3.1 Baseline flow

Figure 5(a) shows a schlieren image of the unmodified baseline interaction, which is seen to be globally stable in the high-speed video recording. The standard pattern of a separation and reattachment shock bounding an expansion fan suggests the presence of a separated region. The oblique interaction domain spreads from $x=110~\text{mm}$ to $x=140~\text{mm}$ with the inviscid shock reflection located at $x=135~\text{mm}$ . At the top of the image, a strong expansion fan emanates from the downstream wedge corner, impinging on the tunnel floor at approximately $x=200~\text{mm}$ , roughly $10\unicode[STIX]{x1D6FF}$ behind the inviscid shock reflection location. The streamwise velocity variation measured along $y=15~\text{mm}$ on the central symmetry plane (marked by a white dot-dashed line in figure 5 a) is shown in figure 5(b). The flow is initially at the incoming free-stream velocity ( $u_{1}$ ), and then decelerates through the incident shock. The following two deceleration steps bounding a re-acceleration region correspond to the typical separated interaction wave pattern depicted in the schlieren photograph. The LDA measurements indicate that the reattachment shock is weak and quite smeared out, even in the outer inviscid flow. The boundary layer profiles measured at two streamwise positions along the centreline are presented in figure 6. Here the inflow boundary layer obtained at $x=80~\text{mm}$ , shown in figure 6(a,b), is 6.1 mm thick, and the incompressible integral values are: displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }=0.84~\text{mm}$ , momentum thickness $\unicode[STIX]{x1D703}=0.61~\text{mm}$ and shape factor $H_{i}=1.37$ . There is a well-defined log region and the agreement with the analytical profile is excellent throughout. A rapid thickening of the boundary layer can be observed across the interaction. The boundary layer measured at $x=160~\text{mm}$ (figure 6 c,d), around $4\unicode[STIX]{x1D6FF}$ downstream of the inviscid shock reflection, is 7.6 mm thick, with a displacement thickness of 2.14 mm, momentum thickness of 1.17 mm and a shape factor of 1.83.

Figure 7.

Interpretation of flow topology on the tunnel floor for baseline flow by (a) surface oil-flow visualisation; (b) schematic illustration of separation sizes extracted from the oil flow. Limiting streamlines and the footprint of the separation are suggested for clarity; the vertical line indicates the inviscid shock location.

Figure 8.

Surface pressure distribution through baseline interaction on the tunnel floor: (a) PSP map; (b) static wall pressure distribution along the tunnel floor centreline.

Figure 7(a) shows the oil-flow visualisation on the tunnel floor. The corner separation regions are indicated with a red line, while the black lines mark the centreline separation and reattachment locations. Limiting streamlines are also shown for clarity, with the surface flow topology presented and annotated schematically in figure 7(b). Here the inviscid shock location, from an extrapolation of the incident shock seen in figure 5(a), is marked by a vertical line. The skin friction lines suggest that the centreline separation topology falls into the ‘owl-face of the first kind’ category, with two saddle points (S) and a pair of foci (F) clearly identifiable. The sizeable separation region in the centre of the tunnel floor exhibits a relatively two-dimensional pattern for much of its spanwise extent. However, towards the sidewalls the streamwise separation length reduces and the separated region is bounded by distinct foci on either side. On the centreline the separation starts at $x=112~\text{mm}$ , measuring 25 mm in streamwise direction (termed $L_{sep}$ ). The spanwise separation width $L_{f}$ is defined as the distance between the two foci and covers 62 mm (54 % of working section width). In each corner region, the presence of reversed flow is suggested by the displacement of the incoming streamlines, originating approximately 18 mm ahead of the centreline separation. The corner interaction ‘footprints’ are seen to extend to approximately 22 mm from the sidewall (20 % of full span). Although critical points are not clearly seen in the corners, the recirculating streamlines suggest the presence of a focus and the formation of a corner vortex on both sides. A narrow attached flow channel is observed between the central and corner separations.

Figure 9.

