4.1. General flow features

Figure 3 shows the 2-D steady distributions of skin friction coefficient for different ramp angles and wall temperature ratios. Also shown in the figure are the enlarged views near the corner. The separation and reattachment points are indicated by open circles. For a fixed T_w/T ₀, the separation and reattachment points move upstream and downstream, respectively, as the ramp angle is increased. Inside the separation region, there are two local minima of C_f. As reported by Smith & Khorrami (Reference Smith and Khorrami1991), Korolev, Gajjar & Ruban (Reference Korolev, Gajjar and Ruban2002) and Gai & Khraibut (Reference Gai and Khraibut2019), the magnitude of the second minimum increases with α. There is a local skin friction peak between the two minima near the corner, which also increases with α. Beyond a certain ramp angle, the local peak becomes positive, indicating the emergence of secondary separation. For T_w/T ₀ = 0.18, a secondary bubble occurs when α is around 13 °–14 °, whereas the critical value is approximately 14 °–15 ° for T_w/T ₀ = 0.54 and 0.86. In general, the occurrence of the secondary separation is postponed to a larger ramp angle as the wall temperature is increased, which is consistent with the observation of Shvedchenko (Reference Shvedchenko2009). Downstream of the reattachment point, the skin friction rises to its peak value and then decreases gradually. Note that there is a local increase in C_f further downstream for T_w/T ₀ = 0.18, which is caused by the impingement of EW induced by the shock interaction between SS and RS (see figure 1).

Figure 3. Distributions of skin friction coefficient for different ramp angles with the enlarged views near the corner: (a) T_w/T ₀ = 0.18; (b) T_w/T ₀ = 0.54; (c) T_w/T ₀ = 0.86. Open circles: separation and reattachment points. Blue, α = 11 °; green, α = 12 °; orange, α = 13 °; red, α = 14 °; black, α = 15 °.

Figure 4 shows the 2-D steady distributions of surface pressure coefficient for different ramp angles and wall temperature ratios. The surface pressure begins to increase upstream of the separation point controlled by the free-interaction process (Chapman, Kuehn & Larson Reference Chapman, Kuehn and Larson1958). The rise is followed by a plateau region, the value of which increases with α and T_w/T ₀. It is consistent with the free-interaction theory that the plateau pressure is inversely proportional to the Reynolds number based on the distance between the leading edge and separation point. The pressure rises again near the reattachment point and reaches its peak value mainly determined by the oblique shock theory. As the wall temperature is increased, the increase in C_p near reattachment becomes more gradual. For T_w/T ₀ = 0.18, the drop in C_p downstream of the peak is also caused by the impingement of EW. For relatively large ramp angles, there is a small ‘dip’ in the surface pressure between the plateau and peak. Similar features have also been observed by Smith & Khorrami (Reference Smith and Khorrami1991), Korolev et al. (Reference Korolev, Gajjar and Ruban2002) and Gai & Khraibut (Reference Gai and Khraibut2019). As will be seen later, the ‘dip’ is indicative of a local region with pressure gradient near the corner, which becomes stronger as α is increased such that the reverse flow boundary layer cannot resist it and gives rise to secondary separation.

Figure 4. Distributions of surface pressure coefficient for different ramp angles with the enlarged views near the corner: (a) T_w/T ₀ = 0.18; (b) T_w/T ₀ = 0.54; (c) T_w/T ₀ = 0.86. Open circles: separation and reattachment points. Blue, α = 11 °; green, α = 12 °; orange, α = 13 °; red, α = 14 °; black, α = 15 °.

Contours of 2-D density gradient magnitude and pressure (near the compression corner) non-dimensionalized by ${\rho _\infty }u_\infty ^2$ are plotted with the streamlines inside the separation region superimposed in figure 5 for T_w/T ₀ = 0.18. Only the flow fields with the ramp angles of 13 °, 14 °, and 15 ° are presented to focus on the emergence of secondary separation. There is no secondary separation at α = 13 °, and the core of the primary vortex is located above the ramp surface downstream of the corner. At α = 14 °, a secondary vortex occurs beneath the primary bubble near the corner. The secondary bubble grows in both horizontal and vertical extents as α is increased to 15 °, while the primary bubble fragments into two vortices. As demonstrated by Shvedchenko (Reference Shvedchenko2009), Egorov et al. (Reference Egorov, Neiland and Shvedchenko2011) and Gai & Khraibut (Reference Gai and Khraibut2019), further increasing the ramp angle leads to fragmentation into multiple vortices and eventually unsteadiness for 2-D laminar simulations.

