The critical Reynolds number of a laminar incompressible mixing layer from minimal composite theory

Abstract

According to parallel-flow theory based on the Orr-Sommerfeld equation, a mixing layer is unstable at all Reynolds numbers. However this is untenable from energy considerations, which demand that there exist a non-zero Reynolds number below which disturbances cannot extract net energy from the mean flow. It is shown here that a linear stability analysis of similarity solutions of the plane mixing layer, including the effects of flow non-parallelism using the minimal composite theory and the properties of adjoints, following Govindarajan & Narasimha (Theor. Comput. Fluid Dyn. vol. 19, 2005, p. 229) resolves the issue by yielding a non-zero critical Reynolds number for co-flowing streams of any velocity ratio. The critical Reynolds number for the total disturbance kinetic energy is found to be close to 30 for all velocity ratios in the range from zero to unity.