Abstract

The present study deals with numerical simulations of the free and submerged hydraulic jumps over different shapes of roughness in various roughness arrangements and different Froude number conditions. The models were studied using three roughness shapes, i.e. triangular, square and semi-oval for 0.2 < T/I < 0.5, where T and I are height and distance of roughness, respectively. The results showed that the numerical model is fairly well able to simulate the free and submerged jump characteristics. The effect of roughness plays a role in the reduction of the relative maximum velocity which is greater in the submerged jump. The thickness of the boundary layer for both free and submerged jumps decreases with increasing the distance between the roughnesses. Triangular macroroughness has a significant effect on the length of the jump and shortest length with respect to the other shapes. The reduction in the submerged depth ratio and tailwater depth ratio depends mainly on the space of the roughnesses. The highest shear stress and energy loss in both jumps occur in a triangular macroroughness (TR) with T/I = 0.50 compared to other ratios and modes. The numerical results were compared with previous studies and relationships with good correlation coefficients were presented for the mentioned parameters.

HIGHLIGHTS

  • CFD model is fairly well able to simulate the free and submerged jump characteristics.

  • Roughness plays a role in the reduction of the relative maximum velocity, the submerged depth ratio.

  • The thickness of the boundary layer for both free and submerged jumps decreases with increasing the distance between the roughnesses.

  • Shear stress and energy loss in the free and submerged jumps increase in the roughnesses.

NOTATIONS

The following symbols and their meanings are used in this paper:

Q l/s Discharge 
D Gate opening 
E1, E2 Specific energy at the beginning and after the free jump 
E3, E4 Specific energy at the beginning and after the submerged jump 
ΔE Energy loss 
y1 Inlet depth of the hydraulic jump 
y2 Sequent depth of the free jump 
y3 Submerged depth 
y4 Tailwater depth 
 Subcritical depth of the classical hydraulic jump 
Ljf Length of the free jump 
Ljs Length of the submerged jump 
P1,P2 pa Pressure before and after the jump 
M1, M2 kgm/s Momentum before and after the jump 
Fτ Shear force per unit width 
Umax m/s Maximum horizontal velocity 
u1 m/s Inlet horizontal velocity 
g m/s2 Gravitational acceleration 
I Distance of triangular roughness 
T Roughness height 
Fr1 – Inlet Froude number 
Re1 – Inlet Reynolds number 
S – Submergence factor 
ε – Bed shear force coefficient 
t Time 
p pa Pressure 
ρ kg/m3 Mass density of water 
μ Ns/m2 Dynamic viscosity of water 
 m2/s Kinematic viscosity of water 
Q l/s Discharge 
D Gate opening 
E1, E2 Specific energy at the beginning and after the free jump 
E3, E4 Specific energy at the beginning and after the submerged jump 
ΔE Energy loss 
y1 Inlet depth of the hydraulic jump 
y2 Sequent depth of the free jump 
y3 Submerged depth 
y4 Tailwater depth 
 Subcritical depth of the classical hydraulic jump 
Ljf Length of the free jump 
Ljs Length of the submerged jump 
P1,P2 pa Pressure before and after the jump 
M1, M2 kgm/s Momentum before and after the jump 
Fτ Shear force per unit width 
Umax m/s Maximum horizontal velocity 
u1 m/s Inlet horizontal velocity 
g m/s2 Gravitational acceleration 
I Distance of triangular roughness 
T Roughness height 
Fr1 – Inlet Froude number 
Re1 – Inlet Reynolds number 
S – Submergence factor 
ε – Bed shear force coefficient 
t Time 
p pa Pressure 
ρ kg/m3 Mass density of water 
μ Ns/m2 Dynamic viscosity of water 
 m2/s Kinematic viscosity of water 

INTRODUCTION

A hydraulic jump is a rapidly varied flow that dissipates a significant amount of energy by changing the flow regime from supercritical to subcritical in a short length. The most important factor in this phenomenon is the Froude number at the beginning of the jump (Chow 1959). Free and submerged hydraulic jumps are commonly applicable to energy dissipation below hydraulic structures, such as control gates, spillways and weirs. Woodward (1917), Bradley & Peterka (1957), Rajaratnam (1968) and Hager et al. (1990) were among the first researchers to study free jumps. In particular, Rajaratnam (1968) stated in his results that, taking into account macroroughness, the length of the jump is significantly reduced if compared to a smooth one. Also, the submerged jump was investigated by Rao & Rajaratnam (1963) who proposed, using the principles of continuity and momentum size, relationships for the conjugate depth ratio and energy loss. Next, the hydraulic jump downstream of the spillways (Samadi-Boroujeni et al. 2013; AlTalib et al. 2019), at the location of sudden channel cross-section changes (Matin et al. 2008; Hassanpour et al. 2017) and after the gate (Mouaze et al. 2005; Lopardo 2013) was studied. Several studies were carried out to study the hydraulic jump on a rough bed. Ead & Rajaratnam (2002) showed that the shear stress of a rough bed is 10 times that of a smooth one. Dey & Sarkar (2008) stated that the thickness of the inner layer of the horizontal velocity distribution increases with increasing macroroughness. Abbaspour et al. (2009) studied the characteristic of free jump and velocity profiles on the rough bed. The energy dissipation caused by free hydraulic jump (FHJ) for different shapes of roughnesses was investigated by Tokyay et al. (2011), while Akib et al. (2015), Felder & Chanson (2018) and Roushangar & Ghasempour (2019) focused the attention on the resulting depth ratio, the relative length of the jump, the air–water flow properties and the energy dissipation on the rough bed. Habibzadeh et al. (2019) studied the effect of blocks downstream of the gate on the characteristics of the free and submerged jump. Pourabdollah et al. (2019) compared the characteristics of a free and submerged jump in a rough bed with the adverse slope. Numerical methods were used to study the characteristics of the hydraulic jump by various researchers. Federico et al. (2012) used the SPH model, Bayon-Barrachina & Lopez-Jimenez (2015) used the Openfoam model and Witt et al. (2018) used computational fluid dynamics (CFD) methods to study free jump and Shekari et al. (2014) for the submerged jump. Summary of the research background of the characteristics of free and submerged jumps over the smooth and macroroughness is shown in Table 1. Although several studies have been carried out on macroroughnesses, there is still a strong need for fundamental studies on the effects of different shape elements of macroroughness and corresponding characteristics of free and submerged jumps. The present paper aimed at contributing numerically with CFD techniques to enhance the understanding of characteristics of free and submerged jumps, such as velocity field and bed shear stress, sequent and submerged depths ratio, the length of jumps and energy loss in triangular, square and semi-oval macroroughnesses through different hydraulic conditions and various geometrical arrangements.