Figure 5. Contours of density gradient magnitude (left column) and non-dimensional pressure (right column) with streamlines superimposed for T_w/T ₀ = 0.18: (a) α = 13 °; (b) α = 14 °; (c) α = 15 °. Closed circles: separation and reattachment points.

At α = 13 °, a low-pressure region can be seen in the vortex core, which balances the centrifugal force of a fluid element rotating about the core (Jeong & Hussain Reference Jeong and Hussain1995). It can be speculated that the ‘dip’ of surface pressure is the footprint of this low-pressure region on the ramp surface. This low-pressure region also stretches out of the separation region in the form of a weak expansion wave. As α is increased, the low pressure in the primary-vortex core only changes slightly; however, the pressure in the upstream half of the separation region is obviously elevated in accordance with the variation of the plateau pressure (see figure 4). Such a behaviour poses a local streamwise adverse pressure gradient to the reverse flow boundary layer developing from the reattachment point. With the emergence of primary-vortex fragmentation and secondary separation, multiple waves emanate from the separation region.

The corresponding non-dimensionalized pressure gradients in the x and y directions are compared in figure 6 for T_w/T ₀ = 0.18 and different ramp angles. The separation and reattachment points and the dividing streamline are also plotted to mark the extent of the separation region. As α in increased, the streamwise pressure gradient (p_x) inside the separation bubble near the corner becomes progressively intensified. Beyond a certain value, the reverse flow boundary layer separates to form a secondary bubble. Meanwhile, the emergence of secondary separation deforms the primary bubble, resulting in a transverse pressure gradient (p_y). Similar observations were made by Khraibut et al. (Reference Khraibut, Gai, Brown and Neely2017) for a hypersonic leading-edge separation.

Figure 6. Contours of non-dimensional streamwise (left column) and transverse (right column) pressure gradients with dividing streamlines superimposed for T_w/T ₀ = 0.18: (a) α = 13 °; (b) α = 14 °; (c) α = 15 °. Closed circles: separation and reattachment points.

Figure 7 shows the contours of density gradient magnitude and non-dimensionalized streamwise pressure gradient for the hot-wall cases at α = 15 °. For T_w/T ₀ = 0.54, the separation region is significantly enlarged compared with the cold-wall condition. However, the occurrence of the secondary bubble is postponed until α = 15 ° and no fragmentation of the primary bubble is observed. Meanwhile, the core of the secondary bubble is shifted downstream of the corner. For the adiabatic-wall condition (T_w/T ₀ = 0.86), only an incipient secondary separation can be seen. Clearly, the strength of the streamwise pressure gradient in the separation region is weakened by elevating the wall temperature, which is responsible for the delay of secondary separation.

Figure 7. Contours of density gradient magnitude (left column) and non-dimensional streamwise pressure gradient (right column) with streamlines superimposed at α = 15 °: (a) T_w/T ₀ = 0.54 and (b) T_w/T ₀ = 0.86. Closed circles: separation and reattachment points.

To closely examine the reverse flow boundary layer, wall-normal distributions of flow quantities are extracted through the core of the primary vortex located downstream of the corner for different wall temperatures at α = 15 °. Figure 8 plots the profiles of streamwise velocity (u) and its streamwise gradient (u_x) as a function of normal distance from the wall (y_n). The locations of the vortex core are marked by closed circles. There is no clear definition of reverse flow boundary layer. Herein, it is defined as the reverse flow region beneath the vortex core as seen in figure 8(a). A maximum reverse flow velocity can be observed between the vortex core and wall, which increases with increasing wall temperature. Above the vortex core, the rapid change in the streamwise velocity corresponds to the shear layer induced by the separation region. In figure 8(b), there are two local minima separated by the vortex core, representing strong deceleration in the shear layer and reverse flow boundary layer, respectively. The magnitude of the minimum in the reverse flow boundary layer is reduced with increasing wall temperature, which may result from the weakened streamwise pressure gradient.

Figure 8. Distributions of (a) streamwise velocity and (b) streamwise gradient of streamwise velocity extracted along the wall-normal direction through the core of the primary vortex as a function of distance from the wall for different wall temperatures at α = 15 °. Closed circles: vortex core.

4.2. Occurrence of global instability

To determine the global stability of the hypersonic compression corner flows with respect to periodic spanwise perturbations, GSA is performed with the 2-D steady solutions obtained in § 4.1 as the base flows over a wide range of spanwise wavelengths.

