3.1. Similarity profiles

The STBLI separation shear layers are visualized in figure 1 by the region of elevated turbulence in the contours of mean turbulent kinetic energy $\mathrm {TKE}=\overline {u_i' u_i'}/2 {U_e}^{2}$. In each case, the shear layer forms at the foot of the shock and makes an angle to the wall surface. The positions of the shock and the separation dividing streamline are also indicated in figure 1. A shear layer coordinate system ($x,z$) is defined for each case such that the longitudinal $x$-axis extends along the centre of the layer in the direction of its development and the $z$-axis is perpendicular to $x$ in the cross-stream direction. Canonical mixing layers are characterized by a linear growth rate of the layer thickness (Tennekes & Lumley Reference Tennekes and Lumley1972; Lesieur Reference Lesieur1987). If linear growth does in fact occur in the present shear flows, it should be possible to collapse profiles of the mean flow onto a single similarity profile by plotting against the similarity variable $\zeta =z/x$. In doing so, a region of approximate linear growth is found in each of the three STBLI flows.

Figure 1. Mixing layer coordinate system definition for the (a) Mach 3, (b) Mach 7 and (c) Mach 10 compression ramp datasets. Contours are of the turbulent kinetic energy, $\mathrm {TKE}={u_i}' {u_i}'/2 {U_e}^{2}$. The black line is the location of the mean shock front and the magenta line is the mean dividing streamline. Dashed lines indicate the range of similarity.

The mean velocity and mean turbulence stresses are plotted versus $\zeta$ for M3, M7 and M10 in figures 2–4, respectively. Obtaining these profiles required the positioning of the shear layer coordinate system $xz$-axes, the rotation of which was determined by the orientation of the mean velocity field, and the origin by the angle of spread observed in the contour of mean TKE. This manual placement of the mixing layer coordinates is similar to the method used by Dupont et al. (Reference Dupont, Piponniau and Dussauge2019). The position of the $xz$-axes for each case are shown in figure 1. The angles of inclination for the Mach 3, 7 and 10 flows are $12.0^{\circ }$, $8.5^{\circ }$ and $10.0^{\circ }$, respectively. The bold dashed lines in figure 1 indicate the range in $x$ for which a good collapse of the similarity profiles was found. The profiles of figures 2–4 were taken from this range.

Figure 2. Similarity profiles in the Mach 3 STBLI separated shear layer: (a) the mean axial and cross-stream velocity; (b) axial turbulence intensity; (c) cross-stream turbulence intensity; and (d) turbulence shear stress. The bold line is the profile at the ramp corner.

Figure 3. Similarity profiles in the Mach 7 STBLI separated shear layer: (a) the mean axial and cross-stream velocity; (b) axial turbulence intensity; (c) cross-stream turbulence intensity; and (d) turbulence shear stress. The bold line is the profile at the ramp corner.

Figure 4. Similarity profiles in the Mach 10 STBLI separated shear layer: (a) the mean axial and cross-stream velocity; (b) axial turbulence intensity; (c) cross-stream turbulence intensity; and (d) turbulence shear stress. The bold line is the profile at the ramp corner.

The collapsed profiles themselves resemble quite well those of the classic mixing layer. The mean longitudinal velocity profiles show high and low velocities connected by a single inflection point, and the profiles of turbulence stress are approximately Gaussian with the peak coinciding with the location of the inflection point in the mean velocity $U$. Both of these features are typical of the canonical mixing layer and together they produce the Kelvin–Helmholtz inviscid instability (White Reference White1974). Unlike the classic mixing layer similarity solution, the collapsed profiles for all three shear layers appear to be non-symmetric with the turbulence peak (equivalently the inflection point in the mean velocity) biased towards the high-speed side of the layer. It is shown in § 5 that this bias is a result of the proximity of the wall on the low-speed side. The profiles of mean cross-stream velocity show that $W$ is essentially zero across the layer for all three cases indicating that the mean velocity is nearly parallel to the $x$-axis. The minimal variation in $W$ across the layer is also consistent with a reduced entrainment rate, and therefore reduced spreading rate as is expected for highly compressible mixing layers. This point is discussed further in § 3.4.

3.2. Two stream flow properties

Encouraged by the quality of collapse of the profiles as well as their resemblance to the canonical free mixing layer flow, we make an attempt to categorize these STBLI shear layers in the manner of conventional compressible mixing layers. To do so we must describe each shear layer as two streams, a high- and a low-speed stream, each with constant velocity and constant thermodynamic properties. As can be seen in figures 3–5 this will only be an approximation as all profiles deviate from the typical mixing layer solution near the edges of the layer. Spreading occurs at the low-speed end of the profiles due to the presence of the wall and at the high-speed end due to the presence of the separation shock (the location of the shock in $\zeta$ is easily seen in the profiles of $W$). It will be shown, however, that even a rough estimation of the mean properties of the two streams is sufficient for a general comparison to canonical mixing layer data. The estimations of the two stream properties for each shear layer are listed in table 2. The methods for determining the entries of table 2 are discussed below. By convention, properties associated with the high-speed side are indicated with the subscript ‘1’ and the low-speed side with subscript ‘2’.

