하부 테일워터를 위한 USBR 및 쐐기형 배플 블록 분지를 사용한 유압 점프의 수치적 조사

Muhammad Waqas Zaffar; Ishtiaq Hassan; Zulfiqar Ali; Kaleem Sarwar; Muhammad Hassan; Muhammad Taimoor Mustafa; Faizan Ahmed Waris

Abstract

The stilling basin of the Taunsa barrage is a modified form of the United States Bureau of Reclamation (USBR) Type-III basin, which consists of baffle and friction blocks. Studies revealed uprooting of baffle blocks due to their vertical face. Additionally, the literature highlighted issues of rectangular face baffle blocks: less drag, smaller wake area, and flow reattachment. In contrast, the use of wedge-shaped baffle blocks (WSBBs) is limited downstream of open-channel flows. Therefore, this study developed numerical models to investigate the effects of USBR and WSBB basins on the hydraulic jump (HJ) downstream of the Taunsa barrage under lower tailwater conditions. Surface profiles in WSBB and modified USBR basins showed agreement with previous studies, for which the coefficient of determination (R²) reached 0.980 and 0.970, respectively. The HJ efficiencies reached 57.9 and 58.6% in WSBB and modified USBR basins, respectively. The results of sequent depths, roller length, and velocity profiles in the WSBB basin were found more promising than the modified USBR basin, which further confirmed the suitability of the WSBB basin for barrages. Furthermore, WSBB improved flow behaviors in the basin, which showed no fluid reattachment on the sides of WSBB, increased wake regions, and decreased turbulent kinetic energies.

Keywords

barrage, geometry, hydraulic jump, stilling basin, wedge-shaped baffle blocks

INTRODUCTION

Barrages in Pakistan were built about 50–100 years ago and play an important role in the economy. However, as time passed, the stability of these barrages was compromised due to hydraulic and structural deficiencies (Zaidi et al. 2011). Similarly, Taunsa Barrage Punjab, on the mighty river Indus, is one of the major hydraulic structures, which was constructed about 65 years ago. The barrage was designed for a design discharge capacity of 28,313 m³/s, and its stilling basin was a modified form of the United States Bureau of Reclamation (USBR) Type-III basin that consisted of USBR impact friction and baffle blocks. These arrangements dissipate excessive kinetic energy, enhance turbulence, kill rollers, and stabilize the hydraulic jump (HJ) even in case of less tailwater depth. The barrage consists of 64 bays, and the total width of the barrage between the abatements is 1,324.60 m. Of total width, 1,176.5 m is a clear waterway (Zaffar & Hassan 2023a). Figure 1 shows the typical cross-section of the Taunsa barrage.

Figure 1

Typical cross-section of the Taunsa barrage.

Soon after the barrage operation in 1958, multiple problems occurred on the barrage downstream, such as uprooting of the impact baffle blocks due to their vertical face, damage to the basin’s floor, lowering of tailwater levels, and bed retrogression (Zulfiqar & Kaleem 2015). During 1959–1962, repair works were carried out to cater to these issues, but the problems remained persistent. To resolve these issues, the Punjab Government constituted a committee of experts in 1966 and 1973, but no specific measures were taken, and the issues continued to aggravate (Zaidi et al. (2004). Additionally, these traditional impact blocks also face flow reattachment on the sides that decreases the drag force (Frizell & Svoboda 2012). On the contrary, after investigating the wedge-shaped baffle blocks (WSBBs) downstream of pipe outlets, research scholars (Pillai et al. 1989; Verma & Goel 2003; Verma et al. 2004; Goel 2007; Goel 2008; Tiwari et al. 2010) reported that these blocks increased the energy dissipation and created more eddies and wake regions on either side. These studies further mentioned that upon the use of WSBBs, the overall length of stilling basins was also reduced from 15 to 25%.

Energy dissipation is the most common issue faced in the design of hydraulic structures. The kinetic energy typically comes from the upstream of the dams (El Baradei et al. 2022), spillways (Sutopo et al. 2022), chute, sluice gates, and weirs, which are further induced by the HJ and its turbulent structure (Elsaeed et al. 2016). In the HJ, flow suddenly changes from supercritical to subcritical conditions which dissipate the energy of the upstream flow, thereby saving the hydraulic structures from damage. The HJ occurs when Froude Number (Fr) falls below unity, which is the ratio of inertia to gravitational forces that can be calculated by the following equation (Bayon-Barrachina & Lopez-Jimenez 2015; Bayon-Barrachina et al. 2015).

(1)

where v, h, and g are stream-wise velocities, flow depth, and acceleration due to gravity, respectively. Hager & Sinniger (1985) investigated characteristics of the HJ for abrupt changes in horizontal bed and proposed the following equation to compute the efficiency  of HJs.

(2)

where Fr₁ is the Froude number in the supercritical flow before the HJ.

Bakhmeteff & Matzke (1936) developed HJ similarity models and proposed dimensionless Equation (3) for the free surface profile of HJs.

(3)

where  is the water depth at x (h_i) and the variable X is the dimensionless longitudinal coordinates (x), as shown in the following dimensionless equations, respectively.

(4)

(5)

where D is the gate opening and h₁ and h₂ are the water depths in supercritical and subcritical regions, respectively. X₁ and X₂ are the functions of variable X, and their values can be calculated at the toe of the HJ and the end of the roller region, respectively. The components of Equations (4) and (5) are shown in Figure 2.

Figure 2

Schematic diagram for the dimensionless free surface profiles of HJs.

