2.1 Wind tunnel and cavity hardware

Experiments were performed in Sandia’s Trisonic Wind Tunnel (TWT), which is a blowdown-to-atmosphere facility using air as the test gas through a test section enclosed within a pressurized plenum. In its transonic configuration, the test section is a straightforward rectangular duct of dimensions 305 mm $\times$ 305 mm with interchangeable walls. Porous walls ordinarily are used for testing near sonic conditions when blockage relief is needed and solid walls are used when imaging diagnostics require windows. For the present experiments, the test section was configured with porous walls on the top wall and one side wall to alleviate non-physical resonances due to wind tunnel duct modes (Wagner et al. Reference Wagner, Casper, Beresh, Henfling, Spillers and Pruett2015a ); a solid wall with a window for imaging was installed in the other side of the test section. Despite the non-uniform test section, no evidence of flow asymmetry was detected in either pressure or PIV data.

Supersonic experiments were conducted in the TWT’s half-nozzle test section, in which the top wall of each supersonic nozzle is retained and a single lower wall extends the inlet contour of the tunnel before fairing into a flat surface at what previously would have been the test section centreline. This provides a flat plate working surface with convenient optical access within the ‘test rhombus’ in which the supersonic expansion is complete, as opposed to the full test section that requires sting-mounted models for testing. The resulting half-nozzle test section is 152 mm high and 305 mm wide.

In the present study, transonic experiments were conducted at Mach numbers 0.60, 0.80 and 0.94, and supersonic experiments were conducted at Mach numbers 1.5, 2.0 and 2.5. Transonic flow conditions were selected to hold the free stream dynamic pressure $q_{\infty }$ constant at 33 kPa. The supersonic data, on the other hand, were acquired using stagnation pressures designed to achieve a common Reynolds number based on the incoming boundary layer thickness of approximately 400 000, compared to approximately 200 000 for the transonic cases. Supersonic values of $q_{\infty }$ were much higher, between 110 and 133 kPa. Previous PIV measurements and Pitot probe surveys have shown that the incoming 99 %-velocity boundary layer thickness ranges from approximately 10–15 mm for the entire Mach range of the present experiments, which is 40 %–60 % of the cavity depth (see below). Since the boundary layer thickness remains within the same range for all cases and the change in Reynolds number is relatively small for a fully turbulent flow, the difference in flow conditions is not expected to be evident in normalized data. Indeed, no effect has been found on the normalized unsteady pressures (Zhuang et al. Reference Zhuang, Alvi, Alkislar and Shih2003; Gai, Kleine & Neely Reference Gai, Kleine and Neely2015; Wagner et al. Reference Wagner, Casper, Beresh, Hunter, Henfling, Spillers and Pruett2015b ), nor does it significantly alter the velocity field or turbulent stresses (based on supplementary data collected as part of the Beresh et al. (Reference Beresh, Wagner, Pruett, Henfling and Spillers2015a ) experiment).

The wind tunnel stagnation temperature $T_{0}$ is fixed at 321 K $\pm$ 2 K by heating in the storage tanks, and the wall temperature is effectively constant at ambient conditions, $T_{w}=307~\text{K}\pm 3~\text{K}$ . Free stream velocities $U_{\infty }$ were measured from previous PIV experiments as 215, 280, 315, 450, 535 and $605~\text{m}~\text{s}^{-1}$ for the six cases in order of increasing Mach number, estimated to be accurate to within 0.3 %.

The finite-width cavity of the present experiments is simply a rectangular pocket in a plate inserted into the lower test section wall, as shown in figure 1. The cavity geometry is identical for both the supersonic and the transonic configurations, although its streamwise position within the test section varies mildly. The floor of the cavity is a BK7 glass flat to allow the laser sheet for the PIV measurements to enter the test section from below. The cavity has dimensions 127 mm $\times$ 127 mm (5 in. $\times$ 5 in.) with a nominal depth of 25.4 mm (1 in.). In practice, the cavity depth was measured to be 25.9 mm (1.02 in.) for the supersonic configuration but achieved the intended dimension of 25.4 mm for the transonic experiments, due to a discrepancy in the crush of a gasket. In addition to the widest cavity dimensions of 127 mm $\times$ 127 mm, insert blocks can be bolted against the cavity side walls to reduce the cavity width for additional tests. This was used to create cavities of 127 mm $\times$ 76 mm (5 in. $\times$ 3 in.) and 127 mm $\times$ 25 mm (5 in. $\times$ 1 in.); the cavity depth was not changed. The respective length-to-width ratios $L/W$ are 1.00, 1.67 and 5.00; the length-to-depth ratio $L/D$ is 4.90 for the three supersonic conditions and 5.00 for the three transonic conditions. These three cavity configurations henceforth will be denoted the 5 $\times$ 5, the 5 $\times$ 3 and the 5  $\times$  1 cases. The coordinate system was chosen such that $x$ lies in the streamwise direction and $y$ is vertical, positive away from the cavity, with the $z$ coordinate spanwise and right handed. The origin is the spanwise centre of the cavity leading edge.