The velocity $U_1$ and temperature $T_1$ for each case are estimated as the inviscid post-shock solution for the STBLI free stream undergoing a flow deflection equal to the angle of inclination of the $x$-axis. This selection of $U_1$ and $T_1$ stems from the observation that the rotated mean $W$ profiles are essentially zero for all three shear layers. It is worth noting that we found the initial deflection angle of the separation shock, as shown for each case in figure 1, corresponds closely to the resulting wave angle of the oblique shock solution. The high-speed stream Mach number $M_1$ is determined from $U_1$ and $T_1$. The M7 and M10 flows maintain Mach number above 5 downstream of the separation shock and can be considered hypersonic shear layers.

The velocity $U_2$ of the low-speed side is estimated as the minimum in the similarity profiles of $U$ in figures 2–4(a). The low-speed side $T_2$ is likewise determined from the similarity profiles of temperature which are plotted in figure 5. All three cases show a satisfactory collapse of temperature within the previously defined range of approximate similarity. In each shear layer, however, there occurs a ‘hook-off’ of the temperature profiles on the low-speed side. This is due to the constant-temperature boundary condition at the wall. The wall temperature of the Mach 3 case is nearly adiabatic and so the divergence of the profiles in figure 5(a) is minimal. Because the Mach 7 and Mach 10 are both cold-wall simulations, the temperature drops significantly inside the separation bubble as seen in figure 5(b,c). The low-speed $T_2$ is therefore estimated as the maximum value in temperature just before the profiles diverge to meet the wall boundary condition. The two hypersonic shear layers have large temperature ratios such that $T_2$ experiences an increase of over five times $T_1$ for the M7 case and nearly seven times for the M10 case. In comparison, the M3 case $T_2$ is only double the value of $T_1$. Also listed in table 2 is the low-speed stream Mach number $M_2$ calculated from $U_2$ and $T_2$.

Figure 5. Similarity profiles of temperature in the (a) Mach 3, (b) Mach 7 and (c) Mach 10 STBLI separated shear layers.

Because the separated flow is shock-induced, an adverse pressure gradient occurs along the length of the shear layer. Figure 6 shows that the pressure increases nearly linearly along the $x$-axis for all three cases. The reference pressure $p_1$ is the post-shock pressure from the oblique shock solution from which $U_1$ and $T_1$ are obtained. The average rate of pressure increase $\textrm {d}p/{\textrm {d}\kern0.5pt x}$ in units of $p_1/L$ was determined from a linear fit to the data of figure 6. For the Mach 3 flow the pressure increases by nearly 50 % across the region of similarity, while for the Mach 7 and Mach 10 flows the pressure approximately doubles. As a result of the adverse pressure gradient, the mean density plotted versus $\zeta$ does not collapse when normalized by the free stream density as is apparent in figures 7(a), 7(c) and 7(d). The density is seen to increase significantly from the most upstream profile to the most downstream profile. However, a much better collapse is achieved if each individual profile of $\rho$ is non-dimensionalized by the local $\rho _2(x)$. The inverse of $\rho$ non-dimensionalized by $\rho _2(x)$ is plotted in figures 7(b), 7(d) and 7(f). That is, although there is a monotonic increase in density along $x$, the ratio between the two streams is approximately constant. Therefore, only the density ratio $s=\rho _2/\rho _1$ is reported in table 2. The value of local $\rho _2(x)$ was determined from the individual profiles in figure 7(a–c) in a manner similar to the selection of $T_2$ from the profiles of temperature. Note that the density could have equivalently been non-dimensionalized by the local $\rho _1(x)$ to obtain the collapse. We chose to use $\rho _2(x)$ because this quantity was easier to select from figure 7(a–c). The Mach 3 STBLI flow produces a density ratio of approximately $1/2$ across the shear layer while both the Mach 7 and Mach 10 interactions produce a density ratio of $1/3$. Also included in table 2 is the velocity ratio $r=U_2/U_1$ for each case.

Figure 6. Pressure gradient along the shear layer centreline.

Figure 7. Similarity profiles of density in the (a,b) Mach 3, (c,d) Mach 7 and (e,f) Mach 10 STBLI separated shear layers. The profiles in panels (a,c,e) are normalized by the free stream density. Panels (b,d,f) are profiles of the inverse density normalized by the density of the low-speed side, that is, ${\rho }_2 / \rho$. Plotting density in this way shows the variation in $s={\rho }_2 / {\rho }_1$.

3.3. Convective Mach number

Now that the properties of the shear layer high- and low-speed streams are known, the theoretical convective Mach number defined as

(3.1)\begin{equation} M_c = \frac{U_1 - U_2}{a_1 + a_2} \end{equation}

and also the theoretical mixing layer vortex convection velocity $U_{c,i}$ defined as

(3.2)\begin{equation} U_{c,i} = \frac{a_1U_2 + a_2U_1}{a_1+a_2} \end{equation}

can be calculated for these flows. These expressions for the convective Mach number and convective velocity are derived for an isentropic mixing layer where $a_1$ and $a_2$ are the speed of sound in the two streams (Bogdanoff Reference Bogdanoff1983; Papamoscho & Roshko Reference Papamoscho and Roshko1988). The $M_c$ and $U_{c,i}$ are computed for each case and listed in table 3. An interesting feature of the separated STBLI flows is that they produce shear layers with rather high $M_c$ even for the Mach 3 compression ramp flow. All three shear layers are above $M_c=1$. This is an attractive feature considering that the majority of mixing layer data available today, particularly for turbulence statistics, is below $M_c=1$.

Table 3. Averaged mixing layer flow properties.

