The need for better understanding of the low-frequency unsteadiness observed in shock wave/turbulent boundary layer interactions has been driving research in this area for several decades. We present here a large-eddy simulation investigation of the interaction between an impinging oblique shock and a Mach 2.3 turbulent boundary layer. Contrary to past large-eddy simulation investigations on shock/turbulent boundary layer interactions, we have used an inflow technique which does not introduce any energetically-significant low frequencies into the domain, hence avoiding possible interference with the shock/boundary layer interaction system. The large-eddy simulation has been run for much longer times than previous computational studies making a Fourier analysis of the low frequency possible. The broadband and energetic low-frequency component found in the interaction is in excellent agreement with the experimental findings. Furthermore, a linear stability analysis of the mean flow was performed and a stationary unstable global mode was found. The long-run large-eddy simulation data were analyzed and a phase change in the wall pressure fluctuations was found to coincide with the global-mode structure, leading to a possible driving mechanism for the observed low-frequency motions.Keywords shock boundary layer interaction · global mode · compressible turbulence · LES · low-frequency unsteadiness · separation bubble · digital filter · inflow turbulence PACS 47.40.Nm · 47.40.Ki · 47.27.ep

Direct numerical simulations for fully developed channel flow, subjected to oscillatory spanwise wall motion, have been performed and analysed in an effort to illuminate the fundamental mechanisms responsible for the reduction in turbulent friction drag, observed to result from the spanwise wall motion. A range of statistical data are discussed, including second-moment budgets, joint-probability-density functions, enstrophy and energy-spectra maps. Structural features are also investigated by reference to the response of streak properties to the oscillatory forcing. The unsteady cross-flow straining is shown to cause major spanwise distortions in the streak nearwall structures, leading to a pronounced reduction in the wall-normal momentum exchange in the viscous sublayer, hence disrupting the turbulence contribution to the wall shear stress. The response of the streaks, in terms of their periodic reorientation in wall-parallel planes, the decline and recovery of their intensity during the cyclic actuation, and their wall-normal coherence, is shown to be closely correlated with the temporal variation of the shear-strain vector. Furthermore, a modulating 'top-to-bottom' effect, associated with large-scale outer-layer structures, is highlighted and deemed responsible for the observed reduction in the actuation efficiency as the Reynolds number is increased.

A DNS-based study is presented, which focuses on the response of near-wall turbulence and skin friction to the imposition of an oscillatory spanwise wall motion in channel flow.One point of contrast to earlier studies is the relatively high Reynolds number of the flow, namely Re τ =1000. Another is the focus on transients in the drag that are in the form of moderate oscillatory variations in the skin friction and near-wall turbulence around the low-drag state at a sub-optimal actuation period. These conditions allow phase-averaged statistics to be extracted, during the periodic drag decline and rise, that shed light on the interaction between turbulence and the unsteady Stokes strain. Results are presented for, among others, the phase-averaged second-moments of stochastic fluctuations and their budgets, enstrophy components and joint PDFs. The study identifies velocity skewnessthe wall-normal derivative of the angle of the velocity vector -as playing a significant role † Email address for correspondence: l.agostini@imperial.ac.uk 2 L. Agostini, E. Touber and M. A. Leschzinerin the streak-damping process during the drag-reduction phase. Furthermore, the phasewise asymmetry in the skewness is identified as the source of a distinctive hysteresis in all properties, wherein the drag decline progresses over a longer proportion of the actuation cycle than the drag increase. This feature, coupled with the fact that the streakgeneration time scale limits the ability of the streaks to re-establish themselves during the low-skewness phase when the actuation period is sufficiently short, is proposed to drive the drag-reduction process. The observations in the study thus augment a previously identified mechanism proposed by two of the present authors, in which the drag-reduction process was linked to the rate-of-change in the Stokes strain in the upper region of the viscous sublayer where the streaks are strongest. Furthermore, an examination of the stochastic-stress budgets and the enstrophy lead to conclusions contrasting those recently proposed by other authors, according to which the drag-reduction process is linked to increases in enstrophy and turbulence-energy dissipation. It is shown, both for the transient drag-reduction phase and the periodic drag fluctuations around the lowdrag state, that the drag decrease/increase phases are correlated with decreases/increases in both enstrophy and dissipation.

A combined numerical and analytical approach is used to study the low-frequency shock motions observed in shock/turbulent-boundary-layer interactions in the particular case of a shock-reflection configuration. Starting from an exact form of the momentum integral equation and guided by data from large-eddy simulations, a stochastic ordinary differential equation for the reflected-shock-foot low-frequency motions is derived. During the derivation a similarity hypothesis is verified for the streamwise evolution of boundary-layer thickness measures in the interaction zone. In its simplest form, the derived governing equation is mathematically equivalent to that postulated without proof by Plotkin (AIAA J., vol. 13, 1975(AIAA J., vol. 13, , p. 1036. In the present contribution, all the terms in the equation are modelled, leading to a closed form of the system, which is then applied to a wide range of input parameters. The resulting map of the most energetic low-frequency motions is presented. It is found that while the mean boundary-layer properties are important in controlling the interaction size, they do not contribute significantly to the dynamics. Moreover, the frequency of the most energetic fluctuations is shown to be a robust feature, in agreement with earlier experimental observations. The model is proved capable of reproducing available lowfrequency experimental and numerical wall-pressure spectra. The coupling between the shock and the boundary layer is found to be mathematically equivalent to a first-order low-pass filter. It is argued that the observed low-frequency unsteadiness in such interactions is not necessarily a property of the forcing, either from upstream or downstream of the shock, but an intrinsic property of the coupled system, whose response to white-noise forcing is in excellent agreement with actual spectra.Key words: compressible boundary layers, low-dimensional models, shock waves IntroductionIn recent years, shock-wave/turbulent boundary-layer interactions (SBLI) have received renewed interest, thanks to considerable progress in experimental and computational techniques. A principal concern is the occurrence of energetically significant low-frequency shock motions, which in turn can lead to undesirable unsteady pressure loads in practical aerospace applications. The physical mechanisms at the origin of the low-frequency shock motions are not currently understood but a number of tentative explanations have been proposed, usually falling into one of two categories. The first relates the low-frequency motions to specific events or flow structures from the upstream turbulent boundary layer, whereas the second looks for causal mechanisms within the interaction itself (i.e. downstream of the shock). In both cases, the difficulty resides in identifying a mechanism that can span time scales of the order of 10 1 δ 0 /ū 1 to 10 2 δ 0 /ū 1 . With respect to the second category, a variety of mechanisms have been proposed. Piponniau et al. (2009) argue that the mass-entrainment time scale associat...

A turbulent flow past a transverse-oscillating wall is considered. The oscillation parameters correspond to the regime where drag reduction is observed. Streak spacing and streak angle obtained from the generalized optimal perturbation approach are compared with results from direct numerical simulations. Other flow features of the generalized optimal perturbation are compared with conditionally-averaged data extracted from numerical simulations. The generalized optimal perturbation at a given instant in time is found to consist of an infinitely long structure at a certain angle to the main flow direction. This angle varies slowly with time for half a period, and then suddenly jumps to a different value, changing both sign and magnitude. The angle variation is shown to be slow, because there is a short time interval in the oscillation period when a small perturbation of a certain angle grows strongly and then remains dominant for almost the entire half-period. The transient growth mechanism of the generalized optimal perturbation is found to be a combination of the Orr mechanism due to the cross-flow shear, acting at the initial stage, followed by the lift-up mechanism of the velocity component directed along the structure by the wall-normal motion also oriented in the same direction.