Table 1

Specifications of the most studies conducted by other researchers in the present study field

ResearcherModel types
Bed form
Jump types
Froude number rangeOther specification
Exp.Num.smoothroughFreeSubmerged
Ead & Rajaratnam (2002)        4–10 Corrugated beds 
Dey & Sarkar (2008)        2.6–4.9 Horizontal rough beds 
Abbaspour et al. (2009)        3.8–8.6 Sinusoidal corrugated bed 
Tokyay et al. (2011)        2.1–11 Non-protruding rough beds 
Shekari et al. (2014)        3.2–8.2 Used the volume of fluid (VOF) method 
Witt et al. (2018)        2.4–4.8 Bubble clustering an air entraining 
Habibzadeh et al. (2019)        3.5–6.8 Used blocks in downstream 
Pourabdollah et al. (2019)        4.5–9.5 Effect of adverse slopes 
Present study       1.7–9.3 Effect of rough shape 
ResearcherModel types
Bed form
Jump types
Froude number rangeOther specification
Exp.Num.smoothroughFreeSubmerged
Ead & Rajaratnam (2002)        4–10 Corrugated beds 
Dey & Sarkar (2008)        2.6–4.9 Horizontal rough beds 
Abbaspour et al. (2009)        3.8–8.6 Sinusoidal corrugated bed 
Tokyay et al. (2011)        2.1–11 Non-protruding rough beds 
Shekari et al. (2014)        3.2–8.2 Used the volume of fluid (VOF) method 
Witt et al. (2018)        2.4–4.8 Bubble clustering an air entraining 
Habibzadeh et al. (2019)        3.5–6.8 Used blocks in downstream 
Pourabdollah et al. (2019)        4.5–9.5 Effect of adverse slopes 
Present study       1.7–9.3 Effect of rough shape 

FREE AND SUBMERGED HYDRAULIC JUMPS

The hydraulic jump occurs in free surface flows in both free and submerged modes, where in the free type the tailwater depth (y4) is equal to the sequent depth of jump (y2). A submerged hydraulic jump (SHJ) occurs when the tailwater depth in an open-channel flow is larger than the sequent depth of the pre-existing free jump; in this case, the jump moves upstream and becomes submerged, air entrainment reduces and turbulence intensities are smaller than for free jump counterparts (Wu & Rajaratnam 1995). Figure 1 shows a schematic view of free and submerged jumps on a triangular rough bed, along with important hydraulic parameters of the present study. In Figure 1, y1 and y2 are referred to supercritical and subcritical depths of the free jump depth, respectively, and y3 and y4 are related to submerged and tailwater depths of the submerged jump, respectively. Ljf and Ljs are lengths of the free and submerged jump. Also, d is gate opening, and T and I are height and distance of macroroughnesses, respectively.

Figure 1

Schematic of a free and submerged hydraulic jump on different macroroughnesses.

Figure 1

Schematic of a free and submerged hydraulic jump on different macroroughnesses.

A submerged jump is characterized by the supercritical Froude number Fr1 and the submergence factor S, defined as (y4y2)/y2 (Rajaratnam 1965). Obviously, S is equal to zero for the free jump and as S increases above zero, we get a submerged jump of different degrees of submergence. From Figure 1, it is possible to distinguish three regions in a submerged jump: the developing, the developed and the recovering regions (Long et al. 1990). While the developing zone occupies as far as the potential-core zone and includes a supercritical flow region with wall jet characteristics, the developed area extends throughout the length of the roller of the horizontal axis (Ljs), where a big counter-clockwise circulating free surface roller dissipates the hydraulic energy, beyond which the recovering region begins and includes a subcritical flow region.

METHODS AND MATERIALS

Input parameters for numerical models

In this study, the characteristics of free and submerged jump on triangular roughness (TR), square roughness (SR) and semi-oval roughness (OR) were investigated. Numerical simulations were performed in the range of Froude numbers (Fr1) from 1.7 to 9.3 with a fixed gate opening (d) of 5 cm, constant roughness height of 4 cm and different T/I ratios (see Figure 1).

