Lionel Larchevêque
email: lionel.larcheveque@univ-amu.fr

Deepak Prem Ramaswamy
email: d.ramaswamy@aia.rwth-aachen.de

Anne-Marie Schreyer
email: a.schreyer@aia.rwth-aachen.de

Effects of favorable downstream pressure gradients on separated shock-wave/boundary-layer interactions

Keywords: Separated shock-wave/boundary-layer interaction, favourable downstream pressure gradient, low-frequency unsteadiness

Shock-wave/boundary-layer interactions (SWBLI) strongly influence the aerodynamic behavior of many aerospace, propulsion and transportation systems. They have therefore been studied intensively -with different methods, configurations, and test facilities. Regions of favorable pressure gradient are often placed downstream of the SWBLI region., e.g. to prevent wind-tunnel blockage. To be able to better understand their effect and compare cases from varying sources, a joint experimental-numerical study of a supersonic compression/expansion corner flow is carried out. A high-resolution long-time numerical simulation complements particle-image velocimetry measurements, both using the same configuration and conditions. Derived setups without the downstream expansion corner are also explored; both for the same compression-corner angle and for a ramp angle inducing a separation bubble of the same size as for the original compression/expansion configuration. Results demonstrate that the topology and very low-frequency dynamics of the separation bubble developing over the compression corner are altered by the downstream expansion corner. Differences are mainly found for the downstream motion of the shock / shrinking of the separated region and extend downstream of the reattachment point up to the expansion corner.

Introduction

The interaction between shock-waves and turbulent boundarylayers is ubiquitous in modern high-speed aerospace applications, like transonic airfoils, supersonic engine inlets, rocket engines, etc. The resulting shock-wave / boundary-layer interactions (SWBLI) must be given careful consideration in the design process, as they can be detrimental to the safe and reliable operation of the aerospace system via enhanced pressure and thermal loading, profound aerodynamic drag, etc. Therefore, SWBLIs have been an active field of research for more than seven decades, in order to characterise their general spatial and temporal organisation [START_REF] Delery | Shock wave/turbulent boundary layer interaction and its control[END_REF][START_REF] Dolling | Fifty years of shock-wave/boundary-layer interaction research: What next[END_REF][START_REF] Gaitonde | Progress in shock wave/boundary layer interactions[END_REF] A characteristic behaviour of SWBLIs is the inherent lowfrequency unsteadiness of the system, where the shock exhibits streamwise motion at frequencies well below the typical timescales associated with the upstream turbulent boundary layer [START_REF] Clemens | Low-frequency unsteadiness of shock wave/turbulent boundary layer interactions[END_REF]). Several experimental and numerical works indicated sources for the origin of this low-frequency shock motion, and various mechanisms have been proposed. A number of studies have identified strong correlations between the shock-motion and the low-and high-momentum zones observed in the upstream boundary-layer [START_REF] Beresh | Relationship between upstream turbulent boundary-layer velocity fluctuations and separation shock unsteadiness[END_REF][START_REF] Humble | Unsteady aspects of an incident shock wave/turbulent boundary layer interaction[END_REF][START_REF] Ganapathisubramani | Lowfrequency dynamics of shock-induced separation in a compression ramp interaction[END_REF]. On the other hand, several studies have suggested mechanisms relating the unsteadiness to the breathing motion of the separationbubble [START_REF] Piponniau | A simple model for low-frequency unsteadiness in shock-induced separation[END_REF][START_REF] Touber | Large-eddy simulation of lowfrequency unsteadiness in a turbulent shock-induced separation bubble[END_REF][START_REF] Wu | Analysis of shock motion in shockwave and turbulent boundary layer interaction using direct numerical simulation data[END_REF]. It has been argued that the downstream feedback dominates strongly separated interactions, while a combination of both the upstream and downstream influences determine the shock-motion for weakly separated interactions [START_REF] Souverein | Effect of interaction strength on unsteadiness in shock-wave-induced separations[END_REF][START_REF] Clemens | Low-frequency unsteadiness of shock wave/turbulent boundary layer interactions[END_REF].

More recently, studies have employed dynamic mode decomposition (DMD) onto high-fidelity simulation data sets to identify dynamically relevant features associated with this interaction [START_REF] Grilli | Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction[END_REF][START_REF] Nichols | Stability and modal analysis of shock/boundary layer interactions[END_REF][START_REF] Priebe | Low-frequency dynamics in a shock-induced separated flow[END_REF][START_REF] Pasquariello | Unsteady effects of strong shock-wave/boundary-layer interaction at high reynolds number[END_REF]. A strong similarity of the DMD modes and the linear stability modes were reported by [START_REF] Priebe | Low-frequency dynamics in a shock-induced separated flow[END_REF]. Furthermore, the existence of unsteady largescale streamwise structures was seen coupled to the separationbubble dynamics [START_REF] Priebe | Low-frequency dynamics in a shock-induced separated flow[END_REF][START_REF] Pasquariello | Unsteady effects of strong shock-wave/boundary-layer interaction at high reynolds number[END_REF].

Almost all numerical SWBLI literature studies are carried out for canonical geometries, and they deviate from typical experimental geometries, where regions of favourable pressure gradient are often located downstream of shock-wave / boundarylayer interactions to prevent wind-tunnel blockage (see e.g. [START_REF] Bookey | New experimental data of STBLI at DNS/LES accessible reynolds numbers[END_REF]; [START_REF] Ganapathisubramani | Effects of upstream boundary layer on the unsteadiness of shockinduced separation[END_REF]; [START_REF] Schreyer | Experimental investigation of two hypersonic shock/turbulent boundary-layer interactions[END_REF]). Experiments of [START_REF] Grossman | Confinement effects on regular-irregular transition in shock-wave-boundary-layer interactions[END_REF] provide evidence of a direct relationship between the size of the separation bubble and the expansion region downstream; such effects are often not reported in experimental investigations.

Numerical studies [START_REF] Grilli | Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction[END_REF][START_REF] Grilli | Large-eddy simulation of a supersonic turbulent boundary layer over a compression-expansion ramp[END_REF][START_REF] Duan | Direct numerical simulation of a supersonic turbulent boundary layer over a compression-decompression corner[END_REF] performed for a compression-expansion corner documented the frequency content of the favourable pressure gradient region, and confirmed the prevalence of low-frequency unsteadiness also for such geometries. However, little attention has been paid to the frequency link between the SWBLI and the favourable pressure gradient regions. Moreover, as far as the authors know, any possible effects induced by the downstream pressure gradient on the low-frequency dynamics of the separation bubble have not yet been documented.

