3.2. Reynolds-Averaged Navier–Stokes Approach
The early study of Chippada et al. [60
] studied two Froude numbers (
= 2, 4) using the standard standard
model and 2D simulations, together with a Eulerian-Lagrangian flow description helping the determination of the moving free surface.
Zhao and Misra [61
] focused on the mean flow motions of a hydraulic jump using the numerical model RIPPLE [72
]. Two turbulence models were used: the standard
model and the
model of Zhao et al. [61
]. Zhao and Misra focused on weak hydraulic jumps, their main interest being on bores. Model results were compared to the LDV laboratory measurements of Bakunin [73
] and Svendsen et al. [74
] with an overall adequate agreement, despite underestimation of downstream streamwise velocities.
Gonzalez and Bombardelli [62
] simulated the entire two-phase flow of low Froude number hydraulic jumps (mean flow, turbulence, and air entrainment). Air features were incorporated through the free surface. Gonzalez and Bombardelli [62
] compared 2D and 3D RANS and LES models to the experimental observations of Liu et al. [75
], including both mean flow and turbulence, and some qualitative analysis of the air fractions.
Carvalho et al. [14
] carried out a 2D RANS simulation of a hydraulic jump with a higher Froude (
= 6). The jump was contained in a stilling basin downstream of a spillway, and the comparison was conducted against their own experimental data and Hager [76
] empirical relations. The numerical model combined high-order schemes [77
] for the advection terms, a refined VOF algorithm [78
] for free surface tracking, the Fractional Area-Volume Obstacle Representation (FAVOR) method for representing internal obstacles (as defined by Hirt and Sicilian [79
]), and the RNG
turbulence model. The numerical models of Carvalho et al. [14
] were not able to properly represent either the free surface or the pressures. Additionally, shear stresses were underestimated. Nevertheless, the mean velocities were satisfactorily predicted, except in the zone where vortices develop. The maximum values of the velocity and pressure in the flow domain remained below those measured in the experimental installation, but their locations were well predicted.
Abbaspour et al. [63
] used 2D numerical models to study the hydraulic jump over a rough bed using both standard and RNG
models, together with a VOF method to determine the free surface location. Characteristics of the jump, such as water surface profile, length of the jump, velocity profile in different sections, and bed shear stress were evaluated for a range of Froude numbers 4 <
< 8 through a total of 12 simulations. The authors found a close agreement with both the experimental values obtained by Ead and Rajaratnam [80
] in terms of mean relative error: about 1%–8.6% for free surface profiles and about 1%–12% for the length of the jump. The bed shear stresses were computed using the measured depths up- and down-stream of the jump. Differences between total shear stress in experimental and numerical models remained small. Abbaspour et al. [63
] concluded that the best agreement was obtained using the RNG model.
Void fraction predictions in a hydraulic jump were the main goal of Ma et al. [64
], who used a sub-grid air entrainment model coupled with RANS and DES models to study an
= 1.98 hydraulic jump. Sub-grid models arise from the hypothesis that the numerical model alone will not be able to reproduce aeration. Thus, some additional “physics” are input; likewise, a turbulence model aims to represent the small scale fluctuations. Free surface was modeled using a level-set method and turbulent viscosity evaluated using the SST model. Results were compared to the experimental model of Murzyn et al. [81
], concluding that air transported in the shear layer was better reproduced than in the upper region due to RANS modeling limitations to capture free surface fluctuations.
Ebrahimi et al. [65
] carried out 2D simulations (together with the standard
and VOF) of hydraulic jumps on a rough bed for a wide range of Froude numbers (3 <
< 7) across a total of 16 simulations. The authors found a close agreement with the experimental measurements from Elsebaie and Shabayek [82
], with mean relative errors around 0–4.4% for water surface profiles and 0–6.7% for the length of the jump. The model also provided good estimates of the total shear force.
Bayon-Barrachina and Jimenéz [66
] used the open source OpenFOAM code with standard
, and SST
models to study a 3D classic hydraulic jump with
= 6.1. Results were compared to the experimental relations of Hager [76
] and Wu and Rajaratnam [83
], among others, obtaining high accuracies. The authors found that the RNG
model was performing better, followed by the SST
; although in all cases, errors were lower than 4%.
Witt et al. [67
] simulated the air entrainment characteristics of three Froude number hydraulic jumps using the realizable
model, with VOF treatment for the free surface through 2D and 3D simulations. Velocity profiles, void fraction profiles, and Sauter mean diameter were compared to the experimental data from Murzyn et al. [81
], Liu et al. [75
], and Lin et al. [84
], among others. Velocity profiles showed good agreement with the experimental observations for jumps. Void fractions were well predicted with differences rarely over 10%. Witt et al. [67
] observed that at least eight cells per bubble diameter are necessary to reproduce the associated aeration. Furthermore, 3D simulation improved void fraction predictions in the shear layer, and consequently, the average free surface elevation.
A systematic performance analysis of the OpenFOAM and FLOW-3D®
codes was presented by Bayon et al. [5
], investigating a steady hydraulic jump of a Froude number of 6.5. The RNG
model and the VOF method were used. Performance assessment was conducted by comparing several hydraulic jump variables with the literature and their own experimental data [76
]. Numerical uncertainty was assessed following Celik et al. [49
], finding that FLOW-3D®
presented a faster convergence with coarser meshes, whereas OpenFOAM had a more stable solution with refinement. All mean variables studied (not including aeration) obtained accuracies above 90%, except for the roller length, which remained at around 80%, and the negative velocities at the upper jump region. An autocorrelation analysis was conducted, finding good computation of jump toe frequencies, thus showing that RANS models can also reproduce the unsteadiness typical of hydraulic jumps.
