2.1. Inviscid criteria

The von Neumann and detachment criteria are illustrated by shock polar lines, as shown in figure 4($a$). If the reflected shock polar line and incident shock polar line intersect at point A, then the flow deflection angle is the von Neumann criterion $\alpha _{vn}$, which can be obtained from the pressure balance equation:

(2.1)\begin{equation} f_p(M_\infty,{\rm \pi}/2)=f_p(M_\infty,\beta_{vn})f_p(M_{vn},\beta_{vnr}), \end{equation}

where $\beta _{vn}$ is the incident shock angle, $M_{vn}$ is the Mach number behind the incident shock and $\beta _{vnr}$ is the local reflected shock angle, as shown in figure 4($b$). The variables fulfil the following equations:

(2.2)\begin{equation} \left. \begin{gathered} f_\alpha(M_\infty,\beta_{vn})=f_\alpha(M_{vn},\beta_{vnr}),\\ {M_{vn}}^{2}\sin^{2}(\beta_{vn}-\alpha_{vn})=f_M(M_\infty,\beta_{vn}),\\ \tan \alpha_{vn}=f_\alpha (M_\infty,\beta_{vn}), \end{gathered} \right\} \end{equation}

where $f_p$, $f_\alpha$ and $f_M$ are functions expressed as follows:

(2.3)\begin{equation} \left. \begin{gathered} f_p(M,\beta)=\frac{2\gamma}{\gamma+1}M^{2}\sin^{2}\beta-\frac{\gamma-1}{\gamma+1},\\ f_\alpha(M,\beta)=\frac{2M^{2}\sin^{2}\beta-2}{[2+M^{2}(\gamma+1-2\sin^{2}\beta)]\tan\beta},\\ f_M(M,\beta)=\frac{2+(\gamma-1)M^{2}\sin^{2}\beta}{2\gamma M^{2}\sin^{2}\beta-\gamma+1}, \end{gathered} \right\} \end{equation}

Figure 4. Illustration of the shock reflection criteria: ($a$) shock polar lines, ($b$) inviscid RR, ($c$) inviscid MR and ($d$) MR considering separation shock.

The solution of $\alpha _{vn}$ does not exist in the Mach number range of $M_\infty < M_c$, where $M_c$ is a critical Mach number, with $M_c\approx 2.2$. If the reflected shock polar line is tangent to the $y$-axis at point B, as shown in figure 4($a$), then the flow deflection angle is the detachment criterion $\alpha _D$, which can be obtained from the following equations:

(2.4)\begin{equation} \left. \begin{gathered} {M_{D}}^{2}\sin^{2}(\beta_{D}-\alpha_{D})=f_M(M_\infty,\beta_{D}),\\ f_\alpha(M_\infty,\beta_D)=f_\alpha(M_D,\beta_{DM}), \end{gathered} \right\} \end{equation}

where $\beta _D$ is the incident shock angle and $M_D$ is the Mach number behind the incident shock, as shown in figure 4($c$). In addition, $\beta _{DM}$ is a critical shock angle for the attached shock, which can be obtained from the following equation (see Li & Ben-Dor Reference Li and Ben-Dor1996a):

(2.5)\begin{align} \sin^{2}\beta_{DM}=\frac{1}{\gamma{M_D}^{2}} \left\{\frac{\gamma+1}{4}{M_D}^{2}-1 +\left[(\gamma+1)\left(1+\frac{\gamma-1}{2}{M_D}^{2}+\frac{\gamma+1}{16}{M_D}^{4}\right)\right]^{0.5}\right\}. \end{align}

2.2. Criteria considering viscosity

The current study focuses on the shock reflections near the leading edge shown in figure 1. Because separation at a sharp leading edge differs fundamentally from separation of a boundary layer from a flat surface, on the one hand, FIT is employed to approximate the separation shock strength when the flow separates from the boundary layer, as shown in figure 1($b$); on the other hand, the MEP principle is employed to approximate the separation shock strength when the flow separates from the leading edge, as shown in figure 1($c$). Following the previous description of FIT, the separation shock strength can be computed by (1.1), the subsequent section mainly discusses the application of the MEP principle.