Furthermore, high-fidelity numerical studies [START_REF] Wu | Analysis of shock motion in shockwave and turbulent boundary layer interaction using direct numerical simulation data[END_REF][START_REF] Wu | Direct numerical simulation of supersonic turbulent boundary layer over a compression ramp[END_REF][START_REF] Loginov | Large-eddy simulation of shock-wave/turbulent-boundary-layer interaction[END_REF] that aim to understand the mechanisms surrounding shock unsteadiness typically use canonical geometries, while the wind-tunnel experiments that they are based on and use for comparison [START_REF] Zheltovodov | Shock waves/turbulent boundary-layer interactions -fundamental studies and applications[END_REF][START_REF] Bookey | New experimental data of STBLI at DNS/LES accessible reynolds numbers[END_REF], employ a downstream pressure gradient for reasons discussed earlier. Direct comparisons are therefore impeded.

A combined experimental-numerical study with identical geometrical and flow-conditions is therefore essential to isolate the effects of a downstream favourable pressure gradient on mean-flow modifications and the unsteady dynamics of shockwave boundary layer interactions.

The objective of the present study is to characterise the lowfrequency dynamics of a strongly separated shock/separationbubble system in the presence of a favourable downstream pressure gradient. To achieve this goal, we jointly analyze the experimental data of a M ∞ = 2.52 compression-corner interaction followed by an expansion corner [START_REF] Ramaswamy | Control of shock-induced separation of a turbulent boundary layer using air-jet vortex generators[END_REF] and high-resolution large-eddy simulations (LESs) for similar geometrical and flow conditions. Analyses will primarily focus on a +24.0 • compression and -24.0 • expansion case, with additional information derived from a +20.0 • / -20.0 • experimental setup.

For this, as a first step, we compute the same +24.0 • compression corner and -24.0 • expansion corner configuration as studied in the experiments. This baseline computation is hereafter labeled as c24.0 -e24.0. To isolate the effects of flow compression, a second LES computation with a single 24 • compression corner was also carried out (case c24.0). Finally, to exclude the effects of interaction length on the flow unsteadiness [START_REF] Agostini | Zones of influence and shock motion in a shock/boundary-layer interaction[END_REF], we performed a third LES (c21.7) with the corner angle reduced to 21.7 • , in order to obtain the same separation location as in the c24.0 -e24.0 case.

Experimental and numerical setup

Experiments

Experimental investigations were conducted in the trisonic wind tunnel at RWTH Aachen University. The intermittentlyoperating vacuum-storage facility achieves stable run times of 3-4 s. Air as the working fluid is supplied to a settling balloon. The air is dried to avoid condensation effects, keeping the relative humidity below 6 %. The 0.4×0.4 m 2 test section is optically accessible through two circular windows on either side The selected Mach number (M ∞ = 2.52) therefore determines the free-stream unit Reynolds number (Re ∞ = 9.6 • 10 6 1/m).

For the two-component particle-image velocimetry (PIV) measurements, we used a 532 nm Litron NANO-L pulsed PIV laser with a maximum pulse energy of 200 mJ to illuminate the di-ethyl-hexyl-sebacate (DEHS) seeding particles in the flow. Two FlowSense EO 11M cameras (equipped with 4008 px×2672 px CCD sensors and Tamron SP AF 180mm f/3.5 objectives) covered the same field of view, the complete SWBLI region (-4.5δ ≤ x ≤ 4.5δ; δ = 10.4 mm is the incoming boundary-layer thickness), at a spatial resolution of 43 px/mm. The cameras were triggered alternately to increase the overall acquisition rate of the low-speed PIV system to 6 Hz. Further details on the flow facility, wind-tunnel model, and PIV setup can be found in [START_REF] Ramaswamy | Control of shock-induced separation of a turbulent boundary layer using air-jet vortex generators[END_REF].

For focusing-schlieren visualizations, we used the system described by [START_REF] Schauerte | Design of a high-speed focusing schlieren system for complex three-dimensional flows[END_REF]. Images were taken at the model centreline with a Photon FastCam SA5 high-speed camera at a recording frequency of 20 kHz, a resolution of 704× 520 px, and an exposure time of 2 µs. The recorded image sequences allow the extraction in the physical space of the dynamics associated with the low-frequency unsteadiness of the shock/separation-bubble system. Full analyses of the related dynamics in the spectral space with approaches such as Dynamical Mode Decomposition are challenging, however, due to the unresolved, high-energy, mixing layer and turbulent dynamics being folded into the resolved low-frequency range due to aliasing effects. The shock motion and frequency can be faithfully resolved outside of the boundary layer.

Large-eddy simulations

All computations were carried out using the FLU3M code with second-order accuracy both in time and space using the same modelling parameterisation as in [START_REF] Jiang | Velocity spectrum estimation in shock-wave/turbulent boundary-layer interaction[END_REF]. Time integration is implicit with a typical maximum CFL number of 11. The space scheme is hybrid centred-shock capturing discriminated by Ducros' sensor [START_REF] Ducros | Large-eddy simulation of the shock/turbulence interaction[END_REF]. The cutoff ϵ added to the denominator to protect acoustic waves from overdissipation in irrotationnal regions is set such that a plane acoustic wave of frequency U ∞ /δ with a 100 dB amplitude propagating in the potential flow will result in a 0.01 value.

Ducros' sensor value is used as a continuous blending function between the centered scheme and the [START_REF] Roe | Approximate riemann solvers, parameter vectors, and difference schemes[END_REF] scheme with an Harten entropy fix [START_REF] Harten | Self adjusting grid methods for onedimensional hyperbolic conservation laws[END_REF]) and a minmod limiter (see [START_REF] Roe | Characteristic-based schemes for the euler equations[END_REF], for instance). The explicit subgrid scale modelling relies on the selective mixed-scale model by [START_REF] Lenormand | Subgrid-scale models for Large-Eddy Simulation of compressible wall bounded flows[END_REF]. A compressible variant of the synthetic eddy method (SEM) proposed by [START_REF] Jarrin | A synthetic-eddy method for generating inflow conditions for les[END_REF] and extended to wall-bounded flow by [START_REF] Pamiès | Generation of synthetic turbulent inflow data for large eddy simulation of spatially evolving wall-bounded flows[END_REF], is et al. (2012). Compared with this reference, the surface density of synthetic eddies mapping the inflow plane has been raised from 10 to 15, as in [START_REF] Jiang | Velocity spectrum estimation in shock-wave/turbulent boundary-layer interaction[END_REF], to reduce the occurrence of extreme events in the generated random fields.

We derived the present mesh parameters from [START_REF] Jiang | Velocity spectrum estimation in shock-wave/turbulent boundary-layer interaction[END_REF]: for all computations, the inflow is located more than 11.5δ in f low upstream of the interaction region, and the width of the computational domain is set larger than 7 separation-bubble heights. The size of the bubble varies between the cases, resulting in different extensions in the spanwise direction. The total cell counts range between 87 and 181 millions, as shown in table 2.

The typical grid resolutions in wall units are set to ∆x + = 26.5 and ∆z + = 10.5. In the wall-normal direction, the mesh has a typical stretching rate of 3 %, starting from ∆y + wall = 0.7, with 170 cells located within the boundary-layer thickness. The body-fitted mesh is built using the freely available gridgenc tool computing a numerical Schwarz-Christoffel conformal mapping through the CRDT algorithm by [START_REF] Driscoll | Numerical conformal mapping using crossratios and delaunay triangulation[END_REF].

