## Abstract

The hydraulic jump is a phenomenon that occurs in open channels. In past studies, hydraulic jumps over smooth and macrorough beds have been investigated to enhance energy dissipation, but triangular macroroughness, specifically the right-angled triangular macroroughness, has not been dealt with. The objective of this article is to numerically investigate submerged hydraulic jumps over right angle and isosceles triangular macrorough beds. To achieve this, a numerical model based on computational fluid dynamics (CFD) has been utilized. Numerically obtained jump characteristics such as submerged depth ratio, tailwater depth ratio, longitudinal velocity profile, flow pattern in the cavity region, and energy dissipation have been presented in detail. In particular, initial Froude number reduction in both tailwater and submerged depth ratios as well as an increase in the energy dissipation of submerged hydraulic jumps have been noticed on isosceles triangular macroroughness with different arrangements, as compared to smooth beds. The present numerical model has been validated with the experimental model, and the mean error between the two for submerged depth and tailwater depth ratios was found to be below 6%. This confirms the adequacy of the present CFD model in predicting relevant submerged hydraulic jump characteristics over macrorough beds.

## HIGHLIGHTS

Isosceles triangular shapes are more commonly used, and right-angled shapes have not been addressed in the past.

Right-angled triangular macroroughness is added as a new shape of triangular configuration.

Different macroroughness arrangements are tested to enhance the jump characteristics, which have not been extensively explored in the past.

## NOTATION

- CFD
computation fluid dynamics

- RANS
Reynolds-averaged Navier–Stokes

- RNG
re-normalization group

- VOF
volume of fluid

*k*turbulent kinetic energy

*ε*dissipation rate of turbulent kinetic energy

*μ*dynamic viscosity

*υ*kinematic viscosity

*g*gravitational acceleration

*v*depth-averaged velocity at a section where jump starts

*u*time-averaged longitudinal velocity at any section

*U*maximum velocity in the horizontal direction at any section

*Q*discharge

*ρ*fluid density

*y*_{0}opening of sluice gate

*y*_{1}initial depth of water in case of submerged hydraulic jump

*y*_{2}tailwater depth of water in case of a free hydraulic jump

*y*_{3}submerged depth of water in case of a submerged hydraulic jump

*y*_{4}tailwater depth of water in case of a submerged hydraulic jump

*H*_{1},*H*_{2}height of macroroughness elements

*B*base width of macroroughness elements

- Fr
_{1} Froude number of the inlet section of the hydraulic jump

*C*_{r}Courant number

*L*_{j}length of submerged hydraulic jump

*E*_{1}specific energy at the upstream end of the jump

*E*_{2}specific energy at the downstream end of the jump

*R*^{2}coefficient of determination

## INTRODUCTION

Jean–Baptiste Belanger developed the Belanger equation in 1838, which relates the pre- and post-jump depths, also known as sequent depths defined in Equation (1). This equation has been used extensively in open-channel hydraulics and has led to the development of various correlations between flow parameters upstream and downstream of the jump.

*et al.*1990). The definition of the length of the jump is ambiguous, where

*L*is the distance between the toe and the point at which the water surface nearly levels out and the mean surface elevation reaches its maximum (Rajaratnam 1967). The ratio of the sequent depths continues smaller than Belanger's equation because of the frictional forces on the wall (Hager & Bremen 1989).where

_{j}*y*

_{1}and

*y*

_{2}are the flow depths at the upstream of the jump and downstream of the jump, respectively, and and are the upstream and downstream Froude numbers, respectively. As the hydraulic jump is a phenomenon with non-deterministic characteristics, the characteristics of it over macroroughness can be investigated by a mathematical approach such as the stochastic approach (Guo

*et al.*2018; Mousavi

*et al.*2020). In addition, the dissolved and sediment fraction in hydraulic jump can be detected using the approach discussed by Lin

*et al.*(2018). Further to tackle the multi-type uncertainty in the optimization of macroroughness in the case of hydraulic jump may use the concept of decision-making, combination, and prediction approach (Chen

*et al.*2018; Li

*et al.*2018). The past study researched this phenomenon in depth utilizing analytical, experimental, and numerical simulation approaches, which are discussed as follows:

