Simulation of viscous flows with undulatory boundaries. Part I: Basic solver

A numerical method for the simulation of viscous flows with undulatory walls and free surfaces is presented. The simulation domain is discretized by a boundary-fitted and time-dependent grid. The Navier–Stokes equations, subject to fully nonlinear kinematic and dynamic boundary conditions at the free surface and no-slip boundary condition at the wall, are simulated by a hybrid pseudo-spectral and finite difference method in space and a semi-implicit fractional-step method in time. The performance of the method is demonstrated by a series of test cases including flows over wavy boundaries, various surface waves, and interaction between vortices and free surfaces. Validation by convergence test and extensive comparisons with previous theoretical, experimental, and numerical studies indicate the accuracy and efficiency of the method. Finally, a simulation example of turbulence and free surface interaction is presented. Results show that the rich features of the free surface such as surface waves, splats, anti-splats, dimples, and scars are captured accurately. Characteristic vortical structures and variation of turbulence statistics in the near-surface region are also elucidated.