Figure 9 presents the growth rates of the least stable mode as a function of spanwise wavelength for different ramp angles and wall temperature ratios. Open symbols represent stationary modes with ω_r = 0, whereas closed symbols denote oscillating modes with ω_r ≠ 0. For T_w/T ₀ = 0.18, the flow system is globally stable to 3-D perturbations at α = 12 °, although not shown here. At α = 13 °, the separated flow becomes unstable with the largest growth rate occurring at λ/L = 0.168. The captured least stable mode is largely stationary, which is oscillating only for short spanwise wavelengths. As α is further increased to 14 °, the stationary unstable mode is shifted to a larger wavelength. Meanwhile, the growth rate of the oscillating mode increases significantly and dominates over its stationary counterpart. Essentially, they belong to the same branch of eigenvalues. As the spanwise wavelength is decreased, the pure imaginary eigenvalue corresponding to the stationary mode leaves the imaginary axis to form two conjugate eigenvalues. At α = 15 °, the conjugate eigenvalues merge again, resulting in a strongly unstable stationary mode at λ/L = 0.066. For T_w/T ₀ = 0.54 and 0.86, the flows are marginally stable when α = 13 ° and become unstable at α = 14 °. The behaviour of the most unstable mode is similar to that for the cold-wall condition, i.e. the stationary mode is shifted to a longer wavelength as α is increased and an unstable mode occurs at a shorter wavelength and becomes dominant.

Figure 9. Growth rates of the most unstable mode as a function of spanwise wavelength for different ramp angles: (a) T_w/T ₀ = 0.18; (b) T_w/T ₀ = 0.54; (c) T_w/T ₀ = 0.86. Open symbols: stationary. Closed symbols: oscillating modes.

For the cold-wall condition, the flow system is unstable when 12 ° < α < 13 °. For the hot-wall conditions, the critical ramp angles are 13 °–14 °. The GSA reveals that the global instability of the hypersonic compression corner flow is closely linked with the emergence of secondary separation. Elevating the wall temperature stabilizes the flow system, although the length of the separation region is considerably enlarged.

Figure 10 shows the contours of the real part of the spanwise velocity perturbation ŵ/u _∞ corresponding to the short-wavelength unstable modes at their respective most unstable wavelengths (i.e. λ/L = 0.066, 0.114, and 0.209) for different wall temperatures at α = 15 °. Note that the short-wavelength mode dominates over the long-wavelength mode as α is increased to a certain value. The corresponding eigenvalue spectra at these conditions are also plotted. Grid independence was verified by comparing the eigenvalues obtained on the coarse (600 × 300) and fine (800 × 400) grids for T_w/T ₀ = 0.18. The spanwise velocity perturbations are mainly confined to the separation region for different wall temperatures. In the downstream portion of the separation bubble, the spanwise velocity perturbations are highly distorted in a pattern of alternating positive and negative components. For T_w/T ₀ = 0.18, the eigenfunction structurally resembles the GSA result of Sidharth et al. (Reference Sidharth, Dwivedi, Candler and Nichols2017) for a heated compression corner. As seen from the spectrum, two oscillating unstable modes are captured with the non-dimensionalized frequencies fL/u _∞ of 0.225 and 0.578, respectively, where f = ω_r/2${\rm \pi}$. The frequencies agree well with the DNS results of Cao et al. (Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021). As the wall temperature is increased, the growth rate of the most unstable mode decreases, which becomes an oscillating mode for the adiabatic-wall condition (T_w/T ₀ = 0.86). Meanwhile, the two oscillating unstable modes are stabilized and eventually merge with the continuous modes.

Figure 10. Contours of real part of spanwise velocity perturbation (left column) and eigenvalue spectra (right column) for α = 15 °: (a) T_w/T ₀ = 0.18 at λ/L = 0.066; (b) T_w/T ₀ = 0.54 at λ/L = 0.114; (c) T_w/T ₀ = 0.86 at λ/L = 0.209. Closed circles: separation and reattachment points. Open circles in the eigenvalue spectra: eigenvalues obtained on a coarse grid (600 × 300). Open squares in the eigenvalue spectra: eigenvalues obtained on a fine grid (800 × 400).

