2.1 Schlieren visualisations

A schlieren system is utilised to examine the temporal shock location characteristics up to 12.5 kHz. A 1.6 kW Xenon lamp light source is directed towards a slit aperture, located at the focus of a spherical concave mirror with a focal length of 4 m, and the schlieren system is configured to follow a classical Z set-up (Hampel Reference Hampel1984). The long focal length ensures that a high degree of parallelism is maintained between the light rays travelling through the test section. On the other side of the test section, the light rays are collected by a second spherical concave mirror and a knife edge is placed at the focal point. The slit and knife edge are oriented such that variation in the schlieren image light intensity corresponds to the density variation in the wall-normal direction. Furthermore, the knife edge is adjusted such that a good contrast is achieved between the regions with weak and strong density gradients (i.e. regions further away and in the vicinity of the shock). The schlieren visualisations are recorded using a PCO dimax S4 high-speed camera.

Figure 3.

An instantaneous schlieren image of the SWBLI. The dark region corresponds to regions with large magnitude in density gradients (integrated in the spanwise direction) with respect to the wall-normal direction. The dashed line shows the identified reflected shock location, $\widetilde{C}_{R}$.

Figure 3 shows an instantaneous example of an image used to determine the shock location. Here, the dark region corresponds to regions with large magnitudes in density gradients. The variation in brightness occurs due to the refraction of light from differences in the refractive indices resulting from variation in density. Furthermore, the recorded brightness is a cumulative of the density variation across the spanwise direction, and since the shock location is three-dimensional, exhibiting spanwise rippling patterns (Humble et al. Reference Humble, Elsinga, Scarano and Van Oudheusden2009), it appears as multiple dark bands in the recorded image due to spanwise integration of the intensities. To determine the shock location, the schlieren image corresponding to the free-stream region ($z/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}>1.4$) is divided into $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\leqslant -1$ and $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}>-1$ regions. For each region, a straight line is fitted through dark pixels weighted by the streamwise gradient of the intensity values to capture the most downstream and upstream high-density gradient regions for the incident and reflected shocks, respectively. The identified reflected shock location, $\widetilde{C}_{R}$, is shown in figure 3 using a dashed line. Since the shock generator is fixed to the roof of the wind tunnel in the current work (see figure 2), the flow upstream of the shock generator is not uniform in the $y$-direction due to the instantaneously varying spanwise distribution of low- and high-momentum features within the boundary layer. Consequently, the incident shock produced by the roof mounted wedge is also three-dimensional. Therefore, unlike for the configuration where the shock generator is placed in the free stream (e.g. Dupont et al. Reference Dupont, Haddad and Debieve2006) and where the incident shock generated is nearly two-dimensional, the incident shock in the current study appears indistinct in the schlieren image shown in figure 3.

2.2 Velocity fields – wall-normal plane

The flow upstream and downstream of the incident shock foot location, $X_{I}$, is captured using planar PIV. In order to facilitate a large FOV, while still capturing sufficient details of the flow, a multi-camera configuration with eight Imager sCMOS is used, as illustrated in figure 2. This resulted in a combined FOV with dimensions of $0.42~\text{m}\times 0.05~\text{m}$ (corresponding to $30\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\times 4\,\widehat{\unicode[STIX]{x1D6FF}}_{I}$) in the streamwise and wall-normal directions, respectively, while still maintaining a scaling of $25~\unicode[STIX]{x03BC}\text{m}~\text{pixel}^{-1}$. Furthermore, to allow convergence of conditional mean quantities, approximately 4700 statistically independent velocity fields are acquired. It should be noted that, unlike the schlieren visualisations, the illumination for the planar PIV is limited to approximately 1 mm in the spanwise direction. Thus, the instantaneous shock locations are much more well defined (e.g. compare figures 3 and 8), since only a minuscule variation in the shock location occurs over the illuminated spanwise distance for the PIV images. To enable PIV, the flow is illuminated by a double-pulsed laser sheet generated utilising a Quantel EverGreen laser head, rated at $200~\text{mJ}~\text{pulse}^{-1}$. The laser sheet is introduced into the test section perpendicularly to the flat-plate wall through an optical glass window installed on the roof of the test section. To minimise regions of high-intensity pixels in the vicinity of the wall, typical in this laser sheet configuration, the aluminium wall surface within the FOV is polished to limit the amount of scattered light and the cameras positioned to be at a similar height as the wall (Kähler, Scharnowski & Cierpka Reference Kähler, Scharnowski and Cierpka2012). In addition, Di-Ethyl-Hexyl-Sebacat (DEHS) tracer particles with a mean diameter of approximately $1~\unicode[STIX]{x03BC}\text{m}$ (Kähler, Sammler & Kompenhans Reference Kähler, Sammler and Kompenhans2002) are released into the flow upstream of the settling chamber. The time offset between the double images is set at $1~\unicode[STIX]{x03BC}\text{s}$, resulting in a mean particle image shift of 21 pixels in the incoming free-stream region. Note that a large pixel shift is important in reducing the relative uncertainty of the PIV measurements, and results from Scharnowski et al. (Reference Scharnowski, Bross and Kähler2019) suggesting that a mean shift of 20 pixels or more is desirable. The PIV images are processed using Davis 8.3 software from LaVision GMbH, where multi-pass image deformation and Gaussian window weighting are enabled. The initial and final window sizes correspond to $128\times 128$ and $16\times 16$, respectively, with a 50 % overlap between the windows used for the PIV evaluation.