Because the MEP principle is proposed based on the assumption of asymmetric shock–shock interaction, the theoretical model needs to be established for a flow field with a separation bubble moving to a very upstream position and the leading-edge shock being totally replaced by separation shock. Assuming that the influences of the boundary layer and leading-edge shock are sufficiently weak compared with the separation bubble, the shock reflection mainly interacts with incident shock $i$, a known shock generated by a wedge, and separation shock $i_s$ an unknown shock on the bottom plate. If incident shock $i$ remains stable, as shown in figure 5($a$), and the separation shock angle increases from $\beta _s$ to $\beta _s^{\prime }$, then the RR solution on the shock polar line should move from $\alpha _r$ to $\alpha _r^{\prime }$, accompanied by changes in reflected shocks $r_s$ to $r_s^{\prime }$ and $r$ to $r^{\prime }$, whereas in figure 5($b$), because the reflected shock polar lines of the incident shock and separation shock have no intersection, the increase in $\beta _s$ to $\beta _s^{\prime }$ only affects the MR solution of $\alpha _{r2}$ by changing $r_s$ to $r_s^{\prime }$.

Figure 5. Influences of the separation shock strength on the reflected shock strength: ($a$) RR and ($b$) MR.

Because, in the MR configuration, the polar lines show that $\alpha _{r2}$ does not affect $\alpha _{r1}$, the whole flow field can be treated as three flow fields: Flow I, which is an inviscid flow determined by the Mach number and incident shock; Flow II, which is a very strong curved shock; and Flow III, which is a separation flow determined by the Mach number and separation shock, as shown in figure 6($a$). A non-dimensional factor $\ddot {S}$ (see Wang et al. Reference Wang, Xue and Cheng2018; Xue et al. Reference Xue, Schrijer, van Oudheusden, Wang, Shi and Cheng2020) is employed to measure the total entropy production, as follows

(2.6)\begin{equation} \ddot{S}=\frac{\dot{S}}{C_v(\gamma-1)\rho_\infty M_\infty l\sqrt{\gamma R T_\infty}}, \end{equation}

where the variables in the denominator correspond to the incoming flow conditions, $C_v$ is the specific heat capacity at constant volume and $\dot {S}$ is the total entropy production (see Li & Ben-Dor Reference Li and Ben-Dor1996a), such that

(2.7)\begin{equation} \dot{S}=\int_l \rho u\Delta s\,\textrm{d}y, \end{equation}

where $\Delta s$ is the local entropy production of flow crossing a shock wave, and is given by

(2.8)\begin{equation} \Delta s={-}C_v(\gamma-1)\ln(p_{02}/p_{01}). \end{equation}

In (2.8) $p_{01}$ and $p_{02}$ are the total pressures ahead and behind the shock wave, respectively. Assuming that the separation shock is a straight shock at the leading edge, the entropy factor $\ddot {S}_{MR}$ can be derived as follows:

(2.9)\begin{equation} \ddot{S}_{MR}={-}\,f_{\ddot{S}}(M_\infty,\beta_s)\left[1+\frac{f_{\ddot{S}}(M_s,\beta_{sr})}{\ln \,f_{p0}(M_\infty,\beta_s)}\right], \end{equation}

where $\beta _s$ is the separation shock angle, $M_s$ is the Mach number behind the separation shock and $\beta _{sr}$ is the local reflected shock angle of the separation shock, as shown in figure 4($d$). The function $\,f_{\ddot {S}}$ is derived as follows:

(2.10)\begin{equation} \,f_{\ddot{S}}(M,\beta)=\frac{\,f_{\rho}(M,\beta)\ln \,f_{p0}(M,\beta)\sqrt{f_M(M,\beta)f_T(M,\beta)}}{M\sin\beta\cos[\beta-\tan^{{-}1}\,f_{\alpha}(M,\beta)]}, \end{equation}

where $f_M$, $f_\alpha$ and $f_p$ have been given by (2.3), and $f_\rho$, $f_T$ and $\,f_{p0}$ are expressed as follows:

(2.11)\begin{equation} \left. \begin{gathered} \,f_{\rho}(M,\beta)=\frac{(\gamma+1)M^{2}\sin^{2}\beta}{(\gamma-1)M^{2}\sin^{2}\beta+2},\\ f_T(M,\beta)=\frac{f_p(M,\beta)}{\,f_{\rho}(M,\beta)},\\ \,f_{p0}(M,\beta)=[f_p(M,\beta)]^{-({1}/({\gamma-1}))} [\,f_{\rho}(M,\beta)]^{{\gamma}/({\gamma-1})}. \end{gathered} \right\} \end{equation}

Figure 6. Simplified flow configurations and polar lines of possible solutions of separation shock: ($a$) flow configurations and ($b$) polar lines.