Hughes & Flack (1984) created conceptual correlations by investigating hydraulic jumps in a horizontal rectangular flume over an artificial rough bed with smooth side walls. Hager & Bretz (1986) investigated hydraulic jumps across positive and negative steps in a rectangular prismatic channel. Ead & Rajaratnam (2002) carried out experiments on the smooth and sinusoidal corrugated beds, and investigated the characteristics of a free hydraulic jump. Elsebaie & Shabayek (2010) investigated hydraulic jump in a rectangular channel with uneven and smooth beds, with a side slope of 45° for two trapezoidal and triangular macroroughness and 60° for other trapezoidal. Afzal *et al.* (2011) performed hydraulic jumps in a rectangular channel over an uneven bed. Tokyay & Velioğlu (2012) researched hydraulic jumps across smooth and rough beds. Abbaspour *et al.* (2013) studied the hydraulic jump over just a sinusoidal corrugated bed in the rectangular channel using artificial neural networks and genetic programming. Samadi-Boroujeni *et al.* (2013) examined hydraulic jump in a rectangular channel with six smooth and six rough beds. Ahmed *et al.* (2014) experimentally explored the influence of spaced triangular strip corrugated beds on downstream submerged jump characteristics. Bayon-Barrachina & Lopez-Jimenez (2015) numerically modelled the free hydraulic jump in a rectangular channel with a smooth bed using Open FOAM. Sauida (2016) examined the submerged hydraulic jump downstream of multi-vent regulators under various gate scenarios. Hafnaoui *et al.* (2016) investigated hydraulic jump in a prismatic inclined channel with rectangular and triangular corrugation. Abbaspour *et al.* (2016) laid the plates at an angle of 50 and 90° at various distances from the apron in open channels with horizontal and reverse bedding to investigate the energy dissipation using a hydraulic jump. Eshkou *et al.* (2018) executed forced hydraulic jumps in a gradually diverging stilling basin Type III with baffle blocks of varying angles. Felder & Chanson (2018) performed hydraulic jumps in rectangular channels with smooth bed and micro-roughness. Al-Hashimi *et al.* (2019) used the computational fluid dynamics (CFD) approach with ANSYS Fluent to numerically model the flow over submerged weir. Kumar *et al.* (2019) tested hydraulic jumps across three roughness heights of crushed and rounded aggregates, as well as two positive bed slopes. Yildiz *et al.* (2020) numerically investigated both free and submerged hydraulic jumps using ANSYS Fluent. Macián-Pérez *et al.* (2020b) attempted using Open FOAM and FLOW-3D to evaluate a traditional hydraulic jump using CFD. Mahtabi *et al.* (2020) examined free and submerged hydraulic jumps across natural and artificial roughness using a decision tree method and a multi-layer neural network. Ghaderi *et al.* (2020) performed numerical simulation for free and submerged jumps in a rectangular channel with triangular, square, and semi-oval macroroughness. Nikmehr & Aminpour (2020) used trapezoidal roughness at the bed to investigate the hydraulic jump characteristics in a rectangular flume. Maleki & Fiorotto (2021) used an analytical approach to investigate the hydraulic jump stilling basins over an uneven substrate using experimental data. Dasineh *et al.* (2021) explored the free and forced hydraulic jump in a rectangular channel on the triangular platform in different configurations and compared it to the prediction of this numerical model using artificial intelligence approaches. Ghaderi *et al.* (2021) investigated the effects of free and forced hydraulic jumps in a rectangular channel on triangular macroroughness.

Based on the aforementioned literature, it seems that the cavity region between two successive elements type energy dissipators was effective for energy dissipation, and hence, many past studies had investigated rough beds specifically in the form of strip corrugation to create a small bucket type structure. The past study considered various shapes of roughness element and found that the triangular shape yields good results. Here, the authors found that the isosceles triangular shape is more commonly used, and the right-angled shape has not been addressed in previous studies. In addition, there is also a need to reinvestigating different macroroughness arrangements such as T_{1}, T_{2}, T_{3}, and T_{4} (Figure 3(b)) to enhance the jump characteristics, which have not been extensively explored in the past studies.