In Table 2, standard definitions were used for the Froude and Reynolds numbers (Fr1=u1/(gy1)0.5 and Re1=(u1y1)/υ, where g and are gravity acceleration and water kinematic viscosity, respectively. If y1 and u1 are, respectively, the depth and inlet velocity of the supercritical flow just upstream of the jump, the subcritical sequent depth of the classical hydraulic jump () is given by the well-known Belanger equation (Chow 1959; Hager 1993; Vischer & Hager 1995; Ead & Rajaratnam 2002; Tokyay 2005; Abbaspour et al. 2009, 2019; Chanson 2009; Samadi-Boroujeni et al. 2013; Hassanpour et al. 2017; Torkamanzad et al. 2019):
formula
(1)
Table 2

Parameters of the numerical models

Bed typeQ (L/s)I (cm)T (cm)d (cm)y1 (cm)y4 (cm)Fr1SRe1
Smooth 30, 45 – – 1.62–3.84 9.64–32.1 1.7–9.3 0.27–0.56 39,884–59,825 
TR 30, 45 8–12–16–20 1.62–3.84 6.82–30.08 1.7–9.3 0.22–0.45 39,884–59,825 
SR 30, 45 8–12–16–20 1.62–3.84 7.26–30.81 1.7–9.3 0.21–0.49 39,884–59,825 
OR 30, 45 8–12–16–20 1.62–3.84 7.61–31.35 1.7–9.3 0.22–0.44 39,884–59,825 
Bed typeQ (L/s)I (cm)T (cm)d (cm)y1 (cm)y4 (cm)Fr1SRe1
Smooth 30, 45 – – 1.62–3.84 9.64–32.1 1.7–9.3 0.27–0.56 39,884–59,825 
TR 30, 45 8–12–16–20 1.62–3.84 6.82–30.08 1.7–9.3 0.22–0.45 39,884–59,825 
SR 30, 45 8–12–16–20 1.62–3.84 7.26–30.81 1.7–9.3 0.21–0.49 39,884–59,825 
OR 30, 45 8–12–16–20 1.62–3.84 7.61–31.35 1.7–9.3 0.22–0.44 39,884–59,825 

Computational fluid dynamics

Most applications of computational fluid dynamics (CFD) are associated with the simulation of free surface and open-channel flows, overflows, and also the simulation of the hydraulic jumps (e.g., Abbaspour et al. 2009; Bayon et al. 2016; Witt et al. 2018; Ghaderi et al. 2020a). The FLOW-3D® computational package was used. The software uses the Finite Volume Method (FVM) to solve the RANS equations (Reynolds Averaged Navier–Stokes) in a Cartesian, staggered grid and evaluates obstacles to the flow by the Fractional Area/Volume Obstacle Representation (FAVOR) method, presented by Hirt & Sicilian (1985). The continuity and momentum Navier–Stokes equations for 3-D incompressible flow, including FAVOR variables, are written as:
formula
(2)
formula
(3)
formula
(4)
where P is the pressure, Vf is the volume fraction of the fluid in each cell, Ai is the fractional areas open to the flow, fi is the viscous acceleration, Gi is the body acceleration, ρ is water density and τw;i is the wall shear stress in the subscript directions. FLOW-3D® uses an advanced algorithm for tracking free surface flows, called the volume of fluid (VOF) and developed by Hirt & Nichols (1981). The VOF method consists of three main components: the definition of the VOF function, a method to solve the VOF transport equation, and the setting of boundary conditions at the free surface. Within the frame of VOF methods, the interface is determined from the volume fraction F. The F value varies between zero, when the grid cell contains no fluid, and unity when the grid cell is fully occupied with fluid, as shown in the example of Figure 2 (Ghaderi & Abbasi 2019; FLOW-3D® User Manual 2016). A free surface must be in cells having F values between unity and zero. Since F is a step function, the normal direction to the cutting line represents the free surface inside the grid cell, which is perpendicular to the direction of rapid change in F values. The VOF transport equation is expressed by the following equation:
formula
(5)
Figure 2

VOF free surface assessment.

Figure 2

VOF free surface assessment.

Turbulence model

FLOW-3D® offers some turbulence models, such as normal turbulence models, the k–ɛ turbulence model and the RNG turbulence model. In this study, the chosen turbulence model was RNG k–ɛ, because Flow Science Inc. (2016) mentioned that the RNG k–ɛ model has wider applicability than the standard k–ɛ and is usually the best choice. RNG k–ɛ model can simulate the flow with a high number of computational meshes and is more accurate for rapidly strained flows and swirling flows and for lower Reynolds numbers (Re) based on the results of numerical studies by researchers, such as Carvalho & Lemos Ramo (2008); Daneshfaraz et al. (2016); Bayon et al. (2016); Daneshfaraz et al. (2019); Sangsefidi et al. (2019) and Ghaderi et al. (2020b, 2020c, 2020d, 2020e), on the acceptable ability of the RNG k–ɛ turbulence model to simulate hydraulic jump in the stilling basin and flow on hydraulic structures. As a result, the RNG k–ɛ was utilized to model the domain.

This model is a modified version of the k–ɛ standard model, which improves its performance. The adopted scheme is a two-equation model. In particular, the first equation (Equation (6)) expresses the energy in turbulence, which is called turbulent kinetic energy (TKE) (k). The second equation (Equation (7)) is the turbulent dissipation rate (ɛd) which determines the rate of kinetic energy dissipation. These equations are expressed as follows:
formula
(6)
formula
(7)
where Gk is the generation of TKE caused by the average velocity gradient, Gb is the generation of TKE caused by buoyancy, while Sk and Sɛ are source terms. The values of αk and αɛ are the inverse effective Prandtl numbers for k and ɛd, respectively, while μeff is the effective viscosity μeff=μ+μt, being μt the eddy viscosity.
For the above Equation (7):
formula
(8)
formula
(9)

The constant values for this model are (Yakhot & Orszag 1986; Samma et al. 2020): Cμ = 0.0845, C1ɛ= 1.42, C2ɛ = 1.68, C3ɛ= 1.0, σk = 0.7194, σɛ = 0.7194, η0 = 4.38 and β = 0.012.

