Nonlinear Interaction between a Pair of Oblique Modes in a Supersonic Mixing Layer: Long‐Wave Limit

Abstract

The nonlinear interaction between a pair of symmetric, oblique, and spatial instability modes is studied in the long‐wave limit using asymptotic methods. The base flow is taken to be a supersonic mixing layer whose Mach number is such that the corresponding vortex sheet is marginally stable according to Miles’ criterion. It is shown that the amplitude of the mode obeys a nonlinear integrodifferential equation. Numerical solutions of this equation show that, when the obliqueness angle is less than π/4, the effect of the nonlinearity is to enhance the growth rate of the instability. The solution terminates in a singularity at a finite streamwise location. This result is reminiscent of that obtained in the vicinity of the neutral point by other authors in several different types of flows. On the other hand, when the obliqueness angle is more than π/4, the streamwise development of the amplitude is characterized by a series of modulations. This arises from the fact that the nonlinear term in the amplitude equation may be either stabilizing or destabilizing, depending on the value of the streamwise coordinate. However, even in this case the amplitude of the disturbance increases, although not as rapidly as in the case for which the angle is less than π/4. Quite generally then, the nonlinear interaction between two oblique modes in a supersonic mixing layer enhances the growth of the disturbance.