1 Introduction

The substantial impact of shock–boundary-layer interactions (SBLIs) on high-speed flow applications has led to their widespread study using wind tunnel experiments (Babinsky and Harvey 2011). Many such investigations are conducted in supersonic wind tunnels with rectangular cross sections, focusing on the response of the tunnel-wall boundary layers to an incident shock wave. There are two standard two-dimensional nozzle configurations commonly used in these wind tunnels, which are illustrated schematically in Fig. 1 using the canonical oblique-shock reflection as an example of a typical SBLI experiment. A “full” setup (Fig. 1a) consists of two contoured nozzle surfaces symmetric about the tunnel centre-height. Meanwhile, the “half” configuration in Fig. 1b features a curved ceiling nozzle surface and a straight horizontal floor. These two approaches to two-dimensional nozzle design were first described in a textbook by Ferri (1949).

Fig. 1
figure 1

Schematic representation of oblique shock reflections in wind tunnels using a a full nozzle and b a half nozzle

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Both full and half nozzle configurations have generally been considered to be equivalent in the past, which has led researchers to make direct comparisons between data collected across different facilities (Bruce et al 2011). However, there is some evidence that this assumption may not be entirely correct. A recent study by Sabnis and Babinsky (2019) found that pressure gradients in the nozzle region set up vertical secondary flows within the sidewall boundary layers, whose direction depends on the installed nozzle configuration.

For a full nozzle, the expansion through the nozzle region is most severe at the tunnel centre height, with the pressure drop on the curved surfaces at the floor and ceiling lagging behind. Therefore, in any cross-sectional plane, there is a vertical pressure gradient, where the pressure at the top and bottom of the tunnel is higher than at the centre height. The low-momentum flow in the sidewall boundary layers is most susceptible to this pressure gradient, and so a secondary vertical flow is induced in these regions from the channel corners towards the centre height (Fig. 2a). The half nozzle configuration is equivalent to the upper half of the full nozzle. Therefore, the vertical pressure gradient in the nozzle and thus the secondary flows in the sidewall boundary layers are directed from the contoured surface to the flat surface (Fig. 2b).

The secondary flows transport the low-momentum fluid within the sidewall boundary layers and therefore affect the local boundary-layer thickness distribution. Sabnis and Babinsky also noted that the effects of the nozzle-induced secondary flows on the sidewall boundary layers also extend to the corner boundary layers. Specifically, Fig. 2 shows that the corner boundary layers corresponding to the full nozzle arrangement or the contoured side of the half nozzle configuration are significantly thinner than those that have developed along the flat wall of a half nozzle setup (Sabnis and Babinsky 2019). Thus, the installed nozzle geometry can have a profound impact on the momentum contained within the corner regions.

Fig. 2
figure 2

Streamwise velocity distribution across the tunnel cross section, measured by Sabnis and Babinsky (2019) for a a full nozzle and b a half nozzle. The white regions correspond to locations where data could not be obtained. The boundary-layer edge is marked with a solid red line, and the directions of sidewall secondary flows are indicated by arrows

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The observed effect of nozzle geometry on the corner boundary layers is particularly relevant in SBLI experiments because, even in wind tunnels with relatively thin boundary layers, the corner regions are known to strongly influence the wider response of the tunnel boundary layers to an incident shock. It has been shown that the separation behaviour of the floor boundary layer is dependent on the size of corner separation for normal shock interactions (Chriss et al 1990; Bruce et al 2011), oblique shock reflections (Xiang and Babinsky 2019; Vyas et al 2019) and compression corner interactions (Williams and Babinsky 2021). The underlying mechanism is related to the displacement effect from the early separation of the low-momentum corner regions generating a series of compression and expansion waves which propagate away from the corners. These waves modify the pressure gradient profile experienced by the incoming boundary layer, and therefore have a substantial effect on the onset and shape of shock-induced boundary-layer separation across the entire span of the tunnel (Bruce et al 2011; Xiang and Babinsky 2019; Williams and Babinsky 2021).

Fig. 3
figure 3

Tunnel arranged in the two configurations used in this study, a setup A and b setup B. The test section width is 114 mm

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The significant impact of corner separation on the wider flow field has led to consideration of corner effects in recent SBLI research. For example, Panduri and Muruganandam (2022) accounted for corner separation when developing separation control methodologies for normal shock interactions. In addition, experimental research on SBLI unsteadiness has focused on the extent to which corner regions influence the dynamics of the main separation for oblique shock reflections (Rabey et al 2019) and compression corner interactions (Deshpande and Poggie 2021; Williams and Babinsky 2022). The three-dimensional separation behaviour caused by confined shock trains in ducts has also been explained using the corner wave mechanism (Hunt and Gamba 2019). Importantly, it is now broadly understood that computations of wind-tunnel SBLI experiments must incorporate sidewalls if they are to accurately replicate measured separation behaviour (Vyas et al 2019; Poggie and Porter 2019).