Numerical domain

The calibration data provided by Ahmed et al. (2014) allow the comparison of the numerical model and laboratory test results. A specification of the experimental results was noticed in the validity of the numerical model part. For this experiment, a flume with the width, depth and length of, respectively, 0.75, 0.7 and 24.5 m was used (see, for more details, Ahmed et al. 2014). AutoCAD® software is used to make the geometry of the models and performed by inserting an STL (stereolithography) file. According to the experimental conditions, all boundary conditions have been employed. The inlet boundary condition was set as the discharge flow rate (Q) equal to the experimental flow exit discharge. The boundary condition at the downstream end of the domain was described by a pressure boundary condition (P) corresponding to the tailwater depth in the flume. Wall roughness has been neglected due to the small roughness of the material of the experimental facility which was used for validation. The lower Z (Zmin) and both of the side boundaries were treated as rigid wall (W). No-slip conditions were applied at the wall boundaries and friction was neglected. No-slip is defined as zero tangential and normal velocities (u = v = w = 0). With a no-slip boundary, it is assumed that a law-of-the wall type profile exists in the boundary region (FLOW-3D® User Manual 2016). An atmospheric boundary condition is set to the upper boundary of the channel. This allows the flow to enter and leave the domain as null von Neumann conditions are imposed on all variables except for pressure, which is set to zero (i.e. atmospheric pressure). Symmetry boundary condition (S) is used at the inner boundaries as well. Figure 3 shows the computational domain of the present study and the associated boundary conditions.

Figure 3

Boundary conditions on the numerical domain in FLOW-3D.

Figure 3

Boundary conditions on the numerical domain in FLOW-3D.

The spatial domain subject of the present study was meshed using a structured rectangular hexahedral mesh with two different mesh blocks. Hence, a containing mesh block was created for the entire spatial domain, and then, a nested mesh block was built, with refined cells for the area of interest, where the hydraulic jump takes place (see Figure 4). This technique, i.e. a nested mesh block, was adopted from previous studies (see, for example, Choufu et al. (2019), Zahabi et al. (2018) and Ghaderi & Abbasi (2019)). Three different computational meshes were utilized to select the appropriate mesh by utilizing Grid Convergence Index (GCI), which is a widely accepted and recommended method for estimating discretization error that has been applied to several CFD cases (e.g., Bayon et al. 2016; Helal et al. 2020). The analysis was developed following the Richardson extrapolation method (Celik et al. 2008). Three different meshes with fine, medium and coarse cells, consisting of 4,624,586, 2,908,596 and 1,285,482, cells in total, respectively, were used to examine the effect of the grid size on the accuracy of the numerical results. Table 3 summarizes some details of the three computational grids.

Table 3

Characteristics of the meshes tested in the convergence analysis

MeshNested block cell size (cm)Containing block cell size (cm)
0.55 1.10 
0.65 1.30 
0.85 1.70 
MeshNested block cell size (cm)Containing block cell size (cm)
0.55 1.10 
0.65 1.30 
0.85 1.70 
Figure 4

Sketch of mesh setup.

Figure 4

Sketch of mesh setup.

The GCI was determined for the computed y3/y1 ratio at Fr1 = 4.5 obtained from numerical solution on the three grids. Using the Richardson error estimator to compare the three grids (fine, medium and coarse grid), the fine-GCI is defined as:
formula
(10)
where E32=(fs2fs3)/fs2 is the approximate relative error between the medium and fine grids, fs2 and fs3 are medium and fine grid solutions for y3/y1 ratios, respectively, and p is the local order of accuracy. For the three-grid solutions, p is obtained by solving the equation:
formula
(11)
where ɛ is the error between two adjacent meshes, namely ɛ21=fs2fs1 and ɛ32=fs3fs2. The value of r32=G3/G2 is the grid refinement factor between the medium and fine grid. G1, G2 and G3 represent the abbreviations of grids. For the present three-grid comparisons, G1 < G2 < G3. Table 4 shows a summary of the results of the mesh convergence analysis conducted. In this table, the values of GCI21 and GCI32 represent the relative change from medium to coarse and from coarse to medium mesh, respectively.
Table 4

Results of the mesh convergence analysis

ParametersValues
fs1 (–) 7.15 
fs2 (–) 6.88 
fs3 (–) 6.19 
p (–) 5.61 
E32 (%) 10.02 
E21 (%) 3.77 
GCI21 (%) 3.03 
GCI32 (%) 3.57 
GCI32/rpGCI21 0.98 
ParametersValues
fs1 (–) 7.15 
fs2 (–) 6.88 
fs3 (–) 6.19 
p (–) 5.61 
E32 (%) 10.02 
E21 (%) 3.77 
GCI21 (%) 3.03 
GCI32 (%) 3.57 
GCI32/rpGCI21 0.98 

Since the GCI values for the finer grid (GCI21) is small as compared to the coarser grid (GCI32), it can be inferred that the grid-independent solution is nearly achieved and does not require carrying out further mesh refinements. Calculated values of GCI32/rpGCI21 close to 1 indicate that the numerical solutions are within the asymptotic range of convergence. As a result, a mesh consisting of a containing block with a cell size of 1.3 cm and a nested block of 0.65 cm was chosen (see Figure 4).

