 지수적입니다. 흐름. 따라서 월류 방류에 대한 웨어 조도의 영향은 상당한 조도 높이와 상당한 방수 침수 하에서 발생하는 표면 흐름 체제에 대해 무시할 수 없는 것으로 보입니다.

방류계수와 침수 및 조도계수의 확립된 실증식은 방류수 침수와 지표조도를 모두 고려한 제방보 위의 방류량을 예측할 수 있게 합니다.

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Figure 11. Sketch of scour mechanism around USAF under random waves.

Scour Characteristics and Equilibrium Scour Depth Prediction around Umbrella Suction Anchor Foundation under Random Waves

by Ruigeng Hu 1,Hongjun Liu 2,Hao Leng 1,Peng Yu 3 andXiuhai Wang 1,2,*

1College of Environmental Science and Engineering, Ocean University of China, Qingdao 266000, China

2Key Lab of Marine Environment and Ecology (Ocean University of China), Ministry of Education, Qingdao 266000, China

3Qingdao Geo-Engineering Survering Institute, Qingdao 266100, China

*Author to whom correspondence should be addressed.

J. Mar. Sci. Eng. 20219(8), 886; https://doi.org/10.3390/jmse9080886

Received: 6 July 2021 / Revised: 8 August 2021 / Accepted: 13 August 2021 / Published: 17 August 2021

(This article belongs to the Section Ocean Engineering)

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Abstract

A series of numerical simulation were conducted to study the local scour around umbrella suction anchor foundation (USAF) under random waves. In this study, the validation was carried out firstly to verify the accuracy of the present model. Furthermore, the scour evolution and scour mechanism were analyzed respectively. In addition, two revised models were proposed to predict the equilibrium scour depth Seq around USAF. At last, a parametric study was carried out to study the effects of the Froude number Fr and Euler number Eu for the Seq. The results indicate that the present numerical model is accurate and reasonable for depicting the scour morphology under random waves. The revised Raaijmakers’s model shows good agreement with the simulating results of the present study when KCs,p < 8. The predicting results of the revised stochastic model are the most favorable for n = 10 when KCrms,a < 4. The higher Fr and Eu both lead to the more intensive horseshoe vortex and larger Seq.

Keywords: 

scournumerical investigationrandom wavesequilibrium scour depthKC number

1. Introduction

The rapid expansion of cities tends to cause social and economic problems, such as environmental pollution and traffic jam. As a kind of clean energy, offshore wind power has developed rapidly in recent years. The foundation of offshore wind turbine (OWT) supports the upper tower, and suffers the cyclic loading induced by waves, tides and winds, which exerts a vital influence on the OWT system. The types of OWT foundation include the fixed and floating foundation, and the fixed foundation was used usually for nearshore wind turbine. After the construction of fixed foundation, the hydrodynamic field changes in the vicinity of the foundation, leading to the horseshoe vortex formation and streamline compression at the upside and sides of foundation respectively [1,2,3,4]. As a result, the neighboring soil would be carried away by the shear stress induced by vortex, and the scour hole would emerge in the vicinity of foundation. The scour holes increase the cantilever length, and weaken the lateral bearing capacity of foundation [5,6,7,8,9]. Moreover, the natural frequency of OWT system increases with the increase of cantilever length, causing the resonance occurs when the system natural frequency equals the wave or wind frequency [10,11,12]. Given that, an innovative foundation called umbrella suction anchor foundation (USAF) has been designed for nearshore wind power. The previous studies indicated the USAF was characterized by the favorable lateral bearing capacity with the low cost [6,13,14]. The close-up of USAF is shown in Figure 1, and it includes six parts: 1-interal buckets, 2-external skirt, 3-anchor ring, 4-anchor branch, 5-supporting rod, 6-telescopic hook. The detailed description and application method of USAF can be found in reference [13].

Jmse 09 00886 g001 550

Figure 1. The close-up of umbrella suction anchor foundation (USAF).

Numerical and experimental investigations of scour around OWT foundation under steady currents and waves have been extensively studied by many researchers [1,2,15,16,17,18,19,20,21,22,23,24]. The seabed scour can be classified as two types according to Shields parameter θ, i.e., clear bed scour (θ < θcr) or live bed scour (θ > θcr). Due to the set of foundation, the adverse hydraulic pressure gradient exists at upstream foundation edges, resulting in the streamline separation between boundary layer flow and seabed. The separating boundary layer ascended at upstream anchor edges and developed into the horseshoe vortex. Then, the horseshoe vortex moved downstream gradually along the periphery of the anchor, and the vortex shed off continually at the lee-side of the anchor, i.e., wake vortex. The core of wake vortex is a negative pressure center, liking a vacuum cleaner. Hence, the soil particles were swirled into the negative pressure core and carried away by wake vortexes. At the same time, the onset of scour at rear side occurred. Finally, the wake vortex became downflow when the turbulence energy could not support the survival of wake vortex. According to Tavouktsoglou et al. [25], the scale of pile wall boundary layer is proportional to 1/ln(Rd) (Rd is pile Reynolds), which means the turbulence intensity induced by the flow-structure interaction would decrease with Rd increases, but the effects of Rd can be neglected only if the flow around the foundation is fully turbulent [26]. According to previous studies [1,15,27,28,29,30,31,32], the scour development around pile foundation under waves was significantly influenced by Shields parameter θ and KC number simultaneously (calculated by Equation (1)). Sand ripples widely existed around pile under waves in the case of live bed scour, and the scour morphology is related with θ and KC. Compared with θKC has a greater influence on the scour morphology [21,27,28]. The influence mechanism of KC on the scour around the pile is reflected in two aspects: the horseshoe vortex at upstream and wake vortex shedding at downstream.

KC=UwmTD��=�wm��(1)

where, Uwm is the maximum velocity of the undisturbed wave-induced oscillatory flow at the sea bottom above the wave boundary layer, T is wave period, and D is pile diameter.

There are two prerequisites to satisfy the formation of horseshoe vortex at upstream pile edges: (1) the incoming flow boundary layer with sufficient thickness and (2) the magnitude of upstream adverse pressure gradient making the boundary layer separating [1,15,16,18,20]. The smaller KC results the lower adverse pressure gradient, and the boundary layer cannot separate, herein, there is almost no horseshoe vortex emerging at upside of pile. Sumer et al. [1,15] carried out several sets of wave flume experiments under regular and irregular waves respectively, and the experiment results show that there is no horseshoe vortex when KC is less than 6. While the scale and lifespan of horseshoe vortex increase evidently with the increase of KC when KC is larger than 6. Moreover, the wake vortex contributes to the scour at lee-side of pile. Similar with the case of horseshoe vortex, there is no wake vortex when KC is less than 6. The wake vortex is mainly responsible for scour around pile when KC is greater than 6 and less than O(100), while horseshoe vortex controls scour nearly when KC is greater than O(100).

Sumer et al. [1] found that the equilibrium scour depth was nil around pile when KC was less than 6 under regular waves for live bed scour, while the equilibrium scour depth increased with the increase of KC. Based on that, Sumer proposed an equilibrium scour depth predicting equation (Equation (2)). Carreiras et al. [33] revised Sumer’s equation with m = 0.06 for nonlinear waves. Different with the findings of Sumer et al. [1] and Carreiras et al. [33], Corvaro et al. [21] found the scour still occurred for KC ≈ 4, and proposed the revised equilibrium scour depth predicting equation (Equation (3)) for KC > 4.

Rudolph and Bos [2] conducted a series of wave flume experiments to investigate the scour depth around monopile under waves only, waves and currents combined respectively, indicting KC was one of key parameters in influencing equilibrium scour depth, and proposed the equilibrium scour depth predicting equation (Equation (4)) for low KC (1 < KC < 10). Through analyzing the extensive data from published literatures, Raaijmakers and Rudolph [34] developed the equilibrium scour depth predicting equation (Equation (5)) for low KC, which was suitable for waves only, waves and currents combined. Khalfin [35] carried out several sets of wave flume experiments to study scour development around monopile, and proposed the equilibrium scour depth predicting equation (Equation (6)) for low KC (0.1 < KC < 3.5). Different with above equations, the Khalfin’s equation considers the Shields parameter θ and KC number simultaneously in predicting equilibrium scour depth. The flow reversal occurred under through in one wave period, so sand particles would be carried away from lee-side of pile to upside, resulting in sand particles backfilled into the upstream scour hole [20,29]. Considering the backfilling effects, Zanke et al. [36] proposed the equilibrium scour depth predicting equation (Equation (7)) around pile by theoretical analysis, and the equation is suitable for the whole range of KC number under regular waves and currents combined.

S/D=1.3(1−exp([−m(KC−6)])�/�=1.3(1−exp(−�(��−6))(2)

where, m = 0.03 for linear waves.

S/D=1.3(1−exp([−0.02(KC−4)])�/�=1.3(1−exp(−0.02(��−4))(3)

S/D=1.3γKwaveKhw�/�=1.3��wave�ℎw(4)

where, γ is safety factor, depending on design process, typically γ = 1.5, Kwave is correction factor considering wave action, Khw is correction factor considering water depth.

S/D=1.5[tanh(hwD)]KwaveKhw�/�=1.5tanh(ℎw�)�wave�ℎw(5)

where, hw is water depth.

S/D=0.0753(θθcr−−−√−0.5)0.69KC0.68�/�=0.0753(��cr−0.5)0.69��0.68(6)

where, θ is shields parameter, θcr is critical shields parameter.

S/D=2.5(1−0.5u/uc)xrelxrel=xeff/(1+xeff)xeff=0.03(1−0.35ucr/u)(KC−6)⎫⎭⎬⎪⎪�/�=2.5(1−0.5�/��)��������=����/(1+����)����=0.03(1−0.35�cr/�)(��−6)(7)

where, u is near-bed orbital velocity amplitude, uc is critical velocity corresponding the onset of sediment motion.

S/D=1.3{1−exp[−0.03(KC2lnn+36)1/2−6]}�/�=1.31−exp−0.03(��2ln�+36)1/2−6(8)

where, n is the 1/n’th highest wave for random waves

For predicting equilibrium scour depth under irregular waves, i.e., random waves, Sumer and Fredsøe [16] found it’s suitable to take Equation (2) to predict equilibrium scour depth around pile under random waves with the root-mean-square (RMS) value of near-bed orbital velocity amplitude Um and peak wave period TP to calculate KC. Khalfin [35] recommended the RMS wave height Hrms and peak wave period TP were used to calculate KC for Equation (6). References [37,38,39,40] developed a series of stochastic theoretical models to predict equilibrium scour depth around pile under random waves, nonlinear random waves plus currents respectively. The stochastic approach thought the 1/n’th highest wave were responsible for scour in vicinity of pile under random waves, and the KC was calculated in Equation (8) with Um and mean zero-crossing wave period Tz. The results calculated by Equation (8) agree well with experimental values of Sumer and Fredsøe [16] if the 1/10′th highest wave was used. To author’s knowledge, the stochastic approach proposed by Myrhaug and Rue [37] is the only theoretical model to predict equilibrium scour depth around pile under random waves for the whole range of KC number in published documents. Other methods of predicting scour depth under random waves are mainly originated from the equation for regular waves-only, waves and currents combined, which are limited to the large KC number, such as KC > 6 for Equation (2) and KC > 4 for Equation (3) respectively. However, situations with relatively low KC number (KC < 4) often occur in reality, for example, monopile or suction anchor for OWT foundations in ocean environment. Moreover, local scour around OWT foundations under random waves has not yet been investigated fully. Therefore, further study are still needed in the aspect of scour around OWT foundations with low KC number under random waves. Given that, this study presents the scour sediment model around umbrella suction anchor foundation (USAF) under random waves. In this study, a comparison of equilibrium scour depth around USAF between this present numerical models and the previous theoretical models and experimental results was presented firstly. Then, this study gave a comprehensive analysis for the scour mechanisms around USAF. After that, two revised models were proposed according to the model of Raaijmakers and Rudolph [34] and the stochastic model developed by Myrhaug and Rue [37] respectively to predict the equilibrium scour depth. Finally, a parametric study was conducted to study the effects of the Froude number (Fr) and Euler number (Eu) to equilibrium scour depth respectively.

2. Numerical Method

2.1. Governing Equations of Flow

The following equations adopted in present model are already available in Flow 3D software. The authors used these theoretical equations to simulate scour in random waves without modification. The incompressible viscous fluid motion satisfies the Reynolds-averaged Navier-Stokes (RANS) equation, so the present numerical model solves RANS equations:

∂u∂t+1VF(uAx∂u∂x+vAy∂u∂y+wAz∂u∂z)=−1ρf∂p∂x+Gx+fx∂�∂�+1��(���∂�∂�+���∂�∂�+���∂�∂�)=−1�f∂�∂�+��+��(9)

∂v∂t+1VF(uAx∂v∂x+vAy∂v∂y+wAz∂v∂z)=−1ρf∂p∂y+Gy+fy∂�∂�+1��(���∂�∂�+���∂�∂�+���∂�∂�)=−1�f∂�∂�+��+��(10)

∂w∂t+1VF(uAx∂w∂x+vAy∂w∂y+wAz∂w∂z)=−1ρf∂p∂z+Gz+fz∂�∂�+1��(���∂�∂�+���∂�∂�+���∂�∂�)=−1�f∂�∂�+��+��(11)

where, VF is the volume fraction; uv, and w are the velocity components in xyz direction respectively with Cartesian coordinates; Ai is the area fraction; ρf is the fluid density, fi is the viscous fluid acceleration, Gi is the fluid body acceleration (i = xyz).

2.2. Turbulent Model

The turbulence closure is available by the turbulent model, such as one-equation, the one-equation k-ε model, the standard k-ε model, RNG k-ε turbulent model and large eddy simulation (LES) model. The LES model requires very fine mesh grid, so the computational time is large, which hinders the LES model application in engineering. The RNG k-ε model can reduce computational time greatly with high accuracy in the near-wall region. Furthermore, the RNG k-ε model computes the maximum turbulent mixing length dynamically in simulating sediment scour model. Therefore, the RNG k-ε model was adopted to study the scour around anchor under random waves [41,42].

∂kT∂T+1VF(uAx∂kT∂x+vAy∂kT∂y+wAz∂kT∂z)=PT+GT+DiffkT−εkT∂��∂�+1��(���∂��∂�+���∂��∂�+���∂��∂�)=��+��+������−���(12)

∂εT∂T+1VF(uAx∂εT∂x+vAy∂εT∂y+wAz∂εT∂z)=CDIS1εTkT(PT+CDIS3GT)+Diffε−CDIS2ε2TkT∂��∂�+1��(���∂��∂�+���∂��∂�+���∂��∂�)=����1����(��+����3��)+�����−����2��2��(13)

where, kT is specific kinetic energy involved with turbulent velocity, GT is the turbulent energy generated by buoyancy; εT is the turbulent energy dissipating rate, PT is the turbulent energy, Diffε and DiffkT are diffusion terms associated with VFAiCDIS1CDIS2 and CDIS3 are dimensionless parameters, and CDIS1CDIS3 have default values of 1.42, 0.2 respectively. CDIS2 can be obtained from PT and kT.

2.3. Sediment Scour Model

The sand particles may suffer four processes under waves, i.e., entrainment, bed load transport, suspended load transport, and deposition, so the sediment scour model should depict the above processes efficiently. In present numerical simulation, the sediment scour model includes the following aspects:

2.3.1. Entrainment and Deposition

The combination of entrainment and deposition determines the net scour rate of seabed in present sediment scour model. The entrainment lift velocity of sand particles was calculated as [43]:

ulift,i=αinsd0.3∗(θ−θcr)1.5∥g∥di(ρi−ρf)ρf−−−−−−−−−−−−√�lift,i=�����*0.3(�−�cr)1.5���(��−�f)�f(14)

where, αi is the entrainment parameter, ns is the outward point perpendicular to the seabed, d* is the dimensionless diameter of sand particles, which was calculated by Equation (15), θcr is the critical Shields parameter, g is the gravity acceleration, di is the diameter of sand particles, ρi is the density of seabed species.

d∗=di(∥g∥ρf(ρi−ρf)μ2f)1/3�*=��(��f(��−�f)�f2)1/3(15)

where μf is the fluid dynamic viscosity.

In Equation (14), the entrainment parameter αi confirms the rate at which sediment erodes when the given shear stress is larger than the critical shear stress, and the recommended value 0.018 was adopted according to the experimental data of Mastbergen and Von den Berg [43]. ns is the outward pointing normal to the seabed interface, and ns = (0,0,1) according to the Cartesian coordinates used in present numerical model.

The shields parameter was obtained from the following equation:

θ=U2f,m(ρi/ρf−1)gd50�=�f,m2(��/�f−1)��50(16)

where, Uf,m is the maximum value of the near-bed friction velocity; d50 is the median diameter of sand particles. The detailed calculation procedure of θ was available in Soulsby [44].

The critical shields parameter θcr was obtained from the Equation (17) [44]

θcr=0.31+1.2d∗+0.055[1−exp(−0.02d∗)]�cr=0.31+1.2�*+0.0551−exp(−0.02�*)(17)

The sand particles begin to deposit on seabed when the turbulence energy weaken and cann’t support the particles suspending. The setting velocity of the particles was calculated from the following equation [44]:

usettling,i=νfdi[(10.362+1.049d3∗)0.5−10.36]�settling,�=�f��(10.362+1.049�*3)0.5−10.36(18)

where νf is the fluid kinematic viscosity.

2.3.2. Bed Load Transport

This is called bed load transport when the sand particles roll or bounce over the seabed and always have contact with seabed. The bed load transport velocity was computed by [45]:

ubedload,i=qb,iδicb,ifb�bedload,�=�b,����b,��b(19)

where, qb,i is the bed load transport rate, which was obtained from Equation (20), δi is the bed load thickness, which was calculated by Equation (21), cb,i is the volume fraction of sand i in the multiple species, fb is the critical packing fraction of the seabed.

qb,i=8[∥g∥(ρi−ρfρf)d3i]1/2�b,�=8�(��−�f�f)��31/2(20)

δi=0.3d0.7∗(θθcr−1)0.5di��=0.3�*0.7(��cr−1)0.5��(21)

2.3.3. Suspended Load Transport

Through the following transport equation, the suspended sediment concentration could be acquired.

∂Cs,i∂t+∇(us,iCs,i)=∇∇(DfCs,i)∂�s,�∂�+∇(�s,��s,�)=∇∇(�f�s,�)(22)

where, Cs,i is the suspended sand particles mass concentration of sand i in the multiple species, us,i is the sand particles velocity of sand iDf is the diffusivity.

The velocity of sand i in the multiple species could be obtained from the following equation:

us,i=u¯¯+usettling,ics,i�s,�=�¯+�settling,��s,�(23)

where, u¯�¯ is the velocity of mixed fluid-particles, which can be calculated by the RANS equation with turbulence model, cs,i is the suspended sand particles volume concentration, which was computed from Equation (24).

cs,i=Cs,iρi�s,�=�s,���(24)

3. Model Setup

The seabed-USAF-wave three-dimensional scour numerical model was built using Flow-3D software. As shown in Figure 2, the model includes sandy seabed, USAF model, sea water, two baffles and porous media. The dimensions of USAF are shown in Table 1. The sandy bed (210 m in length, 30 m in width and 11 m in height) is made up of uniform fine sand with median diameter d50 = 0.041 cm. The USAF model includes upper steel tube with the length of 20 m, which was installed in the middle of seabed. The location of USAF is positioned at 140 m from the upstream inflow boundary and 70 m from the downstream outflow boundary. Two baffles were installed at two ends of seabed. In order to eliminate the wave reflection basically, the porous media was set at the outflow side on the seabed.

Jmse 09 00886 g002 550

Figure 2. (a) The sketch of seabed-USAF-wave three-dimensional model; (b) boundary condation:Wv-wave boundary, S-symmetric boundary, O-outflow boundary; (c) USAF model.

Table 1. Numerical simulating cases.

Table

3.1. Mesh Geometric Dimensions

In the simulation of the scour under the random waves, the model includes the umbrella suction anchor foundation, seabed and fluid. As shown in Figure 3, the model mesh includes global mesh grid and nested mesh grid, and the total number of grids is 1,812,000. The basic procedure for building mesh grid consists of two steps. Step 1: Divide the global mesh using regular hexahedron with size of 0.6 × 0.6. The global mesh area is cubic box, embracing the seabed and whole fluid volume, and the dimensions are 210 m in length, 30 m in width and 32 m in height. The details of determining the grid size can see the following mesh sensitivity section. Step 2: Set nested fine mesh grid in vicinity of the USAF with size of 0.3 × 0.3 so as to shorten the computation cost and improve the calculation accuracy. The encryption range is −15 m to 15 m in x direction, −15 m to 15 m in y direction and 0 m to 32 m in z direction, respectively. In order to accurately capture the free-surface dynamics, such as the fluid-air interface, the volume of fluid (VOF) method was adopted for tracking the free water surface. One specific algorithm called FAVORTM (Fractional Area/Volume Obstacle Representation) was used to define the fractional face areas and fractional volumes of the cells which are open to fluid flow.

Jmse 09 00886 g003 550

Figure 3. The sketch of mesh grid.

3.2. Boundary Conditions

As shown in Figure 2, the initial fluid length is 210 m as long as seabed. A wave boundary was specified at the upstream offshore end. The details of determining the random wave spectrum can see the following wave parameters section. The outflow boundary was set at the downstream onshore end. The symmetry boundary was used at the top and two sides of the model. The symmetric boundaries were the better strategy to improve the computation efficiency and save the calculation cost [46]. At the seabed bottom, the wall boundary was adopted, which means the u = v = w= 0. Besides, the upper steel tube of USAF was set as no-slip condition.

3.3. Wave Parameters

The random waves with JONSWAP wave spectrum were used for all simulations as realistic representation of offshore conditions. The unidirectional JONSWAP frequency spectrum was described as [47]:

S(ω)=αg2ω5exp[−54(ωpω)4]γexp[−(ω−ωp)22σ2ω2p]�(�)=��2�5exp−54(�p�)4�exp−(�−�p)22�2�p2(25)

where, α is wave energy scale parameter, which is calculated by Equation (26), ω is frequency, ωp is wave spectrum peak frequency, which can be obtained from Equation (27). γ is wave spectrum peak enhancement factor, in this study γ = 3.3. σ is spectral width factor, σ equals 0.07 for ω ≤ ωp and 0.09 for ω > ωp respectively.

α=0.0076(gXU2)−0.22�=0.0076(���2)−0.22(26)

ωp=22(gU)(gXU2)−0.33�p=22(��)(���2)−0.33(27)

where, X is fetch length, U is average wind velocity at 10 m height from mean sea level.

In present numerical model, the input key parameters include X and U for wave boundary with JONSWAP wave spectrum. The objective wave height and period are available by different combinations of X and U. In this study, we designed 9 cases with different wave heights, periods and water depths for simulating scour around USAF under random waves (see Table 2). For random waves, the wave steepness ε and Ursell number Ur were acquired form Equations (28) and (29) respectively

ε=2πgHsT2a�=2���s�a2(28)

Ur=Hsk2h3w�r=�s�2ℎw3(29)

where, Hs is significant wave height, Ta is average wave period, k is wave number, hw is water depth. The Shield parameter θ satisfies θ > θcr for all simulations in current study, indicating the live bed scour prevails.

Table 2. Numerical simulating cases.

Table

3.4. Mesh Sensitivity

In this section, a mesh sensitivity analysis was conducted to investigate the influence of mesh grid size to results and make sure the calculation is mesh size independent and converged. Three mesh grid size were chosen: Mesh 1—global mesh grid size of 0.75 × 0.75, nested fine mesh grid size of 0.4 × 0.4, and total number of grids 1,724,000, Mesh 2—global mesh grid size of 0.6 × 0.6, nested fine mesh grid size of 0.3 × 0.3, and total number of grids 1,812,000, Mesh 3—global mesh grid size of 0.4 × 0.4, nested fine mesh grid size of 0.2 × 0.2, and total number of grids 1,932,000. The near-bed shear velocity U* is an important factor for influencing scour process [1,15], so U* at the position of (4,0,11.12) was evaluated under three mesh sizes. As the Figure 4 shown, the maximum error of shear velocity ∆U*1,2 is about 39.8% between the mesh 1 and mesh 2, and 4.8% between the mesh 2 and mesh 3. According to the mesh sensitivity criterion adopted by Pang et al. [48], it’s reasonable to think the results are mesh size independent and converged with mesh 2. Additionally, the present model was built according to prototype size, and the mesh size used in present model is larger than the mesh size adopted by Higueira et al. [49] and Corvaro et al. [50]. If we choose the smallest cell size, it will take too much time. For example, the simulation with Mesh3 required about 260 h by using a computer with Intel Xeon Scalable Gold 4214 CPU @24 Cores, 2.2 GHz and 64.00 GB RAM. Therefore, in this case, considering calculation accuracy and computation efficiency, the mesh 2 was chosen for all the simulation in this study.

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Figure 4. Comparison of near-bed shear velocity U* with different mesh grid size.

The nested mesh block was adopted for seabed in vicinity of the USAF, which was overlapped with the global mesh block. When two mesh blocks overlap each other, the governing equations are by default solved on the mesh block with smaller average cell size (i.e., higher grid resolution). It is should be noted that the Flow 3D software used the moving mesh captures the scour evolution and automatically adjusts the time step size to be as large as possible without exceeding any of the stability limits, affecting accuracy, or unduly increasing the effort required to enforce the continuity condition [51].

3.5. Model Validation

In order to verify the reliability of the present model, the results of present study were compared with the experimental data of Khosronejad et al. [52]. The experiment was conducted in an open channel with a slender vertical pile under unidirectional currents. The comparison of scour development between the present results and the experimental results is shown in Figure 5. The Figure 5 reveals that the present results agree well with the experimental data of Khosronejad et al. [52]. In the first stage, the scour depth increases rapidly. After that, the scour depth achieves a maximum value gradually. The equilibrium scour depth calculated by the present model is basically corresponding with the experimental results of Khosronejad et al. [52], although scour depth in the present model is slightly larger than the experimental results at initial stage.

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Figure 5. Comparison of time evolution of scour between the present study and Khosronejad et al. [52], Petersen et al. [17].

Secondly, another comparison was further conducted between the results of present study and the experimental data of Petersen et al. [17]. The experiment was carried out in a flume with a circular vertical pile in combined waves and current. Figure 4 shows a comparison of time evolution of scour depth between the simulating and the experimental results. As Figure 5 indicates, the scour depth in this study has good overall agreement with the experimental results proposed in Petersen et al. [17]. The equilibrium scour depth calculated by the present model is 0.399 m, which equals to the experimental value basically. Overall, the above verifications prove the present model is accurate and capable in dealing with sediment scour under waves.

In addition, in order to calibrate and validate the present model for hydrodynamic parameters, the comparison of water surface elevation was carried out with laboratory experiments conducted by Stahlmann [53] for wave gauge No. 3. The Figure 6 depicts the surface wave profiles between experiments and numerical model results. The comparison indicates that there is a good agreement between the model results and experimental values, especially the locations of wave crest and trough. Comparison of the surface elevation instructs the present model has an acceptable relative error, and the model is a calibrated in terms of the hydrodynamic parameters.

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Figure 6. Comparison of surface elevation between the present study and Stahlmann [53].

Finally, another comparison was conducted for equilibrium scour depth or maximum scour depth under random waves with the experimental data of Sumer and Fredsøe [16] and Schendel et al. [22]. The Figure 7 shows the comparison between the numerical results and experimental data of Run01, Run05, Run21 and Run22 in Sumer and Fredsøe [16] and test A05 and A09 in Schendel et al. [22]. As shown in Figure 7, the equilibrium scour depth or maximum scour depth distributed within the ±30 error lines basically, meaning the reliability and accuracy of present model for predicting equilibrium scour depth around foundation in random waves. However, compared with the experimental values, the present model overestimated the equilibrium scour depth generally. Given that, a calibration for scour depth was carried out by multiplying the mean reduced coefficient 0.85 in following section.

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Figure 7. Comparison of equilibrium (or maximum) scour depth between the present study and Sumer and Fredsøe [16], Schendel et al. [22].

Through the various examination for hydrodynamic and morphology parameters, it can be concluded that the present model is a validated and calibrated model for scour under random waves. Thus, the present numerical model would be utilized for scour simulation around foundation under random waves.

4. Numerical Results and Discussions

4.1. Scour Evolution

Figure 8 displays the scour evolution for case 1–9. As shown in Figure 8a, the scour depth increased rapidly at the initial stage, and then slowed down at the transition stage, which attributes to the backfilling occurred in scour holes under live bed scour condition, resulting in the net scour decreasing. Finally, the scour reached the equilibrium state when the amount of sediment backfilling equaled to that of scouring in the scour holes, i.e., the net scour transport rate was nil. Sumer and Fredsøe [16] proposed the following formula for the scour development under waves

St=Seq(1−exp(−t/Tc))�t=�eq(1−exp(−�/�c))(30)

where Tc is time scale of scour process.

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Figure 8. Time evolution of scour for case 1–9: (a) Case 1–5; (b) Case 6–9.

The computing time is 3600 s and the scour development curves in Figure 8 kept fluctuating, meaning it’s still not in equilibrium scour stage in these cases. According to Sumer and Fredsøe [16], the equilibrium scour depth can be acquired by fitting the data with Equation (30). From Figure 8, it can be seen that the scour evolution obtained from Equation (30) is consistent with the present study basically at initial stage, but the scour depth predicted by Equation (30) developed slightly faster than the simulating results and the Equation (30) overestimated the scour depth to some extent. Overall, the whole tendency of the results calculated by Equation (30) agrees well with the simulating results of the present study, which means the Equation (30) is applicable to depict the scour evolution around USAF under random waves.

4.2. Scour Mechanism under Random Waves

The scour morphology and scour evolution around USAF are similar under random waves in case 1~9. Taking case 7 as an example, the scour morphology is shown in Figure 9.