Habibzadeh et al. (2012) conducted experiments to investigate the role of baffle blocks for submerged HJs and energy dissipation downstream of low-head hydraulic structures. Chachereau & Chanson (2011) and Wang & Chanson (2015) investigated free surface profiles and turbulent fluctuation within the HJ for a wide range of initial Froude number (Fr₁). The velocity profiles showed a wall jet-like profile, and turbulence intensities were high due to the fluctuations at the free surface. Maleki & Fiorotto (2021) developed a semi-empirical method to investigate HJs on a rough bed. The results showed that the characteristic length scale was linearly changing with Fr₁. Macián-Pérez et al. (2020b) carried out experiments on the USBR-II stilling basin to investigate the characteristics of HJs. The results of sequent depths and HJ efficiency agreed with the experimental studies, which reached 99.2 and 97%, respectively. The results of the dimensionless free surface profile also agreed with the previous studies for which the value of the coefficient of determination (R²) reached 0.979. Murzyn & Chanson (2009) conducted experiments for a wide range of Fr₁ up to 8.3 to investigate bubbly and turbulence structure within the HJ. The results showed that void fraction (C_max) and bubble frequency (F_max) were found in the developing region, and vertical interfacial velocity agreed with the wall jet-like profile. Qasim et al. (2022) conducted experiments on bed discordance downstream of different weirs. The results indicated that as the bed discordance increased, the dimensionless flow depth decreased downstream of the discordance, which increased the Froude number. The results further showed that as the configuration of bed discordance was changed, the free surface profiles were also changed, which affected the flow depths and velocity profiles. Bhosekar et al. (2014) conducted experiments to investigate the characteristics of discharge downstream of orifice spillways. The results showed that free surface profiles were not elliptical due to the flat curve near the gate opening. The results further indicated that the flat curve developed a negative pressure on roof profiles, which reduced the discharge capacity of the spillway.

To stabilize the HJ in the stilling basins, different shapes of baffle blocks are employed, i.e., baffle blocks (Habibzadeh et al. 2012), friction blocks (Chaudary & Sarwar 2014), end sill (Mansour et al. 2004) and vertical sill (Alikhani et al. 2010), splitter blocks (Verma & Goel 2003), curved (Eloubaidy et al. 1999), T-shaped and triangular (Tiwari & Goel 2016), and WSBB (Pillai et al. 1989; Goel 2007; Goel 2008). These arrangements control the HJs in case of fewer tailwater depths (Peterka 1984) and minimize the erosion downstream of structures (Zaffar et al. 2023). Sayyadi et al. (2022) investigated HJ characteristics for negative steps in the stilling basin. The results showed that the negative step increased the energy dissipation up to 11%. Pillai et al. (1989) compared three different stilling basins for the Fr₁ up to 4.5. The results showed that the stilling basin with the WSBB reduced the scour and overall length of the basin. Goel (2008), Goel (2007), and Tiwari et al. (2010) conducted experiments to investigate HJ characteristics downstream of square and circular pipe outlets using WSBBs. The results showed that as compared to the impact USBR-VI basin, the WSBB basin spread the fluid efficiently in the lateral direction and reduced the basin length up to 50%.

In the former section, the experimental studies on HJs, velocity distribution, free surface profiles, and turbulent kinetic energy (TKE) are discussed, which could be assisted by Computational Fluid Dynamics (CFD) models (Ghaderi et al. 2020). Furthermore, over these hydraulic structures, the flow is very complex and associated with secondary currents, which characterized it as highly turbulent in all directions. Hence, using laboratory and field experiments, it is hard to accurately measure the free surface profile, velocities, secondary currents, and TKE over these hydraulic structures (Jothiprakash et al. 2015). Furthermore, physical experiments and on-site measurements are usually expensive and time-consuming. In contrast, the improvements in computational speed, storage, and turbulence modeling have made CFD a viable complementary investigation tool for hydraulic modeling (Ghaderi et al. 2021). Consequently, the use of numerical modeling tools such as Open Foam (Bayon-Barrachina & Lopez-Jimenez 2015), ANSYS Fluent (Aydogdu et al. 2022), and FLOW-3D (Hirt & Sicilian 1985) has become prevalent to get hydraulic characteristics of grade-control structures. Such modeling tools are helpful, especially when the basic fundamental equations are unable to provide desired outputs, like in the case of multifaceted geometries (Herrera-Granados & Kostecki 2016). So far, many researchers have employed numerical models in the hydraulic investigations of HJs and energy dissipation, but only a few of the latest studies are highlighted here. FLOW-3D numerical models were employed to investigate the HJ (Zaffar & Hassan 2023a) and baffle blocks (Zaffar & Hassan 2023b) for different stilling basins of the Taunsa barrage. These studies focused on velocity distribution, TKE, free surface profiles, energy loss energy, and the effects of baffle blocks on the HJ characteristics. Macián-Pérez et al. (2020a) carried out a numerical investigation on a high Reynolds of 210,000 to study the HJ characteristics. Upon comparison, the FLOW-3D model showed 93% accuracy in the roller length of HJs. The results also indicated 94.2 and 94.3% accuracy for sequent depths and HJ efficiency, respectively. Nikmehr & Aminpour (2020) examined the HJ characteristics on rough beds using FLOW-3D, and compared results with the experiments. The results indicated that roughness height and its distance affected the HJ length. Gadge et al. (2018) conducted a numerical study to investigate the impact of roof profiles on the discharge capacity of orifice spillways and validated the models with experimental results. The study revealed that in addition to the pond level and height of orifice (d), the bottom and roof profiles also affected the discharge coefficient (C_d).

From the literature review, it is found that only a few studies are conducted on the flow characteristics downstream of the Taunsa barrage (Zaidi et al. 2004, 2011; Chaudhry 2010). These studies were carried out in laboratory flume and investigated the effects of tailwater on the location of HJs. However, the studies were lacking in providing the data for other essential hydraulic parameters, i.e., velocity distribution, free surface profiles, TKE, and relative energy loss in the stilling basin. On the contrary, the literature has revealed many experimental and numerical studies on different shapes of baffle blocks downstream of open-channel flow, but the use of WSBB downstream of river diversion barrage is found limited. In the previous studies (Pillai et al. 1989; Verma & Goel 2003; Verma et al. 2004; Goel 2008, 2007; Tiwari et al. 2010), these blocks have only been tested downstream of pipe outlet basins for the initial Froude number of 4.5. Therefore, in the present study, FLOW-3D numerical models are developed to investigate the effects of presently available USBR baffle blocks in the stilling basin of the Tuansa barrage. Due to the uprooting problems of these blocks, the study also investigates the suitability of WSBBs downstream of the studied barrage and draws a comparison between the results of modified USBR and WSBB basins. In this study, based on results from the literature, WSBB with a vertex angle of 150° and cutback angle of 90° is applied for Fr₁ up to 6.64. The main objective of this study is to investigate HJs and flow behavior with USBR baffle blocks and WSBB downstream of an investigated barrage at 44 m³/s discharge. At 44 m³/s discharge, the numerical models are operated at the minimum tailwater level of 129.10 m, and investigated free surface profiles, sequent depths, roller lengths, HJ efficiency, velocity profile, and TKE in the two different stilling basins.