Structure of the wave pattern induced by the corner separation illustrated by: (a) PSP map on the right sidewall; (b) static wall pressure distribution measured by pressure tappings at $z=57~\text{mm},y=20~\text{mm}$ .

The static pressure on the tunnel floor is presented in figure 8, as a PSP map (figure 8 a) and the distribution along the centreline (figure 8 b). The simplified separation topology is superimposed on the PSP map (figure 8 a) to allow a comparison with the oil-flow visualisation. It can be seen that there is a uniform region of low pressure upstream of the oblique interaction. This is followed by a rapid pressure rise starting near the separation onset. This separation shock footprint presents little spanwise variation, except for the additional pressure smearing as the sidewalls are approached. Comparing the pressure distribution with the separation boundaries, it can be seen that the more upstream pressure rise near the sidewalls corresponds well to the onset of corner separation. This smeared-out corner pressure distribution is therefore likely to be linked to the corner separation.

The PSP map shown in figure 9(a) presents the pressure distribution on the right sidewall. Here, no quantitative data can be extracted due to a combination of more prominent image noise (poor UV light access) and thermal effects. As identified from the oil-flow image, the corner separation on the sidewall extends approximately 18 mm away from the tunnel floor, with the separation onset located at $x=87~\text{mm}$ . The wall pressure measured with taps located in 3 mm intervals along the white dashed line at $y=20~\text{mm}$ is shown in figure 9(b). It is observed that the pressure near the onset of corner separation first increases and then reduces again. This suggests that the wave pattern, caused by the corner separation, consists of a compression fan followed by expansion waves of equal strength. The presence and strength of the expansion waves have been overlooked in previous studies (Burton & Babinsky Reference Burton and Babinsky2012; Eagle & Driscoll Reference Eagle and Driscoll2014).

Figure 10.

Schematic illustration of corner effects on baseline flow: (a) diagram of corner wave structure; (b) diagram of SBLI structure on the central symmetry plane.

Figure 10(a) schematically illustrates the waves induced by corner separation. It is proposed that the separation acts like an aerodynamic ‘bump’, which first deflects the supersonic flow away from the corner, generating compression waves, and then returns the flow towards the streamwise direction, which is the cause of the observed expansion waves. An adverse and then favourable pressure gradient is therefore produced, which can alter the pressure distribution elsewhere in the flow field as the corner waves propagate downstream. For further analysis of the oblique SBLI flow field, it is important to trace the pattern of these waves generated by corner separation, as this helps to determine which and how other flow regions are influenced by corner effects. Of particular interest is the location of the most upstream leading compression wave along the floor. The PSP image on the tunnel floor (figure 8 a) shows this wave very clearly in the region ahead of the SBLI. Once the wave enters the interaction zone, its footprint is, however, no longer easily identifiable. It is expected that the corner waves will change direction as they penetrate into non-uniform regions and interact with other waves. However, these changes of direction are relatively small. Therefore, a rough approach to determine the upstream boundary of the corner influence region is to define a ‘corner shock footprint’ which combines the leading compression into a single line. The location and direction of this line are determined from the PSP map on the floor and, for simplicity, the footprint is assumed to follow a straight line across the interaction domain (until the central span is reached where it will cross with the equivalent wave from the opposite corner). When this footprint is applied to the oil-flow image, as shown in figure 12, it is observed that the lines generally pass through the point where inflow skin friction lines near the sides of the tunnel are initially deflected towards the centre by the displacement effect of corner separation.

The modification of the pressure gradient by the corner waves appears to be the primary mechanism behind the link between the corner separation size and the other areas of separation. As shown in figure 9(b), the pressure downstream of the expansion fan is approximately the same as that of the inflow, indicating that the pressure rise induced by the corner shock is offset by the following expansion waves. For the region close to the corner separation, a milder APG is hence produced due to a combination of an earlier onset of pressure rise caused by corner shock and an unchanged total shock-induced pressure rise as in the two-dimensional oblique SBLI because of the cancellation effect of the subsequent expansion. This reduced APG, as evidenced by the tunnel floor pressure distribution (figure 8 a), is the most likely cause of the attached channel observed between the corner and central separations.