The considerable influence of corner separation on the wider flow field therefore implies that the nozzle geometry of a wind tunnel, through its effect on the corner boundary layers, could have an impact on any SBLI studies. Nevertheless, the installed nozzle configuration is generally not taken into account when comparing experimental data between different facilities or with computations. For example, although Vyas et al (2019) do capture knowledge of the relevant nozzle geometry in their large-eddy simulations, other computational meshes designed to replicate experiments often do not include the nozzle region (Rabey et al 2019; Poggie and Porter 2019). In order to investigate the effects of different nozzle configurations on SBLI research, the current paper therefore studies the differences between the boundary-layer separation caused by an oblique shock reflection at Mach 2.5 when using two different nozzle geometries in the same facility.

Geometric constraints of the tunnel structure prevent a full nozzle setup from being used for this experiment and so a slightly different approach is adopted. It is noted that the half nozzle configuration contains both “types” of corner region, in particular the corner boundary layers that developed along the contoured surface are equivalent to those on a full nozzle (Sabnis and Babinsky 2019). The SBLI in a full nozzle geometry can therefore be replicated by inverting the half nozzle such that the oblique shock impinges on the tunnel wall corresponding to the contoured surface. The response of the wall boundary layers to the oblique shock reflection are therefore compared between a standard and an inverted half nozzle geometry in order to isolate the effect of nozzle geometry on SBLI experiments. This methodology has the considerable advantage that the geometries within the test section are identical, the only difference between the cases being in the upstream nozzle geometry.

2 Research methodology

2.1 Wind tunnel setup

Experiments are performed in Supersonic Wind Tunnel No. 1 at the Cambridge University Engineering Department. This is a blow-down facility, driven by a high-pressure reservoir. From the settling chamber, the flow is accelerated to high subsonic speeds by a 37:1 contraction, which also incorporates a round-to-rectangular geometry transition. Interchangeable, two-dimensional nozzle blocks then further accelerate the flow to the required Mach number, which can be between 0.7 and 3.5. For the current study, the nominal freestream Mach number is fixed at M\(_{\infty } = 2.5\). The stagnation pressure is set to \(380\pm 1\) kPa and the operating stagnation temperature is measured as \(285\pm 5\) K; this corresponds to a unit Reynolds number of approximately \(40\times 10^6\) m\(^{-1}\).

The rectangular test section of the tunnel has a width of 114 mm and a height of 86 mm. The coordinate system convention is shown in Fig. 3, with the origin at the end of the nozzle on the centre span of the tunnel floor. x represents the streamwise direction; y indicates the floor-normal direction; and z is the spanwise coordinate, such that \(z = \pm 57\) mm correspond to the tunnel sidewalls. The measurements in the current study focus on the turbulent naturally-grown boundary layers on the tunnel floor.

The two nozzle geometries studied are asymmetric, half setups, which consist of one contoured and one flat surface. Setup A, shown in Fig. 3a, is a half nozzle arrangement in its conventional orientation, with the contoured surface on the ceiling. Meanwhile, setup B has an inverted half nozzle setup so that the contoured surface is on the floor (Fig. 3b).

A full-span wedge on the tunnel ceiling produces a flow deflection angle of 8\(^\circ\) and thus generates an oblique shock. In order to prevent tunnel unstart, the wedge is retracted during startup and deployed once supersonic flow has been established. The oblique shock wave then impinges on the turbulent naturally-grown boundary layer on the tunnel floor. This is approximately 6 mm thick and has a Reynolds number based on incompressible displacement thickness of around Re\(_{\delta _i^*}\) = 24,000.

Note that the wedge is placed below the upper surface of the tunnel—this arrangement enables the ceiling boundary layer to disappear into the gap and ensures that the shock is generated in clean flow. Figure 3 indicates that this gap for setup B (8 mm) is larger than for setup A (5 mm). This is made necessary by differences in the upper corner flows between the two setups. The 5 mm gap in setup A is sufficient to capture the ceiling boundary layer across the tunnel span. However, the flat ceiling nozzle for setup B is associated with thicker sidewall and corner boundary layers. Therefore, a wedge placed 5 mm below the ceiling would still be in boundary layers across much of the tunnel span. Instead, by dropping the wedge a further 3 mm, the shock is generated in clean core flow.