Figure 1. The $127\times 127~\text{mm}^{2}$ ( $5\times 5~\text{in.}^{2}$ ) cavity installed into the floor of the transonic test section with the PIV laser sheet.

2.2 High-frequency pressure sensors

Dynamic pressure measurements were made at multiple locations along the fore and aft walls of the cavity using piezoresistive sensors (Kulite XCQ-062-50A or equivalent) whose frequency response is flat to approximately 50 kHz. This exceeds the frequency of the highest detectable cavity resonance tones by nearly an order of magnitude. In the present case, data are shown only from a sensor placed at the spanwise centre of the cavity aft wall, which is representative of the aeroacoustic environment throughout the cavity. The sensors were powered by a voltage amplifier (Endevco Model 136) that also amplified the resulting pressure signals and passed them to a low-pass filter (Krohn-Hite Model 3384) with a cutoff frequency of 50 kHz. Data were digitized at 14 bits and 200 kHz (National Instruments PXI 6133). The measurement capability is described in more detail in Wagner et al. (Reference Wagner, Casper, Beresh, Hunter, Henfling, Spillers and Pruett2015b ) although a slightly different cavity geometry was tested.

Pressure data are shown as power spectra computed using Welch’s periodogram method, Blackman windows, a 50 % overlap between data segments and a frequency resolution of 10 Hz. Spectral amplitudes are given normalized by dynamic pressure and the repeatability has been shown to be within 0.5 % of full scale.

2.3 Particle image velocimetry

Two data sets have been collected, both described previously (Beresh et al. Reference Beresh, Wagner, Pruett, Henfling and Spillers2015a ,Reference Beresh, Wagner, Pruett, Henfling and Spillers b ). Despite any inference based on publication dates, the supersonic measurements were collected several years prior to the transonic measurements and were somewhat less mature.

In PIV experiments, the TWT is seeded by a thermal smoke generator using a mineral oil base (Corona Vi-Count 5000) whose output has been previously measured in situ by tracking the particle response across a shock wave to show a particle size of $0.7{-}0.8~\unicode[STIX]{x03BC}\text{m}$ . Stokes numbers have been estimated as at most 0.05 based on a posteriori measurements of typical cavity shear layer eddies, which is sufficiently small to render particle lag errors negligible. Inspection of PIV images of the recirculation region in the cavity shows that, although the particle density drops at the highest Mach numbers, sufficient particles are retained for high vector quality in all regions.

4.1 Mean velocity field

A representative mean velocity field of the cavity flow is shown in figure 3. This is the case of the 5 $\times$ 5 cavity at Mach 0.8, which is well measured by the dual stereoscopic PIV system and broadly indicative of the flow for all cases. The white box drawn in figure 3 shows the more limited field of view of the supersonic stereoscopic measurements in subsequent figures. The vector field captures the shear layer over the top of the cavity, its impingement on the aft wall and the recirculation of flow near the floor of the cavity. Mean reverse velocities reach approximately $-0.25U_{\infty }$ . The recirculation region is seen to dominate the cavity flow field and extend over nearly its entire streamwise length. As noted in Beresh et al. (Reference Beresh, Wagner, Pruett, Henfling and Spillers2015a ), the general flow structure along the streamwise centre plane of the cavity strongly resembles similar measurements both incompressible and subsonic compressible (e.g. Zhuang et al. Reference Zhuang, Alvi, Alkislar and Shih2006; Haigermoser et al. Reference Haigermoser, Vesely, Novara and Onorato2008; Murray et al. Reference Murray, Sallstrom and Ukeiley2009) even though such experiments were conducted in a two-dimensional cavity whereas the present study uses a finite-width geometry.

Figure 3. Mean velocity field for the 5 $\times$ 5 cavity at Mach 0.8, which is representative of the general flow structure for all cases. The white box shows the field of view of the supersonic stereoscopic measurements in subsequent figures.

4.2 Streamwise turbulence intensity

Figure 4 shows the streamwise turbulence intensity of the cavity for the 5 $\times$ 5 case and four of the six Mach numbers. Superposed on these contours are streamlines derived from the mean velocity field to identify the overall flow field structure. Mach 0.8 represents the three subsonic cases, since the non-dimensional flow field was found to not change significantly from Mach 0.6 to Mach 0.94. The Mach 0.8 plot is formed from the dual stereoscopic measurements, but the three supersonic plots are composites of the two-dimensional field of view and the stereoscopic field of view. Combining them in this manner expands the coverage of the cavity from the available measurements. This approach is useful only for the streamwise velocity component as the vertical component is biased for the two-dimensional measurements and of course the spanwise component is absent.

Figure 4. Streamwise turbulence intensity fields for the 5 $\times$ 5 cavity with superposed streamlines derived from the mean velocity field; (a) Mach 0.8; (b) Mach 1.5; (c) Mach 2; (d) Mach 2.5.

