Shock-induced leading-edge separation in hypersonic flows

Abstract

We investigate the leading-edge separation, also called negligible boundary-layer thickness separation, induced by an impinging shock on a sharp flat plate. The canonical impinging shock wave/boundary-layer interaction configuration consisting of a wedge and a plate wall, placed in hypersonic free stream, is used for the investigations. We first construct a theoretical model for the leading-edge separated flow field (LESF) which predicts separation bubble geometry and surface pressure distribution as a function of three parameters: free-stream Mach number, wedge angle and the reattached flow turning angle. Markedly different predictions of the separated flow field are obtained for an oblique and a near normal reattachment on the plate surface. Experiments in a shock tunnel at a nominal Mach $6$ flow, with an impinging shock generated by a wedge of angle $26.6^{\circ }$, are used to validate the model. Schlieren flow visualization using a high-speed camera and surface pressure measurements using fast response piezoelectric sensors are the diagnostics employed. For a range of shock impingement locations, the LESFs are observed to be geometrically similar and in good agreement with the LESF model. When the impingement location gets closer to the leading edge, it is observed from experiments that the flow field is no longer geometrically similar, and the separation angle increases as the impingement gets closer to the leading edge beyond the range of similarity. The work thereby offers an elaborate description of the leading-edge separated flow when shock impingement occurs near the plate leading edge.

1. Introduction

The interaction between a shock wave and a boundary layer, termed shock wave/boundary-layer interaction (SWBLI), is commonly encountered in high-speed flows. Examples include: a stationary shock wave interacting with the nozzle boundary layer in the over-expanded operation of a rocket nozzle; a ramp shock impinging on a cowl plate boundary layer in the operation of a scramjet engine; a normal shock wave interacting with the boundary layer of an airfoil operating at transonic speeds. In situations involving SWBLIs, the solid body feels a pressure increase in the direction of the flow caused by means of its inherent geometrical configuration (e.g. cavity, compression corner) or by means of an external agent (e.g. shock impingement). In any case, the pressure increase in the (outer) inviscid flow translates to a positive pressure gradient in the (inner) viscous boundary layer of the solid body, causing a flow retardation which may lead to the phenomenon of flow separation. Shock-induced separation is generally a complex, unsteady phenomenon; the source of unsteadiness is not just the incoming turbulence in the boundary layer (Ganapathisubramani, Clemens & Dolling 2007) but also the inherent separation bubble dynamics (Wu & Martin 2008). Due to the inherent unsteadiness, laminar bubbles are also observed to exhibit unsteadiness (Loth & Matthys 1995). There are very few theoretical studies concerning shock-induced separation for relatively simple steady (time-averaged), two-dimensional (2-D) flow fields (Gadd 1957; Chapman, Kuehn & Larson 1958; Stewartson & Williams 1969); scaling laws for the length of 2-D shock-induced laminar separation were obtained by integrating the triple deck formulation of the boundary layer (Burggraf 1975; Davis & Sturtevant 2000). If the shock induces boundary-layer separation, it can lead to some interesting and adverse consequences such as separation bubble unsteadiness (Clemens & Narayanaswamy 2014), thermal loads (Borovoy et al. 2013), performance loses and unstarts (Wagner et al. 2009), often requiring flow control to improve the performance of the systems where they occur. This has motivated decades of extensive studies on SWBLI, the parameters affecting them and their control (Délery & Marvin 1986; Dolling 2001; Babinsky & Harvey 2011; Clemens & Narayanaswamy 2014; Gaitonde 2015).

Among the different SWBLI situations, a particularly intriguing phenomenon of flow separation is reported to occur at the leading edge of the flat plate in a supersonic/hypersonic free stream. The earliest analysis of leading-edge/zero-boundary-layer-thickness separation in a supersonic free stream was presented by Chapman (1956) and Chapman et al. (1958). The theoretical predictions of the gas pressure inside the separation bubble in (1.1) agreed well the experiments, thereby giving confidence in the theoretical calculations of leading-edge separation flows. It must be noted that the predictions concern subsonic and supersonic flows for which the isentropic compression assumption at reattachment is valid. The configurations (near leading-edge compression corner/shock reflection; sharp cornered leading edge) in which supersonic leading-edge separation is expected are shown in figure 1.

Figure 1. Schematic diagrams (not to scale) of some leading-edge separation situations in a supersonic free-stream flow: (a) ‘tick’ model; (b) compression corner model; (c) sharp-corner model; and (d) shock reflection model.

The ‘tick’ configuration (cf. figure 1a) has received greater attention among the leading-edge separation configurations. The theoretical calculation of the heat-transfer rate in a separated laminar region was performed by Chapman (1956). The calculations considered a general zero-boundary-layer-thickness case at the position of separation, a laminar mixing layer and a low velocity (‘dead-air’) region of constant pressure and temperature enclosed by a solid surface and a viscous mixing layer. Chapman et al. (1958) performed theoretical analysis in order to determine the dead-air pressure in a separated region for the case of zero-boundary-layer-thickness separation. The results of the laminar mixing layer theory (Chapman 1950) were used, which defined a velocity ratio $\bar {u}_{*}$ that is the ratio between the velocity of the dividing streamline in the laminar mixing layer and the velocity of the free-stream flow. The velocity ratio is nearly a constant $\bar {u}_{*}=0.589$ and is largely independent of Reynolds number or distance from separation. Then, a mass balance argument was used in calculating the dead-air pressure: the mass flux scavenged from the dead-air zone by the mixing layer is balanced by the mass flux reversed back into the dead-air zone by the pressure rise through the reattachment zone.

A closed-form solution for the dead-air pressure (or separation pressure) $p_{d}$ as a function of the static pressure $p'$ and the Mach number $M'$ immediately downstream of the reattachment zone was found to be

(1.1)\begin{equation} \frac{p_{d}}{p'} = \left[\frac{1 + \dfrac{\gamma - 1}{2} M'^{2}}{1 + \dfrac{\gamma - 1}{2} \dfrac{M'^{2}}{\left( 1 - \bar{u}_{*}^{2} \right) }} \right]^{\gamma/ \left( \gamma - 1\right)}, \end{equation}

where $\gamma = 1.40$ is the specific heat ratio of air. The theoretical calculations of the dead-air pressure ratio in (1.1) were only applicable when the flow reattachment occurred through an isentropic process, typically for relatively small values of $p'/p_{d}$. The value of $p_{d}/p'$ from (1.1) is found to be independent of the Reynolds number and the model shape, which agreed well with the experiments. Chapman et al. (1958) also introduced an efficiency factor in (1.1) which can account for non-isentropic compression, although their estimations were made only using the assumption of isentropic compression of the dividing streamline, i.e. using (1.1) as it is. However, the postulated factor is rather an a posteriori estimate, which can only be determined after finding the relation between $p_{d}$ and $p'$. In recent years, advancements in experimental diagnostics and computations have significantly resolved the flow over the ‘tick’ configuration. Strong to moderate wall cooling effects, sharp and blunt leading-edge configurations, eddy structures within the primary recirculation flow and slightly downstream displacement of the separation location from the leading edge can be found in the works of Khraibut et al. (2017) and Prakash et al. (2019). Particularly, the works by Khraibut et al. (2017) and Prakash et al. (2019) concern hypersonic separation for which the isentropic process at reattachment is not valid.

The leading-edge separation in compression corner models (cf. figure 1b), including those with inclined upstream walls, was investigated by Brower (1961). The flow field is analysed by the extension of laminar mixing layer theory. The theoretical investigations were carried out for a free-stream Mach number range 1–10 and compression corner angles $0^{\circ }$–$45^{\circ }$. It was found that not every combination of the free-stream Mach number and the compression angle yielded a solution. Interestingly, for free-stream Mach numbers below $1.27$, no leading-edge separation solution exists for any value of the compression corner angle. The lower and upper limits of the solutions are obtained for a leading-edge Mach wave and for the maximum strength of the reattachment shock. The solutions to leading-edge separation flow fields were found to be independent of the free-stream Reynolds number and had a satisfactory comparison from previous wind-tunnel experimental data.

Supersonic separated flow over a flat nosed configuration (cf. figure 1c) has not been studied extensively in the literature. The experiments and computations of a 2-D supersonic flow past a square-nose flat plate at Mach $1.96$ were performed by Sharp (1959). The boundary layer on the frontal side of the plate is not able to negotiate the sharp right-angle corner and therefore separates, reattaching at some downstream location of the plate. From the experiments, the wall pressure measurements downstream of the square corner and the ordinates of the separation bubble and the bow wave profile were obtained. Computations were performed using the method of characteristics in the supersonic field outside the separation bubble, by treating the shear layer as a boundary line dividing the separation bubble and the outer inviscid flow. The shape of the sonic line was found to be especially sensitive in the analysis. The analysis was also applied to a rounded leading edge which was found to completely eliminate the separation bubble and the subsequent recompression.