The different wedge position has one main effect–the shock in setup B impinges on the floor boundary layer approximately 6 mm (about one local boundary-layer thickness) upstream compared to setup A. The growth in boundary-layer displacement thickness over this streamwise lengthscale is only about 1.5% (Sabnis et al 2019), and so the difference in impingement location is not thought to be significant. Nevertheless, any direct comparison between the two setups does require an alternative coordinate system. We therefore define \({\tilde{x}}\) as the streamwise coordinate relative to the inviscid shock-reflection location. In other words, \({\tilde{x}} = x -141\) mm for setup A and \({\tilde{x}} = x - 135\) mm for setup B.

2.2 Measurement techniques

Several techniques are used in combination to probe the flowfield. A z-type schlieren system with a horizontal knife-edge enables visualisation of spanwise-averaged density gradients associated with flow features, such as boundary layers and pressure waves.

Fig. 4
figure 4

Schlieren images with the empty tunnel for a setup A, and b setup B. The solid lines indicate the location of LDV measurements in Figs. 5 and 6. The prominent diagonal line in both images is a weak Mach wave which originates at a junction between constituent liner blocks

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Fig. 5
figure 5

The streamwise velocity (u) in 15 mm\(\times 15\) mm regions around the bottom tunnel corners for a setup A and b setup B. These are measured using LDV at \(x = 120\) mm. The directions of the sidewall secondary flows are identified by solid arrows. The dashed lines indicate the location of the profile measured in Fig. 6b

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The topology of the separated flowfield is surveyed using the time-averaged skin-friction lines from surface oil-flow visualisation. This technique involves coating the tunnel floor with a mixture made from paraffin, finely-powdered titanium dioxide, oleic acid and lubricating oil. This is an intrusive method, and there is a small error due to oil-flow producing an inaccurate indication of separation location (by about 0.2 boundary-layer thicknesses = 1.2 mm) (Squire 1961).

In order to ensure that the oil-flow patterns captured after tunnel shutdown are the same as the target flow field during wind tunnel operation, the images are compared with video recordings of the runs when the oil is still moving. In addition, these video recordings enable the direction of oil flow in different regions to be unambiguously established and thus separation and reattachment lines can be reliably identified. The structure and size of the separated regions derived from the oil-flow patterns are observed to be consistent across several repeat runs. Therefore, comparisons of separation size between different setups using oil-flow visualisation are considered to be reliable.

Steady-state surface pressure measurements are taken using pressure-sensitive paint (PSP). The surface of interest is sprayed with a special polymer binder seeded with luminescent molecules. When irradiated by UV light, the resultant luminescence intensity ratio (\(I_{\text {ref}} / I(p,T)\)) is dependent on the local pressure, as specified by the Stern–Volmer relation (Gregory et al 2008):

$$\begin{aligned} \frac{I_{\text {ref}}}{I(p,T)} = A(T) + B(T) \frac{p}{p_{\text {ref}}} \end{aligned}$$
(1)

The luminescence is recorded using a Nikon D7000 camera, and the reference condition is taken with the wind tunnel off (\(p_{\text {ref}}=101\) kPa everywhere). The pressure in the separated flowfield varies from \(29-57\) kPa; this range of pressures is sufficiently large to provide reliable measurements (Sajben 1993). In order to determine the values of A(T) and B(T) in the Stern–Volmer relation, in-situ calibration is performed using five 0.3 mm diameter static pressure taps connected to a differential pressure transducer (error: \(\pm 1\%\)) (Colliss et al 2016). This calibration enables absolute pressure values on the target surface to be extracted from the measured light intensity. A comparison between static tap pressures and the calibrated PSP data places an error bound of 3% on these measurements. However, in regions where the thermal properties change (e.g. filler material or attachment screws) or where reflections from the tunnel walls distort the measured luminescence, the calibration is no longer valid and a much greater error is observed.

The streamwise and floor-normal flow velocities, u and v, respectively, are measured by two-component laser Doppler velocimetry (LDV). The flow is seeded with paraffin in the settling chamber; previous measurements of particle lag through a normal shock have placed the seeding droplet diameter in the range \(200-500\) nm (Colliss et al 2016). The measured velocities have an error of 1% and 14% for u and v, respectively; there are contributions from the number density of seeding particles and from the laser optics. Boundary-layer traverses are carried out with resolution \(\varDelta y \approx 0.1\) mm. The ellipsoidal probe volume spans 0.1 mm in the streamwise direction and 2 mm in the spanwise direction.