2.2.1 Fast-response pressure transducers

In addition to velocity fields, which are recorded at approximately 10 Hz, simultaneous wall pressure measurements are acquired at 100 kHz utilising three fast-response Kulite XCQ-062 pressure transducers with a range of $\pm 350~\text{kPa}$. These transducers are typically able to resolve fluctuating pressure values up to 60 kHz, or approximately one boundary layer turnover time (i.e. $U_{\infty }/(f_{U}\,\widehat{\unicode[STIX]{x1D6FF}}_{I})\sim O(1)$, where $f_{U}$ denotes the upper bound of the resolvable frequency). The pressure taps are located at a small spanwise offset (${\approx}1~\text{mm}$) to the laser sheet, in order to avoid saturation of camera pixels from the scattered light, and are streamwise aligned. The locations of the pressure transducers $\mathtt{P}_{1}$–$\mathtt{P}_{3}$ with respect to the mean incident and reflected shock locations are as indicated in figure 4. The fast-response pressure transducers are calibrated by applying a known pressure onto the backside of the membranes via reference tubes connected to a General Electric PACE 5000 pressure controller. The voltages from each pressure transducer are recorded at nine different pressures in the range $\pm 70~\text{kPa}$, and a linear fit determined between the pressure and the mean recorded voltage. The linear fit is then utilised to convert the recorded voltage values from the pressure transducer during the measurement to the pressure values. The uncertainty associated with the measured pressure values due to nonlinearity, non-repeatability and hysteresis is expected to be less than $\pm 1.8~\text{kPa}$.

Figure 4.

The location of fast-response pressure transducers and wall-parallel PIV planes with respect to the mean incident ($C_{I}$) and reflected ($C_{R}$) shock locations. The colour iso-contours correspond to the mean streamwise velocity field. The pressure transducers $\mathtt{P}_{1}$–$\mathtt{P}_{3}$ are denoted by ○ symbols and are located at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx -4$, $-1.5$ and 0, respectively; while the wall-parallel PIV planes Ⓐ and Ⓑ are at $z/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx 0.4$ and 1.7.

2.3 Velocity fields – wall-parallel plane

In addition to the wall-normal PIV plane, wall-parallel measurements are undertaken at $z/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx 0.4$ and 1.7, as illustrated in figure 4. Both wall-parallel planes are intersected by the incident and reflected shocks, and the planes are configured so that the plane closer to the wall resides within the boundary layer developing over the flat wall, while the plane away from the wall is at the free-stream region. The combined FOV of 0.065 m × 0.16 m ($5\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\times 11\,\widehat{\unicode[STIX]{x1D6FF}}_{I}$) in the streamwise and spanwise directions, respectively, with a scaling of $400~\unicode[STIX]{x03BC}\text{m}~\text{pixel}^{-1}$ is obtained using two Imager sCMOS cameras. The laser, tracer particles and the PIV software used are the same as that for the wall-normal PIV. The initial and final window sizes are set to $64\times 64$ pixels and $24\times 24$  pixel, respectively, with an initial constant window shift to account for the mean velocity along the plane (Raffel et al. Reference Raffel, Willert, Scarano, Kähler, Wereley and Kompenhans2018). In addition, the time offset between double images is set at $2~\unicode[STIX]{x03BC}\text{s}$, resulting in a mean particle image shift of 26 and 30 pixels for the two wall-parallel planes, while all other settings are kept the same. Unlike the wall-normal plane PIV, a mirror-like surface is undesirable for the wall-parallel plane PIV, in terms of minimising the scattered light captured by the camera. Hence, the pressure tap ports were covered and the wall surface painted matt black. Consequently, the wall pressure signals are not available for the current wall-parallel PIV measurements.

Figure 5.

Streamwise–wall-normal view of shock wave boundary layer interaction in a shock reflection configuration. The flow is from left to right and (a) and (b) correspond to the mean and instantaneous streamwise velocity fields.

3.1 Mean and variance profiles

In order to establish whether the boundary layer upstream of the shock is in a canonical state, figure 6(a) shows a comparison between the van Driest (Reference van Driest1951) transformed $U$ profile at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx -7.5$ from the current work against the incompressible canonical boundary layer profile obtained in the high Reynolds number wind tunnel at the University of Melbourne (Baidya et al. Reference Baidya, Philip, Hutchins, Monty and Marusic2017). The van Driest transformation takes into account the influence of wall-normal gradient in density that arises in the compressible flows, thereby allowing comparisons against the incompressible cases where extensive datasets are available in the literature. Note that the time-averaged density, $\unicode[STIX]{x1D70C}(z)$, required for the van Driest transformation is estimated from the mean velocity profile following the temperature–velocity relation of Walz (Reference Walz1959) and the ideal gas law. In order to calculate the mean friction velocity, $U_{\unicode[STIX]{x1D70F}}$, the transformed profile is fitted against a composite profile (Chauhan, Monkewitz & Nagib Reference Chauhan, Monkewitz and Nagib2009), as done for the incompressible dataset. In figure 6, superscript ‘+’ is used to denote length (e.g. $z^{+}=zU_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}_{w}$) and velocity (e.g. $U^{+}=U/U_{\unicode[STIX]{x1D70F}}$) scaled by the corresponding viscous scales (where $\unicode[STIX]{x1D708}_{w}$ corresponds to the kinematic viscosity of the fluid at the wall). A good agreement observed between the two profiles shown in figure 6(a), indicates that the incoming compressible turbulent boundary layer is in a canonical state.

