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. Reference Chapman, Kuehn and Larson1958; Délery & Marvin Reference Délery and Marvin1986).

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 (Reference Edney1968) 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. (Reference Chapman, Kuehn and Larson1958) 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. (Reference Chapman, Kuehn and Larson1958). The theory by Chapman et al. (Reference Chapman, Kuehn and Larson1958) 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 (Reference Délery and Marvin1986), 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 (Reference Sriram2013), Sriram & Jagadeesh (Reference Sriram and Jagadeesh2014, Reference Sriram and Jagadeesh2015), Srinath (Reference Srinath2015) and Sriram et al. (Reference Sriram, Srinath, Devaraj and Jagadeesh2016). 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. Reference Liepmann, Roshko and Dhawan1951; Gadd et al. Reference Gadd, Holder and Regan1954; Chapman et al. Reference Chapman, Kuehn and Larson1958; Hakkinen et al. Reference Hakkinen, Greber, Trilling and Abarbanel1959; Brower Reference Brower1961; Stollery & Bates Reference Stollery and Bates1974; Délery & Marvin Reference Délery and Marvin1986; Katzer Reference Katzer1989; Délery & Dussauge Reference Délery and Dussauge2009). 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 Reference Mallinson, Gai and Mudford1997; Matheis & Hickel Reference Matheis and Hickel2015; Verma & Chidambaranathan Reference Verma and Chidambaranathan2015).

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 (Reference Brower1961) 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. (Reference Chapman, Kuehn and Larson1958), Brower (Reference Brower1961) and Srinath, Sriram & Jagadeesh (Reference Srinath, Sriram and Jagadeesh2017) 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. Reference Sriram, Srinath, Devaraj and Jagadeesh2016) 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 $\theta _{w}$.

4.1. 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 Reference Hayes and Probstein1959) 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 Reference Délery and Marvin1986) 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 (Reference Sriram and Jagadeesh2014, Reference Sriram and Jagadeesh2015) and Sriram et al. (Reference Sriram, Srinath, Devaraj and Jagadeesh2016).

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. (Reference Sriram, Srinath, Devaraj and Jagadeesh2016). 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. (Reference Sriram, Srinath, Devaraj and Jagadeesh2016) 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. Reference Sriram, Srinath, Devaraj and Jagadeesh2016).

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 Reference Délery and Marvin1986), 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. Reference Chapman, Kuehn and Larson1958).

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. (Reference Sriram, Srinath, Devaraj and Jagadeesh2016).

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 (Reference Mahapatra and Jagadeesh2009) 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.