The fundamental structure of the cavity flow is invariant with Mach number for the $u^{\prime }$ fields in figure 4. The size and position of the mean recirculation region does not appear to change appreciably, although closer inspection suggests that it shrinks in size slightly as the Mach number rises with the reduction in length isolated to its upstream extent. The thickness of the shear layer based on $u^{\prime }$ is constant with Mach number. The peak magnitudes of the streamwise turbulence intensity also remain constant within the shear layer, although at Mach 2.5 the magnitudes are lowered somewhat very near the cavity floor where reverse velocities will be greatest. Therefore, at most, compressibility induces only minor changes to the flow field of the 5 $\times$ 5 cavity when the streamwise component of turbulence intensity is considered.

Analogous plots are shown in figure 5 for the 5 $\times$ 3 cavity. Here, a composite plot is available only for the Mach 1.5 case, as two-component data were not acquired more generally because of the more limited depth to which the measurement could reach for this geometry. Instead, solely the stereoscopic field of view is shown for Mach 2 and Mach 2.5.

Figure 5. As figure 4, but for the 5 $\times$ 3 cavity.

A comparison to figure 4 can establish how the cavity width influences the flow field structure. First, unlike the 5 $\times$ 5 configuration, the Mach number influences the shear layer and recirculation region in the 5 $\times$ 3 cavity. The supersonic cases suggest that the recirculation region has a greater upstream extent than the Mach 0.8 case, although the depiction of the streamlines may be influenced by the more limited field of view. For all Mach numbers, the 5 $\times$ 3 cavity has a recirculation region situated nearer the cavity floor than the 5 $\times$ 5 cavity, which is the principal difference in the mean flow field between these two cavity widths. In addition, the subsonic cases show a recirculation region more limited in extent for the 5 $\times$ 3 cavity than the 5 $\times$ 5 (Beresh et al. Reference Beresh, Wagner, Pruett, Henfling and Spillers2015b ) with an aft position within the cavity. In supersonic conditions it is longer and shifted farther upstream (Beresh et al. Reference Beresh, Wagner, Pruett, Henfling and Spillers2015a ), and it displays a noticeable aftward movement for the two highest Mach numbers, which did not occur for the 5  $\times$ 5 cavity.

Contrary to the 5 $\times$ 5 cavity, the shear layer in the 5 $\times$ 3 cavity appears to thin with rising Mach number in figure 5. At the same time, the magnitude of $u^{\prime }$ diminishes in both the shear layer and the recirculation region, which is in contrast to the 5 $\times$ 5 cavity. These trends constitute compressibility effects on the flow field, and their presence is dependent upon the cavity width.

The shear layer is difficult to observe for the 5 $\times$ 1 cavity due to the limited visibility into the cavity, and the recirculation region cannot be seen at all. Just enough of the shear layer can be viewed to suggest that it may thin at Mach 2 and 2.5 and that $u^{\prime }$ diminishes a small but significant degree. For brevity, these plots are omitted due to the reduced spatial coverage, but they will be shown subsequently for the vertical component of the turbulence intensity where the results are more conclusive within the limited field of view.

The present turbulence data may be compared to several previous investigations that have reported similar measurements in a cavity, though flow conditions and geometry differ. The most nearly matched to the present configuration is the work described in Ukeiley & Murray (Reference Ukeiley and Murray2005) and Murray et al. (Reference Murray, Sallstrom and Ukeiley2009), which report a maximum streamwise turbulence intensity in the cavity shear layer of 0.25 at Mach 0.2, in reasonable agreement with the present value of 0.22 at Mach 0.8 in the 5 $\times$ 5 cavity. Similar measurements have been made in incompressible cavity flows of other $L/D$ ratios, yielding streamwise turbulence intensities from 0.13 to 0.23 (Grace, Dewar & Wroblewski Reference Grace, Dewar and Wroblewski2004; Haigermoser et al. Reference Haigermoser, Vesely, Novara and Onorato2008; Basley et al. Reference Basley, Pastur, Lusseyran, Faure and Delprat2011). All these experiments studied a two-dimensional cavity, and as the present measurements have shown, the width of the cavity affects the turbulence levels on the centreline. Therefore, the range of peak turbulence values is not unexpected and in agreement with the present work as well as can be considered plausible.

4.3 Vertical turbulence intensity

Some difference in behaviour can be observed in the 5 $\times$ 5 cavity where the vertical component of the turbulence intensity is concerned, shown in figure 6. The magnitudes of $v^{\prime }$ are lower than those of $u^{\prime }$ , which is readily evident since both are plotted on the same contour scale. The maximum measured value is 0.16, in excellent agreement with the Mach 0.73 data of Murray et al. (Reference Murray, Sallstrom and Ukeiley2009) and consistent with two other incompressible experiments (Haigermoser et al. Reference Haigermoser, Vesely, Novara and Onorato2008; Basley et al. Reference Basley, Pastur, Lusseyran, Faure and Delprat2011). As the Mach number rises, $v^{\prime }$ falls distinctly in the shear layer. This is in contrast to $u^{\prime }$ in figure 4, in which the shear layer did not display a reduction as the Mach number was increased. The $v^{\prime }$ component also diminishes with Mach number in the recirculation region, which for $u^{\prime }$ was only marginally observable at Mach 2.5.