The measured boundary-layer data are fitted to theoretical profiles. A Sun and Childs (1973) fit, adapted to include a van Driest compressibility correction, is used for the outer layer; this combines a log-law of the wall region with a Coles wake function. The viscous sublayer is modelled using a Musker (1979) fit. These fitted profiles are then used to calculate characteristic boundary-layer integral parameters. This avoids errors caused by poor measurement resolution near the wall and therefore provides a more accurate estimate of integral boundary-layer parameters. The boundary-layer properties are determined in their incompressible forms, as these are less sensitive to variations in Mach number and require fewer assumptions to calculate from raw velocity data. The LDV data obtained in this study typically has around 40 measurement points within the boundary layer, the closest data point to the wall is at around \(y^+=80\), and the uncertainty in wall location is \(\varDelta y_\text {wall}/\delta = 0.005\). This corresponds to an estimated error in integral parameters of around 5% for an equilibrium turbulent boundary layer (Titchener et al 2015).

3 Results and discussion

3.1 Empty wind tunnel tests

Prior to investigating the flow field with an oblique shock reflection, preliminary tests are performed in an empty wind tunnel, with no shock-generating wedge and a straight floor and ceiling. Schlieren images are shown in Fig. 4a and b, for setups A and B, respectively. At first glance, the flows in these two setups appear to be very similar in nature. A prominent oblique wave, originating from a junction between constituent liner blocks, is visible in both setups but previous tests in the same facility have shown that this wave is weak in nature and does not perturb the wall boundary layers (Sabnis et al 2019).

Despite their similar appearance, the flow fields are subtly different for the two nozzle arrangements due to the presence of nozzle-dependent secondary flows in the sidewall boundary layers. These vertical flows, indicated schematically in Fig. 2, result in distinct corner boundary layers for the two setups. This difference is highlighted by the streamwise velocity distributions in the corner regions measured at the streamwise location indicated in Fig. 4. In particular, the data in Fig. 5 show that setup A has much thicker corner boundary layers than setup B, even though the floor boundary-layer thickness is the same for both.

Fig. 6
figure 6

Comparison of boundary-layer profiles, measured using LDV at \(x = 120\) mm: a on the tunnel centre span, and b at \(z = 52\) mm. The solid lines in a represent the profiles fitted to boundary-layer models, which are used to determine integral quantities

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Figure 6a suggests that the floor boundary-layer profiles on the tunnel centre span are similar but not identical between the two setups. The small discrepancy in profile shape is likely due to differences in flow development on a flat surface versus a curved surface, which also impact the pressure gradients experienced in the two cases. Nevertheless, these differences are minor and the boundary-layer profiles do match closely between the two setups. As expected from the comparable floor boundary-layer thicknesses in the schlieren images (Fig. 4), the boundary-layer parameters corresponding to the velocity profiles are also very similar. Indeed, Table 1 shows that the boundary-layer displacement thickness and shape factor are in agreement to within 13% and 5%, respectively.

Corresponding velocity measurements at \(z=52\) mm, within the corner boundary layer, are shown in Fig. 6b. The profile for setup A lies within the sidewall boundary layer for the entire traverse distance whereas the data for setup B enters the core flow. Note that the non-equilibrium nature of the boundary layer here means that it is not possible to calculate integral parameters through profile fitting. Therefore, in order to achieve a meaningful comparison, the integral parameters are instead calculated by integrating the velocity data directly using the trapezoidal rule. This approach is known to cause a significantly increased uncertainty of approximately 15% in boundary-layer quantities (Titchener et al 2015). Nevertheless, Table 1 shows that, within the specified uncertainty limits, the direct integration method produces boundary-layer parameters which are consistent with the equivalent values obtained from modelling, for the centre-span profiles in Fig. 6a. As a result, the direct integration method can be instructive in comparing different boundary-layer profiles.

The integral parameters derived from direct integration are compared between the two setups in Table 1. At \(z = 52\) mm, the displacement and momentum thicknesses for setup A are about 2.2 times larger than the equivalent values for setup B, whereas the corresponding difference at the centre-span is only 20%. Alongside the significantly different velocity profiles between setups A and B in Fig. 6b, it is therefore clear that the differences between the two setups are significantly more pronounced in the corner regions than on the tunnel centre line.

As a consequence, when the setup is extended to introduce an oblique shock reflection, setup A and setup B can be considered to be two configurations with approximately the same core flow and floor boundary layer but distinct corner boundary-layer structures.

Table 1 Incompressible floor boundary-layer parameters, measured at \(x = 120\) mm for experimental profiles presented in Fig. 6a (\(z = 0\) mm) and Fig. 6b (\(z = 52\) mm). Integral parameters are calculated using direct integration and, for the centre-span profiles, using the specified boundary-layer modelling approach, based on Sun & Childs (1973) combined with Musker (1979) models
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3.2 Flow field with an oblique shock reflection

Fig. 7
figure 7

Schlieren image of the oblique shock–boundary-layer interaction for a setup A, and b setup B. The horizontal line denotes the LDV traverse location for Fig. 8

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Figure 7 shows schlieren images of the flow for both setups A and B. In both cases, the wedge is visible at the top of the image. The flow deflection caused by the leading edge and by the bottom corner of the wedge generate an oblique shock and an expansion fan, respectively. The latter wave system, denoted the ‘wedge expansion’, complicates the flow field, so we restrict our attention to the features upstream of the first expansion wave.

