Boundary Effects on Exact Solutions of the Lagrangian-Averaged Navier–Stokes-α Equations

Abstract

In order to clarify the behavior of solutions of the Lagrangian-averaged Navier–Stokes-α (LANS-α) equations in the presence of solid walls, we identify a variety of exact solutions of the full equations and their boundary layer approximations. The solutions demonstrate that boundary conditions suggested for the LANS-α equations in the literature(1) for a bounded domain do not apply in a semi-infinite domain. The convergence to solutions of the Navier–Stokes equations as α → 0 is elucidated for infinite-energy solutions in a semi-infinite domain, and non-uniqueness of these solutions is discussed. We also study the boundary layer approximation of LANS-α equations, denoted the Prandtl-α equations, and report solutions for turbulent jets and wakes. Our version of the Prandtl-α equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of Cheskidov.(2)