In addition, $M_s$ and $\beta _{sr}$ fulfil the pressure balance equation, and we write

(2.12)\begin{equation} f_p(M_\infty,\beta_{srs})=f_p(M_\infty,\beta_s)f_p(M_s,\beta_{sr}), \end{equation}

where $\beta _{srs}$ is the strong shock angle of the Mach stem close to the interaction point, which fulfils the following equations:

(2.13)\begin{equation} \left. \begin{gathered} {M_s}^{2}\sin^{2}[\beta_s-\tan^{{-}1}f_\alpha(M_\infty,\beta_s)]=f_M(M_\infty,\beta_s),\\ \tan^{{-}1}f_\alpha(M_\infty,\beta_s)+\tan^{{-}1}f_\alpha(M_\infty,\beta_{srs}) =\tan^{{-}1}f_\alpha(M_s,\beta_{sr}). \end{gathered} \right\} \end{equation}

If the possible flow deflection angle $\alpha _s$ of separation flow resides in the range of 0 to $\alpha _M$, as shown in figure 6($b$), the possible solution of reflected shock $\alpha _{r2}$ ranges from $-\alpha _M$ to $\alpha _M$. Because the Mach stem is a very strong shock, it can be treated as a normal shock for computing the entropy, which is only related to the incoming Mach number. Notably, the incident shock is a known shock, and the separation shock angle $\beta _s$ depends on $M_\infty$ and $\alpha _s$. Thus, the relation between $\ddot {S}_{MR}$ and $\alpha _s$ is established, based on which, the flow deflection angle $\alpha _s$ can be determined by $\ddot {S}_{MR}=minimum$ according to the MEP principle:

(2.14)\begin{equation} \left. \begin{gathered} \frac{\partial\ddot{S}_{MR}}{\partial\alpha_s}=0,\\ \frac{\partial^{2}\ddot{S}_{MR}}{\partial\alpha_s^{2}}\geq 0. \end{gathered} \right\} \end{equation}

The entropy factor line is helpful for understanding the prediction of the deflection angle in separation flow by the MEP principle. Three examples of $M_\infty =2$, 2.5 and 3 are shown in figure 7, in which relations between flow deflection angle $\alpha _s$ and entropy factor $\ddot {S}_{MR}$ are illustrated. The three minimum values of $\ddot {S}_{MR}$ which fulfil (2.14) for $M_\infty =2$, 2.5 and 3 exist at $\alpha _s=14.4^{\circ }$, $16.4^{\circ }$ and $17.1^{\circ }$, respectively, corresponding to $\beta _s=44.6^{\circ }$, $38.5^{\circ }$ and $34.5^{\circ }$. Comparing the three angles to the separation shock angles close to the interaction point of MR measured on the pictures in several recent studies: in the study of Tao et al. (Reference Tao, Fan and Zhao2014), the measured angle is $\beta _s\approx 34^{\circ }$ for $M_\infty =3$ and in the study of Matheis & Hickel (Reference Matheis and Hickel2015), the measured angles are $\beta _s\approx 41^{\circ }$ for $M_\infty =2$ and $\beta _s\approx 34^{\circ }$ for $M_\infty =3$; in the study of Grossman & Bruce (Reference Grossman and Bruce2018), the measured angle is $\beta _s\approx 42^{\circ }$ for $M_\infty =2$; and in the study of Xue et al. (Reference Xue, Schrijer, van Oudheusden, Wang, Shi and Cheng2020), the measured angles are $\beta _s\approx 43^{\circ }$ for $M_\infty =2$ and $\beta _s\approx 37^{\circ }$ for $M_\infty =2.5$. The comparisons are summarised in table 1. It is observed that the theoretical results of the MEP principle are close to the results in the literature. However, the angles in the literature are smaller than the current results because all the shock reflections in the literature were strongly affected by the boundary layer, which did not fulfil the assumption of the MEP principle.