Therefore, the objective of this study is to investigate numerically the effects of different bed macroroughness configurations with their unique arrangements on submerged hydraulic jump characteristics such as tail water and submerged depth ratio, including velocity profiles. By examining these parameters, the study aims to provide further insights into the behaviour of submerged hydraulic jumps over macroroughness.

### Submerged jump

## METHODOLOGY

### Experimental background

An experimental setup for a hydraulic jump typically involves a horizontal rectangular channel with a specific length, width, and depth for a constant flow rate of water through the channel. The flow rate can be controlled using a flow control valve or a pump, while a flowmeter can be used to measure the flow rates. The bed of the channel can be made of different materials and configured in different ways to simulate bed roughness. The channel can also have different inlet and outlet configurations such as a sharp-crested weir to induce the hydraulic jump.

*B*= 0.07 m,

*H*

_{1}= 0.1 m, and

*H*

_{2}= 0.05 m (Figure 3(b)) was laid to the full width of the flume. The length of bed macroroughness remains fixed for all the experimental runs. To investigate the flow field and other parameters of the jump, different arrangements of bed roughness were considered as shown in Figure 3. A total of runs (for each type of bed condition) on the smooth and macrorough bed were performed on a mild bed slope with different Fr

_{1}ranging from 4.0 to 8.5 by varying the discharge and sluice gate opening. Flow velocities were measured with the help of a digital pitot tube at sections 1 and 2 as shown in Figure 1. Details of flume arrangements are presented in Figure 3(a).

### Numerical simulation

Numerical simulation is the process of using computer algorithms to replicate the behaviour of a real-world system or process, typically in the form of mathematical models or equations. It involves breaking down complex phenomena into smaller components that can be mathematically modelled and then using numerical methods to solve these models and generate data that can be used to understand and analyse the behaviour of the system.

Numerical simulation is used in a wide range of fields, including physics, engineering, chemistry, biology, and economics. Some common applications include simulating the behaviour of fluids, predicting weather patterns, modelling the spread of infectious diseases, and designing and testing new products.

The accuracy and reliability of numerical simulations depend on a variety of factors, including the quality of the underlying mathematical models, the accuracy of the numerical methods used to solve these models, and the quality and quantity of the input data used to initialize the simulation. As a result, numerical simulation is an iterative process that often involves refining and improving the models and methods over time.

### Computational fluid dynamics

To conduct a CFD investigation of hydraulic jump characteristics, one needs to model the flow using appropriate numerical methods. Some of the most common CFD models and their simplified equations are Reynolds-averaged Navier–Stokes (RANS) equations, which is a time-averaged equation for the motion of fluids that accounts for the turbulence effects by introducing additional terms that describe the turbulent viscosity and diffusion. Bayon-Barrachina & Lopez-Jimenez (2015); Dasineh *et al.* (2021); Ghaderi *et al.* (2021) used CFD for the use of the sophisticated algorithm volume of fluid (VOF) model for tracking the free surface of the flow. Here, numerical simulations were worked out using ANSYS Fluent 2022 R1, which is a well-known licensed software. Software is based on finite volume methods in the Cartesian form to solve the RANS equation. Two-phase (i.e., gas and liquid) for the present study was considered, where air is one of the forms of gas, and water is one of the forms of liquid.

In the given context, the variables and symbols mentioned earlier are defined as follows:

: Velocity vector in a three-dimensional space, where *i* represents the Cartesian coordinates (*x*, *y*, *z*).

: Position vector in a three-dimensional space, where *i* represents the Cartesian coordinates (*x*, *y*, *z*).

: Time variable. : Pressure of the fluid. *ρ*: Density of the fluid.

: Viscous stress tensor. The subscript *ij* denotes the components of the tensor.

It is a mathematical construct used to describe the internal forces within a fluid caused by its viscosity.

