The stability behavior of a laminar boundary layer with shock boundary layer interaction and small amplitude disturbances is investigated by linear stability theory and direct numerical simulation. By a complex interaction of several physical properties, the impinging shock wave locally influences stability behavior of the boundary layer, dependent on its shock strength, applied disturbance frequency, and disturbance propagation angle with respect to the flow direction (obliqueness angle). Due to the displacement of the boundary layer near shock impingement and the according Reynolds number effect in this area, the boundary layer is locally destabilized. The displacement of the boundary layer also produces an increase of the thickness of local regions of relative supersonic speed, which promotes second mode instability. For the results obtained by direct numerical simulation nonparallel effects could be identified and quantified. Taking these nonparallel effects into account, linear stability theory is able to represent the stability behavior of wall distant disturbance amplitude maxima having small obliqueness angles for the cases investigated here. For larger obliqueness angles and disturbance amplitudes at or close to the wall the agreement between linear stability theory and direct numerical simulation declines considerably.

Two-dimensional direct numerical simulations and linear stability theory investigations have been carried out for a compression ramp at Maϭ4.8 and compared to earlier results of a laminar boundary layer with impinging shock wave. The inflow parameters in both flows were identical; the ramp angle of the compression corner was chosen to cause a separation bubble, which has exactly the same length compared to the case with impinging shock. It turned out, that the two cases are almost identical for the base flow properties. This is in accordance with similarity assumptions, e.g., free interaction theory, which for smaller Reynolds numbers states, that the boundary layer should be independent of the sort of shock-boundary layer interaction. However, linear stability theory results differ near the corner and the impinging shock, respectively. Direct numerical simulations of small-amplitude disturbances, which were introduced into the laminar boundary layer, also behave in a very similar way. Amplitude distributions exhibit the same characteristics. The according distributions of the ramp flow have slightly larger amplitudes than the case with impinging shock.

The two-way coupling mechanisms in particle-laden mixing layers are investigated, with and without particle settling, and with an emphasis on the resulting modifications to the fluid vorticity field. The governing equations are interpreted with respect to the production and cancellation of vorticity. These mechanisms are shown to be related to the misalignment of the concentration gradient and the slip velocity, as well as to the difference in fluid and particle vorticities. Preliminary insight into the physics is obtained from an analysis of the unidirectional base flow. For this model problem, the conditions are established under which the particle velocity remains a single-valued function of space for all times. The resulting simplified set of two-way-coupled equations governing the vorticity of the fluid and particulate phases, respectively, is solved numerically. The formation of a decaying travelling wave solution is demonstrated over a wide range of parameters. Interestingly, the downward propagation of the fluid vorticity field is not accomplished through convection, but rather by the production and loss of vorticity on opposite sides of the mixing layer. For moderate settling velocities, the simulation results reveal an optimal coupling mechanism between the fluid and particle vorticities at intermediate values of the mass loading parameter. For large settling velocities and intermediate mass loadings, more than one local maximum is seen to evolve in the vorticity field. A scaling law for the downward propagation rates of the vorticity fronts is derived.Two-dimensional particle-laden mixing layers are investigated by means of a mixed Lagrangian–Eulerian approach which is based on the vorticity variable. For uniformly seeded mixing layers, the simulations confirm some of the features observed by Druzhinin (1995b) for the model problem of a two-way-coupled particle-laden Stuart vortex, as well as by Dimas & Kiger (1998) in a linear stability analysis. For small values of the Stokes number, a mild destabilization of the mixing layer is observed. At moderate and large Stokes numbers, on the other hand, the transport of vorticity from the braids into the core of the evolving Kelvin–Helmholtz vortices is seen to be slowed by the two-way coupling effects. As a result, the particle ejection from the vortex cores is weakened. For constant mass loadings, the two-way coupling effects are strongest at intermediate Stokes number values. For moderately large Stokes numbers, the formation of two bands of high particle concentration is observed in the braids, which reflects the multi-valued nature of the particle velocity field. For mixing layers in which only one stream is seeded, the particle concentration gradient across the mixing layer leads to strong vorticity production and loss, which results in an effective net motion of the vortex in the flow direction of the seeded stream. Under particle settling, the vortex propagates downward as well. For the parameter range explored here, its settling velocity agrees well with the scaling law derived from the unidirectional flow analysis.

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