Figure 6(b) shows comparisons between incompressible and compressible turbulent stress profiles. Note that, here, $\unicode[STIX]{x1D6FF}_{99}$ corresponds to the $z$ location where $U=0.99U_{\infty }$ for the $U$ profile at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx -7.5$, while the turbulent stress profiles have been pre-multiplied by a factor corresponding to the ratio between the mean density profile and the mean density at the wall, $\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{w}$, to account for the influence of the wall-normal density gradient in the compressible turbulent boundary layer. Note that, here an additional factor of two is applied to the wall-normal turbulence intensity and the Reynolds shear stress to enhance the readability of the stress values. Furthermore, unlike the mean velocity, the measured turbulent stress profiles are dependent on the spatial resolution of the sensor utilised (Ligrani & Bradshaw Reference Ligrani and Bradshaw1987; Lee et al. Reference Lee, Monty and Hutchins2016). The integration window used for PIV evaluation has dimensions of 130 viscous units × 130 viscous units, while the laser sheet thickness is estimated to be approximately 0.5–1 mm (Grayson et al. Reference Grayson, de Silva, Hutchins and Marusic2018), which corresponds to 150–300 viscous units. In contrast, the incompressible dataset is obtained utilising a cross-wire with dimensions of 13 viscous units × 13 viscous units, with a wire spacing of 7 viscous units. Thus, the turbulent stress profiles from the current study are expected to be more attenuated due to spatial averaging over a larger viscous-scaled dimension compared to the incompressible dataset, as observed for the $\overline{w^{2}}$ and $\overline{uw}$ profiles. Based on invariance of the small scales when scaled in viscous units, an estimate of the attenuation in $\overline{u^{2}}$ and $\overline{w^{2}}$ due to mismatched spatial resolution can be obtained (Lee et al. Reference Lee, Monty and Hutchins2016), and the estimated missing energy is indicated by the curly braces at $z/\unicode[STIX]{x1D6FF}_{99}\approx 0.15$ (see figure 6b, $\unicode[STIX]{x0394}\overline{u^{2}}|_{SR^{+}}$ and $\unicode[STIX]{x0394}\overline{w^{2}}|_{SR^{+}}$). The results suggest a slight (${\approx}10\,\%$) overestimation of the $\overline{u^{2}}$ statistics and a good agreement of the $\overline{w^{2}}$ statistics for the current experiment once the missing energy $\unicode[STIX]{x0394}\overline{u^{2}}|_{SR^{+}}$ and $\unicode[STIX]{x0394}\overline{w^{2}}|_{SR^{+}}$ are added to account for the finite spatial resolution effects; however, the observed differences are within the experimental uncertainty.

Figure 6.

Wall-normal profiles of the (a) first- and (b) second-order moments at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx -7.5$ (the streamwise locations contributing to the statistics are shown using curly brace above the ● symbol in figure 5). $U_{VD}$ denotes the van Driest transformed $U$ profile. Solid symbols: current experiment. Open symbols: $Re_{\unicode[STIX]{x1D70F}}\approx 5000$ incompressible canonical turbulent boundary layer data (Baidya et al. Reference Baidya, Philip, Hutchins, Monty and Marusic2017). The symbols represent: ○, ●, mean streamwise velocity ($U$); ▫, ▪, streamwise turbulence intensity ($\overline{u^{2}}$); ♢, ♦, wall-normal turbulence intensity ($\overline{w^{2}}$); and ▵, ▴, Reynolds shear stress ($-\overline{uw}$). The $\unicode[STIX]{x0394}\overline{u^{2}}|_{SR^{+}}$ and $\unicode[STIX]{x0394}\overline{w^{2}}|_{SR^{+}}$ indicate the unresolved missing energy at $z/\unicode[STIX]{x1D6FF}_{99}\approx 0.15$ for the current dataset relative to the incompressible dataset due to larger sensor dimensions in viscous units, based on the formulation by Lee et al. (Reference Lee, Monty and Hutchins2016).

Table 2.

Boundary layer parameters upstream of the shock. Here, ${\mathcal{W}}_{x}$, ${\mathcal{W}}_{y}$ and ${\mathcal{W}}_{z}$ correspond to the expected spatial resolutions upstream of the shock in the $x$, $y$ and $z$ directions, respectively, based on the PIV window size and the laser thickness.