Figure 7. Comparison between current theoretical results and those in the literature: ($a$) $M_\infty =2$ (see Xue et al. Reference Xue, Schrijer, van Oudheusden, Wang, Shi and Cheng2020), ($b$) $M_\infty =2.5$ (see Xue et al. Reference Xue, Schrijer, van Oudheusden, Wang, Shi and Cheng2020) and ($c$) $M_\infty =3$ (see Matheis & Hickel Reference Matheis and Hickel2015).

Table 1. Comparison between current theoretical results of separation shock angle in MR and those measured in the literature.

Because the incident shock–separation shock interaction shown in figure 1($c$) could be assumed to be an asymmetric shock–shock interaction, a detachment criterion for the RR-to-MR transition of the asymmetric shock–shock interaction can be determined (see Li, Chpoun & Ben-Dor Reference Li, Chpoun and Ben-Dor1999). The theoretical lines, including the von Neumann criterion $\alpha _{vn}$, detachment criterion $\alpha _D$ and the MEP criterion $\alpha _D^{\prime }$ lines, are summarised in figure 8. A critical Mach number $M_\infty \approx 2.2$ can be easily observed. On the one hand, when $M_\infty <2.2$, the von Neumann criterion $\alpha _{vn}$ is not available, while the detachment criterion $\alpha _D$ is larger than MEP criterion $\alpha _D^{\prime }$, which means that MR might exist in $\alpha _D^{\prime }<\alpha <\alpha _D$. On the other hand, when $M_\infty >2.2$, $\alpha _D$ is smaller than $\alpha _D^{\prime }$, meaning that RR could exist in $\alpha _D<\alpha <\alpha _D^{\prime }$ and the RR-to-MR transition possibly occurs at $\alpha >\alpha _D$. For high Mach numbers, the distinction between $\alpha _D$ and $\alpha _D^{\prime }$ increasingly grows, and the flow deflection angle might be very large when the transition occurs, while much less information is available in the literature for the RR-to-MR transition on a plate in hypersonic flows. Therefore, the current experiments are performed in $M_\infty =5$, 6, 7 and 8 flows, of which the experimental results agree well with the theoretical results, and the flow deflection angle of the RR-to-MR transition is very close to the MEP criterion $\alpha _D^{\prime }$, as shown in figure 8. The current experiments are discussed in the following section.

Figure 8. Comparison between theoretical lines and experimental results.

3.1. Experimental set-up

Experiments were performed in the NHW hypersonic wind tunnel of Nanjing University of Aeronautics and Astronautics. The wind tunnel runs in a ‘blow-down-to-vacuum’ mode, which can be used to perform tests at 0.04–1.0 MPa total pressure, 288–685 K plenum temperature and $6.47\times 10^{5}$–$2.24\times 10^{7}\ \textrm {m}^{-1}$ unit Reynolds number with 7–10 s of running time (see Wang, Xue & Tian Reference Wang, Xue and Tian2017). The converging–diverging nozzle (exit diameter of 500 mm) mounted in the test section is interchangeable and provides nominal free-stream Mach numbers from 4 to 8. Two glass windows (diameter of 300 mm) are embedded in two sides of the test section walls for optical access. In addition, nozzles for Mach numbers 5, 6, 7 and 8 are employed in the current experiments.

The test model is a rotatable wedge ($40^{\circ }$ apex angle and 140 mm width) over a surface plate (160 mm width and 700 mm length) with a sharp leading edge ($15^{\circ }$ apex angle) for generating incident shock and separation shock, in which the flow deflection angle can be continuously changed from $0^{\circ }$ to $40^{\circ }$ through a high-precision stepping motor, as shown in figure 9. Nine Kulite XTEL-190M fast-response transducers are mounted along the central line of the surface plate and wedge bottom, which acquire data at a rate of 50 kHz with a 10 s sampling time using data acquisition cards. An NAC (NAC Image Technology) Hotshot high-speed camera operating at a frame rate of 5 kHz with a 6 s sampling time and a resolution of $600\times 438$ pixels is employed to take schlieren images.

Figure 9. Test model: ($a$) real model in a wind tunnel and ($b$) illustration of the rotatable wedge.