MATERIALS AND METHODS

Existing and proposed stilling basins, appurtenances

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The present numerical models are developed downstream of the Taunsa barrage, Pakistan. The stilling basin of the barrage includes USBR impact friction and baffle blocks (Zaffar & Hassan 2023b; Zaffar et al. 2023). In the basin, floor level and weir crest are fixed at 126.79 and 130.44 m, respectively. The slopes of upstream and downstream weir glacis are maintained at 1:3 and 1:4 (H:V), respectively. In both the studied basins, the blocks are installed 14.63 m away from the centerline of the crest and are placed in a staggered position. The overall length and height of the USBR blocks is 1.37 m as shown in Figures 3(a) and 3(b). Additionally, between the two staggered rows of baffle blocks, a 1.37-m distance is maintained, while the top width of all the USBR blocks is 0.46 m, which is angled at 45° from the rear side. On the other hand, in the WSBB basin, WSBB is placed at the locations of impact USBR baffle blocks. Furthermore, in both the studied basins, two staggered rows of friction blocks are also placed at the basin’s end about 28.95 m away from the weir’s crest. These friction blocks are 1.37 m long, 1.22 m wide, and 1.37 m high. The top surface of these blocks is identical to their bottom. The overall length, width, and height of the WSBB are kept at 1.37 m, and a detailed geometry of the investigated WSBB can be seen in Figures 3(c) and 3(d). Currently, for the investigated WSBB, a vertex angle of 150° and a cutback angle of 90° are employed.

Figure 3

Baffle block geometry for the basins: (a) top view of USBR baffle blocks, (b) isometric view representing front of USBR baffle blocks, (c) top view of WSBBs and (d) isometric view representing front of WSBBs.

Numerical model implementation

Environmental flows are governed by the laws of physics and represented by Navier–Stokes Equations (NSEs), which are inherently nonlinear, time-dependent, and contain three-dimensional partial deferential schemes (Viti et al. 2018). These partial differential equations explain the procedures of continuity, momentum, heat, and mass transfer. For one- and two-dimensional models, these equations can be solved analytically, while for the solution of three-dimensional models, CFD models are employed to discretize the NSEs. In these models, flow equations, i.e., NSEs and continuity equations, are discretized in each cell. Generally, these models start with a mesh, which further contains multiple interconnected cells in the employed mesh blocks. These meshes subdivide the physical space into small volumes, which are associated with several nodes. The values of unknown parameters are stored on these nodes, such as velocity, temperature, and pressure. Different numerical techniques are available to discretize the NSEs, i.e., Direct Numerical Simulation (DNS) (Jothiprakash et al. 2015), Large Eddy Simulation (LES) (Ghosal & Moin 1995), and Reynold Averaged Navier–Stokes (RANS) Equation (Kamath et al. 2019). However, as compared to DNS and LES models, due to less computation cost and simulation time, the RANS model is frequently used in river and hydraulic investigations. Using the RANS model, two additional variables are generated, for which turbulence closure models are usually employed (Carvalho et al. 2008). These models find closure by averaging the Reynolds stress terms in NSEs and append additional variables for turbulent viscosity and transport equations.

Presently, FLOW-3D models are developed to investigate the effects of different shapes of baffle blocks on HJ downstream of the river diversion barrage. The models employ RANS equations to solve algorithms and equations of incompressible fluid in each computational cell. To further address the additional terms, i.e., Reynolds stresses and turbulent viscosity, the Renormalization group (RNG K–ɛ) method is applied. For the discretization of RANS and other algorithms, at present, the Volume of Fluid (VOF) method (finite volume method (FVM)) is employed, while the equations of the controlled volume are formulated with area and volume porosity functions. This formulation is called the ‘Fractional Area/Volume Obstacle Representation’ (FAVOR) method (Hirt & Sicilian 1985). The proceeding section describes the equations used for the present models.

Assuming the flow is steady and neglecting the fluctuation of specific weight (⁠ = 0), Equations (6) and (7) are used for the turbulent flow (Carvalho et al. 2008).

(6)

(7)

Following the above equations, it is apparent that these relationships consist of three momentum and one continuity equation, in which there are 10 unknowns (p, u, v, w, and six Reynolds stress components). In the present study, the flow is considered incompressible, which implies the following equation to solve the flow domain (Viti et al. 2018).

(8)

In Equations (6)–(8) u, v, and w, are velocity components in x, y, and z directions, respectively.  and p are total pressure and fluid density while the terms  are known as the Reynolds stresses. A_x, A_y, and A_z are flow areas while R,⁠, and R_SOR are the model’s coefficient, flow generic property, and mass source term, respectively.

Turbulence modeling and free surface tracking

Six turbulence models are available in FLOW-3D, which employs numerous equations to solve the closure problems. Among various models, the two-equation turbulence models such as standard K–ɛ (Bradshaw 1997, RNG K–ɛ (Yakhot et al. 1991), and K–ω (Wilcox 2008) are widely used in hydraulic investigations.

The standard K–ɛ and RNG K–ɛ models solve transport equations for TKE and its dissipation. The formulation of both models is identical, but the former derives model coefficients empirically. However, RNG K–ɛ applies a statistical approach to derive the transport equations explicitly, which has shown better capability in low-turbulence and high-shear regions (Macián-Pérez et al. 2020a, 2020b) than the standard K–ɛ model. In contrast, the K–ω model was implemented for stream-wise pressure gradient and near-wall boundaries (Wilcox 2008). The model replaces turbulent dissipation rate with turbulent frequency. In the K–ω model, the values of TKE and turbulent frequency are specified at the inlet boundaries. The above-mentioned turbulence models were investigated by Macián-Pérez et al. (2020b) for flow behaviors in the USBR-II stilling basin and the study indicated that the RNG K–ɛ model showed better accuracy. Also, RNG K–ɛ (Macián-Pérez et al. 2020a) showed more promising results for a grid convergence index (GCI), free surface, roller lengths, and HJ efficiency. Additionally, the RNG K–ɛ model predicted good results in flow rate, free surface profiles, and velocity profiles. Based on bibliographical results, this study also employed the RNG K–ɛ model, for which the following transport equations (Equations (9) and (10)) are utilized for TKE (K) and its dissipation (ɛ), respectively (Macián-Pérez et al. 2020a).

