3.1. Lower limit $g_{\textit{min}}^{(\textit{RR})}$ in the $M_{0}{-}\theta _{w}$ plane

The lower limit for RR, denoted $g_{\textit{min}}^{(\textit{RR})}$ , is obtained by (2.14) with $A=B= 0$ and with $\phi$ computed by (2.2), where $\beta _{12}$ is the shock angle of the reflected shock for RR. Note that the expression for $g_{\textit{min}}^{(\textit{RR})}$ is exact and is displayed by Wang & Wu (2026) within the dual-solution domain. Here, in figure 5, we display $g_{\textit{min}}^{(\textit{RR})}$ in the entire space $0\lt \theta _{w}\lt \theta _{w}^{(D)}$ , where $\theta _{w}^{(D)}=\theta _{w}^{(D)}(M_{0})$ is the detachment condition.

Figure 5.

Contour lines (solid) of $g_{\textit{min}}^{(\textit{RR})}$ in the $M_{0}{-}\theta _{w}$ plane. Dashed lines represent critical conditions.

It is seen that for $\theta _{w}\rightarrow 0$ , $g_{\textit{min}}^{(\textit{RR})}$ approaches $1$ , meaning that for very weak shock waves (for $\theta _{w}\rightarrow 0$ ), RR can occur for all $g$ and for all $M_{0}$ with $ 2.3\lt M_{0}\lt 7$ . When $\theta _{w}$ increases for fixed $M_{0}$ , the lower limit $g_{\textit{min}}^{(\textit{RR})}$ first decreases then increases, and the minimum value of $g_{\textit{min}}^{(\textit{RR})}$ occurs at $\theta_w=\theta _{w,m}$ , where $ \theta _{w,m}=\theta _{w,m}(M_{0})$ is the value of $\theta _{w}$ at which

(3.1) \begin{align} \frac {\partial g_{\textit{min}}^{(\textit{RR})}(M_{0},\theta _{w})}{\partial \theta _{w}}=0. \end{align}

For convenience, the critical line $\theta _{w}=\theta _{w,m}(M_{0})$ will be called the dividing line for monotonicity. As shown in figure 5, there is a point, denoted $P$ , with $M_{0}=M_{P}\approx 3.25$ ; when $ M_{0}\lt M_{P}$ , the dividing line for monotonicity is below the von Neumann condition $\theta _{w}^{(N)}=\theta _{w}^{(N)}(M_{0})$ , while for $ M_{0}\gt M_{P}$ , the dividing line for monotonicity is above the von Neumann condition $\theta _{w}^{(N)}=\theta _{w}^{(N)}(M_{0})$ .

3.2. Lower limit $g_{\textit{min}}^{(\textit{MR})}$ in the $M_{0}{-}\theta _{w}$ plane

The lower limit for MR, denoted $g_{\textit{min}}^{(\textit{MR})}$ , is obtained by (2.14), with $A$ and $B$ given by the calibrated formulas (2.19) and (2.20), respectively. The quantity $\phi$ is computed by (2.2), where the $\beta _{12}$ is the shock angle of the reflected shock that is given by the triple-point solution.

Figure 6 displays $g_{\textit{min}}^{(\textit{MR})}$ in the $M_{0}{-}\theta _{w}$ plane, for $\theta _{w}^{(N)}\lt \theta _{w}\lt \theta _{\textit{max}}$ , where $\theta _{\textit{max}}=\theta _{\textit{max}}(M_{0})$ is the detached angle for incident shock. Unlike $g_{\textit{min}}^{(\textit{RR})}$ , which varies non-monotonically for increasing $ \theta _{w}$ , here $g_{\textit{min}}^{(\textit{MR})}$ increases monotonically with $\theta _{w}$ when $\theta _{w}$ varies from $\theta _{w}^{(N)}$ to $\theta _{w}^{(D)}$ .

Figure 6.

Contour lines (solid) of $g_{\textit{min}}^{(\textit{MR})}$ in the $M_{0}{-}\theta _{w}$ plane. Dashed lines represent critical conditions.

Figure 6. Long description

A line graph with contour lines and critical conditions. The horizontal axis is labeled M 0 ranging from 2.5 to 7.0. The vertical axis is labeled theta w in degrees ranging from 15 to 40. Solid lines represent contour lines with values 0.16, 0.21, 0.26, 0.31, 0.36, 0.41, 0.46, 0.51, 0.56, 0.61, 0.66, 0.71, and 0.76. Dashed lines represent critical conditions labeled theta w max, theta w D, and theta w N. The contour lines show an increasing trend as M 0 increases.

At fixed Mach number, the value of $g_{\textit{min}}^{(\textit{MR})}$ is larger for larger wedge angle. For instance, at $M_{0}=4$ , we have $g_{\textit{min}}^{(\textit{MR})}\approx 0.23$ for $\theta _{w}\rightarrow \theta _{w}^{(N)}$ and $g_{\textit{min}}^{(\textit{MR})}\approx 0.69$ for $\theta _{w}\rightarrow \theta _{\textit{max}}$ . At fixed wedge angle, the value of $g_{\textit{min}}^{(\textit{MR})}$ is larger for smaller Mach number.