Because Mach waves will be propagating inward and reducing the region of the interaction that can be considered two-dimensional flow, as shown in figure 9($b$), $\bar {W}$ is defined to represent the percentage of two-dimensional flow. In addition, two non-dimensional variables $\bar {H}$ and $\bar {X}$ are defined to describe the vertical and horizontal distances between the wedge and the plate, respectively:

(3.1)\begin{equation} \left. \begin{gathered} \bar{W}=\Delta w/W_{plate},\\ \bar{H}=\Delta h/W_{wedge},\\ \bar{X}=\Delta x/W_{wedge}, \end{gathered} \right\} \end{equation}

where $\Delta w$, $\Delta h$, $\Delta x$, $W_{plate}$ and $W_{wedge}$ are marked in figure 9($b$). The shock impingement location on the plate is not in the same position owing to the continuously increasing incident shock angle during one test, thus $\bar {W}$ changes from a lower value to a higher one with the interaction region moving upstream.

The current experiments were performed in a low-enthalpy wind tunnel (the total temperature was set at $600\pm 15\ \textrm {K}$ for each test), thus the real gas effects could be neglected. The free-stream turbulence intensities of the wind tunnel for Mach number 5, 6, 7 and 8 are 0.022, 0.023, 0.025 and 0.025, respectively. More flow conditions are summarised in table 2. An algorithm program for schlieren image quantisation based on the grey level (see Xue, Wang & Cheng Reference Xue, Wang and Cheng2018) is employed to compute the separation shock angles, and experimental uncertainties are characterised by the standard deviation. The experimental uncertainties are contributed by various factors, e.g. the free-stream turbulence, white noise and the possible error of image quantisation (affected by image quality).

Table 2. Flow conditions of the experiments.

3.2. Evolution of the RR-to-MR transition near the leading edge

The evolution of the flow field with increasing incident shock angle is shown in figure 10, where the free-stream Mach number is $M_\infty =5$, and the unit Reynolds number is $Re=7.02\times 10^{6}\ \textrm {m}^{-1}$, $\bar {H}=0.464$ and $\bar {X}=0.286$. Eight pressure transducers named T1 to T8 are mounted on the plate in the interaction region, and one named T0 is mounted on the wedge bottom behind the incident shock. The transducer positions are marked in the schlieren images, as shown in figure 10. The shock reflection is undoubtedly affected by the leading-edge shock and boundary layer when $\alpha =25^{\circ }$, which corresponds to the flow pattern shown in figure 1($b$). The separation shock angle $\beta _{s.Exp}\approx 16.1^{\circ }\pm 0.6^{\circ }$ obtained from schlieren images is between $\beta _{s.FIT0}\approx 15.2^{\circ }$ and $\beta _{s.FIT1}\approx 17.6^{\circ }$ computed by (1.1) with $F(\bar {x})_{lam0}\approx 0.81$ and $F(\bar {x})_{lam1}\approx 1.47$, respectively, indicating that FIT is applicable well when the flow on the plate separates from the boundary layer. In the laminar separation region ahead of the shock impingement location, the static pressure on the plate, T1 to T4 shown in figure 10($a$), stay in low values and fluctuate with small amplitudes (small standard deviations), while the amplitudes grow distinctly at T5 to T8 with a sharp pressure increase trend, demonstrating that the shock impingement location reduces flow stability and damages the laminar boundary layer. The separation bubble moves continuously upstream with increasing $\alpha$, and the separation shock seems to interact with the leading-edge shock $l_e$ when $\alpha =30^{\circ }$, as shown in figure 10($c$). The shock impingement location also moves upstream, causing T1 to T4 increase in turn with relatively large amplitudes, and T5 to T8 decrease in turn with relatively small amplitudes. Finally, $l_e$ is totally replaced by $i_s$ for $\alpha >33^{\circ }$ because $i_s$ is a stronger shock wave than $l_e$. Then, the separation bubble moves to the most upstream position and replaces the boundary layer, which corresponds to the flow pattern shown in figure 1($c$). Meanwhile, the shock impingement location moves to the T1 and T2 positions, and leads to very large amplitudes of T1 and T2 pressures and relatively small amplitudes of T3 to T8, indicating that a reattached flow follows the interaction region. The RR-to-MR transition occurs at $\alpha \approx 36.1^{\circ }$ with the appearance of a very short Mach stem $m$.