Figure 10(b) shows schematically the influence of the corner wave pattern on the flow in the central symmetry plane. Here the corner shocks and the following expansion waves arrive well behind the oblique interaction, with the streamwise intersection point of the corner shocks estimated to be at $x=160~\text{mm}$ . The SBLI is therefore not influenced by corner effects, which explains the localised quasi-two-dimensional separation topology seen in figure 7.

The results suggest that for an oblique SBLI the influence of corner effects depends on the relative position of corner waves with respect to the main interaction. For the actual region of corner flow, where the floor and sidewall boundary layers meet, separation is generally observed to occur earlier than elsewhere. In the central region, the corner waves are too far downstream to influence the interaction. In the region between the corner separations and the centre, the oncoming flow along the floor first meets the corner waves before encountering the main interaction. This has the effect to smear out the pressure rise through the SBLI, leading to a reduction or even absence of separation.

To test the above arguments, experiments are performed, where the size of corner separations is deliberately altered, thus changing the positions of corner waves relative to the impinging shock.

3.2 Interaction with reduced corner separations

Figure 11.

Schlieren photograph with micro-VGs ( $x_{vg}=50~\text{mm}$ ) and corner suction ( $x_{slot}=100~\text{mm}$ ) employed in tandem to reduce corner effects.

Figure 12.

Surface flow visualisation on the tunnel floor for the experiment with combined control applied: (a) oil-flow image; (b) PSP map.

Figure 11 shows a schlieren image for a flow where a combination of micro-VGs ( $x_{vg}=50~\text{mm}$ ) and corner suction ( $x_{slot}=100~\text{mm}$ ) has been used to reduce the size of corner separation. Here, no prominent difference from the baseline flow can be observed except for some evidence of an expansion fan originating from the suction slots and the oblique shock waves created by the leading and trailing edges of the VGs. The main SBLI appears unaffected by the changes to the corner flow.

Figure 12(a) provides the oil-flow visualisation. It can be seen that the corner separations are considerably reduced, spanning only 8 mm, which is a 65 % decrease from the baseline. The three-dimensionality of the overall flow field is also reduced. The spanwise extent of the central separation $L_{f}$ is now 72 mm (16 % greater than that of baseline) while $L_{sep}$ remains unchanged at 25 mm. This is also reflected in the surface pressure distribution presented in figure 12(b), which exhibits a wider region with little spanwise variation at the start of the interaction and reduced pressure smearing near the sidewalls. The corner shock footprint traced by the PSP map suggests that the waves cross at $x=181~\text{mm}$ , 21 mm downstream from the uncontrolled case. It is also noteworthy that on the oil-flow image this footprint passes through the two focal points bounding the centreline separation, and that the region in between the corner waves is characterised by relatively straight separation and reattachment lines. The central separation topology can still be classified as an ‘owl-face of the first kind’.

A configuration employing micro-VGs at $x_{vg}=50~\text{mm}$ alone is also tested, resulting in a flow field somewhere between the previous two cases, and the relevant geometric measurements are also included in table 1. Tables 2 and 3 summarise the boundary layer profile data upstream and downstream of the interaction for all cases discussed so far. It can be seen that there is very little difference, confirming that corner effects did not change the flow on the centreline in these locations. The main effect of reducing the size of corner separation is therefore an increase in the uniform main interaction region while the central separation length is unaffected. Such quasi-two-dimensional behaviour has also been observed in the large-eddy simulations of Wang et al. (Reference Wang, Sandham, Hu and Liu2015) for relatively large wind-tunnel widths, where the corner shocks determined by the pressure gradient contours are shown to reach the symmetry plane well behind the interaction zones.

Table 1.

Summary of separation-related parameters for reduced corner effects.

Table 2.

Boundary layer parameters at $x=80~\text{mm}$ along the tunnel floor centreline for baseline and the experiments with reduced corner separations.