High-fidelity CFD data are used by Wang & Wu (2026) to assess the correctness of the semi-empirical roadmap to determine $ g_{\textit{min}}^{(\textit{MR})}$ . These data are used here to assess the accuracy of (2.14), with $A$ and $B$ given by the calibrated formulas (2.19) and (2.20), respectively.

Table 1 gives three cases for which the present algebraic model defined by (2.14), (2.19) and (2.20) is compared with the CFD data and semi-empirical roadmap data of Wang & Wu (2026). The relative difference between the present algebraic model and the CFD data is below 6 % for all cases.

Table 1.

A comparison of $g_{\textit{min}}$ from different estimations. Here, CFD data are $g_{min}$ by CFD of Wang & Wu (2026), roadmap data are $g_{min}$ by the roadmap of Wang & Wu (2026), and the present model is $g_{min}$ by (2.14), (2.19) and (2.20).

Table 1. Long description

A table comparing different estimation models for three cases. The table has five columns: Case, M 0, θ w, CFD data, Roadmap data, Present model, and Relative difference. It has three rows of data. Row 1: Case 1, M 0, θ w 4, 25 degrees, CFD data 0.4225 ± 0.0025, Roadmap data 0.417, Present model 0.414, Relative difference 2.0 percent. Row 2: Case 2, M 0, θ w 3, 21 degrees, CFD data 0.445 ± 0.005, Roadmap data 0.442, Present model 0.419, Relative difference 5.8 percent. Row 3: Case 3, M 0, θ w 5, 25 degrees, CFD data 0.355 ± 0.005, Roadmap data 0.358, Present model 0.344, Relative difference 3.1 percent.

3.3. The dual-solution stability gap $g_{\textit{min}}^{(\textit{MR})}-g_{\textit{min}}^{(\textit{RR})}$ in the $M_{0}{-}\theta _{w}$ plane

The dual-solution stability gap (Wang & Wu 2026) is computed by

(3.2) \begin{align} \Delta g_{\textit{min}}=g_{\textit{min}}^{(\textit{MR})}-g_{\textit{min}}^{(\textit{RR})}, \end{align}

where $g_{\textit{min}}^{(\textit{RR})}$ is given in § 3.1, and $g_{\textit{min}}^{(\textit{MR})}$ is given in § 3.2. Note that $g_{\textit{min}}^{(\textit{RR})}$ is defined for $\theta _{w}\lt \theta _{w}^{(D)}$ , and $g_{\textit{min}}^{(\textit{MR})}$ is defined for $\theta _{w}\gt \theta _{w}^{(N)}$ , so $\Delta g_{\textit{min}}$ is defined in the dual-solution domain $\theta _{w}^{(N)}\lt \theta _{w}\lt \theta _{w}^{(D)}$ .

Figure 7 displays $\Delta g_{\textit{min}}=g_{\textit{min}}^{(\textit{MR})}-g_{\textit{min}}^{(\textit{RR})}$ , in the dual-solution domain. For $\theta _{w}\rightarrow \theta _{w}^{(N)}$ , we have $\Delta g_{\textit{min}}\rightarrow 0$ , meaning that there is no dual-solution stability gap for $\theta _{w}$ approaching the von Neumann condition.

Figure 7.

Contour lines (solid) of $\Delta g_{\textit{min}}=g_{\textit{min}}^{(\textit{MR})}-g_{\textit{min}}^{(\textit{RR})}$ in the dual-solution domain plane. Dashed lines represent critical conditions.

Figure 7. Long description

A line graph with contour lines and critical conditions in a dual-solution domain plane. The horizontal axis is labeled M 0 and ranges from 2.5 to 7.0. The vertical axis is labeled theta w in degrees and ranges from 16 to 30. Solid contour lines are labeled with values ranging from 0.02 to 0.22. Dashed lines represent critical conditions. A closed curve is indicated at the top right, with a maximum point Q marked. The graph shows a vanishing point and trends of increasing and decreasing values.

It is interesting to note that there exists a maximum point for $\Delta g_{\textit{min}}$ within the dual-solution domain. This maximum point, denoted $Q$ in figure 7, lies at

(3.3) \begin{align} M_{0}\approx 6.2,\quad\theta _{w}\approx 28.8^\circ, \end{align}

and the maximal value of $\Delta g_{\textit{min}}$ is

(3.4) \begin{align} {\rm max} (\Delta g_{\textit{min}})=0.225\text{.} \end{align}

The contour lines with constant $\Delta g_{\textit{min}}$ form closed curves enclosing point $Q$ . The existence of this maximum point means that the range of $g$ with stable dual solution is the largest there.

