2.1. A lower bound for $g_{\textit{min}}^{(\textit{MR})}$ corresponding to the minimum Mach stem height

Now, we derive a lower bound which can be used as an intermediate value to prove the inequality $g_{\textit{min}}^{(\textit{MR})}\gt g_{\textit{min}}^{(RR)}$ or $H_{R,min }^{(\textit{MR})}\gt H_{R,min }^{(RR)}$ through transitivity.

Consider MR at $H_{R}=H_{R,min }$ , as shown in figure 2. The slipline from the triple point $T$ intersects the leading characteristic line of the trailing-edge expansion fan at point O. The tangent of the slipline at the triple point intersects the leading characteristic line of the trailing-edge expansion fan at point $N^{\prime }$ .

Figure 2.

Schematic display for MR with various notations needed in the proof. The minimum Mach stem height ( $H_{NT}$ ) is the Mach stem height at $H_{N}=0$ .

Let the height of $N^{\prime }$ with respect to the reflecting surface be $ H_{N^{\prime }}$ . The Mach stem height $H_{T}$ can be considered as composed of two segments

(2.1) \begin{equation} H_{T}=H_{TN}+H_{N} ,\end{equation}

where $H_{N}=H_{N^{\prime }}$ , so

(2.2) \begin{align} H_{T}=H_{TN}+H_{N^{\prime }}. \end{align}

Using the geometric relations displayed in figure 2, it can be shown that the condition such that the reflected shock grazes the trailing edge leads to

(2.3) \begin{equation} \frac {{{H_{A1}}-}H_{TN}}{{\tan {\beta _{01}}}}+\frac {H_{R1}{-{H_{TN}}}}{{ \tan (\beta _{12}^{MR}-{\theta _{w}})}}=\frac {{{H_{A1}}-{H_{R1}}}}{{\tan { \theta _{w}}}} .\end{equation}

The total length of the horizontal line MN’, where M is the intersection of the line NN’ and the vertical line passing the vertex A, is equal to the wedge horizontal length plus the horizontal projection of the leading characteristic line RN’, so

(2.4) \begin{equation} \frac {{{H_{A1}}-}H_{TN}}{{\tan {\beta _{01}}}}+\frac {H_{TN}}{{\tan {\delta _{s}}}}=\frac {{{H_{A1}}-{H_{R1}}}}{{\tan {\theta _{w}}}}+\frac {{{H_{R1}}}}{{ \tan ({\mu _{2}}+{\delta _{s}})}} ,\end{equation}

where ${{\mu _{2}=\arcsin }}({1}/{M_{2}})$ is the Mach angle of the leading characteristics of the trailing-edge expansion fan and ${\delta _{s}}$ is the slipline angle as marked in figure 2.

Eliminating $H_{TN}$ in (2.3) and (2.4) gives the following expression for $ {{{H_{R1}}}}/{{{H_{A1}}}}$ :

(2.5) \begin{equation} \frac {{{H_{R1}}}}{{{H_{A1}}}}=\frac {{\dfrac {\psi {{t_{2}}}}{{\sin {\theta _{w} }}}+\dfrac {{{t_{1}}}}{{\tan {\theta _{w}}}}-\tan {\delta _{s}}}}{{{t_{2}}+ \dfrac {\psi {{t_{2}}}}{{\sin {\theta _{w}}}}+\dfrac {{{t_{1}}}}{{\tan {\theta _{w}}}}-\dfrac {{{t_{1}}}}{{\tan ({\mu _{2}}+{\delta _{s}})}}}} ,\end{equation}

where ${t_{1}}=\tan {\beta _{01}}\tan {\delta _{s}}$ , ${t_{2}}=\tan {\beta _{01}}-\tan {\delta _{s}}$ and $\psi$ is defined by $\psi = ({\sin (\beta _{12}^{MR}-\theta _{w})\sin (\beta _{01}-\theta _{w})})/({\sin (\beta _{01}+\beta _{12}^{MR}-\theta _{w})})$ . Here, $\beta _{12}^{MR}$ is the shock angle of the reflected shock and $\beta _{01}$ is the shock angle of the incident shock, and both shock angles can be determined by the triple point solution.

According to (1.2), we have

(2.6) \begin{equation} g_{\textit{min}}^{MR}=\frac {{{H_{R1}}+H}_{N}}{{{H_{A1}}+H}_{N}} .\end{equation}

Since $H_{N}\gt 0$ , we have

(2.7) \begin{equation} g_{\textit{min}}^{MR}\gt g_{\textit{min}}^{(LB)} ,\end{equation}

where

(2.8) \begin{equation} g_{\textit{min}}^{(LB)}=\frac {{{H_{R1}}}}{{{H_{A1}}}} \end{equation}

is the lower bound of $g_{\textit{min}}^{MR}$ . Here, $ {{{H_{R1}}}}/{{{H_{A1}}}}$ is given by (2.5).