The mean flow parameters upstream of the shock are summarised in table 2, where $T_{w}$ corresponds to the temperature at the wall. The momentum thickness is defined as

(3.1)$$\begin{eqnarray}\unicode[STIX]{x1D703}=\int _{0}^{\infty }\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}_{\infty }}\frac{U}{U_{\infty }}\left(1-\frac{U}{U_{\infty }}\right)\text{d}z,\end{eqnarray}$$

while the friction and momentum thickness Reynolds numbers are given by $Re_{\unicode[STIX]{x1D70F}}=(\unicode[STIX]{x1D6FF}_{99}U_{\unicode[STIX]{x1D70F}})/\unicode[STIX]{x1D708}_{w}$ and $Re_{\unicode[STIX]{x1D703}}=(\unicode[STIX]{x1D703}U_{\infty })/\unicode[STIX]{x1D708}_{w}$, respectively. In compressible flows, an alternative to $Re_{\unicode[STIX]{x1D703}}$, $Re_{\unicode[STIX]{x1D6FF}_{2}}=(\unicode[STIX]{x1D70C}_{\infty }\unicode[STIX]{x1D703}U_{\infty })/\unicode[STIX]{x1D707}_{w}$, where $\unicode[STIX]{x1D707}_{w}$ corresponds to the dynamic viscosity of the fluid at the wall, is more widely cited since the Reynolds number based on the conventional momentum thickness ceases to be an effective specifying parameter for the compressible flows (Fernholz & Finley Reference Fernholz and Finley1980).

Figures 7(a–f) and 7(g–l) show the streamwise evolution of the velocity deficit ($U_{\infty }-U$) and the streamwise turbulence intensity ($\overline{u^{2}}$), respectively. Note that, for convergence reasons, the profiles at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-7.5$ and 10 have been obtained by spatial averaging the velocity vectors that lie within $\pm 0.5\,\widehat{\unicode[STIX]{x1D6FF}}_{I}$ in the $x$ direction. At far upstream and downstream locations from the shock, the streamwise change in $U$ and $\overline{u^{2}}$ over one boundary layer thickness is minimal. Thus, by considering the velocity vectors that lie within $\pm 0.5\,\widehat{\unicode[STIX]{x1D6FF}}_{I}$, the total number of independent samples used to compute the statistics can be increased, which reduces the uncertainty associated with the computed value. For $x$ locations in the vicinity of the shock (i.e. $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-2.5$, 0, 2.5 and 5), a large change in $U$ and $\overline{u^{2}}$ can occur over a small $x$ displacement and therefore only temporal averaging is performed for these profiles. The profiles are shown normalised by the local boundary layer thickness, $\unicode[STIX]{x1D6FF}_{99,x}$, and the local edge velocity, $U_{\infty ,x}$.

Figure 7.

The streamwise evolution of (a–f) the velocity deficit and (g–l) the streamwise turbulence intensity. Panels (a–f) correspond to locations $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-7.5$, $-2.5$, 0, 2.5, 5 and 10, as indicated by the corresponding symbols in figure 5(a), and this is repeated for (g–l). The open symbols in (b,f,h,l) corresponds to the $U$ and $\overline{u^{2}}$ profiles at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-7.5$.

Immediately downstream of where the interaction region starts (▴), the $\overline{u^{2}}$ profiles in the vicinity of the wall are modified while the upstream canonical profiles are retained for the $z$ locations further away from the wall (see figure 7h). This is because the reflected shock that forms upstream of the incident shock foot location is at an acute angle with respect to the $x$ direction and therefore the interaction region is first encountered at the vicinity of the wall as the flow travels downstream (see figure 5a). At the end of the interaction region (◂), the $U$ and $\overline{u^{2}}$ profiles have been extensively modified throughout the entire boundary layer thickness. The canonical $U$ profile is fuller compared to the $U$ profile at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=0$, while the $u$ fluctuations are much more energetic at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=0$ compared to $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-7.5$. Notably, a similar trend in the deviation of $U$ and $\overline{u^{2}}$ profiles occurs for the rough-wall turbulent boundary layer compared to the canonical smooth-wall profiles (Krogstad, Antonia & Browne Reference Krogstad, Antonia and Browne1992). Downstream of $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=0$, the flow recovers towards a canonical state, and the $U$ and $\overline{u^{2}}$ statistics approach the upstream canonical values. However, even at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=10$ the recovery is not complete, and differences exist compared to $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-7.5$ (see figure 7f,l). Since a collapse of the velocity deficit and streamwise turbulence intensity is expected in the outer scaling for a canonical boundary layer flow (Fernholz & Finleyt Reference Fernholz and Finleyt1996; Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015), the results suggest that a complete recovery of the first- and second-order statistics requires in excess of 10 boundary layer thickness downstream from the SWBLI.

