Direct spatial resonance in the compressible boundary layer on a rotating-disk

This paper focuses on a resonance mechanism that can lead to significant disturbance amplification at conditions which are sub-critical to nonlinear instabilities. Particularly, direct spatial resonance instability is investigated, which is present in the basic three-dimensional viscous compressible boundary-layer flow due to a rotating-disk. Within this purpose, the linearized system of stability equations is treated numerically making use of a spectral Chebyshev collocation method. The analysis provides critical resonant Reynolds numbers above which growth occurs. Amplitudes of the response of the degeneracies decaying rapidly due to their high damping rates are shown to exist for small enough Reynolds numbers while the flow is still in the laminar state. If the flow is restricted to the incompressible case, the results of Turkyilmazoglu and Gajjar (in Sadhana Acad P Engs 25:601-617, 2000) are completely reproduced. The influences of compressibility are then explored by means of varying the Mach and Prandtl numbers in the cases of heating/cooling the wall as well as the isothermal wall. In general, compressibility effects are found strongly in favor of stabilizing as the Mach number increases, while a strong destabilization is observed by lowering the critical values of Reynolds numbers in the cases of wall heating and insulation. The modal interaction and coalescence of the eigenmodes calculated here create local algebraic growth by rapid development of relatively large amplitudes which might then provide the onset of nonlinear effects followed by transition.