Examination of Computational Precision versus Modeling Complexity for Open Channel Flow with Hydraulic Jump

In this work, for flow with a hydraulic jump, the predictive capabilities of popular hydraulic models (HEC-RAS and WSPG) are validated with the published results from the three dimensional Computational Fluid Dynamics (CFD) model (OpenFOAM). The analysis is performed for flows with a Froude number of 6.125 and Reynolds number of 3.54 × 105. While the hydraulic models solve the one-dimensional energy equation, in the CFD model solution of the three dimensional Reynolds averaged Navier-Stokes (RANS) equations, with a turbulence model, is used. As the results indicate, although the hydraulic models can satisfactorily predict the location of the steady-state jump, the length of the hydraulic jump (i.e. distance from the toe of the jump to a location in tail water zone) and other jump characteristics are better simulated by the CFD model. The solution from hydraulic models is sensitive to the channel bottom roughness value.


Introduction
Hydraulic jump in open channels occurs when flow transits from supercritical to subcritical. The nondimensional Froude number (F r ) which is the ratio of inertial to gravitational force determines if the flow is supercritical (F r > 1) or subcritical (F r < 1). Flow in this transitional region is turbulent and accompanied by air entrainment coupled with vortex development. In civil engineering applications, a hydraulic jump is created to dissipate the energy in the flow. Any stan- The availability of increased computation resources has given rise to Computational Fluid Dynamics (CFD) software, and these algorithms are providing opportunities for researchers to capture the physics of flow at microscales. CFD tools focus on solving the three dimensional Navier-Stokes equations across varying spatial and time scales. The CFD models are providing new windows of opportunity to better understand the flow in complex situations in engineering and science disciplines. Application of CFD models needs significant amounts of dedicated computational resources. However, since they can better model the flow, the involved computational costs are a fraction of the prototype physical models, which will continue to motivate the practioner audience to use CFD models. The code in each of these models solves a system of equations based upon conservation of mass, energy, and momentum, typically using either finite difference, finite volume or finite element numerical techniques. Popular CFD models include OpenFOAM, FLOW-3D, and TUFLOW.
In this work, we compare the performance characteristics of the two popular hydraulic models HEC-RAS and WSPG with the CFD OpenFOAM, for a flow situation with a hydraulic jump. The outline of this paper is as follows. In the literature review, the focus is on publications that used OpenFOAM for modeling hydraulic jump and other civil engineering applications. All three models have been briefly described. In the review of the CFD OpenFOAM, the focus was laid on its focus on turbulence modeling components. The target test problem and the associated boundary conditions are next detailed. The CFD results of Bayon et al. [1] have been used as reference data for validating the performance of the selected hydraulic models. In the results section, the depth profiles from the three models are analyzed and result from sensitivity analysis on channel bottom roughness coefficient illustrated.

Literature Review
Bayon et al. [1] simulated the salient characteristics in a hydraulic jump (Froude T. V. Hromadka II, P. Rao Journal of Water Resource and Protection number = 6.1) by solving the three-dimensional equations using OpenFOAM.
The turbulence was modeled using three Reynolds-averaged Navier-Stokes (RANS) models: Standard k-epsilon, RNG k-epsilon, and SST k-omega. Various jump characteristics like sequent depths, efficiency, roller length, free surface profile, were compared with published studies. Bayon et al. [2] used OpenFOAM to model the characteristics of a hydraulic jump and compared the model results with their experimental data. While using experimental data across a rectangular channel in a laboratory setting has been a common practice, they conducted experiments on a prototype channel to simulate diversion of flow due to the construction of high-speed rail infrastructure. Their prototype channel was configured to simulate a combination of curved transition, stilling basin, weir, and a stabilization reach. The air-water interface was defined using a Eulerian volume method. Using their experimental data as a benchmark, they compared the surface profile among different turbulence models at the target stilling basin. the recirculation area of the jump. The authors concluded that the k-ε turbulence model is not appropriate for solving flows with jumps, and they recommended using other turbulence models like k-ω. Romagnoli et al. [6] modeled the hydraulic jump created by a sluice gate by solving the RANS equations with interFOAM solver. Their numerical values for the streamwise velocity and turbulent energy are close to the experimental data.
Martins et al. [7] simulated both using OpenFOAM and experimentally, hydraulic jump in a gully located in a rectangular channel. Their focus was to analyze the structural part of the gully, as the standard drainage models use simplified models. The results provide a complete three-dimensional insight into the hydraulic behavior of the flow inside the gully. Lopes et al. [8] investigated the abil-

Numerical Models
The salient characteristics of the three numerical models considered in this study are listed below.