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Figure 9. Scour morphology under different times for case 7.

From Figure 9, at the initial stage (t < 1200 s), the scour occurred at upstream foundation edges between neighboring anchor branches. The maximum scour depth appeared at the lee-side of the USAF. Correspondingly, the sediments deposited at the periphery of the USAF, and the location of the maximum accretion depth was positioned at an angle of about 45° symmetrically with respect to the wave propagating direction in the lee-side of the USAF. After that, when t > 2400 s, the location of the maximum scour depth shifted to the upside of the USAF at an angle of about 45° with respect to the wave propagating direction.

According to previous studies [1,15,16,19,30,31], the horseshoe vortex, streamline compression and wake vortex shedding were responsible for scour around foundation. The Figure 10 displays the distribution of flow velocity in vicinity of foundation, which reflects the evolving processes of horseshoe vertex.

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Figure 10. Velocity profile around USAF: (a) Flow runup and down stream at upstream anchor edges; (b) Horseshoe vortex at upstream anchor edges; (c) Flow reversal during wave through stage at lee side.

As shown in Figure 10, the inflow tripped to the upstream edges of the USAF and it was blocked by the upper tube of USAF. Then, the downflow formed the horizontal axis clockwise vortex and rolled on the seabed bypassing the tube, that is, the horseshoe vortex (Figure 11). The Figure 12 displays the turbulence intensity around the tube on the seabed. From Figure 12, it can be seen that the turbulence intensity was high-intensity with respect to the region of horseshoe vortex. This phenomenon occurred because of drastic water flow momentum exchanging in the horseshoe vortex. As a result, it created the prominent shear stress on the seabed, causing the local scour at the upstream edges of USAF. Besides, the horseshoe vortex moved downstream gradually along the periphery of the tube and the wake vortex shed off continually at the lee-side of the USAF, i.e., wake vortex.

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Figure 11. Sketch of scour mechanism around USAF under random waves.

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Figure 12. Turbulence intensity: (a) Turbulence intensity of horseshoe vortex; (b) Turbulence intensity of wake vortex; (c) Turbulence intensity of accretion area.

The core of wake vortex is a negative pressure center, liking a vacuum cleaner [11,42]. Hence, the soil particles were swirled into the negative pressure core and carried away by wake vortex. At the same time, the onset of scour at rear side occurred. Finally, the wake vortex became downflow at the downside of USAF. As is shown in Figure 12, the turbulence intensity was low where the downflow occurred at lee-side, which means the turbulence energy may not be able to support the survival of wake vortex, leading to accretion happening. As mentioned in previous section, the formation of horseshoe vortex was dependent with adverse pressure gradient at upside of foundation. As shown in Figure 13, the evaluated range of pressure distribution is −15 m to 15 m in x direction. The t = 450 s and t = 1800 s indicate that the wave crest and trough arrived at the upside and lee-side of the foundation respectively, and the t = 350 s was neither the wave crest nor trough. The adverse gradient pressure reached the maximum value at t = 450 s corresponding to the wave crest phase. In this case, it’s helpful for the wave boundary separating fully from seabed, which leads to the formation of horseshoe vortex with high turbulence intensity. Therefore, the horseshoe vortex is responsible for the local scour between neighboring anchor branches at upside of USAF. What’s more, due to the combination of the horseshoe vortex and streamline compression, the maximum scour depth occurred at the upside of the USAF with an angle of about 45° corresponding to the wave propagating direction. This is consistent with the findings of Pang et al. [48] and Sumer et al. [1,15] in case of regular waves. At the wave trough phase (t = 1800 s), the pressure gradient became positive at upstream USAF edges, which hindered the separating of wave boundary from seabed. In the meantime, the flow reversal occurred (Figure 10) and the adverse gradient pressure appeared at downstream USAF edges, but the magnitude of adverse gradient pressure at lee-side was lower than the upstream gradient pressure under wave crest. In this way, the intensity of horseshoe vortex behind the USAF under wave trough was low, which explains the difference of scour depth at upstream and downstream, i.e., the scour asymmetry. In other words, the scour asymmetry at upside and downside of USAF was attributed to wave asymmetry for random waves, and the phenomenon became more evident for nonlinear waves [21]. Briefly speaking, the vortex system at wave crest phase was mainly related to the scour process around USAF under random waves.

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Figure 13. Pressure distribution around USAF.

4.3. Equilibrium Scour Depth

The KC number is a key parameter for horseshoe vortex emerging and evolving under waves. According to Equation (1), when pile diameter D is fixed, the KC depends on the maximum near-bed velocity Uwm and wave period T. For random waves, the Uwm can be denoted by the root-mean-square (RMS) value of near-bed velocity amplitude Uwm,rms or the significant value of near-bed velocity amplitude Uwm,s. The Uwm,rms and Uwm,s for all simulating cases of the present study are listed in Table 3 and Table 4. The T can be denoted by the mean up zero-crossing wave period Ta, peak wave period Tp, significant wave period Ts, the maximum wave period Tm, 1/10′th highest wave period Tn = 1/10 and 1/5′th highest wave period Tn = 1/5 for random waves, so the different combinations of Uwm and T will acquire different KC. The Table 3 and Table 4 list 12 types of KC, for example, the KCrms,s was calculated by Uwm,rms and Ts. Sumer and Fredsøe [16] conducted a series of wave flume experiments to investigate the scour depth around monopile under random waves, and found the equilibrium scour depth predicting equation (Equation (2)) for regular waves was applicable for random waves with KCrms,p. It should be noted that the Equation (2) is only suitable for KC > 6 under regular waves or KCrms,p > 6 under random waves.

Table 3. Uwm,rms and KC for case 1~9.

Table

Table 4. Uwm,s and KC for case 1~9.

Table

Raaijmakers and Rudolph [34] proposed the equilibrium scour depth predicting model (Equation (5)) around pile under waves, which is suitable for low KC. The format of Equation (5) is similar with the formula proposed by Breusers [54], which can predict the equilibrium scour depth around pile at different scour stages. In order to verify the applicability of Raaijmakers’s model for predicting the equilibrium scour depth around USAF under random waves, a validation of the equilibrium scour depth Seq between the present study and Raaijmakers’s equation was conducted. The position where the scour depth Seq was evaluated is the location of the maximum scour depth, and it was depicted in Figure 14. The Figure 15 displays the comparison of Seq with different KC between the present study and Raaijmakers’s model.

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Figure 14. Sketch of the position where the Seq was evaluated.

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Figure 15. Comparison of the equilibrium scour depth between the present model and the model of Raaijmakers and Rudolph [34]: (aKCrms,sKCrms,a; (bKCrms,pKCrms,m; (cKCrms,n = 1/10KCrms,n = 1/5; (dKCs,sKCs,a; (eKCs,pKCs,m; (fKCs,n = 1/10KCs,n = 1/5.

As shown in Figure 15, there is an error in predicting Seq between the present study and Raaijmakers’s model, and Raaijmakers’s model underestimates the results generally. Although the error exists, the varying trend of Seq with KC obtained from Raaijmakers’s model is consistent with the present study basically. What’s more, the error is minimum and the Raaijmakers’s model is of relatively high accuracy for predicting scour around USAF under random waves by using KCs,p. Based on this, a further revision was made to eliminate the error as much as possible, i.e., add the deviation value ∆S/D in the Raaijmakers’s model. The revised equilibrium scour depth predicting equation based on Raaijmakers’s model can be written as

S′eq/D=1.95[tanh(hD)](1−exp(−0.012KCs,p))+ΔS/D�eq′/�=1.95tanh(ℎ�)(1−exp(−0.012��s,p))+∆�/�(31)

As the Figure 16 shown, through trial-calculation, when ∆S/D = 0.05, the results calculated by Equation (31) show good agreement with the simulating results of the present study. The maximum error is about 18.2% and the engineering requirements have been met basically. In order to further verify the accuracy of the revised model for large KC (KCs,p > 4) under random waves, a validation between the revised model and the previous experimental results [21]. The experiment was conducted in a flume (50 m in length, 1.0 m in width and 1.3 m in height) with a slender vertical pile (D = 0.1 m) under random waves. The seabed is composed of 0.13 m deep layer of sand with d50 = 0.6 mm and the water depth is 0.5 m for all tests. The significant wave height is 0.12~0.21 m and the KCs,p is 5.52~11.38. The comparison between the predicting results by Equation (31) and the experimental results of Corvaro et al. [21] is shown in Figure 17. From Figure 17, the experimental data evenly distributes around the predicted results and the prediction accuracy is favorable when KCs,p < 8. However, the gap between the predicting results and experimental data becomes large and the Equation (31) overestimates the equilibrium scour depth to some extent when KCs,p > 8.

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Figure 16. Comparison of Seq between the simulating results and the predicting values by Equation (31).

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Figure 17. Comparison of Seq/D between the Experimental results of Corvaro et al. [21] and the predicting values by Equation (31).

In ocean environment, the waves are composed of a train of sinusoidal waves with different frequencies and amplitudes. The energy of constituent waves with very large and very small frequencies is relatively low, and the energy of waves is mainly concentrated in a certain range of moderate frequencies. Myrhaug and Rue [37] thought the 1/n’th highest wave was responsible for scour and proposed the stochastic model to predict the equilibrium scour depth around pile under random waves for full range of KC. Noteworthy is that the KC was denoted by KCrms,a in the stochastic model. To verify the application of the stochastic model for predicting scour depth around USAF, a validation between the simulating results of present study and predicting results by the stochastic model with n = 2,3,5,10,20,500 was carried out respectively.

As shown in Figure 18, compared with the simulating results, the stochastic model underestimates the equilibrium scour depth around USAF generally. Although the error exists, the varying trend of Seq with KCrms,a obtained from the stochastic model is consistent with the present study basically. What’s more, the gap between the predicting values by stochastic model and the simulating results decreases with the increase of n, but for large n, for example n = 500, the varying trend diverges between the predicting values and simulating results, meaning it’s not feasible only by increasing n in stochastic model to predict the equilibrium scour depth around USAF.

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Figure 18. Comparison of Seq between the simulating results and the predicting values by Equation (8).

The Figure 19 lists the deviation value ∆Seq/D′ between the predicting values and simulating results with different KCrms,a and n. Then, fitted the relationship between the ∆S′and n under different KCrms,a, and the fitting curve can be written by Equation (32). The revised stochastic model (Equation (33)) can be acquired by adding ∆Seq/D′ to Equation (8).

ΔSeq/D=0.052*exp(−n/6.566)+0.068∆�eq/�=0.052*exp(−�/6.566)+0.068(32)

S′eq¯/D=S′eq/D+0.052*exp(−n/6.566)+0.068�eq′¯/�=�eq′/�+0.052*exp(−�/6.566)+0.068(33)

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Figure 19. The fitting line between ∆S′and n.

The comparison between the predicting results by Equation (33) and the simulating results of present study is shown in Figure 20. According to the Figure 20, the varying trend of Seq with KCrms,a obtained from the stochastic model is consistent with the present study basically. Compared with predicting results by the stochastic model, the results calculated by Equation (33) is favorable. Moreover, comparison with simulating results indicates that the predicting results are the most favorable for n = 10, which is consistent with the findings of Myrhaug and Rue [37] for equilibrium scour depth predicting around slender pile in case of random waves.

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Figure 20. Comparison of Seq between the simulating results and the predicting values by Equation (33).

In order to further verify the accuracy of the Equation (33) for large KC (KCrms,a > 4) under random waves, a validation was conducted between the Equation (33) and the previous experimental results of Sumer and Fredsøe [16] and Corvaro et al. [21]. The details of experiments conducted by Corvaro et al. [21] were described in above section. Sumer and Fredsøe [16] investigated the local scour around pile under random waves. The experiments were conducted in a wave basin with a slender vertical pile (D = 0.032, 0.055 m). The seabed is composed of 0.14 m deep layer of sand with d50 = 0.2 mm and the water depth was maintained at 0.5 m. The JONSWAP wave spectrum was used and the KCrms,a was 5.29~16.95. The comparison between the predicting results by Equation (33) and the experimental results of Sumer and Fredsøe [16] and Corvaro et al. [21] are shown in Figure 21. From Figure 21, contrary to the case of low KCrms,a (KCrms,a < 4), the error between the predicting values and experimental results increases with decreasing of n for KCrms,a > 4. Therefore, the predicting results are the most favorable for n = 2 when KCrms,a > 4.

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Figure 21. Comparison of Seq between the experimental results of Sumer and Fredsøe [16] and Corvaro et al. [21] and the predicting values by Equation (33).

Noteworthy is that the present model was built according to prototype size, so the errors between the numerical results and experimental data of References [16,21] may be attribute to the scale effects. In laboratory experiments on scouring process, it is typically impossible to ensure a rigorous similarity of all physical parameters between the model and prototype structure, leading to the scale effects in the laboratory experiments. To avoid a cohesive behaviour, the bed material was not scaled geometrically according to model scale. As a consequence, the relatively large-scaled sediments sizes may result in the overestimation of bed load transport and underestimation of suspended load transport compared with field conditions. What’s more, the disproportional scaled sediment presumably lead to the difference of bed roughness between the model and prototype, and thus large influences for wave boundary layer on the seabed and scour process. Besides, according to Corvaro et al. [21] and Schendel et al. [55], the pile Reynolds numbers and Froude numbers both affect the scour depth for the condition of non fully developed turbulent flow in laboratory experiments.

4.4. Parametric Study

4.4.1. Influence of Froude Number

As described above, the set of foundation leads to the adverse pressure gradient appearing at upstream, leading to the wave boundary layer separating from seabed, then horseshoe vortex formatting and the horseshoe vortex are mainly responsible for scour around foundation (see Figure 22). The Froude number Fr is the key parameter to influence the scale and intensity of horseshoe vortex. The Fr under waves can be calculated by the following formula [42]

Fr=UwgD−−−√�r=�w��(34)

where Uw is the mean water particle velocity during 1/4 cycle of wave oscillation, obtained from the following formula. Noteworthy is that the root-mean-square (RMS) value of near-bed velocity amplitude Uwm,rms is used for calculating Uwm.

Uw=1T/4∫0T/4Uwmsin(t/T)dt=2πUwm�w=1�/4∫0�/4�wmsin(�/�)��=2��wm(35)

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Figure 22. Sketch of flow field at upstream USAF edges.

Tavouktsoglou et al. [25] proposed the following formula between Fr and the vertical location of the stagnation y

yh∝Fer�ℎ∝�r�(36)

where e is constant.

The Figure 23 displays the relationship between Seq/D and Fr of the present study. In order to compare with the simulating results, the experimental data of Corvaro et al. [21] was also depicted in Figure 23. As shown in Figure 23, the equilibrium scour depth appears a logarithmic increase as Fr increases and approaches the mathematical asymptotic value, which is also consistent with the experimental results of Corvaro et al. [21]. According to Figure 24, the adverse pressure gradient pressure at upstream USAF edges increases with the increase of Fr, which is benefit for the wave boundary layer separating from seabed, resulting in the high-intensity horseshoe vortex, hence, causing intensive scour around USAF. Based on the previous study of Tavouktsoglou et al. [25] for scour around pile under currents, the high Fr leads to the stagnation point is closer to the mean sea level for shallow water, causing the stronger downflow kinetic energy. As mentioned in previous section, the energy of downflow at upstream makes up the energy of the subsequent horseshoe vortex, so the stronger downflow kinetic energy results in the more intensive horseshoe vortex. Therefore, the higher Fr leads to the more intensive horseshoe vortex by influencing the position of stagnation point y presumably. Qi and Gao [19] carried out a series of flume tests to investigate the scour around pile under regular waves, and proposed the fitting formula between Seq/D and Fr as following

lg(Seq/D)=Aexp(B/Fr)+Clg(�eq/�)=�exp(�/�r)+�(37)

where AB and C are constant.

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Figure 23. The fitting curve between Seq/D and Fr.

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Figure 24. Sketch of adverse pressure gradient at upstream USAF edges.

Took the Equation (37) to fit the simulating results with A = −0.002, B = 0.686 and C = −0.808, and the results are shown in Figure 23. From Figure 23, the simulating results evenly distribute around the Equation (37) and the varying trend of Seq/D and Fr in present study is consistent with Equation (37) basically, meaning the Equation (37) is applicable to express the relationship of Seq/D with Fr around USAF under random waves.

4.4.2. Influence of Euler Number

The Euler number Eu is the influencing factor for the hydrodynamic field around foundation. The Eu under waves can be calculated by the following formula. The Eu can be represented by the Equation (38) for uniform cylinders [25]. The root-mean-square (RMS) value of near-bed velocity amplitude Um,rms is used for calculating Um.

Eu=U2mgD�u=�m2��(38)

where Um is depth-averaged flow velocity.

The Figure 25 displays the relationship between Seq/D and Eu of the present study. In order to compare with the simulating results, the experimental data of Sumer and Fredsøe [16] and Corvaro et al. [21] were also plotted in Figure 25. As shown in Figure 25, similar with the varying trend of Seq/D and Fr, the equilibrium scour depth appears a logarithmic increase as Eu increases and approaches the mathematical asymptotic value, which is also consistent with the experimental results of Sumer and Fredsøe [16] and Corvaro et al. [21]. According to Figure 24, the adverse pressure gradient pressure at upstream USAF edges increases with the increasing of Eu, which is benefit for the wave boundary layer separating from seabed, inducing the high-intensity horseshoe vortex, hence, causing intensive scour around USAF.

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Figure 25. The fitting curve between Seq/D and Eu.

Therefore, the variation of Fr and Eu reflect the magnitude of adverse pressure gradient pressure at upstream. Given that, the Equation (37) also was used to fit the simulating results with A = 8.875, B = 0.078 and C = −9.601, and the results are shown in Figure 25. From Figure 25, the simulating results evenly distribute around the Equation (37) and the varying trend of Seq/D and Eu in present study is consistent with Equation (37) basically, meaning the Equation (37) is also applicable to express the relationship of Seq/D with Eu around USAF under random waves. Additionally, according to the above description of Fr, it can be inferred that the higher Fr and Eu both lead to the more intensive horseshoe vortex by influencing the position of stagnation point y presumably.

5. Conclusions

A series of numerical models were established to investigate the local scour around umbrella suction anchor foundation (USAF) under random waves. The numerical model was validated for hydrodynamic and morphology parameters by comparing with the experimental data of Khosronejad et al. [52], Petersen et al. [17], Sumer and Fredsøe [16] and Schendel et al. [22]. Based on the simulating results, the scour evolution and scour mechanisms around USAF under random waves were analyzed respectively. Two revised models were proposed according to the model of Raaijmakers and Rudolph [34] and the stochastic model developed by Myrhaug and Rue [37] to predict the equilibrium scour depth around USAF under random waves. Finally, a parametric study was carried out with the present model to study the effects of the Froude number Fr and Euler number Eu to the equilibrium scour depth around USAF under random waves. The main conclusions can be described as follows.(1)

The packed sediment scour model and the RNG k−ε turbulence model were used to simulate the sand particles transport processes and the flow field around UASF respectively. The scour evolution obtained by the present model agrees well with the experimental results of Khosronejad et al. [52], Petersen et al. [17], Sumer and Fredsøe [16] and Schendel et al. [22], which indicates that the present model is accurate and reasonable for depicting the scour morphology around UASF under random waves.(2)

The vortex system at wave crest phase is mainly related to the scour process around USAF under random waves. The maximum scour depth appeared at the lee-side of the USAF at the initial stage (t < 1200 s). Subsequently, when t > 2400 s, the location of the maximum scour depth shifted to the upside of the USAF at an angle of about 45° with respect to the wave propagating direction.(3)

The error is negligible and the Raaijmakers’s model is of relatively high accuracy for predicting scour around USAF under random waves when KC is calculated by KCs,p. Given that, a further revision model (Equation (31)) was proposed according to Raaijmakers’s model to predict the equilibrium scour depth around USAF under random waves and it shows good agreement with the simulating results of the present study when KCs,p < 8.(4)

Another further revision model (Equation (33)) was proposed according to the stochastic model established by Myrhaug and Rue [37] to predict the equilibrium scour depth around USAF under random waves, and the predicting results are the most favorable for n = 10 when KCrms,a < 4. However, contrary to the case of low KCrms,a, the predicting results are the most favorable for n = 2 when KCrms,a > 4 by the comparison with experimental results of Sumer and Fredsøe [16] and Corvaro et al. [21].(5)

The same formula (Equation (37)) is applicable to express the relationship of Seq/D with Eu or Fr, and it can be inferred that the higher Fr and Eu both lead to the more intensive horseshoe vortex and larger Seq.

Author Contributions

Conceptualization, H.L. (Hongjun Liu); Data curation, R.H. and P.Y.; Formal analysis, X.W. and H.L. (Hao Leng); Funding acquisition, X.W.; Writing—original draft, R.H. and P.Y.; Writing—review & editing, X.W. and H.L. (Hao Leng); The final manuscript has been approved by all the authors. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Fundamental Research Funds for the Central Universities (grant number 202061027) and the National Natural Science Foundation of China (grant number 41572247).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

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Hu, R.; Liu, H.; Leng, H.; Yu, P.; Wang, X. Scour Characteristics and Equilibrium Scour Depth Prediction around Umbrella Suction Anchor Foundation under Random Waves. J. Mar. Sci. Eng. 20219, 886. https://doi.org/10.3390/jmse9080886

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Hu R, Liu H, Leng H, Yu P, Wang X. Scour Characteristics and Equilibrium Scour Depth Prediction around Umbrella Suction Anchor Foundation under Random Waves. Journal of Marine Science and Engineering. 2021; 9(8):886. https://doi.org/10.3390/jmse9080886Chicago/Turabian Style

Hu, Ruigeng, Hongjun Liu, Hao Leng, Peng Yu, and Xiuhai Wang. 2021. “Scour Characteristics and Equilibrium Scour Depth Prediction around Umbrella Suction Anchor Foundation under Random Waves” Journal of Marine Science and Engineering 9, no. 8: 886. https://doi.org/10.3390/jmse9080886

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Study on Hydrodynamic Performance of Unsymmetrical Double Vertical Slotted Barriers

침수된 강성 식생을 갖는 개방 수로 흐름의 특성에 대한 3차원 수치 시뮬레이션

A 3-D numerical simulation of the characteristics of open channel flows with submerged rigid vegetation

Journal of Hydrodynamics volume 33, pages833–843 (2021)Cite this article

Abstract

이 백서는 Flow-3D를 적용하여 다양한 흐름 배출 및 식생 시나리오가 흐름 속도(세로, 가로 및 수직 속도 포함)에 미치는 영향을 조사합니다.

실험적 측정을 통한 검증 후 식생직경, 식생높이, 유량방류량에 대한 민감도 분석을 수행하였다. 종방향 속도의 경우 흐름 구조에 가장 큰 영향을 미치는 것은 배출보다는 식생 직경에서 비롯됩니다.

그러나 식생 높이는 수직 분포의 변곡점을 결정합니다. 식생지 내 두 지점, 즉 상류와 하류의 횡속도를 비교하면 수심에 따른 대칭적인 패턴을 확인할 수 있다. 식생 지역의 가로 및 세로 유체 순환 패턴을 포함하여 흐름 또는 식생 시나리오와 관계없이 수직 속도에 대해서도 동일한 패턴이 관찰됩니다.

또한 식생의 직경이 클수록 이러한 패턴이 더 분명해집니다. 상부 순환은 초목 캐노피 근처에서 발생합니다. 식생지역의 가로방향과 세로방향의 순환에 관한 이러한 발견은 침수식생을 통한 3차원 유동구조를 밝혀준다.

This paper applies the Flow-3D to investigate the impacts of different flow discharge and vegetation scenarios on the flow velocity (including the longitudinal, transverse and vertical velocities). After the verification by using experimental measurements, a sensitivity analysis is conducted for the vegetation diameter, the vegetation height and the flow discharge. For the longitudinal velocity, the greatest impact on the flow structure originates from the vegetation diameter, rather than the discharge. The vegetation height, however, determines the inflection point of the vertical distribution. Comparing the transverse velocities at two positions in the vegetated area, i.e., the upstream and the downstream, a symmetric pattern is identified along the water depth. The same pattern is also observed for the vertical velocity regardless of the flow or vegetation scenario, including both transverse and vertical fluid circulation patterns in the vegetated area. Moreover, the larger the vegetation diameter is, the more evident these patterns become. The upper circulation occurs near the vegetation canopy. These findings regarding the circulations along the transverse and vertical directions in the vegetated region shed light on the 3-D flow structure through the submerged vegetation.

Key words

  • Submerged rigid vegetation
  • longitudinal velocity
  • transverse velocity
  • vertical velocity
  • open channel

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Figure 2 Modeling the plant with cylindrical tubes at the bottom of the canal.

Optimized Vegetation Density to Dissipate Energy of Flood Flow in Open Canals

열린 운하에서 홍수 흐름의 에너지를 분산시키기 위해 최적화된 식생 밀도

Mahdi Feizbahr,1Navid Tonekaboni,2Guang-Jun Jiang,3,4and Hong-Xia Chen3,4
Academic Editor: Mohammad Yazdi

Abstract

강을 따라 식생은 조도를 증가시키고 평균 유속을 감소시키며, 유동 에너지를 감소시키고 강 횡단면의 유속 프로파일을 변경합니다. 자연의 많은 운하와 강은 홍수 동안 초목으로 덮여 있습니다. 운하의 조도는 식물의 영향을 많이 받기 때문에 홍수시 유동저항에 큰 영향을 미친다. 식물로 인한 흐름에 대한 거칠기 저항은 흐름 조건과 식물에 따라 달라지므로 모델은 유속, 유속 깊이 및 수로를 따라 식생 유형의 영향을 고려하여 유속을 시뮬레이션해야 합니다. 총 48개의 모델을 시뮬레이션하여 근관의 거칠기 효과를 조사했습니다. 결과는 속도를 높임으로써 베드 속도를 감소시키는 식생의 영향이 무시할만하다는 것을 나타냅니다.

Abstract

Vegetation along the river increases the roughness and reduces the average flow velocity, reduces flow energy, and changes the flow velocity profile in the cross section of the river. Many canals and rivers in nature are covered with vegetation during the floods. Canal’s roughness is strongly affected by plants and therefore it has a great effect on flow resistance during flood. Roughness resistance against the flow due to the plants depends on the flow conditions and plant, so the model should simulate the current velocity by considering the effects of velocity, depth of flow, and type of vegetation along the canal. Total of 48 models have been simulated to investigate the effect of roughness in the canal. The results indicated that, by enhancing the velocity, the effect of vegetation in decreasing the bed velocity is negligible, while when the current has lower speed, the effect of vegetation on decreasing the bed velocity is obviously considerable.

1. Introduction

Considering the impact of each variable is a very popular field within the analytical and statistical methods and intelligent systems [114]. This can help research for better modeling considering the relation of variables or interaction of them toward reaching a better condition for the objective function in control and engineering [1527]. Consequently, it is necessary to study the effects of the passive factors on the active domain [2836]. Because of the effect of vegetation on reducing the discharge capacity of rivers [37], pruning plants was necessary to improve the condition of rivers. One of the important effects of vegetation in river protection is the action of roots, which cause soil consolidation and soil structure improvement and, by enhancing the shear strength of soil, increase the resistance of canal walls against the erosive force of water. The outer limbs of the plant increase the roughness of the canal walls and reduce the flow velocity and deplete the flow energy in vicinity of the walls. Vegetation by reducing the shear stress of the canal bed reduces flood discharge and sedimentation in the intervals between vegetation and increases the stability of the walls [3841].

One of the main factors influencing the speed, depth, and extent of flood in this method is Manning’s roughness coefficient. On the other hand, soil cover [42], especially vegetation, is one of the most determining factors in Manning’s roughness coefficient. Therefore, it is expected that those seasonal changes in the vegetation of the region will play an important role in the calculated value of Manning’s roughness coefficient and ultimately in predicting the flood wave behavior [4345]. The roughness caused by plants’ resistance to flood current depends on the flow and plant conditions. Flow conditions include depth and velocity of the plant, and plant conditions include plant type, hardness or flexibility, dimensions, density, and shape of the plant [46]. In general, the issue discussed in this research is the optimization of flood-induced flow in canals by considering the effect of vegetation-induced roughness. Therefore, the effect of plants on the roughness coefficient and canal transmission coefficient and in consequence the flow depth should be evaluated [4748].

Current resistance is generally known by its roughness coefficient. The equation that is mainly used in this field is Manning equation. The ratio of shear velocity to average current velocity  is another form of current resistance. The reason for using the  ratio is that it is dimensionless and has a strong theoretical basis. The reason for using Manning roughness coefficient is its pervasiveness. According to Freeman et al. [49], the Manning roughness coefficient for plants was calculated according to the Kouwen and Unny [50] method for incremental resistance. This method involves increasing the roughness for various surface and plant irregularities. Manning’s roughness coefficient has all the factors affecting the resistance of the canal. Therefore, the appropriate way to more accurately estimate this coefficient is to know the factors affecting this coefficient [51].

To calculate the flow rate, velocity, and depth of flow in canals as well as flood and sediment estimation, it is important to evaluate the flow resistance. To determine the flow resistance in open ducts, Manning, Chézy, and Darcy–Weisbach relations are used [52]. In these relations, there are parameters such as Manning’s roughness coefficient (n), Chézy roughness coefficient (C), and Darcy–Weisbach coefficient (f). All three of these coefficients are a kind of flow resistance coefficient that is widely used in the equations governing flow in rivers [53].

The three relations that express the relationship between the average flow velocity (V) and the resistance and geometric and hydraulic coefficients of the canal are as follows:where nf, and c are Manning, Darcy–Weisbach, and Chézy coefficients, respectively. V = average flow velocity, R = hydraulic radius, Sf = slope of energy line, which in uniform flow is equal to the slope of the canal bed,  = gravitational acceleration, and Kn is a coefficient whose value is equal to 1 in the SI system and 1.486 in the English system. The coefficients of resistance in equations (1) to (3) are related as follows:

Based on the boundary layer theory, the flow resistance for rough substrates is determined from the following general relation:where f = Darcy–Weisbach coefficient of friction, y = flow depth, Ks = bed roughness size, and A = constant coefficient.