Figure 10. Evolution of the flow field with increasing incident shock angle ($M_\infty =5$, $Re=7.02\times 10^{6}\ \textrm {m}^{-1}$, $\bar {H}=0.464$ and $\bar {X}=0.286$): ($a$) $\alpha =24^{\circ }$, ($b$) $\alpha =28^{\circ }$, ($c$) $\alpha =30^{\circ }$, ($d$) $\alpha =33^{\circ }$, ($e$) $\alpha =35^{\circ }$, ($f$) $\alpha =36^{\circ }$, ($g$) $\alpha =37^{\circ }$, ($h$) $\alpha =38^{\circ }$ and ($i$) $\alpha =39^{\circ }$.

The comparisons between the theoretical results and experiments are summarised in figure 11. As shown in figure 11($a$), the minimum value of $\ddot {S}_{MR}$ exists at $\alpha _{s.MEP}\approx 16.4^{\circ }$, corresponding to $\beta _{s.MEP}\approx 25.8^{\circ }$, which agrees well with the separation shock angle $\beta _{s.Exp}\approx 25.2^{\circ }\pm 0.9^{\circ }$ obtained from the schlieren images. Figure 11($b$) demonstrates that the separation shock angle in MR is distinctly larger than that in RR, and that the theoretical model of FIT is applicable to flow pattern before the separation bubble moves to the leading edge while the MEP principle is applicable to the flow pattern after the separation bubble moves to the leading edge.

Figure 11. Comparison between theoretical results and experiments ($M_\infty =5$, $Re=7.02\times 10^{6}\ \textrm {m}^{-1}$, $\bar {H}=0.464$ and $\bar {X}=0.286$): ($a$) entropy factor line and ($b$) shock polar lines.

Figures 12 and 13 show the comparison between RR-to-MR and MR-to-RR transitions. Based on the schlieren images, both transitions occur at $\alpha \approx 36.1^{\circ }$, and shock reflections are almost characterised by the same configurations when the test model generates the same flow deflection angle. Figure 13 gives the time histories of pressures by Kulite transducers and corresponding data analysis with $\alpha$ increasing and decreasing. During the test, the wedge angle changes almost linearly, from $24^{\circ }$ to $40^{\circ }$ or from $40^{\circ }$ to $24^{\circ }$ within 2 s. As shown in figure 13($a$), the incident shock strength, represented by T0, increases almost linearly, driving the separation region to move upstream and downstream, resulting in gradual changes for T1 to T8. When the transducers experience upstream movement of the separation bubble, the pressures increase in turn and then decrease in turn, with the peaks growing higher. When the transducers experience downstream movement of the separation bubble, the reversible process is nearly symmetric. Therefore, the shock reflection of the RR-to-MR or MR-to-RR transition near the leading edge does not present a distinct hysteresis.

Figure 12. Evolution of the flow field with $\alpha$ increasing and decreasing ($M_\infty =5$, $Re=7.02\times 10^{6}\ \textrm {m}^{-1}$, $\bar {H}=0.464$ and $\bar {X}=0.286$): ($a$) $\alpha =35^{\circ }$, ($b$) $\alpha =36^{\circ }$, ($c$) $\alpha =36.1^{\circ }$, ($d$) $\alpha =37^{\circ }$, ($e$) $\alpha =38^{\circ }$, ($f$) $\alpha =38^{\circ }$, ($g$) $\alpha =37^{\circ }$, ($h$) $\alpha =36.1^{\circ }$, ($i$) $\alpha =36^{\circ }$ and ($j$) $\alpha =35^{\circ }$.

Figure 13. Static pressure on plate and corresponding data analysis with $\alpha$ increasing and decreasing ($M_\infty =5$, $Re=7.02\times 10^{6}\ \textrm {m}^{-1}$, $\bar {H}=0.464$ and $\bar {X}=0.286$): ($a$) time histories of Kulite transducers (T0 to T8), ($b$) time–frequency map of wavelet transform (WT) for illustrating PSD and ($c$) time–position map for illustrating standard deviation (STDEV).