To our knowledge, there are no detailed theoretical or experimental studies on the leading-edge flow separation caused by means of an impinging shock (cf. figure 1d). Despite the possibility of impinging shock-induced leading-edge separation in scenarios such as intakes at off-design Mach numbers, it has not received scholarly attention. This has motivated the present study. There is a general consensus from previous studies that the separation length increases with increasing impinging shock strength (Elfstrom 1972), decreases with free-stream Mach number (Hayakawa & Squire 1982) and increases with free-stream Reynolds number (Needham & Stollery 1966), except for the transitional boundary layer.

Délery & Marvin (1986) classified SWBLIs into ‘weak’ and ‘strong’ interactions based on the strength of the shock interacting with the boundary layer, such as the impinging shock. When the impinging shock strength is lower than the incipient pressure ratio, the SWBLI is called a ‘weak’ interaction. In such a scenario, the flow alterations are largely felt within the boundary layer while the external flow is unaffected, with very little deviation from the inviscid shock reflection scenario. The boundary layer separates when the impinging shock strength is higher than the incipient pressure ratios, resulting in a ‘strong’ interaction. A situation occurs when the impinging shock is just strong enough for the onset of flow separation, that is, the case in which the minimum shear stress is zero; the pressure jump across the shock which is just required for the onset of separation is called the incipient separation pressure. For impinging shock strengths of the orders of incipient separation pressures, the separation lengths are found to scale with the boundary-layer thickness (Liepmann, Roshko & Dhawan 1951; Gadd, Holder & Regan 1954; Bogdonoff & Kepler 1955; Chapman et al. 1958; Hakkinen et al. 1959; Katzer 1989). However, for impinging shocks which are considerably stronger than incipient separation strength, the separation lengths could be much larger than the boundary-layer thickness. In such a case it could be possible to reduce the separated shear layer into separation lines (dividing outer flow and dead air), and treat it as an inviscid problem where we only deal with the complex wave interactions. The complex wave interactions were described in Lees & Reeves (1964) and Green (1970), and later adapted by Délery & Marvin (1986) to form an inviscid flow model (cf. figure 2). The model consists of a dead-air region having a constant pressure whose free boundary starts from the separation location. The impinging shock, modified by the interaction of the separation shock, hits the separated shear layer and reflects as an expansion fan. The expansion fan alleviates the shock pressure jump causing the free boundary of the dead-air region to end at the reattachment location. It is important to note that the separation for the case of zero-boundary-layer thickness was not considered in figure 2.

Figure 2. Schematic drawing of an inviscid flow model as described by Lees & Reeves (1964) and Green (1970), and later adopted by Délery & Marvin (1986). The model describes the separated flow caused by a ‘strong’ SWBLI. The locations S and R denote the locations of separation and reattachment, respectively. Note that separation does not occur at zero-boundary-layer thickness and reattachment is always parallel to the wall.

From shock tunnel experiments by Sriram & Jagadeesh (2014) and Sriram et al. (2016), such large separation bubbles near the leading edge, having a separation length comparable to the distance of shock impingement from the leading edge, were observed to follow a Reynolds number-independent scaling law given by

(1.2)\begin{equation} \frac{L}{x'} M_{\infty}^{3} \propto \frac{p'}{p_{\infty}}, \end{equation}

where $L$ is the separation length; $x'$ and $p'$ are the reattachment distance from the leading edge and the reattachment pressure; and $M_{\infty }$ and $p_{\infty }$ are the free-stream Mach number and free-stream static pressure. The ‘large’ separation bubbles were also observed to be taller (based on the observed height of the shock impingement point on the separated shear layer) and had an upstream skewness (larger proportion of the bubble length being upstream of the shock impingement location on the separated shear layer) characteristic of hypersonic separated flows. The motivation behind the inviscid modelling of the leading-edge separation flow (LESF) comes from the Reynolds number-independent scaling as given in (1.2). When we have leading-edge separation, we expect the scaled separation length to always be unity (scaling with the distance of reattachment from the leading edge); this means that the scaling law effectively reduces to the inviscid relation between the pressure ratio and Mach number. This motivates us towards the possibility of an inviscid model for the shock-induced leading-edge separation.

In the present work we use a perfect fluid model to determine the pressure in the separated bubble, but this does not mean isentropic compression at reattachment, as assumed by the Chapman et al. (1958) model. We rather have a shock at reattachment due to which we have an entropy increase. This model is similar to the one suggested by Délery & Marvin (1986) where the reattachment pressure is equal to the inviscid shock reflection pressure.

Leading-edge separation studies can find applications in scramjet engine inlets. The air compression in the engine inlet is achieved by the generation of the ramp shock and its subsequent reflection from the cowl plate. At design conditions, the ramp shock impinges exactly at the cowl leading edge. However, at off-design conditions (higher Mach number) the shock impinges on the boundary layer developing downstream of cowl leading edge. The ramp shock–cowl plate interaction (Mahapatra & Jagadeesh 2009) typically occurs at the cowl plate leading edge, called the shock-on-lip configuration, which can cause a leading-edge separation. The present work can shed some light on the separated flow behaviour and therefore the separation bubble characteristics can be well anticipated.

In this paper, we investigate the phenomenon of shock-induced leading-edge separation in hypersonic flows, as shown in figure 3. First, in § 2, we describe the shock tunnel facility, hypersonic free-stream conditions and the test models used in the experiments. The subsequently presented results are twofold: in § 3, we construct a LESF theoretical model for the leading-edge separation. The separation bubble geometry which is constructed from three parameters: free-stream Mach number, wedge angle and the reattached flow turning angle, is detailed here. The second part of our study is presented in § 4 and focusses on the experiments involving leading-edge separation in a shock tunnel facility. Experimentally, the separation flow fields are analysed using schlieren visualizations and surface pressure measurements. The results from the LESF model are compared with the experiments, thereby validating the theoretical model. The geometric similarity of the separation flow fields is carefully looked into as the shock impingement location is moved close to the leading edge of the plate wall. The insights from the theoretical modelling and the experiments shed some light on the ‘large’ separation bubbles typically encountered in scramjet engine inlets.

Figure 3. A schlieren photograph of a shock-induced leading-edge separation. The shock waves, expansion fans and shear layers are labelled. O, A and C denote the locations of the leading edge, shock–shock interaction and reattachment, respectively.

2. Experimental facility and test models

Experiments are performed in the Hypersonic Shock Tunnel 2 (HST2) facility at the Laboratory for Hypersonic and Shockwave Research (LHSR), IISc, Bengaluru. The HST2 is a conventional shock tunnel consisting of the following components in order: a shock tube of 50 mm internal diameter, with driver and driven sections of length 2 and 5.12 m, respectively separated by a metal diaphragm; a conical divergent nozzle of fixed exit diameter of 300 mm, and replaceable throat section between the nozzle and shock tube (which fixes the Mach number); a test section of 450 mm length and $300\,{\rm mm}\times 300$ mm square cross-section; and a dump tank of 2 $m^{3}$ volume. The end of the shock tube is fitted with two piezoelectric pressure sensors which are used to measure the reservoir flow properties of the shocked air prior to the nozzle entry. The end of the shock tube and the nozzle entry are separated by a paper diaphragm. A Pitot pressure measurement at the test section and the reservoir pressure measurement at the shock tube end are used to characterize the free-stream conditions in HST2. Furthermore, the test section is installed with transparent glass windows on either side such that flow visualizations (e.g. schlieren, shadowgraphy) over the test models can be performed. The detailed working of the HST2 can be found in the works of Sriram (2013) and Srinath (2015). The free-stream conditions for the present experiments are shown in table 1. We also validate the free-stream Mach number and static pressure by measuring the wedge angle, wedge shock angle and the dead air pressure in the experiments, details of which are given in Appendix A.

Table 1. The free-stream conditions in HST2.

We remark that the Pitot rake measurements by Kulkarni (2007) and Mahapatra (2010) in HST2 showed that the diameter of the core flow was larger than the model diameter (typically approximately 100 mm) and that the stagnation pressure was constant in this region. Therefore, the length scales of the models (i.e. wedge and flat plate) and their positions relative to the nozzle exit are such that the free stream is not significantly affected by the conicity of the flow. The free-stream uncertainties are estimated following the methodology of Moffat (1988). They are calculated to be $\pm$1.86 % in free-stream Mach number, $\pm$6.54 % in free-stream static pressure, $\pm$6.07 % in static temperature, $\pm$9.71 % in free-stream unit Reynolds number and $\pm$6.54 % in total enthalpy. The detailed uncertainty calculations can be found in the work of Srinath (2015). Here, it must be stated that, at low enthalpies of $H_{0} \approx 1$ MJ kg$^{-1}$, high-temperature effects (e.g. dissociation, chemical reactions) in air are not expected to occur. Under the present experimental conditions, the STN code (Shock Tube Nozzle calculations for equilibrium air by Krek & Jacobs 1993) estimates an insignificant change in the specific heat ratio of the free-stream air ($\gamma \approx 1.40$), which suggests air behaves as an ideal gas at the enthalpies at which the present studies were conducted.