On the other hand, the relationship between the Darcy–Weisbach coefficient of friction and the shear velocity of the flow is as follows:

By using equation (6), equation (5) is converted as follows:

Investigation on the effect of vegetation arrangement on shear velocity of flow in laboratory conditions showed that, with increasing the shear Reynolds number (), the numerical value of the  ratio also increases; in other words the amount of roughness coefficient increases with a slight difference in the cases without vegetation, checkered arrangement, and cross arrangement, respectively [54].

Roughness in river vegetation is simulated in mathematical models with a variable floor slope flume by different densities and discharges. The vegetation considered submerged in the bed of the flume. Results showed that, with increasing vegetation density, canal roughness and flow shear speed increase and with increasing flow rate and depth, Manning’s roughness coefficient decreases. Factors affecting the roughness caused by vegetation include the effect of plant density and arrangement on flow resistance, the effect of flow velocity on flow resistance, and the effect of depth [4555].

One of the works that has been done on the effect of vegetation on the roughness coefficient is Darby [56] study, which investigates a flood wave model that considers all the effects of vegetation on the roughness coefficient. There are currently two methods for estimating vegetation roughness. One method is to add the thrust force effect to Manning’s equation [475758] and the other method is to increase the canal bed roughness (Manning-Strickler coefficient) [455961]. These two methods provide acceptable results in models designed to simulate floodplain flow. Wang et al. [62] simulate the floodplain with submerged vegetation using these two methods and to increase the accuracy of the results, they suggested using the effective height of the plant under running water instead of using the actual height of the plant. Freeman et al. [49] provided equations for determining the coefficient of vegetation roughness under different conditions. Lee et al. [63] proposed a method for calculating the Manning coefficient using the flow velocity ratio at different depths. Much research has been done on the Manning roughness coefficient in rivers, and researchers [496366] sought to obtain a specific number for n to use in river engineering. However, since the depth and geometric conditions of rivers are completely variable in different places, the values of Manning roughness coefficient have changed subsequently, and it has not been possible to choose a fixed number. In river engineering software, the Manning roughness coefficient is determined only for specific and constant conditions or normal flow. Lee et al. [63] stated that seasonal conditions, density, and type of vegetation should also be considered. Hydraulic roughness and Manning roughness coefficient n of the plant were obtained by estimating the total Manning roughness coefficient from the matching of the measured water surface curve and water surface height. The following equation is used for the flow surface curve:where  is the depth of water change, S0 is the slope of the canal floor, Sf is the slope of the energy line, and Fr is the Froude number which is obtained from the following equation:where D is the characteristic length of the canal. Flood flow velocity is one of the important parameters of flood waves, which is very important in calculating the water level profile and energy consumption. In the cases where there are many limitations for researchers due to the wide range of experimental dimensions and the variety of design parameters, the use of numerical methods that are able to estimate the rest of the unknown results with acceptable accuracy is economically justified.

FLOW-3D software uses Finite Difference Method (FDM) for numerical solution of two-dimensional and three-dimensional flow. This software is dedicated to computational fluid dynamics (CFD) and is provided by Flow Science [67]. The flow is divided into networks with tubular cells. For each cell there are values of dependent variables and all variables are calculated in the center of the cell, except for the velocity, which is calculated at the center of the cell. In this software, two numerical techniques have been used for geometric simulation, FAVOR™ (Fractional-Area-Volume-Obstacle-Representation) and the VOF (Volume-of-Fluid) method. The equations used at this model for this research include the principle of mass survival and the magnitude of motion as follows. The fluid motion equations in three dimensions, including the Navier–Stokes equations with some additional terms, are as follows:where  are mass accelerations in the directions xyz and  are viscosity accelerations in the directions xyz and are obtained from the following equations:

Shear stresses  in equation (11) are obtained from the following equations:

The standard model is used for high Reynolds currents, but in this model, RNG theory allows the analytical differential formula to be used for the effective viscosity that occurs at low Reynolds numbers. Therefore, the RNG model can be used for low and high Reynolds currents.

Weather changes are high and this affects many factors continuously. The presence of vegetation in any area reduces the velocity of surface flows and prevents soil erosion, so vegetation will have a significant impact on reducing destructive floods. One of the methods of erosion protection in floodplain watersheds is the use of biological methods. The presence of vegetation in watersheds reduces the flow rate during floods and prevents soil erosion. The external organs of plants increase the roughness and decrease the velocity of water flow and thus reduce its shear stress energy. One of the important factors with which the hydraulic resistance of plants is expressed is the roughness coefficient. Measuring the roughness coefficient of plants and investigating their effect on reducing velocity and shear stress of flow is of special importance.

Roughness coefficients in canals are affected by two main factors, namely, flow conditions and vegetation characteristics [68]. So far, much research has been done on the effect of the roughness factor created by vegetation, but the issue of plant density has received less attention. For this purpose, this study was conducted to investigate the effect of vegetation density on flow velocity changes.

In a study conducted using a software model on three density modes in the submerged state effect on flow velocity changes in 48 different modes was investigated (Table 1).

Table 1 

The studied models.

The number of cells used in this simulation is equal to 1955888 cells. The boundary conditions were introduced to the model as a constant speed and depth (Figure 1). At the output boundary, due to the presence of supercritical current, no parameter for the current is considered. Absolute roughness for floors and walls was introduced to the model (Figure 1). In this case, the flow was assumed to be nonviscous and air entry into the flow was not considered. After  seconds, this model reached a convergence accuracy of .

Figure 1 

The simulated model and its boundary conditions.

Due to the fact that it is not possible to model the vegetation in FLOW-3D software, in this research, the vegetation of small soft plants was studied so that Manning’s coefficients can be entered into the canal bed in the form of roughness coefficients obtained from the studies of Chow [69] in similar conditions. In practice, in such modeling, the effect of plant height is eliminated due to the small height of herbaceous plants, and modeling can provide relatively acceptable results in these conditions.

48 models with input velocities proportional to the height of the regular semihexagonal canal were considered to create supercritical conditions. Manning coefficients were applied based on Chow [69] studies in order to control the canal bed. Speed profiles were drawn and discussed.

Any control and simulation system has some inputs that we should determine to test any technology [7077]. Determination and true implementation of such parameters is one of the key steps of any simulation [237881] and computing procedure [8286]. The input current is created by applying the flow rate through the VFR (Volume Flow Rate) option and the output flow is considered Output and for other borders the Symmetry option is considered.

Simulation of the models and checking their action and responses and observing how a process behaves is one of the accepted methods in engineering and science [8788]. For verification of FLOW-3D software, the results of computer simulations are compared with laboratory measurements and according to the values of computational error, convergence error, and the time required for convergence, the most appropriate option for real-time simulation is selected (Figures 2 and 3 ).

Figure 2 

Modeling the plant with cylindrical tubes at the bottom of the canal.

Figure 3 

Velocity profiles in positions 2 and 5.

The canal is 7 meters long, 0.5 meters wide, and 0.8 meters deep. This test was used to validate the application of the software to predict the flow rate parameters. In this experiment, instead of using the plant, cylindrical pipes were used in the bottom of the canal.

The conditions of this modeling are similar to the laboratory conditions and the boundary conditions used in the laboratory were used for numerical modeling. The critical flow enters the simulation model from the upstream boundary, so in the upstream boundary conditions, critical velocity and depth are considered. The flow at the downstream boundary is supercritical, so no parameters are applied to the downstream boundary.

The software well predicts the process of changing the speed profile in the open canal along with the considered obstacles. The error in the calculated speed values can be due to the complexity of the flow and the interaction of the turbulence caused by the roughness of the floor with the turbulence caused by the three-dimensional cycles in the hydraulic jump. As a result, the software is able to predict the speed distribution in open canals.

2. Modeling Results

After analyzing the models, the results were shown in graphs (Figures 414 ). The total number of experiments in this study was 48 due to the limitations of modeling.


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Figure 4 

Flow velocity profiles for canals with a depth of 1 m and flow velocities of 3–3.3 m/s. Canal with a depth of 1 meter and a flow velocity of (a) 3 meters per second, (b) 3.1 meters per second, (c) 3.2 meters per second, and (d) 3.3 meters per second.

Figure 5 

Canal diagram with a depth of 1 meter and a flow rate of 3 meters per second.

Figure 6 

Canal diagram with a depth of 1 meter and a flow rate of 3.1 meters per second.

Figure 7 

Canal diagram with a depth of 1 meter and a flow rate of 3.2 meters per second.

Figure 8 

Canal diagram with a depth of 1 meter and a flow rate of 3.3 meters per second.


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Figure 9 

Flow velocity profiles for canals with a depth of 2 m and flow velocities of 4–4.3 m/s. Canal with a depth of 2 meters and a flow rate of (a) 4 meters per second, (b) 4.1 meters per second, (c) 4.2 meters per second, and (d) 4.3 meters per second.

Figure 10 

Canal diagram with a depth of 2 meters and a flow rate of 4 meters per second.

Figure 11 

Canal diagram with a depth of 2 meters and a flow rate of 4.1 meters per second.

Figure 12 

Canal diagram with a depth of 2 meters and a flow rate of 4.2 meters per second.

Figure 13 

Canal diagram with a depth of 2 meters and a flow rate of 4.3 meters per second.


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Figure 14 

Flow velocity profiles for canals with a depth of 3 m and flow velocities of 5–5.3 m/s. Canal with a depth of 2 meters and a flow rate of (a) 4 meters per second, (b) 4.1 meters per second, (c) 4.2 meters per second, and (d) 4.3 meters per second.

To investigate the effects of roughness with flow velocity, the trend of flow velocity changes at different depths and with supercritical flow to a Froude number proportional to the depth of the section has been obtained.

According to the velocity profiles of Figure 5, it can be seen that, with the increasing of Manning’s coefficient, the canal bed speed decreases.

According to Figures 5 to 8, it can be found that, with increasing the Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of the models 1 to 12, which can be justified by increasing the speed and of course increasing the Froude number.

According to Figure 10, we see that, with increasing Manning’s coefficient, the canal bed speed decreases.

According to Figure 11, we see that, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of Figures 510, which can be justified by increasing the speed and, of course, increasing the Froude number.

With increasing Manning’s coefficient, the canal bed speed decreases (Figure 12). But this deceleration is more noticeable than the deceleration of the higher models (Figures 58 and 1011), which can be justified by increasing the speed and, of course, increasing the Froude number.

According to Figure 13, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of Figures 5 to 12, which can be justified by increasing the speed and, of course, increasing the Froude number.

According to Figure 15, with increasing Manning’s coefficient, the canal bed speed decreases.

Figure 15 

Canal diagram with a depth of 3 meters and a flow rate of 5 meters per second.

According to Figure 16, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of the higher model, which can be justified by increasing the speed and, of course, increasing the Froude number.

Figure 16 

Canal diagram with a depth of 3 meters and a flow rate of 5.1 meters per second.

According to Figure 17, it is clear that, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of the higher models, which can be justified by increasing the speed and, of course, increasing the Froude number.

Figure 17 

Canal diagram with a depth of 3 meters and a flow rate of 5.2 meters per second.

According to Figure 18, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of the higher models, which can be justified by increasing the speed and, of course, increasing the Froude number.

Figure 18 

Canal diagram with a depth of 3 meters and a flow rate of 5.3 meters per second.

According to Figure 19, it can be seen that the vegetation placed in front of the flow input velocity has negligible effect on the reduction of velocity, which of course can be justified due to the flexibility of the vegetation. The only unusual thing is the unexpected decrease in floor speed of 3 m/s compared to higher speeds.


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Figure 19 

Comparison of velocity profiles with the same plant densities (depth 1 m). Comparison of velocity profiles with (a) plant densities of 25%, depth 1 m; (b) plant densities of 50%, depth 1 m; and (c) plant densities of 75%, depth 1 m.

According to Figure 20, by increasing the speed of vegetation, the effect of vegetation on reducing the flow rate becomes more noticeable. And the role of input current does not have much effect in reducing speed.


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Figure 20 

Comparison of velocity profiles with the same plant densities (depth 2 m). Comparison of velocity profiles with (a) plant densities of 25%, depth 2 m; (b) plant densities of 50%, depth 2 m; and (c) plant densities of 75%, depth 2 m.

According to Figure 21, it can be seen that, with increasing speed, the effect of vegetation on reducing the bed flow rate becomes more noticeable and the role of the input current does not have much effect. In general, it can be seen that, by increasing the speed of the input current, the slope of the profiles increases from the bed to the water surface and due to the fact that, in software, the roughness coefficient applies to the channel floor only in the boundary conditions, this can be perfectly justified. Of course, it can be noted that, due to the flexible conditions of the vegetation of the bed, this modeling can show acceptable results for such grasses in the canal floor. In the next directions, we may try application of swarm-based optimization methods for modeling and finding the most effective factors in this research [27815188994]. In future, we can also apply the simulation logic and software of this research for other domains such as power engineering [9599].


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Figure 21 

Comparison of velocity profiles with the same plant densities (depth 3 m). Comparison of velocity profiles with (a) plant densities of 25%, depth 3 m; (b) plant densities of 50%, depth 3 m; and (c) plant densities of 75%, depth 3 m.

3. Conclusion

The effects of vegetation on the flood canal were investigated by numerical modeling with FLOW-3D software. After analyzing the results, the following conclusions were reached:(i)Increasing the density of vegetation reduces the velocity of the canal floor but has no effect on the velocity of the canal surface.(ii)Increasing the Froude number is directly related to increasing the speed of the canal floor.(iii)In the canal with a depth of one meter, a sudden increase in speed can be observed from the lowest speed and higher speed, which is justified by the sudden increase in Froude number.(iv)As the inlet flow rate increases, the slope of the profiles from the bed to the water surface increases.(v)By reducing the Froude number, the effect of vegetation on reducing the flow bed rate becomes more noticeable. And the input velocity in reducing the velocity of the canal floor does not have much effect.(vi)At a flow rate between 3 and 3.3 meters per second due to the shallow depth of the canal and the higher landing number a more critical area is observed in which the flow bed velocity in this area is between 2.86 and 3.1 m/s.(vii)Due to the critical flow velocity and the slight effect of the roughness of the horseshoe vortex floor, it is not visible and is only partially observed in models 1-2-3 and 21.(viii)As the flow rate increases, the effect of vegetation on the rate of bed reduction decreases.(ix)In conditions where less current intensity is passing, vegetation has a greater effect on reducing current intensity and energy consumption increases.(x)In the case of using the flow rate of 0.8 cubic meters per second, the velocity distribution and flow regime show about 20% more energy consumption than in the case of using the flow rate of 1.3 cubic meters per second.

Nomenclature

n:Manning’s roughness coefficient
C:Chézy roughness coefficient
f:Darcy–Weisbach coefficient
V:Flow velocity
R:Hydraulic radius
g:Gravitational acceleration
y:Flow depth
Ks:Bed roughness
A:Constant coefficient
:Reynolds number
y/∂x:Depth of water change
S0:Slope of the canal floor
Sf:Slope of energy line
Fr:Froude number
D:Characteristic length of the canal
G:Mass acceleration
:Shear stresses.

Data Availability

All data are included within the paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China under Contract no. 71761030 and Natural Science Foundation of Inner Mongolia under Contract no. 2019LH07003.

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Strain rate magnitude at the free surface, illustrating Kelvin-Helmoltz (KH) shear instabilities.

On the reef scale hydrodynamics at Sodwana Bay, South Africa

Environmental Fluid Mechanics (2022)Cite this article

Abstract

The hydrodynamics of coral reefs strongly influences their biological functioning, impacting processes such as nutrient availability and uptake, recruitment success and bleaching. For example, coral reefs located in oligotrophic regions depend on upwelling for nutrient supply. Coral reefs at Sodwana Bay, located on the east coast of South Africa, are an example of high latitude marginal reefs. These reefs are subjected to complex hydrodynamic forcings due to the interaction between the strong Agulhas current and the highly variable topography of the region. In this study, we explore the reef scale hydrodynamics resulting from the bathymetry for two steady current scenarios at Two-Mile Reef (TMR) using a combination of field data and numerical simulations. The influence of tides or waves was not considered for this study as well as reef-scale roughness. Tilt current meters with onboard temperature sensors were deployed at selected locations within TMR. We used field observations to identify the dominant flow conditions on the reef for numerical simulations that focused on the hydrodynamics driven by mean currents. During the field campaign, southerly currents were the predominant flow feature with occasional flow reversals to the north. Northerly currents were associated with greater variability towards the southern end of TMR. Numerical simulations showed that Jesser Point was central to the development of flow features for both the northerly and southerly current scenarios. High current variability in the south of TMR during reverse currents is related to the formation of Kelvin-Helmholtz type shear instabilities along the outer edge of an eddy formed north of Jesser Point. Furthermore, downward vertical velocities were computed along the offshore shelf at TMR during southerly currents. Current reversals caused a change in vertical velocities to an upward direction due to the orientation of the bathymetry relative to flow directions.

Highlights

  • A predominant southerly current was measured at Two-Mile Reef with occasional reversals towards the north.
  • Field observations indicated that northerly currents are spatially varied along Two-Mile Reef.
  • Simulation of reverse currents show the formation of a separated flow due to interaction with Jesser Point with Kelvin–Helmholtz type shear instabilities along the seaward edge.

지금까지 Sodwana Bay에서 자세한 암초 규모 유체 역학을 모델링하려는 시도는 없었습니다. 이러한 모델의 결과는 규모가 있는 산호초 사이의 흐름이 산호초 건강에 어떤 영향을 미치는지 탐색하는 데 사용할 수 있습니다. 이 연구에서는 Sodwana Bay의 유체역학을 탐색하는 데 사용할 수 있는 LES 모델을 개발하기 위한 단계별 접근 방식을 구현합니다. 여기서 우리는 이 초기 단계에서 파도와 조수의 영향을 배제하면서 Agulhas 해류의 유체역학에 초점을 맞춥니다. 이 접근법은 흐름의 첫 번째 LES를 제시하고 Sodwana Bay의 산호초에서 혼합함으로써 향후 연구의 기초를 제공합니다.

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Figure 1 | Laboratory channel dimensions.

강화된 조도 계수 및 인버트 레벨 변화가 있는 90도 측면 턴아웃에서의 유동에 대한 실험적 및 수치적 연구

Experimental and numerical study of flow at a 90 degree lateral turnout with enhanced roughness coefficient and invert level changes

Maryam Bagheria, Seyed M. Ali Zomorodianb, Masih Zolghadrc, H. Md. Azamathulla d,*
and C. Venkata Siva Rama Prasade
a Hydraulic Structures, Department of Water Engineering, Shiraz University, Shiraz, Iran
b Department of Water Engineering, College of Agriculture, Shiraz University, Shiraz, Iran
c Department of Water Sciences Engineering, College of Agriculture, Jahrom University, Jahrom, Iran
d Civil & Environmental Engineering, The University of the West Indies, St. Augustine Campus, Port of Spain, Trinidad
e Department of Civil Engineering, St. Peters Engineering College, Hyderabad, India
*Corresponding author. E-mail: azmatheditor@gmail.com

ABSTRACT

측면 분기기(흡입구)의 상류측에서 유동 분리는 분기기 입구에서 맴돌이 전류를 일으키는 중요한 문제입니다. 이는 흐름의 유효 폭, 분기 용량 및 효율성을 감소시킵니다. 따라서 분리구역의 크기를 파악하고 그 크기를 줄이기 위한 방안을 제시하는 것이 필수적이다.

본 연구에서는 분리 구역의 크기를 줄이기 위한 방법으로 분출구 입구에 7가지 유형의 조면화 요소와 4가지 다른 방류가 있는 3가지 다른 베드 인버트 레벨의 설치(총 84회 실험)를 조사했습니다. 또한 3D 전산 유체 역학(CFD) 모델을 사용하여 분리 구역의 흐름 패턴과 치수를 평가했습니다.

결과는 조도 계수를 향상시키면 분리 영역 치수를 최대 38%까지 줄일 수 있는 반면 드롭 구현 효과는 사용된 조도 계수에 따라 이 영역을 다르게 축소할 수 있음을 보여주었습니다. 두 방법을 결합하면 분리 구역 치수를 최대 63%까지 줄일 수 있습니다.

Flow separation at the upstream side of lateral turnouts (intakes) is a critical issue causing eddy currents at the turnout entrance. It reduces the effective width of flow, turnout capacity and efficiency. Therefore, it is essential to identify the dimensions of the separation zone and propose remedies to reduce its dimensions.

Installation of 7 types of roughening elements at the turnout entrance and 3 different bed invert levels, with 4 different discharges (making a total of 84 experiments) were examined in this study as a method to reduce the dimensions of the separation zone. Additionally, a 3-D Computational Fluid Dynamic (CFD) model was utilized to evaluate the flow pattern and dimensions of the separation zone.

Results showed that enhancing the roughness coefficient can reduce the separation zone dimensions up to 38% while the drop implementation effect can scale down this area differently based on the roughness coefficient used. Combining both methods can reduce the separation zone dimensions up to 63%.

Key words

discharge ratio, flow separation zone, intake, three dimensional simulation

Experimental and numerical study of flow at a 90 degree lateral turnout with enhanced
roughness coefficient and invert level changes
Experimental and numerical study of flow at a 90 degree lateral turnout with enhanced roughness coefficient and invert level changes
Figure 1 | Laboratory channel dimensions.
Figure 1 | Laboratory channel dimensions.
Figure 2 | Roughness plates.
Figure 2 | Roughness plates.
Figure 4 | Effect of roughness on separation zone dimensions.
Figure 4 | Effect of roughness on separation zone dimensions.
Figure 10 | Comparision of the vortex area (software output) for three roughnesses (0.009, 0.023 and 0.032).
Figure 10 | Comparision of the vortex area (software output) for three roughnesses (0.009, 0.023 and 0.032).
Figure 11 | Comparison of vortex area in 3D mode (tecplot output) with two roughnesses (a) 0.009 and (b) 0.032.
Figure 11 | Comparison of vortex area in 3D mode (tecplot output) with two roughnesses (a) 0.009 and (b) 0.032.
Figure 12 | Velocity vector for flow condition Q¼22 l/s, near surface.
Figure 12 | Velocity vector for flow condition Q¼22 l/s, near surface.

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Figure 14. Defects: (a) Unmelt defects(Scheme NO.4);(b) Pores defects(Scheme NO.1); (c); Spattering defect (Scheme NO.3); (d) Low overlapping rate defects(Scheme NO.5).

Molten pool structure, temperature and velocity
flow in selective laser melting AlCu5MnCdVA alloy

용융 풀 구조, 선택적 온도 및 속도 흐름 레이저 용융 AlCu5MnCdVA 합금

Pan Lu1 , Zhang Cheng-Lin2,6,Wang Liang3, Liu Tong4 and Liu Jiang-lin5
1 Aviation and Materials College, Anhui Technical College of Mechanical and Electrical Engineering, Wuhu Anhui 241000, People’s
Republic of China 2 School of Engineering Science, University of Science and Technology of China, Hefei Anhui 230026, People’s Republic of China 3 Anhui Top Additive Manufacturing Technology Co., Ltd., Wuhu Anhui 241300, People’s Republic of China 4 Anhui Chungu 3D Printing Institute of Intelligent Equipment and Industrial Technology, Anhui 241300, People’s Republic of China 5 School of Mechanical and Transportation Engineering, Taiyuan University of Technology, Taiyuan Shanxi 030024, People’s Republic of
China 6 Author to whom any correspondence should be addressed.
E-mail: ahjdpanlu@126.com, jiao__zg@126.com, ahjdjxx001@126.com,tongliu1988@126.com and liujianglin@tyut.edu.cn

Keywords

SLM, molten pool, AlCu5MnCdVA alloy, heat flow, velocity flow, numerical simulation

Abstract

선택적 레이저 용융(SLM)은 열 전달, 용융, 상전이, 기화 및 물질 전달을 포함하는 복잡한 동적 비평형 프로세스인 금속 적층 제조(MAM)에서 가장 유망한 기술 중 하나가 되었습니다. 용융 풀의 특성(구조, 온도 흐름 및 속도 흐름)은 SLM의 최종 성형 품질에 결정적인 영향을 미칩니다. 이 연구에서는 선택적 레이저 용융 AlCu5MnCdVA 합금의 용융 풀 구조, 온도 흐름 및 속도장을 연구하기 위해 수치 시뮬레이션과 실험을 모두 사용했습니다.

그 결과 용융풀의 구조는 다양한 형태(깊은 오목 구조, 이중 오목 구조, 평면 구조, 돌출 구조 및 이상적인 평면 구조)를 나타냈으며, 용융 풀의 크기는 약 132 μm × 107 μm × 50 μm였습니다. : 용융풀은 초기에는 여러 구동력에 의해 깊이 15μm의 깊은 오목형상이었으나, 성형 후기에는 장력구배에 의해 높이 10μm의 돌출형상이 되었다. 용융 풀 내부의 금속 흐름은 주로 레이저 충격력, 금속 액체 중력, 표면 장력 및 반동 압력에 의해 구동되었습니다.

AlCu5MnCdVA 합금의 경우, 금속 액체 응고 속도가 매우 빠르며(3.5 × 10-4 S), 가열 속도 및 냉각 속도는 각각 6.5 × 107 K S-1 및 1.6 × 106 K S-1 에 도달했습니다. 시각적 표준으로 표면 거칠기를 선택하고, 낮은 레이저 에너지 AlCu5MnCdVA 합금 최적 공정 매개변수 창을 수치 시뮬레이션으로 얻었습니다: 레이저 출력 250W, 부화 공간 0.11mm, 층 두께 0.03mm, 레이저 스캔 속도 1.5m s-1 .

또한, 실험 프린팅과 수치 시뮬레이션과 비교할 때, 용융 풀의 폭은 각각 약 205um 및 약 210um이었고, 인접한 두 용융 트랙 사이의 중첩은 모두 약 65um이었다. 결과는 수치 시뮬레이션 결과가 실험 인쇄 결과와 기본적으로 일치함을 보여 수치 시뮬레이션 모델의 정확성을 입증했습니다.

Selective Laser Melting (SLM) has become one of the most promising technologies in Metal Additive Manufacturing (MAM), which is a complex dynamic non-equilibrium process involving heat transfer, melting, phase transition, vaporization and mass transfer. The characteristics of the molten pool (structure, temperature flow and velocity flow) have a decisive influence on the final forming quality of SLM. In this study, both numerical simulation and experiments were employed to study molten pool structure, temperature flow and velocity field in Selective Laser Melting AlCu5MnCdVA alloy. The results showed the structure of molten pool showed different forms(deep-concave structure, double-concave structure, plane structure, protruding structure and ideal planar structure), and the size of the molten pool was approximately 132 μm × 107 μm × 50 μm: in the early stage, molten pool was in a state of deep-concave shape with a depth of 15 μm due to multiple driving forces, while a protruding shape with a height of 10 μm duo to tension gradient in the later stages of forming. The metal flow inside the molten pool was mainly driven by laser impact force, metal liquid gravity, surface tension and recoil pressure. For AlCu5MnCdVA alloy, metal liquid solidification speed was extremely fast(3.5 × 10−4 S), the heating rate and cooling rate reached 6.5 × 107 K S−1 and 1.6 × 106 K S−1 , respectively. Choosing surface roughness as a visual standard, low-laser energy AlCu5MnCdVA alloy optimum process parameters window was obtained by numerical simulation: laser power 250 W, hatching space 0.11 mm, layer thickness 0.03 mm, laser scanning velocity 1.5 m s−1 . In addition, compared with experimental printing and numerical simulation, the width of the molten pool was about 205 um and about 210 um, respectively, and overlapping between two adjacent molten tracks was all about 65 um. The results showed that the numerical simulation results were basically consistent with the experimental print results, which proved the correctness of the numerical simulation model.

Figure 1. AlCu5MnCdVA powder particle size distribution.
Figure 1. AlCu5MnCdVA powder particle size distribution.
Figure 2. AlCu5MnCdVA powder
Figure 2. AlCu5MnCdVA powder
Figure 3. Finite element model and calculation domains of SLM.
Figure 3. Finite element model and calculation domains of SLM.
Figure 4. SLM heat transfer process.
Figure 4. SLM heat transfer process.
Figure 14. Defects: (a) Unmelt defects(Scheme NO.4);(b) Pores defects(Scheme NO.1); (c); Spattering defect (Scheme NO.3); (d) Low
overlapping rate defects(Scheme NO.5).
Figure 17. Two-pass molten tracks overlapping for Scheme NO.2.
Figure 17. Two-pass molten tracks overlapping for Scheme NO.2.

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Development of macro-defect-free PBF-EB-processed Ti–6Al–4V alloys with superior plasticity using PREP-synthesized powder and machine learning-assisted process optimization

Development of macro-defect-free PBF-EB-processed Ti–6Al–4V alloys with superior plasticity using PREP-synthesized powder and machine learning-assisted process optimization

Yunwei GuiabKenta Aoyagib Akihiko Chibab
aDepartment of Materials Processing, Graduate School of Engineering, Tohoku University, 6-6 Aramaki Aza Aoba, Aoba-ku, Sendai, 980-8579, Japan
bInstitute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan

Received 14 October 2022, Revised 23 December 2022, Accepted 3 January 2023, Available online 5 January 2023.Show lessAdd to MendeleyShareCite

https://doi.org/10.1016/j.msea.2023.144595Get rights and content

Abstract

The elimination of internal macro-defects is a key issue in Ti–6Al–4V alloys fabricated via powder bed fusion using electron beams (PBF-EB), wherein internal macro-defects mainly originate from the virgin powder and inappropriate printing parameters. This study compares different types powders by combining support vector machine techniques to determine the most suitable powder for PBF-EB and to predict the processing window for the printing parameters without internal macro-defects. The results show that powders fabricated via plasma rotating electrode process have the best sphericity, flowability, and minimal porosity and are most suitable for printing. Surface roughness criterion was also applied to determine the quality of the even surfaces, and support vector machine was used to construct processing maps capable of predicting a wide range of four-dimensional printing parameters to obtain macro-defect-free samples, offering the possibility of subsequent development of Ti–6Al–4V alloys with excellent properties. The macro-defect-free samples exhibited good elongation, with the best overall mechanical properties being the ultimate tensile strength and elongation of 934.7 MPa and 24.3%, respectively. The elongation of the three macro-defect-free samples was much higher than that previously reported for additively manufactured Ti–6Al–4V alloys. The high elongation of the samples in this work is mainly attributed to the elimination of internal macro-defects.