For mixing layers with $M_c$ above 1 it is likely that shock waves exist in one or both sides of the mixing layer, thus negating the isentropic assumption in the derivation of (3.1) and (3.2). We will show later in § 4.2 that the theoretical $U_{c,{i}}$ in table 3 is quite different from the convection velocity determined from enhanced two-point correlations.

3.4. Spreading rate

Despite its known limitations as a scaling parameter, the convective Mach number defined by (3.1) is currently the most widely accepted metric in the literature for classifying the compressibility effects of mixing layers (Smits & Dussauge Reference Smits and Dussauge2006). One such classification is the observed significant decrease in layer spreading rate with increasing $M_c$. Smits & Dussauge (Reference Smits and Dussauge2006) presented a compilation of compressible mixing layer spreading rate data, expressed as a fraction of the spreading rate of an equivalent incompressible mixing layer with the same values of $r$ and $s$, and plotted these versus $M_c$ (see figure 6.6 in the reference). Included in the data compilation are the classic Langley curve (Birch & Eggers Reference Birch and Eggers1972; Kline et al. Reference Kline, Cantwell and Lilley1980), the semiempirical curve by Dimotakis (Reference Dimotakis1991) and the linear stability analysis prediction of spreading rate decrease with $M_c$ by Day et al. (Reference Day, Reynolds and Mansour1998). The data show that the spreading rate can decrease by as much as 50 % to 80 % from the incompressible case for $M_c$ above 0.5. For the current data, normalized spreading rate predictions from the classic Langley curve are approximately 0.55 for the Mach 3 flow and 0.40 for both the Mach 7 and Mach 10 cases as determined by the values of $M_c$ in table 3.

For the STBLI shear layers, the spreading rate of vorticity thickness $\delta ' = \textrm {d}\delta _\omega /{\textrm {d}\kern0.5pt x}$ where $\delta _\omega = {\rm \Delta} U / {\max }(\textrm {d}U/\textrm {d}z)$ can be estimated using the two different methods outlined by Dupont et al. (Reference Dupont, Piponniau and Dussauge2019). The first of these uses a comparison of the normalized $\overline {u'^{{2}}}$ similarity profile with the same from an incompressible mixing layer. Here the two-stream mixing layer data of Mehta & Westphal (Reference Mehta and Westphal1984) (figure 5b in the reference) is used. This first method assumes that the shape of the $\overline {u'^{{2}}}$ profile as well as the ratio of $(\textrm {d}\delta _\omega / {\textrm {d}\kern0.5pt x}) / \textrm {d}\zeta$ do not differ between the compressible and the incompressible cases. The second method involves fitting a Gaussian curve to the profiles of turbulent shear stress. Both methods provide consistent results. These are listed in table 4.

Table 4. Spreading rate estimates and comparison with incompressible theory.

A theoretical estimate of the spreading rate for an incompressible mixing layer with non-zero density ratio $s$ can be determined from the relation derived by Papamoscho & Roshko (Reference Papamoscho and Roshko1988),

(3.3)\begin{equation} \delta'_{{inc}} = \delta'_{{ref}} \frac{(1-r)(1+\sqrt{s})}{(1+r\sqrt{s})}. \end{equation}

In (3.3), $\delta '_{{ref}}$ is a reference spreading rate from an incompressible mixing layer with $s=1$ and $U_2=0$ and is typically taken to be equal to 0.16 (Smits & Dussauge Reference Smits and Dussauge2006). The STBLI shear layer spreading rates are also listed in table 4 as a fraction of the corresponding incompressible estimate. The spreading rate ratios are less than unity, however, they are approximately 50 % to 70 % higher than the Langley curve predictions. The Langley curve, however, is based primarily on data for single or two stream coflowing mixing layers. It has been shown that the spreading rate can be significantly greater for counter-current mixing layers (Strykowski et al. Reference Strykowski, Krothapalli and Jendoubi1996; Forliti, Tang & Strykowski Reference Forliti, Tang and Strykowski2005) and also for mixing layers subjected to adverse pressure gradients (König, Schulülter & Fiedler Reference König, Schulülter and Fiedler1998). Although there is significantly less data on the counter-current mixing layer than the coflowing configuration, one notable work is that of Strykowski et al. (Reference Strykowski, Krothapalli and Jendoubi1996). The authors performed a series of counter-flowing axisymmetric jet experiments at $M_c \approx 1$ with varying reverse flow strength. They showed that the spreading rates were consistently 60 % greater than the case of a single stream jet. Another consideration is that (3.3) might not be an accurate approximation for the spreading rate of incompressible counter-current mixing layers. Strykowski et al. (Reference Strykowski, Krothapalli and Jendoubi1996), however, also showed that (3.3) was valid for their experimental data if the reverse flow strength did not exceed $r<-0.1$. Even so, the disagreement for $r<-0.1$ was cited by the authors as possibly due to an artefact of their jet nozzle. At any rate, the shear layer data in table 4 clearly shows a decrease in spreading rate from the M3 case at $M_c \approx 1$ to the M7 and M10 cases at $M_c \approx 2$.