The Pitot signals are shown in figure 4. It is observed that the test time is around 600 $\mathrm {\mu }$s (between 3.0 and 3.6 ms in figure 4), and the different signals demonstrate repeatability. Since the test times are extremely short, the time resolution of the separated flow field is very crucial. The evolution of the shock-induced large separated flow field within the short shock tunnel test time was discussed by Sriram et al. (2016). It was reported that within around 300 $\mathrm {\mu }$s after a steady free stream is established, a statistically steady separated flow field evolved. Hence, the steady separated flow measurements can be performed within the 600 $\mathrm {\mu }$s available steady free stream. The steady separated flow features can be verified in the transient pressure signals on the plate wall (cf. figure 18) and the schlieren time evolution of flow field (cf. figure 19) plots.

Figure 4. The pressure signals from a Pitot probe mounted in the test section. The three signals are shown here to illustrate repeatability.

The test models used in the present experiments are: a wedge (impinging shock generator) of angle $26.6^{\circ }$ relative to the free-stream direction and a plate wall of length 200 mm having a sharp leading edge (bluntness radius $r_{b}<0.1$ mm). The wedge and the plate wall each have a span of 120 mm. Figure 5 shows photographic images of the test models used. The effects of model span, for both the wedge and the flat plate, on the separated flow field are given in Appendix D. The wedge and the plate are held in their respective positions as shown in figure 6 with the use of a fixture. A vertical gap of 30 mm is maintained between the wedge and the plate wall for the air flow. Larger vertical gap distances tend to curve the (impinging) wedge shock before its impingement. Smaller vertical gap distances could result in multiple shock reflections with the wall or shear layer on the top, resulting in complex system of shocks and back pressure fluctuations. The horizontal positions of the plate leading edge, relative to the wedge leading edge, are chosen in the range 10–60 mm so as to attain leading-edge separation flows. Since the relative vertical gaps of the models are fixed, the model configurations are solely determined by the horizontal positions. Consequently, the configurations are named corresponding to the inviscid shock impingement locations from the plate leading edge, denoted by $X_{ir}$ (cf. figure 24). For example, if a model configuration produces a (inviscid) shock impingement around 40 mm from the plate leading edge, it will be referred to as the $X_{ir} \approx 40$ mm configuration. The subscript ‘ir’ refers to inviscid (shock) reflection. We also perform experiments with a wedge of angle $21.80^{\circ }$ at nominal Mach numbers of $6$ and $8$ to further validate the LESF model.

Figure 5. Photographs of the test models used in the present study. (a) Side view of the wedge; (b) top view of the plate wall; (c) side view of the plate wall.

Figure 6. Two-dimensional drawing of a typical model arrangement. Wedge (top) and plate wall (bottom) are the test models used, each having a span of 120 mm (not shown in the figure).

Time resolved surface pressure measurements are carried out using piezoelectric pressure sensors (uncertainty of $\pm$1 %) which are flush mounted along the centreline of the plate wall. The pressure sensor centres are placed at $30$, $38$, $46$, $54$, $62$ and $70$ mm from the leading edge of the plate. Some experiments are also performed with Kulite pressure sensors (uncertainty of $\pm$0.5 %) measuring wall pressures at 15 mm from the leading edge of the plate. The sensors, which are flush mounted to the plate wall, connect externally to a data acquisition system (by National Instruments) operating at 1 MHz. The flow field in the test section is visualized using a $z$-type schlieren arrangement (Settles 2001) using a high-speed Phantom $\text {V310}$ camera operating at 20 kHz with a spatial resolution of $518\,{\rm px}\times 288$ px. The pixel resolution in the schlieren images was around 0.18 mm. The schlieren visualizations are configured with a horizontal knife edge which is used to resolve vertical gradients in the flow.

3. Modelling

Figure 7 illustrates the LESF model whose motivation comes from the Reynolds number independence of our previous scaling laws (Sriram & Jagadeesh 2014; Sriram et al. 2016) and from a separated flow model from previous literature (Délery & Marvin 1986). We remark that the LESF model developed in the present study is very similar to the inviscid model of Délery & Marvin (1986) (cf. figure 2), except that (i) in our case we consider a hypersonic Mach number as opposed to a supersonic Mach number in their case, (ii) we consider flow separation at the leading edge and (iii) we consider the possibility of ‘normal’ flow (through a normal shock) at reattachment.

Figure 7. Schematic drawing (not to scale) of the LESF model. The model describes a 2-D steady state inviscid flow field over the plate wall. Regions 2–6 are obtained after passing through a combination of shock (thick lines), expansion (dotted lines) and/or slip lines (dashed lines). The separation bubble $\triangle \text {OBC}$ is characterized by the bubble pressure $p_{d}$, the separation length $L$, the upstream skewness $\alpha$ and the separation height $H$.

3.1. Theoretical framework

In the LESF model, a wedge of angle $\theta _{w} > 0^{\circ }$ is placed in a free-stream flow (cf. figure 7); Mach number $M_{1} = M_{\infty }$, static pressure $p_{1} = p_{\infty }$ and flow direction $\theta _{1} = 0^{\circ }$; where it generates a wedge shock SW1.

The shock SW1 is of strength (static pressure jump) $p_{2}/p_{1}$ and deflects the free-stream flow from $\theta _{1}$ to $\theta _{2} = -\theta _{w}$ towards a plate wall. The shock SW1 causes a SWBLI over the plate wall which results in the flow separation at the leading-edge location O. The leading-edge separation causes generation of a separation shock SW2 and a slip line SL1 at location O.

The shock SW2 is of strength $p_{3}/p_{1}$ and deflects the free-stream flow from $\theta _{1}$ to $\theta _{3}$ away from the plate wall. Here, the separation shock angle is denoted by $\beta _{s}$ and the slip line by $\theta _{s} = \theta _{3}$. A slip line is a dividing streamline which allows tangential velocity jumps while maintaining the local flow direction and static pressure values across its boundary (a concept previously used in the works of Chapman et al. 1958; Délery & Marvin 1986).

The slip line SL1 divides the supersonic flow in region3 from a dead gas flow in the separation bubble. The static pressure in the separation bubble is constant given by ${p_{d} = p_{3}}$.

Downstream of the flow, a Type-I shock–shock interaction (refer to Edney (1968) for shock–shock interactions) is assumed between the shock SW1 and shock SW2, which meet at point A and the formation of two additional shocks SW3 and SW4 and a slip-line SL2 ensues. The slip line SL2 maintains the local flow direction $\theta _{4}$ and the local static pressure value $p_{4}$ while having tangential velocity jumps across its boundary.

The shock SW4 interacts with the separation bubble $\triangle \text {OBC}$ at location B to result in an expansion fan E1, turning SL1 towards the plate. The turned slip line is denoted as SL3. The constant pressure condition in the separation bubble $\triangle \text {OBC}$ is the reason for the shock SW4 to reflect as an expansion fan E1, and therefore turn the slip line SL1 into SL3 in the direction of the wall.

The slip line SL3 divides the supersonic flow in region5 from the dead gas flow in the separation bubble while maintaining the static pressure $p_{5}= p_{d}$ across its boundary. The slip line SL3 meets the wall at location C and disappears. As the supersonic flow in region 5 approaches the plate wall, a reattachment shock SW5 forms at location C, which aims to change the flow direction parallel to the wall. The (apparent) point of intersection between the slip line SL3 and the shock SW5 is the reattachment location C.

Immediately downstream of the reattachment location C, the plate wall experiences a large reattachment pressure ${ p'} = p_{6}$ acting over it, and this jump in pressure to peak values signifies the re-attachment of the separated flow. This large reattachment pressure continues to be felt downstream of the reattachment location unless perturbed by other wave interactions/viscous layers (e.g. expansion fan from the wedge, growing boundary layer of the plate wall). In the LESF model, we investigate the flow field from the plate leading edge up to immediately after the reattachment location and detail the structure of the SWBLI (see figure 7) for a leading-edge separation situation.

3.2. Reattachment flow fields: oblique and normal

The reattachment of the flow is crucial in the LESF model. Figure 8 shows two reattachment flow fields: oblique ($\kappa = 0$) and normal ($\kappa = 1$), where $\kappa$ is a Boolean variable that is either 0 or 1. The terms oblique and normal reattachment should be interpreted as oblique shock and normal shock reattachment.

Figure 8. Schematic drawings (not to scale) of the two reattachment flow fields: (a) oblique ($\kappa = 0$) and (b) normal ($\kappa = 1$) reattachment. For the case of a normal reattachment, the bow shock behaves as a normal shock in the close vicinity of location C. Note that the schematic drawings are only simplifications of real flows near reattachment.

