Characteristics of free and submerged hydraulic jumps over different macroroughnesses

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

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.

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. L_jf and L_js are lengths of the free and submerged jump. Also, d is gate opening, and T and I are height and distance of macroroughnesses, respectively.

A submerged jump is characterized by the supercritical Froude number Fr₁ and the submergence factor S, defined as (y₄ − y₂)/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 (L_js), 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.

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).

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.

The constant values for this model are (Yakhot & Orszag 1986; Samma et al. 2020): C_μ = 0.0845, C_1ɛ= 1.42, C_2ɛ = 1.68, C_3ɛ= 1.0, σ_k = 0.7194, σ_ɛ = 0.7194, η₀ = 4.38 and β = 0.012.

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 (Z_min) 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.

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.

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/r^pGCI₂₁ 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).

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).

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

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.

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.

The dimensionless values (U_m/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 (U_m/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 (U_m/u₁) with the results by Shekari et al. (2014) because this researcher studied only the smooth bed.

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.

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 k–ɛ 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.

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