Figure 11 shows the contours of the real part of $\hat{w}/{u_\infty }$ and the eigenvalue spectra corresponding to the long-wavelength unstable modes at their respective most unstable wavelengths (i.e. λ/L = 0.349, 0.503 and 0.628) for different wall temperatures at α = 15 °. The spanwise velocity perturbation is mostly confined to the separation region with an opposite sign in the upstream and downstream halves, which is similar to the GSA result of Sidharth et al. (Reference Sidharth, Dwivedi, Candler and Nichols2018) for a double-wedge configuration. In contrary to the short-wavelength scenario, only one stationary unstable mode can be identified.

Figure 11. Contours of real part of spanwise velocity perturbation (left column) and eigenvalue spectra (right column) for α = 15 °: (a) T_w/T ₀ = 0.18 at λ/L = 0.349; (b) T_w/T ₀ = 0.54 at λ/L = 0.503; (c) T_w/T ₀ = 0.86 at λ/L = 0.628. Closed circles: separation and reattachment points.

To further understand the physical mechanisms that drive the global instability of the separation bubble, an energy budget analysis is performed for the short-wavelength unstable modes corresponding to different wall temperatures as shown in figure 10 at α = 15 °. According to previous GSA studies on laminar flow separation (Theofilis et al. Reference Theofilis, Hein and Dallmann2000; Sidharth et al. Reference Sidharth, Dwivedi, Candler and Nichols2018; Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021), the global instability induces three-dimensionality in the separation bubble and alters its topological structure, which indicates that the disturbance field is mainly characterized by velocity perturbations. Hence, only the kinetic disturbance energy is considered here, which contributes more than 70 % of the total disturbance energy quantified by the Chu energy norm (Chu Reference Chu1965).

The linearized governing equation of velocity perturbations is given by

(4.1)\begin{equation}\frac{{\partial {{u^{\prime}}_i}}}{{\partial t}} + {\bar{u}_j}\frac{{\partial {{u^{\prime}}_i}}}{{\partial {x_j}}} ={-} \frac{{{{(\rho {u_j})}^\prime }}}{{\bar{\rho }}}\frac{{\partial {{\bar{u}}_i}}}{{\partial {x_j}}} - \frac{1}{{\bar{\rho }}}\frac{{\partial p^{\prime}}}{{\partial {x_i}}} - \frac{1}{{\bar{\rho }}}\frac{{\partial {{\tau ^{\prime}}_{ij}}}}{{\partial {x_j}}},\end{equation}

where indices i = 1, 2 and 3 represent the x, y and z directions, respectively. The specific kinetic disturbance energy is defined by

(4.2)\begin{equation}{e^{\prime}_k} = \frac{1}{2}u^{\prime}{u^{\prime}_i}^{\dagger} ,\end{equation}

where † is used to denote the complex conjugate. The governing equation of ${e^{\prime}_k}$ can be obtained by taking the dot product of (4.1) and ${u^{\prime}_i}^{\dagger}$ as

(4.3)\begin{equation}\frac{{\partial {{e^{\prime}}_k}}}{{\partial t}} = Re \left[ { - {{\bar{u}}_j}{{u^{\prime}}_i}^{\dagger} \frac{{\partial {{u^{\prime}}_i}}}{{\partial {x_j}}} - {{u^{\prime}}_i}^{\dagger} \frac{{{{(\rho {u_j})}^\prime }}}{{\bar{\rho }}}\frac{{\partial {{\bar{u}}_i}}}{{\partial {x_j}}} - \frac{{{{u^{\prime}}_i}^{\dagger} }}{{\bar{\rho }}}\frac{{\partial p^{\prime}}}{{\partial {x_i}}} - \frac{{{{u^{\prime}}_i}^{\dagger} }}{{\bar{\rho }}}\frac{{\partial {{\tau^{\prime}}_{ij}}}}{{\partial {x_j}}}} \right],\end{equation}

where Re denotes the real part of a complex number. Note that the perturbed variables can be written in a modal form as given by (3.3). One can readily obtain the following expression:

(4.4)\begin{equation}{e^{\prime}_k} = {\hat{e}_k}\ \textrm{exp}(2{\omega _i}t),\end{equation}

where ${\hat{e}_k}$ is the eigenfunction of the kinetic disturbance energy. Substituting (4.4) into (4.3) leads to

