Instability of the realizable k-ε turbulence model beneath surface waves

Recent research has proved that most widely used two-equation turbulence closure models are unconditionally unstable in regions of nearly potential flow having finite strain, as commonly found beneath non-breaking surface waves. In this work, we extend such analysis to consider the popular realizable k-epsilon turbulence model. It is proved that this model, unlike all others thus far analyzed, is only conditionally unstable in such regions due to the addition of viscosity in the epsilon dissipation term. A method for formally stabilizing the model in the problematic regions is likewise developed. The results of the analysis, using both standard and stabilized turbulence closures, are confirmed via numerical simulations of progressive surface wave trains using a computational fluid dynamics model. Important qualitative differences are likewise demonstrated in simulations involving spilling breaking waves. (In the above, k is the turbulent kinetic energy density, and epsilon is the turbulence dissipation rate.)