2.2. Proof of $g_{\textit{min}}^{(\textit{MR})}\gt g_{\textit{min}}^{(LB)}\gt g_{\textit{min}}^{(RR)}$

For RR (see figure 1 a), Li & Ben-Dor (Reference Li and Ben-Dor1997) give

(2.9) \begin{equation} H_{R,min }^{(RR)}=\phi ^{RR}w ,\end{equation}

with

(2.10) \begin{equation} \phi ^{RR}=\frac {\sin (\beta _{12}^{RR}-\theta _{w})\sin (\beta _{01}-\theta _{w})}{\sin (\beta _{01}+\beta _{12}^{RR}-\theta _{w})}\text{.} \end{equation}

Here, $\beta _{12}^{RR}$ is the shock angle of the reflected shock.

At $H_{R}=H_{R,min }^{(RR)}$ , we have $w=w_{\textit{min}}$ where $w_{\textit{min}}$ is given by (1.5). Thus (2.9) should be replaced by

(2.11) \begin{align} H_{R,min }^{(RR)}=\phi ^{RR}w_{\textit{min}} ,\end{align}

and, when (1.5) is used, we get

(2.12) \begin{align} g_{\textit{min}}^{(RR)}=\phi ^{RR}\frac {1-g_{\textit{min}}^{(RR)}}{\sin \theta _{w}} ,\end{align}

which can be solved to give

(2.13) \begin{equation} g_{\textit{min}}^{(RR)}=\frac {\phi ^{RR}}{\sin \theta _{w}+\phi ^{RR}}\text{.} \end{equation}

Once the lower bound $g_{\textit{min}}^{(LB)}$ is obtained by (2.8), we can get the relative lower bound $\Delta g_{\textit{min}}^{(LB)}$ through

(2.14) \begin{equation} \Delta g_{\textit{min}}^{(LB)}=g_{\textit{min}}^{(LB)}-g_{\textit{min}}^{(RR)}\text{.} \end{equation}

The algorithm to find the lower bound $g_{\textit{min}}^{(LB)}$ involves flow parameters that are determined by shock wave relations. For this purpose, we first introduce the classical shock wave expressions.

We use $u$ to denote the upstream flow condition of an oblique shock wave, and $d$ to denote the downstream flow condition. The shock angle $\beta _{ud}$ is related to the flow deflection angle $\theta _{ud}$ through the shock angle relation

(2.15) \begin{equation} \tan \theta _{ud}=f_{\theta }(M_{u},\beta _{ud}) ,\end{equation}

where

(2.16) \begin{align} f_{\theta }(M,\beta)=\frac {2\left ( {M}^{2}{\sin }^{2}{\beta }-1\right) }{ \left ( {M}^{2}\left ( \gamma +\cos {2\beta }\right) +2\right) \tan {\beta }}. \end{align}

For given $M_{u}\gt 1$ and $\theta _{ud}$ (below the detached angle), the relation (2.15) admits two solutions for $\beta _{ud}$ ; the smaller one is called the weak solution, and the larger one is the strong solution.

The Mach number $M_{u}$ is then given by the shock relation for Mach number

(2.17) \begin{equation} M_{d}=f_{M}(M_{u},\beta _{ud}) ,\end{equation}

where

(2.18) \begin{align} f_{M}(M,{\beta)}=\sqrt {\frac {\left ( \gamma -1\right) {M}^{2}+2}{2\gamma {M} ^{2}{\sin }^{2}{\beta }-\left ( \gamma -1\right) }+\frac {2{M}^{2}{\cos }^{2}{ \beta }}{\left ( \gamma -1\right) {M}^{2}{\sin }^{2}{\beta }+2}} .\end{align}

Also, the pressure is given by the shock relation for pressure

(2.19) \begin{equation} p_{d}=p_{u}f_{p}(M_{u},\beta _{ud}) ,\end{equation}

where

(2.20) \begin{align} f_{p}(M,{\beta)}=1+\frac {2\gamma }{\gamma +1}\left ( {M}^{2}{\sin }^{2}{ \beta }-1\right). \end{align}

Now we outline the algorithm to find the lower bound $g_{\textit{min}}^{(LB)}$ .

Step 1. Consider any point $M_{0}$ and $\theta _{w}$ inside the dual-solution domain. Use the oblique wave expression to find the solution in region (1) of figure 2. Find the shock angle $\beta _{01}$ by solving $\tan \theta _{w}=f_{\theta }(M_{0},\beta _{01})$ for the weak solution. Find $M_{1}$ and $ p_{1}$ by solving $M_{1}=f_{M}(M_{0},\beta _{01})$ and $ p_{1}=p_{0}f_{p}(M_{0},\beta _{01})$

Step 2. Find $g_{\textit{min}}^{RR}$ through (2.13). For this purpose, we first solve for the two shock theory for RR to find $\beta _{_{12}}^{RR}$ . The shock angle $\beta _{_{12}}^{RR}$ is obtained by solving $\tan \theta _{w}=f_{\theta }(M_{1},{\beta _{_{12}}^{RR}})$ for the weak solution. The expression (2.10) for $\phi ^{RR}$ that appears in (2.13) can then be computed, so we obtain $g_{\textit{min}}^{(RR)}$ through (2.13).