All simulations have been run for 2.2 million time steps, thus providing data sampled over a physical duration of 0.22 s. The corresponding number of typical low-frequency cycles distributed over this duration varies, but is over 50 for all computations (see Tab. 2). This run time ensures good statistical convergence for all spectral estimators in all cases.

Validation

The mean-velocity and Reynolds-stress profiles feeding the SEM boundary conditions were obtained from a preliminary LES of a M ∞ = 2.52 flat-plate boundary layer undergoing transition. The location at which these profiles were sampled has been adjusted so that the SWBLI computation matches the boundary-layer parameters listed in Tab. 1 at location x/δ = -4.3.

Comparisons of the streamwise evolution of the friction coefficient, momentum thickness, and shape factor between the preliminary LES including the transition process and the SEMfed LES demonstrate that the latter is able to recover the expected mean distributions in less than 11δ in f low . At the most downstream locations for which the boundary layer remains unaffected by the adverse pressure gradient, namely between 11.5 δ and 14 δ away from the inflow depending on the flow case, differences between the two LES are lower than 1% for the mean velocity profile, the various boundary layer thicknesses, as well as for the shape factor. They rise up to 2.1% at most for the friction coefficient.

It may be noted that O (10δ) relaxation distances are in par with those found in [START_REF] Mo | Direct numerical simulation of hypersonic wall-bounded turbulent flows: An improved inflow boundary condition and applications[END_REF] for a supersonic boundary layer with the Turbulence Fluctuation Library approach. They are, however, significantly lower than the ones described in recent DNS works by [START_REF] Ceci | Numerical tripping of high-speed turbulent boundary layers[END_REF] and [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers: effect of spatial evolution and reynolds number[END_REF], using either the recycling or Digital-Filter (DF) methods, the latter being conceptually close to SEM. In fact, the main limitation of the DF method described in [START_REF] Ceci | Numerical tripping of high-speed turbulent boundary layers[END_REF] and of the SEM method is similar: a high level of noise generated at the inflow. In the present work, this noise results in an overestimation of the wall-pressure fluctuation by 3.8 dB with respect to LES encompassing the transition at the location where the two computations are compared. The overestimation is essentially associated with the U ∞ /δ frequency range, making it unlikely to interfere with the low-frequency dynamics of the separated region.

The higher relaxation distances from the inflow condition described in [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers: effect of spatial evolution and reynolds number[END_REF] and [START_REF] Ceci | Numerical tripping of high-speed turbulent boundary layers[END_REF] mostly stem from the fact that several demanding metrics are used in these studies, based for instance on the von Kármán integral equation. Considering the friction coefficient, the SEM method results in an underestimation of 2.8% with respect to the experiments at x/δ = -4.3, 14δ downstream of the inflow, whereas such a threshold is reached within ≃ 15 -20 δ in [START_REF] Ceci | Numerical tripping of high-speed turbulent boundary layers[END_REF]. This difference does not imply that the present SEM method is significantly more efficient than the DF approach: in DNS, the unrealistic inflow fluctuations have to cascade down to the Kolmogorov scales before being dissipated and to then make the physical regularization of the flowfield possible. In the present LES framework, the dissipation takes place in the inertial subrange thanks to the subgrid model, making the cascade time, and, most probably, the recovery distance, shorter.

Beyond the skin friction coefficient, the overall velocity profiles are in very good agreement between the LES and the experiments, as visible in Fig. 1. The main discrepancy occurs in the streamwise fluctuation profile above the log layer: the LES fails to reproduce the secondary peak associated with very large structures that is emerging in the experiments due to the high Reynolds number. The reason for this difference is that the inflow boundary is located O(10δ) upstream of the sampling station, i.e. not far enough for structures with a larger length scale to sustain.

The preliminary LES of a laminar boundary layer undergoing transition does not exhibit such an artifact, as visible in the lower plot of Fig. 1. The upper plot, showing Van Driesttransformed profiles, however, demonstrates that the underestimation of the streamwise fluctuations in the upper range of the logarithmic region and above does not translate in a significant alteration of the mean velocity profile at those locations. Moreover, [START_REF] Jiang | Velocity spectrum estimation in shock-wave/turbulent boundary-layer interaction[END_REF] found that the underestimation of the streamwise fluctuations, associated with the range S t δ < 0.1, was restricted to the incoming boundary layer and the mixing layer developing over the separated flow, without significantly distorting the velocity spectra in the recirculating region. It is therefore unlikely that this artifact could result in a misprediction of the size of the separated region. Lastly, the present SWBLIs induce large-to-massive separations, which makes the influence of the very large structures of the incoming boundary layer on the low-frequency motion of the shock, as highlighted by [START_REF] Ganapathisubramani | Effects of upstream boundary layer on the unsteadiness of shockinduced separation[END_REF], less significant.

For pragmatic reasons, we chose to stick to the O(10δ) SEM methodology. This choice implies the assumption that the slight local deficit in streamwise fluctuation will have no impact on the low-frequency dynamics of the separated region, which is the main source of interest of the present paper. We base this assumption on extended experience in meticulous comparisons between LES and unsteady experimental data [START_REF] Agostini | Zones of influence and shock motion in a shock/boundary-layer interaction[END_REF][START_REF] Jiang | Velocity spectrum estimation in shock-wave/turbulent boundary-layer interaction[END_REF]. Nonetheless, this point has to be confirmed by taking advantage of the experimental database. In return, the extended physical time enables advanced analyses in the frequency space with a high level of statistical confidence.

We compare the PIV and LES data in the interaction region for the reference case c24.0 -e24.0. The global structure of the compression/expansion corner flow, visualized using the isolevels of the streamwise velocity and streamlines, is shown in Fig. 2. The LES accurately reproduces all flow features from the separation point to the relaxation region downstream of the reattachment -which is significantly altered by the expansion corner. The most noticeable discrepancy between experiment and simulation is an underestimation by about 7 % of the extent of the separated region by the latter. Contrary to RANS computations, LESs with accurate incoming mean velocity profiles and adequate resolution generally result in very good predictions of the size of the separation bubble. The identified discrepancy is therefore more likely related to the influence of the sidewall boundary layers in the wind tunnel. Sidewall flows have been demonstrated to increase the size of the separated region in the mid-span plane of shock-reflections compared with spanwise homogeneous (periodic) computations, see for instance [START_REF] Garnier | Stimulated detached eddy simulation of threedimensional shock/boundary layer interaction[END_REF]. [START_REF] Wang | Numerical study of oblique shock-wave/boundary-layer interaction considering sidewall effects[END_REF] showed that this threedimensional flow structure is induced by the swept SWBLI formed on the sidewalls by the incident shock.