Similarly as for $g_{\textit{min}}^{(\textit{RR})}$ , for which there is a dividing line for monotonicity (figure 5), here there is also a dividing line for monotonicity (marked by black circles in figure 7). Since $ g_{\textit{min}}^{(\textit{MR})}$ is monotonic according to figure 6, the non-monotonicity of $\Delta g_{\textit{min}}=g_{\textit{min}}^{(\textit{MR})}-g_{\textit{min}}^{(\textit{RR})}$ follows naturally from the non-monotonicity of $g_{\textit{min}}^{(\textit{RR})}$ .

Wang & Wu (2026) have given a formal proof of $\Delta g_{\textit{min}}\gt 0$ except near the left tip-horn region of the dual-solution domain. Using the approximate Mach stem height model of Bai & Wu (Reference Bai and Wu2021), this tip-horn region still exists according to Wang & Wu (2026, figure 13). Here, by using the calibrated Mach stem height model, we have $\Delta g_{\textit{min}}\gt 0$ in the entire dual-solution domain (except $\theta _{w}\rightarrow \theta _{w}^{(N)}$ ).

To assess the accuracy, we still consider the three cases shown in table 1. The comparison is given in table 2, where we also give an estimation using an approximate Mach stem height model in the expression (2.14) for $g_{\textit{min}}^{(\textit{MR})}$ . It is seen that for cases 1 and 3, the present model compares well with CFD data, but for case 2, the error is relatively large (this is probably because the absolute value of $\Delta g_{\textit{min}}$ is small). For an approximate Mach stem height model, the relative errors are large for all three cases.

Table 2.

A comparison of $\Delta g_{\textit{min}}$ from different estimations. Approximate model is the model using the Mach stem height model of Bai & Wu (Reference Bai and Wu2021); see Appendix A of Wang & Wu (2026).

Table 2. Long description

A table comparing different estimation models for three cases. The table has five columns: Case, CFD data, Roadmap data, Approximate model, and Present model. It has three rows with different cases. Row 1: Case 1, M 0, 4, theta w, 25 degrees, CFD data, 0.1835 plus or minus 0.0025, Roadmap data, 0.178, Approximate model, 0.120, 34.6 percent, Present model, 0.175, 4.6 percent. Row 2: Case 2, M 0, 3, theta w, 21 degrees, CFD data, 0.094 plus or minus 0.005, Roadmap data, 0.091, Approximate model, 0.03, 68.1 percent, Present model, 0.068, 27.7 percent. Row 3: Case 3, M 0, 5, theta w, 25 degrees, CFD data, 0.182 plus or minus 0.005, Roadmap data, 0.185, Approximate model, 0.12, 34.1 percent, Present model, 0.171, 6.0 percent.

4.1. Upper limit $g_{\textit{max}}^{(\textit{RR})}$ in the $M_{0}{-}\theta _{w}$ plane

The upper limit for RR, denoted $g_{\textit{max}}^{(\textit{RR})}$ , is obtained by (2.15) with $A=B= 0$ and with $\psi$ computed by (2.8). Note that the expression for $g_{\textit{max}}^{(\textit{RR})}$ is exact and has already been displayed by Bai (Reference Bai2023). Here, for the purpose of comparing with lower limit behaviour, in figure 8, we display $ g_{\textit{min}}^{(\textit{RR})}$ in the space $0\lt \theta _{w}\lt \theta _{w}^{(D)}$ , where $ \theta _{w}^{(D)}=\theta _{w}^{(D)}(M_{0})$ is the detachment condition.

Figure 8.

Contour lines (solid) of $g_{\textit{max}}^{(\textit{RR})}$ in the $M_{0}{-}\theta _{w}$ plane. Dashed lines represent critical conditions.

Unlike $g_{\textit{min}}^{(\textit{RR})}$ , which varies non-monotonically for increasing $ \theta _{w}$ , here $g_{\textit{max}}^{(\textit{RR})}$ increases monotonically with $\theta _{w}$ when $\theta _{w}$ varies from $\theta _{w}^{(D)}$ to $0$ . Similarly as $g_{\textit{min}}^{(\textit{RR})}$ , for $\theta _{w}\rightarrow 0$ and for all $M_{0}$ with $2.3\lt M_{0}\lt 7$ , $g_{\textit{max}}^{(\textit{RR})}$ approaches $1$ , meaning that for very weak shock waves (for $\theta _{w}\rightarrow 0$ ), RR without interference can occur for all $g$ .

4.2. Upper limit $g_{\textit{max}}^{(\textit{MR})}$ in the $M_{0}{-}\theta _{w}$ plane

The upper limit for MR, denoted $g_{\textit{max}}^{(\textit{MR})}$ , is obtained by (2.15) with $A$ and $B$ given by the calibrated formulas (2.19) and (2.20), respectively, and with $\psi$ computed by (2.8).