Table 3.

Boundary layer parameters at $x=160~\text{mm}$ along the tunnel floor centreline for baseline and the experiments with reduced corner separations.

3.3 Interaction with increased corner separations

Figure 13.

Schlieren photographs with corner obstacle introduced to increase corner effects: (a) streamwise position of the block front edge at $x_{block}=120~\text{mm}$ ; (b) streamwise position of the block front edge at $x_{block}=75~\text{mm}$ .

Figure 14.

Surface oil-flow visualisation on the tunnel floor for the oblique SBLIs with increased corner effects: (a)  $x_{block}=130~\text{mm}$ ; (b)  $x_{block}=125~\text{mm}$ ; (c)  $x_{block}=120~\text{mm}$ ; (d)  $x_{block}=115~\text{mm}$ ; (e)  $x_{block}=105~\text{mm}$ ; (f)  $x_{block}=95~\text{mm}$ ; (g)  $x_{block}=90~\text{mm}$ ; (h)  $x_{block}=85~\text{mm}$ ; (i)  $x_{block}=85~\text{mm}$ with elongated blocks at a length of 40 mm; and (j)  $x_{block}=75~\text{mm}$ . The approximate footprint of the corner shock is indicated by red lines.

Figure 13 shows schlieren images for two experiments where corner blocks are applied to increase the size of corner separations. The horizontal white dot-dashed lines represent the position at which streamwise LDA traverses are conducted. Figure 13(a) presents the flow with the corner obstacles placed across the interaction domain at $x_{block}=120~\text{mm}$ . A much larger interaction length with an onset $20~\text{mm}$ upstream of the unmodified SBLI is observed, and the boundary layer downstream of the interaction appears to be thicker than in the baseline flow. Both observations suggest that the oblique shock-induced separation has increased in size. In contrast, when the blocks are located even further upstream, at $x_{block}=75~\text{mm}$ (figure 13 b), the separation shock is more pronounced while the size of the interaction zone exhibits no prominent difference compared to the undisturbed SBLI.

Table 4.

Summary of separation-related parameters for increased corner effects.

A series of oil-flow images with various corner-block positions are shown in figure 14. Compared to the baseline surface streamline pattern (figure 7 a), it can be seen that for all corner-block locations the corner separations are significantly increased in size by up to 50 %. As the blocks move upstream, the onset of corner separation shifts forwards as well and the corner shock footprints (and the approximate crossing points) are found to move accordingly (as indicated by the red lines). This is accompanied by significant changes in the size and shape of centreline separation, as shown by the geometric parameters in table 4.

Figure 14(a–g) shows that when the corner waves cross inside the separated region, the centreline separation length $L_{sep}$ experiences a sharp increase. More specifically, the value of $L_{sep}$ reaches a maximum when the intersection point approaches the reattachment position of the unmodified SBLI at $x=137~\text{mm}$ , as shown in figure 14(c). Here the separated flow reattaches further downstream than in any of the other tests. This is followed by a progressive decrease in $L_{sep}$ as the corner shock intersection point moves upstream. Owing to the more prominent displacement effect caused by enlarged corner separations, corner shocks are located increasingly closer to the mid-span symmetry line and hence affect a larger proportion of the flow field. This is reflected by a change in the spanwise extent of the central separation, which clearly shows a trend of decreasing as the corner shocks move upstream. The central separation topology is also observed to transit from ‘owl-face of the first kind’ to the more complicated ‘second kind’ (as seen in figure 15 a,b), with the approximate footprint of the corner shocks once again crossing through the focal points identified at the edges of centreline separation.

