A conservative scheme for simulation of free-surface turbulent and wave flows

A numerical scheme with good conservation properties is developed for the simulation of free-surface turbulent and viscous wave flows using a surface-fitted curvilinear grid. The Navier–Stokes equations are written in a strong conservative formulation with respect to the curvilinear coordinates, and are discretized by a pseudo-spectral method in the horizontal directions and a finite-difference method in the vertical direction. Large-eddy simulation (LES) is implemented with the conservative scheme to extend the simulation capability to turbulent flows with higher Reynolds numbers. Fully nonlinear kinematic and dynamic boundary conditions are implemented at the free surface. The numerical scheme is validated using a variety of wave and vortical flow test cases. The results show good agreement with previous theoretical and numerical predictions, whereas the present scheme achieves significant improvement in the conservation of mass and momentum over the non-conservative scheme developed by Yang & Shen [1]. Meanwhile, the present conservative scheme is found to be more stable than the non-conservative scheme for the simulation of sideband waves and broadband waves. The effect of viscous dissipation on the long-term nonlinear wave evolution is also captured by the present scheme. The ability of the present scheme for simulating long-term wave-current-turbulence interaction is demonstrated by the computation of Langmuir circulation, for which the non-conservative scheme produces significant errors in mass and momentum conservation and the simulation fails. Flow features of the Langmuir circulation, such as the counter-rotating vortices and converging-diverging zones, have been successfully captured with our numerical scheme. The turbulence statistics also agree with the characteristics of Langmuir circulation.