Step 3. Find $g_{\textit{min}}^{(LB)}$ through (2.8). To do this, we first need to find the triple point solution. Using the flow parallel conditions $\theta _{12}=\theta _{w}-\delta _{s}$ , $\theta _{03}=\delta _{s}$ and the pressure balance relation $p_{3}^{MR}=p_{2}^{MR}$ , we solve

(2.21) \begin{align} \tan \theta _{12}=f_{\theta }(M_{1},{\beta _{_{12}}^{MR}})\text{, } p_{2}^{MR}=p_{1}f_{p}(M_{1},\beta _{12}^{MR})\text{ (weak solution)} ,\end{align}

and

(2.22) \begin{equation} \tan \theta _{03}=f_{\theta }(M_{0},{\beta _{_{03}}^{MR}})\text{, } p_{3}^{MR}=p_{0}f_{p}(M_{0},\beta _{03}^{MR})\text{ (strong solution)} .\end{equation}

Then, (2.5) is used for $ {{{H_{R1}}}}/{{{H_{A1}}}}$ . Finally, we get $g_{\textit{min}}^{(LB)}$ through (2.8).

Step 4. Finally, we get $\Delta g_{\textit{min}}^{(LB)}$ through (2.14). For other $M_{0}$ and $\theta _{w}$ , go to step 1 and repeat step 2 to step 4.

The lower bound $\Delta g_{\textit{min}}^{(LB)}$ thus obtained for $M_{0}$ and $ \theta _{w}$ lying inside the dual-solution domain is shown in figure 3. We see that $\Delta g_{\textit{min}}^{(LB)}$ is away from zero in the dual-solution domain except for the tip horn region (OPQ) which is near the left boundary of this dual-solution domain.

Thus we have shown that the inequality $\Delta g_{\textit{min}}^{(LB)}\gt 0$ and thus

(2.23) \begin{equation} g_{\textit{min}}^{(LB)}\gt g_{\textit{min}}^{(RR)} \end{equation}

holds within the dual-solution domain except in the tip horn region. Note that tip horn region is small.

Figure 3

Contourlines of the relative lower bound $\Delta g_{\textit{min}}^{(LB)}$ in the dual-solution domain. The tip horn region is defined as the region OPQ with $\Delta g_{\textit{min}}^{(LB)}\rightarrow 0$ .

By transitivity through (2.7) and (2.23), the inequality

(2.24) \begin{equation} g_{\textit{min}}^{(\textit{MR})}\gt g_{\textit{min}}^{(RR)} \end{equation}

holds and thus $H_{R,min }^{(\textit{MR})}\gt H_{R,min }^{(RR)}$ holds, within the dual-solution domain, except inside the tip horn region.

3.1. A road map to find $g_{\textit{min}}^{(\textit{MR})}$ and $\Delta g_{\textit{min}}$

For MR with Mach stem height $H_{T}$ (see figure 1 b), Li & Ben-Dor (Reference Li and Ben-Dor1997) give the following expression for $H_{R,min }^{(\textit{MR})}$ :

(3.1) \begin{equation} H_{R,min }^{(\textit{MR})}=H_{T}+\phi ^{MR}w ,\end{equation}

where

(3.2) \begin{equation} \phi ^{MR}=\frac {\sin (\beta _{12}^{MR}-\theta _{w})\sin (\beta _{01}-\theta _{w})}{\sin (\beta _{01}+\beta _{12}^{MR}-\theta _{w})}\text{.} \end{equation}

Here, $\beta _{12}^{MR}$ is the shock angle of the reflected shock in case of MR.

However, (3.1) cannot be directly applied here. The reason is that $H_{T}=H_{T,min }$ and $w=w_{\textit{min}}$ (see (1.5) for $w_{\textit{min}}$ ) at $H_{R}=H_{R,min }^{(\textit{MR})}$ , so expression (3.1) should be replaced by

(3.3) \begin{equation} H_{R,min }^{(\textit{MR})}=H_{T,min }+\phi ^{MR}w_{\textit{min}} .\end{equation}

We define the relative Mach stem height as $h_{T}= {H_{T}}/{H_{A}}$ . Using $H_{T,min }=h_{T,min }H_{A}$ and (1.5) in (3.3), and replacing $H_{R,min }^{(\textit{MR})}$ by $g_{\textit{min}}^{(\textit{MR})}H_{A}$ , we obtain the following expression for $g_{\textit{min}}^{(\textit{MR})}$ :

(3.4) \begin{equation} g_{\textit{min}}^{(\textit{MR})}=h_{T,min }+\frac {\phi ^{MR}}{\sin \theta _{w}}\left ( 1-g_{\textit{min}}^{(\textit{MR})}\right) ,\end{equation}

where $\phi ^{MR}$ is given by (3.2).