Figure 6. Vertical turbulence intensity fields for the 5 $\times$ 5 cavity with superposed streamlines derived from the mean velocity field; (a) Mach 0.8; (b) Mach 1.5; (c) Mach 2; (d) Mach 2.5.

Figure 7 shows that the magnitudes of $v^{\prime }$ are lower for the 5 $\times$ 3 cavity than the 5 $\times$ 5 cavity, which previously has been noted (Beresh et al. Reference Beresh, Wagner, Pruett, Henfling and Spillers2015a ,Reference Beresh, Wagner, Pruett, Henfling and Spillers b ). For this cavity width, the compressibility effect on $v^{\prime }$ is even more striking. Intensity magnitudes fall further with increasing Mach number than they did for the 5 $\times$ 5 cavity and the shear layer extent based on $v^{\prime }$ is notably thinner as well. Turbulence magnitudes also drop sharply within the recirculation region at higher Mach numbers.

Figure 7. As figure 6, but for the 5 $\times$ 3 cavity.

Data from the 5 $\times$ 1 cavity are shown for $v^{\prime }$ in figure 8 despite the limited measurement penetration into the depths of the cavity. The intensity magnitude of this component of turbulence is similar to the 5 $\times$ 5 cavity at Mach 0.8. At Mach 1.5, the intensity magnitude rises slightly, which also was observed for $u^{\prime }$ (but not shown in a figure). Then, as the Mach number increases to 2 and 2.5, $v^{\prime }$ drops markedly, whereas it displayed only a small decrease in $u^{\prime }$ . At these two highest Mach numbers, the shear layer thins sufficiently that its bottom edge becomes visible even within the limited field of view. No comment on the influence of compressibility on the recirculation region is possible.

Figure 8. As figure 6, but for the 5 $\times$ 1 cavity.

4.4 Spanwise turbulence intensity

The spanwise component of the turbulence intensity behaves identically to the vertical component, save that magnitudes are approximately 25 %–30 % greater. The spatial distribution of the turbulence intensity is no different and reduced magnitudes also occur for the 5 $\times$ 3 cavity in comparison to the 5 $\times$ 5 and 5 $\times$ 1 cavities. The compressibility effect is present for the 5  $\times$ 3 cavity, in which magnitudes of $w^{\prime }$ in the shear layer and recirculation region fall as the Mach number reaches 2 and 2.5. Even the small increase in magnitude at Mach 1.5 above the Mach 0.8 levels for the 5 $\times$ 1 cavity is observed. Since no new insights are found from $w^{\prime }$ , these figures are omitted for brevity. Some quantities derived from this component will be included in § 5.

4.5 Turbulent shear stress

The primary component (streamwise-vertical) of the turbulent shear stress, $u^{\prime }v^{\prime }$ , is shown in figure 9 for the 5 $\times$ 5 cavity. As expected, $u^{\prime }v^{\prime }$ is elevated within the shear layer. It is not as prominent in the recirculation region as the normal turbulent stresses were, particularly near the cavity floor. As the Mach number is increased, no significant reduction in turbulent shear stress magnitude is evident, although perhaps a minimal decrease in shear stress may be present for the Mach 2.5 case.

Figure 9. Primary turbulent shear stress fields for the 5 $\times$ 5 cavity with superposed streamlines derived from the mean velocity field; (a) Mach 0.8; (b) Mach 1.5; (c) Mach 2; (d) Mach 2.5.

Conversely, compressibility effects on $u^{\prime }v^{\prime }$ are distinct for the 5 $\times$ 3 cavity in figure 10. The Mach 1.5 magnitudes and extent are perhaps slightly smaller than Mach 0.8, but at Mach 2 and 2.5, $u^{\prime }v^{\prime }$ drops markedly in the shear layer and leaves little impression within the recirculation region either. The shear layer is observed to thin as well.

Figure 10. As figure 9, but for the 5 $\times$ 3 cavity.

Again, data from the 5 $\times$ 1 cavity are omitted. The results for this configuration mimic those of $v^{\prime }$ shown in figure 8, in which the magnitude of $u^{\prime }v^{\prime }$ increases at Mach 1.5 compared to Mach 0.8, then drops slightly at Mach 2 and 2.5 while the extent of the shear layer shrinks to within the limited field of view. Therefore, a compressibility effect can be observed for the 5 $\times$ 1 cavity, but not as strongly as for the 5 $\times$ 3 cavity.