For compression-corner SWBLI, interactions between the sidewall flow and the recompression shock take place downstream of the separation point. Compression-corner interactions are therefore expected to be less prone to sidewall effects than shock-reflection SWBLI. However, [START_REF] Poggie | Numerical simulation of sidewall influence on supersonic compression ramp interaction[END_REF] have reported increases of the interaction length in compression corners similar to the ones found in shock reflections. Such small differences in separation length can easily be compensated by rescaling the separated region with the actual interaction length L int , defined as the distance between the corner and the extrapolation of the head shock down to the wall x head . We estimated L int from the experiments and the c24.0 -e24.0 LES. The relaxation region downstream of reattachment scales more naturally with the distance between the compression and expansion corners, which is identical for the experiments and computation.

This dual-length scale dependence is taken into account by considering a composite reference length scale L LES for the computation. In the separated region, the actual interaction length L LES int is considered: L LES = L LES int . Downstream, from the reattachment point up to the expansion corner, the length scale L LES is increased linearly until it matches the experimental interaction length L Exp int at the corner: L LES = L LES int + x-x reat x expa -x reat L Exp int -L LES int . Downstream of the corner, the same lengthscale is considered for the LES and the experiments: L LES = L Exp int . This way, the usual normalization is considered for the interaction region, and we are able to recover the same normalized location of the expansion corner in both the experiments and the LES.

x / L y / L -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (a) u x / L y / L -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (b) u ′ x / L y / L -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (c) v ′ x / L y / L -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (d) sign(u ′ v ′ ) × √ |u ′ v ′ |
The normalized streamwise evolution of the mean and fluctuating velocity profiles is plotted in Fig. 3. It is clearly visible that the LES accurately reproduces the flow developing over the compression corner / expansion corner geometry. The most noticeable difference occurs once again in the streamwise velocity fluctuation profiles (see Fig. 3(b)), where the peak level at the heart of the mixing layer developing over the separated region is underestimated by about 15 %. Such an underestimation was already observed in [START_REF] Jiang | Velocity spectrum estimation in shock-wave/turbulent boundary-layer interaction[END_REF] and was traced back through spectral analyses to the lack of very large structures in the incoming boundary layer. The same explanation most probably holds here.

The validation process was expanded to gain further information on the flow dynamics by comparing the Proper Orthogonal Decomposition (POD) of the 2D (u, v) velocity fields obtained from the experiments and the LES. POD has been computed using the snapshot method from 782 snapshots with a sampling rate of 6 Hz (experiments) and 88800 snapshots with a sampling rate of 400 kHz (LES). Note that the highest available sampling rate has been used in the latter case to make it possible to compute spectral estimates from the temporal modes over the full frequency range. Maps of normalized 2D turbulent kinetic energy are plotted in Fig. 4 for the six most energetic modes found in the experiments (left column) or the c24.0 -e24.0 LES (center column).

The relative energy contents of the modes are also reported in Fig. 4. The modes obtained from the experiments carry relatively overal more energy than their numerical counterparts. This difference can be explained with the differing vertical extents of the domains over which the inner product is computed in the experiments and the numerical simulation (the streamwise extent is similar and its upstream and downstream bounds, set from the experiments, are denoted by a dashed line onto the LES maps). Moreover, levels of normalized turbulent kinetic energy (obtained by dividing the non-normalized spacial modes by their respective eigenvalues) are lower for the experiments than for the simulation (see Fig. 4). Hence, the non-normalized fluctuations associated with each mode are similar in absolute energy content.

The two first modes are similar in the experiments and the LES, demonstrating that the dominant dynamics of the flow are well predicted by the simulation. The third modes sightly differ in the location of the intermediate structure. This is most probably caused by the slight difference in size of the separated region between the experiments and the computation, considering that the mode is driven both by the (variable) size of the separation bubble and the (fixed) extent of the ramp.

Higher-order modes significantly differ between LES and experiment. They occasionally bear a resemblance in the separated region, but differ further downstream. This behavior is possibly due to the difference in separation length/ramp length ratio: compare for instance modes 6 from the experiment and mode 5 from the numerical simulation. We will show that the corner indeed induces compression as well as truncation in space for the modes at the end of the next section.

Results and discussion

Equivalent compression corner flows

Comparing the c24.0 -e24.0 computations with the two other computations, for which the downstream expansion corner has been removed, helps assessing the influence of the downstream favourable pressure gradient on the separated region. The skin-friction and wall-pressure coefficient distributions are displayed in Fig. 5 for the three computations. They demonstrate that case c21.7 (reduced ramp angle) indeed results in the same separation location as the c24.0 -e24.0 case, but that the lack of a downstream favourable pressure gradient leads to a slightly delayed reattachment. In contrast, the c24.0 case exhibits a very large separation region inducing a true pressure plateau. The expansion corner in the nominal setup has indeed a significant impact on the mean development of the SWBLI and leads to a shrinking of the separation bubble by more than 30 %.

Spectral analysis

On the basis of the scaling well acknowledged for turbulent SWBLIs [START_REF] Dupont | Space and time organization in a shock-induced separated boundary layer[END_REF], we expected that the c24.0 -e24.0 and c21.7 cases would exhibit similar low frequency dynamics (associated with the breathing of the separated region), since they have the same interaction lengths. Such dynamics are indeed evidenced by the wall pressure spectrum at x * = (xx head ) /L int = 0, in the vicinity of the separation point. Standard turbulent SWBLIs exhibit a clustering of the lowfrequency power around a Strouhal number based on the length of the interaction region of S t L = ( f L int ) /U ∞ ≃ 0.03. Both the c21.7 and c24.0 cases indeed exhibit such a clustering, as visible in Fig. 6, for which the premultiplied wall-pressure power spectra have been normalized by the power content of the lowfrequency band 0 ≤ S t L ≤ 0.5 to make comparison easier. The c24.0 -e24.0 case, however, exhibits a narrower low frequency distribution whose centre is shifted up by 50 % because of the disappearance of the energy content in the lower part of the low-frequency range. The lack of dynamic content at the lowest frequencies persists until the region downstream of the expansion corner, as visible in the center plot of Fig. 7. The frequency range over S t L = 0.03 (denoted by the horizontal dashed line) dominates the streamwise distribution of the norm of the wall pressure cross-spectrum between the current location and the beginning of the interaction. The norm was normalised by the local pressure variance to highlight the energetically prevailing linear coupling. The single corner cases, on the other hand, exhibit significant linear coupling for frequencies below S t L = 0.03 downstream of the corner (located at x * = 1), as visible in the upper and lower maps of Fig. 7.

As mentioned in Sec. 2.1, high-speed schlieren recordings are the only experimentally available time-resolved measurements from whose low-frequency dynamics can be assessed. Dynamic mode decomposition (DMD) has been applied to the schlieren-image sequences by [START_REF] Schauerte | Design of a high-speed focusing schlieren system for complex three-dimensional flows[END_REF] to obtain a first estimate of the low-frequency shock motion. However, the sampling rate of 20 kHz results in severe aliasing when moving to the spectral space because both the mixing layer dynamics and the turbulent structures from the incoming boundary layer have typical frequencies higher than the Nyquist frequency of 10 kHz. Such spectral distortions make it difficult to assess the complete related dynamics from DMD (or any kind of time-only Fourier decomposition) for an alteration of the spectral power distribution with respect to the canonical turbulent SWBLI in the low-frequency range, which would be similar to the one seen in the c24.0 -e24.0 LES.