Near wall treatment

Usually, all the turbulence models are quite accurate and valid only to fully turbulent flows. However, near the wall, the flow is almost laminar and the turbulent stress hardly works, especially for the viscous sublayer region (Garcia Perez & Vakkilainen 2019). As a result, the traditional turbulence models do not work well here. Currently, there are two different methods (near-wall modeling method and wall function method) to solve the problem. Both methods involve a dimensionless distance y+. The ratio of turbulent and laminar influences in a cell, y+, is defined as:
formula
(12)
where yp is the distance of the first node from the wall, u* is the shear velocity of the wall () and υ is the kinematic viscosity of the fluid. The near-wall modeling method directly utilizes the low Reynolds number turbulence model with many fine grids near the wall to deduce parameters in the viscous sublayer. It requires that the grid point of the first layer should be arranged within the viscous layer (y+ < 1). The wall function method does not directly deduce the viscous sublayer but arranges the grid point of the first layer within the log law region (30 < y+ < 200 ∼ 400) and then relates the viscous layer to the log law region with empirical formulas (Salim & Cheah 2009). Compared with the near-wall modeling method, the wall function method does not need to specifically compact the grid near the wall but saves much more computation time with higher efficiency. In view of the unique grid generation technology in the FLOW-3D software, as shown in Figure 4, the grid can be flexibly adjusted to embed into the walls of the numerical model. In addition, as shown in Table 5, two different grid systems are arranged to fill the blocks to evaluate the effect of the grid size upon the numerical results. It can be noticed that the dimensionless y+ in all grid systems ranges from 72 to 135.72 and confirms the requirement of the wall function method. The inlet Reynolds number varies between 39,884 and 59,825.
Table 5

Grid details in numerical domain

BlockMax cell size (cm)Min cell size (cm)Near wall distance (cm)Range of dimensionless distance y+
Containing and nested block 1.3*1.3*1.3 0.65*0.65*0.65 0.5 72 < y+ <135.72 
BlockMax cell size (cm)Min cell size (cm)Near wall distance (cm)Range of dimensionless distance y+
Containing and nested block 1.3*1.3*1.3 0.65*0.65*0.65 0.5 72 < y+ <135.72 

Stability condition

A stability criterion similar to the Courant number is a function of time step size. The time step was calculated over each cell with the help of the CFL (Courant–Friedrichs–Lewy) criterion. During the iteration, the time step size was controlled by both of the stability and convergence criterion, which leads to time steps between 0.001 and 0.0016 s. The evolution in time was used as a relaxation to the final steady state. The steady-state convergence of the solutions was checked through monitoring the flow discharge variations at the inlet and outlet boundaries during the simulations. Figure 5 shows that t = 16 s is appropriate to achieve a stable steady-state condition for the adopted two discharges, i.e. Q = 0.03 and 0.045 m3/s. The computational time for the simulations was between 14 and 18 h using a personal computer with eight cores of a CPU (Intel Core i7-7700 K @ 4.20 GHz and 16 GB RAM).

Figure 5

Time variation of computational fluid dynamics (CFD) flow discharge in the inlet and outlet boundaries. (a) Q = 0.03 m3/s, (b) Q = 0.045 m3/s.

Figure 5

Time variation of computational fluid dynamics (CFD) flow discharge in the inlet and outlet boundaries. (a) Q = 0.03 m3/s, (b) Q = 0.045 m3/s.

RESULTS AND DISCUSSIONS

The validity of the FLOW-3D® model

A first analysis of the performance of the numerical model and laboratory test results carried out over basic variables is summarized in Table 6.

Table 6

Basic flow variables for the numerical simulation and physical models (Ahmed et al. 2014)

ModelsBed typeQ (l/s)d (cm)y1 (cm)u1 (m/s)F1
Numerical and physical models Smooth 45 0.05 1.62–3.83 1.04–3.70 1.7–9.3 
ModelsBed typeQ (l/s)d (cm)y1 (cm)u1 (m/s)F1
Numerical and physical models Smooth 45 0.05 1.62–3.83 1.04–3.70 1.7–9.3 
A quantitative evaluation of the computed and measured submerge ratio (y3/y1) values for submerged jump and jump depths ratio (y2/y1) values for free jump comparisons is made by using mean square error (MSE) and mean absolute relative error (MARE).
formula
(13)
formula
(14)
where Oi and Pi are measured and computed values, respectively, and N is the total number of data. Table 7 gives the results for MSE and MARE using Equations (13) and (14) for different inflow Froude numbers on the smooth bed.
Table 7

MSE and MARE values for submerge ratio y3/y1 values on the smooth bed

ModelFr1y3/y1- measured valuesy3/y1- computed valuesy2/y1- measured valuesy2/y1- computed valuesMSE (–) y3/y1MARE (%) y3/y1MSE (–) y2/y1MARE (%) y2/y1
Smooth bed 1.7 1.98 1.90 1.94 1.98 0.006 4.04 0.002 2.06 
2.5 3.20 3.16 2.98 3.09 0.016 1.25 0.012 3.70 
4.5 6.88 7.25 5.81 5.91 0.136 5.37 0.010 1.72 
6.1 10.02 9.19 8.08 8.17 0.689 8.28 0.010 1.11 
9.3 15.67 15.34 12.58 12.69 0.109 2.10 0.012 0.87 
Mean 0.188 4.21 0.010 1.89 
ModelFr1y3/y1- measured valuesy3/y1- computed valuesy2/y1- measured valuesy2/y1- computed valuesMSE (–) y3/y1MARE (%) y3/y1MSE (–) y2/y1MARE (%) y2/y1
Smooth bed 1.7 1.98 1.90 1.94 1.98 0.006 4.04 0.002 2.06 
2.5 3.20 3.16 2.98 3.09 0.016 1.25 0.012 3.70 
4.5 6.88 7.25 5.81 5.91 0.136 5.37 0.010 1.72 
6.1 10.02 9.19 8.08 8.17 0.689 8.28 0.010 1.11 
9.3 15.67 15.34 12.58 12.69 0.109 2.10 0.012 0.87 
Mean 0.188 4.21 0.010 1.89 

Regarding the overall mean values of MSE and MARE in Table 7, it can be concluded that there is a good agreement between numerical and laboratory results. The mean maximum error is 4.21%, which confirms the ability of the numerical model to predict the specifications of the free and submerged jumps.

