On the Existence and Uniqueness of Weak Solutions for a Vorticity Seeding Model

Abstract

In this paper we study the Navier--Stokes equations with a Navier-type boundary condition that has been proposed as an alternative to common near wall models. The boundary condition we study, involving a linear relation between the tangential part of the velocity and the tangential part of the Cauchy stress-vector, is related to the vorticity seeding model introduced in the computational approach to turbulent flows. The presence of a pointwise nonvanishing normal flux may be considered as a tool to avoid the use of phenomenological near wall models in the boundary layer region. Furthermore, the analysis of the problem is suggested by recent advances in the study of large eddy simulation. In the two-dimensional case, by using rather elementary tools, we prove existence and uniqueness of weak solutions. The asymptotic behavior of the solution, with respect to the averaging radius $\delta,$ is also studied. In particular, we prove convergence of the solutions toward the corresponding solutions of the Navier--Stokes equations with the usual no-slip boundary conditions, as the small parameter $\delta$ goes to zero.