Three-Dimensional Unsteady RANS Modeling of Discontinuous Gravity Currents in Rectangular Domains

Discontinuous gravity currents in rectangular channels are modeled numerically by solving the 3D unsteady Reynolds-averaged Navier-Stokes (URANS) equations closed with a buoyancy corrected low-Reynolds number (LRN) $k\text{-}\epsilon$ model using second-order accurate finite-volume numerics. It is shown that, on moderately fine computational meshes (with $\sim {10}^6$ grid nodes) and with careful modeling of the near-wall flow, the URANS model can capture the essential large-scale 3D dynamics of gravity current flows, which previously had been resolved only by DNS and/or large-eddy simulation (LES) on very fine computational meshes. These 3D dynamics include the onset of the well-known lobe-and-cleft instability at the current head, the onset of large-scale Kelvin-Helmholtz billows at the head of the gravity current, and the breakdown of the interfacial billows in the rear part of the current head due to intense three-dimensional mixing. The computed results underscore the importance of careful modeling of the near wall flow in URANS simulations. The standard $k\text{-}\epsilon$ model with wall functions fails to capture the aforementioned complex 3D dynamics, which are only resolved by the LRN $k\text{-}\epsilon$ model on grids that resolve the near-wall region. Furthermore, numerical experiments show that including in the simulation the lateral no-slip end walls of the channel has a profound effect on the accuracy of the computed solutions. End-wall effects enhance the three-dimensionality of the flow, result in increased mixing of the dense and the ambient fluids behind the head of gravity currents, and yield results in good agreement with measurements. On the other hand, when end-wall effects are omitted, by imposing periodicity in the spanwise direction, three-dimensional mixing is suppressed and the breakdown of interfacial billows is significantly underestimated. Grid sensitivity studies are also carried out using three successively refined meshes and show that the URANS LRN model yields grid-converged solutions at affordable computational resources. As URANS modeling requires only a fraction of the computational cost of DNS or LES with near-wall resolution, the present results underscore the potential of unsteady statistical turbulence models for predicting and elucidating the physics of gravity current flows in complex geometries and at Reynolds numbers of engineering relevance.