4.1. Enhanced correlation method

A schematic of the shear layer in the compression ramp STBLI flow is given in figure 8. In panel (a) is shown a model of the spatial development of the mixing layer structures as they convect along the $x$-axis. It is assumed that the vortices convect at a constant velocity $U_c$ and that they follow one after the other at fairly regular intervals. It is also assumed that they do not stray too far from the shear layer centreline. At reattachment the vortices are shed into the downstream flow. These assumptions are based on observations of the temporally resolved and spatially resolved LES data and are verified in § 4.3 by the instantaneous vortex visualizations. If the flow is probed at a stationary point in the shear layer, the resulting time signals can be converted to spatially ‘frozen’ turbulence via Taylor's hypothesis. This is drawn schematically in figure 8(b). The average signature of the frozen turbulence can be determined from the cross-correlations of the time signals of mass flux and pressure fluctuations in the following way. Consider for example the time signal of pressure taken from a point along the centreline of the shear layer. As a vortex core convects past the probe location there will be a negative fluctuation in the pressure. Likewise, in-between successive vortices there will be a positive pressure fluctuation from the stagnation point in the convective reference frame. In a similar way, the time signal of longitudinal mass fluctuations $(\rho u)'$ taken near the bottom edge of the shear layer will give information on the aperiodic signature of the passing vortices due to the orientation of the vortex rotation. Taking the cross-correlation between the centreline $p'$ and the bottom edge $(\rho u)'$ time signals produces a sinusoidal signature, the period of which is equal to the average time between successive vortices as they convect past the probe points. Although not shown in the schematic of figure 8, similar arguments can be made for cross-correlations between centreline $p'$ and centreline cross-stream momentum $(\rho w)'$. Here we consider both $R_{(\rho u)'p'}$ and $R_{(\rho w)'p'}$.

Figure 8. Schematic of the vortex structures (a) in the spatially developing shear layer in the separated compression ramp STBLI flow and (b) time signals taken from within the shear layer converted to ‘frozen’ vortices by using Taylor's hypothesis.

A similar cross-correlation method was demonstrated by Kiya & Sasaki (Reference Kiya and Sasaki1983) and also Cherry, Hillier & Latour (Reference Cherry, Hillier and Latour1984) for an incompressible separation shear layer, and Samimy et al. (Reference Samimy, Reeder and Elliott1992) for compressible mixing layers. There are a couple of points to be made regarding the cross-correlation method used here. First, the signal of longitudinal momentum fluctuations could be taken from either the top or bottom edge of the shear layer. Kiya & Sasaki (Reference Kiya and Sasaki1983), for example, used the high-speed edge. In this analysis, the bottom edge was chosen so as to avoid the separation shock. Second, autocorrelations of pressure with itself will also give a periodic correlation curve as demonstrated by Kiya & Sasaki (Reference Kiya and Sasaki1983), Cherry et al. (Reference Cherry, Hillier and Latour1984) and Samimy et al. (Reference Samimy, Reeder and Elliott1992). Here cross-correlations of $p'$ and $(\rho u)'$, and also $p'$ and $(\rho w)'$, were used in order to couple the mass flux and pressure field events, that is, to ensure that a pressure fluctuation is accompanied by a corresponding mass fluctuation. We found this strategy also ensures a more robust selection method for the enhanced correlation technique to be described shortly. Last, it was found that nearly identical correlation signatures were achieved when correlating velocity with pressure compared with correlating mass flux with pressure. In general the mass flux correlations provided a stronger signature and so only the mass flux and pressure correlations are included in this paper.

The details of the correlation method are as follows. Before calculating the cross-correlations, the signals of pressure and velocity are first bandpass-filtered in time. Fully separated STBLI flows are characterized by frequency spectra consisting of three distinct broadband ranges of energized turbulence motions. These are associated with (i) the inherent low-frequency unsteadiness of the separated flow; (ii) the mixing layer vortices; and (iii) the fine scale boundary layer turbulence. To demonstrate these frequency bands, premultiplied power spectral density (PSD) of wall pressure taken in the upstream boundary layer, the mean separation point, the ramp corner and the mean reattachment point are plotted in figure 9 for each of the three STBLI flows. These spectra were calculated using Welch's method with eight time segments with 50 % overlap and then bin sampled with a bin width of 0.1 in the log scale. From these spectra it is possible to make out the shifts in the distribution of turbulence energy as the flow progresses through the separated region. In the boundary layer, only the high-frequency turbulence exists with little to no energy present at the lowest frequencies. The undisturbed boundary layer turbulence is generally centred at $St_\delta = f \delta _{bl}/U_e=1$ and experiences a shift to $St_L=1$ downstream of the shock. This shift in the boundary layer turbulence can be seen when comparing the broadband energy peaks between the most upstream and most downstream spectra. The low-frequency oscillations of the shock appear in the separation spectra. It is well documented in the literature that the low-frequency oscillations in quasi-2-D separated STBLI flows occur at $St_L=f L/U_e$ of the order of 0.01 (among many references, see for example Dupont, Haddad & Debiève (Reference Dupont, Haddad and Debiève2006) for reflected shock interactions and Priebe & Martín (Reference Priebe and Martín2012) for compression ramp interactions). The relative strength of the low-frequency oscillations diminishes downstream, however, there still remains elevated energy at these frequencies in the corner and reattachment spectra. Although not appearing as a distinct peak in the premultiplied PSD, a substantial increase in energy at frequencies of approximately $St_L=0.5$ occurs in the corner and reattachment spectra when compared with the first two spectra profiles. The increase in energy content at these intermediate frequencies is attributed to the development of the mixing layer turbulence (Dupont et al. Reference Dupont, Haddad and Debiève2006). Because the three energized frequency ranges are separate from each other, even more so as the ratio $L/\delta _{{bl}}$ increases, it is possible to filter out both the low-frequency oscillations and the fine-scale boundary layer turbulence from the mixing layer time signals. Therefore, a bandpass filter is designed for each case to retain frequencies between $St_L=0.3$ and $St_\delta =0.2$. Note that the low-frequency cutoff scales on $L$ and the high-frequency cutoff on $\delta _{bl}$. Correlation curves of $R_{(\rho u)'p'}$ and $R_{(\rho w)'p'}$ from bandpass-filtered time signals taken from the corner profile in each of the Mach 3, 7 and 10 flows are plotted in figure 10(a–c). The corner profile refers to the slice through the mixing layer that intersects the ramp corner as drawn in figure 8(a). The signals of $p'$ and $(\rho w)'$ are taken at $\zeta =0$ on the $x$-axis and $(\rho u)'$ along the bottom edge of the shear layer at $\zeta =-0.06$. The time axis is oriented so that a positive time shift indicates a motion of the fluid, $(\rho u)'$ or $(\rho w)'$, that occurs before the correlated fluctuation in pressure. Time is non-dimensionalized by the preshock free stream velocity $U_e$ and separation length $L$. Both the $R_{(\rho u)'p'}$ and $R_{(\rho w)'p'}$ curves are sinusoidal and are almost perfectly out of phase with each other. Only $R_{(\rho u)'p'}$ for the Mach 10 case fails to have a noticeable signature. A decrease in the mixing layer structure correlation level with increasing $M_c$ was also observed by Samimy et al. (Reference Samimy, Reeder and Elliott1992). The Mach 7 and Mach 10 flows, both $M_c \approx 2.0$, have a noticeably smaller amplitude than the Mach 3 with $M_c \approx 1.0$. The approximate period of the correlations is $2 L/U_e$ which is consistent with the expected $St_L=0.5$ for the mixing layer frequencies.