The maximum non-dimensional turbulent shear stress in the 5 $\times$ 5 cavity was measured as 0.015, compared to a value of 0.022 in Murray et al.’s (Reference Murray, Sallstrom and Ukeiley2009) Mach 0.73 experiment in a two-dimensional cavity of similar $L/D$ . Forestier, Jacquin & Geffroy (Reference Forestier, Jacquin and Geffroy2003) acquired Mach 0.8 data in a much deeper cavity and reported a maximum turbulent shear stress of approximately 0.02. Incompressible values have ranged from 0.008 to 0.025 (Grace et al. Reference Grace, Dewar and Wroblewski2004; Haigermoser et al. Reference Haigermoser, Vesely, Novara and Onorato2008; Basley et al. Reference Basley, Pastur, Lusseyran, Faure and Delprat2011).

The remaining two components of the turbulent shear stress are not discussed because their magnitudes are low and the data noisy, which makes it problematic to draw any meaningful conclusions.

6.1 Cavity flow measurements

Studies of free shear layers have shown that compressibility leads to a thinning of the vertical extent and a reduction of growth rate (see the discussion in the introduction). Much of the seminal research has determined the thickness of a free shear layer using a threshold rule, in which the thickness of the shear layer is determined between fixed percentage levels of each of the two free stream values. In the case of a cavity shear layer, an ambiguity arises from the presence of the recirculation region on the lower side rather than the uniform stream of a free shear layer. This presents a complication in the selection of an appropriate boundary value, which furthermore will vary along the streamwise axis. A more rigorously defined shear layer scale may be found by using the momentum thickness, which integrates the velocity profile analogously to the boundary layer momentum thickness. This occasionally has been used in cavity flows (Forestier et al. Reference Forestier, Jacquin and Geffroy2003; Larchevêque et al. Reference Larchevêque, Sagaut, Thien-Hiep and Comte2004) but suffers from an ill-defined lower boundary where the velocity does not trend towards zero owing to the recirculation region.

A better approach in cavity flows is the vorticity thickness (Rowley et al. Reference Rowley, Colonius and Basu2002), $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}=(U_{1}-U_{2})/(\text{d}U/\text{d}y)_{max}$ . The use of the velocity gradient presents an unambiguous measure of the thickness without need to consider a boundary threshold. However, a value must be assumed for the lower stream, conventionally achieved simply by setting $U_{2}=0$ (Haigermoser et al. Reference Haigermoser, Vesely, Novara and Onorato2008; Kang, Lee & Sung Reference Kang, Lee and Sung2008; Murray et al. Reference Murray, Sallstrom and Ukeiley2009; Bian et al. Reference Bian, Driscoll, Elbing and Ceccio2011; Crook et al. Reference Crook, Lau and Kelso2013). Of course, $U_{1}=U_{\infty }$ . The vorticity thickness has the additional advantage for the present experiment that it does not require measurement of the velocity field all the way to the floor of the cavity. The maximum velocity gradient can be expected near the centre of the shear layer and no further penetration into the cavity is necessary. This means that the two-component measurements from the supersonic experiments and the stereoscopic measurements for the 5 $\times$ 1 cavity are all sufficient to provide an objective measure of the shear layer thickness.

Figure 15(a) shows the vorticity thicknesses for the 5 $\times$ 5 cavity at all six Mach numbers as a function of streamwise position in the cavity. Data are available at every $x$ location even for the supersonic cases because the two-component measurements cover the full length of the cavity at this width. Figure 15(b) provides the equivalent data for the 5 $\times$ 3 cavity but here two-component data exist only for Mach 1.5 amongst the supersonic cases. Therefore, only a limited portion of the cavity has been measured using the stereoscopic data for Mach 2 and 2.5. Finally, figure 15(c) shows the vorticity thicknesses for the 5 $\times$ 1 cavity, which has no two-component measurements extending the field of view for the supersonic cases.

Figure 15. Vorticity thickness of the cavity shear layer. Black lines indicate slopes of the region 1 and region 2 growth rates. (a) 5 $\times$ 5 cavity; (b) 5 $\times$ 3 cavity; (c) 5 $\times$ 1 cavity.