Horizontal velocity distributions

The longitudinal velocity profiles on the macroroughnesses (TR, OR and SR) and smooth bed at Fr1 = 4.5 after creating steady-state conditions in FLOW-3D are shown in Figure 6. It can be seen that close to the sluice gate, due to the high velocity of flow, the water depth decreases initially before it again increases to the tailwater depth y2 for FHJ and y4 for the SHJ. A large counter-clockwise roller exists in the upstream flow field. The zero-velocity line, where a free surface roller circulating as big counter-clockwise, dissipates the hydraulic energy and finally reaches the downstream end. From there, the velocity profile becomes more uniform. As in a fixed initial Froude number (Fr1), the flow velocity decreases in the macroroughness, occurs faster than the smooth bed and the length of jump decreases. The peak of the velocity profile closer to the bed shows that the flow in this location is strongly influenced by the bed. Also, the flow velocity near the bed reduces and in the distance between the roughnesses becomes negative. This negative velocity created in the distance between the macroroughnesses (TR, OR and SR) increases with increasing the distance between the roughnesses so that for T/I = 0.20, it is more than the other modes as well as it is observed that the triangular macroroughness gives the highest negative velocity with respect to the other two shapes. This is due to a clockwise recirculation zone and eddy flow exists between the roughnesses. A comparison of velocity profiles in free and submerged jumps shows that the thickness of the negative velocity layer near the flow surface is greater in submerged jumps. According to Figure 6, the horizontal velocity distribution has recovered by the time that the flow arrives at the next roughness when the distance between the macroroughnesses (TR, OR and SR) is long enough. But, if this distance is short, the flow arrives at the next roughness without adequate recovery of the horizontal velocity distribution. Hence, with an increasing number of macroroughnesses (TR, OR and SR), the rate of increase of the frictional coefficient decreases.

Figure 6

Typical longitudinal velocity profiles for the different models at Fr1 = 4.5.

Figure 6

Typical longitudinal velocity profiles for the different models at Fr1 = 4.5.

Figure 7 shows the dimensionless relative maximum horizontal velocity (Um/u1) versus (X/y1) for the different roughness elements and for free and submerged hydraulic jumps. In Figure 7, X represents the distance from the beginning of the hydraulic jump. It can be seen that the relative maximum velocity in the macroroughness is less than in the smooth one and the effect of roughness plays a role in the reduction of the relative maximum velocity which is greater in the submerged jump. The difference of the values (Um/u1) for the investigated roughness shape (TR, OR and SR) in T/I= 0.25 and 0.50 is not high under similar conditions and slightly lower for the TR, while the SR gives values of velocity close to the smooth bed. The boundary layer thickness () at each section of the hydraulic jump was equal to the depth at which the maximum velocity occurred. Therefore, the changes of the dimensionless boundary layer thickness (δ/y1) versus (X/y1) are shown in Figure 8. It is observed that for both free and submerged jumps, with increasing distance from the beginning of the hydraulic jump, the position of maximum velocity (i.e., the thickness of the inner layer of the horizontal velocity distribution) increases. Also, the thickness of the boundary layer decreases with increasing distance between the roughnesses on the macroroughnesses (TR, OR and SR). The lowest value of δ/y1 is related to T/I = 0.25, while its highest value is related to T/I = 0.50 in different modes. The triangular macroroughness shows the maximum values of the boundary layer thickness. Based on the above findings, for macroroughnesses (TR, OR and SR) and for all T/I ratios, Um/u1 and δ/y1 of free and submerged jumps can be estimated through the following equations, respectively:
formula
(15)
formula
(16)
formula
(17)
formula
(18)
Figure 7

Variations of (Um/u1) vs. (X/y1) for different roughness shape.

Figure 7

Variations of (Um/u1) vs. (X/y1) for different roughness shape.

Figure 8

Variations of (δ/y1) versus (X/y1) for different roughness shape.

Figure 8

Variations of (δ/y1) versus (X/y1) for different roughness shape.

The dimensionless values (Um/u1) and (δ/y1) in free and submerged jumps over the smooth and rough bed are shown in Figure 9. The results of the present study are compared with the experimental data by Abbaspour et al. (2009), Shekari et al. (2014) and Pourabdollah et al. (2019). For the smooth bed, the maximum flow velocity at the beginning of the jump for the free jump is more than the submerged jump, but at the end of the jump, these values are higher for the submerged jump. In the rough bed, the dimensionless ratio (Um/u1) at a specified X in the submerged jump is higher than in the free jump. This finding agrees with previous research by Pourabdollah et al. (2019). In addition, for both smooth and rough bed in the submerged jump due to increasing depth and eddy flow, the maximum velocity distance from the bed is reduced and the boundary layer thickness is less than free jump. The results of the present study are in good agreement with the data by Abbaspour et al. (2009) and Pourabdollah et al. (2019) and there is a discrepancy in (Um/u1) with the results by Shekari et al. (2014) because this researcher studied only the smooth bed.

Figure 9

Variations of the dimensionless parameters (Um/u1) and (δ/y1) in free and submerged jumps.