Figure 9. Premultiplied PSD of wall pressure signals from the (a) Mach 3, (b) Mach 7 and (c) the Mach 10 data. Spectra are shown for the upstream boundary layer (solid bold), separation point (dotted), corner (dashed bold) and reattachment (solid).

Figure 10. Cross-correlations between bandpass-filtered mixing layer centreline pressure and mass flux signals at the corner profile of each dataset. Averaged over full time signal (a–c) and enhanced average (d–f). (a,d) M3, (b,e) M7 and (c,f) M10.

Although a distinct sinusoidal signature is visible in the full time signal correlations, the overall magnitude of the correlation is rather low particularly for M7 and M10. In order to obtain a stronger signature of the vortex events, an enhanced correlation method is used. The conditional averaging technique used here is similar to the method of Brown & Thomas (Reference Brown and Thomas1977) for the detection of hairpin packets in a turbulent boundary layer. The strategy of Brown & Thomas assumes that the hairpin packet, or in this case the mixing layer vortex, is a specific isolated event occurring in the flow and that the corresponding fluid motion, or pressure fluctuation, associated with that event produces a specific signature in the time signal. Time signals of relevant fluid properties can be broken up into shorter segments and the cross-correlation computed for each of the shortened segments. If a vortex occurs in a given segment, the cross-correlation curve of that segment will produce the ‘signature’ of the vortex event. The enhanced correlation, therefore, is the average over all of the short-signal correlations that show the vortex signature.

For the detection of the mixing layer vortices, the time signals of $p'$, $(\rho u)'$ and $(\rho w)'$ are broken up into $N$ segments of length $6.5 U_e/L$, or twice the wavelength of the bandpass filter low-frequency cutoff. Successive time segments are taken with 50 % overlap. We assume that the signature of the mixing layer vortices has the same form as the full time signal correlations. The criteria for the selection of the enhanced correlations are such that the segment correlations ${R^{n}}_{(\rho u)'p'}$ and ${R^{n}}_{(\rho w)'p'}$ simultaneously have maxima and minima in the same location as, but at least twice the magnitude of, the full time average signature. More specifically, a correlation is retained if (i) $\max ({R^{n}}_{(\rho u)'p'}) \geqslant 2 \max (R_{(\rho u)'p'})$ and (ii) $\min ({R^{n}}_{(\rho w)'p'}) \leqslant 2 \min (R_{(\rho w)'p'})$ both in the range $-2 \leqslant {\rm \Delta} t(U_e/L) \leqslant 0$ for $n \in N$. The enhanced results for the corner profile are shown in figure 10(d–f). For the Mach 3, approximately 30 % of the time segments met the criteria, and approximately 20 % for the Mach 7 and Mach 10 flows. A distinct wavelength appears in the enhanced correlation for all three cases including the Mach 10 $R_{(\rho u)'p'}$.

4.2. Vortex convection velocity

The enhanced correlation technique was repeated for several stations along the $x$-axis. The frequency in $St_L$ determined from the $R_{(\rho u)'p'}$ enhanced correlation curve time period are plotted in figure 11(a) versus $X/L$. Although not shown, the time period selected from the enhanced correlations of $R_{(\rho w)'p'}$ produces similar frequencies to those from $R_{(\rho u)'p'}$. The frequency is approximately constant through the region of similarity for each case. Dupont et al. (Reference Dupont, Haddad and Debiève2006) also showed that the shear layer frequency plateaus at a constant $St_L=0.5$ in the separated flow of their reflected shock STBLI with free stream Mach number of 2.3. They also showed that this frequency was independent of the incident shock angle.

