Second Order Adaptive Boundary Conditions for Exterior Flow Problems: Non-Symmetric Stationary Flows in Two Dimensions

Abstract

We consider the problem of solving numerically the stationary incompressible Navier-Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a …nite sub-domain, which leads to the problem of …nding so called "arti…cial boundary conditions" to replace the boundary conditions at in…nity. To solve this problem we construct -by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke -a smooth divergence free vector …eld depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to nonsymmetric ‡ows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent way as an integral part of the solution process. When compared with our previous paper where …rst order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitude.Mathematics Subject Classi…cation (2000). 76D05, 76D25, 76M10, 41A60, 35Q35.

Existence of stationary solutions of the Navier–Stokes equations in the presence of a wall

“…We now prove that for > 3 any -solution provides a classical solution (u; p) to (1), (2), (3). In what follows F is a smooth source term of compact support.…”

“…More precisely, the equations (15), (17), (18) are equivalent to (12), the equations (16), (19), (20) are equivalent to (13), the equations (21), (22) are equivalent to (10) and (9) and (23) is equivalent to (2). Equation (11) de…nes !…”

“…But, if done by brute force, such computations are excessively costly even with today's computers. In [1,2,14,15], the second author and his collaborators have developed techniques that lead for similar problems to an overall gain of computational e¢ ciency of typically several orders of magnitude. These techniques use as an input a precise asymptotic description of the ‡ow.…”

“…which have to be solved together with divergence free condition (2) in the domain = + nB, subject to the boundary conditions (3), (4) and uj @B = e 1 :…”

“…Theorem 1 For all F 2 C 1 c ( + ) with F su¢ ciently small in a sense to be de…ned below, there exist a vector …eld u = (u 1 ; u 2 ; u 3 ) 2 H 1 ( + ) and a function p satisfying the Navier-Stokes equations (1), (2) in in [21] will also be explored in our forthcoming work [8]. The proof that the functions F, obtained from the original exterior problem by the truncation procedure, are su¢ ciently small to apply the present theorem will also be given in a subsequent paper.…”