Three-dimensional tertiary motions in a plane shear layer

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Abstract
The nonlinear evolution of instabilities of a plane-parallel shear flow with an inflection-point profile is studied. The particular example of the cubic-profile flow generated in an inclined layer heated from above and cooled from below is chosen because it exhibits supercritical bifurcations for secondary and tertiary flows. Since the limit of small Prandtl number is assumed, buoyancy effects caused by temperature perturbations are negligible. The analysis describes first the transition to transverse roll-like vortices which become unstable at slightly supercritical Grashof numbers to a vortex-pairing instability with alternating pairing in the spanwise direction. Three-dimensional finite-amplitude solutions for this tertiary mode of motion are computed and discussed. Finally the question of the stability of the tertiary flow is addressed.

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