According to Hornung & Robinson (Reference Hornung and Robinson1982), $ {H_{T}}/{w}=h ( M_{0},\theta _{w},( {H_{R}}/{w}))$ , which can be equivalently written as

(3.5) \begin{equation} h_{T}=h_{T}\left ( M_{0},\theta _{w},g\right) \end{equation}

due to (1.4). Note that the functional form also depends on $\gamma$ , the ratio of the specific heats. Here, we do not consider varying $\gamma$ so this parameter is considered as a constant in (3.5).

Putting $g=g_{\textit{min}}^{(\textit{MR})}$ and $h_{T}=h_{T,min }$ in (3.5), we have

(3.6) \begin{equation} h_{T,min }=h_{T}\left ( M_{0},\theta _{w},g_{\textit{min}}^{(\textit{MR})}\right) .\end{equation}

Since $h_{T,min }$ and $ {\phi ^{MR}}/{\sin \theta _{w}}$ are functions of $M_{0}$ and $\theta _{w}$ , expression (3.4) means that $g_{\textit{min}}^{(\textit{MR})}$ is a function of $M_{0}$ and $\theta _{w}$ .

Then (3.4) can be solved for $g_{\textit{min}}^{(\textit{MR})}$ if we know expression (3.6).

3.2. Efficient method for $g_{\textit{min}}^{(\textit{MR})}$ and $\Delta g_{\textit{min}}$ when $h_{T}$ is a linear function of $g$

Li & Ben-Dor (Reference Li and Ben-Dor1997) stated that the calculations of Vuillon et al. (Reference Vuillon, Zeitoun and Ben-Dor1995) indicated that, within the forth-order approximation, the trajectory of the Mach stem foot (similar to point G in figure 1 b ) is a straight line, meaning that the Mach stem height may decrease linearly with $H_{R}$ . Schotz et al.(Reference Schotz, Levy, Ben-Dor and Igra1997, equation (21)) and Bai & Wu (Reference Bai and Wu2021) also suggested equivalent linear variation.

Bai & Wu (Reference Bai and Wu2021) considered several conditions and showed, through high fidelity numerical simulation, that the relative Mach stem height can be fitted by a linear formula, i.e.

(3.7) \begin{equation} h_{T}=Ag+B \end{equation}

in terms of the present notations. The linear formula fits the numerical results with $R^{2}$ above $0.99$ , where $R^{2}$ stands for the linear correlation coefficient.

Only one condition considered by Bai & Wu (Reference Bai and Wu2021) lies within the dual-solution domain. Here, we added two more cases in the dual-solution domain, with $A$ and $B$ determined by numerical simulation using the Fluent package with the choice of second-order Roe scheme in the same way as Bai & Wu (Reference Bai and Wu2021) but we increase the number grid points in the horizontal direction since the geometry is longer in the present calculation. The condition by Bai & Wu (Reference Bai and Wu2021) and the two new cases are shown in table 1.

Note that Bai & Wu (Reference Bai and Wu2021) suggested the grid number after grid convergence study. Here, we follow the same approach and display in figure 4 the relative Mach stem heights obtained by numerical simulation with four different values of $g$ . The numerical results are marked with black circles, the additional marks in figure 4 show that numerical results with a grid of $400\times 800$ points are acceptable since further refining the grid does not change the Mach stem height noticeably.

Similarly to case 1 studied by Bai & Wu (Reference Bai and Wu2021), the normalised Mach stem height is almost linear with respect to $g$ . In figure 4, a straight line $e_{1}e_{2}$ is marked across the numerical data and this straight line is given by (3.7), with $A$ and $B$ shown in table 1. For case 3, we also have similar behaviour and the fitted coefficients $A$ and $B$ are shown in table 1. In all cases, the linear form (3.7) holds with $R^{2}\gt 0.99$ .

Table 1.

The coefficients $A$ and $B$ for three cases. Case 1 is from Bai & Wu (Reference Bai and Wu2021). Case 2 is from the present calculation with $g=0.45,0.50,0.55,0.60$ . Case 3 is from the present calculation with $g=0.45,0.50,0.55,0.60$ .

Figure 4.

Numerical results of the normalised Mach stem heights as functions of $g$ for case 2.The line $e_{1}e_{2}$ is a straight line passing through Computational Fluid Dynamics (CFD) data. Open circles are CFD data using different grids.

In conditions such that the relative Mach stem height follows the linear form (3.7), we have

(3.8) \begin{equation} h_{T,min }=Ag_{\textit{min}}^{(\textit{MR})}+B \end{equation}

so that the expression (3.4) can be solved to give the following explicit expression for $g_{\textit{min}}^{(\textit{MR})}$ :

(3.9) \begin{equation} \overline {g}_{\textit{min}}^{{\small (MR)}}=\frac {\dfrac {\phi ^{MR}}{\sin \theta _{w}}+B}{1-A+\dfrac {\phi ^{MR}}{\sin \theta _{w}}}, \end{equation}

where $\overline {g}_{\textit{min}}^{(\textit{MR})}$ denotes $g_{\textit{min}}^{(\textit{MR})}$ when (3.8) is applied.