The vorticity thickness profiles for the 5 $\times$ 5 cavity of figure 15(a) show several features consistent with previous cavity studies. The shear layer properties may be broken into three regions based on vorticity thickness behaviour (Forestier et al. Reference Forestier, Jacquin and Geffroy2003), although some earlier studies have regarded these as only two regions by excluding the rapid decay in thickness near the aft wall (Larchevêque et al. Reference Larchevêque, Sagaut, Thien-Hiep and Comte2004; Brès & Colonius Reference Brès and Colonius2008; Bian et al. Reference Bian, Driscoll, Elbing and Ceccio2011). Regardless of this viewpoint, the first region is characterized by fast growth of the shear layer, then the growth slows significantly to mark the onset of the second region, which extends until the aft wall influence is felt. The rapid initial growth of the shear layer is fed by the cavity’s acoustic resonances exciting the Kelvin–Helmholtz instability (Gharib & Roshko Reference Gharib and Roshko1987; Rowley et al. Reference Rowley, Colonius and Basu2002). At some point, this instability saturates and the growth of the shear layer tapers off to a lower but approximately constant value, accounting for region 2 (Larchevêque et al. Reference Larchevêque, Sagaut, Thien-Hiep and Comte2004). An outlying simulation finds a higher growth rate for the second stage (Brès & Colonius Reference Brès and Colonius2008). Some incompressible or weakly compressible cavity flows have measured only a single region of growth, but these appear to occur predominately when conditions create self-sustained oscillations but do not induce resonance (Ashcroft & Zhang Reference Ashcroft and Zhang2005; Haigermoser et al. Reference Haigermoser, Vesely, Novara and Onorato2008; Kang et al. Reference Kang, Lee and Sung2008; Crook et al. Reference Crook, Lau and Kelso2013). Single-stage growth has been found in resonating compressible flows as well (Rowley et al. Reference Rowley, Colonius and Basu2002; Murray et al. Reference Murray, Sallstrom and Ukeiley2009) and may be a function of a cavity length insufficient to reach saturation of the instability (Rowley et al. Reference Rowley, Colonius and Basu2002). Given this diversity of past experiences and flow conditions, the present case is typical of compressible cavity flows, but no unique representative flow field may be identified.

Figure 15(a) reveals clear differences in the growth of the 5 $\times$ 5 cavity shear layer as a function of Mach number. The three transonic cases all are essentially identical, but compressibility effects become evident for the supersonic cases. In region 1, increasing supersonic Mach number leads to a reduction of the shear layer growth rate. Conversely, region 2 appears to display less dependence on Mach number, but the overall thickness of the shear layer remains reduced at higher Mach numbers due to the continuing impact of region 1. The behaviour in region 1 is precisely that which has been well established from numerous compressible free shear layer experiments. No effect comparable to region 2 is found from free shear layer studies since the latter do not exhibit saturation of the shear layer instability as may occur in cavity flows.

Figure 15 includes sample uncertainty estimates for the vorticity thickness at two representative Mach numbers and in each region. The uncertainties of the vorticity thicknesses were propagated from the velocity uncertainties using a Taylor expansion. This results in a large uncertainty downstream where the gradient in the mean velocity profile is smaller, which appears overly conservative in region 2 given that it greatly exceeds the random variations in $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ that are observed. The uncertainty is larger still for the supersonic cases because of the fewer number of vector fields and therefore greater convergence uncertainty. However, the data appear better behaved than is suggested by the uncertainty estimates, which do not account for trends through numerous data points. The variation of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ as a function of Mach number is quite distinct despite lying within the uncertainty band and is highly likely to be significant.

Data for the 5 $\times$ 3 cavity are supplied in figure 15(b) and they exhibit some differences in comparison to the 5 $\times$ 5 cavity. Region 1 measurements are not available for the two higher Mach numbers, but the transonic data alone suggest a different trend. Whereas the 5 $\times$ 5 cavity did not show any change in the region 1 growth rate for the three transonic cases, the 5 $\times$ 3 cavity suggests a diminishing growth rate as the transonic Mach number is increased. The single available supersonic case continues this trend. In region 2, growth rates are all approximately equal, as was also the case for the 5 $\times$ 5 cavity. The lower overall vorticity thicknesses for Mach 2 and Mach 2.5 in region 2, despite common slopes, suggests that the growth rate in region 1 must have been lower still for these cases.

The behaviour of the 5 $\times$ 1 cavity is again different. Here, the region 1 growth rates are identical for the three transonic cases, but the region 2 growth rates appear to show lower values for the supersonic cases as compared to transonic. Moreover, region 1 terminates earlier than for the 5 $\times$ 5 and 5 $\times$ 3 cavities, suggesting that the instability saturates more rapidly. The 5 $\times$ 1 data do not extend quite as far downstream as for the two wider cavities because the limited field of view interferes with evaluation of the vorticity thickness near the aft wall.

The growth rates may be found by determining the slopes of the vorticity thickness plots in each of the first two regions, with the results presented in figure 16. Region 1 is defined as $0.4<x/D<1.7$ for the 5 $\times$ 5 and 5 $\times$ 3 cavities but shortened to $0.4<x/D<1.2$ for the 5 $\times$ 1 cavity, based on linearity of the plots in figure 15. Region 2 is $2.3<x/D<3.9$ in all cases. A simple least-squares fit was used to calculate the shear layer growth rate $\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/\text{d}x$ . Uncertainties were estimated based upon the quality of the fit, which in turn is a function of the degree of scatter of the data points.

Figure 16. Growth rates of the cavity shear layer for each of the cavity widths. Supersonic data in region 2 are shown in solid lines for stereoscopic measurements and a dashed line for two-component measurements.