4.1 Influences of the oscillating incident shock

In most of the past works examining a shock reflection SWBLI configuration, the incident shock is generated utilising a two-dimensional obstacle placed in the free-stream region (e.g. Dupont et al. Reference Dupont, Haddad and Debieve2006), leading to a relatively fixed incident shock. However, in many practical applications, a series of shock reflections are generated from two opposing walls. Hence, the reflected shock from one side of the wall can impinge on the opposing wall, leading to cases where the incident shocks themselves oscillate. To investigate the influence of oscillating incident shock, the shock generator is fixed to the roof of the wind tunnel in the current work, as illustrated in figure 2. Therefore, the incoming flow to the shock generator is unsteady and contains a wide range of turbulent scales from the boundary layer developing on the roof, resulting in an oscillating incident shock. Furthermore, high-speed schlieren visualisation indicate that the incident shock oscillations are dominated by time scales that are smaller than $3\,\widehat{\unicode[STIX]{x1D6FF}}_{I}/U_{\infty }$, which corresponded to the smallest resolvable time step from the high-speed visualisation (see § 5 for further details).

Figure 9 shows the joint probability density function ( j.p.d.f.) of the incident and reflected shock angles. To identify the shock locations, a linear fit is performed through the $\text{d}\widetilde{U}/\text{d}x$ minima in the free-stream region. The j.p.d.f. shown in figure 9 resembles an upright ellipse, an indication that the two variables are linearly independent with a correlation coefficient approximately equal to 0.08. The independence of $\unicode[STIX]{x1D6FD}_{I}$ and $\unicode[STIX]{x1D6FD}_{R}$ is contrary to the inviscid theory, where $\unicode[STIX]{x1D6FD}_{R}$ is expected to be more perpendicular to the wall when $\unicode[STIX]{x1D6FD}_{I}$ increases, suggesting that the instantaneous location of the oscillating incident shock has a minimal influence on the instantaneous reflected shock location, at least in a linear sense (there may be a nonlinear influence as further discussed in § 6.1).

4.2 Influences of the incoming boundary layer

Notably, in figure 8 the incoming boundary layer is thicker when the reflected shock is located upstream of the mean location compared to the case when it is located downstream of the mean location. This suggests that the reflected shock location is influenced by the state of the incoming boundary layer as proposed by Ünalmis & Dolling (Reference Ünalmis and Dolling1998), Beresh et al. (Reference Beresh, Clemens and Dolling2002), Humble et al. (Reference Humble, Elsinga, Scarano and Van Oudheusden2009) and others. To further assess the relation between the state of the incoming boundary layer and the reflected shock location, a subset of the data where the incident shock lies in the range $\pm 0.25^{\circ }$ of the mean position (corresponding to 15 % of the total dataset), and therefore remains relatively fixed, is used for the subsequent analysis presented in this section.

Figure 10.

(a) Two-point correlation coefficient between local boundary layer thickness and reflected shock location. Three choices for the measure of local boundary layer thickness are shown: ○, instantaneous displacement thickness; ▵, $z_{75}$, i.e. $z$ locations where $\widetilde{U}=0.75U_{\infty }$; and ♢, upper TNTI envelope. (b) Instantaneous and (c) temporal-averaged streamwise velocity fields, where the colour contours are as in figure 5. The corresponding displacement thickness, $z_{75}$ and upper TNTI envelope are indicated by the dashed (– – –), dash-dotted (— ⋅ —) and dotted ($\cdots \cdots$) lines.

Figure 10(a) shows the correlation coefficient between the instantaneous boundary layer thicknesses and the reflected shock location at various $x$ locations. Note that, through the interaction region the edge velocity changes as a function of the streamwise location, resulting in difficulties when defining a boundary layer thickness that remains consistent across all $x$ locations. In figure 10(a), three choices are considered (their relative locations are demonstrated in figure 10b,c). The circles (○) show the correlation coefficient when the instantaneous displacement thickness, $\widetilde{\unicode[STIX]{x1D6FF}}^{\ast }=\int 1-(\unicode[STIX]{x1D70C}_{upstream}\widetilde{U}\,)/(\unicode[STIX]{x1D70C}_{\infty }U_{\infty })\,\text{d}z$, is considered as a measure of the boundary layer thickness. Note that, here, $\unicode[STIX]{x1D70C}_{upstream}$ corresponds to the mean density estimated from the mean velocity profile upstream of the shock following Walz’s (Reference Walz1959) relation. The triangles (▵) consider the $z$ locations where the velocity is 75 % of the incoming free-stream velocity as a measure of the boundary layer thickness. Lastly, the diamonds (♢) consider the upper envelope of the turbulent/non-turbulent interface (TNTI) as a measure of the boundary layer thickness. Note that, here, the TNTI location is determined from the velocity field based on the local kinetic energy criterion proposed by Reuther & Kähler (Reference Reuther and Kähler2018), where a value close to zero is expected in the free-stream region while a non-zero value occurs within the boundary layer. For full details of the TNTI detection procedure utilised for the current study, the reader is referred to Baidya et al. (Reference Baidya, Scharnowski, Bross and Kähler2019).

All three choices for the boundary layer thickness used in figure 10(a) indicate that the reflected shock location is well correlated ($|{\mathcal{R}}_{N}|>0.4$) with the instantaneous boundary layer thickness immediately upstream of the SWBLI region. In addition, the instantaneous boundary layer thickness is influenced by the presence of the $\unicode[STIX]{x1D6FF}$-scaled structures. Hence, the passage of low-/high-momentum large-scale turbulent features through the interaction region leads to the reflected shock to move upstream/downstream of the mean position, respectively.

Figure 11.