The wave pattern in both setups is representative of a typical two-dimensional separated oblique shock–boundary-layer interaction (Babinsky and Harvey 2011). The incident shock causes the incoming floor boundary layer to separate. The flow deflection at the upstream edge of the separation produces a separation shock, the curvature of the separation bubble (where the incident shock reflects off the constant-pressure shear layer) causes a series of expansion waves, and the turning of the flow as it reattaches results in a smeared reattachment shock.

Comparing Fig. 7a and b, the incident shock, separation shock, expansion fan and reattachment shock are all at approximately the same angles and relative position for the two setups. The only obvious difference between the two schlieren images is the absolute location of the interaction. This is due to the 3 mm difference in vertical position of the wedge between the two setups.

Fig. 8
figure 8

Streamwise LDV traverse at \(y = 15\) mm and \(z = 0\) mm, for both setup A and setup B. The figure shows a streamwise velocity (u), and b the local flow angle, measured upwards from horizontal

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The data collected from a streamwise LDV traverse 15 mm from the tunnel floor on the centre span are shown in Fig. 8. For both setups, the streamwise velocity, u, shows a profile which agrees with the waves captured in Fig. 7. The velocity drops as it passes through the incident shock and the flow deflects by \(8.2\pm 1.1^{\circ }\) in setup A, and by \(7.6\pm 1.1^{\circ }\) in setup B. In both cases, these error bounds encompass the flow deflection angle of \(8^{\circ }\) set by the wedge. The following two deceleration regions, which bound an area of re-acceleration, correspond to the separation shock and the reattachment shock, respectively. The reattachment shock appears to be weak in strength and smeared, even outside the boundary layer. Note that the flow has not yet fully returned to horizontal by the time the expansion fan from the wedge trailing edge arrives. However, up to this point, the flow is indeed being turned towards horizontal by weak waves generated due to the displacement effect as the boundary layer recovers.

Whilst the velocity profiles for both setups in Fig. 8 are similar, there are some differences. The incident shock is detected at the same streamwise position relative to the inviscid shock reflection location (\({\tilde{x}} = -30\) mm), although the thinner incoming boundary layer in setup A means that this shock wave appears slightly further upstream than setup B in relative terms. Figure 8 also shows that the separation shock is substantially further upstream for setup A than for setup B. An apparent conclusion of this difference is that setup A has an earlier separation compared to setup B. However, such an analysis requires caution because the exact streamwise position for a given pressure wave at \(y=15\) mm is affected not only by the streamwise position of the wave’s origin but also by the wave’s angle (determined by how much the flow deflects) and the vertical position of the wave’s origin (i.e. the distance of the sonic line from the floor).

The similarities in the incoming boundary-layer velocity profiles (Fig. 6a) indicate that the sonic line for the origin of the separation shock is likely to be at a similar height from the wall for both setups. Therefore, measurement of a more upstream separation shock in setup A is consistent with either a more upstream separation point and/or a steeper wave angle, indicating a more severe flow deflection at separation. Meanwhile, the expansion fan measurements also being more upstream in setup A provides some evidence that the separation bubble in this flow is larger in height, since the incident shock has reflected from the sonic line further from the wall. It is not possible, however, to make any similar inferences about the angle or streamwise origin of the reattachment shock, because the measured position of this wave at \(y=15\) mm is also determined by the height of the sonic line (i.e. the state of the boundary layer) which could be substantially different for the two setups downstream of the interaction.

Fig. 9
figure 9

Oil-flow image of the separated flow field on the tunnel floor with a setup A, and b setup B. The regions of separation are highlighted and the inviscid shock reflection line is marked by a dotted line. Numerical values for the labelled parameters (streamwise and spanwise extents of central and corner separations) are listed in Table 2

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Fig. 10
figure 10

Static pressure distribution on the tunnel floor for a setup A, and b setup B: the vertical blue/red dotted line shows the inviscid shock reflection location, the light blue dashed line shows separation regions extracted from oil flow and the grey dots mark the locations of pressure taps. c Pressure measurements at \(z = -1\) mm, including a comparison with pressure taps (dashed line shows the inviscid pressure distribution)

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A more direct comparison of separation extent between the two setups is provided by the surface oil-flow visualisation on the tunnel floor in Fig. 9. Several separated regions, identified from a close study of the skin-friction line topology, are highlighted in the figure. This topology appears to be largely similar between setup A and setup B. In both cases, there is significant separation on the centre span, which is relatively two-dimensional. Separated by a narrow channel of attached flow are regions of corner separation on each side. The spatial extents of the separated regions are listed in Table 2 for both setups.