Introduction

Additive manufacturing (AM) technologies can rapidly manufacture complex or custom parts, reducing process steps and saving manufacturing time [[1], [2], [3], [4]], and are widely used in the aerospace, automotive, and other precision industries [5,6]. Powder bed fusion using an electron beam (PBF-EB) is an additive manufacturing method that uses a high-energy electron beam to melt metal powders layer by layer to produce parts. In contrast to selective laser melting, PBF-EB involves the preparation of samples in a high vacuum environment, which effectively prevents the introduction of impurities such as O and N. It also involves a preheating process for the print substrate and powder, which reduces residual thermal stress on the sample and subsequent heat treatment processes [[2], [3], [4],7]. Due to these features and advantages, PBF-EB technology is a very important AM technology with great potential in metallic materials. Moreover, PBF-EB is the ideal AM technology for the manufacture of complex components made of many alloys, such as titanium alloys, nickel-based superalloys, aluminum alloys and stainless steels [[2], [3], [4],8].

Ti–6Al–4V alloy is one of the prevalent commercial titanium alloys possessing high specific strength, excellent mechanical properties, excellent corrosion resistance, and good biocompatibility [9,10]. It is widely used in applications requiring low density and excellent corrosion resistance, such as the aerospace industry and biomechanical applications [11,12]. The mechanical properties of PBF-EB-processed Ti–6Al–4V alloys are superior to those fabricated by casting or forging, because the rapid cooling rate in PBF-EB results in finer grains [[12], [13], [14], [15], [16], [17], [18]]. However, the PBF-EB-fabricated parts often include internal macro-defects, which compromises their mechanical properties [[19], [20], [21], [22]]. This study focused on the elimination of macro-defects, such as porosity, lack of fusion, incomplete penetration and unmelted powders, which distinguishes them from micro-defects such as vacancies, dislocations, grain boundaries and secondary phases, etc. Large-sized fusion defects cause a severe reduction in mechanical strength. Smaller defects, such as pores and cracks, lead to the initiation of fatigue cracking and rapidly accelerate the cracking process [23]. The issue of internal macro-defects must be addressed to expand the application of the PBF-EB technology. The main studies for controlling internal macro-defects are online monitoring of defects, remelting and hot isostatic pressing (HIP). The literatures [24,25] report the use of infrared imaging or other imaging techniques to identify defects, but the monitoring of smaller sized defects is still not adequate. And in some cases remelting does not reduce the internal macro-defects of the part, but instead causes coarsening of the macrostructure and volatilization of some metal elements [23]. The HIP treatment does not completely eliminate the internal macro-defects, the original defect location may still act as a point of origin of the crack, and the subsequent treatment will consume more time and economic costs [23]. Therefore, optimizing suitable printing parameters to avoid internal macro-defects in printed parts at source is of great industrial value and research significance, and is an urgent issue in PBF-EB related technology.

There are two causes of internal macro-defects in the AM process: gas pores trapped in the virgin powder and the inappropriate printing parameters [7,23]. Gui et al. [26] classify internal macro-defects during PBF-EB process according to their shape, such as spherical defects, elongated shape defects, flat shape defects and other irregular shape defects. Of these, spherical defects mainly originate from raw material powders. Other shape defects mainly originate from lack of fusion or unmelted powders caused by unsuitable printing parameters, etc. The PBF-EB process requires powders with good flowability, and spherical powders are typically chosen as raw materials. The prevalent techniques for the fabrication of pre-alloyed powders are gas atomization (GA), plasma atomization (PA), and the plasma rotating electrode process (PREP) [27,28]. These methods yield powders with different characteristics that affect the subsequent fabrication. The selection of a suitable powder for PBF-EB is particularly important to produce Ti–6Al–4V alloys without internal macro-defects. The need to optimize several printing parameters such as beam current, scan speed, line offset, and focus offset make it difficult to eliminate internal macro-defects that occur during printing [23]. Most of the studies [11,12,22,[29], [30], [31], [32], [33]] on the optimization of AM processes for Ti–6Al–4V alloys have focused on samples with a limited set of parameters (e.g., power–scan speed) and do not allow for the guidance and development of unknown process windows for macro-defect-free samples. In addition, process optimization remains a time-consuming problem, with the traditional ‘trial and error’ method demanding considerable time and economic costs. The development of a simple and efficient method to predict the processing window for alloys without internal macro-defects is a key issue. In recent years, machine learning techniques have increasingly been used in the field of additive manufacturing and materials development [[34], [35], [36], [37]]. Aoyagi et al. [38] recently proposed a novel and efficient method based on a support vector machine (SVM) to optimize the two-dimensional process parameters (current and scan speed) and obtain PBF-EB-processed CoCr alloys without internal macro-defects. The method is one of the potential approaches toward effective optimization of more than two process parameters and makes it possible for the machine learning techniques to accelerate the development of alloys without internal macro-defects.

Herein, we focus on the elimination of internal macro-defects, such as pores, lack of fusion, etc., caused by raw powders and printing parameters. The Ti–6Al–4V powders produced by three different methods were compared, and the powder with the best sphericity, flowability, and minimal porosity was selected as the feedstock for subsequent printing. The relationship between the surface roughness and internal macro-defects in the Ti–6Al–4V components was also investigated. The combination of SVM and surface roughness indices (Sdr) predicted a wider four-dimensional processing window for obtaining Ti–6Al–4V alloys without internal macro-defects. Finally, we investigated the tensile properties of Ti–6Al–4V alloys at room temperature with different printing parameters, as well as the corresponding microstructures and fracture types.

Section snippets

Starting materials

Three types of Ti–6Al–4V alloy powders, produced by GA, PA, and PREP, were compared. The particle size distribution of the powders was determined using a laser particle size analyzer (LS230, Beckman Coulter, USA), and the flowability was measured using a Hall flowmeter (JIS-Z2502, Tsutsui Scientific Instruments Co., Ltd., Japan), according to the ASTM B213 standard. The powder morphology and internal macro-defects were determined using scanning electron microscopy (SEM, JEOL JCM-6000) and X-ray 

Comparison of the characteristics of GA, PA, and PREP Ti–6Al–4V powders

The particle size distributions (PSDs) and flowability of the three types of Ti–6Al–4V alloy powders produced by GA, PA, and PREP are shown in Fig. 2. Although the average particle sizes are similar (89.4 μm for GA, 82.5 μm for PA, and 86.1μm for PREP), the particle size range is different for the three types of powder (6.2–174.8 μm for GA, 27.3–139.2 μm for PA, and 39.4–133.9 μm for PREP). The flowability of the GA, PA, and PREP powders was 30.25 ± 0.98, 26.54 ± 0.37, and 25.03 ± 0.22 (s/50

Conclusions

The characteristics of the three types of Ti–6Al–4V alloy powders produced via GA, PA, and PREP were compared. The PREP powder with the best sphericity, flowability, and low porosity was found to be the most favorable powder for subsequent printing of Ti–6Al–4V alloys without internal macro-defects. The quantitative criterion of Sdr <0.015 for even surfaces was also found to be applicable to Ti–6Al–4V alloys. The process maps of Ti–6Al–4V alloys include two regions, high beam current/scan speed 

Uncited references

[55]; [56]; [57]; [58]; [59]; [60]; [61]; [62]; [63]; [64]; [65].

CRediT authorship contribution statement

Yunwei Gui: Writing – original draft, Visualization, Validation, Investigation. Kenta Aoyagi: Writing – review & editing, Supervision, Resources, Methodology, Funding acquisition, Conceptualization. Akihiko Chiba: Supervision, Funding acquisition.

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 study was based on the results obtained from project JPNP19007, commissioned by the New Energy and Industrial Technology Development Organization (NEDO). This work was also supported by JSPS KAKENHI (Proposal No. 21K03801) and the Inter-University Cooperative Research Program (Proposal nos. 18G0418, 19G0411, and 20G0418) of the Cooperative Research and Development Center for Advanced Materials, Institute for Materials Research, Tohoku University. It was also supported by the Council for

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Sketch of approach channel and spillway of the Kamal-Saleh dam

CFD modeling of flow pattern in spillway’s approach channel

Sustainable Water Resources Management volume 1, pages245–251 (2015)Cite this article

Abstract

Analysis of behavior and hydraulic characteristics of flow over the dam spillway is a complicated task that takes lots of money and time in water engineering projects planning. To model those hydraulic characteristics, several methods such as physical and numerical methods can be used. Nowadays, by utilizing new methods in computational fluid dynamics (CFD) and by the development of fast computers, the numerical methods have become accessible for use in the analysis of such sophisticated flows. The CFD softwares have the capability to analyze two- and three-dimensional flow fields. In this paper, the flow pattern at the guide wall of the Kamal-Saleh dam was modeled by Flow 3D. The results show that the current geometry of the left wall causes instability in the flow pattern and making secondary and vortex flow at beginning approach channel. This shape of guide wall reduced the performance of weir to remove the peak flood discharge.

댐 여수로 흐름의 거동 및 수리학적 특성 분석은 물 공학 프로젝트 계획에 많은 비용과 시간이 소요되는 복잡한 작업입니다. 이러한 수력학적 특성을 모델링하기 위해 물리적, 수치적 방법과 같은 여러 가지 방법을 사용할 수 있습니다. 요즘에는 전산유체역학(CFD)의 새로운 방법을 활용하고 빠른 컴퓨터의 개발로 이러한 정교한 흐름의 해석에 수치 방법을 사용할 수 있게 되었습니다. CFD 소프트웨어에는 2차원 및 3차원 유동장을 분석하는 기능이 있습니다. 본 논문에서는 Kamal-Saleh 댐 유도벽의 흐름 패턴을 Flow 3D로 모델링하였다. 결과는 왼쪽 벽의 현재 형상이 흐름 패턴의 불안정성을 유발하고 시작 접근 채널에서 2차 및 와류 흐름을 만드는 것을 보여줍니다. 이러한 형태의 안내벽은 첨두방류량을 제거하기 위해 둑의 성능을 저하시켰다.

Introduction

Spillways are one of the main structures used in the dam projects. Design of the spillway in all types of dams, specifically earthen dams is important because the inability of the spillway to remove probable maximum flood (PMF) discharge may cause overflow of water which ultimately leads to destruction of the dam (Das and Saikia et al. 2009; E 2013 and Novak et al. 2007). So study on the hydraulic characteristics of this structure is important. Hydraulic properties of spillway including flow pattern at the entrance of the guide walls and along the chute. Moreover, estimating the values of velocity and pressure parameters of flow along the chute is very important (Chanson 2004; Chatila and Tabbara 2004). The purpose of the study on the flow pattern is the effect of wall geometry on the creation transverse waves, flow instability, rotating and reciprocating flow through the inlet of spillway and its chute (Parsaie and Haghiabi 2015ab; Parsaie et al. 2015; Wang and Jiang 2010). The purpose of study on the values of velocity and pressure is to calculate the potential of the structure to occurrence of phenomena such as cavitation (Fattor and Bacchiega 2009; Ma et al. 2010). Sometimes, it can be seen that the spillway design parameters of pressure and velocity are very suitable, but geometry is considered not suitable for conducting walls causing unstable flow pattern over the spillway, rotating flows at the beginning of the spillway and its design reduced the flood discharge capacity (Fattor and Bacchiega 2009). Study on spillway is usually conducted using physical models (Su et al. 2009; Suprapto 2013; Wang and Chen 2009; Wang and Jiang 2010). But recently, with advances in the field of computational fluid dynamics (CFD), study on hydraulic characteristics of this structure has been done with these techniques (Chatila and Tabbara 2004; Zhenwei et al. 2012). Using the CFD as a powerful technique for modeling the hydraulic structures can reduce the time and cost of experiments (Tabbara et al. 2005). In CFD field, the Navier–Stokes equation is solved by powerful numerical methods such as finite element method and finite volumes (Kim and Park 2005; Zhenwei et al. 2012). In order to obtain closed-form Navier–Stokes equations turbulence models, such k − ε and Re-Normalisation Group (RNG) models have been presented. To use the technique of computational fluid dynamics, software packages such as Fluent and Flow 3D, etc., are provided. Recently, these two software packages have been widely used in hydraulic engineering because the performance and their accuracy are very suitable (Gessler 2005; Kim 2007; Kim et al. 2012; Milési and Causse 2014; Montagna et al. 2011). In this paper, to assess the flow pattern at Kamal-Saleh guide wall, numerical method has been used. All the stages of numerical modeling were conducted in the Flow 3D software.

Materials and methods

Firstly, a three-dimensional model was constructed according to two-dimensional map that was prepared for designing the spillway. Then a small model was prepared with scale of 1:80 and entered into the Flow 3D software; all stages of the model construction was conducted in AutoCAD 3D. Flow 3D software numerically solved the Navier–Stokes equation by finite volume method. Below is a brief reference on the equations that used in the software. Figure 1 shows the 3D sketch of Kamal-Saleh spillway and Fig. 2 shows the uploading file of the Kamal-Saleh spillway in Flow 3D software.

figure 1
Fig. 1
figure 2
Fig. 2

Review of the governing equations in software Flow 3D

Continuity equation at three-dimensional Cartesian coordinates is given as Eq (1).

vf∂ρ∂t+∂∂x(uAx)+∂∂x(vAy)+∂∂x(wAz)=PSORρ,vf∂ρ∂t+∂∂x(uAx)+∂∂x(vAy)+∂∂x(wAz)=PSORρ,

(1)

where uvz are velocity component in the x, y, z direction; A xA yA z cross-sectional area of the flow; ρ fluid density; PSOR the source term; v f is the volume fraction of the fluid and three-dimensional momentum equations given in Eq (2).

∂u∂t+1vf(uAx∂u∂x+vAy∂u∂y+wAz∂u∂z)=−1ρ∂P∂x+Gx+fx∂v∂t+1vf(uAx∂v∂x+vAy∂v∂y+wAz∂v∂z)=−1ρ∂P∂y+Gy+fy∂w∂t+1vf(uAx∂w∂x+vAy∂w∂y+wAz∂w∂z)=−1ρ∂P∂y+Gz+fz,∂u∂t+1vf(uAx∂u∂x+vAy∂u∂y+wAz∂u∂z)=−1ρ∂P∂x+Gx+fx∂v∂t+1vf(uAx∂v∂x+vAy∂v∂y+wAz∂v∂z)=−1ρ∂P∂y+Gy+fy∂w∂t+1vf(uAx∂w∂x+vAy∂w∂y+wAz∂w∂z)=−1ρ∂P∂y+Gz+fz,

(2)

where P is the fluid pressure; G xG yG z the acceleration created by body fluids; f xf yf z viscosity acceleration in three dimensions and v f is related to the volume of fluid, defined by Eq. (3). For modeling of free surface profile the VOF technique based on the volume fraction of the computational cells has been used. Since the volume fraction F represents the amount of fluid in each cell, it takes value between 0 and 1.

∂F∂t+1vf[∂∂x(FAxu)+∂∂y(FAyv)+∂∂y(FAzw)]=0∂F∂t+1vf[∂∂x(FAxu)+∂∂y(FAyv)+∂∂y(FAzw)]=0

(3)

Turbulence models

Flow 3D offers five types of turbulence models: Prantl mixing length, k − ε equation, RNG models, Large eddy simulation model. Turbulence models that have been proposed recently are based on Reynolds-averaged Navier–Stokes equations. This approach involves statistical methods to extract an averaged equation related to the turbulence quantities.

Steps of solving a problem in Flow 3D software

(1) Preparing the 3D model of spillway by AutoCAD software. (2) Uploading the file of 3D model in Flow 3D software and defining the problem in the software and checking the final mesh. (3) Choosing the basic equations that should be solved. (4) Defining the characteristics of fluid. (5) Defining the boundary conditions; it is notable that this software has a wide range of boundary conditions. (6) Initializing the flow field. (7) Adjusting the output. (8) Adjusting the control parameters, choice of the calculation method and solution formula. (9) Start of calculation. Figure 1 shows the 3D model of the Kamal-Saleh spillway; in this figure, geometry of the left and right guide wall is shown.

Figure 2 shows the uploading of the 3D spillway dam in Flow 3D software. Moreover, in this figure the considered boundary condition in software is shown. At the entrance and end of spillway, the flow rate or fluid elevation and outflow was considered as BC. The bottom of spillway was considered as wall and left and right as symmetry.

Model calibration

Calibration of the Flow 3D for modeling the effect of geometry of guide wall on the flow pattern is included for comparing the results of Flow 3D with measured water surface profile. Calibration the Flow 3D software could be conducted in two ways: first, changing the value of upstream boundary conditions is continued until the results of water surface profile of the Flow 3D along the spillway successfully covered the measurement water surface profile; second is the assessment the mesh sensitivity. Analyzing the size of mesh is a trial-and-error process where the size of mesh is evaluated form the largest to the smallest. With fining the size of mesh the accuracy of model is increased; whereas, the cost of computation is increased. In this research, the value of upstream boundary condition was adjusted with measured data during the experimental studies on the scaled model and the mesh size was equal to 1 × 1 × 1 cm3.

Results and discussion

The behavior of water in spillway is strongly affected by the flow pattern at the entrance of the spillway, the flow pattern formation at the entrance is affected by the guide wall, and choice of an optimized form for the guide wall has a great effect on rising the ability of spillway for easy passing the PMF, so any nonuniformity in flow in the approach channel can cause reduction of spillway capacity, reduction in discharge coefficient of spillway, and even probability of cavitation. Optimizing the flow guiding walls (in terms of length, angle and radius) can cause the loss of turbulence and flow disturbances on spillway. For this purpose, initially geometry proposed for model for the discharge of spillway dam, Kamal-Saleh, 80, 100, and 120 (L/s) were surveyed. These discharges of flow were considered with regard to the flood return period, 5, 100 and 1000 years. Geometric properties of the conducting guidance wall are given in Table 1.Table 1 Characteristics and dimensions of the guidance walls tested

Full size table

Results of the CFD simulation for passing the flow rate 80 (L/s) are shown in Fig. 3. Figure 3 shows the secondary flow and vortex at the left guide wall.

figure 3
Fig. 3

For giving more information about flow pattern at the left and right guide wall, Fig. 4 shows the flow pattern at the right side guide wall and Fig. 5 shows the flow pattern at the left side guide wall.

figure 4
Fig. 4
figure 5
Fig. 5

With regard to Figs. 4 and 5 and observing the streamlines, at discharge equal to 80 (L/s), the right wall has suitable performance but the left wall has no suitable performance and the left wall of the geometric design creates a secondary and circular flow, and vortex motion in the beginning of the entrance of spillway that creates cross waves at the beginning of spillway. By increasing the flow rate (Q = 100 L/s), at the inlet spillway secondary flows and vortex were removed, but the streamline is severely distorted. Results of the guide wall performances at the Q = 100 (L/s) are shown in Fig. 6.

figure 6
Fig. 6

Also more information about the performance of each guide wall can be derived from Figs. 7 and 8. These figures uphold that the secondary and vortex flows were removed, but the streamlines were fully diverted specifically near the left side guide wall.

figure 7
Fig. 7
figure 8
Fig. 8

As mentioned in the past, these secondary and vortex flows and diversion in streamline cause nonuniformity and create cross wave through the spillway. Figure 9 shows the cross waves at the crest of the spillway.

figure 9
Fig. 9

The performance of guide walls at the Q = 120 (L/s) also was assessed. The result of simulation is shown in Fig. 10. Figures 11 and 12 show a more clear view of the streamlines near to right and left side guide wall, respectively. As seen in Fig. 12, the left side wall still causes vortex flow and creation of and diversion in streamline.

figure 10
Fig. 10
figure 11
Fig. 11
figure 12
Fig. 12

The results of the affected left side guide wall shape on the cross wave creation are shown in Fig. 13. As seen from Fig. 3, the left side guide wall also causes cross wave at the spillway crest.

figure 13
Fig. 13

As can be seen clearly in Figs. 9 and 13, by moving from the left side to the right side of the spillway, the cross waves and the nonuniformity in flow is removed. By reviewing Figs. 9 and 13, it is found that the right side guide wall removes the cross waves and nonuniformity. With this point as aim, a geometry similar to the right side guide wall was considered instead of the left side guide wall. The result of simulation for Q = 120 (L/s) is shown in Fig. 14. As seen from this figure, the proposed geometry for the left side wall has suitable performance smoothly passing the flow through the approach channel and spillway.

figure 14
Fig. 14

More information about the proposed shape for the left guide wall is shown in Fig. 15. As seen from this figure, this shape has suitable performance for removing the cross waves and vortex flows.

figure 15
Fig. 15

Figure 16 shows the cross section of flow at the crest of spillway. As seen in this figure, the proposed shape for the left side guide wall is suitable for removing the cross waves and secondary flows.

figure 16
Fig. 16

Conclusion

Analysis of behavior and hydraulic properties of flow over the spillway dam is a complicated task which is cost and time intensive. Several techniques suitable to the purposes of study have been undertaken in this research. Physical modeling, usage of expert experience, usage of mathematical models on simulation flow in one-dimensional, two-dimensional and three-dimensional techniques, are some of the techniques utilized to study this phenomenon. The results of the modeling show that the CFD technique is a suitable tool for simulating the flow pattern in the guide wall. Using this tools helps the designer for developing the optimal shape for hydraulic structure which the flow pattern through them are important.

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  1. Department of Water Engineering, Lorestan University, Khorram Abad, IranAbbas Parsaie, Amir Hamzeh Haghiabi & Amir Moradinejad

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Correspondence to Abbas Parsaie.

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Parsaie, A., Haghiabi, A.H. & Moradinejad, A. CFD modeling of flow pattern in spillway’s approach channel. Sustain. Water Resour. Manag. 1, 245–251 (2015). https://doi.org/10.1007/s40899-015-0020-9

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  • Received28 April 2015
  • Accepted28 August 2015
  • Published15 September 2015
  • Issue DateSeptember 2015
  • DOIhttps://doi.org/10.1007/s40899-015-0020-9

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Keywords

  • Approach channel
  • Kamal-Saleh dam
  • Guide wall
  • Flow pattern
  • Numerical modeling
  • Flow 3D software
    Figure 10. Flow distribution at the approach channel in PMF based on revised plan design. A. Hydarulic model test; B. Numerical simulation; C. Section view.

    Improvement of hydraulic stability for spillway using CFD model

    Hydraulic model test was used to analyze the rapidly varied flow on the spillway. But, it has some shortcomings such as error of scale effect and expensive costs. Recently, through the development of three dimensional computational fluid dynamics (CFD), rapidly varied flow and turbulence can be simulated. In this study, the applicability of CFD model to simulate flow on the spillway was reviewed. The Karian dam in Indonesia was selected as the study area. The FLOW-3d model, which is well known to simulate a flow having a free surface, was used to analyze flow. The flow stability in approach channel was investigated with the initial plan design, and the results showed that the flow in approach channel is unstable in the initial plan design. To improve flow stability in the spillway, therefore, the revised plan design was formulated. The appropriateness of the revised design was examined by a numerical modeling. The results showed that the flow in spillway is stable in the revised design.

    여수로의 급격하게 변화하는 흐름을 분석하기 위해 수리학적 모델 테스트를 사용했습니다. 그러나 스케일 효과의 오차와 고가의 비용 등의 단점이 있다. 최근에는 3차원 전산유체역학(CFD)의 발달로 급변하는 유동과 난류를 모사할 수 있다. 본 연구에서는 여수로의 흐름을 시뮬레이션하기 위한 CFD 모델의 적용 가능성을 검토했습니다. 인도네시아의 Karian 댐이 연구 지역으로 선정되었습니다. 자유표면을 갖는 유동을 모의하는 것으로 잘 알려진 FLOW-3d 모델을 유동해석에 사용하였다. 접근수로의 흐름 안정성은 초기 계획설계와 함께 조사한 결과 초기 계획설계에서 접근수로의 흐름이 불안정한 것으로 나타났다. 따라서 방수로의 흐름 안정성을 향상시키기 위해 수정된 계획 설계가 공식화되었습니다. 수정된 설계의 적합성을 수치모델링을 통해 검토하였다. 결과는 수정된 설계에서 여수로의 흐름이 안정적이라는 것을 보여주었습니다.

    Key words

    Spillway, FLOW-3D, approach channel, flow stability, numerical modeling, hydraulic model test.

    Figure 6. Two dimensional flow velocity distribution at the
approach channel (Flow velocity distribution at depth EL. 68.12 m).
    Figure 6. Two dimensional flow velocity distribution at the approach channel (Flow velocity distribution at depth EL. 68.12 m).
    Figure 7. Flow distribution at the approach channel in PMF.
A. Hydraulic model test; B. Numerial simulatio
C. Cross section view.
    Figure 7. Flow distribution at the approach channel in PMF. A. Hydraulic model test; B. Numerial simulatio C. Cross section view.
    Figure 8. Revised approach channel section.
A. Initial plan design; B. Revised plan design.
    Figure 8. Revised approach channel section. A. Initial plan design; B. Revised plan design.
    Figure 9. Two dimensional flow velocity distribution at the approach channel
based on revised plan design (Flow velocity distribution at depth EL. 68.12 m).
    Figure 9. Two dimensional flow velocity distribution at the approach channel based on revised plan design (Flow velocity distribution at depth EL. 68.12 m).
    Figure 10. Flow distribution at the approach channel in PMF based on revised plan design.
A. Hydarulic model test; B. Numerical simulation; C. Section view.
    Figure 10. Flow distribution at the approach channel in PMF based on revised plan design. A. Hydarulic model test; B. Numerical simulation; C. Section view.

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    Numerical analysis of energy dissipator options using computational fluid dynamics modeling — a case study of Mirani Dam

    전산 유체 역학 모델링을 사용한 에너지 소산자 옵션의 수치적 해석 — Mirani 댐의 사례 연구

    Arabian Journal of Geosciences volume 15, Article number: 1614 (2022) Cite this article

    Abstract

    이 연구에서 FLOW 3D 전산 유체 역학(CFD) 소프트웨어를 사용하여 파키스탄 Mirani 댐 방수로에 대한 에너지 소산 옵션으로 미국 매립지(USBR) 유형 II 및 USBR 유형 III 유역의 성능을 추정했습니다. 3D Reynolds 평균 Navier-Stokes 방정식이 해결되었으며, 여기에는 여수로 위의 자유 표면 흐름을 캡처하기 위해 공기 유입, 밀도 평가 및 드리프트-플럭스에 대한 하위 그리드 모델이 포함되었습니다. 본 연구에서는 5가지 모델을 고려하였다. 첫 번째 모델에는 길이가 39.5m인 USBR 유형 II 정수기가 있습니다. 두 번째 모델에는 길이가 44.2m인 USBR 유형 II 정수기가 있습니다. 3번째와 4 번째모델에는 길이가 각각 48.8m인 USBR 유형 II 정수조와 39.5m의 USBR 유형 III 정수조가 있습니다. 다섯 번째 모델은 네 번째 모델과 동일하지만 마찰 및 슈트 블록 높이가 0.3m 증가했습니다. 최상의 FLOW 3D 모델 조건을 설정하기 위해 메쉬 민감도 분석을 수행했으며 메쉬 크기 0.9m에서 최소 오차를 산출했습니다. 세 가지 경계 조건 세트가 테스트되었으며 최소 오류를 제공하는 세트가 사용되었습니다. 수치적 검증은 USBR 유형 II( L = 48.8m), USBR 유형 III( L = 35.5m) 및 USBR 유형 III 의 물리적 모델 에너지 소산을 0.3m 블록 단위로 비교하여 수행되었습니다( L= 35.5m). 통계 분석 결과 평균 오차는 2.5%, RMSE(제곱 평균 제곱근 오차) 지수는 3% 미만이었습니다. 수리학적 및 경제성 분석을 바탕으로 4 번째 모델이 최적화된 에너지 소산기로 밝혀졌습니다. 흡수된 에너지 백분율 측면에서 물리적 모델과 수치적 모델 간의 최대 차이는 5% 미만인 것으로 나타났습니다.

    In this study, the FLOW 3D computational fluid dynamics (CFD) software was used to estimate the performance of the United States Bureau of Reclamation (USBR) type II and USBR type III stilling basins as energy dissipation options for the Mirani Dam spillway, Pakistan. The 3D Reynolds-averaged Navier–Stokes equations were solved, which included sub-grid models for air entrainment, density evaluation, and drift–flux, to capture free-surface flow over the spillway. Five models were considered in this research. The first model has a USBR type II stilling basin with a length of 39.5 m. The second model has a USBR type II stilling basin with a length of 44.2 m. The 3rd and 4th models have a USBR type II stilling basin with a length of 48.8 m and a 39.5 m USBR type III stilling basin, respectively. The fifth model is identical to the fourth, but the friction and chute block heights have been increased by 0.3 m. To set up the best FLOW 3D model conditions, mesh sensitivity analysis was performed, which yielded a minimum error at a mesh size of 0.9 m. Three sets of boundary conditions were tested and the set that gave the minimum error was employed. Numerical validation was done by comparing the physical model energy dissipation of USBR type II (L = 48.8 m), USBR type III (L =35.5 m), and USBR type III with 0.3-m increments in blocks (L = 35.5 m). The statistical analysis gave an average error of 2.5% and a RMSE (root mean square error) index of less than 3%. Based on hydraulics and economic analysis, the 4th model was found to be an optimized energy dissipator. The maximum difference between the physical and numerical models in terms of percentage energy absorbed was found to be less than 5%.