Because transducer T1 experiences the separation bubble and the shock impingement location during 1.8 s to 3.6 s, the amplitude is distinctly larger than those of T2 to T8, which have been residing in the downstream reattached flow. Notably, although the amplitude of T1 is very large with noise (the pressure sampling frequency is $5\times 10^{4}\ \textrm {Hz}$), no distinct dominant frequency occurs below $10^{4}\ \textrm {Hz}$ according to the fast Fourier transform. From the power spectral density (PSD) map, as shown in figure 13($b$), the pressure of T1 in laminar separation region shows no obvious frequency component when $t<1.8\ \textrm {s}$ and $t>3.6\ \textrm {s}$, and T1 mainly fluctuates during 1.8 s to 3.6 s with most of the frequency components evenly spread in the range of 1 to $10^{4}\ \textrm {Hz}$, indicating that the laminar flow might be damaged by the shock impingement during 1.8 s to 3.6 s. In addition, there is no large scale shock oscillation observed in the flow configuration, thus the separation bubble and separation shock are relatively steady.

The pressure fluctuations of the transducers are related to free-stream turbulence and white noise, while the large amplitudes are mostly affected by the movement of the shock impingement location, which can be observed from the standard deviation of time–position map shown in figure 13($c$). Obviously, the movement of the shock impingement location causes a large pressure amplitude, which grows distinctly larger (T2 position) with the RR-to-MR transition ($\alpha \approx 36^{\circ }$) when the shock impingement location moves upstream, and the reversible process shows the opposite trend. Therefore, the flow instability depends on the location of the incident shock impingement, and it may be increased by the RR-to-MR transition. Because the RR-to-MR transition agreed well with MEP principle, the criterion based on this principle might be useful for predicting the instability of incident shock–separation shock interaction near the leading edge.

3.3. The influences of wedge positions

To investigate the influences of wedge positions on shock reflections, the following tests were conducted at $M_\infty =5$ and $Re=2.2\text {--}2.6\times 10^{6}\ \textrm {m}^{-1}$ with $\bar {X}$ decreasing from 0.429 to $-0.143$, as shown in figure 14, and $\bar {H}$ increasing from 0.179 to 0.607, as shown in figure 15. It can be observed from figures 14($a1$), 14($b1$), 14($c1$), 14($d1$) and 14($e1$) that the separation shock angles change in a small range before $l_e$ coincides with $i_s$, indicating that the separation shock strength exerts weak effects by decreasing $\bar {X}$. In the cases of $\bar {X}=0.429$, $\bar {X}=0.286$ and $\bar {X}=0.143$, RR-to-MR transitions occur at $\alpha \approx 35.7^{\circ }\pm 0.4^{\circ }$, whereas transitions do not occur in the cases of $\bar {X}=0$ and $\bar {X}=-0.143$. Figures 14($d2$) and 14($e2$) show that the separation bubble does not reach the most upstream position and that $l_e$ is not replaced by $i_s$ when $\alpha =35^{\circ }$, meaning that the flows on the plate still separate from the boundary layer rather than the leading edge. Thus, the separation shock is not strong enough to trigger an RR-to-MR transition. A similar phenomenon can be observed from figure 15($d2$) in which the RR-to-MR transition does not occur in the case of $\bar {H}=0.607$ because the separation bubble does not reach the most upstream position. Figures 15($a2$) and 15($a3$) show that the occurrence of the RR-to-MR transition is replaced by an unstart flow field because the model generates too much compression when $\bar {H}$ decreases to 0.179.

Figure 14. Influence of $\bar {X}$ on shock reflections ($M_\infty =5$, $Re=2.2\text {--}2.6\times 10^{6}\ \textrm {m}^{-1}$ and $\bar {H}=0.464$): ($a1$–$a3$) $\bar {X}=0.429$, ($b1$–$b3$) $\bar {X}=0.286$, ($c1$–$c3$) $\bar {X}=0.143$, ($d1$–$d3$) $\bar {X}=0$ and ($e1$–$e3$) $\bar {X}=-0.143$.

Figure 15. Influence of $\bar {H}$ on shock reflections ($M_\infty =5$, $Re=2.2\text {--}2.6\times 10^{6}\ \textrm {m}^{-1}$ and $\bar {X}=0.286$): ($a1$–$a3$) $\bar {H}=0.179$, ($b1$–$b3$) $\bar {H}=0.321$, ($c1$–$c3$) $\bar {H}=0.464$ and ($d1$–$d3$) $\bar {H}=0.607$.

