Three-Dimensional Effects in Boundary-Layer Transition: A High Reynolds Number Weakly-Nonlinear Theory

Abstract

We are concerned here with the development of Tollmien-Schlichting (T-S) waves in an incompressible flat-plate boundary-layer. Suppose that a small-amplitude 2-D T-S wave of frequency ω _*. is introduced by a vibrating ribbon. It decays up to the lower-branch neutral position x₀, and subsequently grows. If the initial amplitude is not too small, nonlinear effects then come into play, and the disturbance acquires significant three-dimensionality. This may take the form of a roughly-periodic spanwise variation of T-S amplitude (“peak-valley splitting”), together with the formation of a mean longditudinal-vortex system, as in the experiments of Klebanoff et al (1962). On the other hand, for lower disturbance amplitudes, the dominant three-dimensional components may be subharmonics of the primary 2-D wave — e.g. Kachanov and Levchenko (1984).