Figure 9 displays $g_{\textit{max}}^{(\textit{MR})}$ in the $M_{0}{-}\theta _{w}$ plane. The behaviour of $g_{\textit{max}}^{(\textit{MR})}$ is much like $g_{\textit{min}}^{(\textit{RR})}$ , the lower limit for RR, in the dual-solution domain. When $\theta _{w}$ increases for fixed $M_{0}$ , the upper limit $g_{\textit{max}}^{(\textit{MR})}$ first decreases then increases, and the minimum value of $g_{\textit{max}}^{(\textit{MR})}$ occurs at $\theta_w=\theta _{w,m}$ , where $\theta _{w,m}=\theta _{w,m}(M_{0})$ is the value of $\theta _{w}$ at which

(4.1) \begin{align} \frac {\partial g_{\textit{max}}^{(\textit{MR})}(M_{0},\theta _{w})}{\partial \theta _{w}}=0. \end{align}

For convenience, the critical line $\theta _{w}=\theta _{w,m}(M_{0})$ will be called the dividing line for monotonicity (marked with black circles). As shown in figure 9, there is a point, denoted $P$ , with $ M_{0}=M_{P}\approx 5.4$ ; when $M_{0}\lt M_{P}$ , the dividing line for monotonicity is below the detachment condition $\theta _{w}^{(D)}=\theta _{w}^{(D)}(M_{0})$ , while for $M_{0}\gt M_{P}$ , the dividing line for monotonicity is above the detachment condition $\theta _{w}^{(D)}=\theta _{w}^{(D)}(M_{0})$ .

Figure 9.

Contour lines (solid) of $g_{\textit{max}}^{(\textit{MR})}$ in the $M_{0}{-}\theta _{w}$ plane. Dashed lines represent critical conditions.

Figure 9. Long description

A line graph with contour lines (solid) and critical conditions (dashed lines) in the plane. The horizontal axis is labeled M 0 and ranges from 2.5 to 7.0. The vertical axis is labeled theta w (degrees) and ranges from 15 to 40 degrees. The contour lines are labeled with values ranging from 0.6 to 0.775. The dashed lines represent critical conditions, including theta max, theta w (D), and theta w (N). The graph shows an increase in theta w as M 0 increases, with specific points marked and labeled.

4.3. Dual-solution interference gap $g_{\textit{max}}^{(\textit{MR})}-g_{\textit{max}}^{(\textit{RR})}$ in the $M_{0}{-}\theta _{w}$ plane

The dual-solution interference gap $\Delta g_{\textit{max}}$ is the difference between $g_{\textit{max}}^{(\textit{MR})}$ and $g_{\textit{max}}^{(\textit{RR})}$ , i.e.

(4.2) \begin{align} \Delta g_{\textit{max}}=g_{\textit{max}}^{(\textit{MR})}-g_{\textit{max}}^{(\textit{RR})}\text{.} \end{align}

This gap is defined in the dual-solution domain since $g_{\textit{max}}^{(\textit{MR})}$ exists for $\theta _{w}\lt \theta _{w}^{(D)}$ , and $g_{\textit{max}}^{(\textit{RR})}$ exists for $\theta _{w}\gt \theta _{w}^{(N)}$ .

Figure 10 displays $\Delta g_{\textit{max}}$ in the dual-solution domain. The contour lines are almost horizontal lines, so $\Delta g_{\textit{max}}$ is insensitive to the Mach number.

Similarly as for $\Delta g_{\textit{min}}$ (figure 7), $\Delta g_{\textit{max}}$ is small at the left-horn corner of the dual-solution domain. Note that $ \Delta g_{\textit{max}}$ may display very small negative values due to rounding errors (figure 11). For $\theta _{w}\rightarrow \theta _{w}^{(N)}$ , we have $\Delta g_{\textit{max}}\rightarrow 0$ , meaning that there is no dual-solution interference gap for $\theta _{w}$ approaching the von Neumann condition.

Figure 10.

Contour lines (solid) of the dual-solution interference gap $\Delta g_{\textit{max}}=g_{\textit{max}}^{(\textit{MR})}-g_{max}^{(\textit{RR})}$ in the dual-solution domain plane. Dashed lines represent critical conditions.

Figure 11.

Contour lines (solid) of $\Delta g_{\textit{max}}=g_{\textit{max}}^{(\textit{MR})}-g_{max}^{(\textit{RR})}$ in the left-horn corner of the dual-solution domain plane. Dashed lines represent critical conditions.

Unlike $\Delta g_{\textit{min}}$ , which has a maximum point, here $\Delta g_{\textit{max}}$ is monotonic. The maximal value of the dual-solution stability gap is greater than $0.22$ , while for $M_{0}\lt 7$ , the maximum value of $\Delta g_{\textit{max}}$ is far less than $0.05$ (for $M_{0}\lt 7$ ), i.e. the dual-solution interference gap is very small, so that once we have expansion fan–incident shock interference for RR, we readily have such interference for MR. The reason for this small gap is that MR with interaction between trailing edge expansion fan and the incident shock occurs at small Mach stem height.

