Direct numerical simulation of laminar breakdown in high-speed, axisymmetric boundary layers

Abstract

The laminar breakdown of high-speed, axisymmetric boundary-layer flow is simulated numerically by solving the compressible Navier-Stokes equations using spectral collocation and high-order compact-difference techniques. Numerical test cases include Mach 4.5 flow along a hollow cylinder and Mach 6.8 flow along a sharp cone. From initial states perturbed by “second-mode” primary and subharmonic (H-type) secondary disturbances, the well-resolved (temporal) calculations proceed well into the laminar breakdown stages, characterized by saturation of the primary and secondary instability waves, explosive growth of higher harmonics, and rapid increase in the wall shear stress. The numerical results qualitatively replicate two previously unexplained phenomena which have been observed in high-speed transition experiments: the appearance of so-called “rope-like waves” and the “precursor transition” effect, in which transitional flow appears to originate near the critical layer well upstream of the transition location at the wall. The numerical results further reveal that neither of these effects can be explained, even qualitatively, by linear stability theory alone. Structures of “rope-like” appearance are shown to arise from secondary instability. Whereas certain features of the precursor transition effect also emerge from secondary instability theory, its nature is revealed to be fundamentally nonlinear.