The most complete data for region 1 are those of the 5 $\times$ 5 cavity. The three transonic cases show minimal variation in growth rate, but once supersonic conditions are reached a considerable reduction in growth rate occurs as the Mach number rises. The 5 $\times$ 3 cavity differs in that the transonic growth rates are lower than those of the 5 $\times$ 5 cavity and show a diminishing trend with increased Mach number even at these weakly compressible conditions. The single data point for a supersonic case in the 5 $\times$ 3 cavity closely matches its 5 $\times$ 5 counterpart. The three available transonic data points for the 5 $\times$ 1 cavity agree well with the 5 $\times$ 5 data.

The region 2 growth rates drop significantly for Mach 1.5 as compared to the transonic cases, but then rise slightly for Mach 2 and 2.5. This is true for all three cavity widths. Overall, no clear or strong trend is evident for region 2 and it is reasonable to conclude that the growth rate is approximately constant for all conditions following saturation of the instability. Yet the shear layer thickness clearly decreases with increasing supersonic Mach number for all three cavity widths, as evidenced in figure 15. This implies that the region 1 growth rate must diminish appreciably as Mach number increases for all three cavity widths, which makes the absence of supersonic measurements in this region unfortunate for the narrower two cavities. Nevertheless, this observation helps confirm that region 1 growth rates fall with rising supersonic Mach numbers even where direct measurements are absent.

Finally, for the 5 $\times$ 5 cavity, region 2 data can be determined from both two-component and stereoscopic measurements. A small absolute difference is observed between the two in figure 16, but trends are consistent and no reconsideration of any conclusions would be warranted.

The region 1 growth rates should be most comparable to the growth rates of free shear layers. The classic compressibility effect found in free shear layers is that increasing Mach number reduces the growth rate, and that trend clearly is replicated for the cavity shear layer in region 1. A comparison of the magnitude of the growth rates is more challenging given the considerable scatter of the data found in the literature. This is the case even for incompressible shear layers. Smits & Dussauge (Reference Smits and Dussauge2006) ultimately concluded that a constant-density free shear layer with $U_{2}=0$ most probably has a growth rate of $\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/\text{d}x=0.16$ , although they note the considerable scatter of approximately 35 % found in the data due to sensitivity of the instability to experimental conditions.

In the present case, the incompressible growth rate is $\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/\text{d}x=0.26$ , based on the relatively constant trend for the three transonic cases in the 5 $\times$ 5 and 5 $\times$ 1 cavities. The trend of the 5 $\times$ 3 cavity can plausibly be found consistent with this value. This growth rate exceeds the free shear layer value of 0.16 by nearly twice its 35 % scatter. Other cavity studies demonstrate even larger scatter. Some incompressible studies are fairly close to the 0.16 value for free shear layers (Ashcroft & Zhang Reference Ashcroft and Zhang2005; Haigermoser et al. Reference Haigermoser, Vesely, Novara and Onorato2008; Murray et al. Reference Murray, Sallstrom and Ukeiley2009). Conversely, Rowley et al. (Reference Rowley, Colonius and Basu2002) and Brès & Colonius (Reference Brès and Colonius2008) find values as low as 0.03 and Kang et al. (Reference Kang, Lee and Sung2008) find the largest known growth rate of 0.37. Certainly, differing experimental conditions play a role, as the growth rate is a function of the cavity length (Sarohia Reference Sarohia1977; Rowley et al. Reference Rowley, Colonius and Basu2002; Brès & Colonius Reference Brès and Colonius2008); with a contrasting opinion by Ashcroft & Zhang (Reference Ashcroft and Zhang2005) and the state of the incoming boundary layer (Rowley et al. Reference Rowley, Colonius and Basu2002; Brès & Colonius Reference Brès and Colonius2008; Haigermoser et al. Reference Haigermoser, Vesely, Novara and Onorato2008). Additionally, Crook et al. (Reference Crook, Lau and Kelso2013) notes growth rate can vary when measurements are made off the centreline in a rectangular cavity.

Differences in growth rates amongst the variety of experiments and computations also may be attributable to differences in the resonance amplitudes or their match with the frequency of the Kelvin–Helmholtz instability. This would lead to variations in the energy forcing the instability growth. Alternatively, the receptivity of the instability may vary amongst the flow fields, as this property is known to be sensitive to boundary conditions (Ragab & Wu Reference Ragab and Wu1989; Morris & Giridharan Reference Morris and Giridharan1991; Day, Reynolds & Mansour Reference Day, Reynolds and Mansour1998; Barone & Lele Reference Barone and Lele2005) and therefore would vary amongst the differing experimental conditions of the cavity studies considered.