(4.5)\begin{equation}2{\omega _i}{\hat{e}_k} = Re \left[ {\underbrace{{ - {{\bar{u}}_j}{{\hat{u}}_i}^{\dagger} \frac{{\partial {{\hat{u}}_i}}}{{\partial {x_j}}}}}_{{Convection}}\underbrace{{ - {{\hat{u}}_i}^{\dagger} \frac{{\widehat {\rho {u_j}}}}{{\bar{\rho }}}\frac{{\partial {{\bar{u}}_i}}}{{\partial {x_j}}}}}_{{Production}}\underbrace{{ - \frac{{{{\hat{u}}_i}^{\dagger} }}{{\bar{\rho }}}\frac{{\partial \hat{p}}}{{\partial {x_i}}}}}_{{Transfer}}\underbrace{{ - \frac{{{{\hat{u}}_i}^{\dagger} }}{{\bar{\rho }}}\frac{{\partial {{\hat{\tau }}_{ij}}}}{{\partial {x_j}}}}}_{{Viscous}}} \right],\end{equation}

where the convection term represents the advection of ${\hat{e}_k}$ by the mean velocity, the production term is the work done by the Reynolds stress on the mean velocity gradient, the transfer term is the energy transport due to velocity and pressure fluctuations and the viscous term accounts for the molecular dissipation effects.

Following Mittal (Reference Mittal2010), we integrate (4.5) over the entire computational domain to obtain the following equation:

(4.6)\begin{equation}{\omega _i} = {\omega _{i,convection}} + {\omega _{i,production}} + {\omega _{i,transfer}} + {\omega _{i,viscous}},\end{equation}

where ω_{i ,convection}, ω_{i ,production}, ω_{i ,transfer} and ω_{i ,viscous} are the contributions to the total growth rate due to different mechanisms. Variations of these growth rate components for the short-wavelength unstable modes are plotted in figure 12 as a function of wall temperature ratio. The total growth rate is mainly contributed by the production term, whereas the convection and transfer terms are negligible. As the wall temperature is increased, the production term is significantly reduced.

Figure 12. Variation of growth rate components for the short-wavelength unstable modes at λ/L = 0.066, 0.114 and 0.209 as a function of wall temperature ratio for α = 15 °.

Figure 13 shows the spatial distributions of the production term for the short-wavelength unstable modes corresponding to different wall temperatures. The production term is localized near the primary-vortex core, which confirms that the instability arises from the separation region. Distributions of the production term are also plotted in this figure in the wall-normal direction through the vortex core. Close examination of the production term reveals that the major contribution comes from the streamwise gradient of the streamwise velocity (u_x), which is consistent with the finding of Sidharth et al. (Reference Sidharth, Dwivedi, Candler and Nichols2018). Comparing with figure 8(b), one can see that it is the deceleration of the reverse flow boundary layer instead of the shear layer that leads to the peak production of the kinetic disturbance energy. Increasing the wall temperature alleviates this deceleration due to the weakened streamwise pressure gradient (see figures 4 and 7), which explains the suppression of global instability. Similar behaviours were observed near the critical ramp angle of global instability. Deceleration of the streamwise velocity in the reverse flow boundary layer becomes stronger with increasing ramp angle, which results in a production term prevailing over the viscous term.

Figure 13. Contours of production term in the kinetic disturbance energy equation with streamlines superimposed (left column) and its distribution extracted along the wall-normal direction through the core of the primary vortex as a function of distance from the wall (right column) for α = 15 °: (a) T_w/T ₀ = 0.18 at λ/L = 0.066; (b) T_w/T ₀ = 0.54 at λ/L = 0.114; (c) T_w/T ₀ = 0.86 at λ/L = 0.209. Closed circles: vortex core.

4.3. Criterion in terms of triple-deck scaling

This study reveals that the occurrence of both secondary separation and global instability are governed by the characteristics of the reverse flow boundary layer. To describe the behaviour of flow separation, an asymptotic theory known as the triple-deck theory was established by Neiland (Reference Neiland1969) and Stewartson & Williams (Reference Stewartson and Williams1969) using the method of matched asymptotic expansions. They divided the boundary layer upstream of separation into three decks, including a viscous lower deck, an inviscid and rotational middle deck and an inviscid and irrotational upper deck. By introducing a specific set of scaled variables, the governing equations in the lower deck were reduced to the incompressible boundary-layer equations, which can be efficiently solved. Most importantly, the characteristic scaling defined by the triple-deck theory provides an effective approach to correlate and interpret numerical and experimental data in terms of a scaled ramp angle α*. According to Stewartson & Williams (Reference Stewartson and Williams1969), α* is defined by

(4.7a,b)\begin{equation}{\alpha ^ \ast } = \frac{{\alpha {Re}_L^{1/4}}}{{{C^{1/4}}{{0.332}^{1/2}}{{(M_\infty ^2 - 1)}^{1/4}}}},\quad C = \frac{{{\mu _w}{T_\infty }}}{{{\mu _\infty }{T_w}}},\end{equation}

where C is the Chapman–Rubesin parameter.