(9)

(10)

where  is the coordinate in the x direction;  is the dynamic viscosity;  is the turbulent dynamic viscosity; K is the turbulent kinetic energy;  is the turbulent dissipation;  is the fluid density, and  is the production of TKE. Finally, the terms⁠, ⁠, ⁠, and  are model parameters whose values are given in Yakhot et al. (1991).

For free surface modeling, the VOF method is employed. VOF applies an additional variable, which is called fraction of fluid (F), in which F represents the proportion of fluid. To compute F in the domain, the following equation (Hirt & Sicilian 1985) is used:

(11)

In FLOW-3D, the fluid fraction (F) in each cell is usually presented by three possibilities:

• (A)F = 0, cell is empty.
• (B)F = 1, a cell is fully occupied by fluid.
• (C)0 < F < 1, cell represents the surface between the two fluids.

One fluid (water) with a free surface is considered in the present models, for which FLOW-3D automatically selects the free surface method from the availableVOF advection scheme. For the free surface tracking, 0.5 value is assigned in each computation cell.

Pressure velocity coupling

One of the major issues in solving the NSEs is pressure–velocity coupling, and for that, a network of algorithms (SIMPLE (Patankar & Spalding 1972) and PISO (ISSA 1985)) has been developed. These above-mentioned algorithms use under- and over-relaxing factors for pressure correction in the continuity and momentum equations, which contain large memory. Additionally, due to the relaxation factors, sometimes the solution becomes unstable and does not find convergence. On the contrary, FLOW-3D employs the Generalized Minimum Residual Method (GMRES) (Joubert 1994) because it possesses good convergence, high speed, and uses less memory. Additionally, GMRES does not apply any relaxation factor and possesses an additional algorithm, ‘Generalized Minimum Residual Solver (GCG),’ to treat the viscous terms.

Model geometry

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The solid geometry of the barrage bay was designed in AutoCAD and converted into a stereo lithography file. Before importing the stereo lithography file into FLOW-3D, it was tested in Netfabb-basic software to remove any holes, facets, and boundary edges. Figures 4(a) and 4(b) show stereo lithography files used for the studied models.

Figure 4

Geometry details of the studied stilling basins: (a) modified USBR baffle block basin and (b) WSBB basin.

Meshing and boundary condition

The structured rectangular hexahedral mesh was employed to resolve the geometry and flow domain. To resolve the flow domain, a coarse mesh block was initiated upstream of the barrage (X_min = 10 m) which ended at the upstream side of the gate (X_max = 32.90 m). However, the fine mesh block was started from X_min = 32.90 m which was extended up to the basin’s end (X_max = 71 m). In total, 60 m of the model domain was simulated, out of which 22.90 m comprised upstream while the remaining included downstream side. For discharge measurement, mesh sensitivity analysis was performed. The blocks with coarse meshes were initially employed to calculate the volume flow rate (Q). The total number of mesh cells used in the coarse mesh block was 1,108,705. Out of 1,108,705 mesh cells, 481,000 cells were employed for the upstream block, while 627,705 mesh cells were utilized for the downstream mesh block. On the contrary, upon the use of fine mesh blocks, a total of 2,991,820 mesh cells were employed, out of which 481,000 cells were contained in the upstream mesh block while 2,510,820 cells were employed on the downstream side. Figure 5 shows mesh grids employed for flow and solid domains.

Figure 5

Meshing setup of the modeling domain.

It is essential to mention that in both meshing scenarios, fine mesh blocks were used on the downstream side of the bay because the focus of the present investigation was made around the baffle blocks and in the HJ regions. The details of mesh cell size and mesh quality indicators for the various mesh blocks are provided in Tables 1 and 2, respectively. Notably, except for discharge analysis, the results of other hydraulic parameters are produced from the fine meshing.

Table 1

Details of mesh blocks and cell sizes

Table 2

Meshing quality indicators for various mesh blocks

A vertical gate of 18.5-m width, 0.53-m length, and 6.10-m height was mounted upstream of the weir crest. Pond levels of 135.93 and 136.24 m were maintained for free and orifice flows, respectively. Table 3 shows the conditions used for models’ operation.

Table 3

Free and gated flow conditions for operation of the simulations (Zaffar & Hassan 2023b)

For the first mesh block, the upstream and downstream boundaries were set as pressure (P), while for the second block upstream boundary was set as symmetry (S). The lateral sides were set as rigid boundaries (W), and no-slip conditions were expressed as zero tangential and normal velocity (u=v=w = 0), where u, v, and w are the velocities in x, y, and z directions, respectively. These boundaries indicate a wall law velocity profile, which further expresses that the average velocity of turbulent flows is proportional to the logarithm of the distance from that point to the fluid boundary. For all variables (except pressure (P) (which was set to zero), upper boundaries (Z_max) were set as atmospheric pressure to allow water to null von Neumann. For both mesh blocks, the lower boundaries (Z_min) were set as walls.

For the present models, the stability and convergence at each iteration were checked by Courant number (Ghaderi et al. 2020), which affected the time steps from 0.06 to 0.0023 and 0.015 to 0.0025 for free and gated flow, respectively. It is worth mentioning here that for the free flow analysis of higher discharge such as 444 m³/s, the steady state solution can only be achieved by mass-averaged fluid kinetic energy (MAFKE) and volume flow rate (VFR) at the inlet and outlet boundaries. Therefore, the time at which the MAFKE and VFR reach the steady state is assigned as the simulation time (T_s) of models. Presently, VFRs at the inlet and outlet boundaries are considered as the stability and convergence indicators. Based on the criterion mentioned above, the present free and gated models achieved hydraulic stability at T_s = 60 s while the actual time (T_a) of models ranged between 30 and 48 h. However, to accommodate free surface fluctuations, the models were run for T_s = 80 s.