In an oblique reattachment (cf. figure 8a), the flow is supersonically turned parallel to the wall by a planar reattachment shock SW5. However, in a normal reattachment (cf. figure 8b), the flow is not able to turn large angles and a detached bow shock SW5 occurs. Downstream of this bow shock, the flow is subsonically turned parallel to the wall. Importantly, the bow shock in the immediate vicinity of the reattachment location C behaves like a normal shock. The detachment distance (i.e. the shock stand-off distance of the bow shock SW5) and the pressure variations in the subsonic region of the bow shock are ignored in the present analysis.

The static pressure in regions 5 and 6 are important parameters for flow reattachment. Note that the static pressure in region 5 is equal to the dead-air pressure of the separation bubble, i.e. $p_{5} = p_{d}$ and the static pressure in region 6 is called the reattachment pressure, i.e. $p_{6} = { p'}$. They are related by the oblique shock relation

(3.1)\begin{equation} \frac{{ p'}}{p_{d}} = \frac{2 \gamma ({M_{5}} \sin {\beta_{56}})^{2} - (\gamma - 1)}{(\gamma + 1)}. \end{equation}

Here, $M_{5}$ is the Mach number in region 5 and $\beta _{56}$ is the angle of the reattachment shock relative to the oncoming flow in region 5. Note that $M_{5}$ and $\beta _{56}$ are only functions of $\gamma$, $M_{\infty }$, $\theta _{w}$ and $p_{d}/p_{\infty }$ resulting from solving two oblique shocks (SW2 and SW4) and an expansion fan (E1) between regions 1, 3, 4 and 5 (cf. figure 7). Therefore, (3.1) essentially gives a relation between the separation pressure $p_{d}$ and the reattachment pressure ${ p'}$, which must be solved numerically since there is no closed-form solution. We form a relation between $p_{d}$ and ${ p'}$ in (3.1) as was done before by Chapman et al. (1958) in (1.1), except that the present scenario concerns the case of non-isentropic compression at reattachment. This is indeed a critical difference between the present approach and the theory by Chapman et al. (1958). The theory by Chapman et al. (1958) considers isentropic recompression of the dividing streamline, imposing the boundary-layer theoretical consideration that the outer flow static pressure at reattachment is the same as total pressure of the dividing streamline. In relating the dead-air pressure with the outer flow conditions after reattachment, the theory assumes isentropic compression of all streamlines, including those in the outer flow. However, at hypersonic Mach numbers this is not the case. In the present model, based on the inviscid picture by Délery & Marvin (1986), the calculations only concern the outer inviscid flow; the entire outer flow at the reattachment location is turned by a reattachment shock which is a non-isentropic phenomenon, and thus $p_{d}$ and ${ p'}$ are related using shock relations. We remark that, for a normal reattachment, $\beta _{56}$ is always equal to $90^{\circ }$.

The occurrence of an oblique or a normal reattachment depends on the strength of the wedge shock. For wedge shocks of low strength i.e. low $p_{2}/p_{\infty }$, an oblique reattachment is expected because the reattachment pressures ratios are relatively low i.e. low ${p'}/p_{\infty }$. However, as the wedge shock strength increases, we expect a transition to a normal reattachment where the reattachment pressure ratios are relatively high. The experimental observation of ‘large’ separation bubbles and a detached shock at reattachment that is strongly curved near the wall was reported by Sriram (2013), Sriram & Jagadeesh (2014, 2015), Srinath (2015) and Sriram et al. (2016). It was hypothesized that the flow upstream of reattachment sees a small normal reattachment shock region which can cause large reattachment pressure ratios.

We note that, for intermediate wedge shock strengths, a near-sonic reattachment, i.e. $M_{6} \approx 1$, would occur where the flow downstream of the reattachment shock is at the attached–detached transition of the reattachment shock. Since the LESF model can produce a dual solution in some cases i.e. an oblique and normal reattachment, we will rely on experimental evidence to determine the correct solution.

We emphasize that, in real flows, the reattachment is a complex process. The reattachment shock is always detached from the plate wall and is curved at the location of reattachment. This is due to the interactions between viscous layers and shear layers at the location of reattachment, which cannot be captured in the LESF model.

3.3. Methodology

In figure 9 we contrast the flow fields and pressure distributions of separated and unseparated flows. For the unseparated flow (cf. figure 9a) there are two characteristic parameters: the shock impingement distance $OR =X_{ir}$ and the inviscid shock reflection pressure ratio $p_{ir}/p_{\infty }$. However, for the separated flow (cf. figure 9b), there are three characteristics parameters: the separation bubble length $OC =L$, dead-air pressure ratio $p_{d}/p_{\infty }$ and reattachment pressure ratio ${ p'}/p_{d}$.

Figure 9. Schematic diagram (not to scale) and wall pressure profiles of (a) unseparated and (b) separated flows. The unseparated flow is characterized by the pressure jump $p_{ir}/p_{\infty }$ and the shock impingement distance $X_{ir}=OR$. The separated flow is characterized by the pressure jumps $p_{d}/p_{\infty }$ and ${ p'}/p_{d}$, and the separation length $L=OC$.

The central goal of the LESF model is to determine $p_{d}/p_{\infty }$ (or equivalently the separated shock angle $\beta _{s}$), so that the geometry of the separation bubble $L/X_{ir}, H/L, \alpha$ can be easily computed. Since $p_{d}$ is related to ${ p'}$ in (3.1), a criterion for ${ p'}$ must be fixed. We assume that the reattachment pressure for the separated flow is equal to the inviscid shock reflection pressure, i.e. ${ p'} = p_{ir}$. The details on $p_{ir}$ are provided in Appendix B. This approximation of the pressure ratio at reattachment is supposedly a valid assumption for ‘weak’ shocks where $p_{2}/p_{1}$ is relatively small, as described in the literature (Liepmann et al. 1951; Gadd et al. 1954; Chapman et al. 1958; Hakkinen et al. 1959; Brower 1961; Stollery & Bates 1974; Délery & Marvin 1986; Katzer 1989; Délery & Dussauge 2009). For ‘strong’ shocks, where $p_{2}/p_{1}$ is relatively large, the reattachment pressure ratio is observed to be lower but of the same order as that of the inviscid shock reflection pressure ratio (Mallinson, Gai & Mudford 1997; Matheis & Hickel 2015; Verma & Chidambaranathan 2015).

Following this, we again state the assumptions in the LESF model:

1. (i) The flow separation occurs at the leading edge of the plate wall.

2. (ii) The reattachment pressure ratio approximately equals the inviscid shock reflection pressure ratio, ${ p'}/p_{\infty } \approx p_{ir}/p_{\infty }$.

3. (iii) The interaction between the wedge shock SW1 and the separation shock SW2 is a Type I interaction.

4. (iv) The working gas is calorically perfect (constant $C_{p}$ and $C_{v}$ everywhere).

Figure 10 shows the flow chart of the iterative method used to calculate flow quantities in uniform regions in figure 7. Appendix C shows the oblique and normal reattachment solutions to a small range of impinging shocks over a plate wall in an air flow. Here, it must be stated that, although the LESF model does not account for any viscosity effects, in real flow situations, separation is a viscous effect. We paraphrase this statement from Brower (1961) since his research also involved leading-edge separation flow fields but over compression corners. For leading-edge separation flows, the separated flow field interestingly happens to be viscosity independent (or equivalently Reynolds number independent) since the complicated development of the boundary layer (e.g. laminar to turbulent boundary-layer transition) and its associated pressure variations from plate leading edge to separation location may be avoided. Previous works on leading-edge separation flows by Chapman et al. (1958), Brower (1961) and Srinath, Sriram & Jagadeesh (2017) show geometrically similar flow fields and an independence of free-stream Reynolds number. We remark that the structure of two separated flows is said to be geometrically similar if it can be perfectly matched by simply enlarging or reducing the geometry.

Figure 10. Double iterative algorithm for solving the separation shock angle in the LESF model. Input parameters are $M_{\infty }, \theta _{w}$ and $\kappa$. The output parameter is $\beta _{s}$, which determines the separation bubble parameters (e.g. $p_{d}/p_{\infty }, L/X_{ir}, H/L$). Note that regions 4a and 4b denote the regions above and below the slip line SL2 in figure 7, respectively. In the calculations, $\gamma = 1.40$ is fixed, which corresponds to air flows.

3.4. Solution range

For a fixed Mach number, the LESF model allows solutions only within a limited range of wedge angles. The minimum wedge angle is obtained when either the separation/reattachment is a Mach wave and the maximum wedge angle is obtained when a possible Type II interaction occurs between the wedge shock and the separation shock (cf. table 2).

Table 2. Criteria at minimum and maximum wedge angles. Here, Type II refers to the nature of interaction between the separation and the wedge shock.