Figure 11. Non-dimensional vortex frequency (a), convection velocity (b) and length scale (c) determined from the enhanced correlations.

The enhanced correlations can be used to determine the actual mixing layer vortex convection velocity $U_c$. At a given position on the $x$-axis, if a time segment is selected by the enhanced correlation criteria, the centreline pressure signal from that time segment can be correlated with the same from an adjacent position along $x$. The convection velocity of that vortex event is then obtained by dividing the distance between the two points in $x$ by the offset in time of the peak in $R_{p'p'}$. The $U_c$ can then be averaged over all enhanced correlation selections. Here the cross-correlation of adjacent pressure signals is used because the theoretical $U_{c,i}$ discussed in § 3.3 is by definition the convection velocity of the stagnation point between successive vortices (Papamoscho & Roshko Reference Papamoscho and Roshko1988; Smits & Dussauge Reference Smits and Dussauge2006). The averaged convection velocity versus $X/L$ is plotted in figure 11(b). For the cross-correlations of pressure, adjacent points are spaced approximately $0.1L$ apart. An average of both the forward adjacent point and the backward adjacent point correlation is used to calculate $U_c$ at each data point plotted in figure 11(b). The convection velocity seems to undergo a gradual transition in the first half of the similarity region, but, for all three flows, $U_c$ levels off at $0.4U_e$ in the second half of the region of similarity. For comparison, $U_{c,i}$ calculated from (3.2) is $0.5U_e$ for the M3 flow and $0.6U_e$ for M7 and M10. A similar comparison was made by Dupont et al. (Reference Dupont, Haddad and Debiève2006) for their Mach $2.3$ reflected shock experiments. They found the phase velocity of wall pressure signals in the frequency range of $0.2 \leqslant St_L \leqslant 0.5$ gave a shear layer convection velocity of approximately $0.3U_e$ compared with the isentropic prediction of $0.5U_e$. In either case of the compression ramp or the reflected shock flow, the theoretical convection velocity significantly overpredicts the measured vortex convection velocity.

The time scale of figure 11(a) and the convection velocities of figure 11(b) can be combined to estimate the spatial wavelength of the frozen vortices. The spatial quantity $St_L^{-1} U_c$ is plotted in figure 11(c) and represents the average distance between successive vortex cores. As with the convection velocity, the wavelength seems to reach a constant value in the second half of the region of similarity, levelling off at approximately $0.8L$ spacing for all three flows.

4.3. 3-D vortex signature

In order to investigate the spatial organization of the large vortex structure in our STBLI mixing layers, the following correlation coefficient is defined:

(4.1)\begin{equation} R_{f'p'} = \frac{\overline{f'(x+{\rm \Delta} x,y+{\rm \Delta} y,z+{\rm \Delta} z)p'(x,y,z_{cl})}}{f'_{{rms}} p'_{{rms}}} , \end{equation}

where $f'$ can refer to either $(\rho u)'$ or $(\rho w)'$, and $p'(x,y,z_{cl})$ is the pressure along the mixing layer centreline. Again Taylor's hypothesis of frozen vortices is used to convert time signals into spatial information and so, in (4.1), we set $x=tU_c$ where $U_c$ is the convection velocity determined from the enhanced correlations described above. The enhanced spatial correlation can be generated in the same manner as the one-dimensional (1-D) correlations by averaging $R_{f'p'}$ over all time segments selected by the previously defined criteria. The enhanced spatial correlations of bandpass-filtered time signals from the corner profiles of the Mach 3, 7 and 10 flows are plotted in figures 12, 13 and 14, respectively. These plots represent the averaged ‘frozen’ spatial waveform of the mixing layer vortices as they convect past the corner profile as drawn schematically in figure 8. In figures 12–14, panel (a) is the enhanced average in the $xz$-plane for ${\rm \Delta} y=0$, and panel (b) is the enhanced average in the $xy$-plane for $R_{(\rho u)' p'}$. Panels (c) and (d) are the same for $R_{(\rho w)' p'}$. The $z$-location of the $xy$-plane is indicated by the solid black line in the corresponding panel (a) and also in panel (c). For $R_{(\rho u)' p'}$ the $xy$-plane is along the mixing layer bottom edge as was defined for the 1-D enhanced correlations. For $R_{(\rho w)' p'}$ the $xy$-plane is along the mixing layer centreline. Note that plotting the values of $R_{f'p'}$ along the line drawn in the $xz$-plane would result in the same 1-D correlation curves as in figure 10(d–f).

Figure 12. Enhanced correlation contours of the Mach 3 convective mixing layer structure. Correlations are of centreline pressure with longitudinal mass flux in (a) $xz$- and (b) $xy$-planes and centreline pressure with cross-stream mass flux in (c) $xz$- and (d) $xy$-planes. The horizontal dotted line in the $xz$-planes indicates the location of the corresponding $xy$-plane. Time correlations are converted to spatial information using the mixing layer convection velocity (i.e. $x/L = tU_c / L)$.

