In summary, this chapter first established the ability of LES to properly reproduce the observed low-frequency shock motions. It is therefore assumed that the simulations cap- ture the underlying key mechanisms at the origin of the low-frequency shock motions. Moreover, the good agreement with the wall-pressure experimental spectra suggests that the observed motions are unlikely to be solely caused by experimental artifacts. The region under the reflected shock exhibits the most significant low-frequency fluctua- tions. These are found to be broadband and to cover at least one decade of frequencies (around St = 0.03) about two orders of magnitude below the turbulence-related fluc- tuations. The choice of L and ¯u1 for the definition of the Strouhal number appears
justified in the light of the three flow cases considered in this chapter. The wall-pressure fluctuations attributed to the low frequency shock motions and to the turbulence are both significantly contributing to the signal variance, making the distinction between the two difficult. In all cases considered the most energetic low-frequency wall-pressure fluctuations were localised in the first part of the interaction whereas the energy was more evenly distributed between the low frequencies and the higher turbulence-related frequencies in the second section.
The dispersion relation of the wall-pressure fluctuations was characterised by organ- ised convective structures such as shear-layer vortical structures and acoustic-wave prop- agation on top of more broadband dispersions which are characteristic of the turbulence. Upstream convecting pressure waves could be detected in the first third of the interac- tion with wavelengths exceeding the interaction length. Of particular interest was the phase evolution of such low-frequency pressure waves where a phase jump about one third down the interaction region could be detected, potentially indicating the source point of these waves. Following Hudy et al. (2003), the observed phase structure may be related to a global mode such as the one described in chapter 5.
Although the mean separation bubbles of the flow cases considered in this chapter were shallow (with heights not exceeding the beginning of the log-law region), the proba- bility of encountering locally reversed flow could still be around 10% aty/δ0∼1/3. One
main challenge in performing time-resolved simulations of SBLI is the significant dynam- ical range required to resolve both the turbulence and the broadband low-frequency. One direct consequence is that achieving well-converged spectra for the low frequencies is still prohibitive in terms of computational cost. The short-signal effects were considered in this chapter and it was shown that at least an order of fifty low-frequency cycles are needed to achieve acceptable levels of convergence. An additional constraint is the choice of computational spanwise extent which can affect both the time-averaged flow fields (see chapter 4) and the amplitude of the low-frequency dynamics. It was argued that the computational domain should at least be wider than the interaction length itself, greatly
adding to the computational cost of resolving the low-frequency motions.
In cases where the computational domain was sufficiently wide, cells of reversed flow with preferential dimensions could form. These cells were found to be capable of persist- ing for more than 102δ
0/u¯1 with spanwise extents of the order of the interaction length.
They were seen to meander in the spanwise direction which can occasionally lead to the merging of two cells. The presence of such structures and their spanwise motions has direct consequences on the interpretation of fixed-point wall-pressure spectra, for example where the low-frequency end of the spectrum can be artificially underestimated while the high-frequency end is artificially enhanced. This can account for some but not all the differences between the large- and wide-span wall-pressure spectra.
This chapter was also concerned with the choice of the digital-filter approach described in chapter 2 to generate the inflow turbulence. It was shown that this approach could prevent long coherent structures from developing at the inflow plane. Consequently, the length of the coherent structures approaching the interaction region was mainly gov- erned by the computational extent available upstream of interaction. In the present work this was set to be about 10δ0. The main idea behind such a choice was to pre-
vent long coherent structures such those described in Ganapathisubramani et al. (2007b) from interfering with the interaction. The present chapter provides evidence of a flow deprived of upstream long coherent structures, but still exhibiting low-frequency shock motions. This leads us to suggest that although long-coherent upstream structures are likely to enhance the low-frequency shock motions they are not necessary to the exis- tence of the aforementioned shock motions.
Finally, the motions of the reflected shock were extracted from the simulation data. It was shown that the observed transverse waves along the shock have two main origins: direct perturbations by the upstream acoustic field and reflections in the form of trans- verse waves by the acoustic field from the bottom side of the shock, impinging at the critical angle studied in Robinet and Casalis (2001). Moreover, the spectral analysis of the shock motions confirmed that the oblique shock acts like a low-pass filter with the oscillations at the lowest frequencies propagating along the shock more easily than at the highest frequencies. Based on the shock-foot displacement, conditionally-averaged fields could be computed and phase-averaged flow fields could be defined. These phase averages were used to estimate the kinetic energy of the low-frequency motions. In the case of the narrow-span LES of the IUSTI case, we found that the low-frequency motions could account for up to 30% of the total energy of the fluctuations, which is in agreement with the findings from the wall-pressure data analysis.
Guided by the aforementioned LES data, the following chapter will propose an ana- lytic approach to the problem of the reflected-shock low-frequency motions. As we shall see, the derivations will lead to a stochastic ordinary differential equation which will then be discussed in light of the presented numerical and experimental observations.
The variety of mechanisms proposed in the literature as being potentially responsible for the low-frequency shock motions, together with the subsequent debate about the merits of one approach relative to another is symptomatic of the difficulty one has in identifying and then separating individual events from a (supposedly) non-linear (chaotic) system, where actual causal events may well be impossible to detect. Instead of attempting to check the relevance of one assumed mechanism against numerical/experimental data with the inherent complexity of extracting this from fully turbulent flow, it could be more useful to identify the properties of the dynamical system arising from the cou- pling between the shock and the boundary layer. To some extent, this is the approach followed by Plotkin (1975), who postulated that the shock displacement was obeying a first-order stochastic Ordinary Differential Equation (ODE) with an associated charac- teristic timescale. Plotkin shows that such a mathematical model is capable of reproduc- ing the wall-pressure low-frequency spectrum. This interesting point has been verified in two subsequent papers by Poggie and Smits (2001, 2005). Two main reasons why Plotkin’s model has not been widely adopted are: (i) it is a postulate and therefore lacks a physical basis for its ability to reproduce experimental wall-pressure spectra; (ii) it is impractical since the key parameter, the characteristic timescale of the ODE, needs to be determined a posteriori from existing data.
Nevertheless, it is intriguing that a relatively simple ODE is capable of reproduc- ing the low-frequency spectra. The mathematical implications of this observation have been considered only at a superficial level. For example, one can read that Plotkin’s model is a mathematical explanation of how relatively broadband perturbations, caused by the incoming turbulence, can lead to relatively low-frequency motions; or that it assumes that the restoring mechanism ensuring the shock stability is linear. But there are more subtle implications. First, the analytical expression given by Plotkin for the spectrum is based on the response to white noise, meaning that the model does not assume as an input a turbulent signal but instead one which is equally composed of high and low frequencies. Second, while it is true that the postulated governing equation is linear, it is possible that the time constant associated with the restoring mechanism already incorporates non-linear interactions between a velocity fluctuation and the cou- pled shock/boundary-layer system. This latter point is clearly indicated by Poggie and Smits (2001).
This chapter aims at deriving an equation describing the shock low-frequency motions, in the spirit of Plotkin’s pioneering work, but from a completely different approach. The same case of a shock-impingement configuration as described in the previous chapters is chosen but this work could be extended to compression-ramp flows in the future. A com- bined LES/analytical approach is used, where the LES results are extensively employed to support and guide each step of the derivations. The chapter is organised as follows. The first section presents the derivations of the shock-foot dynamical equation, the con- stituents of which are then modelled in the subsequent section. The closed form of the model is then summarised and its solutions to particular forcing examined. Finally, the last section discusses the low-frequency shock motions in the light of the model.