VOF tracking equations:

In the aforementioned equation, *F* is the fraction function that determines whether a cell is empty or filled. It is a function of position and represents the fraction of the cell occupied by a fluid. If a cell is empty (i.e., no fluid is present), then *F* = 0. This indicates that the fraction of the cell volume occupied by the fluid is zero. If a cell is filled (i.e., fluid is present), then *F* = 1 (Canonsburg 2013; Choufu *et al.* 2019). This indicates that the fraction of the cell volume occupied by the fluid is 1, representing a fully filled cell. The purpose of the fraction function *F* is to determine the distribution of fluid and the location of the free surface within a computational domain. By assigning different values to *F*, it is possible to determine the intermediate amounts of fluid present in a cell, allowing for the determination of the free surface position.

Typically, the free surface is determined at a position where the fraction function has an intermediate value, often *F* = 0.5. This means that at the free surface, half of the cell volume is occupied by the fluid and the other half is empty. However, it is worth noting that the choice of the specific intermediate amount for determining the free surface position can vary depending on the specific application or user preferences.

### Geometry and mesh

Geometry was prepared by keeping the experimental flume configuration in mind to avoid any type of mismatch (Figure 7(a)). In the next steps, mesh analysis was carried out in a meshing setup, where proper names were provided to different faces. In addition, inflation layers at the bed of the flume were provided to enhance the flow behaviour.

### Turbulence model

ANSYS provides various turbulence models that can be used for simulating turbulent fluid flows. Turbulence models are mathematical formulations that simulate the effects of turbulence on fluid flow, such as eddies and vortices. The most commonly used turbulence models in ANSYS are the RANS models, which assume that the turbulence is statistically steady and can be averaged over time. These models include the k-ε model, the Spalart–Allmaras model, and the Reynolds Stress model, among others.

The software provides the facility of different turbulent models, but the exact and appropriate model depends on the phenomenon of the study. The k-*ε* model is one of the most widely used RANS models and is based on the assumption that the turbulent kinetic energy (TKE) (*k*) and its dissipation rate (*ε*) are the two main variables that determine the turbulence characteristics of the flow. Bayon-Barrachina & Lopez-Jimenez (2015) investigated the efficiency of re-normalization group (RNG) *k*-*ε*, standard *k*-*ε*, and shear stress transport (SST) *k*-*ω* turbulent models for the study of hydraulic jump over a smooth bed and stated that the most accurate model for hydraulic jump analysis is RNG *k*-*ε* followed by SST *k*-*ω* and standard *k*-*ε*. Ghaderi *et al.* (2021) said that RNG *k*-*ε* is similar to the standard *k*-*ε* model, even if it contains some refinement. Nikmehr & Aminpour (2020) proved that the RNG *k*-*ε* model efficiently predicts the hydraulic jump over a corrugated bed also. The previous studies (Salmasi & Samadi 2018; Hien & Duc 2020; Macián-Pérez *et al.* 2020a; Yildiz *et al.* 2020; Yamini *et al.* 2021) employed the turbulent *k*-*ε* model in their research. The RNG *k*-*ε* approach uses statistical techniques to obtain the averaged equations for two turbulence quantities: TKE (*k*) and its dissipation rate (*ε*). One benefit of this model is that, compared to the standard k-*ε*, it typically yields superior results when simulating whirling flows (Canonsburg 2013; Bayon-Barrachina & Lopez-Jimenez 2015). On the basis of the past pieces of literature, the authors attempted to employ a two-equation RNG *k*-*ε* turbulence model for the CFD technique to investigate the flow field of the submerged hydraulic jump throughout the analysis.

The variables and symbols in the aforementioned equation are defined as follows:

*k*: TKE, which represents the energy associated with the turbulent fluctuations in the fluid flow. It is a measure of the intensity of turbulence in the flow.*ε*: dissipation rate of TKE. It represents the rate at which turbulent energy is converted into heat through viscous dissipation.*t*: time variable.*ρ*: density of the fluid.: coordinate in the

*i*-axis of a Cartesian reference system.*μ*: dynamic viscosity of the fluid.: turbulent eddy dynamic viscosity. It is a model parameter that represents the effective viscosity associated with turbulence effects.

: production term for TKE. It represents the rate at which TKE is generated within the flow due to the action of turbulent stresses.

: buoyancy effect term. It represents the influence of buoyancy forces on the turbulent flow.