5.1. Geometric limits along the $g$ -axis

We have given in figures 5, 6, 8 and 9 the contour lines for the critical values $ g_{\textit{min}}^{(\textit{RR})}$ , $g_{\textit{min}}^{(\textit{MR})}$ , $g_{\textit{max}}^{(\textit{RR})}$ and $g_{\textit{max}}^{(\textit{MR})}$ , respectively. The relative positions of these critical values of $ g$ are displayed on the $g$ -axis in figure 12.

Figure 12.

Six limits of $g$ along the $g$ -axis, plus a potential limit for no reflection.

Figure 12 also marks the geometric transition criterion $g_{\textit{tr}}$ , at which the shock angle ( $\beta _{G}$ ) of the incident shock at the reflecting surface reaches its local von Neumann condition, due to weakening of the incident shock by the trailing edge expansion fan. At $g=g_{\textit{tr}}$ , the shock angle is the one at the von Neumann condition, i.e.

(5.1) \begin{align} \beta _{G}=\beta ^{(N)}(M_{0}), \end{align}

and the flow deflection angle at this point is determined by $\theta _{G}=\theta ^{(N)}(M_{0})$ , where $\tan \theta ^{(N)}=f_{\theta }(M_{0},\beta _{G})$ . The shock reflection has the configuration as displayed in figure 13. Above $g=g_{\textit{tr}}$ , MR will necessarily transit to RR. Since this transition is caused by a geometric set-up, it is called the geometric transition limit. Using methods of characteristics, Bai (Reference Bai2023, figure 6a) and Baby, Paramanantham & Rajesh (Reference Baby, Paramanantham and Rajesh2024, figure10b) obtained the values of $g_{\textit{tr}}$ in the $M_{0}{-}\theta _{w}$ plane.

In figure 12, we also marked a potential limit $g_{rv}$ above which there is no reflection. The possibility of the existence of such a critical value seems to have not been studied before, and will be considered shortly in § 5.4.

The relative positions of the various critical values displayed in figure 12 have not been given arbitrarily. They are based on the predicted values at the maximum point for the dual-solution domain, i.e. at $ M_{0}=6.2$ and $\theta _{w}=28.8^\circ$ . Table 3 summarises the predicted values.

Table 3.

Critical values of $g$ at the maximal point $M_{0}=6.2$ and $ \theta _{w}=28.8^\circ$ .

Table 3. Long description

A table comparing critical values for different scenarios. The table has three rows and four columns. The columns are labeled g min, g max, and g tr. The rows are labeled RR, MR, and Gap between MR and RR. Row 1: RR, g min 0.1652, g max 0.5528, g tr 0.715. Row 2: MR, g min 0.3908, g max 0.5885, g tr Null. Row 3: Gap between MR and RR, g min 0.2260, g max 0.0357, g tr Null.

Figure 13.

Schematic display of the incident shock when $g=g_{\textit{tr}}$ .

In the previous sections, we have only given schematic displays of shock reflections at various critical values of $g$ . In figure 14, we give the Mach number contour lines obtained by numerical simulation (performed by solving the Euler equations in gas dynamics, using the same approach as used by Wang & Wu 2026), for the conditions displayed in table 3 to show the shock reflection patterns.

Figure 14.

Mach number contour lines at the maximum point $M_{0}=6.2$ and $ \theta _{w}=28.8^\circ$ , for (a) $g=g_{\textit{min}}^{(\textit{RR})}$ , (b) $g=g_{\textit{min}}^{(\textit{MR})}$ , (c) $g=g_{\textit{max}}^{(\textit{RR})}$ , (d) $g=g_{\textit{max}}^{(\textit{MR})}$ , (e) $g=g_{\textit{tr}}$ ,.

Figure 14. Long description

Panel A: A diagram showing a simple triangular shape with straight lines. Panel B: A diagram with a triangular shape and multiple contour lines radiating from the right angle. Panel C: A diagram with a triangular shape and contour lines forming complex patterns near the right angle. Panel D: A diagram with a triangular shape and contour lines spreading out from the right angle. Panel E: A diagram with a triangular shape and contour lines radiating from the right angle, labeled with an angle of approximately 29 degrees.