One way to discriminate between original and aliased dynamics in the frequency space is to take advantage of their respective distinct distributions in space. In the present work, this has been achieved by means of spectral Proper Orthogonal Decomposition (sPOD). sPOD allows to decompose the unsteady flow into a sum of decoupled (orthogonal) modes in both the frequency and space domains. Refer to [START_REF] Schmidt | Guide to spectral proper orthogonal decomposition[END_REF] for a practical introduction to this decomposition.

We applied sPOD to pseudo-schlieren fields extracted from the unsteady c24.0 -e24.0 LES database with both the original sampling rate (400 kHz) and with a 20 times rawer subsampling to match the sampling rate of the experiments. Pseudoschlieren data have been computed by averaging the vertical density gradient in the spanwise direction over a width of approximately one boundary-layer thickness, reproducing the estimated depth of field of the experimental pictures. A mask has been applied to the flow field to remove the region downstream of the Mach wave issued from the compression corner, in order to highlight the dynamics associated with the motion of the head shock. The Welch-snapshot approach of the sPOD was retained to minimise the computational burden: the whole time signal has been split into 112 segments overlapping by 50 %, whose individual duration roughly corresponds to 2 typical low-frequency timescales. This approach ensures both a reasonable statistical convergence and a frequency resolution suitable to analyse the low-frequency dynamics.

We plotted the premultiplied sPOD spectra encompassing all the modes obtained from the c24.0 -e24.0 original database and from the subsampled base in the top part of Fig. 8 in red and orange, respectively. A vanishing bump in the low-frequency range is discernible for the original sampling rate. The associated energy, however, is way too low to allow the reliable identification of a typical frequency scale, contrary to the plots in Fig. 6. The reason for this behavior is that the global sPOD spectrum encompasses all density-gradient fluctuations found in the unmasked part of the field of view. It includes the lowfrequency shock motion that was (locally) dominant in Fig 6, but also the density variation and radiated pressure waves stemming from the mixing layer dynamics and from the boundary layer turbulence, which both carry way more overall energy. Since the latter are associated with frequency ranges higher than half of the sub-sampling rate, spectral folding associated with the aliasing completely distorts the global spectrum obtained from the sub-sampled database. As a consequence, it is no longer possible to shed light on distinct low-frequency physics.

In the sPOD framework, however, fluctuations at a single frequency are decomposed into several modes with distinct distributions in space. It is unlikely that the aliased high-frequency structures share common space modulations with the low-frequency shock motion. It is therefore possible to distinguish between true and aliased fluctuations by considering modes on an individual basis, possibly using their structure in space as a selection criterion.

The sPOD spectrum of the first (dominant) modes obtained from the c24.0 -e24.0 database at the original sampling rate (blue plot in the top part of Fig. 8) exhibits a clear bump in the low-frequency range, whose center matches the one at S t L ≃ 0.055 found in Fig. 6. Moreover, the associated distribution in space, not shown here for the sake of conciseness, is dominated by the region of the head shock. This observation confirms that the mode is indeed associated to the low-frequency shock motion, which is in turn related to the breathing of the separated region, as shown by [START_REF] Piponniau | A simple model for low-frequency unsteadiness in shock-induced separation[END_REF] amongst others.

The same sPOD procedure was also applied to the two other LES databases with full ramps (c21.7 and c24.0). Note that for the c24.0 database, the individual size of the Fourier-transformed segment has been increased by 50 % to take into account the lower typical frequency associated with larger separated regions. The number of segments, and therefore modes, has in turn been reduced by 50 %. The spectra of the dominant sPOD modes obtained from the original c21.7 and c24.0 databases are not displayed here for the sake of conciseness. It has nonetheless been checked that they display low-frequency bumps whose centers are shifted down to S t L ≃ 0.03, as in Fig. 6.

For the full-ramp cases, two distinct modes have space distributions compatible with the physics of the low-frequency shock motion for each frequency in the low-frequency range, whereas a single mode was found in the c24.0 -e24.0 sPOD.

The sPOD spectra of the dominant modes obtained from the subsampled databases are not significantly distorted by aliasing, as visible in the cyan plot in the upper Fig. 8. As expected, aliased energy is distributed over low-frequency sPOD modes of higher ranks. The possibility to reliably identify the typical Strouhal number associated with the low-frequency range from the experimental schlieren visualizations sampled at 20 kHz has thus been demonstrated. Consequently, we applied sPOD to the experimental schlieren database for the original 24°compression/expansion corner, as well as for another experimental case with 20°compression and expansion and the same horizontal extent of the compression ramp. Note that for those decompositions, four distinct sPOD modes with the expected distribution in space (not shown for brevity) have been identified in the low frequency range, possibly because of optical aberrations.

The cumulative sPOD spectra obtained for a 20 kHz sampling rate, encompassing the first dominant modes for the c24.0-e24.0 LES, the first two dominant modes for the c21.7 and c24.0 cases, and the first four dominant modes for the two experimental configurations are displayed in the bottom part of Fig. 8. They confirm that the experiments share the same power depletion of the very low frequency range as identified in the c24.0 -e24.0 LES in comparison with the computations for non-truncated compression ramps.

There are indications that the depletion is slightly less severe in the 20°experimental case. This observation may be explained by a lower constraint on the separated region induced by the downstream favorable pressure gradient because of the the relatively larger expansion ramp with respect to the extent of the separated region; the latter is reduced because of the milder initial adverse pressure gradient.

Proper orthogonal decomposition 3.3.1. Velocity-based POD

Although the low-frequency dynamics of the separated region appear to be significantly affected by a downstream favourable gradient exerted by the expansion corner, it is unclear how this gradient modifies the frequency-space structure of the interaction and downstream relaxation regions. Therefore, we computed time-resolved POD modes for the c21.7 case to complement the corresponding POD modes for the c24.0 -e24.0 LES as a first attempt to relate flow entities coherent in space with given frequency distributions.

Turbulent kinetic energy maps for the six first modes obtained from the c21.7 computation are plotted in the right column of Fig. 4. The three first modes appear to be elongated versions of the three first modes of the c24.0 -e24.0 case, while modes 5 and 6 look similar to modes 5 and 4 in the experiments. It should, however, be considered that modes 2, 3 and 4 of the c21.7 case have to be associated with modes 1, 2 and 3 of the c24.0 -e24.0 case. This illustrates the difficulty to match the c21.7 and c24.0 -e24.0 modes; it is challenging to infer the kind of distortion in space induced by the expansion corner.