As the crossing point of corner shocks shifts ahead of the separated zone (figure 14 h–j), the trend of reducing separation length continues, even beyond the baseline. For the most upstream location of the corner blocks, the centreline separation length reduces to 9 mm, which is a 64 % decrease from the baseline value. Closer inspection reveals that the reduction in $L_{sep}$ is due to a delayed separation onset, while the reattachment location remains approximately unchanged from the undisturbed interaction. The topological structure of the central separation is seen to exhibit a more complicated pattern. As shown in figure 15(c), an additional pair of saddle points (S $_{1}$ , S $_{2}$ ) and foci (F $_{1}$ , F $_{2}$ ), respectively, are observed in the upstream portion, while the rest of the flow field closely resembles an ‘owl-face of the second kind’. The more extensive corner separations deflect the flow near the sidewalls toward the mid-span more strongly. This, along with a reduced centreline separation distance, gives rise to an additional pair of vortices, which act to help the flow accumulated in the central region lift off and escape from the tunnel floor. In this case, the spanwise width $L_{f}$ is defined as the distance between F $_{3}$ and F $_{4}$ , the two focal points inside the shock-induced separated region.

Figure 15.

Surface flow topology of the centreline separation underneath the oblique SBLIs with increased corner effects: (a) crossing point of corner shocks downstream from the separation onset of baseline flow; (b) crossing point at the separation onset of baseline flow; (c) crossing point upstream of the separation onset of baseline flow.

In figure 16 the static wall pressure distribution on the tunnel floor is shown for the two corner-block experiments with reduced $L_{sep}$ . Compared to the baseline flow (figure 8 a), the corner shock footprint covers a larger area ahead of the main interaction. As a result of the more upstream position of corner shocks, there is no longer a uniform flow region directly ahead of the interaction domain.

Figure 16.

Static wall pressure distribution $P/P_{0}$ for the tunnel floor for the oblique SBLIs with increased corner effects: (a) $x_{block}=85~\text{mm}$ ; (b) $x_{block}=75~\text{mm}$ . The approximate footprint of the corner shock is indicated by red lines.

Figure 17.

Static wall pressure distribution on the tunnel floor centreline.

Figure 17 compares selected wall pressure distributions along the tunnel floor centreline. The $x_{block}=85~\text{mm}$ case shows an increased upstream influence, with the pressure rise occurring approximately 12 mm earlier than in the baseline flow. This is followed by a less steep APG across the interaction and an unchanged total shock-induced pressure rise. For the $x_{block}=75~\text{mm}$ experiment, the upstream influence is further increased with the onset of pressure rise located 20 mm upstream compared to the baseline, exhibiting a larger interaction length with a noticeably milder APG. If the centre-span flow behaved quasi-two-dimensionally, the larger interaction length would be associated with an increase in centreline separation. This, however, contradicts the considerably reduced $L_{sep}$ shown in the oil-flow images (figure 14 h,j), suggesting that the increase in pressure smearing is not caused by the local separation ‘bubble’ but rather is a consequence of the changes to the corner wave positions and their influence on the centreline flow.

The inflow boundary layer parameters along the tunnel floor centreline are listed in table 5. The data for the configuration $x_{block}=75~\text{mm}$ are not available, as this block position obstructs the laser light path. In all cases, the inflow boundary layer parameters are almost unchanged from the baseline. This confirms that the corner obstacles, located well away from the tunnel mid-span, do not affect the centre-span flow ahead of the interaction.

Figure 18.

Variation in streamwise velocity at $y=15~\text{mm}$ , $z=0~\text{mm}$ .

Table 5.

Boundary layer parameters at $x=80~\text{mm}$ along the tunnel floor centreline for the experiments with increased corner separations.

Table 6.

Boundary layer parameters at $x=160~\text{mm}$ along the tunnel floor centreline for the experiments with increased corner separations.

Table 6 compares the boundary layer parameters measured 25 mm downstream of the inviscid shock reflection point. A much thicker boundary layer is seen in the $x_{block}=120~\text{mm}$ experiment (also observed in the schlieren visualisation in figure 13 a). This is in accordance with the delayed reattachment process and the largest separation length $L_{sep}$ among all the tests. Lower incompressible shape factors are observed for the experiments with the corner blocks applied at $x_{block}=85~\text{mm}$ and $x_{block}=75~\text{mm}$ , which indicate fuller boundary layer profiles recovering quicker from the effect of the oblique interaction. This is most likely a consequence of the reduced centreline separation.