Figure 13. Enhanced correlation contours of the Mach 7 convective mixing layer structure. Correlations are of centreline pressure with longitudinal mass flux in (a) $xz$- and (b) $xy$-planes and centreline pressure with cross-stream mass flux in (c) $xz$- and (d) $xy$-planes. The horizontal dotted line in the $xz$-planes indicates the location of the corresponding $xy$-plane. Time correlations are converted to spatial information using the mixing layer convection velocity (i.e. $x/L = tU_c / L)$.

Figure 14. Enhanced correlation contours of the Mach 10 convective mixing layer structure. Correlations are of centreline pressure with longitudinal mass flux in (a) $xz$- and (b) $xy$-planes and centreline pressure with cross-stream mass flux in (c) $xz$- and (d) $xy$-planes. The horizontal dotted line in the $xz$-planes indicates the location of the corresponding $xy$-plane. Time correlations are converted to spatial information using the mixing layer convection velocity (i.e. $x/L = tU_c / L)$.

The form of the mixing layer vortices as determined from the 2-D correlation plots is a streamwise periodic structure that exists all through the cross-stream width of the mixing layer. In $R_{(\rho u)'p'}$, the sign of the periodic correlation is reversed in bands both above and below the mixing layer edges. These bands coincide with the position of the separation shock and the reverse flow, respectively. In the $xz$-plane, the coherent structures are tilted ‘forward’ in the correlations of streamwise mass flux and tilted ‘backwards’ in the correlation of the cross-stream mass flux. The horizontal axis in figures 12–14 is oriented so that positive ${\rm \Delta} x$ is ‘downstream’ and negative ${\rm \Delta} x$ is ‘upstream’. In the $xy$-plane, an obvious oblique pattern occurs and the mixing layer structures do not appear as 2-D bands in the spanwise direction. This obliqueness in the average signature is consistent with compressible mixing layer research showing increased spanwise variation of the large mixing layer vortices with elevated convective Mach number (Sandham & Reynolds Reference Sandham and Reynolds1991; Clemens & Mungal Reference Clemens and Mungal1992, Reference Clemens and Mungal1995; Rossmann et al. Reference Rossmann, Mungal and Hanson2002).

The interpretation of the correlation contour plots can be aided by considering the vector field defined by the magnitude of $R_{(\rho u)'p'}$ and $R_{(\rho w)'p'}$. Assuming that a negative fluctuation in pressure coincides with a vortex core, a plot of the vector field defined by $-(R_{(\rho u)'p'}, R_{(\rho w)'p'})$ will provide information on the average motion about a mixing layer vortex centre. These are plotted in figure 15. Also plotted in figure 15 are the location of the mixing layer centreline and the inclination angles of the isolines of zero correlation from figures 12–14. The point of crossing of the zero-correlation isolines can be interpreted as the centre of the vortex. A similar correlation vector plot was used by Kiya & Sasaki (Reference Kiya and Sasaki1983) for an incompressible separation shear layer. Unlike in Kiya & Sasaki, no clear rotational motion is observed around the vortex centre in figure 15. Instead, a saddle point occurs. The vector plot shows that the cross-stream momentum flux is positive to the left of the vortex core and negative to the right, as one would expect based on the (clockwise) orientation of the vortex roll-up. The vectors on the top and bottom of the vortex centre, however, are in the opposite orientation from what is expected. The interpretation of this stems from the fact that the density in the low-speed side of the layer is a factor of two less than on the high-speed side for the Mach 3 flow and a factor of four for the Mach 7 and 10 flows. The rotation of the vortex brings the low-momentum, low-density fluid into the high-speed, high-density side causing a negative streamwise correlation component to the left of the vortex centre. The opposite occurs for fluid being pulled from the high-speed side into the low-speed side to the right of the vortex centre.

Figure 15. Vector fields of $-(R_{(\rho u)'p'},R_{(\rho w)'p'})$ from figures 12–14 for (a) M3, (b) M7 and (c) M10. The vector field gives information on the averaged mass flux motion about a negative fluctuation in pressure. The horizontal line indicates the location of the mixing layer centreline. The inclination from vertical of the coherent structures as determined from the isoline of zero correlation in $R_{(\rho u)'p'}$ and $R_{(\rho w)'p'}$ are also indicated.

Visualizations of the actual mixing layer vortices helps in asserting the interpretation of the enhanced correlation plots. Flow visualizations of individual mixing layer vortices in the raw data of the separated STBLI flows is made particularly difficult by the environment in which they reside. One must be able to separate specifically the mixing layer rollers from (i) the smaller scale vortical hairpin vortices in the incoming boundary layer turbulence and (ii) the separation shock which sits very close to the high speed side of the mixing layer, as was shown in figure 1. We found that vortex detection methods based on the eigenvalues of the velocity divergence, such as swirl strength, were more problematic concerning the first issue. Vorticity methods, on the other hand, are dominated by the strong shear in the separation shock. Ultimately we found the method developed by Graftieaux, Michard & Grosjean (Reference Graftieaux, Michard and Grosjean2001) to be the most robust for isolating the mixing layer vortices in the raw data. The Graftieaux method is based on the topology of the velocity field rather than on derivative quantities. It effectively searches the flow for points about which there is a net circulating motion and, because it uses a summation over a search window, it also acts as a spatial filter. This method was successively used by Dupont et al. (Reference Dupont, Piponniau, Sidorenko and Debiève2008) to identify mixing layer vortices in particle image velocimetry (known as PIV) data from their separated reflected shock STBLI experiments.

