Weak Dirichlet boundary conditions for wall-bounded turbulent flows

In turbulence applications, strongly imposed no-slip conditions often lead to inaccurate mean flow quantities for coarse boundary-layer meshes. To circumvent this shortcoming, weakly imposed Dirichlet boundary conditions for fluid dynamics were recently introduced in [Y. Bazilevs, T.J.R. Hughes, Weak imposition of Dirichlet boundary conditions in fluid mechanics, Comput. Fluids 36 (2007) 12-26]. In the present work, we propose a modification of the original weak boundary condition formulation that consistently incorporates the well-known "law of the wall". To compare the different methods, we conduct numerical experiments for turbulent channel flow at Reynolds number 395 and 950. In the limit of vanishing mesh size in the wall-normal direction, the weak boundary condition acts like a strong boundary condition. Accordingly, strong and weak boundary conditions give essentially identical results on meshes that are stretched to better capture boundary layers. However, on uniform meshes that are incapable of resolving boundary layers, weakly imposed boundary conditions deliver significantly more accurate mean flow quantities than their strong counterparts. Hence, weakly imposed boundary conditions present a robust technique for flows of industrial interest, where optimal mesh design is usually not feasible and resolving boundary layers is prohibitively expensive. Our numerical results show that the formulation that incorporates the law of the wall yields an improvement over the original method.