3.3. Predicted values of $\overline {g}_{\textit{min}}^{(\textit{MR})}$ and $\triangle \overline {g}_{\textit{min}}$ for three cases

For a given condition ( $M_{0},\theta _{w}$ ), if (3.7) holds and if the exact values of $A$ and $B$ are known (e.g. from high fidelity numerical simulation), then (3.9) yields the required values of $g_{\textit{min}}^{(\textit{MR})}=\overline {g}_{\textit{min}}^{{\small (MR)}}$ , and hence the required values of $\triangle g_{\textit{min}}=\triangle \overline {g}_{\textit{min}}$ , where

(3.10) \begin{equation} \triangle \overline {g}_{\textit{min}}=\overline {g}_{\textit{min}}^{(\textit{MR})}-g_{\textit{min}}^{(RR)} ,\end{equation}

with $g_{\textit{min}}^{(RR)}$ computed by (2.13). The quantity $g_{\textit{min}}^{(RR)}$ computed by the exact expression (2.13) is shown in figure 5.

Figure 5.

Contourlines of $g_{\textit{min}}^{(RR)}$ in the dual-solution domain. This value is given by the exact formula (2.13).

Table 2 gives $\overline {g}_{\textit{min}}^{(\textit{MR})}$ and $\triangle \overline {g}_{\textit{min}}$ thus obtained.

Table 2.

Predicted values of $\overline {g}_{\textit{min}}^{(\textit{MR})}$ and $\triangle \overline {g}_{\textit{min}}$ for the three cases shown in table 1.

Now let us give some details for case 1. Using (2.13), we get, for $M_{0}=4$ , $\theta _{w}=25^{o}$ ,

(3.11) \begin{equation} g_{\textit{min}}^{(RR)}=0.239 .\end{equation}

With $A=-1.031$ and $B=0.723$ from table 1, we get from (3.9) the value

(3.12) \begin{equation} \overline {g}_{\textit{min}}^{(\textit{MR})}=0.417 .\end{equation}

Hence, for case 1,

(3.13) \begin{align} \triangle \overline {g}_{\textit{min}}=0.178 .\end{align}

3.4. Numerical validation

Now we perform numerical simulation, using the same numerical approach as stated above, to check the predicted quantities of $\overline {g}_{\textit{min}}^{(\textit{MR})}$ and $\triangle \overline {g}_{\textit{min}}$ . To do this, we seek a $g=g_{min,CFD}^{l}$ value near $\overline {g}_{\textit{min}}^{(\textit{MR})}$ for which we get a unstable MR numerical solution, and a $g=g_{min,CFD}^{u}$ value near $\overline {g}_{\textit{min}}^{(\textit{MR})}$ for which we get a stable MR numerical solution. Once $g_{\textit{min} ,\textit{CFD}}^{u}$ is close to $g_{\textit{min} ,\textit{CFD}}^{l}$ , we may say that $g_{\textit{min}}^{(\textit{MR})}$ predicted by numerical simulation is

(3.14) \begin{align} g_{\textit{min} ,\textit{CFD}}=\frac {1}{2}(g_{\textit{min} ,\textit{CFD}}^{u}+g_{\textit{min} ,\textit{CFD}}^{l}). \end{align}

The difference between the predicted value $\overline {g}_{\textit{min}}^{(\textit{MR})}$ and $g_{\textit{min} ,\textit{CFD}}$ is defined by

(3.15) \begin{align} \varepsilon =\frac {g_{\textit{min} ,\textit{CFD}}-\overline {g}_{\textit{min}}^{(\textit{MR})}}{\overline {g}_{\textit{min}}^{(\textit{MR})}} .\end{align}

The values of $g_{\textit{min} ,\textit{CFD}}^{l}$ , $g_{\textit{min} ,\textit{CFD}}^{u},g_{\textit{min} ,\textit{CFD}}$ and $\varepsilon$ are shown in table 3. The Mach number contourlines at $g=g_{\textit{min} ,\textit{CFD}}^{l}$ and $g=$ $g_{\textit{min} ,\textit{CFD}}^{u}$ are displayed in figures 6, 7 and 8 for cases 1, 2 and 3. Note that for $g=g_{\textit{min} ,\textit{CFD}}^{l}$ the flow is unstable, so figures 6(a), 7(a) and 8(a) are a snapshot of the time-dependent result.

Table 3.

The values of $g_{\textit{min} ,\textit{CFD}}^{l}$ , $g_{\textit{min} ,\textit{CFD}}^{u}$ , $g_{\textit{min} ,\textit{CFD}}$ and $ \varepsilon$ for the three cases.