At $g=g_{\textit{min}}^{(\textit{RR})}$ , the reflected shock indeed grasps the trailing edge, as can be seen from figure 14(a). At $g=g_{\textit{min}}^{(\textit{MR})}$ , the reflected shock has moved upstream of the trailing edge, as shown in figure 14(b) (a snapshot of the unsteady solution). At $g=g_{\textit{max}}^{(\textit{RR})}$ , the leading characteristics of the trailing edge expansion fan indeed intersects the incident shock at the reflecting surface, as is clear from figure 14(c). At $g=g_{\textit{max}}^{(\textit{MR})}$ , the leading characteristics of the trailing edge expansion fan indeed intersect the triple point, as is clear from figure 14(d). At $g=g_{\textit{tr}}$ , the incident shock reflects at a shock angle that is near the von Neumann condition ( $\beta ^{(N)}(6.2)\approx 29^\circ$ ), as shown in figure 14(e). To complement the result shown in figure 14(b), we have performed a further computation for $g=0.41$ , with which we have been able to obtain stable MR, as shown in figure 15. The fact that we have the unstable solution shown in figure 14(b) for $g=0.3908$ and the stable solution shown in figure 15 for $g=0.41$ means that the precise value of $g_{\textit{min}}^{(\textit{MR})}$ should satisfy $0.3908\lt g_{\textit{min}}^{(\textit{MR})}\lt 0.41$ . Thus the predicted $g_{\textit{min}}^{(\textit{MR})}$ is different from the real value by an error as small as $\approx 5\,\%$ .

Figure 15.

Mach number contour lines at the maximum point $M_{0}=6.2$ and $ \theta _{w}=28.8^\circ$ , for $g=0.41$ .

5.2. Various gaps along the $g$ -axis

The distance between adjacent geometric limits defines meaningful gaps, as displayed in figure 16, which shows eight gaps, plus a potential gap with no reflection (see § 5.4), as follows.

Figure 16.

A full geometric picture defining six limits of $g$ and eight gaps along the $g$ -axis, plus a potential gap for vanished reflection.

1. (i) The unstart-prone reflection gap $\Delta g_{us}$ , corresponding to $ 0\lt g\lt g_{\textit{min}}^{(\textit{RR})}$ . In this gap, the reflected shock will touch the lower wedge surface, leading to unstart flow for MR (Vuillon et al. Reference Vuillon, Zeitoun and Ben-Dor1995), or shock reflection on the lower wedge surface for RR. For the latter case, the boundary layer in real flow will be subjected to shock wave/boundary layer interaction leading to possibly unsteady flow separation.

2. (ii) The dual-solution stability gap $\Delta g_{\textit{min}}$ : this is the gap between $g_{\textit{min}}^{(\textit{RR})}$ and $g_{\textit{min}}^{(\textit{MR})}$ , for which stable RR may transit to unstable MR under suitable disturbance (Wang & Wu 2026).

3. (iii) The dual-solution interference gap $\Delta g_{\textit{max}}$ : this is the gap between $g_{\textit{max}}^{(\textit{RR})}$ and $g_{\textit{max}}^{(\textit{MR})}$ . Within this gap, the incident shock interacts with the trailing edge expansion fan for RR, and not for MR.

4. (iv) The geometric transition gap $\Delta g_{\textit{tr}}$ : this is the gap between $ g_{\textit{max}}^{(\textit{MR})}$ and $g_{\textit{tr}}$ . When $g\gt g_{\textit{max}}^{(\textit{MR})}$ , the weakening of the incident shock forces an MR to transit to RR when $g\rightarrow g_{\textit{tr}}$ .

5. (v) The clean RR gap $\Delta g_{c}^{(\textit{RR})}$ : this is the gap between $g_{\textit{min}}^{(\textit{RR})}$ and $g_{\textit{max}}^{(\textit{RR})}$ . Within this gap, RR has no interaction between the incident shock and the trailing edge expansion fan, and the reflected shock will not grasp the trailing edge. More details are given in § 5.3.

6. (vi) The clean MR gap $\Delta g_{c}^{(\textit{MR})}$ : this is the gap between $g_{\textit{min}}^{(\textit{MR})}$ and $g_{\textit{max}}^{(\textit{MR})}$ . Within this gap, MR has no interaction between the incident shock and the trailing edge expansion fan, and the reflected shock will not grasp the trailing edge, i.e. MR can be stable. More details are given in § 5.3.

7. (vii) The robust reflection gap $\Delta g_{rb}$ : this is the gap between $ g_{\textit{min}}^{(\textit{MR})}$ and $g_{\textit{max}}^{(\textit{RR})}$ . The meaning of robust reflection will be described in § 5.3.

8. (viii) The always RR gap $\Delta g_{\textit{ARR}}$ : this is the gap between $g_{\textit{tr}}$ and $ g_{rv}$ , where we only have RR.

9. (ix) Potential gap with no reflection $\Delta g_{\textit{nr}}$ : this is the gap between the hypothetical $g_{rv}$ and $g=1$ .

The existence of $g_{rv}$ will be discussed shortly in § 5.4.