Models’ verification and validation

Analysis of design discharge

For performance assessment of the numerical models, H_e/H_d = 0.998 (Johnson & Savage 2006; Gadge et al. 2019; Zaffar & Hassan 2023a, 2023b) was implemented for free flow analysis, where H_e and H_d are effective and designed heads, respectively. This was the design discharge of the Taunsa barrage, for which the models were operated on the pond and tailwater levels of 135.93 and 133.8 m, respectively, as provided in Table 3.

Gated flow modeling

Computational discharge is of paramount importance in hydraulic modeling, and the following discharge formula is used for gated flow operations (Gadge et al. 2018). To model 44 m³/s of discharge, gate opening and designed head for orifice are set at D = 0.280 m and H_d = 136.24 m, respectively. Figure 6 illustrates the typical cross-section for gated flow operations.

(12)

where Q (m³/s) is discharged through the orifice opening, A (m²) is the area of the orifice, g (m/s²) is the acceleration due to gravity and h_c is the centerline head (h_c = H_d–D/2). The values of coefficients of discharge (C_d) used and simulated were 0.816 and 0.819, respectively, and were found well within the range of C_d values calculated by Bhosekar et al. (2014).

Figure 6

Cross-section showing gated flow through the orifice.

RESULTS AND DISCUSSION

Discharges and flow evolution

Figure 7 shows the time instant of flow evolution for designed discharge. At the start of the simulation, due to the inlet velocity, the free surface was found to be changed. However, when it reached a steady state, a fully developed flow on the downstream glacis was achieved, as shown in Figure 7. A stable free jump was observed at T_s = 60 s for free and gated flow, and the free surface on the downstream side was found to be stable with little fluctuation. The accuracy of FLOW-3D simulations was checked by comparing those discharges with designed values. At T_s = 80 s, the modified USBR baffle block basin underestimated the discharge, which reached 440.48 m³/s and displayed a 0.80% error. Similarly, at T_s = 80 s, for the WSBB basin, the model produced 440.17 m³/s discharge, in which the maximum error reached 0.893%. However, upon the use of coarse meshing, the errors in the computed discharge were increased, which reached −5 and −4% in modified USBR and WSBB basins, respectively. The free flow analysis of modified USBR and WSBB models showed acceptable validation with the designed flow, which allowed us to run the models for orifice discharge.

Figure 7

Flow evolution at the designed flow: USBR basin (a–c) and WSBB basin (d–f).

For the gated flow, on using fine meshing with similar meshing and boundary conditions, the modified USBR basin produced 44.14 m³/s of discharge, for which the maximum error reached 0.32%. However, in the WSBB basin, the numerical model underestimated the flow, which displayed only a 1.14% error. The evolutionary process of gated flows is shown in Figure 8. Based on the validation results of free and gated flows, further analysis was performed on the characteristics of HJs in the two different basins, and their results were compared with the relevant literature.

Figure 8

Evolutionary process of orifice discharge: USBR basin (a–c) and WSBB basin (d–f).

Free surface profiles

Due to the limited results of investigated hydraulic parameters on the studied barrage, the models’ results are compared with the previous relevant experimental and numerical studies. For such comparison, the models require some similarity in boundary and initial conditions, as obtained from Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015). Similar to the studies by Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015), in the present gated models, the upstream and downstream initial conditions are set to fluid elevation, i.e., pond and tailwater level, with hydrostatic pressure boundaries, while upstream boundaries are set to atmospheric pressure. Additionally, the sides and bottom are set to wall boundaries as described in Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015). However, the present models differ from the basin appurtenances as the compared studies have investigated HJ and other parameters on the horizontal flat beds. Furthermore, Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015) have investigated hydraulic parameters such as sequent depths, roller lengths, free surface profiles, energy dissipation, and TKE for Fr₁ of 6.10 and 3.8 < Fr₁ < 8.5, respectively. Similar to the above-mentioned study, the present modified USBR and WSBB basins are investigated for the Fr₁ of 6.5 and 6.64, respectively. Hence, to confirm the results of free surface profiles, the studies of Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015) are utilized, for which the relevant discussion is made in the proceeding paragraphs.

Using Equation (3) of Bakhmeteff & Matzke (1936), the free surface profiles of HJs were obtained by the VOF method. Figure 9 compares the results of free surface profiles with Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015). The results of the present model agreed well with Bayon-Barrachina & Lopez-Jimenez (2015) (coefficient of determination (R²) = 0.992), for which the value of R² in WSBB and USBR basins reached 0.980 and 0.970, respectively. Similarly, after comparing free surface profiles with Wang & Chanson (2015), the results of the present model were found to be more promising. However, as compared to the USBR basin, the results of free surface profiles in the WSBB basin showed more agreement with the compared studies, as can be seen in Figure 9.

Figure 9

Comparison of dimensionless free surface profiles of HJs with the literature.

To further assess the performance and gain deeper insight into the models’ efficiency, residual plots are drawn for the investigated hydraulic parameters. These errors referred to the difference between the observed (literature) and predicted data, which monitored the regression quality (Hassanpour et al. 2021). At a 5% level of significance, a homo-scedasticity analysis measured the residual errors, and the results of predicted residual errors were compared with the previous study.

Figure 10 compares the residual errors of free surface profiles of present models with Wang & Chanson (2015) and Bayon-Barrachina & Lopez-Jimenez (2015). Notably, the solid horizontal line in Figure 10 is the agreement line. From Figure 8(b), the maximum residual errors of free surface profiles in the experimental study of Wang and Chanson ranged between −0.2 and 0.13, while from the agreement line, the maximum negative and positive residual errors in the numerical study of Bayon-Barrachina & Lopez-Jimenez (2015) reached −0.12 to 0.20. In comparison to the previous studies, at the start of the regression line, the residual errors in the present models indicated a random scattered pattern below the agreement line, which further showed that the residual errors in the present and compared models were not normally distributed. After X = 0.3, above the agreement line, the distribution pattern of residual errors was also not normal. The stilling basin with USBR baffle blocks indicated the maximum negative and positive residual values of −0.160 to 0.10, respectively, while −0.184 to 0.124 values of residuals were noticed in the WSBB basin. Furthermore, from Figure 10, it is evident from residual analysis that the free profiles of HJs within modified USBR and WSBB basins have followed the trend of previous studies and the residuals of the present model are found less, which indicates reasonable accuracy of the models. Overall, the regression analysis of free surface profiles both for the present and compared studies revealed a curvilinear pattern that showed a heteroscedasticity residual.