For setup A, Fig. 9a shows that the central separation begins 23 mm upstream of the inviscid shock impingement location, extends 26.4 mm in the streamwise direction and covers 68% of the tunnel span. Meanwhile, the regions of corner separation begin about 24 mm further upstream than the central separation and each cover about 17% of the tunnel span. These measures of the separated flow field agree extremely well with those obtained from similar experiments in the same facility by Xiang and Babinsky (2019). On the other hand, the oil-flow visualisation for setup B (Fig. 9b) has a central separation which extends 16.6 mm in the streamwise direction and which covers 69% of the tunnel span. The regions of corner separation begin roughly 19 mm further upstream than the central separation and each cover about 10% of the tunnel span.

A comparison between the two oil-flow images in Fig. 9 highlights some additional differences between the two setups. Setup B has a corner separation which starts about 5 mm further downstream than in setup A. The corner separation is also roughly 40% less wide for setup B than for setup A. Furthermore, the width of the corner separation in Fig. 9a for setup A remains close to its maximum extent up to \({\tilde{x}}=60\) mm, the end of the measurement region. On the other hand, the corner separation in setup B has a footprint on the tunnel floor which becomes much narrower towards the downstream region of Fig. 9b. A final key difference between the two setups is the streamwise length of the central separation, which is about 35% shorter in setup B than in setup A.

Table 2 Separation parameters determined from oil-flow visualisation presented in Fig. 9a and b for setup A and for setup B, respectively. The parameters are defined in Fig. 9
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The steady-state surface pressure distribution, measured using pressure-sensitive paint, is presented in Fig. 10a and b for setup A and for setup B, respectively. In both cases, there is a uniform pressure region upstream of the interaction. Across the SBLI, there is a rapid pressure rise. This is relatively two-dimensional across much of the tunnel span. However, the pressure rise does start noticeably further upstream near the corners and is therefore smeared over a greater streamwise distance, causing a less severe pressure gradient. For both setups, the location and size of the pressure rise correlates well with the separated regions identified from oil-flow visualisation (Fig. 9).

A more direct comparison between the two pressure distributions is obtained by studying the pressure profile on the tunnel centre span (Fig. 10c). The sharp pressure rise corresponding to the separation shock is very similar in both cases. Downstream of the separation shock, instead of a plateau underneath the separation bubble followed by a second pressure rise, the measurements show a smeared, more gradual increase in pressure. The pressure rise occurs over a smaller streamwise distance in setup B than in setup A. This behaviour corresponds to a shorter interaction length, in agreement with the shorter central separation extent determined in Fig. 9 using oil flow. Note that both profiles, especially setup B, exceed the pressure rise expected for inviscid, two-dimensional flow, which suggests that additional compression is provided by the corner waves which can be visualised in Fig. 10a and b.

3.3 Discussion of flow physics

In both setups, the separated flow field exhibits departures from a two-dimensional separation due to corner effects, which are consistent with the mechanisms proposed by Xiang and Babinsky (2019). Figure 9 shows that the corner separation begins ahead of the central separated region. Pressure waves produced by the displacement effect of this corner separation are evident in the surface pressure distribution presented in Fig. 10. These waves modify the pressure gradient imposed by the incident shock. For example, the narrow channels of attached flow between the central and corner separations in Fig. 9 are known to be associated with the smearing of the overall pressure rise when the corner waves are ahead of the interaction Xiang and Babinsky (2019).

Fig. 11
figure 11

Comparison of floor separation topology between setup A (blue) and setup B (red). The topology is determined from the oil-flow visualisation presented in Fig. 9a (setup A) and in Fig. 9b (setup B). The onset of corner separation is marked by arrows, and the solid line indicates the inviscid shock reflection location

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Fig. 12
figure 12

Schematic illustration of the footprint of hypothesised corner waves on the tunnel floor for a setup A, and b setup B. Red lines correspond to compression waves and blue lines denote expansion waves. Separated regions are marked with a solid line and the dashed line is the inviscid shock reflection location

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Fig. 13
figure 13

Comparison of the floor boundary-layer profile for setup A (blue) and setup B (red). Measurements taken on the centre span at: a \({\tilde{x}} = -40\) mm (upstream of the interaction), and b \({\tilde{x}} = 20\) mm (downstream of the interaction). The solid lines represent the profiles fitted to boundary-layer models, which are used to determine integral quantities

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The main differences between the two setups are summarised in Fig. 11, which shows a comparison of separated regions, determined by the oil-flow visualisation in Fig. 9. Setup B exhibits a corner separation which starts further downstream and covers a smaller spanwise extent than setup A. Since setup B is the case with thinner corner boundary layers and thus has greater flow momentum in this region (Figs. 5 and 6), the observations in Fig. 11 are consistent with intuitive expectations that a higher-momentum corner boundary layer would be more resistant to separation and so would exhibit a less severe corner separation.