The triple-deck theory was initially applied to incipient or small-scale separation. As α* is increased, a pressure plateau was formed by solving the triple-deck equations (Rizzetta, Burggraf & Jenson Reference Rizzetta, Burggraf and Jenson1978). Smith & Khorrami (Reference Smith and Khorrami1991) suggested that there was a critical value of α* beyond which a singularity developed near reattachment in the triple-deck solution. However, Korolev et al. (Reference Korolev, Gajjar and Ruban2002) found that the computational domain used by Smith & Khorrami (Reference Smith and Khorrami1991) was insufficient to resolve the flow problem. Secondary separation was observed near the corner inside the separation region for a large α*. They also confirmed that the scaled length of the separation region was linearly proportional to α*^3/2 for a large separation region, as suggested by Burggraf (Reference Burggraf1975). It was indicated that the triple-deck theory can be extended to moderate- to large-scale separation problems.

In the present study, the secondary separation occurs when α* is 3.73–4.92, which is consistent with the findings of Smith & Khorrami (Reference Smith and Khorrami1991), Korolev et al. (Reference Korolev, Gajjar and Ruban2002), Shvedchenko (Reference Shvedchenko2009) and Gai & Khraibut (Reference Gai and Khraibut2019). Note that different definitions of scaled ramp angle were used in these studies. The lower and upper boundaries correspond to the cold- and adiabatic-wall conditions, respectively. A similar trend was observed by Egorov et al. (Reference Egorov, Neiland and Shvedchenko2011). Then, a criterion can be established in terms of α* to predict the stability boundary, i.e. 3.44 < α* < 4.59. Again, a higher wall temperature results in a larger critical ramp angle.

Figure 14 presents the variation of the critical scaled ramp angle as a function of wall temperature ratio. The error bars represent an uncertainty of 1 °, which is the minimum increment in ramp angle considered in this study. Additional simulations and GSA were performed for T_w/T ₀ = 0.18 with Re_L = 2.1 × 10⁵ and 8.4 × 10⁵, respectively. For the lower Reynolds number, the flow becomes globally unstable when α is 15 °–16 °, whereas secondary separation occurs when α is 16 °–17 °. For the larger Reynolds number, the critical ramp angles for global instability and secondary separation are 10 °–11 ° and 11 °–12 °, respectively. As expected, the resulting scaled ramp angles are almost overlapped, which indicates that the effect of the Reynolds number is well accounted for.

Figure 14. The critical scaled ramp angle for global instability as a function of wall temperature ratio. Open symbols: globally stable. Closed symbols: globally unstable.

Also shown in the figure are existing theoretical, numerical and experimental data collected from the literature for hypersonic compression corner and double-wedge flows with the free-stream Mach numbers ranging from 5 to 11.63 and the unit Reynolds numbers from 4.0 × 10⁵ m⁻¹ to 13.6 × 10⁶ m⁻¹. The detailed flow conditions are provided in table 1. For the double-wedge flows, α represents the angle between the first and second wedges, and the scaled ramp angle is evaluated using the flow properties behind the oblique shock induced by the first wedge. For the theoretical studies, it is straightforward to determine whether a flow system is globally unstable based on the GSA predictions. For example, the Mach 5 double-wedge flow considered by Sidharth et al. (Reference Sidharth, Dwivedi, Candler and Nichols2018) was found to be marginally stable when the turn angle is 7.8 ° and unstable for 10 °. For the computational fluid dynamics simulations, a flow is considered to be globally unstable only if 3-D streamwise streaks can be formed with no external disturbance. As for the experimental data, an observation of streamwise streaks is a necessary but not sufficient condition for global instability, as demonstrated by Dwivedi et al. (Reference Dwivedi, Sidharth, Nichols, Candler and Jovanović2019). Here, only those experimental data that have been confirmed by GSA are included. The proposed criterion clearly demarcates the stability boundary. Extension to shock impingement on a flat plate and axisymmetric flows will be the focus of a future study.

Table 1. Flow conditions of the theoretical, numerical and experimental data collected in figure 14.

Furthermore, it is important to note that no experimental evidence of secondary separation has been found. The speculation that the secondary separation leads to three-dimensionality and unsteadiness remains to be experimentally verified.