Correlations with downstream flow features

The incoming boundary layer is not the only place where coherent structures occur. The idea that the vortical structures emerging from the shear layer could play an impor- tant role in the interaction is appealing. For example, Dussauge et al. (2006) write that the “eddies in the separated zone may be the source of excitation”. Pirozzoli and Grasso (2006) go further and, based on a DNS of an impinging shock at Mach 2.25 and Reδ2 = 3725, argue that the interaction mechanism works as follows: large coherent

structures are shed close to the average separation point from the mixing layer, interact- ing with the incident shock to produce acoustic disturbances that propagate upstream (in the subsonic layer), thus inducing an oscillatory motion of the separation point and a subsequent branching and flapping motion of the reflected shock, enhancing the for- mation of discrete vortices. The large-scale low-frequency unsteadiness would then be

sustained by an acoustic resonance mechanism. The authors relate the aforementioned mechanism to the one responsible for the generation of tones in cavity flows. To support this, they developed a simplified model for the acoustic resonance and were successful at predicting the characteristic tones for the interaction they studied. Although appeal- ing, this mechanism has not been explicitly confirmed using experimental or numerical data. In fact, the possibility that a resonance can occur is bound to the sensitive and selective nature of the shear layer. The receptivity of the particular shear layer to an acoustic field must be addressed. As mentioned earlier, Borodai and Moser (2001) have demonstrated the possible decoupling between the acoustic field and the turbulence so that the effect of the acoustic field on the turbulence could be neglected. Moreover, it is important to note that the integration time obtained by Pirozzoli and Grasso (2006) was much too short to cover any low-frequency oscillation, making the interpretation of the correlation functions subject to caution. Finally, the idea of a resonance-based mechanism seems rather surprising as it appears to oppose the experimental evidences of broadband oscillations.

Piponniau et al. (2009) have also considered the shear-layer as a key for the low- frequency shock motions and proposed a model based on the mass-entrainement timescale associated with the separation bubble and the developping mixing layer. The resulting timescale is of the same order of magnitude as the dominant shock-motion timescale and the model suggests that the main parameter controling the low-frequency shock motions is the spreading rate of the compressible mixing layer. Therefore, the authors argue that the low-frequency motions are closely related to the presence of a separated region down- stream of the shock and that the geometry of the flow configuration (i.e. corner flow or shock reflection) does not influence them much, as long as the mean separation-bubble height is sufficiently large. Moreover, the authors find that the characteristic frequency of the shock motions are affected by the shock intensity p2/p1 and not directly related to any time scale from the upstream boundary layer.

Yet more alternative approaches have been suggested in the literature. One particu- larly interesting approach is to look at possible hydrodynamic instabilities. Boin et al. (2006); Boin and Robinet (2004) argue that the unsteadiness is intrinsic to the dynamics of the separated zone. They show that laminar SBLI can, under some assumptions, be the place of unsteady self-sustained low-frequency dynamics and that it is not necessary to have upstream disturbances to generate the unsteady motion. Their 3D calculations show that before becoming unsteady (when gradually increasing the shock angle), the SBLI goes through a phase were the flow becomes three-dimensional and stationary and that this state is unstable and leads to fully 3D and unsteady flows. They base their scenario on Dallman’s conjecture (Dallmann, 1988), which states that before unsteady vortex shedding occurs, multiple recirculation zones occur inside the primary bubble

which finally leads to a global structural flow change with “multiple structurally unsta- ble saddle-to-saddle connections”. Theofilis et al. (2000) linked the 3D global instabilities in incompressible flows to Dallman’s conjecture. Boin and Robinet (2004) believe that this extends to supersonic flows and SBLI and that it could explain the first stage of the establishment of the unsteady low frequency. In a relativelly recent paper, Robinet (2007) performs a BiGlobal analysis of the laminar shock-reflection case and reports that there exists a global mode (in the BiGlobal sense) for sufficiently strong shock strengths. The most unstable mode is reported to be three-dimensional with a wavelength scaling on the separation length while the 2D mode is found to be stable.

Finally, it is recalled that despite the numerous studies cited above, there remain uncertainties regarding possible external sources of unsteadiness. Dolling (2001) men- tions possible stagnation chamber resonances, or vortices embedded with the test section (like the “span-wise tornadoes” mentioned by Dussauge et al., 2006) that could lead to the low-frequency unsteadiness. In fact, this led some researchers to believe that small changes in the incoming boundary layer thickness could be the cause of the unsteadiness (McClure, 1992; ¨Unalmis and Dolling, 1994). Beresh et al. (2002), Chan (1994) and Dupont et al. (2006) could not find such correlations but the latter authors could not rule out the idea that there could be “side wall effects” in their results. Similarly, numer- ical simulations suffer from the inevitable need for boundary conditions. In particular, LES and DNS of turbulent wall-bounded flows necessitate time-varying inflow boundary conditions which often introduce characteristic frequencies that could be of the same order of magnitude as the observed low-frequency oscillations. Would the simulations and experiments be both wrong for different reasons but still give comparable results? This is believed to be unlikely. Nevertheless, for the present simulations, a large amount of time was devoted in order to define inflow conditions that could not introduce any particular low frequency in the computational domain. This is thought to be the right start for numerical investigations on SBLI unsteadiness.