: dilatation oscillation effect term. It accounts for the effects of compression and expansion of fluid elements on the turbulence.

and : source terms for TKE (

*k*) and dissipation rate of TKE (*ε*), respectively. These terms represent additional contributions to the rate of change of*k*and*ε*in the turbulence model.: model parameters in the RNG

*k*-*ε*turbulence model. These parameters have specific values assigned to them to govern the behaviour and accuracy of the turbulence model. In the RNG*k*-*ε*model, the specific values for these parameters were 0.084, 1.42, 1.68, 1.0, 0.7194, and 0.7194, respectively. These variables and symbols were used in the RNG*k*-*ε*turbulence model (Canonsburg 2013), which was a specific turbulence model within CFD simulations. The model equations and terms involving these variables were utilized to describe the behaviour and evolution of turbulence in the fluid flow, providing insights into turbulence-related phenomena and their impact on the overall flow characteristics.

### Boundary conditions

In a hydraulic jump simulation, boundary conditions were used to define the flow conditions at the inlet and outlet of the computational domain. The boundary condition is essential in hydraulic jump analysis as the flume has different surfaces like an inlet, outlet, wall, and top. Choosing the appropriate boundary conditions depends on the specific hydraulic jump being simulated and the level of accuracy required. It is important to carefully select the boundary conditions to ensure that the simulation accurately represents the real-world hydraulic jump.

Hydraulic jump is known to be an open-channel flow, and for this, ANSYS Fluent has provided a toggle key for open-channel flow, which had been chosen for the simulation purpose in the present study. To simulate the hydraulic jump numerically, the options available in software as ‘pressure inlet’ and ‘pressure outlet’ were chosen for flume configuration. The wall was treated as a no-slip wall, which assumes that the flow velocity at the wall is zero and a wall roughness constant was taken equal to 0.5. Initial conditions at the inlet and outlet were set as similar to experimental conditions.

### Near-wall treatment

A critical part of CFD is near-wall treatment. It involves precisely modelling and resolving the flow behaviour near solid boundaries, as flow characteristics in this region can have a substantial impact on the overall simulation results. Wall functions are empirical correlations that estimate flow variables along the wall using information from the adjoining cell. By avoiding the resolution of the near-wall region, these functions reduce computational effort. It is important to create a mesh capable of precisely predicting the velocity gradient over the boundary layer. These criteria cannot be met for turbulent flows in complex geometry since it would necessitate an extremely fine mesh resolution near the wall, which would significantly increase the time necessary to solve the problem. The mesh should not be over-refined because the wall function resolves the flow near the wall, which may result in less accurate results (Bayon-Barrachina & Lopez-Jimenez 2015). To address this requirement, a wall function was developed that enables the usage of a ‘relatively’ larger mesh near the wall.

The non-dimensional distance between the wall and the first node away from the wall is given by *y*^{+}, which is an essential aspect of wall functions (Canonsburg 2013). The turbulent boundary layer is categorised into three sections based on *y*^{+}: laminar sub-layer (*y*^{+} < 5), transition layer (5 < *y*^{+} < 30), and turbulent (*y*^{+} > 30). Near-wall treatment options in ANSYS Fluent for the *k*-*ε* turbulence model incorporate normal wall function, non-equilibrium wall function, enhanced wall treatment, and others. The present study used the standard wall function as it was the most commonly used wall function in ANSYS Fluent because it provides reasonably accurate predictions for the majority of high Reynolds numbers, wall-bounded flows and it lowers computational effort by eliminating the resolution of the near-wall region. The dimensionless *y*^{+} value for all the grid systems fell between 46 and 113, which supported the need for the wall function technique. The range of inlet Reynolds numbers was 31,654–52,861.

### Discretization scheme

ANSYS Fluent uses various discretization schemes to solve the governing equations of fluid flow numerically. These discretization schemes are used to transform the partial differential equations (PDEs) that describe the fluid flow into a set of algebraic equations that can be solved using a computer. The most commonly used discretization schemes in ANSYS Fluent are finite volume and finite element methods. Finite volume methods are used to discretize the governing equations on a mesh of control volumes, while finite element methods are used to discretize the equations on a mesh of finite elements. In finite volume methods, the control volume is a small, discrete region of space that encloses a small portion of the fluid being simulated. The PDEs are then integrated over the control volume and approximated using numerical methods to obtain an algebraic equation for each control volume. These equations are then assembled into a global matrix equation that can be solved using numerical methods.