    Keywords

    • Numerical modeling
    • Spillway
    • Hydraulic jump
    • Energy dissipation
    • FLOW 3D

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    다공층에 대한 돌발 댐 붕괴의 3차원 유동 수치해석 시뮬레이션

    A. Safarzadeh1*, P. Mohsenzadeh2, S. Abbasi3
    1 Professor of Civil Eng., Water Engineering and Mineral Waters Research Center, Univ. of Mohaghegh Ardabili,Ardabil, Iran
    2 M.Sc., Graduated of Civil-Hydraulic Structures Eng., Faculty of Eng., Univ. of Mohaghegh Ardabili, Ardabil, Iran
    3 M.Sc., Graduated of Civil -Hydraulic Structures Eng., Faculty of Eng., Univ. of Mohaghegh Ardabili, Ardabil, Iran Safarzadeh@uma.ac.ir

    Highlights

    유체 이동에 의해 생성된 RBF는 Ls-Dyna에서 Fluent, ICFD ALE 및 SPH 방법으로 시뮬레이션되었습니다.
    RBF의 과예측은 유체가 메인 도메인에서 고속으로 분리될 때 발생합니다.
    이 과잉 예측은 요소 크기, 시간 단계 크기 및 유체 모델에 따라 다릅니다.
    유체 성능을 검증하려면 최대 RBF보다 임펄스가 권장됩니다.

    Abstract

    Dam break is a very important problem due to its effects on economy, security, human casualties and environmental consequences. In this study, 3D flow due to dam break over the porous substrate is numerically simulated and the effect of porosity, permeability and thickness of the porous bed and the water depth in the porous substrate are investigated. Classic models of dam break over a rigid bed and water infiltration through porous media were studied and results of the numerical simulations are compared with existing laboratory data. Validation of the results is performed by comparing the water surface profiles and wave front position with dam break on rigid and porous bed. Results showed that, due to the effect of dynamic wave in the initial stage of dam break, a local peak occurs in the flood hydrograph. The presence of porous bed reduces the acceleration of the flood wave relative to the flow over the solid bed and it decreases with the increase of the permeability of the bed. By increasing the permeability of the bed, the slope of the ascending limb of the flood hydrograph and the peak discharge drops. Furthermore, if the depth and permeability of the bed is such that the intrusive flow reaches the rigid substrate under the porous bed, saturation of the porous bed, results in a sharp increase in the slope of the flood hydrograph. The maximum values of the peak discharge at the end of the channel with porous bed occurred in saturated porous bed conditions.

    댐 붕괴는 경제, 보안, 인명 피해 및 환경적 영향으로 인해 매우 중요한 문제입니다. 본 연구에서는 다공성 기재에 대한 댐 파괴로 인한 3차원 유동을 수치적으로 시뮬레이션하고 다공성 기재의 다공성, 투과도 및 다공성 층의 두께 및 수심의 영향을 조사합니다. 단단한 바닥에 대한 댐 파괴 및 다공성 매체를 통한 물 침투의 고전 모델을 연구하고 수치 시뮬레이션 결과를 기존 실험실 데이터와 비교합니다. 결과 검증은 강체 및 다공성 베드에서 댐 파단과 수면 프로파일 및 파면 위치를 비교하여 수행됩니다. 그 결과 댐파괴 초기의 동적파동의 영향으로 홍수수문곡선에서 국부첨두가 발생하는 것으로 나타났다. 다공성 베드의 존재는 고체 베드 위의 유동에 대한 홍수파의 가속을 감소시키고 베드의 투과성이 증가함에 따라 감소합니다. 베드의 투수성을 증가시켜 홍수 수문곡선의 오름차순 경사와 첨두방류량이 감소한다. 더욱이, 만약 층의 깊이와 투과성이 관입 유동이 다공성 층 아래의 단단한 기질에 도달하는 정도라면, 다공성 층의 포화는 홍수 수문곡선의 기울기의 급격한 증가를 초래합니다. 다공층이 있는 채널의 끝단에서 최대 방전 피크값은 포화 다공층 조건에서 발생하였다.

    Keywords

    Keywords: Dams Break, 3D modeling, Porous Bed, Permeability, Flood wave

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    Flow Field in a Sloped Channel with Damaged and Undamaged Piers: Numerical and Experimental Studies

    Flow Field in a Sloped Channel with Damaged and Undamaged Piers: Numerical and Experimental Studies

    Ehsan OveiciOmid Tayari & Navid Jalalkamali
    KSCE Journal of Civil Engineering volume 25, pages4240–4251 (2021)Cite this article

    Abstract

    본 논문은 경사가 완만한 수로에서 손상되거나 손상되지 않은 교각 주변의 유동 패턴을 분석했습니다. 실험은 길이가 12m이고 기울기가 0.008인 직선 수로에서 수행되었습니다. Acoustic Doppler Velocimeter(ADV)를 이용하여 3차원 유속 데이터를 수집하였고, 그 결과를 PIV(Particle Image Velocimetry) 데이터와 분석하여 비교하였습니다.

    다중 블록 옵션이 있는 취수구의 퇴적물 시뮬레이션(SSIIM)은 이 연구에서 흐름의 수치 시뮬레이션을 위해 통합되었습니다. 일반적으로 비교에서 얻은 결과는 수치 데이터와 실험 데이터 간의 적절한 일치를 나타냅니다. 결과는 모든 경우에 수로 입구에서 2m 거리에서 기복적 수압 점프가 발생했음을 보여주었습니다.

    경사진 수로의 최대 베드 전단응력은 2개의 손상 및 손상되지 않은 교각을 설치하기 위한 수평 수로의 12배였습니다. 이와 같은 경사수로 교각의 위치에 따라 상류측 수위는 수평수로의 유사한 조건에 비해 72.5% 감소한 반면, 이 감소량은 경사면에서 다른 경우에 비해 8.3% 감소하였다. 채널 또한 두 교각이 있는 경우 최대 Froude 수는 수평 수로의 5.7배였습니다.

    This paper analyzed the flow pattern around damaged and undamaged bridge piers in a channel with a mild slope. The experiments were carried out on a straight channel with a length of 12 meters and a slope of 0.008. Acoustic Doppler velocimeter (ADV) was employed to collect three-dimensional flow velocity data, and the results were analyzed and compared with particle image velocimetry (PIV) data. Sediment Simulation in Intakes with Multiblock option (SSIIM) was incorporated for the numerical simulation of the flow in this study. Generally, the results obtained from the comparisons referred to the appropriate agreement between the numerical and the experimental data. The results showed that an undular hydraulic jump occurred at a distance of two meters from the channel entrance in every case; the maximum bed shear stress in the sloped channel was 12 times that in a horizontal channel for installing two damaged and undamaged piers. With this position of the piers in the sloped channel, the upstream water level underwent a 72.5% reduction compared to similar conditions in a horizontal channel, while the amount of this water level decrease was equal to 8.3% compared to the other cases in a sloped channel. In addition, with the presence of both piers, the maximum Froude number was 5.7 times that in a horizontal channel.

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    하류하천의 영향 최소화를 위한 보조 여수로 최적 활용방안 검토

    The Optimal Operation on Auxiliary Spillway to Minimize the Flood Damage in Downstream River with Various Outflow Conditions

    하류하천의 영향 최소화를 위한 보조 여수로 최적 활용방안 검토

    Hyung Ju Yoo1, Sung Sik Joo2, Beom Jae Kwon3, Seung Oh Lee4*

    유 형주1, 주 성식2, 권 범재3, 이 승오4*

    1Ph.D Student, Dept. of Civil & Environmental Engineering, Hongik University
    2Director, Water Resources & Environment Department, HECOREA
    3Director, Water Resources Department, ISAN
    4Professor, Dept. of Civil & Environmental Engineering, Hongik University

    1홍익대학교 건설환경공학과 박사과정
    2㈜헥코리아 수자원환경사업부 이사
    3㈜이산 수자원부 이사
    4홍익대학교 건설환경공학과 교수

    ABSTRACT

    최근 기후변화로 인해 강우강도 및 빈도의 증가에 따른 집중호우의 영향 및 기존 여수로의 노후화에 대비하여 홍수 시 하류 하천의 영향을 최소화할 수 있는 보조 여수로 활용방안 구축이 필요한 실정이다. 이를 위해, 수리모형 실험 및 수치모형 실험을 통하여 보조 여수로 운영에 따른 흐름특성 변화 검토에 관한 연구가 많이 진행되어 왔다. 그러나 대부분의 연구는 여수로에서의 흐름특성 및 기능성에 대한 검토를 수행하였을 뿐 보조 여수로의 활용방안에 따른 하류하천 영향 검토 및 호안 안정성 검토에 관한 연구는 미비한 실정이다. 이에 본 연구에서는 기존 여수로 및 보조 여수로 방류 조건에 따른 하류영향 분석 및 호안 안정성 측면에서 최적 방류 시나리오 검토를 3차원 수치모형인 FLOW-3D를 사용하여 검토하였다. 또한 FLOW-3D 수치모의 수행을 통한 유속, 수위 결과와 소류력 산정 결과를 호안 설계허용 기준과 비교하였다. 수문 완전 개도 조건으로 가정하고 계획홍수량 유입 시 다양한 보조 여수로 활용방안에 대하여 수치모의를 수행한 결과, 보조 여수로 단독 운영 시 기존 여수로 단독운영에 비하여 최대유속 및 최대 수위의 감소효과를 확인하였다. 다만 계획홍수량의 45% 이하 방류 조건에서 대안부의 호안 안정성을 확보하였고 해당 방류량 초과 경우에는 처오름 현상이 발생하여 월류에 대한 위험성 증가를 확인하였다. 따라서 기존 여수로와의 동시 운영 방안 도출이 중요하다고 판단하였다. 여수로의 배분 비율 및 총 허용 방류량에 대하여 검토한 결과 보조 여수로의 방류량이 기존 여수로의 방류량보다 큰 경우 하류하천의 흐름이 중심으로 집중되어 대안부의 유속 저감 및 수위 감소를 확인하였고, 계획 홍수량의 77% 이하의 조건에서 호안의 허용 유속 및 허용 소류력 조건을 만족하였다. 이를 통하여 본 연구에서 제안한 보조 여수로 활용방안으로는 기존 여수로와 동시 운영 시 총 방류량에 대하여 보조 여수로의 배분량이 기존 여수로의 배분량보다 크게 설정하는 것이 하류하천의 영향을 최소화 할 수 있는 것으로 나타났다. 그러나 본 연구는 여수로 방류에 따른 대안부에서의 영향에 대해서만 검토하였고 수문 전면 개도 조건에서 검토하였다는 한계점은 분명히 있다. 이에 향후에는 다양한 수문 개도 조건 및 방류 시나리오를 적용 및 검토한다면 보다 효율적이고, 효과적인 보조 여수로 활용방안을 도출이 가능할 것으로 기대 된다.

    키워드 : 보조 여수로, FLOW-3D, 수치모의, 호안 안정성, 소류력

    1. 서 론

    최근 기후변화로 인한 집중호우의 영향으로 홍수 시 댐으로 유입되는 홍수량이 설계 홍수량보다 증가하여 댐 안정성 확보가 필요한 실정이다(Office for Government Policy Coordination, 2003). MOLIT & K-water(2004)에서는 기존댐의 수문학적 안정성 검토를 수행하였으며 이상홍수 발생 시 24개 댐에서 월류 등으로 인한 붕괴위험으로 댐 하류지역의 극심한 피해를 예상하여 보조여수로 신설 및 기존여수로 확장 등 치수능력 증대 기본계획을 수립하였고 이를 통하여 극한홍수 발생 시 홍수량 배제능력을 증대하여 기존댐의 안전성 확보 및 하류지역의 피해를 방지하고자 하였다. 여기서 보조 여수로는 기존 여수로와 동시 또는 별도 운영하는 여수로로써 비상상황 시 방류 기능을 포함하고 있고(K-water, 2021), 최근에는 기존 여수로의 노후화에 따라 보조여수로의 활용방안에 대한 관심이 증가하고 있다. 따라서 본 연구에서는 3차원 수치해석을 수행하여 기존 및 보조 여수로의 방류량 조합에 따른 하류 영향을 분석하고 하류 호안 안정성 측면에서 최적 방류 시나리오를 검토하고자 한다.

    기존의 댐 여수로 검토에 관한 연구는 주로 수리실험을 통하여 방류조건 별 흐름특성을 검토하였으나 최근에는 수치모형 실험결과가 수리모형실험과 비교하여 근사한 것을 확인하는 등 점차 수치모형실험을 수리모형실험의 대안으로 활용하고 있다(Jeon et al., 2006Kim, 2007Kim et al., 2008). 국내의 경우, Jeon et al.(2006)은 수리모형 실험과 수치모의를 이용하여 임하댐 바상여수로의 기본설계안을 도출하였고, Kim et al.(2008)은 가능최대홍수량 유입 시 비상여수로 방류에 따른 수리학적 안정성과 기능성을 3차원 수치모형인 FLOW-3D를 활용하여 검토하였다. 또한 Kim and Kim(2013)은 충주댐의 홍수조절 효과 검토 및 방류량 변화에 따른 상·하류의 수위 변화를 수치모형을 통하여 검토하였다. 국외의 경우 Zeng et al.(2017)은 3차원 수치모형인 Fluent를 활용한 여수로 방류에 따른 흐름특성 결과와 측정결과를 비교하여 수치모형 결과의 신뢰성을 검토하였다. Li et al.(2011)은 가능 최대 홍수량(Probable Maximum Flood, PMF)조건에서 기존 여수로와 신규 보조 여수로 유입부 주변의 흐름특성에 대하여 3차원 수치모형 Fluent를 활용하여 검토하였고, Lee et al.(2019)는 서로 근접해있는 기존 여수로와 보조여수로 동시 운영 시 방류능 검토를 수리모형 실험 및 수치모형 실험(FLOW-3D)을 통하여 수행하였으며 기존 여수로와 보조 여수로를 동시운영하게 되면 배수로 간섭으로 인하여 총 방류량이 7.6%까지 감소되어 댐의 방류능력이 감소하였음을 확인하였다.

    그러나 대부분의 여수로 검토에 대한 연구는 여수로 내에서의 흐름특성 및 기능성에 대한 검토를 수행하였고. 이에 기존 여수로와 보조 여수로 방류운영에 따른 하류하천의 흐름특성 변화 및 호안 안정성 평가에 관한 추가적인 검토가 필요한 실정이다. 따라서 본 연구에서는 기존 여수로 및 보조 여수로 방류 조건에 따른 하류하천의 흐름특성 및 호안 안정성분석을 3차원 수치모형인 FLOW-3D를 이용하여 검토하였다. 또한 다양한 방류 배분 비율 및 허용 방류량 조건 변화에 따른 하류하천의 흐름특성 및 소류력 분석결과를 호안 설계 허용유속 및 허용 소류력 기준과 비교하여 하류하천의 영향을 최소화 할 수 있는 최적의 보조 여수로 활용방안을 도출하고자 한다.

    2. 본 론

    2.1 이론적 배경

    2.1.1 3차원 수치모형의 기본이론

    FLOW-3D는 미국 Flow Science, Inc에서 개발한 범용 유체역학 프로그램(CFD, Computational Fluid Dynamics)으로 자유 수면을 갖는 흐름모의에 사용되는 3차원 수치해석 모형이다. 난류모형을 통해 난류 해석이 가능하고, 댐 방류에 따른 하류 하천의 흐름 해석에도 많이 사용되어 왔다(Flow Science, 2011). 본 연구에서는 FLOW-3D(version 12.0)을 이용하여 홍수 시 기존 여수로의 노후화에 대비하여 보조 여수로의 활용방안에 대한 검토를 하류하천의 호안 안정성 측면에서 검토하였다.

    2.1.2 유동해석의 지배방정식

    1) 연속 방정식(Continuity Equation)

    FLOW-3D는 비압축성 유체에 대하여 연속방정식을 사용하며, 밀도는 상수항으로 적용된다. 연속 방정식은 Eqs. (1)(2)와 같다.

    (1)

    ∇·v=0

    (2)

    ∂∂x(uAx)+∂∂y(vAy)+∂∂z(wAz)=RSORρ

    여기서, ρ는 유체 밀도(kg/m3), u, v, w는 x, y, z방향의 유속(m/s), Ax, Ay, Az는 각 방향의 요소면적(m2), RSOR는 질량 생성/소멸(mass source/sink)항을 의미한다.

    2) 운동량 방정식(Momentum Equation)

    각 방향 속도성분 u, v, w에 대한 운동방정식은 Navier-Stokes 방정식으로 다음 Eqs. (3)(4)(5)와 같다.

    (3)

    ∂u∂t+1VF(uAx∂u∂x+vAy∂v∂y+wAz∂w∂z)=-1ρ∂p∂x+Gx+fx-bx-RSORρVFu

    (4)

    ∂v∂t+1VF(uAx∂u∂x+vAy∂v∂y+wAz∂w∂z)=-1ρ∂p∂y+Gy+fy-by-RSORρVFv

    (5)

    ∂w∂t+1VF(uAx∂u∂x+vAy∂v∂y+wAz∂w∂z)=-1ρ∂p∂z+Gz+fz-bz-RSORρVFw

    여기서, Gx, Gy, Gz는 체적력에 의한 가속항, fx, fy, fz는 점성에 의한 가속항, bx, by, bz는 다공성 매체에서의 흐름손실을 의미한다.

    2.1.3 소류력 산정

    호안설계 시 제방사면 호안의 안정성 확보를 위해서는 하천의 흐름에 의하여 호안에 작용하는 소류력에 저항할 수 있는 재료 및 공법 선택이 필요하다. 국내의 경우 하천공사설계실무요령(MOLIT, 2016)에서 계획홍수량 유하 시 소류력 산정 방법을 제시하고 있다. 소류력은 하천의 평균유속을 이용하여 산정할 수 있으며, 소류력 산정식은 Eqs. (6)(7)과 같다.

    1) Schoklitsch 공식

    Schoklitsch(1934)는 Chezy 유속계수를 적용하여 소류력을 산정하였다.

    (6)

    τ=γRI=γC2V2

    여기서, τ는 소류력(N/m2), R은 동수반경(m), γ는 물의 단위중량(10.0 kN/m3), I는 에너지경사, C는 Chezy 유속계수, V는 평균유속(m/s)을 의미한다.

    2) Manning 조도계수를 고려한 공식

    Chezy 유속계수를 대신하여 Manning의 조도계수를 고려하여 소류력을 산정할 수 있다.

    (7)

    τ=γn2V2R1/3

    여기서, τ는 소류력(N/m2), R은 동수반경(m), γ는 물의 단위중량(10.0 kN/m3), n은 Manning의 조도계수, V는 평균유속(m/s)을 의미한다.

    FLOW-3D 수치모의 수행을 통하여 하천의 바닥 유속을 도출할 수 있으며, 본 연구에서는 Maning 조도계수롤 고려하여 소류력을 산정하고자 한다. 소류력을 산정하기 위해서 여수로 방류에 따른 대안부의 바닥유속 변화를 검토하여 최대 유속 값을 이용하였다. 최종적으로 산정한 소류력과 호안의 재료 및 공법에 따른 허용 소류력과 비교하여 제방사면 호안의 안정성 검토를 수행하게 된다.

    2.2 하천호안 설계기준

    하천 호안은 계획홍수위 이하의 유수작용에 대하여 안정성이 확보되도록 계획하여야 하며, 호안의 설계 시에는 사용재료의 확보용이성, 시공상의 용이성, 세굴에 대한 굴요성(flexibility) 등을 고려하여 호안의 형태, 시공방법 등을 결정한다(MOLIT, 2019). 국내의 경우, 하천공사설계실무요령(MOLIT, 2016)에서는 다양한 호안공법에 대하여 비탈경사에 따라 설계 유속을 비교하거나, 허용 소류력을 비교함으로써 호안의 안정성을 평가한다. 호안에 대한 국외의 설계기준으로 미국의 경우, ASTM(미국재료시험학회)에서 호안블록 및 식생매트 시험방법을 제시하였고 제품별로 ASTM 시험에 의한 허용유속 및 허용 소류력을 제시하였다. 일본의 경우, 호안 블록에 대한 축소실험을 통하여 항력을 측정하고 이를 통해서 호안 블록에 대한 항력계수를 제시하고 있다. 설계 시에는 항력계수에 의한 블록의 안정성을 평가하고 있으나, 최근에는 세굴의 영향을 고려할 수 있는 호안 안정성 평가의 필요성을 제기하고 있다(MOLIT, 2019). 관련된 국내·외의 하천호안 설계기준은 Table 1에 정리하여 제시하였고, 본 연구에서 하천 호안 안정성 평가 시 하천공사설계실무요령(MOLIT, 2016)과 ASTM 시험에서 제시한 허용소류력 및 허용유속 기준을 비교하여 각각 0.28 kN/m2, 5.0 m/s 미만일 경우 호안 안정성을 확보하였다고 판단하였다.

    Table 1.

    Standard of Permissible Velocity and Shear on Revetment

    Country (Reference)MaterialPermissible velocity (Vp, m/s)Permissible Shear (τp, kN/m2)
    KoreaRiver Construction Design Practice Guidelines
    (MOLIT, 2016)
    Vegetated5.00.50
    Stone5.00.80
    USAASTM D’6460Vegetated6.10.81
    Unvegetated5.00.28
    JAPANDynamic Design Method of Revetment5.0

    2.3. 보조여수로 운영에 따른 하류하천 영향 분석

    2.3.1 모형의 구축 및 경계조건

    본 연구에서는 기존 여수로의 노후화에 대비하여 홍수 시 보조여수로의 활용방안에 따른 하류하천의 흐름특성 및 호안안정성 평가를 수행하기 위해 FLOW-3D 모형을 이용하였다. 기존 여수로 및 보조 여수로는 치수능력 증대사업(MOLIT & K-water, 2004)을 통하여 완공된 ○○댐의 제원을 이용하여 구축하였다. ○○댐은 설계빈도(100년) 및 200년빈도 까지는 계획홍수위 이내로 기존 여수로를 통하여 운영이 가능하나 그 이상 홍수조절은 보조여수로를 통하여 조절해야 하며, 또한 2011년 기존 여수로 정밀안전진단 결과 사면의 표층 유실 및 옹벽 밀림현상 등이 확인되어 노후화에 따른 보수·보강이 필요한 상태이다. 이에 보조여수로의 활용방안 검토가 필요한 것으로 판단하여 본 연구의 대상댐으로 선정하였다. 하류 하천의 흐름특성을 예측하기 위하여 격자간격을 0.99 ~ 8.16 m의 크기로 하여 총 격자수는 49,102,500개로 구성하였으며, 여수로 방류에 따른 하류하천의 흐름해석을 위한 경계조건으로 상류는 유입유량(inflow), 바닥은 벽면(wall), 하류는 수위(water surface elevation)조건으로 적용하도록 하였다(Table 2Fig. 1 참조). FLOW-3D 난류모형에는 혼합길이 모형, 난류에너지 모형, k-ϵ모형, RNG(Renormalized Group Theory) k-ϵ모형, LES 모형 등이 있으며, 본 연구에서는 여수로 방류에 따른 복잡한 난류 흐름 및 높은 전단흐름을 정확하게 모의(Flow Science, 2011)할 수 있는 RNG k-ϵ모형을 사용하였고, 하류하천 호안의 안정성 측면에서 보조여수로의 활용방안을 검토하기 위하여 방류시나리오는 Table 3에 제시된 것 같이 설정하였다. Case 1 및 Case 2를 통하여 계획홍수량에 대하여 기존 여수로와 보조 여수로의 단독 운영이 하류하천에 미치는 영향을 확인하였고 보조 여수로의 방류량 조절을 통하여 호안 안정성 측면에서 보조 여수로 방류능 검토를 수행하였다(Case 3 ~ Case 6). 또한 기존 여수로와 보조 여수로의 방류량 배분에 따른 하류하천의 영향 검토(Case 7 ~ Case 10) 및 방류 배분에 따른 허용 방류량을 호안 안정성 측면에서 검토를 수행하였다(Case 11 ~ Case 14).

    수문은 완전개도 조건으로 가정하였으며 하류하천의 계획홍수량에 대한 기존 여수로와 보조여수로의 배분량을 조절하여 모의를 수행하였다. 여수로는 콘크리트의 조도계수 값(Chow, 1959)을 채택하였고, 댐 하류하천의 조도계수는 하천기본계획(Busan Construction and Management Administration, 2009) 제시된 조도계수 값을 채택하였으며 FLOW-3D의 적용을 위하여 Manning-Strickler 공식(Vanoni, 2006)을 이용하여 조도계수를 조고값으로 변환하여 사용하였다. Manning-Strickler 공식은 Eq. (8)과 같으며, FLOW-3D에 적용한 조도계수 및 조고는 Table 4와 같다.

    (8)

    n=ks1/68.1g1/2

    여기서, kS는 조고 (m), n은 Manning의 조도계수, g는 중력가속도(m/s2)를 의미한다.

    시간에 따라 동일한 유량이 일정하게 유입되도록 모의를 수행하였으며, 시간간격(Time Step)은 0.0001초로 설정(CFL number < 1.0) 하였다. 또한 여수로 수문을 통한 유량의 변동 값이 1.0%이내일 경우는 연속방정식을 만족하고 있다고 가정하였다. 이는, 유량의 변동 값이 1.0%이내일 경우 유속의 변동 값 역시 1.0%이내이며, 수치모의 결과 1.0%의 유속변동은 호안의 유속설계기준에 크게 영향을 미치지 않는다고 판단하였다. 그 결과 모든 수치모의 Case에서 2400초 이내에 결과 값이 수렴하는 것을 확인하였다.

    Table 2.

    Mesh sizes and numerical conditions

    MeshNumbers49,102,500 EA
    Increment (m)DirectionExisting SpillwayAuxiliary Spillway
    ∆X0.99 ~ 4.301.00 ~ 4.30
    ∆Y0.99 ~ 8.161.00 ~ 5.90
    ∆Z0.50 ~ 1.220.50 ~ 2.00
    Boundary ConditionsXmin / YmaxInflow / Water Surface Elevation
    Xmax, Ymin, Zmin / ZmaxWall / Symmetry
    Turbulence ModelRNG model
    Table 3.

    Case of numerical simulation (Qp : Design flood discharge)

    CaseExisting Spillway (Qe, m3/s)Auxiliary Spillway (Qa, m3/s)Remarks
    1Qp0Reference case
    20Qp
    300.58QpReview of discharge capacity on
    auxiliary spillway
    400.48Qp
    500.45Qp
    600.32Qp
    70.50Qp0.50QpDetermination of optimal division
    ratio on Spillways
    80.61Qp0.39Qp
    90.39Qp0.61Qp
    100.42Qp0.58Qp
    110.32Qp0.45QpDetermination of permissible
    division on Spillways
    120.35Qp0.48Qp
    130.38Qp0.53Qp
    140.41Qp0.56Qp
    Table 4.

    Roughness coefficient and roughness height

    CriteriaRoughness coefficient (n)Roughness height (ks, m)
    Structure (Concrete)0.0140.00061
    River0.0330.10496
    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F1.jpg
    Fig. 1

    Layout of spillway and river in this study

    2.3.2 보조 여수로의 방류능 검토

    본 연구에서는 기존 여수로와 보조 여수로의 방류량 배분에 따른 하류하천 대안부의 유속분포 및 수위분포를 검토하기 위해 수치모의 Case 별 다음과 같이 관심구역을 설정하였다(Fig. 2 참조). 관심구역(대안부)의 길이(L)는 총 1.3 km로 10 m 등 간격으로 나누어 검토하였으며, Section 1(0 < X/L < 0.27)은 기존 여수로 방류에 따른 영향이 지배적인 구간, Section 2(0.27 < X/L < 1.00)는 보조 여수로 방류에 따른 영향이 지배적인 구간으로 각 구간에서의 수위, 유속, 수심결과를 확인하였다. 기존 여수로의 노후화에 따른 보조 여수로의 방류능 검토를 위하여 Case 1 – Case 6까지의 결과를 비교하였다.

    보조 여수로의 단독 운영 시 기존 여수로 운영 시 보다 하류하천의 대안부의 최대 유속(Vmax)은 약 3% 감소하였으며, 이는 보조 여수로의 하천 유입각이 기존 여수로 보다 7°작으며 유입하천의 폭이 증가하여 유속이 감소한 것으로 판단된다. 대안부의 최대 유속 발생위치는 하류 쪽으로 이동하였으며 교량으로 인한 단면의 축소로 최대유속이 발생하는 것으로 판단된다. 또한 보조 여수로의 배분량(Qa)이 증가함에 따라 하류하천 대안부의 최대 유속이 증가하였다. 하천호안 설계기준에서 제시하고 있는 허용유속(Vp)과 비교한 결과, 계획홍수량(Qp)의 45% 이하(Case 5 & 6)를 보조 여수로에서 방류하게 되면 허용 유속(5.0 m/s)조건을 만족하여 호안안정성을 확보하였다(Fig. 3 참조). 허용유속 외에도 대안부에서의 소류력을 산정하여 하천호안 설계기준에서 제시한 허용 소류력(τp)과 비교한 결과, 유속과 동일하게 보조 여수로의 방류량이 계획홍수량의 45% 이하일 경우 허용소류력(0.28 kN/m2) 조건을 만족하였다(Fig. 4 참조). 각 Case 별 호안설계조건과 비교한 결과는 Table 5에 제시하였다.

