Solutions of the Navier–Stokes Equation at Large Reynolds Number

Abstract

The problem of two-dimensional incompressible laminar flow past a bluff body at large Reynolds number $( R )$ is discussed. The governing equations are the Navier–Stokes equations. For $R = \infty $, the Euler equations are obtained. A solution for R large should be obtained by a perturbation of an Euler solution. However, for given boundary conditions, the Euler solution is not unique. The solution to be perturbed is the relevant Euler solution, namely the one which is the Euler limit of the Navier–Stokes solution with the same boundary conditions. For certain semi-infinite or streamlined bodies, the relevant Euler solution represents potential flow. For flow inside a closed domain a theorem of Prandtl states the relevant Euler solution has constant vorticity in each vortex. In many cases it can be determined by simultaneously considering the boundary layer equations. For flow past a bluff body, the relevant Euler solution is not known, although the free streamline flow for which the free streamline detaches smoothly from the body is a likely candidate. Even if this is correct, many unsolved problems remain. Various scalings have to be used for various regions of the flow. Possibilities of scaling for the various regions are discussed here. Special attention is paid to the region near the point of separation. A famous paper by Goldstein asserts that for an adverse smooth pressure gradient, the solution of the boundary layer equations can, in general, not be continued beyond the point of separation. Subsequent attempts by many authors to overcome the difficulty of continuation have failed. A very promising theory, going beyond conventional boundary layer theory, has recently been put forward independently by Sychev and Messiter. They assume that separation takes place in a sublayer whose thickness and length tend to zero as R tends to infinity. The pressure gradient in the sublayer is self-induced and is positive upstream of the point of separation and zero downstream. Their theory does not contradict experiments and numerical calculations, which may be reliable up to, say, $R = 100$, but it also shows that in this context, 100 may not be regarded as a large Reynolds number. The sublayer has the same scaling in orders of R as the sublayer at the trailing edge of a plate, found earlier by Stewartson and Messiter in studying the matching of the boundary layer solution on the plate with the Goldstein wake solution downstream of the trailing edge.