OpenFOAM
Open Solving these require using a turbulence model. The standard k-ε [16], k-ω [17] and the k-ω Shear Stress Transport (SST) model [18] have been widely used. The equations in the k-ε model can be written as [1] ( ) ( ) where k is turbulent kinetic energy, ε is the dissipation rate of k, t is time, ρ is density, 1 x is the coordinate in the i axis, µ is dynamic viscosity, t µ is turbulent dynamic viscosity, k P is the production of turbulent kinetic energy, b P is the buoyancy effect, M Y is the dilatation effect, and K S and S ε are the moduli of mean rate-of-strain tensor. In Direct numerical simulation (DNS), the full Navier-Stokes equations are solved by resolving the whole range of spatial and temporal scales of turbulence.
Although DNS appears to be the preferred approach, the limiting factor in using this is the expensive computing cost. This is due to the small mesh size that is required for capturing turbulence which occurs at varying spatial scales coupled with the use of higher-order accurate numerical techniques. The dnsFoamsolver facilitates DNS simulations in OpenFOAM.

HEC-RAS
HEC-River Analysis System (RAS) facilitates one-dimensional steady flow, one and two-dimensional unsteady flow, sediment transport/mobile bed computations, and water temperature/water quality modeling [20]. The one-dimensional river analysis components are steady flow water surface profile computations and unsteady flow simulation. In steady-state mode, the energy equation between successive cross-sections is solved. Energy losses are evaluated using friction and contraction/expansion coefficients. The momentum equation may be used in situations where the water surface profile is rapidly varied, as in the present case. HEC-RAS has been widely used across various hydraulic simulations, and its results often act as a benchmark for other models.

Water Surface Pressure Gradient (WSPG) model is perhaps the first numerical model that was developed by the Los Angeles County Department of Public
Works. It solves the Bernoulli energy equation between any two cross-sections, using the standard step method [21]. The program computes uniform and non-uniform steady flow water surface profiles. As part of the solution, it can compute the jump characteristics.

Application
As mentioned earlier, we have used the results of Bayon et al. [1] as the benchmark data for validating the hydraulic models. While we refer readers (for a detailed Journal of Water Resource and Protection understanding of the equations, modeling approach, parameters, results) to their work, some salient aspects of their effort that will suffice for our current model comparison effort are summarized below. They developed a three-dimensional computational fluid dynamics model (using OpenFOAM) for analyzing hydraulic jumps in a horizontal smooth rectangular prismatic channel. The Reynolds-averaged Navier-Stokes (RANS) equations were solved. The tested turbulence models were the Standard k-ε, RNG k-ε, and SST k-ω. Sensitivity tests by varying the mesh size, turbulence parameters, and boundary condition location were conducted. Four different mesh sizes ranging from 7:00-8:75 mm, were used. These mesh sizes resulted in total cells ranging across 3.47 to 6.51 million in the computational domain. Since they assumed the channel to be smooth, their model did not consider the channel bottom roughness coefficient. The variables that they studied include sequent depths, efficiency, roller length, free surface profile, and the turbulence model accuracy.

Boundary Conditions
The boundary conditions used in the model are stated below  Figure 1 is a plot of the stationary depth profile along the length of the channel for the three subject models. In the hydraulic models, the channel length was divided into 100 cross-sections, and a manning's roughness coefficient of 0.013 T. V. Hromadka II, P. Rao Journal of Water Resource and Protection was used. While for both the HEC-RAS and WSPG models, the jump forms across two adjacent nodes, the result of the CFD model is more in agreement with the reported experimental data, where the jump forms over an elongated reach. Although all the three turbulence models performed well, RNG k-ε solution was identified as more accurate, and hence we chose it for validating the hydraulic model's output. As shown by Bayon et al. [1], the CFD model can capture additional details of the flow that occur along the other two dimensions.

Results
These include bubble breakup and coalescence, fluid mixing, free-surface turbulent interactions, surface wave formation and breaking processes. Although the considered hydraulic models cannot capture these details, the location of the jump which is an important variable in design computations is reasonably predicted. The sensitivity of the end solution to the roughness coefficient for the RAS and WSPG modela is shown in Figure 2 and Figure 3.

Conclusion
The article compared the predicted steady-state flow profile of a hydraulic jump.
The solution from two one-dimensional hydraulic models was compared with a published benchmark outcome, produced from a three dimensional CFD model.
The CFD model solved the three-dimensional RANS equations using computational model OpenFOAM. The hydraulic models solved the standard Bernoulli's energy equation. Based on the outcomes from all three models, it can be concluded that 1) the hydraulic model depth profiles are similar to the CFD outcome 2) the hydraulic models fail to adequately predict the length of the jump and 3) the solution from hydraulic models is sensitive to channel roughness value.

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.