Figure 10

Residual error diagram of free surface profiles of HJs with the literature.

Sequent depth ratio

In 1840, Belanger developed a famous equation for the sequent depth of HJs in smooth rectangular channels, which is widely used by numerous researchers to validate the results. Similarly, in the laboratory experimentation for different upstream and downstream tailwater water levels, Hager & Bremen (1989) developed a relationship of sequent depth (y₂/y₁) against a wide range of initial Froude numbers (Fr₁). For the present models, 6.5 and 6.64 values of Fr₁ are obtained in WSBB and USBR basins, respectively, and their relationship with sequent depths is developed as shown in Figure 11(a). The results of y₂/y₁ against Fr₁ are compared with the experiments of other authors (Belanger 1841; Hager & Bremen 1989; Kucukali & Chanson 2008) and with the numerical study of Bayon-Barrachina & Lopez-Jimenez (2015).

Figure 11

(a) Comparison of sequent depth ratio with previous studies and (b) comparison of residual errors of sequent depths with the literature.

The sequent depths obtained from WSBB and modified USBR basins were 8.96 and 8.68, respectively, which were found to agree with the experimental results of other studies (Hager & Bremen 1989 and Belanger 1841), as shown in Figure 11(a). The results were also compared with the experiments of Kucukali & Chanson (2008), in which the value of Fr₁ was 6.9. However, the Fr₁ values obtained from present numerical models were 6.5 and 6.64 within WSBB and USBR basins, respectively. The comparison indicated that the present models overestimated the sequent depths, for which the errors reached 8.6 and 5.7% in WSBB and modified USBR basins, respectively. Furthermore, upon comparison with Bayon-Barrachina & Lopez-Jimenez (2015), results showed that present models underestimated the sequent depths, for which the maximum errors reached −13.2 and −9.8% errors in WSBB and USBR basins, respectively.

Figure 11(b) shows the comparison of residual errors of sequent depths with the previous studies. The maximum positive and negative residual values of 0.250 and −0.222 were found in WSBB and USBR baffle block stilling basins, respectively. The pattern of residual errors indicated an equal variance along the agreement line, which showed normal destitution of residual errors, thereby a homoscedastic pattern was noticed. The residual errors of sequent depths in both the tested basins were found within the ranges of Bayon-Barrachina & Lopez-Jimenez (2015) and Kucukali & Chanson (2008).

Roller length

Figure 12 compares the roller length (L_r/d₁) of two different stilling basins with the previous studies, where L_r is the roller length of HJs and d₁ is the initial flow depth before the HJ. Following Figure 10, the results showed that both the stilling basins produced almost similar roller lengths. Furthermore, in comparison to the previous studies, the relationship between roller lengths (Lr/d₁) and Fr₁ was found close to Kucukali and Chanson’s experiments (2008). However, the comparison with other studies indicated that the present models underestimated the roller lengths. The reason for reduced roller lengths was the effects of the basin’s appurtenance, which controlled the HJ lengths, as shown in Figures 13(a) and 13(b) (encircled regions). However, as compared to the modified USBR basin, the roller length in the WSBB basin was found to be less.

Figure 12

Comparison of roller lengths of HJ and initial Froude number with previous studies.

Figure 13

Roller lengths and energy dissipators: (a) WSBB basin and (b) modified USBR basin.

Figure 14 compares the residual errors for the roller lengths of present basins with the previous studies. The analysis showed a random distribution of residual errors and indicated homo-scedasticity of residuals. The results of residual errors for both the tested basins showed a good agreement with the compared studies and remained within the range of their residual errors. However, as compared to the USBR basin, the residual errors in the WSBB basin were found less, which reached −1.38.

Figure 14

Comparison of residual errors for the roller lengths with the previous studies.

HJ efficiency

The efficiency of the HJ is the ratio of energy loss to the upstream hydraulic head. Flow depth (h_i), velocity (v_i), and acceleration due to gravity (g) are the variables of HJ efficiency. The following equation was used to measure the efficiency  of the HJ (Bayon-Barrachina & Lopez-Jimenez 2015).

(13)

where H₁ and H₂ are the specific energy heads upstream and downstream of HJs, respectively.

The results of numerical models showed 57.9 and 58.6% efficiencies in WSBB and modified USBR basins, respectively. The efficiencies for both the basins were also computed by Equation (2) and the results indicated 61.2 and 61.9% efficiencies for WSBB and modified USBR basins, respectively. The comparison further revealed that the present model underestimated the efficiencies, which reached the maximum errors of 5.41 and 5.45% in WSBB and modified USBR basins, respectively.

Figure 15(a) compares the efficiencies of present models with the previous studies. Upon comparing with Wu & Rajaratnam (1996), the results showed that the present model underestimated efficiencies for which the errors reached 6.6 and 5.54% in WSBB and modified USBR basins, respectively. Similarly, after comparing with Kucukali & Chanson (2008), the present model also showed a reduction in the HJ efficiencies for which the maximum errors reached 5.07 and 3.99% in WSBB and modified USBR basins, respectively. However, after comparing with Bayon-Barrachina & Lopez-Jimenez (2015), the results showed good agreement and indicated only 0.34 and 1.37% errors in WSBB and modified USBR basins, respectively. Overall, based on the bibliographic comparison, the overall accuracy of the present models for energy dissipation reached 93%.

Figure 15

(a) Comparison of HJ efficiency and (b) comparison of residual errors for the hydraulic jump efficiency.