The LESF model cannot admit wedge angles smaller than $\theta _{w,min}$ since the separation/reattachment shock strength cannot be lower than that of a Mach wave. For the maximum value, we limit the wedge angles by $\theta _{w,max}$ at which a Type I to Type II transition can possibly occur between the wedge shock and the separation shock. An estimate of $\theta _{w,max}$ is obtained from the shock polar plots which are commonly used to quantify shock–shock interactions. Figure 11 shows examples of wedge shock–separation shock interactions at free-stream Mach 5. The interaction for minimum and maximum wedge angles for an oblique reattachment is shown in figures 11(a) and 11(b). And figures 11(c) and 11(d) are similar plots for a normal reattachment. The locus line plot is shown in figure 11(e), which spans the solution for all possible wedge shock–separation shock interactions for an oblique and a normal reattachment. The locus lines begin at $\theta _{w} = \theta _{w, min}$ and end at $\theta _{w} = \theta _{w, max}$. The maximum wedge angle $\theta _{w, max}$ is defined at the triple point i.e. when the wedge shock polar, separation shock polar and the free-stream shock polar intersect at a single point. We do not consider solutions outside of the free-stream shock polar as they would possibly lead to a Type II interaction (Sriram et al. 2016) where the LESF model would not work.

Figure 11. Shock polar plots of SW1–SW2 interactions. SW1, SW2 and free-stream shock polars are indicated by dashed, dotted and continuous (thin) lines. The wedge angles are indicated by open circles, the separation shock angles by open squares and the SW1–SW2 shock polar intersections by black stars. (a,c) The interactions at $\theta _{w,min}$ and $\theta _{w,max}$ at $\kappa =1$. (b,d) The interactions at $\theta _{w,min}$ and $\theta _{w,max}$ at $\kappa =0$. (e) The locus of the SW1–SW2 interactions is denoted by LM and PQ (thick lines) for $\kappa = 0$ and $\kappa = 1$ respectively. The coordinates at each point in either LM or PQ denote the flow deflection and the pressure ratio in region 4 (cf. figure 7). For all cases in this figure, $M_{\infty } = 5$.

Figure 12 shows the plots of the minimum, maximum and detached wedge angles which are functions of free-stream Mach number. The detached wedge angle $\theta _{w}^{D}$ is another critical wedge angle which marks the transition between the regular–irregular reflection on a plate wall (cf. Appendix B). In figure 12, we find that $\theta _{w,max} < \theta _{w}^{D}$. Furthermore, a region in the range $16^{\circ } < \theta _{w} < 19^{\circ }$ is identified where no oblique/normal solution exists. We suppose that the wedge angles are intermediate in this region resulting in neither a ‘weak’ nor a ‘strong’ impinging shock. A near-sonic reattachment is very likely to occur in this region which is thought to transition towards a normal reattachment.

Figure 12. Plots of critical wedge angles $\theta _{w,min}$, $\theta _{w,max}$ and $\theta _{w}^{D}$ as a function of $M_{\infty }$ for (a) $\kappa = 0$ and(b) $\kappa = 1$. The shaded grey regions show the range of wedge angles where solutions to the LESF model exist.

3.5. Separation bubble characteristics

The separation bubble in the LESF model is characterized by the dead gas pressure $p_{d}$, separation length $L$, upstream skewness $\alpha$ and separation height $H$. An important feature of the LESF model is the geometrical similarity of the SWBLI. Since the separated flow variables (e.g. $p_{d}/p_{\infty }$, $\beta _{s}$, $L/X_{ir}$) depend only on inviscid flow variables (e.g. $M_{\infty }$, $\theta _{w}$), the resulting leading-edge separation flows are expected to be geometrically similar and hence do not depend on the location of shock impingement $X_{ir}$. The consequence of this is that the leading-edge separated flows are Reynolds number independent.

3.5.1. Oblique reattachment $\kappa = 0$

Figure 13 shows the separation bubble characteristics for the case of an oblique reattachment. Here, we remark that the flow immediately downstream of the reattachment shock (region 6 in figure 8a) is typically supersonic. When the wedge angle tends to its minimum value $\theta _{w} \rightarrow \theta _{w, min}$, we find that $p'/p_{d} \rightarrow 1$, that is, the reattachment shock tends to a Mach wave, while $p_{d}/p_{\infty } > 1$, that is, the separation shock is stronger than a Mach wave. This necessarily means near zero flow deflection across the reattachment shock, resulting in an open separation $L/X_{ir} \rightarrow \infty$. And since $p_{d}/p_{\infty } > 1$, we have a finite $H/X_{ir}$ and thus $H/L \rightarrow 0$. That is to say, we have an open and shallow separation when $\theta _{w} \rightarrow \theta _{w, min}$. We remark that this extreme geometry is a theoretical limit when the reattachment shock tends to a Mach wave. However, as the wedge angle increases, we see that the separation and reattachment shock strengths gradually increase with the condition that the separation shock strength is always greater than the reattachment shock strength $p_{d}/p_{\infty } > { p'}/p_{d}$. The open and shallow separation bubble is gradually relieved as the wedge angle increases since the reattachment shock strength gradually increases. When the wedge angle tends to the maximum value $\theta _{w} \rightarrow \theta _{w, max}$, the separation and reattachment shocks strengths reach their respective maximum values while a stabilization occurs in the separation bubble geometry. The stabilization can be seen in figures 13(c), 13(d) and 13(e) where a gradual collapse of the curves occurs near the maximum wedge angle. When the wedge angles are greater than the maximum value $\theta _{w} > \theta _{w, max}$, we would expect a gradual transition to a near-sonic reattachment.

Figure 13. Separation bubble characteristics for an oblique reattachment $\kappa = 0$. Plots of (a) $p_{d}/p_{\infty }$,(b) ${ p'}/p_{d}$, (c) $\alpha$, (d) $H/L$, (e) $L/X_{ir}$ as a function of $\theta _{w}$.

3.5.2. Normal reattachment $\kappa = 1$

Figure 14 shows the separation bubble characteristics for a normal reattachment. When the wedge angle tends to its minimum value $\theta _{w} \rightarrow \theta _{w, min}$, we find that $p_{d}/p_{\infty } \rightarrow 1$, that is, the separation shock tends to a Mach wave, while $p'/p_{d} \gg 1$, that is, the reattachment shock strength is relatively high. This necessarily means near zero flow deflection across the separation shock, and thus a shallow separation $H/L \rightarrow 0$. Here, we do not find an open separation since $p'/p_{d} \gg 1$, that is, the flow upstream of the reattachment is directed towards the plate wall. Again, this extreme geometry is a theoretical limit, but for the case when the separation shock tends to a Mach wave. However, as the wedge angle increases, we see that the separation shock strength increases while the reattachment shock strength decreases with the condition that separation shock strength is always lower than the reattachment shock strength $p_{d}/p_{\infty } < { p'}/p_{d}$. The extreme geometry of the separation bubble is gradually relieved as the wedge angle increases since the separation shock strength gradually increases. When the wedge angle tends to the maximum value $\theta _{w} \rightarrow \theta _{w, max}$, the separation shock reaches its respective maximum value while the reattachment shock reaches its respective minimum value. The separation bubble does not stabilize at the maximum wedge angle, since variations can be seen in figures 14(c), 14(d) and 14(e) near the maximum wedge angle. An important feature of the normal reattachment is that it depicts the separation bubble length nearly equal to the shock impingement location from the leading edge $L/X_{ir} \approx 1$.

Figure 14. Separation bubble characteristics for a normal reattachment $\kappa = 1$. Plots of (a) $p_{d}/p_{\infty }$, (b) ${ p'}/p_{d}$, (c) $\alpha$, (d) $H/L$, (e) $L/X_{ir}$ as a function of $\theta _{w}$.

4. Experiments

Figure 15 shows the schlieren images of the steady SWBLIs at Mach $5.52$ with a $26.6^{\circ }$ wedge for various shock impingement locations on the flat plate (given in terms of distance from leading edge). There are a number of observations that can be made from the figure and we discuss each of them separately.

Figure 15. LESF fields at Mach $5.52$ with $26.6^{\circ }$ wedge. The shock impingement location is varied in the range 10–60 mm from the plate leading edge: (a) $X_{ir} \approx 60$ mm; (b) $X_{ir} \approx 50$ mm; (c) $X_{ir} \approx 40$ mm; (d) $X_{ir} \approx 30$ mm; (e) $X_{ir} \approx 20$ mm; ( f) $X_{ir} \approx 10$ mm.