Figure 9

Variations of the dimensionless parameters (Um/u1) and (δ/y1) in free and submerged jumps.

Bed shear stress

The roughness bed increases the bed shear stress and reduces the sequent water depth and the length of the hydraulic jump (Ead & Rajaratnam 2002; Samadi-Boroujeni et al. 2013; Pourabdollah et al. 2019). As a result, it leads to a reduction in the length of the stilling basin. The shear stress coefficient (ɛ) is calculated as follows (Rajaratnam 1965):
formula
(19)
where Fτ is shear force per unit width and can be obtained using the following equations for the free and submerged jumps, respectively:
formula
(20)
formula
(21)
where F1=0.5γ, F2=0.5γ, F3=0.5γ and F4=0.5γ represent, respectively, the hydrostatic pressure forces imposed on the control volume at sections 1, 2, 3 and 4 that are related to y1, y2, y3 and y4. The symbols β1, β2 and β4 are the momentum correction factors at sections 1, 2 and 4, which were assumed to be equal to 1 (Pourabdollah et al. 2019). γ is the specific weight, q is the flow rate in width unit, and V1, V2 and V4 are the average velocities at the initial jump (section 1), at sections 2 and 4, respectively. The value of the bed shear force coefficient was determined by using Equation (19). Figure 10 shows the variations of shear stress coefficient (ɛ) of the free and submerged jumps as a function of the inlet Froude number (Fr1). The numerical results of the shear stress coefficient were compared with Ead & Rajaratnam (2002) (Equations (22) and (23) for the hydraulic jump on the smooth bed and macroroughnesses, respectively) and Izadjoo & Shafai Bajestan (2007) (experimental data and Equation (24)) for free jump, and Pourabdollah et al. (2019) (Equation (25)) for the submerged jump, respectively:
formula
(22)
formula
(23)
formula
(24)
formula
(25)
It can be seen from Figure 10 that the shear stress coefficient (ɛ) generally increases with the increasing inlet Froude number (Fr1). The value of ɛ of the free and submerged jumps on the macroroughnesses (TR, OR and SR) is more than the smooth bed. The highest shear stress occurs in a triangular macroroughness (TR) with T/I = 0.50 for both the free and the submerged jumps compared to other ratios (T/I = 0.33, 0.25 and 0.20) and roughnesses (OR and SR). Also, the value of the shear stress coefficient in the submerged jump was more for the smooth bed and less for the free jump.
Figure 10

Variation of shear force coefficient (ɛ) versus different inlet Froude number (Fr1).

Figure 10

Variation of shear force coefficient (ɛ) versus different inlet Froude number (Fr1).

In general, based on the results drawn from this study, the following equations for the shear stress coefficient in macroroughnesses (TR, OR and SR) for all ratios and roughnesses with a good correlation coefficient were obtained for the free and submerged jumps, respectively:
formula
(26)
formula
(27)

Sequent depth ratio and submerged depth ratio

Figure 11 shows the values of (y2/y1) versus (Fr1) and a comparison between numerical data and experimental results obtained from Tokyay (2005) in the free jump is performed. As can be seen, the sequent depth ratio (y2/y1), which somehow represents the height of the jump, is directly related to the changes in the inlet Froude number (Fr1) and the distance of roughness element, and by increasing these parameters, the value (y2/y1) is increased. In other words, the reduction of the boundary layer thickness will further increase the effect of roughness and intensify the reduction of the sequent depth ratio. Also, this slight decrease can be attributed to the increase in the flow separation and recirculation vortex moving between the roughnesses, which increases with increasing the inlet Froude number. The greatest reduction of sequent depth ratio (y2/y1) occurs in triangular macroroughnesses for T/I = 0.5 compared to other shapes and ratios.

Figure 11

Sequent depth ratio (y2/y1) versus (Fr1).

Figure 11

Sequent depth ratio (y2/y1) versus (Fr1).

One of the important characteristics of the submerged jump is the submergence depth on the gate (y3) which is needed to calculate the initial energy of the submerged jump. Therefore, the submerged depth ratio (y3/y1), as well as the tailwater depth ratio of the submerged jump (y4/y1) versus the inlet Froude number (Fr1), are shown in Figure 12. According to Figure 12, by increasing the inlet Froude number, both parameters (y3/y1) and (y4/y1) are increased for the different distances between the roughnesses. The results of the present study have good agreement with previous research by Ahmed et al. (2014). The results showed that the reduction in the values of y3/y1 and y4/y1 depends mainly on the space of the roughnesses, in which the vortex flow forms. For this reason, for T/I = 0.5, the values of y3/y1 and y4/y1 decrease by about 20.87% and 23.34% as a mean, respectively. The values of the ratio of these depths in the macroroughnesses are always less than in the smooth bed. The following equations describe the relationship of the sequent depth, submerged depth and tailwater depth ratios versus Froude number in macroroughnesses with all ratios for free and submerged jumps, respectively:
formula
(28)
formula
(29)
formula
(30)
Figure 12

Submerged depth ratio (y3/y1) and tailwater depth ratio (y4/y1) versus Fr1.

Figure 12

Submerged depth ratio (y3/y1) and tailwater depth ratio (y4/y1) versus Fr1.