Figure 15(b) indicates the residual errors of  for modified USBR and WSBB basins and compares the errors with the previous experimental and numerical studies. Upon comparison with the modified USBR basin and with the literature, the HJ efficiency in the WSBB basin showed a close agreement with the zero residual line, for which the maximum error reached 0.001. On the other hand, the residual error in the modified USBR basin reached 0.003. Figure 14 also showed that the maximum residual error in the HJ efficiencies was found to be less than that was observed in previous studies, which remained within the limits of the compared studies (Wu & Rajaratnam 1996; Kucukali & Chanson 2008; Bayon-Barrachina & Lopez-Jimenez 2015).

Velocity distribution

Velocity distribution was measured at different flow depths to obtain vertical velocity profiles in two different stilling basins. Figure 16 shows the typical profiles in HJs, where (δ) is the y value at which the maximum velocity (U_max) occurs, while (b) is the length scale where u = 0.5U_max and ∂u/∂y < 0 (Ead & Rajaratnam 2002; Nasrabadi et al. 2012). The results indicated that in both basins, the velocity profiles showed a wall jet-like structure (Ead & Rajaratnam 2002). The results further showed that as the distance from the HJ-initiating locations was increased the maximum velocity decreased, thereby boundary growth layers were also decreased.

Figure 16

Typical velocity profile in HJs (Ead & Rajaratnam 2002).

Figures 17 and 18 show that due to the supercritical velocity, a contracted jet was impinging near the beds of basins, and velocity decreased in upper fluid regions. The sections A-A and B-B in the basins indicated reverse flow and eddies in the HJs. The results of the upper fluid region of the HJ indicated typical backward velocity profiles as described by other authors (Ead & Rajaratnam 2002; Nasrabadi et al. 2012). Over time, these reverse fluid circulations were found to be stabilized and showed stagnation zones (Yamini et al. 2022). The analysis further showed a recirculation region within the HJ, and the maximum backward velocity profiles were found in the developed regions. The results also showed that after the jump termination, the negative velocity profiles converted into forward velocity profiles, as can be seen in sections C-C of Figures 17 and 18. Additionally, the results showed that after the WSBB and USBR baffle blocks, the velocity near the bed decreased and became positive at the free surface.

Figure 17

2D illustration of vertical velocity profiles in the USBR basin.

Figure 18

2D illustration of vertical velocity profiles in the WSBB basin.

Figure 19 shows the vertical velocity profile in the HJs at five different horizontal sections. The dimensionless plots between (y/b) and (U/U_max) illustrated that the velocity profiles followed the wall jet-like structure and were found to be agreed with Ead & Rajaratnam (2002), where y was the flow depth, b was the length scale, and U_max was the maximum velocity in the vertical section. The results also showed that in the HJ regions, both stilling basins produced identical structures of forward velocity profiles as can be seen in Figure 19(a) and 19(b).

Figure 19

Dimensionless velocity profiles: (a) WSBB basin and (b) modified USBR basin.

From Figure 19, results showed that as the distance from the HJ toe increased, the vertical distance of U_max and inner layer thickness also increased. The analysis further indicated that as the distance from the initial location of HJs increased, the position of U_max was increased, which leveled off after the HJ, as can be seen in sections (C-C) of Figures 17 and 18. In both stilling basins, at X = 2 m, from the HJ initial location, the forward velocity profiles were found well agreed with the profile of Ead & Rajaratnam (2002) and the values of R² reached 0.937 and 0.887 for WSBB and modified USBR basins, respectively, as shown in Figures 18(a) and 18(b), respectively. However, at X = 5.4 m, as compared to velocity (U/U_max = 0.36) in the modified USBR basin, the results showed less forward velocity (U/U_max = 0.21) in the WSBB basin at the upper fluid region.

Figure 20(a) shows the residual errors of velocity profiles in the WSBB basin. At x = 2 m, x = 3.2 m, and x = 4.3 m, the residual errors were found close to the agreement line. At the above-mentioned locations in the WSBB basin, the residual errors were also found less than Ead & Rajaratnam (2002). At x = 5.4 m in the WSBB basin, the maximum positive and negative residual errors of velocity profiles were 0.179 and −0.371, respectively, which were less than those that were found in the modified USBR basin. Figure 20(b) compares residual errors of velocity profiles in the modified USBR basin with the literature. It was evident from the residual diagrams that as the stream-wise distance from the jump-initiating location was increased, the maximum positive and negative errors also increased. The maximum positive and negative residual errors in the USBR basin were found at x = 5.4 m, which reached 1.139 and −1.352, respectively, and showed deviation from Ead & Rajaratnam (2002).

Figure 20

Comparison of residual errors with previous studies: (a) WSBB and (b) modified USBR basins.

Turbulent kinetic energy

TKE is the averaged velocity value in x, y, and z directions and describes energy dissipation at two different flow sections. The root mean square values of the velocity fluctuations are used to compute TKE. By considering the successive velocity values, the root mean square velocity (U_rms) can be computed by the following equation (Gray et al. 2005).

(14)

where u₁, u₂, and u₃ are successive velocities in the flow direction. Now, TKE can be calculated by the following equation.

(15)

where u_rms, v_rms, and w_rms are the root mean square velocities in x, y and z directions, respectively. Figure 21 shows the depth-wise (X–Y) TKE in a modified USBR basin. The results showed that the maximum TKE was found within and foreside of HJs while after the HJ, TKE continued to decline up to the end of the basin. Near the foreside of basins, flows were found to be strongly turbulent, dissipating most of the TKEs. Figure 19 shows the distribution of TKEs at seven vertical sections (X–Y), i.e., at Z = 0 m, 0.47 m, 0.93 m, 1.39 m, 1.85 m, 2.51 m, and free surface. At Z = 0 m, the maximum TKE in the USBR basin was found in the HJ region, which reached 0.30 J/kg, as shown in Figure 21(a). It is observed that behind the baffle blocks, TKEs were reduced due to eddies and fluid circulations, which dissipated the TKEs. Due to the impact of supercritical flows, a small number of eddies and fluid circulations were also noticed in front of the baffle blocks. The results showed that at the basin’s floor, TKEs traveled up to X = 21 m from the HJ-initiating location. Figures 21(b)–21(d) show that as the vertical distance from the basin’s floor was increased, the TKEs also increased, while their magnitude in the longitudinal direction was found to be reduced. The maximum TKEs at Z = 0.93 m, 1.39 m, and 1.85 m were 4.2, 4.5, and 3.6 J/kg, respectively. The results further indicated that as compared to the floor level, the TKEs from the central fluid depth to the free surface were found to be increased and their distribution in horizontal and lateral directions also increased. In the modified USBR basin, the maximum TKEs were noted at the toe of the HJ, which gradually reduced as the flow moved downstream, reaching 0.1 J/kg at the basin end, as shown in Figure 21(g).