4.1. Incipient vs ‘large’ separation

From the inviscid calculations, the peak pressure ratio imposed on the plate wall due to the shock impingement is estimated to be around $55$ (cf. figure 25). The severity of this imposed pressure ratio can be understood in relation to the incipient pressures necessary to separate the flow. We evaluate the typical values of the incipient separation pressures in our experiments. For this we assume that the shock impingement occurs in the range $X_{ir} \approx 10\unicode{x2013}60$ mm, as in the present experiments. We then use the expression for the boundary-layer thickness (Hayes & Probstein 1959) which is given by (4.1)

(4.1)\begin{equation} \frac{\delta}{x} \sim \frac{\gamma - 1}{2} \frac{M_{\infty}^{2} \sqrt{C_{\infty}}}{\sqrt{Re_{x,\infty}}}, \end{equation}

where $\delta$ is the boundary-layer thickness at a distance $x$ from the plate leading edge, $C_{\infty }$ is the Chapman–Rubesin constant which is estimated using a Sutherland's law viscosity model at wall and free-stream locations and is of order $C_{\infty } \sim 1$ and $Re_{x,\infty }$ is the Reynolds number based on free-stream conditions and the distance $x$ from the plate leading edge. Please note that (4.1) is the boundary-layer growth expression in the absence of shock impingement. Using (4.1) we evaluate the incipient pressure. The expression for the incipient pressure (Délery & Marvin 1986) is given by (4.2)

(4.2)\begin{equation} \frac{\mathcal{P}}{p_{\infty}} \sim 1 + \mathcal{F} \gamma \frac{M_{\infty}^{2}}{\sqrt[4]{M_{\infty}^{2} - 1}} \sqrt{\frac{C_{f}}{2}}, \end{equation}

where $\mathcal {P}$ is the pressure at the onset of separation, $\mathcal {F}$ is the universal correlation function which for laminar flows is of order $\mathcal {F} \sim 1$ and $C_{f}$ is the skin friction which depends on the boundary-layer thickness, analogous to the skin friction coefficient obtained through the (laminar) Blasius solution. We summarize the orders of magnitude of relevant pressure ratios and length scales in table 3.

Table 3. Imposed pressure ratios and separation length scales for the incipient situation and present experiments. The $\delta$ and $\mathcal {P}$ values are evaluated at $x = X_{ir}\approx 10\unicode{x2013}60$ mm.

Even with hypersonic Mach numbers and high skin friction near the leading edge, the shock is orders of magnitude stronger than the incipient separation pressure at impingement locations. The LESF model results have already shown ‘large bubbles’ which are of length comparable to the distance of shock impingement, similar to what was reported previously by Sriram & Jagadeesh (2014, 2015) and Sriram et al. (2016).

The present experiments also showcase such ‘large bubbles’ for strong impinging shocks. The lengths of the ‘large bubbles’ are experimentally determined simply by locating the reattachment. The methodology to obtain the reattachment locations is detailed by Sriram et al. (2016). The uncertainty in the reattachment location is $\pm$2.5 mm.

4.2. Separation bubble geometry

Interestingly, the separation bubble geometries at $X_{ir} \approx 60$, $50$ and 40 mm are observed to be geometrically similar (cf. figures 15(a), 15(b) and 15(c)). The separated shock angle is nearly a constant with a value of around $20.0^{\circ }$ for the three impingement locations. This separation shock angle has a very good agreement with the theoretical value of around $19.4^{\circ }$ using the LESF model. A comparison of the SWBLI between experimental and LESF model at $X_{ir} \approx 40$ mm is shown in figure 16.

Figure 16. Comparison of the separation bubble geometry between the experiments and the LESF model. The shock pair is a result of the wedge shock and separation shock interactions. The location of the leading edge of the plate and the location of the reattachment are marked O and C, respectively. The model assumes $\kappa =1$. In the figure, $M_{\infty } = 5.52$, $\theta _{w} = 26.6^{\circ }$ and $X_{ir} \approx 40$ mm.

It is noteworthy that the experimental and theoretical SWBLI are in close agreement with the exception of the flow at the reattachment location C. The shock waves, slip lines and expansion fan from the model are represented as continuous, dash-dotted and dotted lines, respectively, in figure 16. The separation bubble is the triangular region enclosed by the shear layer, the reattachment shear layer and the plate wall. The separated shock angle relative to the free stream compares well with the experiment and model. The separated slip line in the model lies well within the growing shear layer in the experiments. The wedge shock–separation shock intersection, a Type I interaction, is slightly displaced by around 2 mm downstream of flow in the model because of a slightly larger wedge shock angle measured in the experiments $|\beta _{w}| \approx 38.5^{\circ }$ when compared with $|\beta _{w}| \approx 36.9^{\circ }$ using the $\theta _{w}\mbox {-}\beta _{w}\mbox {-}M_{\infty }$ oblique shock relation. Due to this small difference in $\beta _{w}$, the flow structures predicted by the model appear slightly elongated downstream of the flow. There is, however, a main difference in the reattachment shock clearly evident in figure 16 which is not captured by the model. The curving of the reattachment shock seen in experiments does not match with the normal reattachment shock angle in the model. The reason for this discrepancy is the curving of the reattachment shock due to the reattachment of the shear layer, which is not accounted for by the LESF model. In real flows, the shear layer reattachment causes the outer supersonic flow (immediately upstream of the reattachment and near the sonic line of the boundary layer) to be progressively ramped by a series of compression waves that focus into the reattachment shock. The interactions of the expansion fan (the expansion fan emanating from the reflection of the shock wave from the region of constant pressure, and the expansion fan emanating from the trailing edge of the wedge) also curve the reattachment shock downstream of the flow, but this is a secondary effect. The expansion fan effects emanating from the separation bubble apex (E1 in figure 7) and from the wedge trailing edge (peak pressure drop in figure 25b) may cause the curving of the reattachment shock in real flows. The turning of the reattaching shear layer from the plate wall is indeed a complex process in real flows which we do not account in the LESF model. Although the shock curvature can be only obtained from the experiments, the overall SWBLI pattern seems to resemble a near normal reattachment flow field. The separation length $L$ measures at around 42.8 and 41.1 mm in the experiment and the LESF model respectively. The separation lengths are slightly larger compared with the inviscid shock impingement distance form the leading edge $L > X_{ir}$ (see figure 14c). The skewness $\alpha$ and the separation bubble height ratio $H/L$ are also in good agreement between the experiments and the LESF model. We find that the near-sonic reattachment LESF model does not agree well with the flow field shown in figure 16. Table 4 shows the separation bubble characteristics between experiments and the LESF model.

Table 4. Comparison of separation bubble parameters between experiments and LESF model ($\kappa = 1$): separation shock angle $\beta _{s}$, reattachment location $x'$, detachment length ratio $L/X_{ir}$, detachment bubble height ratio $H/L$ and detachment bubble skewness $\alpha$ are used for comparison. For all cases, $M_{\infty } = 5.52$ and $\theta _{w} = 26.6^{\circ }$.

For $X_{ir} \approx 40, 50$ and 60 mm the experiments show geometric similarity. First, we make a quick visual inspection of the structure of SWBLI in figures 15(a), 15(b) and 15(c) and find that they look geometrically similar. Note that the structures of the separated flows are said to be geometrically similar if they can be perfectly matched by simply enlarging or reducing the geometry. We also perform image processing to identify the shock coordinates and measure the angle of the separated shocks to obtain a quantitative comparison. The separated shock angles shown in table 4 are nearly the same for the different shock impingement locations, justifying the geometric similarity. Later, we show the wall pressure measurements which further justify the geometric similarity. Based on this evidence, we reason that there is a geometric similarity in the structure of SWBLI at $X_{ir} \approx 40, 50$ and 60 mm.

Figure 17 not only shows the geometric similarity for $X_{ir} \approx 40, 50$ and 60 mm, but also shows very good agreement with the LESF model. Note that the geometric similarity does not imply a fixed length scale, but a fixed geometry (i.e. fixed angles and length ratios).

Figure 17. Separation bubble geometry between the experiments and the LESF model at shock impingement locations (a) $X_{ir} \approx 60$ mm, (b) $X_{ir} \approx 50$ mm and (c) $X_{ir} \approx 40$ mm. The figures clearly show a geometric similarity of the SWBLI at different shock impingements. The LESF model agrees well with the experiments. The shock wave, slip line and expansion fan of the LESF model are denoted by continuous, dashed and dotted lines, respectively. The model assumes $\kappa =1$. For all cases in the figure, $M_{\infty } = 5.52$, $\theta _{w} = 26.6^{\circ }$.