ANSYS Fluent also offers different discretization schemes for the spatial and temporal derivatives of the governing equations. For example, for the spatial derivatives, Fluent offers first-order upwind, second-order upwind, and second-order central differencing schemes. These schemes differ in the way they approximate the spatial derivatives, and they have different levels of accuracy and stability. Choosing the appropriate discretization schemes depends on the specific fluid flow being simulated and the desired level of accuracy and computational cost. It is important to carefully select the discretization schemes to ensure that the simulation accurately represents the real-world fluid flow.

Discretization is more influencing parameters in the case of numerical simulation to obtain the desired results. Bayon-Barrachina & Lopez-Jimenez (2015) stated that the RANS model, upwind schemes have less instability, and hence, the present study used the upwind schemes. The Courant number is a dimensionless parameter used in numerical simulations that represents the ratio of the time step to the distance between computational grid points. It was used to determine the stability of the simulation, as a Courant number that is too high can lead to numerical instability. In the present case of the submerged hydraulic jump simulation, the maximum Courant number used was 0.45, which was within the stable range for this type of simulation. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) pressure–velocity coupling scheme was used, which was a widely used algorithm for solving fluid flow problems. The density and momentum equations were discretized using second-order upwind and first-order upwind schemes, respectively.

### Stability and convergence criteria

Stability and convergence are critical factors in CFD modelling as they ensure the accuracy and reliability of the simulation results. Stability criteria in CFD modelling refer to the ability of the numerical solution to remain bounded as the simulation progresses. Convergence criteria refer to the ability of the numerical solution to converge to a solution that accurately represents the real-world fluid flow. In CFD modelling, the solution is considered converged when the numerical solution changes negligibly with each iteration. Convergence can be monitored by examining the residuals of the governing equations, which represent the error between the numerical solution and the exact solution. The most common convergence criterion is based on the magnitude of the residuals. When the residuals drop below a specified tolerance, the solution is considered converged.

For the present study to examine the convergence criteria, residuals were set to 10^{−6}. The stability of the numerical model for the hydraulic jump was quite difficult as the hydraulic jump was abrupt in nature. Here, the stability of the jump was achieved by the hit and trial method – first, by measuring the discharge equilibrium curve for the inlet and outlet, and second, by knowing the inlet and outlet pressure distribution. The stable conditions of the jump were noticed after 10 s of time steps.

### Mesh sensitivity

Mesh sensitivity analysis is a statistical technique used to evaluate the sensitivity of a model output to variations in input parameters. It involves varying one or more input parameters over a predefined range of values, running the model for each set of input parameters, and analysing the resulting output to determine which input parameter has the most significant impact on the output. Mesh sensitivity analysis typically involves creating a grid of input parameter values and running the model for each combination of values in the grid. The output from each model run is then analysed using statistical methods to determine the sensitivity of the output to each input parameter.

*et al.*(2008) and second on the tailwater depth ratio as adopted by Bayon-Barrachina & Lopez-Jimenez (2015). Five mesh sizes of coarse to fine variety were considered. In the initial stage of the numerical simulation, a coarse mesh size of 50 mm was selected to plot the longitudinal velocity profile of submerged hydraulic jump at

*x*= 1.33 m in the direction of flow. Numerically plotted data were compared with the experimental measured data, shown in in Figure 4. From Figure 4, it can be inferred that the 50 mm grid size is not sufficient to obtain the desired result, and hence, finer mesh sizes such as 30, 10, 7, and 5 mm were tested to obtain the desired result. Mesh sensitivity analysis result reflected that 7 and 5 mm mesh size has almost same data points, and hence, further reduction in the mesh size was not executed. The maximum percentage error of 5.35% was reported by comparing the numerically obtained longitudinal velocity profile with experimental findings. Similarly, the numerically obtained the tailwater depth ratio (

*y*

_{4}/

*y*

_{1}) for Fr

_{1}= 4.6 is compared with the experimental findings and achieved the least percentage error of 1.41% at 7 mm mesh size, which is tabulated in Table 1. Therefore, to avoid the computational cost, 7 mm mesh size was used throughout the simulation work.