Witt et al. [68
] extended the analysis of the work of Witt et al. [67
], studying some air-water mean and turbulent flow features, showing that satisfactory prediction of turbulence quantities can also be achieved through RANS modeling.
Harada and Li [69
] modeled a 2D hydraulic jump of
= 5.8 using the standard
model and the VOF method. Air concentrations were compared to the experimental model of Kucukali and Chanson [91
], obtaining errors below 10% roller length against Hager [76
Valero et al. [6
] studied the flow field (RNG
model and the VOF method) inside a United States Bureau of Reclamation (USBR) Type III stilling basin and compared numerical results, contemplating eight Froude numbers (in the range of 3.12–9.52), to the data of Frizell and Svoboda [7
]. Good agreement was obtained for sequent depth relations and sweep-off point. The numerical model allowed insight into the distribution of forces on the different elements of the basin, thus explaining the experimental observation of Frizell and Svoboda [7
] on the capacity of the Type III to hold inside jumps even for very low tailwater levels.
shows the accuracy of previous RANS models with standard
, and the SST
turbulence models. In Figure 3
, the results are also classified by Froude number; following the jump typology of Bradley and Peterka [92
] (see Part 1). The RNG
model comparatively provided the best performances for all four jump parameters, despite is also showing the largest scatter. The standard
model provided more accurate results in terms of the roller length and the free surface.
3.3. Detached Eddy Simulation and Large Eddy Simulation
Ma et al. [64
], besides RANS simulations, used also the DES approach to predict air entrainment in a hydraulic jump of
= 1.98. By comparing the numerical results with the experimental data of Murzyn et al. [81
], Ma et al. [64
] found that the averaged DES results predicted the observed void fraction profiles in both the lower shear layer and the upper roller region, while the RANS approach missed the latter. This is due to the better capabilities of DES to capture strong fluctuations. Nonetheless, Bayon et al. [5
] showed that the RANS model can reproduce satisfactorily big-scale fluctuations inside hydraulic jumps.
Recently, Jesudhas et al. [70
] studied an
= 8.5 hydraulic jump using a DES approach together with a VOF and a high-resolution capturing scheme. Not only were the numerical results compared to past experimental literature (among others, [84
])—with excellent agreement—but also new insights into the turbulence structure were presented, as for instance how the developing shear layer leads to intense free surface deformations. The study of Jesudhas et al. [70
] proved that DES may suffice to study turbulent features.
Concerning the LES part of the study of Gonzalez and Bombardelli [62
], satisfactory agreement was found in terms of mean flow, turbulence, and air entrainment. However, LES outcomes showed similar patterns in the time-averaged flow field as those obtained with the
Large eddy simulation of the hydraulic jump was also executed by Lubin et al. [71
]. The numerical results were then compared with experimental data collected on a physical model. Important handicaps were noticed as a result of the strong interface deformations, break-ups, and high shear levels. The flow did not tend to a stationary state because of some numerical diffusion at the free surface.
3.5. Smoothed Particle Hydrodynamics
Lopez et al. [57
] investigated the capabilities of Smoothed Particle Hydrodynamics (SPH) to reproduce a mobile hydraulic jump for different Froude numbers. Numerical simulations were compared to their own physical model. Results showed good agreement for
< 5. Applying the standard
model, a considerable improvement in the model simulating hydraulic jumps with
> 5 was also obtained. The main disadvantage of this approach is the doubling of the computational time. Finally, it was found that SPH provides correct estimates of the average pressures at the boundaries, but exhibits large dispersion for instantaneous water depths.
De Padova et al. [58
] carried out different studies on the numerical modeling of the hydraulic jump using the SPH method and a weakly-compressible XSPHscheme, which includes a mixing-length turbulence model and a two-equation turbulence model to represent turbulent stresses. First, they reproduced the formation of different undular jumps [58
]. Instantaneous and time-averaged flow variables were compared with acoustic Doppler velocity measurements for
equal to 3.9 and 8.3. The SPH model slightly overestimated the measurements of the velocity, in the zone of the lateral shock wave and in the vortex zone near the lateral wall. The predicted free surface elevations and velocity profiles showed a satisfactory agreement with measurements and most of the particular features of the flow, such as the trapezoidal shape of the wave front and the flow separations at the toe of the oblique shock wave along the side walls. The numerical results were compared with their own experimental investigations. The comparison proved that, when simulating hydraulic jumps without a surface roller, a simple turbulence model based on the mixing-length hypothesis suffices to yield results that, in terms of water depths and average velocity predictions, were like those obtained using the standard
De Padova et al. [59
] investigated the oscillating characteristics and cyclic mechanisms in hydraulic jumps. Both the mixing-length turbulence model and the standard
model were able to predict some features of the flow, but comparison with experimental results showed that the mixing-length results were closer to the experimental data [96
] than those from the
in, for instance, predicting the spectra of the surface elevations upstream and downstream of the jump.