According to table 3, $\overline {g}_{\textit{min}}^{(\textit{MR})}$ is very close to $g_{\textit{min} ,\textit{CFD}}$ , with a relative difference less than $2\,\%$ . Consider for instance case 1, where the predicted value for $g_{\textit{min}}^{(\textit{MR})}$ is $\overline {g}_{\textit{min}}^{(\textit{MR})}=0.417$ (see (3.12)), while the numerical simulation gives $g_{\textit{min} ,\textit{CFD}}$ $=0.4225\pm 0.0025$ . The difference is $\varepsilon =1.\, 32\,\% \pm 0.60\,\%$ . Note that $g_{\textit{min} ,\textit{CFD}}^{l}$ and $g_{\textit{min} ,\textit{CFD}}^{u}$ are both (slightly) larger than $\overline {g}_{\textit{min}}^{(\textit{MR})}$ (possibly due to numerical error). For case 2 and case 3, we have $g_{\textit{min} ,\textit{CFD}}^{l}\lt \overline {g}_{\textit{min}}^{(\textit{MR})}\lt$ $g_{\textit{min} ,\textit{CFD}}^{u}$ .

Figure 6.

Mach number contours for case 1: (a) $g=0.420$ , (b) $g=0.425$ .

Figure 7.

Mach number contours for case 2: (a) $g=0.440$ , (b) $g=0.450$ .

Figure 8.

Mach number contours for case 3: (a) $g=0.350$ , (b) $g=0.360$ .

4.1. Dynamic transition problem

We have shown in § 3 that, within a large part of the dual-solution domain, the supercritical geometric threshold $g_{\textit{min}}^{(\textit{MR})}$ is larger than the subcritical one $g_{\textit{min}}^{(RR)}$ , i.e. there exists a dual-solution stability gap $\triangle g_{\textit{min}}=g_{\textit{min}}^{(\textit{MR})}-g_{\textit{min}}^{(RR)}\gt 0$ within the dual-solution domain except for its tip horn region. This means that, at any point ( $M_{0},\theta _{w}$ ) within this region, if the relative trailing-edge height $g$ satisfies

(4.1) \begin{equation} g_{\textit{min}}^{(RR)}\lt g\lt g_{\textit{min}}^{(\textit{MR})}, \end{equation}

then the RR solution is stable, and the MR solution is unstable. This would mean a new dynamic transition possibility: stable RR may transit to unstable MR when there is large-amplitude disturbance.

Before this is made clearer, we recall what happens in conventional dynamic transition, for which stable RR transits to stable MR by large-amplitude disturbance, for $g$ outside the dual-solution stability gap.

It is known that, in the dual-solution domain, RR may transit to MR if the amplitude of the disturbance exceeds a certain level (Ivanov et al. Reference Ivanov, Klemenkov, Kudryavtsev, Fomin and Kharitonov1997, Reference Ivanov, Markelov, Kudryavtsev and Gimelshein1998, Reference Ivanov, Kudryavtsev and Khotyanovskii2000). The dynamic transition process can be understood by following the time history of this transition using numerical simulation. Various forms of disturbance have been considered. Ivanov et al. (Reference Ivanov, Klemenkov, Kudryavtsev, Fomin and Kharitonov1997) considered disturbances in the form of strong short-time changes in the free-stream velocity. Kudyavtev et al. (Reference Kudryavtsev, Khotyanovsky, Ivanov and Vandromme2002) used three types of upstream perturbation: inlet pressure wave disturbances (including shock waves and rarefaction waves), inlet contact discontinuity disturbances and localised density disturbances. All these types of disturbances have been shown to be able to cause dynamic transition from RR to MR.

The dynamic transition process has been also studied theoretically. Mouton & Hornung (Reference Mouton and Hornung2007) assumed a single but evolutionary MR, which satisfies a steady flow model when the reference frame is attached to the moving triple point, and built a dynamic transition model that tracks the growth of Mach stem height during the transition. Li et al. (Reference Li, Gao and Wu2011) followed the work of Mouton & Hornung (Reference Mouton and Hornung2007) and included more fine structures than a single unsteady MR to build a dynamic transition model. They also used a local discontinuity to force the transition from RR to MR, and the initial-period Riemann solution of the local discontinuity interacted with the initial RR to evolve the flow to steady MR. He & Shi (Reference He and Shi2025) studied dynamic transition from RR to MR under velocity perturbation, for $H_{R}\gt H_{R,max }$ . The evolution of the Mach stem height was predicted theoretically, when the effect of the trailing-edge expansion fan was considered.

The present study is concerned with dynamic transition from RR to MR due to local disturbance. Felthun & Skews (Reference Felthun and Skews2002) and Felthun & Skews (Reference Felthun and Skews2004) examined the transition dynamics between RR and MR, when the wedge angle is changing in time. Margha et al. (Reference Margha, Hamada and Eltaweel2023) investigated the dynamic transition of unsteady supersonic flow from MR to RR over a horizontally moving wedge. Naidoo & Skews (Reference Naidoo and Skews2011) studied dynamic transition from RR to MR when the wedge is rapidly rotating. Naidoo & Skews (Reference Naidoo and Skews2014) examined the effects of wedge rotation on dynamic transition from MR to RR. Goyal et al. (Reference Goyal, Sameen, Jayachandran and Rajesh2021) also studied the dynamic effects in the transition from RR to MR when the wedge is rotating, notably, they considered the effect of wedge rotation speeds on the transition conditions, the growth rate of the Mach stem and motion of the reflection point.