5.3. Clean and robust reflection gaps

Clean shock reflection (CSR) is defined here as the reflection for which the incident shock is not perturbed by the trailing edge expansion fan, and the reflected shock does not grasp the trailing edge. In terms of the range of $ g$ , the CSR gap is defined as $g_{\textit{min}}\lt g\lt g_{\textit{max}}$ . In terms of the distance between $g_{\textit{min}}$ and $g_{\textit{max}}$ , this CSR gap is defined by

(5.2) \begin{align} \Delta g_{c}=g_{\textit{max}}-g_{\textit{min}}. \end{align}

Figure 17 displays, for RR, the CSR gap $\Delta g_{c}$ in the $ M_{0}{-}\theta _{w}$ plane. It is seen that when $\theta _{w}\rightarrow 0$ , the CSR gap satisfies $\Delta g_{c}\rightarrow 0$ , meaning that for very weak shock waves the range of $g$ with CSR is narrow. At fixed $ M_{0}$ , $\Delta g_{c}$ has a maximal value at some $\theta _{w}=\theta _{w,m}$ , which is always below the von Neumann condition.

Figure 17.

Contour lines (solid) of $\Delta g_{c}^{(\textit{RR})}=g_{\textit{max}}^{(\textit{RR})}-g_{\textit{min}}^{(\textit{RR})}$ in the $M_{0}{-}\theta _{w}$ plane. Dashed lines represent critical conditions.

Figure 18 displays, for MR, the CSR gap $\Delta g_{c}$ in the $ M_{0}{-}\theta _{w}$ plane. It is seen that when $\theta _{w}\rightarrow \theta _{\textit{max}}$ , the CSR gap satisfies $\Delta g_{c}\rightarrow 0$ , meaning that for very strong shock waves, the range of $g$ with CSR is narrow. At fixed $M_{0}$ , $\Delta g_{c}$ increases monotonically for $ \theta _{w}$ varying from $\theta _{\textit{max}}$ to $\theta _{w}^{(N)}$ . Thus the maximum values of $\Delta g_{c}$ occur near $\theta _{w}\rightarrow \theta _{w}^{(N)}$ for MR, unlike RR for which the maximum value of $ \Delta g_{c}$ occurs at $\theta _{w}\rightarrow \theta _{w}^{(D)}$ .

Figure 18.

Contour lines (solid) of $\Delta g_{c}^{(\textit{MR})}=g_{\textit{max}}^{(\textit{MR})}-g_{\textit{min}}^{(\textit{MR})}$ in the $M_{0}{-}\theta _{w}$ plane. Dashed lines represent critical conditions.

Figure 18. Long description

A line graph with contour lines (solid) and critical conditions (dashed lines) in the plane. The horizontal axis is labeled M 0 ranging from 2.5 to 7.0. The vertical axis is labeled theta w (degrees) ranging from 15 to 40. The graph includes multiple solid contour lines labeled with values such as 0.01, 0.06, 0.11, 0.16, 0.21, 0.26, 0.31, 0.36, 0.41, and 0.46. The dashed lines represent critical conditions labeled as theta max, theta w superscript D, and theta w superscript N. An arrow indicates an increase in a specific direction.

The robust reflection gap $\Delta g_{rb}=g_{\textit{max}}^{(\textit{RR})}-g_{\textit{min}}^{(\textit{MR})}$ , as shown in figure 16, defines the range of $g$ with which the incident shock is free of interaction with the trailing edge expansion fan, and the reflected shock has no risk of grasping the trailing edge, whenever the reflection is RR and MR, or may undergo transition between RR and MR. This gap, computed by using the values of $g_{\textit{min}}^{(\textit{MR})}$ displayed in figure 6 and $g_{\textit{max}}^{(\textit{RR})}$ displayed in figure 8, is displayed in figure 19.

Figure 19.

The robust reflection gap $\Delta g_{c}=g_{\textit{max}}^{(\textit{RR})}-g_{\textit{min}}^{(\textit{MR})}$ in the dual-solution domain.

Comparing figures 7, 10 and 19, we see that the robust reflection gap is the largest, while the dual-solution interference gap is the smallest.

5.4. The potential existence of $g_{rv}$ and $\Delta g_{\textit{nr}}$

Now consider the possibility $g=g_{rv}$ , beyond which there is no reflection. This means that at $g=g_{rv}$ , we would have $\beta _{G}=\mu (M_{0})$ , i.e. at the reflecting point, the incident shock degenerates to a Mach wave.

Figure 20(a) is a schematic display of the hypothetical configuration at $g=g_{rv}$ . The incident shock just degenerates to a Mach wave at the reflecting surface, and the reflected wave is also a Mach wave. Figure 20(b) is a schematic display of the hypothetical configuration of no reflection for $g\gt g_{rv}$ . There is a hypothetical death point ( $D$ ) that marks the end of the incident shock, below which the incident shock degenerates to a Mach wave. One would ask if the existence of $g_{rv}$ with $g_{rv}\lt 1$ can be theoretically justified.

Figure 20.

Schematic display of vanishing reflection, for (a) $g=g_{rv}$ , (b) $ g\gt g_{rv}$ .