    하류하천의 수위도 기존 여수로 운영 시 보다 보조 여수로 단독 운영 시 최대 수위(ηmax)가 약 2% 감소하는 효과를 보였으며 최대 수위 발생위치는 수충부로 여수로 방류시 처오름에 의한 수위 상승으로 판단된다. 기존 여수로의 단독운영(Case 1)의 수위(ηref)를 기준으로 보조 여수로의 방류량이 증가함에 따라 수위는 증가하였으나 계획홍수량의 58%까지 방류할 경우 월류에 대한 안정성(ηmax/ηref<0.97(=기설제방고))은 확보되었다(Fig. 5 참조). 그러나 계획홍수량 조건에서는 월류에 대한 위험성이 존재하기 때문에 기존여수로와 보조여수로의 적절한 방류량 배분 조합을 도출하는 것이 중요하다고 판단되어 진다.

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F2.jpg
    Fig. 2

    Region of interest in this study

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F3.jpg
    Fig. 3

    Maximum velocity and location of Vmax according to Qa

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F4.jpg
    Fig. 4

    Maximum shear according to Qa

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F5.jpg
    Fig. 5

    Maximum water surface elevation and location of ηmax according to Qa

    Table 5.

    Numerical results for each cases (Case 1 ~ Case 6)

    CaseMaximum Velocity
    (Vmax, m/s)
    Maximum Shear
    (τmax, kN/m2)
    Evaluation
    in terms of Vp
    Evaluation
    in terms of τp
    1
    (Qa = 0)
    9.150.54No GoodNo Good
    2
    (Qa = Qp)
    8.870.56No GoodNo Good
    3
    (Qa = 0.58Qp)
    6.530.40No GoodNo Good
    4
    (Qa = 0.48Qp)
    6.220.36No GoodNo Good
    5
    (Qa = 0.45Qp)
    4.220.12AccpetAccpet
    6
    (Qa = 0.32Qp)
    4.040.14AccpetAccpet

    2.3.3 기존 여수로와 보조 여수로 방류량 배분 검토

    기존 여수로 및 보조 여수로 단독운영에 따른 하류하천 및 호안의 안정성 평가를 수행한 결과 계획홍수량 방류 시 하류하천 대안부에서 호안 설계 조건(허용유속 및 허용 소류력)을 초과하였으며, 처오름에 의한 수위 상승으로 월류에 대한 위험성 증가를 확인하였다. 따라서 계획 홍수량 조건에서 기존 여수로와 보조 여수로의 방류량 배분을 통하여 호안 안정성을 확보하고 하류하천에 방류로 인한 피해를 최소화할 수 있는 배분조합(Case 7 ~ Case 10)을 검토하였다. Case 7은 기존 여수로와 보조여수로의 배분 비율을 균등하게 적용한 경우이고, Case 8은 기존 여수로의 배분량이 보조 여수로에 비하여 많은 경우, Case 9는 보조 여수로의 배분량이 기존 여수로에 비하여 많은 경우를 의미한다. 최대유속을 비교한 결과 보조 여수로의 배분 비율이 큰 경우 기존 여수로의 배분량에 의하여 흐름이 하천 중심에 집중되어 대안부의 유속을 저감하는 효과를 확인하였다. 보조여수로의 방류량 배분 비율이 증가할수록 기존 여수로 대안부 측(0.00<X/L<0.27, Section 1) 유속 분포는 감소하였으나, 신규여수로 대안부 측(0.27<X/L<1.00, Section 2) 유속은 증가하는 것을 확인하였다(Fig. 6 참조). 그러나 유속 저감 효과에도 대안부 전구간에서 설계 허용유속 조건을 초과하여 제방의 안정성을 확보하지는 못하였다. 소류력 산정 결과 유속과 동일하게 보조 여수로의 방류량이 기존 여수로의 방류량 보다 크면 감소하는 것을 확인하였고 일부 구간에서는 허용 소류력 조건을 만족하는 것을 확인하였다(Fig. 7 참조).

    따라서 유속 저감효과가 있는 배분 비율 조건(Qa>Qe)에서 Section 2에 유속 저감에 영향을 미치는 기존 여수로 방류량 배분 비율을 증가시켜 추가 검토(Case 10)를 수행하였다. 단독운영과 비교 시 하류하천에 유입되는 유량은 증가하였음에도 불구하고 기존 여수로 방류량에 의해 흐름이 하천 중심으로 집중되는 현상에 따라 대안부의 유속은 단독 운영에 비하여 감소하는 것을 확인하였고(Fig. 8 참조), 호안 설계 허용유속 및 허용 소류력 조건을 만족하는 구간이 발생하여 호안 안정성도 확보한 것으로 판단되었다. 최종적으로 각 Case 별 수위 결과의 경우 여수로 동시 운영을 수행하게 되면 대안부 전 구간에서 월류에 대한 안정성(ηmax/ηref<0.97(=기설제방고))은 확보하였다(Fig. 9 참조). 각 Case 별 대안부에서 최대 유속결과 및 산정한 소류력은 Table 6에 제시하였다.

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F6.jpg
    Fig. 6

    Maximum velocity on section 1 & 2 according to Qa

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F7.jpg
    Fig. 7

    Maximum shear on section 1 & 2 according to Qa

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F8.jpg
    Fig. 8

    Velocity results of FLOW-3D (a: auxiliary spillway operation only , b : simultaneous operation of spillways)

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F9.jpg
    Fig. 9

    Maximum water surface elevation on section 1 & 2 according to Qa

    Table 6.

    Numerical results for each cases (Case 7 ~ Case 10)

    Case (Qe &amp; Qa)Maximum Velocity (Vmax, m/s)Maximum Shear
    (τmax, kN/m2)
    Evaluation in terms of VpEvaluation in terms of τp
    Section 1Section 2Section 1Section 2Section 1Section 2Section 1Section 2
    7
    Qe : 0.50QpQa : 0.50Qp
    8.106.230.640.30No GoodNo GoodNo GoodNo Good
    8
    Qe : 0.61QpQa : 0.39Qp
    8.886.410.610.34No GoodNo GoodNo GoodNo Good
    9
    Qe : 0.39QpQa : 0.61Qp
    6.227.330.240.35No GoodNo GoodAcceptNo Good
    10
    Qe : 0.42QpQa : 0.58Qp
    6.394.790.300.19No GoodAcceptNo GoodAccept

    2.3.4 방류량 배분 비율의 허용 방류량 검토

    계획 홍수량 방류 시 기존 여수로와 보조 여수로의 배분 비율 검토 결과 Case 10(Qe = 0.42Qp, Qa = 0.58Qp)에서 방류에 따른 하류 하천의 피해를 최소화시킬 수 있는 것을 확인하였다. 그러나 대안부 전 구간에 대하여 호안 설계조건을 만족하지 못하였다. 따라서 기존 여수로와 보조 여수로의 방류 배분 비율을 고정시킨 후 총 방류량을 조절하여 허용 방류량을 검토하였다(Case 11 ~ Case 14).

    호안 안정성 측면에서 검토한 결과 계획홍수량 대비 총 방류량이 감소하면 최대 유속 및 최대 소류력이 감소하고 최종적으로 계획 홍수량의 77%를 방류할 경우 하류하천의 대안부에서 호안 설계조건을 모두 만족하는 것을 확인하였다(Fig. 10Fig. 11 참조). 각 Case 별 대안부에서 최대 유속결과 및 산정한 소류력은 Table 7에 제시하였다. 또한 Case 별 수위 검토 결과 처오름으로 인한 대안부 전 구간에서 월류에 대한 안정성(ηmax/ηref<0.97(=기설제방고))은 확보하였다(Fig. 12 참조).

    Table 7.

    Numerical results for each cases (Case 11 ~ Case 14)

    Case (Qe &amp; Qa)Maximum Velocity
    (Vmax, m/s)
    Maximum Shear
    (τmax, kN/m2)
    Evaluation in terms of VpEvaluation in terms of τp
    Section 1Section 2Section 1Section 2Section 1Section 2Section 1Section 2
    11
    Qe : 0.32QpQa : 0.45Qp
    3.634.530.090.26AcceptAcceptAcceptAccept
    12
    Qe : 0.35QpQa : 0.48Qp
    5.745.180.230.22No GoodNo GoodAcceptAccept
    13
    Qe : 0.38QpQa : 0.53Qp
    6.704.210.280.11No GoodAcceptAcceptAccept
    14
    Qe : 0.41QpQa : 0.56Qp
    6.545.240.280.24No GoodNo GoodAcceptAccept
    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F10.jpg
    Fig. 10

    Maximum velocity on section 1 & 2 according to total outflow

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F11.jpg
    Fig. 11

    Maximum shear on section 1 & 2 according to total outflow

    /media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F12.jpg
    Fig. 12

    Maximum water surface elevation on section 1 & 2 according to total outflow

    3. 결 론

    본 연구에서는 홍수 시 기존 여수로의 노후화로 인한 보조 여수로의 활용방안에 대하여 하류하천의 호안 안정성 측면에서 검토하였다. 여수로 방류로 인한 하류하천의 흐름특성을 검토하기 위하여 3차원 수치모형인 FLOW-3D를 활용하였고, 여수로 지형은 치수능력 증대사업을 통하여 완공된 ○○댐의 제원을 이용하였다. 하류하천 조도 계수 및 여수로 방류량은 하천기본계획을 참고하여 적용하였다. 최종적으로 여수로 방류로 인한 하류하천의 피해를 최소화 시킬 수 있는 적절한 보조 여수로의 활용방안을 도출하기 위하여 보조 여수로 단독 운영과 기존 여수로와의 동시 운영에 따른 하류 하천의 흐름특성 및 소류력의 변화를 검토하였다.

    수문은 완전 개도 상태에서 방류한다는 가정으로 계획 홍수량 조건에서 보조 여수로 단독 운영 시 하류하천 대안부의 유속 및 수위를 검토한 결과 기존 여수로 단독운영에 비하여 최대 유속 및 최대 수위가 감소하는 것을 확인할 수 있었으며, 이는 보조 여수로 단독 운영 시 하류하천으로 유입각도가 작아지고, 유입되는 하천의 폭이 증가되기 때문이다. 그러나 계획 홍수량 조건에서 하천호안 설계기준에서 제시한 허용 유속(5.0 m/s)과 허용 소류력(0.28 kN/m2)과 비교하였을 때 호안 안정성을 확보하지 못하였으며, 계획홍수량의 45% 이하 방류 시에 대안부의 호안 안정성을 확보하였다. 수위의 경우 여수로 방류에 따른 대안부에서 처오름 현상이 발생하여 월류에 대한 위험성을 확인하였고 이를 통하여 기존 여수로와의 동시 운영 방안을 도출하는 것이 중요하다고 판단된다. 따라서 기존 여수로와의 동시 운영 측면에서 기존 여수로와 보조 여수로의 배분 비율 및 총 방류량을 변화시켜가며 하류 하천의 흐름특성 및 소류력의 변화를 검토하였다. 배분 비율의 경우 기존 여수로와 보조 여수로의 균등 배분(Case 7) 및 편중 배분(Case 8 & Case 9)을 검토하여 보조 여수로의 방류량이 기존 여수로의 방류량보다 큰 경우 하류하천의 중심부로 집중되어 대안부의 최대유속, 최대소류력 및 최대수위가 감소하는 것을 확인하였다. 이를 근거로 기존 여수로의 방류 비율을 증가(Qe=0.42Qp, Qa=0.58Qp)시켜 검토한 결과 대안부 일부 구간에서 허용 유속 및 허용소류력 조건을 만족하는 것을 확인하였다. 이를 통하여 기존 여수로와 보조 여수로의 동시 운영을 통하여 적절한 방류량 배분 비율을 도출하는 것이 방류로 인한 하류하천의 피해를 저감하는데 효과적인 것으로 판단된다. 그러나 설계홍수량 방류 시 전 구간에서 허용 유속 및 소류력 조건을 만족하지 못하였다. 최종적으로 전체 방류량에서 기존 여수로의 방류 비율을 42%, 보조 여수로의 방류 비율을 58%로 설정하여 허용방류량을 검토한 결과, 계획홍수량의 77%이하로 방류 시 대안부의 최대유속은 기존여수로 방류의 지배영향구간(section 1)에서 3.63 m/s, 기존 여수로와 보조 여수로 방류의 영향구간(section 2)에서 4.53 m/s로 허용유속 조건을 만족하였고, 산정한 소류력도 각각 0.09 kN/m2 및 0.26 kN/m2로 허용 소류력 조건을 만족하여 대안부 호안의 안정성을 확보하였다고 판단된다.

    본 연구 결과는 기후변화 및 기존여수로의 노후화로 인하여 홍수 시 기존여수로의 단독운영으로 하류하천의 피해가 발생할 수 있는 현시점에서 치수증대 사업으로 완공된 보조 여수로의 활용방안에 대한 기초자료로 활용될 수 있고, 향후 계획 홍수량 유입 시 최적의 배분 비율 및 허용 방류량 도출에 이용할 수 있다. 다만 본 연구는 여수로 방류에 따른 제방에 작용하는 수충력은 검토하지 못하고, 허용 유속 및 허용소류력은 제방과 유수의 방향이 일정한 구간에 대하여 검토하였다. 또한 여수로 방류에 따른 대안부에서의 영향에 대해서만 검토하였고 수문 전면 개도 조건에서 검토하였다는 한계점은 분명히 있다. 이에 향후에는 다양한 수문 개도 조건 및 방류 시나리오를 적용 및 검토하여 보다 효율적이고, 효과적인 보조 여수로 활용방안을 도출하고자 한다.

    Acknowledgements

    본 결과물은 K-water에서 수행한 기존 및 신규 여수로 효율적 연계운영 방안 마련(2021-WR-GP-76-149)의 지원을 받아 연구되었습니다.

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    Korean References Translated from the English

    1 건설교통부·한국수자원공사 (2004). 댐의 수문학적 안정성 검토 및 치수능력증대방안 기본계획 수립 보고서. 세종: 국토교통부.

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    Fig. 5. The predicted shapes of initial breach (a) Rectangular (b) V-notch. Fig. 6. Dam breaching stages.

    Investigating the peak outflow through a spatial embankment dam breach

    공간적 제방댐 붕괴를 통한 최대 유출량 조사

    Mahmoud T.GhonimMagdy H.MowafyMohamed N.SalemAshrafJatwaryFaculty of Engineering, Zagazig University, Zagazig 44519, Egypt

    Abstract

    Investigating the breach outflow hydrograph is an essential task to conduct mitigation plans and flood warnings. In the present study, the spatial dam breach is simulated by using a three-dimensional computational fluid dynamics model, FLOW-3D. The model parameters were adjusted by making a comparison with a previous experimental model. The different parameters (initial breach shape, dimensions, location, and dam slopes) are studied to investigate their effects on dam breaching. The results indicate that these parameters have a significant impact. The maximum erosion rate and peak outflow for the rectangular shape are higher than those for the V-notch by 8.85% and 5%, respectively. Increasing breach width or decreasing depth by 5% leads to increasing maximum erosion rate by 11% and 15%, respectively. Increasing the downstream slope angle by 4° leads to an increase in both peak outflow and maximum erosion rate by 2.0% and 6.0%, respectively.

    유출 유출 수문곡선을 조사하는 것은 완화 계획 및 홍수 경보를 수행하는 데 필수적인 작업입니다. 본 연구에서는 3차원 전산유체역학 모델인 FLOW-3D를 사용하여 공간 댐 붕괴를 시뮬레이션합니다. 이전 실험 모델과 비교하여 모델 매개변수를 조정했습니다.

    다양한 매개변수(초기 붕괴 형태, 치수, 위치 및 댐 경사)가 댐 붕괴에 미치는 영향을 조사하기 위해 연구됩니다. 결과는 이러한 매개변수가 상당한 영향을 미친다는 것을 나타냅니다. 직사각형 형태의 최대 침식율과 최대 유출량은 V-notch보다 각각 8.85%, 5% 높게 나타났습니다.

    위반 폭을 늘리거나 깊이를 5% 줄이면 최대 침식률이 각각 11% 및 15% 증가합니다. 하류 경사각을 4° 증가시키면 최대 유출량과 최대 침식률이 각각 2.0% 및 6.0% 증가합니다.

    Keywords

    Spatial dam breach; FLOW-3D; Overtopping erosion; Computational fluid dynamics (CFD)

    1. Introduction

    There are many purposes for dam construction, such as protection from flood disasters, water storage, and power generationEmbankment failures may have a catastrophic impact on lives and infrastructure in the downstream regions. One of the most common causes of embankment dam failure is overtopping. Once the overtopping of the dam begins, the breach formation will start in the dam body then end with the dam failure. This failure occurs within a very short time, which threatens to be very dangerous. Therefore, understanding and modeling the embankment breaching processes is essential for conducting mitigation plans, flood warnings, and forecasting flood damage.

    The analysis of the dam breaching process is implemented by different techniques: comparative methods, empirical models with dimensional and dimensionless solutions, physical-based models, and parametric models. These models were described in detail [1]Parametric modeling is commonly used to simulate breach growth as a time-dependent linear process and calculate outflow discharge from the breach using hydraulics principles [2]. Alhasan et al. [3] presented a simple one-dimensional mathematical model and a computer code to simulate the dam breaching process. These models were validated by small dams breaching during the floods in 2002 in the Czech Republic. Fread [4] developed an erosion model (BREACH) based on hydraulics principles, sediment transport, and soil mechanics to estimate breach size, time of formation, and outflow discharge. Říha et al. [5] investigated the dam break process for a cascade of small dams using a simple parametric model for piping and overtopping erosion, as well as a 2D shallow-water flow model for the flood in downstream areas. Goodarzi et al. [6] implemented mathematical and statistical methods to assess the effect of inflows and wind speeds on the dam’s overtopping failure.

    Dam breaching studies can be divided into two main modes of erosion. The first mode is called “planar dam breach” where the flow overtops the whole dam width. While the second mode is called “spatial dam breach” where the flow overtops through the initial pilot channel (i.e., a channel created in the dam body). Therefore, the erosion will be in both vertical and horizontal directions [7].

    The erosion process through the embankment dams occurs due to the shear stress applied by water flows. The dam breaching evolution can be divided into three stages [8][9], but Y. Yang et al. [10] divided the breach development into five stages: Stage I, the seepage erosion; Stage II, the initial breach formation; Stage III, the head erosion; Stage IV, the breach expansion; and Stage V, the re-equilibrium of the river channel through the breach. Many experimental tests have been carried out on non-cohesive embankment dams with an initial breach to examine the effect of upstream inflow discharges on the longitudinal profile evolution and the time to inflection point [11].

    Zhang et al. [12] studied the effect of changing downstream slope angle, sediment grain size, and dam crest length on erosion rates. They noticed that increasing dam crest length and decreasing downstream slope angle lead to decreasing sediment transport rate. While the increase in sediment grain size leads to an increased sediment transport rate at the initial stages. Höeg et al. [13] presented a series of field tests to investigate the stability of embankment dams made of various materials. Overtopping and piping were among the failure tests carried out for the dams composed of homogeneous rock-fill, clay, or gravel with a height of up to 6.0 m. Hakimzadeh et al. [14] constructed 40 homogeneous cohesive and non-cohesive embankment dams to study the effect of changing sediment diameter and dam height on the breaching process. They also used genetic programming (GP) to estimate the breach outflow. Refaiy et al. [15] studied different scenarios for the downstream drain geometry, such as length, height, and angle, to minimize the effect of piping phenomena and therefore increase dam safety.

    Zhu et al. [16] examined the effect of headcut erosion on dam breach growth, especially in the case of cohesive dams. They found that the breach growth in non-cohesive embankments is slower than cohesive embankments due to the little effect of headcut. Schmocker and Hager [7] proposed a relationship for estimating peak outflow from the dam breach process.(1)QpQin-1=1.7exp-20hc23d5013H0

    where: Qp = peak outflow discharge.

    Qin = inflow discharge.

    hc = critical flow depth.

    d50 = mean sediment diameter.

    Ho = initial dam height.

    Yu et al. [17] carried out an experimental study for homogeneous non-cohesive embankment dams in a 180° bending rectangular flume to determine the effect of overtopping flows on breaching formation. They found that the main factors influencing breach formation are water level, river discharge, and embankment material diameter.

    Wu et al. [18] carried out a series of experiments to investigate the effect of breaching geometry on both non-cohesive and cohesive embankment dams in a U-bend flume due to overtopping flows. In the case of non-cohesive embankments, the non-symmetrical lateral expansion was noticed during the breach formation. This expansion was described by a coefficient ranging from 2.7 to 3.3.

    The numerical models of the dam breach can be categorized according to different parameters, such as flow dimensions (1D, 2D, or 3D), flow governing equations, and solution methods. The 1D models are mainly used to predict the outflow hydrograph from the dam breach. Saberi et al. [19] applied the 1D Saint-Venant equation, which is solved by the finite difference method to investigate the outflow hydrograph during dam overtopping failure. Because of the ability to study dam profile evolution and breach formation, 2D models are more applicable than 1D models. Guan et al. [20] and Wu et al. [21] employed both 2D shallow water equations (SWEs) and sediment erosion equations, which are solved by the finite volume method to study the effect of the dam’s geometry parameters on outflow hydrograph and dam profile evolution. Wang et al. [22] also proposed a second-order hybrid-type of total variation diminishing (TVD) finite-difference to estimate the breach outflow by solving the 2D (SWEs). The accuracy of (SWEs) for both vertical flow contraction and surface roughness has been assessed [23]. They noted that the accuracy of (SWEs) is acceptable for milder slopes, but in the case of steeper slopes, modelers should be more careful. Generally, the accuracy of 2D models is still low, especially with velocity distribution over the flow depth, lateral momentum exchange, density-driven flows, and bottom friction [24]. Therefore, 3D models are preferred. Larocque et al. [25] and Yang et al. [26] started to use three-dimensional (3D) models that depend on the Reynolds-averaged Navier-Stokes (RANS) equations.

    Previous experimental studies concluded that there is no clear relationship between the peak outflow from the dam breach and the initial breach characteristics. Some of these studies depend on the sharp-crested weir fixed at the end of the flume to determine the peak outflow from the breach, which leads to a decrease in the accuracy of outflow calculations at the microscale. The main goals of this study are to carry out a numerical simulation for a spatial dam breach due to overtopping flows by using (FLOW-3D) software to find an empirical equation for the peak outflow discharge from the breach and determine the worst-case that leads to accelerating the dam breaching process.

    2. Numerical simulation

    The current study for spatial dam breach is simulated by using (FLOW-3D) software [27], which is a powerful computational fluid dynamics (CFD) program.

    2.1. Geometric presentations

    A stereolithographic (STL) file is prepared for each change in the initial breach geometry and dimensions. The CAD program is useful for creating solid objects and converting them to STL format, as shown in Fig. 1.

    2.2. Governing equations

    The governing equations for water flow are three-dimensional Reynolds Averaged Navier-Stokes equations (RANS).

    The continuity equation:(2)∂ui∂xi=0

    The momentum equation:(3)∂ui∂t+1VFuj∂ui∂xj=1ρ∂∂xj-pδij+ν∂ui∂xj+∂uj∂xi-ρu`iu`j¯

    where u is time-averaged velocity,ν is kinematic viscosity, VF is fractional volume open to flow, p is averaged pressure and -u`iu`j¯ are components of Reynold’s stress. The Volume of Fluid (VOF) technique is used to simulate the free surface profile. Hirt et al. [28] presented the VOF algorithm, which employs the function (F) to express the occupancy of each grid cell with fluid. The value of (F) varies from zero to unity. Zero value refers to no fluid in the grid cell, while the unity value refers to the grid cell being fully occupied with fluid. The free surface is formed in the grid cells having (F) values between zero and unity.(4)∂F∂t+1VF∂∂xFAxu+∂∂yFAyv+∂∂zFAzw=0

    where (u, v, w) are the velocity components in (x, y, z) coordinates, respectively, and (AxAyAz) are the area fractions.

    2.3. Boundary and initial conditions

    To improve the accuracy of the results, the boundary conditions should be carefully determined. In this study, two mesh blocks are used to minimize the time consumed in the simulation. The boundary conditions for mesh block 1 are as follows: The inlet and sides boundaries are defined as a wall boundary condition (wall boundary condition is usually used for bound fluid by solid regions. In the case of viscous flows, no-slip means that the tangential velocity is equal to the wall velocity and the normal velocity is zero), the outlet is defined as a symmetry boundary condition (symmetry boundary condition is usually used to reduce computational effort during CFD simulation. This condition allows the flow to be transferred from one mesh block to another. No inputs are required for this boundary condition except that its location should be defined accurately), the bottom boundary is defined as a uniform flow rate boundary condition, and the top boundary is defined as a specific pressure boundary condition with assigned atmospheric pressure. The boundary conditions for mesh block 2 are as follows: The inlet is defined as a symmetry boundary condition, the outlet is defined as a free flow boundary condition, the bottom and sides boundaries are defined as a wall boundary condition, and the top boundary is defined as a specific pressure boundary condition with assigned atmospheric pressure as shown in Fig. 2. The initial conditions required to be set for the fluid (i.e., water) inside of the domain include configuration, temperature, velocities, and pressure distribution. The configuration of water depends on the dimensions and shape of the dam reservoir. While the other conditions have been assigned as follows: temperature is normal water temperature (25 °c) and pressure distribution is hydrostatic with no initial velocity.

    2.4. Numerical method

    FLOW-3D uses the finite volume method (FVM) to solve the governing equation (Reynolds-averaged Navier-Stokes) over the computational domain. A finite-volume method is an Eulerian approach for representing and evaluating partial differential equations in algebraic equations form [29]. At discrete points on the mesh geometry, values are determined. Finite volume expresses a small volume surrounding each node point on a mesh. In this method, the divergence theorem is used to convert volume integrals with a divergence term to surface integrals. After that, these terms are evaluated as fluxes at each finite volume’s surfaces.

    2.5. Turbulent models

    Turbulence is the chaotic, unstable motion of fluids that occurs when there are insufficient stabilizing viscous forces. In FLOW-3D, there are six turbulence models available: the Prandtl mixing length model, the one-equation turbulent energy model, the two-equation (k – ε) model, the Renormalization-Group (RNG) model, the two-equation (k – ω) models, and a large eddy simulation (LES) model. For simulating flow motion, the RNG model is adopted to simulate the motion behavior better than the k – ε and k – ω.

    models [30]. The RNG model consists of two main equations for the turbulent kinetic energy KT and its dissipation.εT(5)∂kT∂t+1VFuAx∂kT∂x+vAy∂kT∂y+wAz∂kT∂z=PT+GT+DiffKT-εT(6)∂εT∂t+1VFuAx∂εT∂x+vAy∂εT∂y+wAz∂εT∂z=C1.εTKTPT+c3.GT+Diffε-c2εT2kT

    where KT is the turbulent kinetic energy, PT is the turbulent kinetic energy production, GT is the buoyancy turbulence energy, εT is the turbulent energy dissipation rate, DiffKT and Diffε are terms of diffusion, c1, c2 and c3 are dimensionless parameters, in which c1 and c3 have a constant value of 1.42 and 0.2, respectively, c2 is computed from the turbulent kinetic energy (KT) and turbulent production (PT) terms.

    2.6. Sediment scour model

    The sediment scour model available in FLOW-3D can calculate all the sediment transport processes including Entrainment transport, Bedload transport, Suspended transport, and Deposition. The erosion process starts once the water flows remove the grains from the packed bed and carry them into suspension. It happens when the applied shear stress by water flows exceeds critical shear stress. This process is represented by entrainment transport in the numerical model. After entrained, the grains carried by water flow are represented by suspended load transport. After that, some suspended grains resort to settling because of the combined effect of gravity, buoyancy, and friction. This process is described through a deposition. Finally, the grains sliding motions are represented by bedload transport in the model. For the entrainment process, the shear stress applied by the fluid motion on the packed bed surface is calculated using the standard wall function as shown in Eq.7.(7)ks,i=Cs,i∗d50

    where ks,i is the Nikuradse roughness and Cs,i is a user-defined coefficient. The critical bed shear stress is defined by a dimensionless parameter called the critical shields number as expressed in Eq.8.(8)θcr,i=τcr,i‖g‖diρi-ρf

    where θcr,i is the critical shields number, τcr,i is the critical bed shear stress, g is the absolute value of gravity acceleration, di is the diameter of the sediment grain, ρi is the density of the sediment species (i) and ρf is the density of the fluid. The value of the critical shields number is determined according to the Soulsby-Whitehouse equation.(9)θcr,i=0.31+1.2d∗,i+0.0551-exp-0.02d∗,i

    where d∗,i is the dimensionless diameter of the sediment, given by Eq.10.(10)d∗,i=diρfρi-ρf‖g‖μf213

    where μf is the fluid dynamic viscosity. For the sloping bed interface, the value of the critical shields number is modified according to Eq.11.(11)θ`cr,i=θcr,icosψsinβ+cos2βtan2φi-sin2ψsin2βtanφi

    where θ`cr,i is the modified critical shields number, φi is the angle of repose for the sediment, β is the angle of bed slope and ψ is the angle between the flow and the upslope direction. The effects of the rolling, hopping, and sliding motions of grains along the packed bed surface are taken by the bedload transport process. The volumetric bedload transport rate (qb,i) per width of the bed is expressed in Eq.12.(12)qb,i=Φi‖g‖ρi-ρfρfdi312

    where Φi is the dimensionless bedload transport rate is calculated by using Meyer Peter and Müller equation.(13)Φi=βMPM,iθi-θ`cr,i1.5cb,i

    where βMPM,i is the Meyer Peter and Müller user-defined coefficient and cb,i is the volume fraction of species i in the bed material. The suspended load transport is calculated as shown in Eq.14.(14)∂Cs,i∂t+∇∙Cs,ius,i=∇∙∇DCs,i

    where Cs,i is the suspended sediment mass concentration, D is the diffusivity, and us,i is the grain velocity of species i. Entrainment and deposition are two opposing processes that take place at the same time. The lifting and settling velocities for both entrainment and deposition processes are calculated according to Eq.15 and Eq.16, respectively.(15)ulifting,i=αid∗,i0.3θi-θ`cr,igdiρiρf-1(16)usettling,i=υfdi10.362+1.049d∗,i3-10.36

    where αi is the entrainment coefficient of species i and υf is the kinematic viscosity of the fluid.