The Graftieaux method is a vortex search method in a 2-D velocity vector field. If $P$ is a point in the flow, $S$ is a specified area surrounding $P$ and $M$ is a point inside $S$, the vector detector ${\varGamma }_1$ is defined by

(4.2)\begin{equation} {\varGamma}_1(P) = \frac{1}{S} \int_{\boldsymbol{M} \in S} \frac{(\boldsymbol{PM} \times \boldsymbol{U}_{\boldsymbol{M}}) \boldsymbol{\cdot} \boldsymbol{y}}{|\boldsymbol{PM}| \boldsymbol{\cdot} |\boldsymbol{U}_{\boldsymbol{M}}|} \textrm{d}S = \frac{1}{S} \int_{S} \sin \theta_{\boldsymbol{M}} \,\textrm{d}S , \end{equation}

where $\boldsymbol {PM}$ is the vector connecting points $P$ and $M$. The velocity vector at point $\boldsymbol {M}$ is $\boldsymbol {U}_{\boldsymbol {M}}$ and $\theta$ is the angle between the vectors $\boldsymbol {PM}$ and $\boldsymbol {U}_{\boldsymbol {M}}$. The parameter $\varGamma _1$ will take on values between -1 and 1 where the sign depends on the direction of rotation. It can be shown that a vortex exists at $\boldsymbol {P}$ if $|\varGamma _1| > 2 / {\rm \pi}$. For a square interrogation area with N equally spaced discreet points inside the area $S$, (4.2) can be re-expressed as

(4.3)\begin{equation} {\varGamma}_1(P) = \frac{1}{S} \int_{S} \sin \theta_{\boldsymbol{M}} \,\textrm{d}S = \frac{1}{N} \sum_{N} \sin \theta_{\boldsymbol{M}}. \end{equation}

The bandpass-filtered time signals of velocity from the corner profile of the ramp grids were again converted to space signals via the convection velocity of § 4.2. Thus the 3-D velocity field on which $\varGamma _1$ operates was generated. The 2-D velocity vector $\boldsymbol {U}_{\boldsymbol {M}}$ is defined as $(u-U_c,w)$ and the Graftieaux vortex detector was applied throughout the volume but always in the $xz$-plane. A square interrogation window of size $0.5\delta _{bl}$ was used throughout. The results are plotted in figures 16–18. The contour of $\varGamma _1$ in the streamwise–spanwise plane sliced along the mixing layer centre ($\zeta =0$) is plotted for a time segment equivalent to $8L$ in length that was randomly selected from the full time signal. This provides a top view of the instantaneous frozen mixing layer structures. In the inset of figures 16–18(b) is shown a side view of the structures. The location in the span of the 3-D volume of $\varGamma _1$ is indicated by the dashed line in the $xy$-plane contour. Similar plots are provided for arbitrarily selected time segments from the M3, M7 and M10 data.

Figure 16. Contour of the vortex detector variable $\varGamma _1$ for the convective frozen flow from the corner grid plane of the Mach 3 flow. (a) The $xy$-plane sliced through the mixing layer centre and (b) the $xz$-plane sliced through the section indicated by the dashed lines and arrows in panel (a). Dotted diagonal lines indicate the vortex angle predicted by (4.4).

Figure 17. Contour of the vortex detector variable $\varGamma _1$ for the convective frozen flow from the corner grid plane of the Mach 7 flow. (a) The $xy$-plane sliced through the mixing layer centre and (b) the $xz$-plane sliced through the section indicated by the dashed lines and arrows in panel (a). Dotted diagonal lines indicate the vortex angle predicted by (4.4).

Figure 18. Contour of the vortex detector variable $\varGamma _1$ for the convective frozen flow from the corner grid plane of the Mach 10 flow. (a) The $xy$-plane sliced through the mixing layer centre and (b) the $xz$-plane sliced through the section indicated by the dashed lines and arrows in panel (a). Dotted diagonal lines indicate the vortex angle predicted by (4.4).

From the top view, one can immediately observe the spanwise angular pattern in the vortices as is consistent with the 2-D correlation plots of figures 12–14. From the top plan view, the M3 vortices are visually more coherent than the M7 and M10 flows. Also, in the side view, the M3 vortices appear more regular and resemble a sinusoidal wavy interface between the high- and low-speed sides of the mixing layer. The vortex cores appear to occur predominantly at the upslope of the wave. A similar pattern is seen in the $xz$-plane slice of the M7 and M10 flows although, in general, the M3 flow is apparently more regular.

With regard to the spanwise oblique angle observed in both the enhanced correlation contours and the instantaneous vortex visualizations, it is interesting to consider the compressible mixing layer inviscid linear stability analysis by Sandham & Reynolds (Reference Sandham and Reynolds1990, Reference Sandham and Reynolds1991). These authors showed that an oblique unstable mode becomes dominant over the 2-D mode for $M_c > 0.6$. Furthermore, they found that the angle $\alpha$ measured from the 2-D mode increased with increasing $M_c$ by

(4.4)\begin{equation} M_c \cos{\alpha} \approx 0.6. \end{equation}

For the current STBLI shear layers, $\alpha = 54^{\circ }$ for the M3 flow and $72^{\circ }$ for M7 and M10. These angles are indicated by the diagonal dotted lines drawn in the top-view contours of $\varGamma _1$ in figures 16–18(a) and prove to be a close representation of the actual structure occurring in these flows.