Joint probability function of the reflected shock location and TNTI upper envelope, $z_{TNTI}$. The j.p.d.f. at (a) $(x-X_{O})=-2\,\widehat{\unicode[STIX]{x1D6FF}}_{I}$, (b) $(x-X_{O})=0$ and (c) $(x-X_{O})=2\,\widehat{\unicode[STIX]{x1D6FF}}_{I}$ is shown.

To further quantify how the two-point correlation coefficient, ${\mathcal{R}}_{N}$, changes as a function of the streamwise location, figure 11(a–c) shows the j.p.d.f. of the reflected shock location and TNTI upper envelope at $(x-X_{O})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-2$, 0 and 2, respectively. Here, $X_{O}$ corresponds to the $x$ location where the mean locations of the incident and reflected shock intersect (i.e. point $O$ in figure 8). In the vicinity of $x=X_{O}$, the TNTI upper envelope correlates most strongly with the reflected shock (${\mathcal{R}}_{N}\approx -0.5$), leading to a j.p.d.f. resembling an inclined elongated ellipse. Upstream of the SWBLI region, the degree of correlation drops rapidly and ${\mathcal{R}}_{N}\approx -0.1$ is reached at $(x-X_{O})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-2$, and consequently, the j.p.d.f. is circular in shape. In contrast, the drop in correlation downstream of the shock is much more gradual and ${\mathcal{R}}_{N}\approx -0.3$ is reached at $(x-X_{O})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=2$. Hence, the j.p.d.f. at $(x-X_{O})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=2$ is more elliptical than at $(x-X_{O})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-2$. The gradual approach to ${\mathcal{R}}_{N}=0$ suggests that the instantaneous velocity profiles several boundary layer thicknesses downstream of SWBLI are modified by the presence of the shock, which is consistent with the long recovery observed for the mean and variance statistics in § 3.

Figure 12.

Conditional statistics based on the $\widetilde{C}_{R}$ shock foot (streamwise) location. The left (a,c) and right (b,d) columns show the ensemble-averaged streamwise velocity and Reynolds shear stress, respectively. The top row (a,b) corresponds to instances when the shock foot is upstream from the mean value, while the bottom row (c,d) corresponds to the downstream instances. The colour and line contour shown in (a) and (c) are at levels 0.2 to 0.8 in steps of 0.2, where the dotted lines indicate the contours of ensemble-averaged $\widetilde{U}$ over all instances when the location of $\widetilde{C}_{I}$ is fixed. The dashed lines in (b) and (d) correspond to $-\langle uw\rangle /U_{\infty }^{2}=-0.2\times 10^{-3}$ contours, while the dotted lines are as in (a) and (c) for the 0.2 $\langle \widetilde{U}\rangle /U_{\infty }$ level. The grey lines denote the mean incident ($C_{I}$) and reflected ($C_{R}$) shock locations.

Figure 12 shows the conditional statistic results based on the streamwise location of the reflected shock foot. Figure 12(a,b) shows the conditional statistics for the third of cases when the shock foot is at the most upstream location. Conversely, figure 12(c,d) corresponds to the third of cases when the shock foot is at the most downstream location. The colour contours shown in figure 12(a,c) corresponds to the ensemble-averaged streamwise velocity, $\langle \widetilde{U}\rangle$, while the dotted lines show iso-contours of the ensemble average from the entire dataset when the incident shock location $\widetilde{C}_{I}$ is fixed. Both the colour and line contours are at levels 0.2 to 0.8 in steps of 0.2. Due to issues with obtaining accurate velocity measurements in the vicinity to the wall from the large FOV configuration utilised in the current study, the conditional statistics shown in figure are only reliable starting from 0.8 mm away from the wall and are unfortunately unable to capture the separation region (however, we are able to observe instances with separation – see figure 8). Nevertheless, other studies where the separation bubble is resolved show that the decrease/increase in the zero-velocity isocontour corresponding to the separation bubble size is accompanied by the decrease/increase of other velocity isocontour levels (Piponniau et al. Reference Piponniau, Dussauge, Debieve and Dupont2009; Grilli et al. Reference Grilli, Schmid, Hickel and Adams2012; Priebe & Martín Reference Priebe and Martín2012). Hence, the higher velocity isocontour level is typically used as a surrogate for the behaviour of the separation bubble when the zero-velocity isocontour is not experimentally accessible (Ganapathisubramani et al. Reference Ganapathisubramani, Clemens and Dolling2007; Ganapathisubramani, Clemens & Dolling Reference Ganapathisubramani, Clemens and Dolling2009). From comparisons of the 0.2 level, it is evident that the region where $\langle \widetilde{U}\rangle <0.2U_{\infty }$ is smaller in (c) compared to (a). Furthermore, comparison of $\langle \widetilde{U}\rangle =0.8U_{\infty }$ at the $x$ locations indicated by arrows shows that the incoming boundary layer is thinner in (c) compared to (a). Since the $\unicode[STIX]{x1D6FF}$-scaled structures play a critical role in determining the instantaneous boundary layer thickness, the present results support the direct link between the passage of very-large-scale structures through the interaction region and the expansion and contraction of the separation bubble proposed by Ganapathisubramani et al. (Reference Ganapathisubramani, Clemens and Dolling2007, Reference Ganapathisubramani, Clemens and Dolling2009). A similar correlation between the passage of very-large-scale structures and the expansion/contraction of the separation bubble (for this particular case the separation bubble was well resolved using a highly magnified FOV PIV, unlike for the current experiment) has also been observed in adverse-pressure-gradient turbulent boundary layer flows (Bross, Fuchs & Kähler Reference Bross, Fuchs and Kähler2019a). Furthermore, a region of strongly negative Reynolds shear stress is observed to move downstream with the shock foot (see the dashed lines relative to dotted lines in figure 12b,d), which notably coincides with the shrinking of $\langle \widetilde{U}\rangle =0.2U_{\infty }$ contour observed at $(x-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx -1.5$. Thus, the data suggest that the expansion and contraction of the separation bubble is strongly influenced by the ejection and sweep mechanisms associated with the low- and high-momentum fluid that dominates the Reynolds shear stress contributions.

