Direct numerical simulations have been performed to study the dynamics of isolated turbulent spots in compressible isothermal-wall boundary layers. Results of a bypass transition scenario at Mach 2, 4 and 6 are presented. At all Mach numbers the evolved spots have a leading-edge overhang, followed by a turbulent core and a calmed region at the rear interface. The spots have an upstream-pointing arrowhead shape when visualized by near-wall slices, but a downstream-pointing arrowhead in slices away from the wall. The lateral spreading of the spot decreases substantially with the Mach number, consistent with a growth mechanism based on the instability of lateral shear layers. Evidence for a supersonic (Mack) mode substructure is found in the Mach 6 case, where coherent spanwise structures are observed under the spot overhang region.

Direct numerical simulations have been conducted to study the passage of a turbulent spot through a shock-induced separation bubble. Localized blowing is used to trip the boundary layer well upstream of the shock impingement, leading to mature turbulent spots at impingement, with a length comparable to the length of the separation zone. Interactions are simulated at free stream Mach numbers of two and four, for isothermal ͑hot͒ wall boundary conditions. The core of the spot is seen to tunnel through the separation bubble, leading to a transient reattachment of the flow. Recovery times are long due to the influence of the calmed region behind the spot. The propagation speed of the trailing interface of the spot decreases during the interaction and a substantial increase in the lateral spreading of the spot was observed. A conceptual model based on the growth of the lateral shear layer near the wingtips of the spot is used to explain the change in lateral growth rate.

The effect of Mach number on the growth of unstable disturbances in a boundary layer undergoing a strong interaction with an impinging oblique shock wave is studied by direct numerical simulation and linear stability theory ͑LST͒. To reduce the number of independent parameters, test cases are arranged so that both the interaction location Reynolds number ͑based on the distance from the plate leading edge to the shock impingement location for a corresponding inviscid flow͒ and the separation bubble length Reynolds number are held fixed. Small-amplitude disturbances are introduced via both white-noise and harmonic forcing and, after verification that the disturbances are convective in nature, linear growth rates are extracted from the simulations for comparison with parallel flow LST and solutions of the parabolized stability equations ͑PSE͒. At Mach 2.0, the oblique modes are dominant and consistent results are obtained from simulation and theory. At Mach 4.5 and Mach 6.85, the linear Navier-Stokes results show large reductions in disturbance energy at the point where the shock impinges on the top of the separated shear layer. The most unstable second mode has only weak growth over the bubble region, which instead shows significant growth of streamwise structures. The two higher Mach number cases are not well predicted by parallel flow LST, which gives frequencies and spanwise wavenumbers that are significantly different from the simulations. The PSE approach leads to good qualitative predictions of the dominant frequency and wavenumber at Mach 2.0 and 4.5, but suffers from reduced accuracy in the region immediately after the shock impingement. Three-dimensional Navier-Stokes simulations are used to demonstrate that at finite amplitudes the flow structures undergo a nonlinear breakdown to turbulence. This breakdown is enhanced when the oblique-mode disturbances are supplemented with unstable Mack modes.

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The complex transition flow physics associated with the merging of turbulent spots in a Mach 2 boundary-layer has been studied using direct numerical simulation. Dynamics of an isolated turbulent spot, merging of laterally displaced spots, and merging of two spots in tandem are considered. The coherent structures associated with the wingtip region of the spot are found to play a major role in destabilising the surrounding laminar fluid. In the merging of laterally displaced spots a strong velocity defect, resulting in unstable inflectional velocity profiles, is observed in the interaction zone. These local inflectional instabilities within the interaction region trigger new large scale coherent structures. During the inline merging, the calmed region behind the tail of the downstream spot is found to suppress the growth of the upstream spot. The upstream spot is ultimately engulfed by the downstream spot.