On the over-production of turbulence beneath surface waves in Reynolds-averaged Navier–Stokes models

Abstract

In previous computational fluid dynamics studies of breaking waves, there has been a marked tendency to severely over-estimate turbulence levels, both pre- and post-breaking. This problem is most likely related to the previously described (though not sufficiently well recognized) conditional instability of widely used turbulence models when used to close Reynolds-averaged Navier–Stokes (RANS) equations in regions of nearly potential flow with finite strain, resulting in exponential growth of the turbulent kinetic energy and eddy viscosity. While this problem has been known for nearly 20 years, a suitable and fundamentally sound solution has yet to be developed. In this work it is demonstrated that virtually all commonly used two-equation turbulence closure models are unconditionally, rather than conditionally, unstable in such regions. A new formulation of the $k$–$\unicode[STIX]{x1D714}$ closure is developed which elegantly stabilizes the model in nearly potential flow regions, with modifications remaining passive in sheared flow regions, thus solving this long-standing problem. Computed results involving non-breaking waves demonstrate that the new stabilized closure enables nearly constant form wave propagation over long durations, avoiding the exponential growth of the eddy viscosity and inevitable wave decay exhibited by standard closures. Additional applications on breaking waves demonstrate that the new stabilized model avoids the unphysical generation of pre-breaking turbulence which widely plagues existing closures. The new model is demonstrated to be capable of predicting accurate pre- and post-breaking surface elevations, as well as turbulence and undertow velocity profiles, especially during transition from pre-breaking to the outer surf zone. Results in the inner surf zone are similar to standard closures. Similar methods for formally stabilizing other widely used closure models ($k$–$\unicode[STIX]{x1D714}$ and $k$–$\unicode[STIX]{x1D700}$ variants) are likewise developed, and it is recommended that these be utilized in future RANS simulations of surface waves. (In the above $k$ is the turbulent kinetic energy density, $\unicode[STIX]{x1D714}$ is the specific dissipation rate, and $\unicode[STIX]{x1D700}$ is the dissipation.)

Numerical simulations of surf zone wave dynamics using Smoothed Particle Hydrodynamics

“…Of these models, CFD models based on full solution of the Reynolds Averaged Navier Stokes equations are most analogous to the present SPH simulations as they both can directly resolve overturning breaking waves. In the context of recent applications of high-resolution mesh-based CFD models (i.e., OpenFOAM) that have been applied to the same experimental test cases (e.g., Larsen and Fuhrman, 2018, Brown et al, 2016, Jacobsen et al, 2012, the present results indicate that the SPH approach can reproduce these surf zone processes with comparable skill. In fact, the results of this work suggest that the SPH approach can help to improve predictions within the crest region of breaking waves, as evident by robust predictions of crossshore mass fluxes and undertow profiles that have been notoriously difficult to predict in meshbased CFD models.…”

“…Brown et al (2016) were able to improve predictions of the undertow profiles by adopting more sophisticated turbulence closure schemes (for example, see Figures 5 and 8 in Brown et al (2016) for comparisons with the spilling and plunging cases, respectively); however, in all cases the model overpredicted the undertow velocities more than in the present study, especially within the inner surf zone region where agreement with the SPH results were excellent. Larsen and Fuhrman (2018) observed similar overpredictions of the undertow in their CFD simulations, which they believed was due to surface rollers travelling too fast due to local underpredictions of eddy viscosities (hence flow resistance) simulated within the upper-most part of the water column, leading to increased onshore flows above the wave trough and increased return flows below the wave trough. While the undertow profiles were generally accurately predicted using the present SPH approach, the small discrepancies observed could likewise be due to some deviations in the vertical turbulent flow structure predicted within the inner surf zone.…”

“…While both studies found that the nonhydrostatic approach could reproduce the main features of the current profiles reasonably well, the results deviated significantly more than in the present study, with both an overprediction of the onshore transport in the crest region (by a factor of 2 for the plunging case) and with a similar overprediction of the offshore undertow transport (i.e., defined by more exaggerated belly-shaped profiles). Several studies have also applied CFD models (i.e., OpenFOAM) to the TK94 experiments, and have often also observed a significant overprediction of the undertow transport compared to the present SPH results, particularly when using conventional RANS-based turbulence closure models (e.g., Jacobsen et al, 2012, Brown et al, 2016, Larsen and Fuhrman, 2018, Devolder et al, 2018. Brown et al (2016) were able to improve predictions of the undertow profiles by adopting more sophisticated turbulence closure schemes (for example, see Figures 5 and 8 in Brown et al (2016) for comparisons with the spilling and plunging cases, respectively); however, in all cases the model overpredicted the undertow velocities more than in the present study, especially within the inner surf zone region where agreement with the SPH results were excellent.…”

“…Although these models can capture the breaking process and the turbulent flow field in detail, comparisons between laboratory experiments and mesh-based RANS solvers have highlighted some of the difficulties in accurately predicting turbulent flow fields within the surf zone. In particular, model predictions have been shown to be sensitive to the turbulence model used (e.g., Brown et al, 2016), which can lead to a significant discrepancies in undertow profiles predicted throughout the surf zone (e.g., Brown et al, 2016, Larsen andFuhrman, 2018).…”

A Solution to the Over-Production of Turbulence Beneath Surface Waves in Rans Models

“…It will be shown that the new stabilized closure eliminates entirely the wide-spread problem of over-predicted pre-breaking turbulence, with corresponding major qualitative improvements in the preto-post breaking evolution of the undertow profiles. Further details are provided in Larsen & Fuhrman (2018).…”