To identify partial dynamics common to both cases, we computed a joint POD encompassing snapshots from both the c21.7 and c24.0 -e24.0 LESs with the same full sampling rate. As expected, modes obtained from the diagonalization of the timecorrelation matrix are mostly dominated by fluctuations associated with one of the single flow cases and match the modes obtained when a single database is considered (see Fig. 4). Most often, however, flow patterns can also be identified for the respective other case, yet with much lower energy. These "ghosts" are related to the flow patterns of another mode, highlighting common dynamics.

Based on this methodology, we conclude that modes 1 to 3 of the c24.0 -e24.0 case are most likely related to modes 2 to 4 of case c21.7, especially for the last pair (c24.0 -e24.0 mode 3 ; c21.7 mode 4). Moreover, the approach allows to clearly infer that the expansion corner causes modes 5 and 6 of the c21.7 computation (corresponding to modes 8 and 9 in the joint POD) to merge into the composite mode 4 (mode 10 in the joint POD), as evidenced in Fig. 9. "Ghosts" of case c24.0 -e24.0 in joint modes 8 and 9 have the same structure in space as mode 10, whereas the "ghost" of the c21.7 LES for mode 10 appears to be a blend of modes 8 and 9. These findings may explain the previous observation that no mode obtained from the experiments matches the c24.0-e24.0 mode 4, whereas c21.7 modes 5 and 6 have matching experimental counterparts. The length scales of the latter two modes, whose length naturally scales with the size of the separation bubble, are compatible with the extent of the compression ramp in the experiments. For case c24.0 -e24.0, the slightly reduced lengthscale, due to the reduced separation length, is no longer compatible with the unchanged extent of the compression ramp, and modes cannot be sustained without merging.

After identifying the links between modes, it is possible to study their temporal behaviour by analyzing the power spectra obtained from the temporal modes. Spectra obtained for the dominant modes are plotted in Fig. 10. The upper plot shows that modes 1 for both computations have a similar frequency distribution and therefore cannot account for the difference in the low-frequency region seen in Fig. 6.

The respective modes 2 exhibit a shift towards higher frequencies from c21.7 to c24.0 -e24.0, that could be compatible with the observations made on the wall-pressure power spectra in Fig. 6. However, as mentioned above, mode 2 for case c24.0 -e24.0 has more likely to be related to mode 3 of case c21.7. Indeed, as visible in the middle plot of Fig. 10, the frequency range associated with c24.0 -e24.0 mode 2 has a slightly better match with c21.7 mode 3 than with c21.7 mode 2, and the frequency ranges of the c24.0 -e24.0 mode 3 and c21.7 mode 4 overlap almost perfectly.

Lastly, the frequency distribution of c24.0 -e24.0 mode 4 is a mix of the distributions of the two related modes of the c21.7 case. A net power loss results in the low-frequency range for the c24.0 -e24.0 single mode compared to the c21.7 pair of modes.

The lack of the lowest frequency range in case c24.0-e24.0 thus appears to result from a shift of the POD modes towards higher frequencies, from the disappearance of low-frequency dominated modes, or from a combination of the two. Further refined analyses are required to clarify that point.

Pressure-based POD

The difficulties in identifying the origin of the depletion in the low-frequency range (shown in Fig. 6) from the above described POD analyses partly stem from the non-dominant turbulent kinetic energy content associated with this range. As a matter of fact, the high level of low-frequency power spectral density found in the pressure spectrum has no direct dynamical significance since it is merely related to time intermittency between pre-and post-shock states, regardless of the amplitude of the shock motion because of the pressure discontinuity. The low-frequency power-spectral density therefore scales with the square of the pressure jump across the shock, which is much higher than the pressure variance associated with the turbulent field, see for instance [START_REF] Agostini | Mechanism of shock unsteadiness in separated shock/boundary-layer interactions[END_REF]. The same intermittency-driven fluctuations also occur for the velocity field in the near-shock region, but here the jump across the shock is of the same order of magnitude as the turbulent fluctuations. Additional low-frequency velocity fluctuations associated with the shock motion are indeed found in the interaction region, as evidenced by the conditional analyses of [START_REF] Piponniau | A simple model for low-frequency unsteadiness in shock-induced separation[END_REF] highlighting a breathing of the separated region. However, these additional fluctuations do not result in a large amount of turbulent kinetic energy since the velocity gradient in the direction normal to the dividing streamline is weak with respect to the variation across the shock. As a consequence, alterations of the velocity field that are related to the aforementioned depletion are unlikely to leave a clear trace in a velocitybased POD.

A POD has therefore been carried out by considering the pressure field at the wall and over four streamwise-wall normal planes evenly distributed in the spanwise direction. We included the same mask as for the sPOD analyses of the schlieren pictures in the definition of the inner product in order to remove possible contributions from the expansion-corner region. The first dominant mode obtained from this decomposition is associated with the streamwise shock motion for all the three computation cases: fluctuations are mostly confined in the excursion region of the head shock and are essentially two-dimensional (see Fig. 11 for the c21.7 and c24.0 -e24.0 LESs).

The power spectra computed from the time coefficients of the first pressure-POD mode are plotted in Fig. 12. These plots are mostly similar to the ones displayed in Fig. 6 up to S t L ≃ 0.2, which confirms that the first mode is encompassing most of the dynamics of the pressure fluctuations at the foot of the head shock -on the basis of which the low-frequency unsteadiness in SWBLI is most often characterized.

It is nonetheless worth noting that the power spectral density distributions for the second and third modes for the c24.0 computation are also dominated by the low-frequency content. The distributions in the low frequency range are narrower than for the first mode and peak at S t L ≃ 0.015. They are complemented by about 15 % of the mode total power distributed in the frequency range 0.2 < S t L < 2. The summed power contribution of these two modes in the low-frequency range corresponds to slightly less than half of the power of the first mode. Their structures in space are three-dimensional and lead to a strong spanwise corrugation of the shock as well as of the reattaching flow. These modes are therefore most likely related to the low-frequency spanwise modulations of the flow by Görtler-like vortices developing in the downstream part of the interaction region, as described by [START_REF] Priebe | Low-frequency dynamics in a shock-induced separated flow[END_REF].

Modes with similar spanwise-corrugated structures also occur for the c21.0 and c24.0 -e24.0 computations, but the spectral power distribution is fully dominated by the 0.2 < S t L < 2 range, with a vanishing summed power at low frequency of at most 10 % of the corresponding power of the first mode. It therefore seems that those modes do not correspond to what is traditionally meant by low-frequency unsteadiness in turbulent SWBLI flow: they appear to be promoted in the low-frequency range in the c24.0 LES because of the unusually large extent of the separated region, whose height is about equal to the incom-ing boundary layer thickness.

Since the first POD modes of the c21.7 and c24.0 -e24.0 setup account for most of the low-frequency fluctuations of pressure, it is possible to extract fluctuations linearly related to the typical low-frequency dynamics for any given quantity by simply multiplying the first eigenvectors of the pressure POD with the snapshots of the quantity under consideration. Note that the "modes" thus obtained do not exhibit the same energy optimality and orthogonality as the true pressure POD modes.

