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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 × 10
^{5}. 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.

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 standard textbook in fluids can provide relevant background theory on hydraulic jumps.

Over the last three decades, numerical modeling of open channel flows with hydraulic jump has drawn the attention of many researchers. The complexity of the solved flow equations ranged from Bernoulli’s energy equation to three dimensional Navier-Stokes equations. Hydraulic models typically solve either the energy equation or shallow water equations. General-purpose hydraulic models are relatively simple to use and can broadly serve the purpose for many applications. Popular one-dimensional hydraulic models include HEC-RAS, WSPG, DHM, MIKE 11, TELEMAC and SWMM among others. For surface water applications where the flow is predominantly in one direction, these models have been shown to provide a reasonably accurate simulation. These models can be applied for testing various “what if” flow scenarios and are not constrained by the required CPU time.

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. [

Bayon et al. [

At the transition of supercritical and subcritical flows, because of the turbulence, pockets of air are captured in water, and they move in recirculatory motion. Simultaneous bubble breakup and their mergers occur in the turbulent shear section of the recirculating region. Large air bubbles, because of velocity gradients, can experience multiple breakups. If the bubble is big, buoyancy lifts it to the surface, while smaller bubbles remain in the lower portions. The volume of the air bubbles changes continuously. Witt et al. [

Bayon et al. [

Romagnoli et al. [

Egea [

Other investigators who have used OpenFOAM for hydraulic applications include the works of Teuber et al. [

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

Open Field Operation And Manipulation (OpenFOAM) is a free, open-source software for CFD. It has an extensive range of features to solve anything from complex fluid flows involving turbulence and multi-phase [

Flow in a hydraulic jump is turbulent. In turbulent flows, the field properties in the vicinity of the jump are random functions of space and time. A feature of the flows is the presence of small-scale, high-frequency random fluctuation, which is superimposed on the main flow that has a primary flow axis. Although the magnitude of these fluctuations is small, they tend to have a major impact on some of the jump characteristics. In applications, where analyzing the turbulent characteristics of the hydraulic jump are essential, application of three dimensional CFD models is required. OpenFOAM provides a variety of turbulence model options from Reynolds-Averaged Navier-Stokes (RANS) to Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS).

Reynolds averaged Navier-Stokes simulation (RANS) also known as Reynolds averaged simulations have been widely used for solving the time-averaged flow equations. The assumption behind the RANS equations is that the time-dependent turbulent velocity fluctuations in Navier-Stokes equations can be separated from the mean flow velocity. The new set of unknowns called the Reynolds stresses are functions of the velocity fluctuations. Solving these require using a turbulence model. The standard k-ε [

∂ ∂ t ( ρ k ) + ∂ ∂ x i ( ρ k μ i ) = ∂ ∂ x j [ ( μ + μ t σ k ) ∂ k ∂ x j ] + P k + P b − ρ ε − Y M − S K (1)

∂ ∂ t ( ρ ε ) + ∂ ∂ x i ( ρ ε μ i ) = ∂ ∂ x j [ ( μ + μ t σ ε ) ∂ ε ∂ x j ] + C 1 ε ε k ( P k + C 3 ε P b ) − C 2 ε ρ ε 2 k + S ε (2)

where k is turbulent kinetic energy, ε is the dissipation rate of k, t is time, ρ is density, x 1 is the coordinate in the i axis, μ is dynamic viscosity, μ t is turbulent dynamic viscosity, P k is the production of turbulent kinetic energy, P b is the buoyancy effect, Y M is the dilatation effect, and S K and S ε are the moduli of mean rate-of-strain tensor. C μ , C 1 ε , C 2 ε , C 3 ε , σ k and are model parameters. Some of the available RAS turbulence models in OpenFOAM, include kEpsilon, kOmega, SSG, LRR, v2f, and RNGkEpsilon. The RANS based models are a good compromise between the accuracy of the end solution and the computational cost. The requirement of a turbulence model in the RANS equations is a weakness. The turbulence models are typically optimized for specific cases, and may not be the best choice for all hydraulic applications. An extended description of RANS equations and turbulence closures can be found in Pope [

The above limitation in RANS equations has given rise to the Large Eddy Simulation (LES) method as an alternative to the RANS equations. In LES, the turbulent scales under a specific filter length scale are modeled (RANS approach), while for the larger ones Navier Stokes equations are resolved (DNS approach). The computational costs associated with LES are higher when compared to RANS models. Some of the available LES turbulence models in OpenFOAM include Smagorinsky, kOmegaSSTDES, WALE, DeardorffDiffStress and dynamicKEqn.

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-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 [

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 [

As mentioned earlier, we have used the results of Bayon et al. [

The boundary conditions used in the model are stated below

· OpenFOAM: Bayon et al. [^{3}/s. At the upstream end, the authors used a flow depth of 0.07 m, and a velocity profile was imposed using a Dirichlet boundary condition. The pressure profile was hydrostatic. The constants in the RANS turbulence modeled were assigned a low value, and a short stretch of the channel was added so that the flow is well developed before the jump forms. At the downstream end, instead of using the subcritical flow depth of 0.553 m, a velocity profile is imposed, so that hydrostatic pressure profile develops. As long as the mass is conserved in the system, this downstream boundary approach will translate to the required subcritical flow depth. The channel bottom was assumed to be smooth. A no-slip condition is imposed at the walls, and roughness is not considered. At the top of the channel surface, an atmospheric boundary condition is imposed which allows fluids to enter and leave the channel. The density and the kinematic viscosity are ρ = 1000 kg/m^{3} and 10^{−6} m/s^{2}.

· HEC-RAS: At the upstream and downstream ends of the domain, the water surface elevations of 0.07 m and 0.553 m were specified. The flow in the channel was 0.177 m^{3}/s.

· WSPG: At the downstream end (system outlet), a flow depth of 0.553 m was assigned. At the upstream end (system headwork), a depth of 0.07 m was specified. The flow in the channel was specified as 0.177 m^{3}/s.

Computational methods have evolved significantly over the last decade with computer capabilities greatly increasing thus enabling the solution of massive matrix systems to be economically solved. As a result, the class of differential equations becoming commonplace for use in the typical analysis have also greatly evolved from the solution of the classic Bernoulli’s energy equation to now the full Navier-Stokes equations. With this evolution of technology, it is important to view the new technology modeling outcomes with respect to the prior more traditional modeling approach outcomes. Such a comparison between modeling technology levels is provided for the situation of a relatively high Froude number flow with a hydraulic jump. Other such comparisons involving other related topics are important to be examined in order to provide continuity between modeling and advances in technology.

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.

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

Hromadka II, T.V. and Rao, P. (2019) Examination of Computational Precision versus Modeling Complexity for Open Channel Flow with Hydraulic Jump. Journal of Water Resource and Protection, 11, 1233-1244. https://doi.org/10.4236/jwarp.2019.1110071