4.3. Transient flow

Figure 18 shows the transient pressure readings on the plate wall at $X_{ir} \approx 40$ mm. Until around 2.4 ms the wall pressure sensors at $30$, $38$, $46$, $54$, $62$ and 70 mm from the plate leading edge read noise having fluctuations of $\pm$0.3 kPa around zero mean value. The wall pressures slowly rise from around 2.4 ms and show a good correspondence to the Pitot pressure (cf. figure 4) which also begins to rise around the same time. The slow rise in the pressure sensors is followed by a steady time. The mean pressure steady times are different for each sensor location. For example, the 30 and 38 mm sensors in figure 18, which read the pressures inside the separation bubble (see figure 15 at $X_{ir} \approx 40$ mm for the separation bubble region), read mean pressure values of 11.2 and 26.4 kPa, respectively, between 3.2 and 3.6 ms. The sensor closest to the reattachment (reattachment location at 42.8 mm, see table 4), i.e. the 46 mm sensor picks up high fluctuations of $\pm$17 kPa and has a mean of around 69 kPa between 3.2 and 3.6 ms. A large amplitude pressure fluctuation is observed near the reattachment location since small movement of the reattachment shock causes significant variations in the wall pressures which are then picked up by the sensor. The wall pressures downstream of reattachment stabilize quickly and read lower pressures because of expansion fan interactions. The wall pressures downstream of reattachment are significantly lower, having values less than 60 kPa with mean pressure steady time between 3.0 and 3.6 ms.

Figure 18. Time history of the wall pressure signals. The wall pressure steady times is between 3.2 and 3.6 ms. The 15 mm pressure sensor reads pressure inside the separation bubble and the 46 mm pressure sensor reads pressure near the reattachment location, which can both be verified by looking at the schlieren image in figure 15(c). For all cases in this figure, $\theta _{w} = 26.6^{\circ }$, $M_{\infty } = 5.52$ and $X_{ir} \approx 40$ mm.

The 600 $\mathrm {\mu }$s Pitot steady time (between 3.0 and 3.6 ms) does not necessarily match with the 400 $\mathrm {\mu }$s wall pressure steady time (between 3.2 and 3.6 ms) since separated flow, in general, require time for establishment. This establishment time was estimated to be around 200 $\mathrm {\mu }$s by Sriram et al. (2016) for similar experimental conditions as in the present study. Thus, using this information, the wall pressure steady time is estimated around 400 $\mathrm {\mu }$s from the available 600 $\mathrm {\mu }$s Pitot steady time. The Pitot steady time and wall pressure steady time fit very well with the time evolution of flow in figure 19 explained later in this section.

Figure 19. Schlieren time evolution of flow field for $X_{ir} \approx 40$ mm at $M_{\infty } = 5.52$ and $\theta _{w} = 26.6^{\circ }$.

The steady flow fields in figure 15 are taken from the schlieren time evolution of flow field during shock tunnel run time. The schlieren images have a temporal resolution of 50 $\mathrm {\mu }$s. The steady flow structures in the schlieren images are visually checked within the duration of Pitot and wall pressure steady time. Figure 19 shows the time evolution of the wedge shock–plate wall interaction for the $X_{ir} \approx 40$ mm impingement location. Both Pitot steady time (between 3.0 and 3.6 ms in figure 4) and wall pressures steady time (between 3.2 and 3.6 ms in figure 18) give a very good correspondence with the flow structures in the schlieren time evolution of the flow field. From three independent information types; Pitot signal, wall pressure signals and the schlieren time evolution images, the shock tunnel test time is estimated, which happens to be 400 $\mathrm {\mu }$s between 3.2 and 3.6 ms in the present experiments.

4.4. Wall pressure distribution

To measure the wall pressures we make use of piezoelectric and Kulite pressure sensors. Kulite pressure sensors can be easily placed very close to the leading edge of the plate, since not very much material is required to support the sensors. They also have a better signal-to-noise ratio compared with PCB pressure sensors, but with the disadvantage of being very delicate and prone to damage.

A comparison of surface pressure distributions, for shock impingements of $X_{ir} \approx 60, 50$ and 40 mm, can also show whether the flow field is geometrically similar. Here, we note that it is difficult to accurately identify regions of constant pressure at each sensor location, e.g. in figure 18. This is mainly due to the pressure oscillations and the transient nature of the flow. However, the reported pressures are time average values after the separated shock seems to have reached the leading edge. The initial transient of the flow is nearly 200 $\mathrm {\mu }$s between 3.0 and 3.2 ms and accounts for the establishment time of the separated flow, although the flow field is visibly steady throughout the Pitot steady time (Sriram et al. 2016).

Figure 20 shows the pressure distribution over the plate wall for three different impingement locations. The wall pressure distributions show some similarities compared with wall pressure distributions in a strong SWBLI (Délery & Marvin 1986), which is far from a shock reflection scenario (cf. figure 24a). The pressure distribution of a real (viscous) flow gives us information on two important quantities: plateau pressure and peak pressure. Plateau pressure, of the same order as separation pressure, is the static pressure inside the separation bubble region which is nearly constant within the bubble. Peak pressure, of the same order as reattachment pressure, is the highest pressure felt over the plate wall. It is to be noted here that the peak pressure, although being a local quantity, is approximated here as the peak pressure from among the signals sensed by the six pressure sensors.

Figure 20. Pressure distribution on the plate wall. The pressure at each sensor location is averaged between 3.2 and 3.6 ms. The mean and error bar values of pressure are based on transient pressure data at each sensor location and at each run; (a) $X_{ir} \approx 60$ mm, (b) $X_{ir} \approx 50$ mm and (c) $X_{ir} \approx 40$ mm. For all cases in this figure, $M_{\infty } = 5.52$ and $\theta _{w}=26.6^{\circ }$.

Present experiments cannot precisely determine the reattachment pressures since no information on the location of zero wall shear stress is obtained. In the experiments, we report plateau pressures of approximately 7.82, 8.60 and 8.17 kPa for $X_{ir} \approx 60$, 50 and 40 mm, respectively.

To address the plateau pressures we performed some experiments using a Kulite sensor at 15 mm from the leading edge (we could place Kulites relatively close to the leading edge unlike the PCB sensors). The near leading-edge portion is always in the upstream portion of the bubble, and we observe the same pressure at 15 mm from the leading edge for all impingement locations, see figure 20. Due to the additional sensor in the upstream portion of the bubble, the averages of the pressure in the bubble are also closer for the different cases. This too suggests that the flow field is geometrically similar.

The difference in the average plateau pressure is due to the downstream most sensor in the bubble being closer to reattachment in a few cases, due to which they measure relatively higher pressure. The geometrical similarity is best noted in the sensors which are in the most upstream location. At 15 mm from the leading edge, the same pressure of 8.0 kPa is observed for all impingement locations, illustrating the geometrical similarity.

The LESF model ($\kappa = 1$) for a similar experimental condition as in figure 20 gives the pressure inside the separation bubble as 7.67 kPa.

The plateau pressure is then followed by a gradual increase to the peak pressure which then subsequently decreases due to expansion fan interactions. This gradual increase and decrease in pressure near the reattachment location can also be seen in the transient pressure signals (cf. figure 18). For $X_{ir} \approx 50$ and 40 mm, the plateau pressure increases to peak pressures of around 69 and 71 kPa, respectively, which then subsequently decreases.

In the case where $X_{ir} \approx 60$ mm, the pressure sensor at 70 mm is located near the neck region (or near reattachment location) and reads an average value of 75 kPa, which is higher than that of sensors at other locations. The present experiments have a limitation in the spatial resolution of the surface pressure measurement, i.e. a minimum sensor-to-sensor distance of 8 mm. The closest sensor that can be placed downstream of 70 mm can only be 78 mm from the leading edge. We argue that a pressure measurement at 78 mm, sufficiently downstream of the neck region, would indicate a lower pressure than the sensor at 70 mm. An example of such lower pressure measurements downstream from the neck region can be seen in other shock impingements, e.g. at $X_{ir} \approx 40$ and $50$ in figure 20. Therefore, we argue that, in the case of $X_{ir} \approx 60$ mm, the last pressure sensor, i.e. the 70 mm pressure sensor, would measure peak pressure compared with the 62 and 78 mm pressure sensors.

We therefore establish a geometrical similarity which can be seen from the shock angles (cf. table 4) and further illustrated from the measured the plateau pressures (cf. figure 20).

The average plateau and peak pressures along with the sensor locations are shown in table 5. A comparison with the LESF model ($\kappa = 1$) is also shown in table 5.

Table 5. Plateau pressure, peak pressure and sensor locations for $X_{ir} \approx [60,50,40]$ mm. Wall pressures in the inviscid model (normal reattachment) are compared with experiments at $M_{\infty } = 5.52$ and $\theta _{w} = 26.6^{\circ }$.

The measured peak pressure ratios in table 5 are around 1.65 times lower than the inviscid shock reflection pressure ratio. The possible reasons for the decrease in peak pressure are attributed to the inexact placement of the reattachment pressure sensors and the finite sensing area associated with the pressure sensor. Firstly, a small displacement in the location of the reattachment pressure sensor can cause a drop in the measurement of the peak pressure. Secondly, the pressure sensors have a finite area (diameter 5 mm) and hence average the pressure in a circular region. In any case, it is important to see that the measured peak pressures are of the same order as the inviscid shock reflection pressure ratio. For other impinging locations $X_{ir} \approx [30,20,10]$ mm, no transient pressure measurements are performed because the resulting small separated regions approach very close to the plate leading edge. We face the issue of mounting pressure sensors very close to the plate leading edge due to insufficient material required to hold the sensors.