As mentioned previously, the shock motion at low frequency has been conditionally related to the breathing of the separated region. One efficient way to characterize the breathing process is to extract the dividing streamline of the separation bubble. This streamline can be easily extracted for time-or conditionallyaveraged flow fields, but is complex to extract from an unsteady three-dimensional flow field, as it requires the sampling of the the whole three-dimensional flow field over which the dividing streamline has to be computed at each time step. The dividing streamline can, however, be linked to the mass fluxes through the mass-conservation equation. Considering this equation is also of interest regarding [START_REF] Piponniau | A simple model for low-frequency unsteadiness in shock-induced separation[END_REF]'s model, which relates the low-frequency unsteadiness with considerations on mass conservation within the separated bubble.

Hereafter, we therefore analyze the global spacial structures of the velocity vector field linearly related to the low-frequency shock motion by considering the mass fluxes integrated along wall-normal grid lines from the wall up to a point x i j , y i j in the flow field. Here, i and j denote indices of the structured grid, which vary in the streamwise and wall-normal directions, respectively. The integrated mass flux I MF i, j then reads:

I MF i, j = l i, j wall ρ u.n dl along a i = constant gridline.(1)
The value of this integral is independent of the integration path from the wall to field point x i j , y i j for a time-averaged flow field homogeneous in the spanwise direction as a consequence of the time-averaged mass-conservation equation. Therefore, there is no mean mass flux across lines of constant IF M x i, j , y i, j , which means that those lines coincide with streamlines of the Favre-averaged velocity field. Such isolines are plotted for the mean flow of the c21.7 and c24.0-e24.0 computations in Fig. 13(a). The plot demonstrates that the mean distribution of the mass flow rate in space is essentially similar upstream of the expansion corner in both cases.

The various properties listed above formally no longer hold for an unsteady flow field because of additional terms in the three-dimensional, time-dependant mass-conservation equation, namely the unsteady volumetric compression/expansion term and the spanwise mass flux. However, the streamwise mass flux dominates over the two aforementioned terms when considering flow fluctuations with large time scales with respect to the convective time scales, i.e. for S t L ≪ 1, which are also mostly two-dimensional. Consequently, lines of constant I MF i, j values become reasonable approximations of lines of constant massflow rate. As visible in Figs. 11 and12, the first mode of the pressure POD meets those criteria. We then computed the density and momentum fluctuations linearly related to this mode as described above. The momentum in the e x -and e z -directions is required to compute I MF from eq. 1. The momentum component in the spanwise e zdirection and the density helped to verify that the ∂ρ ∂t dV and ρ u.e z dS terms can be neglected with respect to I MF i, j , being typically three to four orders of magnitude smaller.

Lastly, the time-dependant I MF i j quantity was computed from the flow field reconstructed by adding the mean quantities to the fluctuations. The extremal kinematics of the I MF associated with the first POD mode can then be highlighted by conditionally averaging the time-dependant I MF flow field over times for which the time coefficients of the first POD mode fall within its 10 % lowest and highest values. In the POD framework, this is obtained by multiplying the time coefficient averaged over the 10 % upper and lower tails of its probability density function with the sum of the mean I MF field and the first I MF "mode".

Isolevels of these conditionally-averaged I MF are plotted in Figs. 13(b) and 13(c) for the c21.7 and c24.0 -e24.0 LESs, respectively. Both setups exhibit a similar farthest upstream excursion of the head shock. As a consequence, the IMF line differences between the two flow fields are similar to the ones seen previously for the mean flow in Fig. 13(a).

In contrast, downstream shock excursions and bubble shrinking are reduced in the c24.0 -e24.0 computation compared with the c21.7 case. Moreover, the reduction in amplitude is not evenly distributed in space. Differences in mass-flux distribution between the most upstream/downstream shock locations are smoothed more rapidly in the region downstream of the Mach wave issued from the reattachment point, but still upof the first characteristic coming from the expansion corner. This feature can be identified better by comparing Figs. 13(c) and 13(d); the latter corresponds to the c21.7 case, for which the amplitude of the lowest values of the time coefficients has been artificially reduced by 35 %, so that the downstream shock excursion matches the one observed in the c24.0 -e24.0 case.

It is nonetheless difficult to assess whether the differences in the low-frequency dynamics of the whole flow field arise from the depletion of the very-low frequency range in the c24.0 -e24.0 case shown in Fig. 6. Indeed, the power spectral distribution of the modes encompass both the very low (S t L < 0.03) and low (0.03 < S t L < 0.2) frequency ranges (see Fig. 12).

This shortcoming can be alleviated by computing a spectral POD of the pressure field, on the basis of which we will be able to discriminate between frequency ranges. The same approach as described in Sec. 3.2 has been retained. Given the duration of the segments from which Fourier transform is performed and the kind of time window used (Welch), the resolving power of the sPOD estimator is about ∆S t L = 0.021. The sPOD modes have therefore been computed for Strouhal numbers S t L = 0.025 and S t L = 0.06, which are sufficiently far away from each other to made the modes distinguishable.

For both Strouhal numbers, the first pressure sPOD mode is both two-dimensional, highly dominant in power, and mostly localized in the region of the head shock. Note that the second mode of the c21.7 case at S t L = 0.06 is also two-dimensional and partly localized in the shock region. However, at these locations, the second mode induces pressure fluctuations 20 time lower than the ones associated with the first mode and is in fact related to two-dimensional flapping in the reattachment region.

As for the POD analyses, the change in global structure of the velocity vector field induced by one sPOD mode can been evaluated by computing I MF for the time-dependent, partially reconstructed flow field obtained by summing the mean field with the fluctuations associated with the sPOD mode. Taking advantage of the linearity of Eq. 1, we computed the sums of the mean I MF and I MF fluctuations associated with a sPOD mode for both flow cases and Strouhal numbers. The resulting isolines are displayed in Fig. 14. Note that the fluctuations were arbitrarily increased by 300 % to make the plots legible: the difference between the lower and upper states is visually increased. This approach is required since we consider a single frequency in the present sPOD, which makes the overall power contribution to the fluctuations rather limited. The plots in Fig. 14 therefore characterize the spacial structure of the fluctuating flow fields rather than being a direct measurement of the mass-flux distributions.

For the lowest Strouhal number (Figs. 14(a) and 14(c)), differences between the two flow cases are similar to the ones identified in the POD analysis: for c24.0-e24.0, both the amplitude of the shock motion is reduced (because of a smaller downstream excursion) and differences between the upper/lower states are vanishing downstream of reattachment but upstream of the expansion corner. Such a downstream damping is, however, found in both flow fields for S t L = 0.06 (see Figs. 14(b) and 14(d)). Moreover, the amplitude of the bubble breathing and shock excursion are almost identical for the two computations.