Figure 20. Long description

Panel A: A schematic diagram showing the vanishing reflection phenomenon. The diagram includes a reflecting surface labeled as G, and various regions labeled as 0, 1, and 2. The diagram shows the interaction of a Mach wave with the reflecting surface. Key points include A, R, and G, with angles labeled as beta 01, i, and mu 1. The vertical axis is labeled H A and the horizontal axis is labeled H rv. Panel B: A schematic diagram showing another configuration of the vanishing reflection phenomenon. This diagram also includes a reflecting surface labeled as G, and regions labeled as 0, 1, and 2. The diagram shows the interaction of a shock wave and a degenerated Mach wave with the reflecting surface. Key points include A, R, and D, with angles labeled as beta 01, i, and mu 1. The vertical axis is labeled H A and the horizontal axis is labeled H rv.

When $g\rightarrow 1$ , the initial point of the incident shock approaches the trailing edge expansion fan infinitely, so it will degenerate into a Mach wave, meaning that there is no shock reflection.

To see whether $g_{rv}\lt 1$ , we display some numerical results with $g$ approaching $1$ . We give in figure 21 the Mach number contour lines for $g=0.82$ , $0.95$ , $0.99$ and $0.995$ , at $M_{0}=6.2$ and $\theta _{w}=28.8^\circ$ .

Compared to figure 14(e), with $g=g_{\textit{tr}}=0.71$ and $\beta _{G}\approx 29^\circ$ (close to the von Neumann condition $\beta ^{(N)}(M_{0})=28.6^\circ$ ), here with $g=0.82$ , the incident shock has shock angle $\beta _{G}\approx 22.3^\circ$ , which is well below the von Neumann condition. For $g=0.95$ , the incident shock has shock angle $\beta _{G}\approx 12.7^\circ$ at the reflecting surface; see figure 21(b). For $g=0.99$ , the incident shock appears to be as weak as a Mach wave, but at the reflecting point, the incident shock has shock angle $\beta _{G}\approx 11.1^\circ$ , which is still well above the Mach angle $\mu =\arcsin ( {1}/{ M_{0}})= 9.28^\circ$ ; see figure 21(c). Figure 21(d) corresponds to $g=0.995$ and shows that $\beta _{G}\approx 10.4^\circ$ , which is close to but still slightly larger than the Mach angle $\mu = 9.28^\circ$ .

Figure 21.

Mach number contourlines at the maximum point $M_{0}=6.2$ and $ \theta _{w}=28.8$ $^{\circ}$ (a) $g=0.82$ , (b) $g=0.95$ , (c) $g=0.99$ , (d) $ g=0.995$ .

Despite it being anticipated that no reflection occurs when $g\rightarrow 1$ , we have not yet observed the appearance of such a limit $g_{\textit{nr}}$ , with $ g_{\textit{nr}}\lt 1$ , through numerical simulation with $g$ as large as $0.995$ .

For a sonic boom, a supersonic vehicle generates shock waves and expansion waves, which interact so that there is a sonic boom even at large distance (Reference CrowCrow1969; Carlson Reference Carlson1978). There appears to be no cut-off point at which the shock degenerates into a Mach wave, but the strength of the shock wave will decay gradually.

For the present problem, one would expect to prove analytically whether there is a reflection vanishing $g_{rv}$ satisfying $g_{rv}\lt 1$ . The problem of interaction between an expansion fan and a shock wave has been studied before (Moeckel Reference Moeckel1952; Rosciszewski Reference Rosciszewski1960; Hasimoto Reference Hasimoto1964; Li & Ben-Dor Reference Li and Ben-Dor1995; Meshkov et al. Reference Meshkov, Omelchenko and Uskov2002; Hillier Reference Hillier2007; Silnikov et al. Reference Silnikov, Chernyshov and Uskov2014; Uskov & Chernyshov Reference Uskov and Chernyshov2014; Bai Reference Bai2023; Baby et al. Reference Baby, Paramanantham and Rajesh2024). When the expansion wave is upstream of the shock, analytical solutions can be found (Silnikov et al. Reference Silnikov, Chernyshov and Uskov2014; Uskov & Chernyshov Reference Uskov and Chernyshov2014). In the present problem, the shock and the expansion fan in interaction are in the same side, and the expansion fan is downstream of the shock wave, so there is no analytical solution. Methods by characteristics (Hillier Reference Hillier2007; Bai Reference Bai2023; Baby et al. Reference Baby, Paramanantham and Rajesh2024) are approximate in nature and do not allow us to make a rigorous mathematical claim that there exists a $ g_{rv}$ with $g_{rv}\lt 1$ . Thus the existence of $g_{rv}$ remains an open question. Hasimoto (Reference Hasimoto1964) derived the curvature of the shock at the starting point of interaction between this shock and a downstream expansion fan from the same side. It is unclear whether this analytical approach can be extended to study the properties of the entire part of the shock under interaction.