4.2. Dual-solution unstable condition and disturbance to inaugurate dynamic transition

Here, we consider dynamic transition in the dual-solution domain, for a geometrical set-up which has a trailing-edge height above the lower limit $H_{R,min }$ for RR, but below the lower limit $H_{R,min }$ for MR. In this situation, RR should transit to unstable MR, unlike the dynamic transition previously studied, in which RR transits to stable MR.

For numerical simulation, we choose the specific condition $M_{0}=4,\theta _{w}=25^{o}$ . We have shown, in § 3.3, that

(4.2) \begin{align} g_{\textit{min}}^{(RR)}=0.239\text{, }g_{\textit{min}}^{(\textit{MR})}=0.417 ,\end{align}

for this condition. Thus, if we choose a $g$ such that $0.239\lt g\lt 0.417$ , then under some disturbance, RR could transit to unstable MR, in contrast to the transition from RR to stable MR studied earlier for which $g$ is above the lower limit. The value $g=0.328$ meets such a condition, i.e. with $g=0.328$ , RR could transit to unstable MR.

Following Kudyavtev et al. (Reference Kudryavtsev, Khotyanovsky, Ivanov and Vandromme2002), we use a contact discontinuity disturbance at the inlet. Starting from the steady numerical solution of RR, we put in the inlet, for a time interval $t_{disturb}$ , a disturbance of the density $\bigtriangleup \rho /\rho _{0}$ . Since this is a contact discontinuity, the pressure and velocity are kept unchanged. This disturbance then causes a disturbance of the Mach number $M^{\prime }=M_{0}\sqrt {1+\bigtriangleup \rho /\rho }$ so the disturbance $\bigtriangleup \rho \lt 0$ induces a decrease of Mach number to be above the detachment condition so that transition from RR to MR is made possible. The disturbance is either given in the entire height of the inlet, or is localised by giving the disturbance only in a height of $({1}/{20})H_{A}$ counting from the reflecting surface (approximately 10 cells). The duration $t_{disturb}$ of disturbance, measured with $\tau _{disturb}= {t_{disturb}}/({H_{A}/a_{0}})$ where $a_{0}$ is the sound speed at the inlet, is also a factor to be considered.

The initial RR result is obtained in a similar way as in the numerical simulation of § 3.2, i.e. we use a second-order Roe scheme in the same way as Bai & Wu (Reference Bai and Wu2021), and the grid number is $400\times 800$ . During transition calculation, unsteady calculation with second-order accuracy is used, with a time step of $\Delta t\lt 10^{-4}({H_{A}}/{a_{0}})$ , following Li et al. (Reference Li, Gao and Wu2011).

Several test cases with various values of these factors are given in table 4. Numerical simulation shows three situations: failure to inaugurate any MR, transition from RR to MR to unstart flow (here called direct transition) and transition from RR to MR to RR (here called inverted transition).

Table 4.

Dynamic transition in the double solution domain with $(M_{0}, \theta _{w})=(4$ , $25^{o})$ (at which $g_{\textit{min}}^{RR}=0.239$ , $g_{\textit{min}}^{MR}=0.417$ ) and $g=0.328$ . Type IV SI means type IV shock interference.

For case 1, the disturbance is applied locally in a stripe close to the reflected surface, with relatively small discontinuity and short duration. We observe no transition, i.e. no MR structure is produced at any time. For other conditions, we have either direct transition or inverted transition.

4.3. Direct dynamic transition type I: from RR to MR to unstart

Compared with case 1, cases 2 and 3 increase the intensity of the density disturbances. We observe `RR to MR to unstart’ transition. For case 4 and case 5, the duration of density is increased compared with case 1, and we also observe `RR to MR to unstart’ transition. Figure 9 displays for case 4 the Mach number at several stages of the transition process.

Figure 9.

Mach number contours for direct dynamic transition type I (case 4 in table 4).

Stage 1: initial RR. The RR result, not displayed, is treated as the initial condition before the upstream disturbance is introduced.

Stage 2: disturbance propagation. The upstream disturbance is generated at the inlet, and then propagates toward the reflecting point. Figure 9(a) is the result at some instance. This upstream disturbance has not yet touched the reflection point so the RR configuration near the reflecting point is not yet affected.

Stage 3: disturbance RR interaction stage. The disturbance reaches the reflecting point and strengthens the incident shock at the reflecting point (figure 9 b). Locally, the shock angle of the incident shock overtakes the detachment condition so the local RR structure transits to MR (figure 9 c).

Stage 4: pseudo-steady MR stage (figure 9 d). The density disturbance has fully transmitted the Mach stem and there remains a pure pseudo-steady MR structure. For conventional dynamic transition as considered by Kudyavtev et al. (Reference Kudryavtsev, Khotyanovsky, Ivanov and Vandromme2002) and Mouton & Hornung (Reference Mouton and Hornung2007), the MR will become stable. Here, since $g$ lies within the dual-solution stability gap, the MR cannot be stabilised, and it will propagate towards the upstream direction.