4.5. Validation of LESF model

We have already validated the LESF model against experiments for the case of $M_{\infty } = 5.52$ and $\theta _{w} = 26.6^{\circ }$. The qualitative comparison is shown in figure 17. The quantitative comparisons are shown in tables 4 and 5. The LESF model is also verified for other wedge angles and free-stream conditions. We perform experiments at Mach $5.97$ and $8.04$ with a $21.80^{\circ }$ wedge that depict a leading-edge separation flow. Note that the free-stream Mach number is varied by using different throat inserts. Figure 21 shows the experimental and LESF model flow field comparison. The LESF model with an oblique reattachment LESF model ($\kappa = 0$) for a $21.80^{\circ }$ wedge shows a very good match with the experiments.

Figure 21. Comparison between the schlieren and LESF flow model using $\kappa = 0$ (a) $M_{\infty } = 5.97$, $\theta _{w} = 21.80^{\circ }$ and $X_{ir} \approx 50$ mm and (b) $M_{\infty } = 8.04$, $\theta _{w} = 21.80^{\circ }$ and $X_{ir} \approx 60$ mm.

Although a normal reattachment solution exists for the flow conditions in figures 21(a) and 21(b), we find that it does not agree well with the experiments. We suppose the $21.80^{\circ }$ wedge angle is an intermediate wedge angle which shows a near-sonic reattachment flow field (i.e. in the vicinity of the attached–detached transition). The comparison of the separated shock angle between experiments and LESF model is given in table 6. For a dual solution, we cannot precisely determine which combination of wedge angle and free-stream Mach number causes a normal or an oblique, the experiments can hint at the reattachment flow fields based on the overall structure of the SWBLI. While the experiments at Mach $5.54$ and with a $26.6^{\circ }$ wedge compare well with a LESF model having a normal reattachment, the experiments at Mach $5.97$, $8.04$ and with a $21.80^{\circ }$ wedge compare well with an LESF model having an oblique reattachment. This does not come as a surprise since the $26.6^{\circ }$ lies very close to $\theta _{w, max}$ whereas $21.80^{\circ }$ lies very close to $\theta _{w,min}$ in figure 12(b). We suppose the near-sonic reattachments to occur at intermediate wedge angles around $\theta _{w} = \theta _{w,min}$ (cf. figure 12b) which is supported by our experiments.

Table 6. Separation shock angle $\beta _{s}$ comparison between present inviscid model and experiments depicting a leading-edge separation.

We then compare the separated shock angle $\beta _{s}$ obtained from the LESF model and from the experiments. The comparison is shown in table 6. For the LESF model, we solve for an oblique reattachment at $\theta _{w}=17^{\circ }$, and a normal reattachment at $\theta _{w}=26.6^{\circ }$. However, we solve for a oblique reattachment at $\theta _{w}=21.8^{\circ }$. We suppose a near-sonic reattachment at intermediate wedge angle since there must be a transitional phase between the attached–detached reattachment. A good agreement in the separation shock angles gives confidence in the LESF model.

4.6. Breaking geometric similarity

We remark that the geometrically similar flow fields occur for a small range in Reynolds number. In our experiments with Mach $5.52$ and a $26.6^{\circ }$ wedge, the geometric similarity is only observed for shock impingement locations 40–60 mm from the leading edge or equivalently in the Reynolds number range $1.92 \times 10^{5}\unicode{x2013}2.87 \times 10^{5}$. However, beyond this Reynolds number range we observe a breakage of geometric similarity for two situations: when shock impingement is either sufficiently far or sufficiently close to the leading edge. The terms sufficiently far and sufficiently close are used here since a precise impingement location causing the transition to geometric similarity breakage is not obtained in the present experiments.

For shock impingement sufficiently far away from the plate leading edge, flow separation occurs at some downstream location from the leading edge. For example, in figure 22 it is evident that the flow separation occurs at some downstream location. The shock impingement (inducing separation) causes the separation location to move upstream of (inviscid) impingement until mass flow balance of the separation bubble is reached; mass entering the bubble due to pressure rise at reattachment is balanced by mass leaving the bubble through mixing (shear) layers. While the overall geometry of the separated flow field is governed by the impinging shock strength, the flow in the onset of separation is governed by the free-interaction theory (Chapman et al. 1958).

Figure 22. Flow separation at a downstream location from the leading edge. The separated flow field is shown for Mach $5.88$ and $31^{\circ }$ wedge at shock impingement location of 73 mm from the leading edge. Image taken from Sriram et al. (2016).

Another breakage of geometric similarity occurs when shock impingement is sufficiently close to the plate leading edge. For example, in the present experiments, for $X_{ir} < 40$ mm from the plate leading edge (see figures 15(d), 15(e) and 15( f)) the breakage of geometric similarity occurs. An increase in the separation shock angle and a Type II wedge shock–separation shock interaction is the characteristic associated with the breakage of the geometric similarity. The reason for this breakage is not clear, although experiments by Mahapatra & Jagadeesh (2009) hint at inlet unstart phenomena in impulse facilities. It was observed that, as the cowl plate length decreased (similar effect to the shock impingement moving close to the plate leading edge), a transition from Type I to Type II shock–shock interaction was observed. The present LESF model does not account for a Type-II wedge shock–separation shock interaction and hence cannot explain the separation bubble geometries for Type-II interactions.

5. Conclusion

In this work, we perform theoretical modelling and experiments of leading-edge separation over a plate wall caused by means of an impinging shock. The considered impinging shock strengths are an order of magnitude larger than the incipient pressures required to cause flow separation. The resultant separation bubbles are ‘large’ (Sriram & Jagadeesh 2014) and scale directly with the distance of shock impingement from the plate leading edge. The theoretical model is iteratively solved for a system of shocks, expansion fan and slip lines. The experiments use schlieren visualizations and surface pressure measurements in order to characterize the separation flow fields.

We first construct a LESF model which derives from the separated flow model proposed by Délery & Marvin (1986). The model describes a 2-D steady state inviscid flow field over a plate wall. The LESF model is bifurcated based on two reattachment flow fields: an oblique reattachment and a normal reattachment. Critical wedge angles $16^{\circ }\unicode{x2013}19^{\circ }$ are identified for free-stream Mach numbers 5–8, where neither an oblique nor a normal reattachment occurs. We speculate a near-sonic reattachment at around the critical wedge angle which bridges the gap between the two reattachments. Contrary to expectations, the LESF model predicts large scale and roughly symmetric separation bubbles for an oblique reattachment when compared with a normal reattachment. This is due to strong separation shocks as compared with the reattachment shocks in oblique reattachment situations. However, in normal reattachment situations the reattachment shocks are much stronger than the separation shocks which give rise to small scale and skewed separation bubbles. Through the LESF model, we provide rationale for two important questions: (a) why is a normal reattachment preferred for strong impinging shocks? and (b) why does the separation length, for strong impinging shocks, scale directly with the distance of shock impingement from the plate leading edge?

Next, we perform experiments of a wedge shock impinging near a plate wall leading edge. The wedge shock is generated by a $26.6^{\circ }$ wedge in a Mach $5.52$ flow. For a small range of shock impingement locations, the separations occur at the plate leading edge and are geometrically similar. The self-similarity is confirmed by the geometrical structure of the SWBLI, the pressures inside the separation bubble and the peak pressure over the plate wall remain nearly constant. The size of the separation bubble, separation bubble pressure, Type I wedge shock–separation shock interaction and other separated flow features predicted by the LESF model have a very good agreement with the experiments. Although, the model fails to predict curvature of the reattachment shock near the reattachment location, the overall flow structure nevertheless resembles a near normal reattachment as compared with an oblique reattachment. At intermediate wedge angles of $21.80^{\circ }$, the flow fields have a good agreement with an oblique reattachment. The geometric self-similarity breaks when shock impingements occur sufficiently close to or sufficiently far away from the plate leading edge, which has been reported in the literature. The reasons for this are attributed to the inlet unstart phenomena in impulse facilities and the free-interaction phenomenon, respectively, which influences the separated flowfield.

The prediction of separated flow fields in a hypersonic free stream has proven challenging owing to the complex interaction patterns formed between the impinging shocks and boundary layers. However, when a strong shock impingement causes leading-edge separation, typical in scramjet engine inlets, the resultant separated flow field can be modelled as inviscid and therefore the separation bubble characteristics can be well anticipated. The separated flow field happens to be viscosity independent since the complicated development of the boundary layer and its associated pressure variations from plate leading edge to separation location are completely avoided (Chapman et al. 1958). We find that the separated flow fields predicted by the LESF model has a very good agreement with the experiments which gives reasonable confidence in the assumptionsused.