The differences in the corner flow behaviour affect the rest of the flow field through the waves generated by the displacement effect of the corner separation. In general, these waves take the form of compression waves followed by expansion waves, as shown for the two setups in Fig. 12. The representation of approximate wave trajectories in this schematic is based on the angle of the corner waves visualised using the PSP data in Fig. 10, and the distribution of compression and expansion waves is chosen to be consistent with the shape of the corner separation footprint on the tunnel floor. Note that the expansion waves in both these setups occur too far downstream to have an effect on the central separation apart from, possibly, in a narrow region near the sidewalls for setup B. Therefore, we need only consider the compression waves for now.

Table 3 Incompressible floor boundary-layer parameters, along the tunnel centre line, both upstream and downstream of the interaction for setups A and B. These correspond to the profiles presented in Fig. 13. Integral parameters are calculated using the specified boundary-layer modelling approach, based on Sun & Childs (1973) combined with Musker (1979) models
Full size table

The streamwise extent of the central separation, \(L_x\), is measured in Fig. 9 to be about 35% shorter for setup B than for setup A. This effect can be explained by the differences in corner separation. Figure 12a shows that in setup A, the more upstream corner separation point causes the compression waves to begin earlier. As a result the footprint of these waves covers a large proportion of the central separated region. These compression waves impose an additional pressure rise on the central separation, particularly around the expected reattachment location. On the other hand, Fig. 12b shows that setup B has a more downstream corner separation. The trajectories of these waves intersect the very edge of the central separation in setup B, and so they have only a minor impact on this separation region. There is therefore a smaller additional pressure rise on the tunnel centre span.

Whilst the compression waves in Fig. 12 appear to intersect just downstream of the reattachment point, they are still able to influence the centre-span pressure profile for two reasons. Firstly, as the flow moves downstream, the corner waves will enter a regime where the local Mach number is reduced due to the impinging shock. Thus, the waves will be refracted to shallower angles and will intersect further upstream than indicated in Fig. 12. Secondly, any pressure rise is thought to be transmitted quite effectively in the spanwise direction within the separation bubble, as evidenced by the spanwise-uniform pressure within the central separation in Fig. 10a and b. Therefore, even on the tunnel centre span the overall pressure rise is increased by the corner compression waves.

Note that the corner waves are difficult to detect directly in centre-span pressure data (Fig. 10c) because: they are much weaker than the main oblique shock reflection; there is significant smearing in both streamwise and spanwise directions; and the accuracy of pressure measurements is limited by the 3% measurement uncertainty for PSP. Nevertheless, the presence of the waves suggested in the schematic images (Fig. 12) is consistent with the detailed centre-span pressure profiles in Fig. 10c. For example, setup B exhibits a short pressure plateau around \({\tilde{x}}/\delta = -0.5\), which coincides with the region between reattachment and the first compression waves in Fig. 12b. There is a subsequent pressure rise in excess of the inviscid value caused by the corner compression waves. In contrast, setup A has a continuous, more gradual pressure rise consistent with corner compression waves arriving at the centre span close to the reattachment location and being more diffuse (Fig. 12a). Indeed, the final corner compression waves for setup A have likely not arrived at the centre span by the end of the measurement region, hence the apparent difference in final pressure values between the two setups in Fig. 10c.

Whatever the subtle differences, the principal effect of the corner waves is that setup A experiences a significant additional pressure rise around reattachment (Fig. 12a), and so the central separation is extended in the streamwise direction. In contrast, the smaller additional pressure rise on the tunnel centre span in setup B (Fig. 12b) results in a shorter streamwise extent of the central separation.

The spanwise extent of the central separation is, however, similar for the two setups with a difference in width, \(L_z\), of only 1.5%. The insensitivity of this parameter to nozzle setup is consistent with two separate effects, both of which are related to the corner wave mechanism presented in Fig. 12. These effects relate to the region where the corner waves are ahead of the SBLI. In this region, the overall pressure rise is smeared. The pressure-smearing in setup A takes place over a larger streamwise distance and so is highly effective in helping the incoming boundary layer to stay attached. On the other hand, the compression waves in setup B are more concentrated and closer to the inviscid shock impingement location, which results in less efficient pressure-smearing. However, the convex nature of the corner separation in setup B causes the expansion waves from the corners to propagate through this region much sooner, and therefore the adverse pressure gradient experienced by the flow is reduced. The two competing mechanisms appear to have approximately the same impact, and so the spanwise extent of the central separation is relatively insensitive to the size of corner separation.