Stage 5: unsteady double MR stage. The reflected shock of the pseudo-steady MR, after grazing the trailing edge, reflects at the lower wedge surface, and creates another pseudo-steady MR structure for the present condition. The lower MR and upper MR both propagate toward the inlet (figure 9 e).

Stage 6: unstart subsonic flow (figure 9 f). The double MR structure has touched the inlet and a shock is formed at the inlet. This shock would become a bow shock once a steady state could be reached. The flow downstream becomes subsonic, and corresponds to what we call unstart flow.

4.4. Direct dynamic transition type II: from RR to MR + type IV shock interference to unstart

For case 6, both the intensity of density disturbance and the duration of disturbance are increased compared with case 1. We observe the so-called direct transition type II, as displayed in figure 10, which shows the Mach number at several stages of the transition process.

Stage 1: initial RR. The RR result, shown in figure 10(a), is treated as the initial condition before the upstream disturbance is introduced.

Figure 10.

Mach number contours for direct dynamic transition (case 6 in table 4).

Stage 2: disturbance propagation. The upstream disturbance is generated at the inlet, and then propagates toward the reflecting point. Figure 10(b) is the result at $t\approx ({1}/{2})t_{disturb}$ . This upstream disturbance has not yet touched the reflection point so the RR configuration near the reflecting point is not yet affected.

Stage 3: disturbance RR interaction stage. The disturbance reaches the reflecting point and strengthens the incident shock at the reflecting point (figure 10 c). For the present condition, this occurs at $t\approx t_{disturb}$ , when the upstream disturbance at the inlet is terminated. Locally, the shock angle of the incident shock overtakes the detachment condition so the local RR structure transits to MR (figure 10 d).

Stage 4: disturbed MR + type VI interference stage (figure 10 e, see figure 11 for a schematic display). Note that type VI interference does not appear in direct dynamic transition type I. The Mach stem of MR is still subjected to the interaction of the upstream density disturbance. This disturbance has a slipline (PM) almost parallel to the reflecting surface. The interaction between this slipline and the Mach stem, at point M, leads to a transmitted slipline (MQ). The type VI interference structure is composed of the reflected shock of MR, and a recompression shock over the turning point S of the slipline of MR. This interference leads to a shock that is one part of the reflected shock of the original RR. The recompression shock is due to the flow, which is initially parallel to the slipline, will be deflected to be parallel to the reflecting surface. Note that type VI interference also appears in the conventional dynamic transition problem studied by Kudyavtev et al. (Reference Kudryavtsev, Khotyanovsky, Ivanov and Vandromme2002) and Li et al. (Reference Li, Gao and Wu2011).

Figure 11.

Schematic display of the MR + type VI interference. The SM interaction means the interaction between the edge of the density disturbance (slipline PM) and the Mach stem, which gives a transmitted slipline (MQ).

Stage 5: pseudo-steady MR. The density disturbance has fully transmitted the Mach stem. Both this disturbance and the type VI interference structure have propagated far downstream, so there remains a pure pseudo-steady MR structure (figure 10 e). For conventional dynamic transition as considered by Kudyavtev et al. (Reference Kudryavtsev, Khotyanovsky, Ivanov and Vandromme2002) and Mouton & Hornung (Reference Mouton and Hornung2007), the MR will become stable. Here, since $g$ lies within the dual-solution stability gap, the MR cannot be stabilised, and it will propagate towards the upstream direction.

Stage 6: unsteady double MR stage. The reflected shock of the pseudo-steady MR, after grazing the trailing edge, reflects at the lower wedge surface, and creates another pseudo-steady MR structure for the present condition. The lower MR and upper MR both propagate toward the inlet (figure 10 f).

Stage 7: unstart subsonic flow (figure 10 g). The double MR structure has touched the inlet and a shock is formed at the inlet. The flow downstream becomes subsonic, and corresponds to what we call unstart flow.

4.5. Inverted transition: from RR to MR to RR

Figure 12 shows the results for case 7. The disturbance is applied to the whole inlet so the entire incident shock will be disturbed. We observed what we call here inverted transition.

Stage 1: RR to MR transition. This incident shock will be strengthened so it causes RR to MR transition once the disturbance has reached the reflecting point. Figure 12(a) displays the result at a moment when RR has transited to MR. The early stage during RR to MR transition is similar to the cases considered above and is not shown here.

Stage 2: weakening MR. The highly disturbed part of the incident shock is weakened by the ending part of the density disturbance, so the Mach stem reduces its height (figure 12 b,c).

Figure 12.

Mach number contours for inverted dynamic transition (case 7 in table 4).

Stage 3: MR back into RR. The transition process is inverted, and the MR transits back into a highly disturbed RR (figure 12 d–f).

Stage 4: stable RR. Finally, we get back the initial RR structure.

The inverted dynamic transition occurs when the density disturbance first strengthens the incident shock (so that detachment condition is reached) and then weakens the incident shock (so that von Neumann condition is reached).