    2.7. Grid type

    Using simple rectangular orthogonal elements in planes and hexahedral in volumes in the (FLOW-3D) program makes the mesh generation process easier, decreases the required memory, and improves numerical accuracy. Two mesh blocks were used in a joined form with a size ratio of 2:1. The first mesh block is coarser, which contains the reservoir water, and the second mesh block is finer, which contains the dam. For achieving accuracy and efficiency in results, the mesh size is determined by using a grid convergence test. The optimum uniform cell size for the first mesh block is 0.012 m and for the second mesh block is 0.006 m.

    2.8. Time step

    The maximum time step size is determined by using a Courant number, which controls the distance that the flow will travel during the simulation time step. In this study, the Courant number was taken equal to 0.25 to prevent the flow from traveling through more than one cell in the time step. Based on the Courant number, a maximum time step value of 0.00075 s was determined.

    2.9. Numerical model validation

    The numerical model accuracy was achieved by comparing the numerical model results with previous experimental results. The experimental study of Schmocker and Hager [7] was based on 31 tests with changes in six parameters (d50, Ho, Bo, Lk, XD, and Qin). All experimental tests were conducted in a straight open glass-sided flume. The horizontal flume has a rectangular cross-section with a width of 0.4 m and a height of 0.7 m. The flume was provided with a flow straightener and an intake with a length of 0.66 m. All tested dams were inserted at various distances (XD) from the intake. Test No.1 from this experimental program was chosen to validate the numerical model. The different parameters used in test No.1 are as follows:

    (1) uniform sediment with a mean diameter (d50 = 0.31 mm), (2) Ho = 0.2 m, (3) Bo = 0.2 m, (4) Lk = 0.1 m,

    (5) XD = 1.0 m, (6) Qin = 6.0 lit/s, (7) Su and Sd = 2:1, (8) mass density (ρs = 2650 kg/m3(9) Homogenous and non-cohesive embankment dam. As shown in Fig. 2, the simulation is contained within a rectangular grid with dimensions: 3.56 m in the x-direction (where 0.66 m is used as inlet, 0.9 m as dam base width, and 1.0 m as outlet), in y-direction 0.2 m (dam length), and in the z-direction 0.3 m, which represents the dam height (0.2 m) with a free distance (0.1 m) above the dam. There are two main reasons that this experimental program is preferred for the validation process. The first reason is that this program deals with homogenous, non-cohesive soil, which is available in FLOW-3D. The second reason is that this program deals with small-scale models which saves time for numerical simulation. Finally, some important assumptions were considered during the validation process. The flow is assumed to be incompressible, viscous, turbulent, and three-dimensional.

    By comparing dam profiles at different time instants for the experimental test with the current numerical model, it appears that the numerical model gives good agreement as shown in Fig. 3 and Fig. 4, with an average error percentage of 9% between the experimental results and the numerical model.

    3. Analysis and discussions

    The current model is used to study the effects of different parameters such as (initial breach shapes, dimensions, locations, upstream and downstream dam slopes) on the peak outflow discharge, QP, time of peak outflow, tP, and rate of erosion, E.

    This study consists of a group of scenarios. The first scenario is changing the shapes of the initial breach according to Singh [1], the most predicted shapes are rectangular and V-notch as shown in Fig. 5. The second scenario is changing the initial breach dimensions (i.e., width and depth). While the third scenario is changing the location of the initial breach. Eventually, the last scenario is changing the upstream and downstream dam slopes.

    All scenarios of this study were carried out under the same conditions such as inflow discharge value (Qin=1.0lit/s), dimensions of the tested dam, where dam height (Ho=0.20m), crest width.

    (Lk=0.1m), dam length (Bo=0.20m), and homogenous & non-cohesive soil with a mean diameter (d50=0.31mm).

    3.1. Dam breaching process evolution

    The dam breaching process is a very complex process due to the quick changes in hydrodynamic conditions during dam failure. The dam breaching process starts once water flows reach the downstream face of the dam. During the initial stage of dam breaching, the erosion process is relatively quiet due to low velocities of flow. As water flows continuously, erosion rates increase, especially in two main zones: the crest and the downstream face. As soon as the dam crest is totally eroded, the water levels in the dam reservoir decrease rapidly, accompanied by excessive erosion in the dam body. The erosion process continues until the water levels in the dam reservoir equal the remaining height of the dam.

    According to Zhou et al. [11], the breaching process consists of three main stages. The first stage starts with beginning overtopping flow, then ends when the erosion point directed upstream and reached the inflection point at the inflection time (ti). The second stage starts from the end of the stage1 until the occurrence of peak outflow discharge at the peak outflow time (tP). The third stage starts from the end of the stage2 until the value of outflow discharge becomes the same as the value of inflow discharge at the final time (tf). The outflow discharge from the dam breach increases rapidly during stage1 and stage2 because of the large dam storage capacity (i.e., the dam reservoir is totally full of water) and excessive erosion. While at stage3, the outflow values start to decrease slowly because most of the dam’s storage capacity was run out. The end of stage3 indicates that the dam storage capacity was totally run out, so the outflow equalized with the inflow discharge as shown in Fig. 6 and Fig. 7.

    3.2. The effect of initial breach shape

    To identify the effect of the initial breach shape on the evolution of the dam breaching process. Three tests were carried out with different cross-section areas for each shape. The initial breach is created at the center of the dam crest. Each test had an ID to make the process of arranging data easier. The rectangular shape had an ID (Rec5h & 5b), which means that its depth and width are equal to 5% of the dam height, and the V-notch shape had an ID (V-noch5h & 1:1) which means that its depth is equal to 5% of the dam height and its side slope is equal to 1:1. The comparison between rectangular and V-notch shapes is done by calculating the ratio between maximum dam height at different times (ZMax) to the initial dam height (Ho), rate of erosion, and hydrograph of outflow discharge for each test. The rectangular shape achieves maximum erosion rate and minimum inflection time, in addition to a rapid decrease in the dam reservoir levels. Therefore, the dam breaching is faster in the case of a rectangular shape than in a V-notch shape, which has the same cross-section area as shown in Fig. 8.

    Also, by comparing the hydrograph for each test, the peak outflow discharge value in the case of a rectangular shape is higher than the V-notch shape by 5% and the time of peak outflow for the rectangular shape is shorter than the V-notch shape by 9% as shown in Fig. 9.

    3.3. The effect of initial breach dimensions

    The results of the comparison between the different initial breach shapes indicate that the worst initial breach shape is rectangular, so the second scenario from this study concentrated on studying the effect of a change in the initial rectangular breach dimensions. Groups of tests were carried out with different depths and widths for the rectangular initial breach. The first group had a depth of 5% from the dam height and with three different widths of 5,10, and 15% from the dam height, the second group had a depth of 10% with three different widths of 5,10, and 15%, the third group had a depth of 15% with three different widths of 5,10, and 15% and the final group had a width of 15% with three different heights of 5, 10, and 15% for a rectangular breach shape. The comparison was made as in the previous section to determine the worst case that leads to the quick dam failure as shown in Fig. 10.

    The results show that the (Rec 5 h&15b) test achieves a maximum erosion rate for a shorter period of time and a minimum ratio for (Zmax / Ho) as shown in Fig. 10, which leads to accelerating the dam failure process. The dam breaching process is faster with the minimum initial breach depth and maximum initial breach width. In the case of a minimum initial breach depth, the retained head of water in the dam reservoir is high and the crest width at the bottom of the initial breach (L`K) is small, so the erosion point reaches the inflection point rapidly. While in the case of the maximum initial breach width, the erosion perimeter is large.

    3.4. The effect of initial breach location

    The results of the comparison between the different initial rectangular breach dimensions indicate that the worst initial breach dimension is (Rec 5 h&15b), so the third scenario from this study concentrated on studying the effect of a change in the initial breach location. Three locations were checked to determine the worst case for the dam failure process. The first location is at the center of the dam crest, which was named “Center”, the second location is at mid-distance between the dam center and dam edge, which was named “Mid”, and the third location is at the dam edge, which was named “Edge” as shown in Fig. 11. According to this scenario, the results indicate that the time of peak outflow discharge (tP) is the same in the three cases, but the maximum value of the peak outflow discharge occurs at the center location. The difference in the peak outflow values between the three cases is relatively small as shown in Fig. 12.

    The rates of erosion were also studied for the three cases. The results show that the maximum erosion rate occurs at the center location as shown in Fig. 13. By making a comparison between the three cases for the dam storage volume. The results show that the center location had the minimum values for the dam storage volume, which means that a large amount of water has passed to the downstream area as shown in Fig. 14. According to these results, the center location leads to increased erosion rate and accelerated dam failure process compared with the two other cases. Because the erosion occurs on both sides, but in the case of edge location, the erosion occurs on one side.

    3.5. The effect of upstream and downstream dam slopes

    The results of the comparison between the different initial rectangular breach locations indicate that the worst initial breach location is the center location, so the fourth scenario from this study concentrated on studying the effect of a change in the upstream (Su) and downstream (Sd) dam slopes. Three slopes were checked individually for both upstream and downstream slopes to determine the worst case for the dam failure process. The first slope value is (2H:1V), the second slope value is (2.5H:1V), and the third slope value is (3H:1V). According to this scenario, the results show that the decreasing downstream slope angle leads to increasing time of peak outflow discharge (tP) and decreasing value of peak outflow discharge. The difference in the peak outflow values between the three cases for the downstream slope is 2%, as shown in Fig. 15, but changing the upstream slope has a negligible impact on the peak outflow discharge and its time as shown in Fig. 16.

    The rates of erosion were also studied in the three cases for both upstream and downstream slopes. The results show that the maximum erosion rate increases by 6.0% with an increasing downstream slope angle by 4°, as shown in Fig. 17. The results also indicate that the erosion rates aren’t affected by increasing or decreasing the upstream slope angle, as shown in Fig. 18. According to these results, increasing the downstream slope angle leads to increased erosion rate and accelerated dam failure process compared with the upstream slope angle. Because of increasing shear stress applied by water flows in case of increasing downstream slope.

    According to all previous scenarios, the dimensionless peak outflow discharge QPQin is presented for a fixed dam height (Ho) and inflow discharge (Qin). Fig. 19 illustrates the relationship between QP∗=QPQin and.

    Lr=ho2/3∗bo2/3Ho. The deduced relationship achieves R2=0.96.(17)QP∗=2.2807exp-2.804∗Lr

    4. Conclusions

    A spatial dam breaching process was simulated by using FLOW-3D Software. The validation process was performed by making a comparison between the simulated results of dam profiles and the dam profiles obtained by Schmocker and Hager [7] in their experimental study. And also, the peak outflow value recorded an error percentage of 12% between the numerical model and the experimental study. This model was used to study the effect of initial breach shape, dimensions, location, and dam slopes on peak outflow discharge, time of peak outflow, and the erosion process. By using the parameters obtained from the validation process, the results of this study can be summarized in eight points as follows.1.

    The rectangular initial breach shape leads to an accelerating dam failure process compared with the V-notch.2.

    The value of peak outflow discharge in the case of a rectangular initial breach is higher than the V-notch shape by 5%.3.

    The time of peak outflow discharge for a rectangular initial breach is shorter than the V-notch shape by 9%.4.

    The minimum depth and maximum width for the initial breach achieve maximum erosion rates (increasing breach width, b0, or decreasing breach depth, h0, by 5% from the dam height leads to an increase in the maximum rate of erosion by 11% and 15%, respectively), so the dam failure is rapid.5.

    The center location of the initial breach leads to an accelerating dam failure compared with the edge location.6.

    The initial breach location has a negligible effect on the peak outflow discharge value and its time.7.

    Increasing the downstream slope angle by 4° leads to an increase in both peak outflow discharge and maximum rate of erosion by 2.0% and 6.0%, respectively.8.

    The upstream slope has a negligible effect on the dam breaching process.

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    Effect of roughness on separation zone dimensions.

    Experimental and numerical study of flow at a 90 degree lateral turnout with enhanced roughness coefficient and invert level changes

    조도 계수 및 역전 수준 변화가 개선된 90도 측면 분출구에서의 유동에 대한 실험적 및 수치적 연구

    Maryam BagheriSeyed M. Ali ZomorodianMasih ZolghadrH. Md. AzamathullaC. Venkata Siva Rama Prasad

    Abstract

    측면 분기기(흡입구)의 상류 측에서 흐름 분리는 분기기 입구에서 와류를 일으키는 중요한 문제입니다. 이는 흐름의 유효 폭, 출력 용량 및 효율성을 감소시킵니다. 따라서 분리지대의 크기를 파악하고 크기를 줄이기 위한 방안을 제시하는 것이 필수적이다. 본 연구에서는 분리 구역의 치수를 줄이기 위한 방법으로 7가지 유형의 거칠기 요소를 분기구 입구에 설치하고 4가지 다른 배출(총 84번의 실험을 수행)과 함께 3개의 서로 다른 베드 반전 레벨을 조사했습니다. 또한 3D CFD(Computational Fluid Dynamics) 모델을 사용하여 분리 영역의 흐름 패턴과 치수를 평가했습니다. 결과는 거칠기 계수를 향상시키면 분리 영역 치수를 최대 38%까지 줄일 수 있는 반면, 드롭 구현 효과는 사용된 거칠기 계수를 기반으로 이 영역을 다르게 축소할 수 있음을 보여주었습니다. 두 가지 방법을 결합하면 분리 영역 치수를 최대 63%까지 줄일 수 있습니다.

    Flow separation at the upstream side of lateral turnouts (intakes) is a critical issue causing eddy currents at the turnout entrance. It reduces the effective width of flow, turnout capacity and efficiency. Therefore, it is essential to identify the dimensions of the separation zone and propose remedies to reduce its dimensions. Installation of 7 types of roughening elements at the turnout entrance and 3 different bed invert levels, with 4 different discharges (making a total of 84 experiments) were examined in this study as a method to reduce the dimensions of the separation zone. Additionally, a 3-D Computational Fluid Dynamic (CFD) model was utilized to evaluate the flow pattern and dimensions of the separation zone. Results showed that enhancing the roughness coefficient can reduce the separation zone dimensions up to 38% while the drop implementation effect can scale down this area differently based on the roughness coefficient used. Combining both methods can reduce the separation zone dimensions up to 63%.

    HIGHLIGHTS

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    • Flow separation at the upstream side of lateral turnouts (intakes) is a critical issue causing eddy currents at the turnout entrance.
    • Installation of 7 types of roughening elements at the turnout entrance and 3 different bed level inverts were investigated.
    • Additionally, a 3-D Computational Fluid Dynamic (CFD) model was utilized to evaluate the flow.
    • Combining both methods can reduce the separation zone dimensions by up to 63%.
    Experimental and numerical study of flow at a 90 degree lateral turnout with enhanced roughness coefficient and invert level changes
    Experimental and numerical study of flow at a 90 degree lateral turnout with enhanced roughness coefficient and invert level changes

    Keywords

    discharge ratioflow separation zoneintakethree dimensional simulation

    INTRODUCTION

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    Turnouts or intakes are amongst the oldest and most widely used hydraulic structures in irrigation networks. Turnouts are also used in water distribution, transmission networks, power generation facilities, and waste water treatment plants etc. The flows that enter a turnout have a strong momentum in the direction of the main waterway and that is why flow separation occurs inside the turnout. The horizontal vortex formed in the separation area is a suitable place for accumulation and deposition of sediments. The separation zone is a vulnerable area for sedimentation and for reduction of effective flow due to a contracted flow region in the lateral channel. Sedimentaion in the entrance of the intake can gradually be transfered into the lateral channel and decrease the capacity of the higher order channels over time (Jalili et al. 2011). On the other hand, the existence of coarse-grained materials causes erosion and destruction of the waterway side walls and bottom. In addition, sedimentation creates conditions for vegetation to take root and damage the waterway cover, which causes water to leak from its perimeter. Therefore, it is important to investigate the pattern of the flow separation area in turnouts and provide solutions to reduce the dimensions of this area.

    The three-dimensional flow structure at turnouts is quite complex. In an experimental study by Neary & Odgaard (1993) in a 90-degree water turnout it was found that the secondary currents and separation zone varies from the bed to the water surface. They also found that at a 90-degree water turnout, the bed roughness and discharge ratio play a critical role in flow structure. They asserted that an explanation of sediment behavior at a diversion entrance requires a comprehensive understanding of 3D flow patterns around the lateral-channel entrance. In addition, they suggested that there is a strong similarity between flow in a channel bend and a diversion channel, and that this similarity can rationalize the use of bend flow models for estimation of 3D flow structures in diversion channels.

    Some of the distinctive characteristics of dividing flow in a turnout include a zone of separation immediately near the entrance of the lateral turnout (separation zone), a contracted flow region in the branch channel (contracted flow), and a stagnation point near the downstream corner of the junction (stagnation zone). In the region downstream of the junction, along the continuous far wall, separation due to flow expansion may occur (Ramamurthy et al. 2007), that is, a separation zone. This can both reduce the turnout efficiency and the effective width of flow while increasing the sediment deposition in the turnout entrance (Jalili et al. 2011). Installation of submerged vanes in the turnout entrance is a method which is already applied to reduce the size of flow separation zones. The separation zone draws sediments and floating materials into themselves. This reduces effective cross-section area and reduces transmission capacity. These results have also been obtained in past studies, including by Ramamurthy et al. (2007) and in Jalili et al. (2011). Submerged vanes (Iowa vanes) are designed in order to modify the near-bed flow pattern and bed-sediment motion in the transverse direction of the river. The vanes are installed vertically on the channel bed, at an angle of attack which is usually oriented at 10–25 degrees to the local primary flow direction. Vane height is typically 0.2–0.5 times the local water depth during design flow conditions and vane length is 2–3 times its height (Odgaard & Wang 1991). They are vortex-generating devices that generate secondary circulation, thereby redistributing sediment within the channel cross section. Several factors affect the flow separation zone such as the ratio of lateral turnout discharge to main channel discharge, angle of lateral channel with respect to the main channel flow direction and size of applied submerged vanes. Nakato et al. (1990) found that sediment management using submerged vanes in the turnout entrance to Station 3 of the Council Bluffs plant, located on the Missouri River, is applicable and efficient. The results show submerged vanes are an appropriate solution for reduction of sediment deposition in a turnout entrance. The flow was treated as 3D and tests results were obtained for the flow characteristics of dividing flows in a 90-degree sharp-edged, junction. The main and lateral channel were rectangular with the same dimensions (Ramamurthy et al., 2007).

    Keshavarzi & Habibi (2005) carried out experiments on intake with angles of 45, 67, 79 and 90 degrees in different discharge ratios and reported the optimum angle for inlet flow with the lowest flow separation area to be about 55 degrees. The predicted flow characteristics were validated using experimental data. The results indicated that the width and length of the separation zone increases with the increase in the discharge ratio Qr (ratio of outflow per unit width in the turnout to inflow per unit width in the main channel).

    Abbasi et al. (2004) performed experiments to investigate the dimensions of the flow separation zone at a lateral turnout entrance. They demonstrated that the length and width of the separation zone decreases with the increasing ratio of lateral turn-out discharge. They also found that with a reducing angle of lateral turnout, the length of the separation zone scales up and width of separation zone reduces. Then they compared their observations with results of Kasthuri & Pundarikanthan (1987) who conducted some experiments in an open-channel junction formed by channels of equal width and an angle of lateral 90 degree turnout, which showed the dimensions of the separation zone in their experiments to be smaller than in previous studies. Kasthuri & Pundarikanthan (1987) studied vortex and flow separation dimensions at the entrance of a 90 degree channel. Results showed that increasing the diversion discharge ratio can reduce the length and width of the vortex area. They also showed that the length and width of the vortex area remain constant at diversion ratios greater than 0.7. Karami Moghaddam & Keshavarzi (2007) analyzed the flow characteristics in turnouts with angles of 55 and 90 degrees. They reported that the dimensions of the separation zone decrease by increasing the discharge ratio and reducing the turnout angle with respect to the main channel. Studies about flow separation zone can be found in Jalili et al. (2011)Nikbin & Borghei (2011)Seyedian et al. (2008).

    Jamshidi et al. (2016) measured the dimensions of a flow separation zone in the presence of submerged vanes with five arrangements (parallel, stagger, compound, piney and butterflies). Results showed that the ratio of the width to the length of the separation zone (shape index) was between 0.2 and 0.28 for all arrangements.

    Karami et al. (2017) developed a 3D computational fluid dynamic (CFD) code which was calibrated by measured data. They used the model to evaluate flow pattern, diversion ratio of discharge, strength of the secondary flow, and dimensions of the vortex inside the channel in various dikes and submerged vane installation scenarios. Results showed that the diversion ratio of discharge in the diversion channel is dependent on the width of the flow separation area in the main channel. A dike, perpendicular to the flow, doubles the ratio of diverted discharge and reduces the suspended sediment load compared with the base-line situation by creating outer arch conditions. In addition, increasing the longitudinal distance between vanes increases the velocity gradient between the vanes and leads to a more severe erosion of the bed near the vanes.Figure 1VIEW LARGEDOWNLOAD SLIDE

    Laboratory channel dimensions.

    Al-Zubaidy & Hilo (2021) used the Navier–Stokes equation to study the flow of incompressible fluids. Using the CFD software ANSYS Fluent 19.2, 3D flow patterns were simulated at a diversion channel. Their results showed good agreement using the comparison between the experimental and numerical results when the k-omega turbulence viscous model was employed. Simulation of the flow pattern was then done at the lateral channel junction using a variety of geometry designs. These improvements included changing the intake’s inclination angle and chamfering and rounding the inner corner of the intake mouth instead of the sharp edge. Flow parameters at the diversion including velocity streamlines, bed shear stress, and separation zone dimensions were computed in their study. The findings demonstrated that changing the 90° lateral intake geometry can improve the flow pattern and bed shear stress at the intake junction. Consequently, sedimentation and erosion problems are reduced. According to the conclusions of their study, a branching angle of 30° to 45° is the best configuration for increasing branching channel discharge, lowering branching channel sediment concentration.

    The review of the literature shows that most of the studies deal with turnout angle, discharge ratio and implementation of vanes as techniques to reduce the area of the separation zone. This study examines the effect of roughness coefficient and drop implementation at the entrance of a 90-degree lateral turnout on the dimensions of the separation zone. As far as the authors are aware, these two variables have never been studied as a remedy to decrease the separation zone dimensions whilst enhancing turnout efficiency. Additionally, a three-dimensional numerical model is applied to simulate the flow pattern around the turnout. The numerical results are verified against experimental data.

    METHOD

    Experimental setup

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    The experiments were conducted in a 90 degree dividing flow laboratory channel. The main channel is 15 m long, 0.5 m wide and 0.4 m high and the branch channel is 3 m long, 0.35 m wide and 0.4 m high, as shown in Figure 1. The tests were carried out at 9.65 m from the beginning of the flume and were far enough from the inlet, so we were sure that the flow was fully developed. According to Kirkgöz & Ardiçlioğlu (1997) the length of the developing region would be approximantly 65 and 72 times the flow depth. In this study, the depth is 9 cm, which makes this condition.

    Both the main and lateral channel had a slope of 0.0003 with side walls of concrete. A 100 hp pump discharged the water into a stilling basin at the entrance of the main flume. The discharge was measured using an ultrasonic discharge meter around the discharge pipe. Eighty-four experiments in total were carried out at range of 0.1<Fr<0.4 (Froude numbers in main channel and upstream of turnout). The depth of water in the main channel in the experiments was 9 cm, in which case the effect of surface tension can be considered; according to research by Zolghadr & Shafai Bejestan (2020) and Zolghadr et al. (2021), when the water depth is more than 6 cm, the effect of surface tension is reduced and can be ignored given that the separation phenomenon occurs in the boundary layer, the height of the roughness creates disturbances in growth and development of the boundary layer and, as a result, separation growth is also faced with disruption and its dimensions grow less compared to smooth surfaces. Similar conditions occur in case of drop implementation. A disturbance occurs in the growth of the boundary layer and as a result the separation zone dimensions decrease. In order to investigate the effect of roughness coefficient and drop implementation on the separation zone dimensions, four different discharges (16, 18, 21, 23 l/s) in subcritical conditions, seven Manning (Strickler) roughness coefficients (0.009, 0.011, 0.017, 0.023, 0.028, 0.030, 0.032) as shown in Figure 2 and three invert elevation differences between the main channel and lateral turnout invert (0, 5 and 10 cm) at the entrance of the turnout were considered. The Manning roughness coefficient values were selected based on available and feasible values for real conditions, so that 0.009 is equivalent to galvanized sheet roughness and selected for the baseline tests. 0.011 is for concrete with neat surface, 0.017 and 0.023 are for unfinished and gunite concrete respectively. 0.030 and 0.032 values are for concrete on irregular excavated rock (Chow 1959). The roughness coefficients were created by gluing sediment particles on a thin galvanized sheet which was installed at the upstream side of the lateral turnout. The values of roughness coefficients were calculated based on the Manning-Strickler formula. For this purpose, some uniformly graded sediment samples were prepared and the Manning roughness coefficient of each sample was determined with respect to the median size (D50) value pasted into the Manning-Strickler formula. Some KMnO4 was sifted in the main channel upstream to visualize and measure the dimensions of the separation zone. Consequently, when KMnO4 approached the lateral turnout a photo of the separation zone was taken from a top view. All the experiments were recorded and several photos were taken during the experiment after stablishment of steady flow conditions. The photos were then imported to AutoCAD to measure the separation zone dimensions. Because all the shooting was done with a high-definition camera and it was possible to zoom in, the results are very accurate.Figure 2VIEW LARGEDOWNLOAD SLIDE

    Roughness plates.

    The velocity values were also recorded by a one-dimensional velocity meter at 15 cm distance from the turnout entrance and in transverse direction (perpendicular to the flow direction).

    The water level was also measured by depth gauges with a accuracy of 0.1 mm, and velocity in one direction with a single-dimensional KENEK LP 1100 with an accuracy of ±0.02 m/s (0–1 m/s), ± 0.04 m/s (1–2 m/s), ± 0.08 m/s (2–4 m/s), ±0.10 m/s (4–5 m/s).

    Numerical simulation

    ListenA FLOW-3D numerical model was utilized as a solver of the Navier-Stokes equation to simulate the three-dimensional flow field at the entrance of the turnout. The governing equations included continuity momentum equations. The continuity equation, regardless of the density of the fluid in the form of Cartesian coordinates x, y, and z, is as follows:

    formula

    (1)where uv, and w represent the velocity components in the x, y, and z directions, respectively; AxAy, and Az are the surface flow fractions in the xy, and z directions, respectively; VF denotes flow volume fraction; r is the density of the fluid; t is time; and Rsor refers to the source of the mass. Equations (2)–(4) show momentum equations in xy and z dimensions respectively :

    formula

    (2)

    formula

    (3)

    formula

    (4)where GxGy, and Gz are the accelerations caused by gravity in the xy, and z directions, respectively; and fxfy, and fz are the accelerations caused by viscosity in the xy, and z directions, respectively.

    The turbulence models used in this study were the renormalized group (RNG) models. Evaluation of the concordance of the mentioned models with experimental studies showed that the RNG model provides more accurate results.

    Two blocks of mesh were used to simulate the main channels and lateral turnout. The meshes were denser in the vicinity of the entrance of the turnout in order to increase the accuracy of computations. Boundary conditions for the main mesh block included inflow for the channel entrance (volumetric flow rate), outflow for the channel exit, ‘wall’ for the bed and the right boundary and ‘symmetry’ for the top (free surface) and left boundaries (turnout). The side wall roughness coefficient was given to the software as the Manning number in surface roughness of any component. Considering the restrictions in the available processor, a main mesh block with appropriate mesh size was defined to simulate the main flow field in the channel, while the nested mesh-block technique was utilized to create a very dense solution field near the roughness plate in order to provide accurate results around the plates and near the entrance of the lateral turnout. This technique reduced the number of required mesh elements by up to 60% in comparison with the method in which the mesh size of the main solution field was decreased to the required extent.

    The numerical outputs are verified against experimental data. The hydraulic characteristics of the experiment are shown in Table 1.Table 1

    Hydraulic conditions of the flow

    Q(L/s)FrY1 (m)Q2/Q1
    16 0.449 0.09 0.22 
    18 0.335 0.09 0.61 
    21 0.242 0.09 0.71 
    23 0.180 0.09 1.04 

    RESULTS AND DISCUSSION

    Experimental results

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    During the experiments, the dimensions of the separation zone were recorded with an HD camera. Some photos were imported to AutoCad software. Then, the separation zones dimensions were measured and compared in different scenarios.

    At the beginning, the flow pattern in the separation zone for four different hydraulic conditions was studied for seven different Manning roughness coefficients from 0.009 to 0.032. To compare the obtained results, roughness of 0.009 was considered as the base line. The percentage of reduction in separation zone area in different roughness coefficients is shown in Figure 3. According to this figure, by increasing the roughness of the turnout side wall, the separation zone area ratio reduces (ratio of separation zone area to turnout area). In other words, in any desired Froud number, the highest dimensions of the separation zone area are related to the lowest roughness coefficients. In Figure 3, ‘A’ is the area of the separation zone and ‘Ai’ represents the total area of the turnout.Figure 3VIEW LARGEDOWNLOAD SLIDE

    Effect of roughness on separation zone dimensions.Figure 4VIEW LARGEDOWNLOAD SLIDE

    Effect of roughness on separation zone dimensions.

    It should be mentioned that the separation zone dimensions change with depth, so that the area is larger at the surface than near the bed. This study measured the dimensions of this area at the surface. Figure 4 show exactly where the roughness elements were located.Figure 5VIEW LARGEDOWNLOAD SLIDE

    Comparison of separation zone for n=0.023 and n=0.032.