The consequence of any differences in the central separation regions between the two setups is assessed by comparing the centre-span floor boundary-layer profile at two separate locations. Figure 13a presents velocity profiles upstream of the inviscid shock reflection location, at \({\tilde{x}} = -40\) mm, equivalent to Fig. 6a but for the shock-generating wedge installed and for exactly the same \({\tilde{x}}\)-locations in both setups. Figure 13a shows that the two setups exhibit very similar profiles. However, there are large differences between the two setups in Fig. 13b, when equivalent measurements are taken at \({\tilde{x}} = 20\) mm, i.e. about three boundary-layer thicknesses downstream of the inviscid shock reflection location.

The fitted boundary-layer profiles in Fig. 13 show good agreement with the velocity measurements. Therefore, the integral parameters calculated with these fitted profiles (Table 3) are expected to be reliable and so the equivalent data using direct integration is not included for these profiles. Consistent with the comparison in Table 1 for measurements in an empty wind tunnel, the incoming boundary-layer in setup A is only 13% thicker, and has a shape factor just 5% larger, than setup B. However, downstream of the SBLI, Table 3 indicates that the equivalent differences in boundary-layer displacement thickness and shape factor are more substantial at 41% and 20%, respectively. These parameters indicate that the downstream boundary layer in setup A is significantly less healthy. It is worth noting, however, that the recovery of a boundary layer downstream of an SBLI is fastest immediately after the interaction (Babinsky and Harvey 2011). Whilst the measurements are taken at the same position relative to the inviscid shock reflection location, this measurement position is further downstream of the reattachment line in setup B than in setup A. Therefore, the boundary layer in setup B has been recovering over a greater distance, which exacerbates any differences between the two setups.

Nevertheless, the differences between the boundary-layer profiles in Fig. 13b show that central separation of setup A, with thicker corner boundary layers, has a more severe impact on the incoming floor boundary layer. Therefore, whilst the differences between the two setups in the footprint of the interaction on the tunnel floor may appear to be small, the effect on the flow is in fact quite noticeable when considering the state of the boundary layer itself.

4 Conclusions

The paper studies the influence of nozzle geometry on the response of the tunnel floor boundary layers at Mach 2.5 to an incident oblique shock reflection produced by an 8 deg full-span wedge. In order to isolate any nozzle geometry effects, the experiment was conducted in a single facility using two different nozzle configurations, one where the floor boundary layer develops along a contoured surface and the other where it grows along a flat wall.

The data collected indicate that the setup with the floor boundary layer developing along a contoured surface results in thinner corner boundary layers which, when they encounter the shock wave, exhibit a corner separation which starts further downstream and covers a smaller spanwise extent. This behaviour can be directly attributed to a higher-momentum corner flow being more resistant to the adverse pressure gradient of the oblique shock.

The differences in corner separation between the two setups also influence the central separated region of the floor boundary layer. The more benign corner separation in the thin corner boundary-layer setup generates fewer compression waves from the displacement effect of the corner separation. The floor boundary layer at the centre span therefore experiences a smaller additional adverse pressure gradient on top of that imposed by the oblique shock. As a result, the streamwise extent of the central separation is reduced by 35%. This consequently results in the centre-span floor boundary layer downstream of the interaction exhibiting a fuller profile and having a displacement thickness 41% smaller.

These results indicate that, whilst full and half nozzle setups have generally been considered to be equivalent in the past, this is not quite correct. This finding highlights the importance of considering the wider flow field when interpreting wind tunnel experiments on SBLIs—it has been known for some time that sidewall effects are significant but the current study suggests that the geometry of the wind tunnel nozzle can also affect separated flow field. It is therefore important to account for nozzle geometry when interpreting data from any experiments which are focused on the response of the tunnel boundary layers to a shock wave. This is especially true when comparing data across facilities, where different nozzle geometries can lead to noticeable differences in flow field. Furthermore, any computations which are quantitatively compared with experimental data require a knowledge of the installed nozzle configuration in order to accurately capture the physical flow.

It is also worth pointing out that the consequences of this research are not entirely restricted to wind tunnels, however. Inlets of supersonic aircraft feature oblique shock waves, which generate vertical pressure gradients that could produce vertical secondary flows in the sidewall boundary layers. These are expected to affect the corner boundary layers and, in turn, the wider flow field. Alongside the different pressure profiles experienced by the floor and ceiling boundary layers, this mechanism could also contribute to additional distortion of the flow entering the engine.