Finite-Amplitude Instability of the Compressible Laminar Wake. Weakly Nonlinear Theory

Abstract

An extension of the Lees-Gold linear theory to a theory of finite-amplitude instability of two-dimensional compressible wake flows is presented. Use is made of the amazing similarity between the experimentally observed transition region of certain high-speed wakes and that of the low-speed Sato-Kuriki wake. A Fourier expansion procedure, similar to that of the Stuart-Watson low-speed finite-amplitude instability theory, is used. The mean flow is distorted by the disturbance correlations. A local analysis of the disturbances is made. The Landau amplitude equation is, as in the incompressible theory, again required to solve the problem. The fundamental disturbance component satisfies the Lees-Lin inviscid (spatial) pressure equation. The first harmonic component also satisfies such an equation but with nonhomogeneous terms due to nonlinear interactions. The fundamental distortion function and the modified disturbance growth rate come from the nonhomogeneous third-order problem. This is solved through the use of the smallness of the amplification rate far downstream for a fixed frequency, which leads to an orthogonality condition appropriate to the compressible problem. The streamwise, exponential growth of the disturbance amplitude is limited by nonlinearity.