    Figure 5 shows images of the separation zone at n=0.023 and n=0.032 as examples, and show that the separation area at n=0.032 is smaller than that of n=0.023.

    The difference between the effect of the two 0.032 and 0.030 roughnesses is minor. In other words, the dimensions of the separation zone decreased by increasing roughness up to 0.030 and then remained with negligable changes.

    In the next step, the effect of intake invert relative to the main stream (drop) on the dimensions of the separation zone was investigated. To do this, three different invert levels were considered: (1) without drop; (2) a 5 cm drop between the main canal and intake canal; and (3) a 10 cm drop between the main canal and intake canal. The without drop mode was considered as the control state. Figure 6 shows the effect of drop implementation on separation zone dimensions. Tables 2 and 3 show the reduced percentage of separation zone areas in 5 and 10 cm drop compared to no drop conditions as the base line. It was found that the best results were obtained when a 10 cm drop was implemented.Table 2

    Decrease percentage of separation zone area in 5 cm drop

    Frn=0.011n=0.017n=0.023n=0.028n=0.030n=0.032
    0.08 10.56 11.06 25.27 33.03 35.57 36.5 
    0.121 7.66 11.14 11.88 15.93 34.59 36.25 
    0.353 1.38 2.63 8.17 14.39 31.20 31.29 
    0.362 11.54 19.56 25.73 37.89 38.31 

    Table 3

    Decrease percentage of separation zone area in 10 cm drop

    Frn=0.011n=0.017n=0.023n=0.028n=0.030n=0.032
    0.047 4.30 8.75 23.47 31.22 34.96 35.13 
    0.119 11.01 13.16 15.02 21.48 39.45 40.68 
    0.348 3.89 5.71 9.82 16.09 29 30.96 
    0.354 2.84 10.44 18.42 25.45 35.68 35.76 

    Figure 6VIEW LARGEDOWNLOAD SLIDE

    Effect of drop implementation on separation zone dimensions.

    The combined effect of drop and roughness is shown in Figure 7. According to this figure, by installing a drop structure at the entrance of the intake, the dimensions of the separation zone scales down in any desired roughness coefficient. Results indicated that by increasing the roughness coefficient or drop implementation individually, the separation zone area decreases up to 38 and 25% respectively. However, employing both techniques simultaneously can reduce the separation zone area up to 63% (Table 4). The reason for the reduction of the dimensions of the separation zone area by drop implementation can be attributed to the increase of discharge ratio. This reduces the dimensions of the separation zone area.Table 4

    Reduction in percentage of combined effect of roughness and 10 cm drop

    Qin=0.011n=0.017n=0.023n=0.028n=0.030n=0.032
    16 32.3 35.07 37.2 45.7 58.01 59.1 
    18 44.5 34.15 36.18 48.13 54.2 56.18 
    21 43.18 32.33 42.30 37.79 57.16 63.2 
    23 40.56 34.5 34.09 46.25 50.12 57.2 

    Figure 7VIEW LARGEDOWNLOAD SLIDE

    Combined effect of roughness and drop on separation zone dimensions.

    This method increases the discharge ratio (ratio of turnout to main channel discharge). The results are compatible with the literature. Some other researchers reported that increasing the discharge ratio can scale down the separation zone dimensions (Karami Moghaddam & Keshavarzi 2007Ramamurthy et al. 2007). However, these researchers employed other methods to enhance the discharge ratio. Drop implementation is simple and applicable in practice, since there is normally an elevation difference between the main and lateral canal in irrigation networks to ensure gravity flow occurance.

    Table 4 depicts the decrease in percentage of the separation zone compared to base line conditions in different arrangements of the combined tests.Figure 8VIEW LARGEDOWNLOAD SLIDE

    Velocity profiles for various roughness coefficients along turnout width.

    A comparison between the proposed methods introduced in this paper and traditional methods such as installation of submerged vanes, and changing the inlet geometry (angle, radius) was performed. Figure 8 shows the comparison of the results. The comparison shows that the new techniques can be highly influential and still practical. In this research, with no change in structural geometry (enhancement of roughness coefficient) or minor changes with respect to drop implementation, the dimensions of the separation zone are decreased noticeably. The velocity values were also recorded by a one-dimensional velocity meter at 15 cm distance from the turnout entrance and in a transverse direction (perpendicular to the flow direction). The results are shown in Figure 9.Figure 9VIEW LARGEDOWNLOAD SLIDE

    Effect of roughness on separation zone dimensions in numerical study.

    Numerical results

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    This study examined the flow patterns around the entrance of a diversion channel due to various wall roughnesses in the diversion channel. Results indicated that increasing the discharge ratio in the main channel and diversion channel reduces the area of the separation zone in the diversion channel.Figure 10VIEW LARGEDOWNLOAD SLIDE

    Comparision of the vortex area (software output) for three roughnesses (0.009, 0.023 and 0.032).A laboratory and numerical error rate of 0.2605 was calculated from the following formula,

    formula

    where Uexp is the experimental result, Unum is the numerical result, and N is the number of data.

    Figure 9 shows the effect of roughness on separation zone dimensions in numerical study. Figure 10 compares the vortex area (software output) for three roughnesses, 0.009, 0.023 and 0.032 and Figure 11 shows the flow lines (tecplot output) that indicate the effect of roughness on flow in the separation zone. Numerical analysis shows that by increasing the roughness coefficient, the dimensions of the separation zone area decrease, as shown in Figure 10 where the separation zone area at n=0.032 is less than the separation zone area at n=0.009.Figure 11VIEW LARGEDOWNLOAD SLIDE

    Comparison of vortex area in 3D mode (tecplot output) with two roughnesses (a) 0.009 and (b) 0.032.Figure 12VIEW LARGEDOWNLOAD SLIDE

    Velocity vector for flow condition Q1/422 l/s, near surface.

    The velocities intensified moving midway toward the turnout showing that the effective area is scaled down. The velocity values were almost equal to zero near the side walls as expected. As shown in Figure 12 the approach vortex area velocity decreases. Experimental and numerical measured velocity at x=0.15 m of the diversion channel compared in Figure 13 shows that away from the separation zone area, the velocity increases. All longitudinal velocity contours near the vortex area are distinctly different between different roughnesses. The separation zone is larger at less roughness both in length and width.Figure 13VIEW LARGEDOWNLOAD SLIDE

    Exprimental and numerical measured velocity.

    CONCLUSION

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    This study introduces practical and feasible methods for enhancing turnout efficiency by reducing the separation zone dimensions. Increasing the roughness coefficient and implementation of inlet drop were considered as remedies for reduction of separation zone dimensions. A data set has been compiled that fully describes the complex, 3D flow conditions present in a 90 degree turnout channel for selected flow conditions. The aim of this numerical model was to compare the results of a laboratory model in the area of the separation zone and velocity. Results showed that enhancing roughness coefficient reduce the separation zone dimensions up to 38% while the drop implementation effect can scale down this area differently based on roughness coefficient used. Combining both methods can reduce the separation zone dimensions up to 63%. Further research is proposed to investigate the effect of roughness and drop implementation on sedimentation pattern at lateral turnouts. The dimensions of the separation zone decreases with the increase of the non-dimensional parameter, due to the reduction ratio of turnout discharge increasing in all the experiments.

    This method increases the discharge ratio (ratio of turnout to main channel discharge). The results are compatible with the literature. Other researchers have reported that intensifying the discharge ratio can scale down the separation zone dimensions (Karami Moghaddam & Keshavarzi 2007Ramamurthy et al. 2007). However, they employed other methods to enhance the discharge ratio. Employing both techniques simultaneously can decrease the separation zone dimensions up to 63%. A comparison between the new methods introduced in this paper and traditional methods such as installation of submerged vanes, and changing the inlet geometry (angle, radius) was performed. The comparison shows that the new techniques can be highly influential and still practical. The numerical and laboratory models are in good agreement and show that the method used in this study has been effective in reducing the separation area. This method is simple, economical and can prevent sediment deposition in the intake canal. Results show that CFD prediction of the fluid through the separation zone at the canal intake can be predicted reasonably well and the RNG model offers the best results in terms of predictability.

    DATA AVAILABILITY STATEMENT

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    All relevant data are included in the paper or its Supplementary Information.

    REFERENCES

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    Figure 8: Instantaneous flow structures extracted using the Q-criterion (Qcriterion=1200) and colored by the magnitude of flow velocity.

    Hybrid modeling on 3D hydraulic features of a step-pool unit

    Chendi Zhang1
    , Yuncheng Xu1,2, Marwan A Hassan3
    , Mengzhen Xu1
    , Pukang He1
    1State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084, China. 2
    College of Water Resources and Civil Engineering, China Agricultural University, Beijing, 100081, China.
    5 3Department of Geography, University of British Columbia, 1984 West Mall, Vancouver BC, V6T1Z2, Canada.
    Correspondence to: Chendi Zhang (chendinorthwest@163.com) and Mengzhen Xu (mzxu@mail.tsinghua.edu.cn)

    Abstract

    스텝 풀 시스템은 계류의 일반적인 기반이며 전 세계의 하천 복원 프로젝트에 활용되었습니다. 스텝 풀 장치는 스텝 풀 기능의 형태학적 진화 및 안정성과 밀접하게 상호 작용하는 것으로 보고된 매우 균일하지 않은 수력 특성을 나타냅니다.

    그러나 스텝 풀 형태에 대한 3차원 수리학의 자세한 정보는 측정의 어려움으로 인해 부족했습니다. 이러한 지식 격차를 메우기 위해 SfM(Structure from Motion) 및 CFD(Computational Fluid Dynamics) 기술을 기반으로 하이브리드 모델을 구축했습니다. 이 모델은 CFD 시뮬레이션을 위한 입력으로 6가지 유속의 자연석으로 만든 인공 스텝 풀 장치가 있는 침대 표면의 3D 재구성을 사용했습니다.

    하이브리드 모델은 스텝 풀 장치에 대한 3D 흐름 구조의 고해상도 시각화를 제공하는 데 성공했습니다. 결과는 계단 아래의 흐름 영역의 분할, 즉 수면에서의 통합 점프, 침대 근처의 줄무늬 후류 및 그 사이의 고속 제트를 보여줍니다.

    수영장에서 난류 에너지의 매우 불균일한 분포가 밝혀졌으며 비슷한 용량을 가진 두 개의 에너지 소산기가 수영장에 공존하는 것으로 나타났습니다. 흐름 증가에 따른 풀 세굴 개발은 점프 및 후류 와류의 확장으로 이어지지만 이러한 증가는 스텝 풀 실패에 대한 임계 조건에 가까운 높은 흐름에서 점프에 대해 멈춥니다.

    음의 경사면에서 발달된 곡물 20 클러스터와 같은 미세 지반은 국부 수력학에 상당한 영향을 주지만 이러한 영향은 수영장 바닥에서 억제됩니다. 스텝 스톤의 항력은 가장 높은 흐름이 사용되기 전에 배출과 함께 증가하는 반면 양력은 더 큰 크기와 더 넓은 범위를 갖습니다. 우리의 결과는 계단 풀 형태의 복잡한 흐름 특성을 조사할 때 물리적 및 수치적 모델링을 결합한 하이브리드 모델 접근 방식의 가능성과 큰 잠재력을 강조합니다.

    Step-pool systems are common bedforms in mountain streams and have been utilized in river restoration projects around the world. Step-pool units exhibit highly non-uniform hydraulic characteristics which have been reported to closely 10 interact with the morphological evolution and stability of step-pool features. However, detailed information of the threedimensional hydraulics for step-pool morphology has been scarce due to the difficulty of measurement. To fill in this knowledge gap, we established a hybrid model based on the technologies of Structure from Motion (SfM) and computational fluid dynamics (CFD). The model used 3D reconstructions of bed surfaces with an artificial step-pool unit built by natural stones at six flow rates as inputs for CFD simulations. The hybrid model succeeded in providing high-resolution visualization 15 of 3D flow structures for the step-pool unit. The results illustrate the segmentation of flow regimes below the step, i.e., the integral jump at the water surface, streaky wake vortexes near the bed, and high-speed jets in between. The highly non-uniform distribution of turbulence energy in the pool has been revealed and two energy dissipaters with comparable capacity are found to co-exist in the pool. Pool scour development under flow increase leads to the expansion of the jump and wake vortexes but this increase stops for the jump at high flows close to the critical condition for step-pool failure. The micro-bedforms as grain 20 clusters developed on the negative slope affect the local hydraulics significantly but this influence is suppressed at pool bottom. The drag forces on the step stones increase with discharge before the highest flow is used while the lift force has a larger magnitude and wider varying range. Our results highlight the feasibility and great potential of the hybrid model approach combining physical and numerical modeling in investigating the complex flow characteristics of step-pool morphology.

    Figure 1: Workflow of the hybrid modeling. SfM-MVS refers to the technology of Structure from Motion with Multi View Stereo. DSM is short for digital surface model. RNG-VOF is short for Renormalized Group (RNG) k-ε turbulence model coupled with Volume of Fluid method.
    Figure 1: Workflow of the hybrid modeling. SfM-MVS refers to the technology of Structure from Motion with Multi View Stereo. DSM is short for digital surface model. RNG-VOF is short for Renormalized Group (RNG) k-ε turbulence model coupled with Volume of Fluid method.
    Figure 2: Flume experiment settings in Zhang et al., (2020): (a) the artificially built-up step-pool model using natural stones, with stone number labelled; (b) the unsteady hydrograph of the run of CIFR (continually-increasing-flow-rate) T2 used in this study.
    Figure 2: Flume experiment settings in Zhang et al., (2020): (a) the artificially built-up step-pool model using natural stones, with stone number labelled; (b) the unsteady hydrograph of the run of CIFR (continually-increasing-flow-rate) T2 used in this study.
    Figure 3: Setup of the CFD model: (a) three-dimensional digital surface model (DSM) of the step-pool unit by structure from motion with multi view stereo (SfM-MVS) method as the input to the 3D computational fluid dynamics (CFD) modeling; (b) extruded bed 160 surface model connected to the extra downstream component (in purple blue) and rectangular columns to fill leaks (in green), with the boundary conditions shown on mesh planes; (c) recognized geometry with mesh grids of two mesh blocks shown where MS is short for mesh size; (d) sampling volumes to capture the flow forces acting on each step stone at X, Y, and Z directions; and (e) an example for the simulated 3D flow over the step-pool unit colored by velocity magnitude at the discharge of 49.9 L/s. The abbreviations for boundary conditions in (b) are: V for specified velocity; C for continuative; P for specific pressure; and W for wall 165 condition. The contraction section in Figure (e) refers to the edge between the jet and jump at water surface.
    Figure 3: Setup of the CFD model: (a) three-dimensional digital surface model (DSM) of the step-pool unit by structure from motion with multi view stereo (SfM-MVS) method as the input to the 3D computational fluid dynamics (CFD) modeling; (b) extruded bed 160 surface model connected to the extra downstream component (in purple blue) and rectangular columns to fill leaks (in green), with the boundary conditions shown on mesh planes; (c) recognized geometry with mesh grids of two mesh blocks shown where MS is short for mesh size; (d) sampling volumes to capture the flow forces acting on each step stone at X, Y, and Z directions; and (e) an example for the simulated 3D flow over the step-pool unit colored by velocity magnitude at the discharge of 49.9 L/s. The abbreviations for boundary conditions in (b) are: V for specified velocity; C for continuative; P for specific pressure; and W for wall 165 condition. The contraction section in Figure (e) refers to the edge between the jet and jump at water surface.
    Figure 4: Distribution of time-averaged velocity magnitude (VM_mean) and vectors in three longitudinal sections. The section at Y = 0 cm goes across the keystone while the other two (Y = -18 and 13.5 cm) are located at the step stones beside the keystone with 265 lower top elevations. Q refers to the discharge at the inlet of the computational domain. The spacing for X, Y, and Z axes are all 10 cm in the plots.
    Figure 4: Distribution of time-averaged velocity magnitude (VM_mean) and vectors in three longitudinal sections. The section at Y = 0 cm goes across the keystone while the other two (Y = -18 and 13.5 cm) are located at the step stones beside the keystone with lower top elevations. Q refers to the discharge at the inlet of the computational domain. The spacing for X, Y, and Z axes are all 10 cm in the plots.
    Figure 5: Distribution of time-averaged flow velocity at five cross sections which are set according to the reference section (x0). The reference cross section x0 is located at the downstream end of the keystone (KS). The five sections are located at 18 cm and 6 cm upstream of the reference section (x0-18 and x0-6), and 2 cm, 15 cm and 40 cm downstream of the reference section (x0+2, x0+15, x0+40). The spacing for X, Y, and Z axes are all 10 cm in the plots.
    Figure 5: Distribution of time-averaged flow velocity at five cross sections which are set according to the reference section (x0). The reference cross section x0 is located at the downstream end of the keystone (KS). The five sections are located at 18 cm and 6 cm upstream of the reference section (x0-18 and x0-6), and 2 cm, 15 cm and 40 cm downstream of the reference section (x0+2, x0+15, x0+40). The spacing for X, Y, and Z axes are all 10 cm in the plots.
    Figure 6: Distribution of the time-averaged turbulence kinetic energy (TKE) at the five cross sections same with Figure 3.
    Figure 6: Distribution of the time-averaged turbulence kinetic energy (TKE) at the five cross sections same with Figure 3.
    Figure 7: Boxplots for the distributions of the mass-averaged flow kinetic energy (KE, panels a-f), turbulence kinetic energy (TKE, panels g-l), and turbulent dissipation (εT, panels m-r) in the pool for all the six tested discharges (the plots at the same discharge are in the same row). The mass-averaged values were calculated every 2 cm in the streamwise direction. The flow direction is from left to right in all the plots. The general locations of the contraction section for all the flow rates are marked by the dashed lines, except for Q = 5 L/s when the jump is located too close to the step. The longitudinal distance taken up by negative slope in the pool for the inspected range is shown by shaded area in each plot.
    Figure 7: Boxplots for the distributions of the mass-averaged flow kinetic energy (KE, panels a-f), turbulence kinetic energy (TKE, panels g-l), and turbulent dissipation (εT, panels m-r) in the pool for all the six tested discharges (the plots at the same discharge are in the same row). The mass-averaged values were calculated every 2 cm in the streamwise direction. The flow direction is from left to right in all the plots. The general locations of the contraction section for all the flow rates are marked by the dashed lines, except for Q = 5 L/s when the jump is located too close to the step. The longitudinal distance taken up by negative slope in the pool for the inspected range is shown by shaded area in each plot.
    Figure 8: Instantaneous flow structures extracted using the Q-criterion (Qcriterion=1200) and colored by the magnitude of flow velocity.
    Figure 8: Instantaneous flow structures extracted using the Q-criterion (Qcriterion=1200) and colored by the magnitude of flow velocity.
    Figure 9: Time-averaged dynamic pressure (DP_mean) on the bed surface in the step-pool model under the two highest discharges, with the step numbers marked. The negative values in the plots result from the setting of standard atmospheric pressure = 0 Pa, whose absolute value is 1.013×105 Pa.
    Figure 9: Time-averaged dynamic pressure (DP_mean) on the bed surface in the step-pool model under the two highest discharges, with the step numbers marked. The negative values in the plots result from the setting of standard atmospheric pressure = 0 Pa, whose absolute value is 1.013×105 Pa.
    Figure 10: Time-averaged shear stress (SS_mean) on bed surface in the step-pool model, with the step numbers marked. The standard atmospheric pressure is set as 0 Pa.
    Figure 10: Time-averaged shear stress (SS_mean) on bed surface in the step-pool model, with the step numbers marked. The standard atmospheric pressure is set as 0 Pa.
    Figure 11: Variation of fluid force components and magnitude of resultant flow force acting on step stones with flow rate. The stone 4 is the keystone. Stone numbers are consistent with those in Fig. 9-10. The upper limit of the sampling volumes for flow force calculation is higher than water surface while the lower limit is set at 3 cm lower than the keystone crest.
    Figure 11: Variation of fluid force components and magnitude of resultant flow force acting on step stones with flow rate. The stone 4 is the keystone. Stone numbers are consistent with those in Fig. 9-10. The upper limit of the sampling volumes for flow force calculation is higher than water surface while the lower limit is set at 3 cm lower than the keystone crest.
    Figure 12: Variation of drag (CD) and lift (CL) coefficient of the step stones along with flow rate. Stone numbers are consistent with those in Fig. 8-9. KS is short for keystone. The negative values of CD correspond to the drag forces towards the upstream while the negative values of CL correspond to lift forces pointing downwards.
    Figure 12: Variation of drag (CD) and lift (CL) coefficient of the step stones along with flow rate. Stone numbers are consistent with those in Fig. 8-9. KS is short for keystone. The negative values of CD correspond to the drag forces towards the upstream while the negative values of CL correspond to lift forces pointing downwards.
    Figure 13: Longitudinal distributions of section-averaged and -integral turbulent kinetic energy (TKE) for the jump and wake vortexes at the largest three discharges. The flow direction is from left to right in all the plots. The general locations of the contraction sections under the three flow rates are marked by dashed lines in figures (d) to (f).
    Figure 13: Longitudinal distributions of section-averaged and -integral turbulent kinetic energy (TKE) for the jump and wake vortexes at the largest three discharges. The flow direction is from left to right in all the plots. The general locations of the contraction sections under the three flow rates are marked by dashed lines in figures (d) to (f).
    Figure A1: Water surface profiles of the simulations with different mesh sizes at the discharge of 43.6 L/s at the longitudinal sections at: (a) Y = 24.5 cm (left boundary); (b) Y = 0.3 cm (middle section); (c) Y = -24.5 cm (right boundary). MS is short for mesh size. The flow direction is from left to right in each plot.
    Figure A1: Water surface profiles of the simulations with different mesh sizes at the discharge of 43.6 L/s at the longitudinal sections at: (a) Y = 24.5 cm (left boundary); (b) Y = 0.3 cm (middle section); (c) Y = -24.5 cm (right boundary). MS is short for mesh size. The flow direction is from left to right in each plot.
    Figure A2: Contours of velocity magnitude in the longitudinal section at Y = 0 cm at different mesh sizes (MSs) under the flow condition with the discharge of 43.6 L/s: (a) 0.50 cm; (b) 0.375 cm; (c) 0.30 cm; (d) 0.27 cm; (e) 0.25 cm; (f) 0.24 cm. The flow direction is from left to right.
    Figure A2: Contours of velocity magnitude in the longitudinal section at Y = 0 cm at different mesh sizes (MSs) under the flow condition with the discharge of 43.6 L/s: (a) 0.50 cm; (b) 0.375 cm; (c) 0.30 cm; (d) 0.27 cm; (e) 0.25 cm; (f) 0.24 cm. The flow direction is from left to right.
    Figure A3: Measurements of water surfaces (orange lines) used in model verification: (a) water surface profiles from both sides of the flume; (b) upstream edge of the jump regime from top view. KS refers to keystone in figure (b).
    Figure A3: Measurements of water surfaces (orange lines) used in model verification: (a) water surface profiles from both sides of the flume; (b) upstream edge of the jump regime from top view. KS refers to keystone in figure (b).
    Figure A15. Figure (a) shows the locations of the cross sections and target coarse grains at Q = 49.9 L/s. Figures (b) to (e) show the distribution of velocity magnitude (VM_mean) in the four chosen cross sections: (a) x0+8.0; (b) x0+14.0; (c) x0+21.5; (d) x0+42.5. G1 to G6 refer to 6 protruding grains in the micro-bedforms in the pool.
    Figure A15. Figure (a) shows the locations of the cross sections and target coarse grains at Q = 49.9 L/s. Figures (b) to (e) show the distribution of velocity magnitude (VM_mean) in the four chosen cross sections: (a) x0+8.0; (b) x0+14.0; (c) x0+21.5; (d) x0+42.5. G1 to G6 refer to 6 protruding grains in the micro-bedforms in the pool.
    Figure A16. The distribution of turbulent kinetic energy (TKE) in the same cross sections as in figure S15: (a) x0+8.0; (b) x0+14.0; (c) x0+21.5; (d) x0+42.5.
    Figure A16. The distribution of turbulent kinetic energy (TKE) in the same cross sections as in figure S15: (a) x0+8.0; (b) x0+14.0; (c) x0+21.5; (d) x0+42.5.

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    Numerical study of the effect of flow velocity and flood roughness components on hydraulic flow performance in composite sections with converging floodplains

    Numerical study of the effect of flow velocity and flood roughness components on hydraulic flow performance in composite sections with converging floodplains

    Authors

    1 Civil Enigneering Department, Lahijan Branch.Islamic Azad University.Lahijan.Iran

    2 Department of Civil Engnieering, University of Qom,Qom,Iran

    3 Civil Engineering Department, Lahijan Branch,Islamic Azad Univeristy,Lahijan,Iran

    Abstract

    홍수와 그 위험을 통제해야 할 필요성은 누구에게도 숨겨져 있지 않습니다. 또한 이 현상으로 인해 다양한 경제, 사회 및 환경 문제가 영향을 받습니다. 홍수 제어 방법의 설계 및 최적 관리의 첫 번째 단계는 홍수 중 하천 거동을 올바르게 식별하는 것입니다.

    홍수 경로 지정, 하상 및 하천 면적 결정 등과 같은 대부분의 하천 엔지니어링 프로젝트에서 하천 단면의 수리학적 매개변수의 평균값을 계산하는 것으로 충분합니다. 오늘날 유체 환경 연구에서 수치 및 분석 방법의 사용이 성장하고 발전했습니다.

    신뢰할 수 있는 결과 생성으로 인해 물리적 모델에 대한 좋은 대안이 될 수 있었습니다. 오늘날 수치 모델의 급속한 발전과 컴퓨터 계산 속도의 증가로 인해 3D 수치 모델의 사용이 선호되며 또한 강의 속도 분포 및 전단 응력을 측정하는 데 시간이 많이 걸리고 비용이 많이 들기 때문에 결과 3D 수치 모델의 가치가 있을 것입니다.

    한편, 본 연구에서는 복합단면에 대해 FLOW-3D 모델을 이용한 종합적인 수치연구가 이루어지지 않았음을 보여주고 있어 적절한 연구기반을 제공하고 있습니다.

    따라서 본 연구의 혁신은 발산 및 수렴 범람원을 동반하는 비 각형 복합 단면에서 흐름의 상태 및 수리 성능에 대한 거칠기와 같은 매개 변수의 영향에 대한 수치 연구입니다.

    수치해석 결과를 검증하기 위해 Younesi(2013) 연구를 이용하였습니다. 이 실험에서는 먼저 고정층이 있는 복합 프리즘 및 비 프리즘 단면의 수리 흐름을 조사한 다음 조건을 유지하면서 프리즘 및 비 프리즘 모드에서 퇴적물 이동 실험을 수행했습니다.

    실험은 15미터 길이의 연구 채널에서 수행되었습니다. 이 운하는 초당 250리터의 시스템에서 재순환을 위해 제공될 수 있는 유속과 0.0088 000의 종경사를 가진 폭 400mm의 두 개의 대칭 범람원이 있는 합성 운하입니다. 범람원의 가장자리는 0.18미터와 같고 주요 운하의 너비는 0.4미터와 같습니다(그림 1).

    본수로의 바닥과 벽을 거칠게 하기 위해 평균직경 0.65mm의 퇴적물을 사용하였으며, 각 단계에서 범람원의 벽과 바닥은 평균직경 0.65, 1.3, 1.78의 퇴적물로 거칠게 하였습다. (mm). 삼각형 오버플로는 운하 상류에서 운하로의 유입량을 측정하는 데 사용됩니다.

    상대깊이 0.15와 0.25, 직경 14mm의 마이크로몰리나 실험과 상대깊이 0.35의 실험에서는 유속을 측정하기 위해 3차원 속도계(ADV)를 사용하였습니다. 수위는 0.1mm의 정확도로 깊이 게이지로 측정 되었습니다.

    본 연구에서는 수면 프로파일의 수치적 모델을 검증하기 위해 실험 0.25-2에서 발산대의 시작, 중간 및 끝에서 세 단면의 평균 깊이 속도 분포 및 경계 전단 응력) -11.3-NP 및 0.25-2-5.7-NP 및 또한 각형 복합 단면의 0.25-2-2 P 테스트가 평가되었습니다.

    각형 합성 단면의 P.20-2-2-P 테스트와 관련된 RMSE 및 NRMSE 지수 값 및 표 (2) 실험 11.3에서 RMSE 및 NRMSE 지수 값 -2-0.25-NP 및 -0.25. 2-5.7-NP가 제공됩니다. 실험 0.25-2-5.7-NP-11.3-2-0.25, NP 및 P.2.0-2-2-P의 평균 깊이 속도의 검증과 관련된 결과가 표시됩니다. 0.25-2-5.7-NP 실험에서 초, 중, 기말 NRMSE의 양은 각각 5.7, 11.8, 10.3%로 계산되었으며, 이는 초급이 우수, 중급이 양호, 최종 성적. 배치. 보시다시피, RMSE 값은 각각 0.026, 0.037 및 0.026으로 계산됩니다.

    실험 11.3-2-0.25, NP에서 초급, 중급 및 최종 수준의 NRMSE 값은 각각 7, 11.2 및 15.4%로 계산되었으며, 이는 초급에서 우수 범주 및 우수 범주에서 중간 및 최종 수준. 가져 가다. 보시다시피, RMSE 값은 각각 0.032, 0.038, 0.04로 계산됩니다. 0.25-2-P 실험에서 NRMSE 값은 1.7%로 계산되어 우수 범주에 속한다. 보시다시피 RMSE 값도 0.004로 계산됩니다. 중간 깊이의 속도 분포와 관련하여 수치 모델은 실험실 결과에 적합하며 접합 영역에 작은 오류만 입력되었다고 말할 수 있습니다. 이는 2차 전지의 이동 결과로 간주될 수 있