Figure 21

Depth-wise distribution of TKEs in the USBR stilling basin at (a) Z = 0 m (floor level), (b) Z = 0.47 m, (c) Z = 0.93 m, (d) Z= 1.39 m, (e) 1.85 m, (f) Z= 2.51 m, and (g) free surface.

In the WSBB basin, at Z = 0 m (floor level), maximum TKEs reached 0.20 J/kg as shown in Figure 22(a). The results showed that as compared to USBR baffle blocks, the WSBBs were spreading the flow more efficiently in the lateral direction. Due to the spreading of fluid in the lateral direction, the results indicated that in the WSBB basin, the TKEs declined earlier in the basin and less energy was reached at the basin’s end. In the WSBB basin, only the TKEs in central fluid depths traveled downstream, which ended at X = 13 m from the toe of HJs. It is worth mentioning here that as compared to the modified USBR basin, the TKEs in the WSBB basin declined earlier, which indicated 8 m less distance than the USBR basin.

Figure 22

Depth-wise distribution of TKEs in the WSBB stilling basin at (a) Z= 0 m (floor level), (b) Z= 0.47 m, (c) Z= 0.93 m, (d) Z= 1.39 m, (e) 1.85 m, (f) Z= 2.51 m, and (g) free surface.

The results further showed that upon the use of WSBBs, no flow reattachment was witnessed on either side of the baffle blocks. Due to reduced reattachment, more wake areas were generated on the side of the WSBB basin, and the results showed agreement with the statement of other authors (Verma & Goel 2003; Verma et al. 2004; Goel & Verma 2006). At Z = 0.47 m, 0.93 m, and 1.39 m, the maximum TKEs were noticed in the HJ region, which reached 4.3, 4.6, and 3.5 J/kg, respectively, as shown in Figure 22(b)–22(d), respectively. In the WSBB basin, after the HJ, the baffle blocks declined the TKEs due to the development of sharp discontinuities in the flow. After Z = 1.39 m, the value of TKEs up to the free surface gradually reduced, as shown in Figure 22(e) and 22(f). Figure 22(g) shows 2D illustrations of TKEs on the free surface, and the results indicate that as compared to the modified USBR basin, the magnitude of TKEs was lower and traveled less distance in the WSBB basin.

CONCLUSIONS

This study developed numerical models on the rigid bed to investigate the effects of USBR and WSBB baffle blocks on the HJ downstream of the river diversion barrage using FLOW-3D. VOF and RNG K–ɛ models were employed to track the free surface and turbulence, respectively. For the proposed new basin (WSBB basin), WSBB with a vertex angle of 150° and cutback of 90° is employed in the baffle block region, while the friction block region remained unchanged. The performance of the two different basins is assessed by HJs and other hydraulic parameters such as free surface profile, sequent depths, roller lengths, HJ efficiency, velocity profile, and TKE. Furthermore, the results of the present modified USBR Type-III and WSBB basins are compared with the relevant literature, for which regression analysis is performed and residual error diagrams are plotted. However, the present models are limited to the single discharge of 44 m³/s and employ only one turbulence model, i.e., RNG K–ɛ. Additionally, the present models were designed for a single bay of the barrage.

• Upon use of fine meshing, in comparison to the designed discharge, the present models showed 0.80 and 0.90% of errors in modified USBR Type-III and WSBB basins, respectively. Similarly, for the gated flow, the results indicated 0.32 and 1.14% errors in the modified USBR Type-III and WSBB basins, respectively.
• After employing regression analysis, the results of free surface profiles showed agreement with the previous studies for which R² reached 0.980 and 0.970 in WSBB and modified USBR basins, respectively. From the results, it can be believed that as compared to the modified USBR Type-III, the newly proposed WSBB basin produced a better free surface profile of HJs.
• Due to the inclusion of the baffle blocks in the studied basins, the roller lengths of HJs were contained efficiently, and thereby, as compared to the literature, lesser roller lengths were observed in the modified USBR Type-III and WSBB basins.
• The overall efficiency of HJs in modified USBR and WSBB basins reached 58.60 and 57.90%, respectively, which showed good agreement with the literature. Based on the results of the efficiency of HJs, the accuracy of the present models reached 93%.
• In the hydraulic regions, the results of dimensionless velocity profiles indicated a wall jet-like structure, which agreed well with the literature. In addition, as compared to the modified USBR Type-III basin, the velocity profiles in the WSBB basin were found to be more promising, for which R² reached 0.937. Additionally, after the HJ, as compared to the USBR Type-III basin, the forward velocity (U/U_max) in the WSBB basin was found to be less. Conclusively, it can be said that in comparison to the modified USBR Type-III basin, at the lower discharges, the WSBB basin decays the velocities more efficiently.
• The results of TKEs indicated that the flow was strongly turbulent near the foreside of the HJs, and the maximum TKEs were noted in the central fluid depths. In the WSBB basin, no fluid reattachment was observed on either side of the baffle blocks, and the results further indicated that as compared to the modified USBR Type-III basin, fewer TKEs were found at the end of the WSBB basin.

Based on the models’ results, the study confirms the suitability of WSBB downstream of the barrage for lower tailwater conditions. From the results, it is believed that FLOW-3D is a very effective and efficient tool for the hydraulic investigation of flow behavior downstream of the barrage. However, in Pakistan, the use of such modeling tools is found very limited, therefore, the study results will help hydraulic and civil engineers to assess different energy dissipation arrangements within the stilling basins and will provide suitable alternative solutions. The present study was limited to the fixed geometry of the WSBB, therefore, it is suggested to investigate HJ and flow characteristics with other vertex and cutback angles. In addition, it is also recommended to study the hydraulics of WSBB downstream of barrages by employing multiple bays of barrage and other turbulence models.

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