Figures 14 and 15 demonstrate that the wedge positions strongly affect the shock reflections on the plate near the leading edge. The RR-to-MR transition does not occur with a relatively small $\bar {X}$ or high $\bar {H}$ because the separation bubble cannot reach the leading-edge position. This can be explained by the theoretical models. On one hand, FIT is applicable when flow on the plate separates from the boundary layer, and the separation shock strength of the FIT result is too weak to trigger an RR-to-MR transition, on the other hand, the separation shock strength of the MEP result is strong enough, while the MEP principle is applicable only if the flow on the plate separates from the leading edge. Therefore, the RR-to-MR transition is much harder to observe when flow separates from the boundary layer rather than from the leading edge.

3.4. The influences of Reynolds and Mach numbers

To investigate the influences of Reynolds and Mach numbers on shock reflections, further tests were conducted at $\bar {H}=0.464$ and $\bar {X}=0.286$ with $Re$ increasing from $1.77\times 10^{6}\ \textrm {m}^{-1}$ to $6.55\times 10^{6}\ \textrm {m}^{-1}$, as shown in figure 16, and $M_\infty$ increasing from 5 to 8, as shown in figure 17. It can be observed from figure 16 that when flow separates from the boundary layer, the separation shock angle is $\beta _{s.Exp}\approx 16.4^{\circ }\pm 0.5^{\circ }$, while it increases to $\beta _{s.Exp}\approx 24.9^{\circ }\pm 0.6^{\circ }$ with the RR-to-MR transitions occurring at $\alpha \approx 35.8^{\circ }\pm 0.5^{\circ }$. Thus, the separation shock angles in the configurations, before and after the separation bubble moves to the leading-edge position, are not as sensitive to the Reynolds number. Figure 17 shows that the separation shock angle decreases with increasing Mach number, and for each Mach number, the separation shock angle of flow separating from the leading edge is distinctly larger than that from the boundary layer. Therefore, the Mach number exerts a much stronger influence on shock reflection near the leading edge than the Reynolds number. The flow deflection angles $\alpha \approx 35.8^{\circ }\pm 0.7^{\circ }$, $37.8^{\circ }\pm 0.7^{\circ }$, $38.1^{\circ }\pm 0.6^{\circ }$ and $38.9^{\circ }\pm 0.6^{\circ }$ of the RR-to-MR transitions for $M_\infty =5$, 6, 7 and 8 agree well with the MEP criterion angles of $\alpha \approx 34.9^{\circ }$, $37.1^{\circ }$, $37.5^{\circ }$ and $38.2^{\circ }$ based on the MEP results, respectively. The RR-to-MR transitions are delayed in turn with increasing Mach number, and the transition does not occur before the separation bubble moves to the leading-edge position. Figure 18 gives a comparison between the theoretical and experimental results, demonstrating that the separation shock angle in the MR configuration for each Mach number agrees well with the MEP result. Because the MEP of shock reflection corresponds to the maximum total pressure recovery, the separation flow seems to choose a configuration that produces the smallest possible flow loss.

Figure 16. Influence of $Re$ on shock reflections ($M_\infty =5$, $\bar {H}=0.464$ and $\bar {X}=0.286$): ($a1$–$a3$) $Re=1.77\times 10^{6}\ \textrm {m}^{-1}$, ($b1$–$b3$) $Re=2.32\times 10^{6}\ \textrm {m}^{-1}$, ($c1$–$c3$) $Re=3.97\times 10^{6}\ \textrm {m}^{-1}$, ($d1$–$d3$) $Re=5.41\times 10^{6}\ \textrm {m}^{-1}$ and ($e1$–$e3$) $Re=6.55\times 10^{6}\ \textrm {m}^{-1}$.

Figure 17. Influence of $M_\infty$ on shock reflections ($\bar {H}=0.464$ and $\bar {X}=0.286$): ($a1$–$a3$) $M_\infty =5$, $Re=6.55\times 10^{6}\ \textrm {m}^{-1}$, ($b1$–$b3$) $M_\infty =6$, $Re=4.95\times 10^{6}\ \textrm {m}^{-1}$, ($c1$–$c3$) $M_\infty =7$, $Re=2.29\times 10^{6}\ \textrm {m}^{-1}$ and ($d1$–$d3$) $M_\infty =8$, $Re=1.38\times 10^{6}\ \textrm {m}^{-1}$.

Figure 18. Comparison between theoretical lines and experimental results: ($a$) $M_\infty =6$, ($b$) $M_\infty =7$ and ($c$) $M_\infty =8$.