6.1 Behaviour of spanwise scales as a function of the streamwise location

The presence of the shock alters the behaviour of the spanwise scales, as evident from figure 17. Here, how these spanwise scales are altered as a function of the streamwise location from the shock front is further investigated by means of cross-spectrum between $u(x_{ref})$ and $u(x_{ref}+\unicode[STIX]{x0394}x)$. The cross-spectrum $\unicode[STIX]{x1D719}_{u_{ref}u_{\unicode[STIX]{x0394}x}}$ is defined as

(6.1)$$\begin{eqnarray}\unicode[STIX]{x1D719}_{u_{ref}u_{\unicode[STIX]{x0394}x}}(x_{ref},\unicode[STIX]{x0394}x;\unicode[STIX]{x1D706}_{y})={\mathcal{F}}_{y}(\langle u(x_{ref},y_{ref})u(x_{ref}+\unicode[STIX]{x0394}x,y_{ref}+\unicode[STIX]{x0394}y)\rangle ),\end{eqnarray}$$

where ${\mathcal{F}}_{y}$ denotes the Fourier transform in the spanwise direction, while $\unicode[STIX]{x0394}x$ and $\unicode[STIX]{x0394}y$ are streamwise and spanwise displacements relative to the reference point $(x_{ref},y_{ref})$.

Figure 18.

The $\unicode[STIX]{x1D719}_{u_{ref}u_{\unicode[STIX]{x0394}x}}$ cross-spectrogram at (a) $(x_{ref}-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-2$, (b) $(x_{ref}-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx -1.25$, (c) $(x_{ref}-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx -0.75$ and (d) $(x_{ref}-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=0.5$. The colour contours are at levels 0.1 to 0.9 in steps of 0.1, while the line contours correspond to $k_{y}\unicode[STIX]{x1D719}_{u_{ref}u_{\unicode[STIX]{x0394}x}}(x_{ref},-\unicode[STIX]{x0394}x;\unicode[STIX]{x1D706}_{y})/\max (k_{y}\unicode[STIX]{x1D719}_{u_{ref}u_{ref}})=0.1$ (i.e. the colour contour 0.1 level reflected about the $\unicode[STIX]{x0394}x=0$ axis). The horizontal dotted lines correspond to $\unicode[STIX]{x1D706}_{y}/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=1$ and $\unicode[STIX]{x1D706}_{y}/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=6$, while the vertical dashed lines denote the mean incident and reflected shock locations.

Figure 18(a–d) shows $\unicode[STIX]{x1D719}_{u_{ref}u_{\unicode[STIX]{x0394}x}}$ cross-spectrogram normalised by $\max (k_{y}\unicode[STIX]{x1D719}_{u_{ref}u_{ref}})$ when the reference $x$ location is upstream, in the vicinity and downstream of the SWBLI region. In addition to colour contours at levels 0.1 to 0.9 in steps of 0.1, the line contours show the lowest colour contour level reflected about the $\unicode[STIX]{x0394}x=0$ axis. Thus, by comparing the 0.1 colour contour and the dash-dotted line, the upstream and downstream $\unicode[STIX]{x1D719}_{u_{ref}u_{\unicode[STIX]{x0394}x}}$ behaviour can be compared. For example, at $(x_{ref}-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=-2$ the cross-spectrogram is nearly symmetric about the reference point for spanwise scales $\unicode[STIX]{x1D706}_{y}/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}<0.6$, since 0.1 level colour and line contours exhibit a good collapse. Also note that for an unmodified boundary layer the cross-spectrogram is near symmetric for all scales as the change with respect to $x$ is minimal.