The relative length of jumps

Figure 13 shows the dimensionless relative length of the jump (/) versus inlet Froude numbers (Fr1) for ratios T/I = 0.2 and 0.5 with different roughness shapes and smooth bed. It shows that in all modes, the length of the jump for the macroroughnesses is less than the smooth bed and for the submerged jump larger than the free jump. This finding agrees with the results obtained by Tokyay et al. (2011). Also, the value of Lj/y2* for T/I = 0.2 is greater than T/I = 0.5. Apart from some cases related to specific Froude numbers, the results show that the relative length of jump generally increases with increasing inlet Froude numbers, particularly for the case of the submerged jump. By comparing the types of roughness shapes, the triangular macroroughnesses have a significant effect on the length of the jump and show the shortest length with respect to other shapes. In Figure 14, the ratio length of the jump (Lj/y1) for triangular macroroughnesses was plotted at 0.2 < T/I < 0.5 versus different inlet Froude numbers (Fr1). It can be seen that with increasing distance between roughnesses, the length of the jump for both jumps increases and acts as a smooth bed. For T/I = 0.5, the ratio length of the jump decreases by about 25.52% as a mean. Therefore, it is recommended that to reduce the length of the jump to achieve the shortest length of the stilling basins, the triangular macroroughnesses with near-roughness elements (T/I = 0.5) can be used in the stilling basins. The following equations show the relationships of the ratio length of jumps versus Froude numbers that were obtained for the free and submerged jumps, respectively, and for macroroughnesses (TR, OR and SR) and all ratios:
formula
(31)
formula
(32)
Figure 13

Variation of / versus Fr1 for different roughness shape.

Figure 13

Variation of / versus Fr1 for different roughness shape.

Figure 14

Variation of Lj/y1 versus Fr1 for triangular macroroughness.

Figure 14

Variation of Lj/y1 versus Fr1 for triangular macroroughness.

Energy loss

The relative energy loss of free and submerged jumps can be calculated as follows (Pourabdollah et al. 2019):
formula
(33)
formula
(34)
where E1, E2, E3 and E4 are specific energies upstream and downstream of the free and submerged jumps, respectively (see Figure 1). Figure 15 shows the relative energy loss of free and submerged jumps versus inlet Froude number (Fr1). It can be seen that the dimensionless ratio ΔE/E1 increases with the increasing the inlet Froude number (Fr1). Figure 15 indicates that the energy loss of the free jump was greater than that of the submerged jump. Also, for the same inlet Froude number, the energy loss on macroroughnesses (TR, OR, SR) is greater than that of the smooth bed. The results of the energy loss compared with previous experimental data. In particular, these findings are in acceptable agreement with the results of Pourabdollah et al. (2019). By increasing the distance between the roughnesses and inlet Froude number, the energy loss decreases for all the macroroughnesses (TR, OR, SR) in both jumps. The highest ΔE/E1 occurs in a triangular macroroughness (TR) with T/I = 0.50 in the free and submerged jumps compared to other ratios and modes. In general, for all macroroughnesses based on the numerical data of the present study, the relationships provided for the free and submerged jumps are expressed as:
formula
(35)
formula
(36)
Figure 15

The energy loss (ΔE/E1) of free and submerged jumps versus inlet Froude number (Fr1).

Figure 15

The energy loss (ΔE/E1) of free and submerged jumps versus inlet Froude number (Fr1).

CONCLUSIONS

In this study, the characteristics of free and submerged hydraulic jumps including horizontal velocity distribution, bed shear stress, sequent and submerged depth ratio, hydraulic jump length and energy loss over the different shapes of macroroughnesses were numerically investigated using FLOW-3D model. To simulate the flow turbulence at hydraulic jumps, the RNG kɛ turbulence model was chosen. The results of this study are briefly summarized as follows:

  • The maximum MSE and MARE errors between numerical results and experimental data of Ahmed et al. (2014) on a smooth bed are 4.21% and 0.188, respectively, that confirms the ability of the numerical model to predict the specifications of free and submerged jumps.

  • The relative maximum horizontal velocity in the macroroughnesses is less than in the smooth bed and the effect of roughness plays a role in the reduction of this parameter and is greater in the submerged jump.

  • The thickness of the boundary layer decreases with increasing distance between the roughnesses and increases with increasing the distance from the beginning of the hydraulic jump.

  • The reduction of the boundary layer thickness will further increase the effect of roughness and intensify the reduction of the sequent depth ratio. The reduction of this ratio (y2/y1) for the triangular macroroughnesses is greater than the square and semi-oval one.

  • The ratio length of the jump for T/I = 0.5 of rough bed decreases by about 25.52% than a smooth bed and increases with increasing distance between roughnesses.

  • By comparing the types of roughness shapes, the triangular macroroughnesses have a significant effect on the length of the jump and lead to the shortest length with respect to the other shapes.

  • The reduction in the submerged depth ratio (y3/y1) and the tailwater depth ratio (y4/y1) depends mainly on the space between the macroroughness, in which the vortex flow forms. For T/I = 0.5, the values of (y3/y1) and (y4/y1) decrease by about 20.87 and 23.34% as a mean, respectively. The values of the ratio of these depths in the macroroughness are always less than in the smooth bed.

  • The value of shear stress coefficient (ɛ), energy loss, the submerged depth ratio (y3/y1), the tailwater depth ratio and, generally, the relative length of jump in free and submerged jumps (y2/y1) or (y4/y1) increases with the increasing inlet Froude number (Fr1). The highest shear stress and energy loss in the free and submerged jumps occur in a triangular macroroughness (TR) with T/I = 0.50 compared to other ratios and modes.

Overall, CFD models may provide very good predictions of characteristics of free and submerged jumps through different hydraulic conditions and various geometrical arrangements. Velocity field, bed shear stress and specifications of the hydraulic jump can be simulated with a numerical solution. However, the analysis of the roughness height on flow field alteration and TKE as a future work remains still an issue to be faced.

DATA AVAILABILITY STATEMENT

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

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