Figure 18 presents the streamwise velocity variation through the impinging SBLI above the boundary layer edge at $y=15~\text{mm}$ , $z=0~\text{mm}$ . The measurements show that the deceleration through the separation shock occurs approximately 10 mm further upstream in the three corner-block tests than in the baseline experiment, in line with an increased interaction length and an earlier onset of pressure rise. For the $x_{block}=120~\text{mm}$ test, the flow deceleration through the reattachment shock is more pronounced, indicating that the strength of the pressure rise is intensified by the corner shock waves which reach the symmetry line at this location (see also the approximate footprint of the corner shocks marked in figure 14 c). The expansion waves generated by the corner separation can be seen in the flow acceleration downstream of the reattachment shock. In the $x_{block}=85~\text{mm}$ experiment where corner shocks arrive ahead of the centreline separation, the flow is decelerated to a lower velocity than the baseline case through the combined effects of the corner and separation shocks. This is followed by a more pronounced re-acceleration caused by the subsequent corner expansion waves, which also meet inside the interaction zone. For the case of $x_{block}=75~\text{mm}$ , an additional velocity reduction occurs ahead of the incident shock. This is due to the corner shocks being shifted further upstream, now located approximately 30 mm (5 times inflow $\unicode[STIX]{x1D6FF}$ ) ahead of the centreline separation onset, as shown in figure 14(j). The following deceleration of the incident shock appears to be almost the same as for the baseline case. A similar intensified re-acceleration is again observed.

Figure 19.

Schematic illustration of corner effects on the oblique SBLIs with increased corner effects along the symmetry line: (a) corner shocks cross inside the centreline separation; (b) corner shocks cross ahead of the centreline separation.

The effect of the corner waves on the SBLI in the central plane is shown schematically in figure 19. Figure 19(a) describes interactions where the corner shock crosses around the reattachment region (e.g. $x_{block}=120~\text{mm}$ ). In this scenario, the reattachment shock is strengthened by the corner shocks. Thus, the pressure rise at reattachment is intensified, which requires an increased separation bubble size to accommodate the greater overall pressure jump across the interaction. The subsequent corner expansion waves, which might relieve the additional pressure rise, however, arrive too far downstream to cancel the effects of the corner shocks before reattachment (as shown in figure 18). This leads to the increased separation length observed in the oil-flow visualisation (figure 14 c) and a downstream shift of the reattachment line compared to the baseline flow. The enlarged centreline separation then gives rise to an increased interaction domain, which explains the earlier onset of deceleration through the separation shock compared to the baseline case (figure 18).

An equivalent schematic illustration for the interaction where the corner shocks are even further upstream and cross ahead of the separated region is provided in figure 19(b). Here the corner shocks reach the centreline before the separation point. The additional pressure rise caused by the corner shock might be expected to increase the central separation, whereas the evidence (figure 14 h,j) shows the opposite behaviour. This can be explained by two factors. First, the corner ‘shock’ is of relatively limited strength and consists of a series of compression waves rather than a single strong shock. This means that the overall pressure jump is smeared over a larger area. Second, the corner shock is followed by expansion waves which now reach the symmetry plane within the interaction zone (as seen in figure 18 by the strong re-acceleration of the flow above the interaction region for $x_{block}=85~\text{mm}$ and $x_{block}=75~\text{mm}$ ). This means that they can now relieve the overall pressure rise, which reduces the separation length.

As the corner expansion waves play a significant role in the physical mechanism proposed for the interactions with reduced $L_{sep}$ , an additional experiment has been conducted with obstacles of a different length to test whether the prominent corner expansion waves seen here are induced by the corner separation or the rear edge of the block. The oil-flow visualisation shows that $L_{sep}$ of the flow disturbed by the elongated obstacles (figure 14 i) is almost the same as that altered by the standard blocks (figure 14 h), confirming the corner expansion fan emanating from the tunnel sidewall is a consequence of the displacement effect of the corner separation.