6.2 Comparison to free shear layers

Part of the reason for any ambiguities in the cavity shear layer growth may be that, thus far, they have been studied in terms of the free stream Mach number rather than the convective Mach number, $M_{c}$ . Studies of free shear layers have well established that use of the convective Mach number of the flow better establishes similarity (Lele Reference Lele1994; Dutton Reference Dutton1997; Smits & Dussauge Reference Smits and Dussauge2006), as originally proposed by Bogdanoff (Reference Bogdanoff1983) and Papamoschou & Roshko (Reference Papamoschou and Roshko1988). The convective Mach number is more difficult to determine in the present case because the slow side of the shear layer is bounded by the recirculation region in the cavity as opposed to a uniform free stream, but the assumption of $U_{2}=0$ that was used to calculate the vorticity thickness may be redeployed here. A similar difficulty is encountered in the shear layer over a base flow and the convective Mach number still can be reasonably estimated to find similarity in the data (Dutton Reference Dutton1997).

To properly assess the effect of compressibility on the shear layer growth rate, it must be normalized to its incompressible value. This poses something of a challenge in the present case given that the experiments were conducted exclusively in the compressible regime and no other known incompressible cavity experiments exactly replicate the current finite-width geometries. This is an important consideration, as the current data have shown that the changes to the cavity width affect the shear layer properties. Moreover, as Murray et al. (Reference Murray, Sallstrom and Ukeiley2009) point out, incompressible cavities of the current dimensional range often do not resonate, and this appears to alter the shear layer growth rate.

Figure 16 suggests that a reasonable choice of the incompressible cavity growth rate in region 1 can be extrapolated as 0.26, as noted in § 6.1. This value is used to normalize the measured compressible shear layer growth rates reported in figure 16. The growth rate of free shear layers as a function of the convective Mach number has been assembled previously by several authors (e.g. Lele Reference Lele1994; Dutton Reference Dutton1997; Smits & Dussauge Reference Smits and Dussauge2006). In each case, the normalized growth rate $\unicode[STIX]{x1D6F7}=(\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/\text{d}x)/(\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/\text{d}x)_{incompressible}$ and is plotted as a function of $M_{c}$ .

Figure 17 reproduces the free shear layer compilation of Smits & Dussauge (Reference Smits and Dussauge2006), which is probably the most complete survey of the available data. Growth rates may differ somewhat from the values reported by the original authors as Smits and Dussauge incorporate subsequent reassessments of the data. The trend of decreasing normalized growth rate with convective Mach number is strongly evident, even within the substantial scatter of the data. Some of this scatter is attributable to the disparate experimental means of measuring the shear layer thickness (Smits & Dussauge Reference Smits and Dussauge2006; Barone, Oberkampf & Blottner Reference Barone, Oberkampf and Blottner2006).

Figure 17. Normalized growth rates of region 1 of the cavity shear layer as a function of convective Mach number, superposed on values from free shear layers. After Smits & Dussauge (Reference Smits and Dussauge2006).

The present cavity data are superposed on the free shear layer data in figure 17. Region 1 growth rates are comparable to those from free shear layers because the physical processes are expected to be similar before the instability saturates in region 2, which is not known to occur in free shear layers. The most complete data are those for the 5 $\times$ 5 cavity and these are observed to faithfully follow the free shear layer trends. The 5 $\times$ 3 cavity data also are well within the bounds of the free shear layer data. The 5 $\times$ 1 case is excluded because region 1 measurements are not available for Mach numbers at which compressibility effects are observed. Smits & Dussauge (Reference Smits and Dussauge2006) noted that shear layer thicknesses determined using the velocity field tend to produce values of $\unicode[STIX]{x1D6F7}$ towards the higher end of the scatter, and in fact the present measurements lie somewhat greater than the general centroid of the data points. Therefore, the classic compressibility effect on free shear layer growth has been confirmed in the present cavity shear layer data as well. It is interesting to note that the cavity shear layer growth rates well match those from free shear layers despite the absence of good agreement in the turbulence quantities analysed in the previous section.

Free shear layer studies have suggested that the compressible reduction of shear layer growth occurs predominantly above $M_{c}=0.6$ , at which point the instability mechanism may shift away from the classic Kelvin–Helmholtz instability (Sandham & Reynolds Reference Sandham and Reynolds1991; Clemens & Mungal Reference Clemens and Mungal1995; Elliott et al. Reference Elliott, Samimy and Arnette1995). This appears to be the case for the present cavity data as well, but only for the 5 $\times$ 5 cavity. The 5 $\times$ 3 cavity is not consistent with this trend and displays a falling growth rate across all measured Mach numbers, which may suggest a different instability is operative for this cavity width. The inference of a stronger spanwise instability in the 5 $\times$ 3 cavity would be supported by the earlier observations regarding turbulent stresses in § 5. Interestingly, figure 15(a,b) shows that the shift from region 1 growth to region 2 growth is consistent across all Mach numbers, which may suggest that the mechanisms for cavity shear layer growth do not exhibit a change around $M_{c}=0.6$ , regardless of the clear suppression of the growth rate.