Mesh . | Mesh size (mm) . | . | . | Percentage error . |
---|---|---|---|---|

1 | 50 | 6.40 | 5.97 | 6.72 |

2 | 30 | 6.40 | 6.06 | 5.32 |

3 | 10 | 6.40 | 6.14 | 4.07 |

4 | 07 | 6.40 | 6.31 | 1.41 |

5 | 05 | 6.40 | 6.31 | 1.41 |

Mesh . | Mesh size (mm) . | . | . | Percentage error . |
---|---|---|---|---|

1 | 50 | 6.40 | 5.97 | 6.72 |

2 | 30 | 6.40 | 6.06 | 5.32 |

3 | 10 | 6.40 | 6.14 | 4.07 |

4 | 07 | 6.40 | 6.31 | 1.41 |

5 | 05 | 6.40 | 6.31 | 1.41 |

## RESULTS AND DISCUSSION

### Validation of numerical simulation

Validation of numerical simulation refers to the process of determining whether the results of a computer simulation accurately represent the real-world phenomena being studied. It involves comparing the results of the simulation to experimental or observational data to assess the accuracy and reliability of the simulation. Accuracy and reliability of the simulation can involve quantifying the differences between the simulation results and the experimental or observational data and identifying areas where the simulation may need to be improved.

*y*

_{3}/

*y*

_{1}) and tail water depth ratio (

*y*

_{4}/

*y*

_{1}) are presented in Figure 5, which indicates a good agreement between the two models, with a mean error less than 6%. This level of agreement confirms the capability of CFD modelling to accurately simulate complex fluid dynamics problems such as submerged hydraulic jump. The derived equations for smooth and macrorough beds are presented in Equations (8)–(11):

### Velocity distribution

In a hydraulic jump, the velocity distribution changes abruptly from a supercritical flow with high velocity to a subcritical flow with lower velocity. The velocity distribution in a hydraulic jump is complex and varies depending on the specific conditions of the flow. In the supercritical flow region upstream of the jump, the velocity is relatively constant and high. At the jump location, the velocity decreases rapidly to a subcritical flow region, where the velocity varies significantly across the depth of the flow. In the subcritical flow region downstream of the jump, the velocity is lower and varies with the depth, with the lowest velocities occurring near the bottom of the flow. This phenomenon is commonly referred to as the ‘recovering zone’ (Figure 1), where the fluid gradually recovers its downstream velocity profile after the hydraulic jump. The recovering zone is characterized by a gradually decreasing flow depth and a gradual increase in velocity as the fluid moves downstream. The extent of the recovering zone depends on several factors such as the initial Froude number (Fr_{1}), bed roughness, and channel geometry. The recovering zone is an important aspect to consider in hydraulic engineering design as it affects the downstream flow conditions and energy dissipation.

*y*ʹ is the vertical distance from the bed level where maximum velocity can be found as

*U*and at a vertical distance of

*b*, and velocity reduces to almost half of the maximum velocity. This is a typical velocity variation in a fully formed roller zone, just above point

*b*velocity stagnation takes place and reverse flow occurs.

Typical pictorial representations of numerically obtained longitudinal velocity profile of submerged hydraulic jump on smooth and macrorough beds are presented as follows.

*et al.*(2021). In Figure 8, the graphical velocity distribution is shown for different sections within the flume, which also confirms the earlier description.

_{1}= 4.6. Here, the vertical axis represents the ratio of

*y*/

*b*, where

*y*is the distance from the bed and

*b*is the length scale at which velocity if half of the maximum velocity. The horizontal axis represents the ratio of

*u*/

*U*, where

*u*is the time-averaged longitudinal velocity at any section and

*U*is the maximum longitudinal velocity at any section.

