Evolution of laminar mixing layer

This investigation is directed at the analysis of the developing flow due to the mixing of two uniform incompressible fluid streams with different velocities. The asymptotic behavior of a laminar mixing layer is a similar solution of the boundary layer equations. Using the method of matched asymptotic expansions of the full Navier-Stokes equations for large Reynolds numbers, equations and solutions of the outer and inner expansions are presented.; From the knowledge of asymptotic expansions, the governing Navier-Stokes equations evolve into the boundary layer equations downstream. The present formulation of the governing Navier-Stokes equations evolves gradually into the boundary layer equations downstream. Without imposing the similarity solution far downstream, the technique shows that the converged Navier-Stokes solution evolves into the laminar similarity solution downstream for reasonable velocity ratios.; The Navier-Stokes solution is obtained based on the physical intuition of the interface trajectory which deflects monotonically downwards toward the slower moving stream, then gradually approaches an asymptotic value downstream without a wiggle. The downstream interface deflection naturally falls out from the Navier-Stokes solution and it is the information left over from the initial mixing region flow. This finding establishes the basis of the missing third boundary condition of the Blasius equation with the downstream interface deflection. This finding also reaffirms a unique third boundary condition for the Blasius equation that originated from the initial mixing flow region, an idea originally proposed by Klemp and Acrivos (1972) and first confirmed by Alston and Cohen (1992). Compared with the downstream interface deflection derived from other third boundary conditions, the present result falls between the von Karman condition and Alston and Cohen's result.; The physical parameters evolve to their similarity form or their asymptotic forms at different rates with a given velocity ratio. It takes longer for the physical variables to evolve into their similarity profiles with decreasing velocity ratio. The longitudinal velocity always approaches the similarity form first. Calculation reveals that the nonsimilar solutions stretch out to a downstream Reynolds number of 103 to 104 for the velocity ratios studied (0.95--0.7).