We therefore conclude that the unsteady alteration of the flowfield structure in the most downstream shock location/ smallest bubble case induced by the corner is indeed related to the depleted very-low frequency range. However, the alteration is not simply due to a diminished amplitude of the dominant mode in that frequency range, since the distribution in space is also distorted in the region located between the reattachment point and the expansion corner. The distortion occurs upstream of the expansion corner, which merely acts as a barrier for upstream propagating influence. We therefore infer that this behavior is induced by a lack of feedback from the relaxing flow downstream of the reattachment point. The upstream feedback that would take place at very low Strouhal number in the c21.7 case can unfortunately not be confirmed by looking at the phase structure of the sPOD mode since the phase is merely constant for the entire region downstream of the reattachment point.

Conclusion and perspectives

Our joint experimental/numerical study has highlighted that a favourable downstream pressure gradient alters the low-frequency dynamics of a separated SWBLI beyond the frequency scaling associated with a change in size of the separated region.

Comparisons of velocity-POD modes obtained for computations with and without the downstream expansion corner demonstrated that the corner induces distortions of the velocity modes, either by contracting them or by merging several modes. It is still unclear, by which of these mechanisms the presence of an expansion corner results in the depletion of the energy content at very low frequencies.

Nonetheless, a POD on the pressure signal revealed that the expansion corner results in a distortion of the fluctuating velocity field associated with the head-shock motion and bubble breathing. The distortion takes place at very low frequency only and is located in the region downstream of the reattachment point, yet significantly upstream of the expansion corner. This distortion is most probably related to a lack of feedback from the very downstream region of the interaction, whose structure is completely annihilated by the expansion flow.

It appears reasonable to postulate that such a feedback is missing because of the identified truncation/merging of POD velocity modes for the velocity field. Confirmation of this point would require cross analyses of fields reconstructed from the velocity (s)POD on the one hand and pressure (s)POD on the other hand, which is beyond the scope of the present work.

Alternatively, it may be fruitful to perform (s)POD on the integrated mass flux directly, since this quantity is sensitive to the low-frequency shock motion/bubble breathing, as well as to convective structures located within the mixing layer and the shedding region in a higher frequency range (see [START_REF] Jiang | Eduction scheme for convective structures in turbulent compressible separated flow[END_REF]. These sPOD analyses would most probably require a non-linear approach to identify links between these different frequency ranges.

From a more practical viewpoint, the fact that a shortened compression ramp induces a distortion of the low-frequency spectral content has to be taken into account when designing an experimental or numerical setup and when analyzing and comparing existing data. The most obvious length scales to consider here are the interaction length, L int , and the distance L down from the reference location of the interaction (shock impingement or corner for shock reflections and compression corners, respectively) to the location where the favourable pressure gradient is established farther downstream. The interaction length can be roughly estimated a priori, for instance using the model by Souverein et al.. An a posteriori assessment of an existing setup is more difficult, since the occurrence of a favourable pressure gradient may reduce the natural interaction length.

In the present study, the reference case c24.0 -c24.0 has a ratio L down /L int close to unity for both the experiment and the LES, while the experimental 20 • case, subject to slightly milder spectral distortion, corresponds to L down /L int ≃ 1.5. On the other hand, the computational domains of cases c21.7 and c24.0 without expansion corner (and thus resulting in non-distorted spectra) extend more than 2.5 L int downstream of the corner.The flow in the most downstream part of this region is altered by mesh stretching and increased numerical dissipation, but any flow structure of possible upstream influence can be accurately sustained in the LES sense over at least 2 L int away from the corner.

It is therefore reasonable to assume, as a rule of thumb, that a L down /L int ≳ 2 should prevent spectral distortions in the lowfrequency range, even if further studies are needed the refine this threshold. Such a criterion would be strongly helpful when comparing experimental measurements and numerical predictions of spectral data, since the former are generally gained for truncated ramps to avoid blockage effects in the wind tunnel, whereas the latter more likely correspond to an "infinite" ramp

Figure 1 :

 1 Figure 1: Van Driest-transformed streamwise velocity profiles in wall unit (top) and velocity fluctuation profiles in Morkovin scaling (bottom) at location x/δ = -4.3

Figure 2 :

 2 Figure 2: Isocontour of streamwise velocity with superimposed streamlines. The red dashed line denotes the 0-isolevel of streamwise velocity; PIV measurements (top) and LES simulation (bottom).

Figure 3 :

 3 Figure 3: Streamwise evolution of the velocity profiles along the interaction region: PIV measurements (•) and c24.0 -e24.0 LES (----).

Figure 4 :Figure 5 :

 45 Figure 4: Isocontours of the 2D turbulent kinetic energy for normalized POD modes obtained from snapshots of streamwise and wall-normal velocities: experiments (left), c24.0 -e24.0 LES (middle) and c21.7 LES (right).

Figure 6 :

 6 Figure 6: Normalized wall pressure power spectra at the beginning of the interaction region (x * = 0).

Figure 7 :Figure 8 :

 78 Figure 7: Streamwise distribution of the normalized cross-spectrum between the local pressure and the pressure at location x * = 0 for the three LES computations.

Figure 9 :

 9 Figure 9: Isocontour of 2D turbulent kinetic energy from various modes of the joint POD: c24.0 -e24.0 plane (left) and c21.7 plane (right).

Figure 10 :

 10 Figure 10: Premultiplied power spectrum of the first POD modes for the c24.0-e24.0 and c21.7 LES.

Figure 11 :

 11 Figure 11: Isolevels of pressure fluctuations of the first mode from the pressure POD for the c21.7 (top) and c24.0 -e24.0 (bottom) LESs.

Figure 12 :

 12 Figure 12: Premultiplied normalized power spectrum of the POD on pressure field: first mode for the c24.0 -e24.0 and c21.7 LES.

  Figure 13: Isolevels of integrated mass flux from the flowfied linearly related to the pressure POD.

St

  Figure 14: Isolevels of integrated mass flux from the flowfied linearly related to the pressure sPOD: sum/difference of mean and arbitrary amplified fluctuations of the first mode.

Table 1 :

 1 Flow parameters

	M ∞	P 0 (Pa)	T 0 (K)	Re θ	δ (mm)
	2.52	9.949×10 4 297	8225	10.4
	of the test section and one on the top wall. The ambient condi-
	tions listed in Tab. 1 set the wind-tunnel stagnation conditions.

Table 2 :

 2 Simulation parameters

	Case	Span	Cells	LF cycles
	c24.0-e24.0	3.5δ	87 M	145
	c24.0	7.4δ	181 M	55
	c21.7	3.5δ	87 M	90
	used to generate inflow turbulence. The adaptation of the latter
	method to supersonic boundary layers is described in Agostini

Acknowledgments

This work was granted access to the HPC resources of IDRIS under allocations A0062A01877, A0082A01877 and A0102A01877 made by GENCI.

This research was partially funded by the German Research Foundation DFG (Emmy Noether Programme, grant SCHR 1566/1-1, project number 326485414).

setup.