The dimensionless velocity distribution graph is typically plotted for the point where the velocity is only half the maximum velocity, which is at a distance of *b* from the jump toe. This is because the velocity distribution is relatively flat in the downstream region and does not show significant variation. To present the dimensionless velocity distribution, additional calculations were performed to normalize the velocity values based on the mean velocity and the height of the jump. For the validation purpose, past studies (Rajaratnam 1967; Jesudhas *et al.* 2018; Ghaderi *et al.* 2021)were taken as references and presented in contrast with present data in Figure 9, and it was found that the trend of dimensionless velocity profiles is more or less similar to the past research and found that the maximum horizontal forward velocity occurred at lesser flow depth in comparison to studies by Rajaratnam (1967) and Jesudhas *et al.* (2018).

### Longitudinal velocity vector variation over rough beds

_{1}and T

_{2}arrangements, whereas a combination of clockwise and anti-clockwise circulation was observed for T

_{3}and T

_{4}arrangements. The formation of a large circulation in the vicinity of the cavity region under the fully developed portion may be the reason for reduction in excess kinetic energy.

In addition, it is interesting to note that the velocity in the cavity region between two successive macroroughness elements was somewhat less than the mean velocity, which suggests that the flow in this region behaves differently from the rest of the flow. Finally, it seems that the presence of the cavity region between two successive elements may affect the behaviour of the flow, especially for higher initial Froude numbers.

### Turbulent kinetic energy

### Energy loss

*et al.*(2020, 2021) found that the energy loss over macroroughness was much higher than the smooth bed. Ghaderi

*et al.*(2020) stated that a free jump dissipates more energy compared to the submerged hydraulic jump over macroroughness. In the present study, kinetic energy at a particular section is calculated using Equation (12), and the relative energy loss through the submerged hydraulic jump is calculated using Equation (13).

*i*’ (i.e.,

*i*= 0, 1, 2, 3, 4). represents relative energy loss through submerged hydraulic jump, whereas and are the specific energies calculated at the upstream and downstream ends of the submerged hydraulic jump respectively. Figure 12 depicts the variation of energy loss over different channel bed conditions.

## CONCLUSION

Overall, the research aimed to enhance the understanding of submerged hydraulic jumps by exploring various flow characteristics such as tailwater depth ratio, submerged depth ratio, including longitudinal velocity profile, and flow pattern in the cavity region over different shapes of macroroughness with their unique arrangements. Based on the numerical investigation, the key findings of the present study are presented as follows:

- 1.
ANSYS Fluent, popular CFD software, is capable of predicting complex phenomena such as submerged hydraulic jump over smooth and macrorough beds.

- 2.
The trend of velocity profiles obtained for submerged hydraulic jumps over macrorough beds is in good agreement with past studies.

- 3.
It is reported that the cavity region between two successive macrorough elements leads to a decrease in both the tailwater and submerged depth ratios, as well as an increase in the energy dissipation of the submerged jump.

- 4.
Clockwise and anti-clockwise circulation of the velocity vector is observed in between the cavity region of macroroughness elements in the case of submerged hydraulic jumps, which may be the reason for the reduction in tailwater depth ratio, submerged depth ratio, and longitudinal velocity.

- 5.
In the cavity region between two successive macroroughness elements, it is found that the velocity of the fluid is quite less than the mean velocity.

- 6.
The right-angle and isosceles triangular macroroughness have almost similar jump characteristics in the case of a submerged hydraulic jump.

- 7.
By varying the arrangements of macroroughness elements, better energy dissipation can be achieved in the case of a submerged hydraulic jump.

- 8.
For the basic parameters of submerged hydraulic jump, such as tailwater depth ratio and submerged depth ratio, the mean error between numerically derived results and experimental findings is determined to be less than 6%.

As a final remark for future research, optimization of possible configurations with their unique arrangements may be investigated. Furthermore, this numerical study may help hydraulic engineers in designing and modelling of energy dissipators.

## ACKNOWLEDGEMENTS

The authors want to express their sincere gratitude to the anonymous reviewers for contributing their time to this article.

## FUNDING

Delhi Technological University has provided the laboratory for the experiments and the ANSYS 2022 R1 license version for the numerical simulations.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.