At spanwise wavelengths larger than $\widehat{\unicode[STIX]{x1D6FF}}_{I}$, figure 18(a–d) suggests that the $u$ fluctuations retain correlation even after being modified by the shock. At $(x_{ref}-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx -1.25$, two persistent modes, $\unicode[STIX]{x1D706}_{y}/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx 1$ and $\unicode[STIX]{x1D706}_{y}/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx 6$, are evident and they remain correlated over longer streamwise distances compared to other spanwise wavelengths. Furthermore, the persistent mode at $\unicode[STIX]{x1D706}_{y}/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx 6$ shifts towards a wider spanwise length scales downstream of the interaction region. Since the boundary layer thickens downstream of the interaction region, the widening of the spanwise length scales may be an indication that these highly persistent modes scale with the local boundary layer thickness. However, it should be noted that the very wide persistent mode is close to the full FOV of the current experiment and therefore the cross-spectra can only be resolved at limited wavelengths at these scales, and experiments with a much wider FOV are necessary to confirm these findings. The dual persistent modes are again evident at $(x_{ref}-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx -0.75$, and the correlations are generally retained longer downstream of $x_{ref}$ compared to upstream where the $u$ fluctuations are modified by the shock. Finally, at $(x_{ref}-X_{I})/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}=0.5$, the small scales exhibit symmetry in the $x$ direction about the reference point, suggesting that these scales depend only on the local conditions, while the larger spanwise scales still retain influence from modifications due to shock and are yet to recover to a canonical state. Note that to verify that the highly persistent modes observed are not an artefact of the wind tunnel, where spanwise variations that are $O(\widehat{\unicode[STIX]{x1D6FF}})$ can be generated within the boundary layer due to slight wind tunnel imperfections (Townsend Reference Townsend1976), further SWBLI experiments with a different incoming boundary layer thickness are necessary.

Figure 19.

(a) Pre-multiplied spanwise spectra of the streamwise shock location captured on the plane $z/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx 1.7$. The solid (——) and dashed (– – –) lines correspond to the incident and reflected shock locations, $x_{I}^{\ast }$ and $x_{R}^{\ast }$, respectively. Also shown in (a) as a dotted ($\cdots \cdots$) line is the pre-multiplied $u$-spectra for the incoming boundary layer at $z/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx 0.4$ (i.e. $u$-spectra corresponding to region $\unicode[STIX]{x2460}$ from figure 17b). (b) Linear coherence spectra of incident and reflected shock locations.

Figure 19(a) shows the pre-multiplied spanwise spectra of the incident (——) and reflected (– – –) shock locations. It should be noted that the incident and reflected shock fluctuations, $x_{I}^{\ast }$ and $x_{R}^{\ast }$, respectively, are based on wall-parallel PIV conducted at $z/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\approx 1.7$. Also shown overlaid is the $u$-spectra of the incoming boundary layer in the outer region of the boundary layer ($\cdots \cdots$). From the figure, it is evident that the spanwise scales of the reflected shock location become increasingly energetic at wider spanwise length scales when $\unicode[STIX]{x1D706}_{y}/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}\gtrsim 4$, while the energetic $u$ motion of the incoming boundary layer remains approximately constant at these length scales. Hence, the increase in the energetic content of $x_{R}^{\ast }$ at these spanwise scales is unlikely to originate from the incoming boundary layer. In contrast to the $u$-spectra, an increased energetic contribution at a wider spanwise scale, similar to that observed for the reflected shock location, is also observed for the incident shock location. Thus, the data suggest that the increasing energetic contribution at a very wide spanwise scale ($\unicode[STIX]{x1D706}_{y}/\,\widehat{\unicode[STIX]{x1D6FF}}_{I}>4$) for the reflected shock is due to influence of increasing energetic content for the incident shock at these scales. Figure 19(b) shows the linear coherence spectra of the incident and reflected shock oscillations. Here, the linear coherence spectra, $\unicode[STIX]{x1D6FE}_{L,x_{I}^{\ast }x_{R}^{\ast }}^{2}$ is defined as

(6.2)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{L,x_{I}^{\ast }x_{R}^{\ast }}^{2}(\unicode[STIX]{x1D706}_{y})=\frac{|\unicode[STIX]{x1D719}_{x_{I}^{\ast }x_{R}^{\ast }}(\unicode[STIX]{x1D706}_{y})|^{2}}{\unicode[STIX]{x1D719}_{x_{I}^{\ast }x_{I}^{\ast }}(\unicode[STIX]{x1D706}_{y})\unicode[STIX]{x1D719}_{x_{R}^{\ast }x_{R}^{\ast }}(\unicode[STIX]{x1D706}_{y})},\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{x_{I}^{\ast }x_{R}^{\ast }}$ corresponds to cross-spectrum of the incident and reflected shock locations while $\unicode[STIX]{x1D719}_{x_{I}^{\ast }x_{I}^{\ast }}$ and $\unicode[STIX]{x1D719}_{x_{R}^{\ast }x_{R}^{\ast }}$ denote the spectra of the incident and reflected shock locations, respectively, and a vertical bar ($|\;|$) designates the modulus. By definition $0\leqslant \unicode[STIX]{x1D6FE}_{L,x_{I}^{\ast }x_{R}^{\ast }}^{2}\leqslant 1$, and here may be interpreted as a square of scale-specific correlation coefficient (Bendat & Piersol Reference Bendat and Piersol2010; Baars, Hutchins & Marusic Reference Baars, Hutchins and Marusic2017). Hence, although the very wide scales in $x_{R}^{\ast }$ are influenced by $x_{I}^{\ast }$, the relation is not linear in nature and therefore results in a near-zero correlation as reported in § 4.1.