2.1. Updated Mach stem height model

Figure 2 is a schematic of Mach reflection, which shows that the flow field mainly includes four structures: the reflected shock ( $\textit{TK}$ ), the transmitted Mach waves ( $\textit{BGKF}$ ), the free part of the slipline ( $\textit{TB}$ ), and the interactive part of the slipline ( $\textit{BG}$ ).

The flow parameters in the vicinity of the triple point ( $T$ ) are associated with the triple-shock theory (von Neumann Reference von Neumann1943). Specifically, as shown in figure 3(a), these parameters include the shock angles ( $\beta _{01}^{T}$ , $\beta _{12}^{T}$ and $\beta _{03}^{T}$ ), the flow deflection angles ( $\theta _{01}^{T}$ , $\theta _{12}^{T}$ and $\theta _{03}^{T}$ ), pressure ( $p_{1}^{T}$ , $p_{2}^{T}$ and $p_{3}^{T}$ ) and Mach numbers ( $M_{1}^{T}$ , $M_{2}^{T}$ and $M_{3}^{T}$ ). The Mach angle $\mu$ is given by $\mu =\arcsin ({1}/{M})$ .

2.1.1. The reflected shock sub-model

As shown in figure 2, the reflected shock is divided into two parts ( $\textit{TF}$ and $\textit{FK}$ ) depending on whether it interacts with the trailing edge expansion fan. The intersection of the reflected shock with the first characteristic line is point $F$ . The intersection of the reflected shock with the last characteristic line is point $K$ . The flow parameters near the reflected shock are illustrated in figure 3(b). Superscripts $(u)$ and $(d)$ denote the upstream and downstream flow parameters of the reflected shock, respectively.

For $\textit{TF}$ , the reflected shock is straight and the shock intensity is constant. Based on geometrical relations, the coordinates of point $F$ are given by

(2.1) \begin{equation} \left \{ \begin{array}{l} x_{F}=\dfrac {H_{\!R}-y_{T}+x_{T}\tan \left ( \beta _{12}^{T}-\theta _{w}\right ) +x_{R}\tan \left ( \mu _{1}^{T}+\theta _{w}\right ) }{\tan \left ( \beta _{12}^{T}-\theta _{w}\right ) +\tan \left ( \mu _{1}^{T}+\theta _{w}\right ) }, \\[15pt] y_{F}=y_{T}+\left ( x_{F}-x_{T}\right ) \tan \left ( \beta _{12}^{T}-\theta _{w}\right )\!. \end{array} \right . \end{equation}

For $\textit{FK}$ , due to its interaction with the expansion wave, the reflected shock is curved. Consider a typical point $f$ on $\textit{FK}$ as shown in figure 3(b).

1. (i) The flow parameters in region (5) are related to those in region (4) by the Prandtl–Meyer relation (B4) and isentropic relations (B5).

2. (ii) The flow parameters in region (6) are related to those in region (4) by shock relations (B1)–(B3).

3. (iii) The flow parameters in region (7) are related to those in region (4) by the Prandtl–Meyer relation (B4) and the isentropic relation (B5), and the flow parameters in region (8) are related to those in region (5) by shock relations (B1)–(B3).

4. (iv) The pressures and flow angles in region (7) and region (5) are equal.

Based on these relations, the shock angle $\beta _{\!f}$ and the downstream flow parameters (the flow deflection angle $\theta _{\!f}^{(d)}$ , the Mach number $M_{\!f}^{(d)}$ and the pressure $p_{\!f}^{(d)}$ ) at point $f$ can be obtained.

To determine the shape of the reflected shock, for an arbitrary point $f$ on the reflected shock, it is also located on the straight characteristic line $Rf$ , thus there are two geometric relations:

(2.2) \begin{equation} y_{\!f}-y_{R}=(x_{\!f}-x_{R})\tan (-\theta _{\!f}^{(u)}-\mu _{\!f}^{(u)}), \end{equation}

(2.3) \begin{equation} {\rm d}y_{\!f}={\rm d}x_{\!f}\tan (\beta _{\!f}-\theta _{\!f}^{(u)}). \end{equation}

These relations give the coordinates of any point $f$ on the curved shock.

2.1.2. The free part of the slipline sub-model

Figure 4.

Flow schematics: (a) slipline and transmitted Mach wave; (b) three families of characteristics $C_{+}$ , $C_{-}$ and $ C_{0}$ .

As shown in figure 4(a), the free part of the slipline ( $\textit{TB}$ ) does not interacts with the transmitted Mach waves. According to Gao & Wu (Reference Gao and Wu2010) and Bai & Wu (Reference Bai and Wu2017), the shape of the free part of the slipline can be determined as follows.

(i) The slipline is curved, with varying slipline angle defined by

(2.4) \begin{equation} \tan \theta _{s}=-\frac {{\rm d}H_{s}}{{\rm d}x}\text{.} \end{equation}

(ii) The average flow parameters in the flow duct below the slipline satisfy the isentropic quasi-one-dimensional flow model

(2.5) \begin{equation} \frac {H_{s}}{H_{T}}=\frac {M_{m}}{M_{s}}\left ( \frac {\vartheta (M_{s})}{ \vartheta (M_{m})}\right ) ^{({\gamma +1})/{\left ( 2(\gamma -1)\right ) }},\quad p_{s}=p_{m}\left ( \frac {\vartheta (M_{m})}{\vartheta (M_{s})} \right ) ^{{\gamma }/({\gamma -1})}, \end{equation}

where $\vartheta (M)=1+(( {\gamma -1})/{2})M^{2}$ , and $M_{m}$ and $p_{m}$ denote the initial average Mach number and pressure of the duct, respectively.

(iii) Secondary Mach waves are generated above the slipline; the flow parameters satisfy the Prandtl–Meyer relation (B4) and isentropic relation (B5) with varying $\theta _{s}$ defined by (2.4).

(iv) The pressure obtained in step (ii) is balanced with $p_{s}$ determined by (2.5). This provides a relation for $\theta _{s}$ , then the shape can be solved by (2.4).

2.1.3. The sub-models of the transmitted Mach wave and the interactive part of the slipline

In two-dimensional steady rotational supersonic flow (region $\textit{FBGK}$ shown in figure 2), there are three families of characteristics (as shown in figure 4 b) along which hold the following compatibility relations (Holt Reference Holt1956; Liepmann & Roshko Reference Liepmann and Roshko1957):

(2.6a) \begin{equation} \begin{array}{llll} -{\rm d}\theta -\varPhi (p,M)\,{\rm d}p=0 & \text{along} & {\rm d}y/{\rm d}x=\tan \left ( -\theta -\mu \right )\!, & \text{(}C_{-}\text{)} \end{array} \end{equation}

(2.6b) \begin{equation} \begin{array}{llll} {\rm d}p-a^{2}\,{\rm d}\rho =0,\quad \rho V\,{\rm d}V+{\rm d}p=0 & \text{along} & {\rm d}y/{\rm d}x=\tan \left ( -\theta \right )\!, & \text{(}C_{0}\text{)} \end{array} \end{equation}

(2.6c) \begin{equation} \begin{array}{llll} -{\rm d}\theta +\varPhi (p,M)\,{\rm d}p=0 & \text{along} & {\rm d}y/{\rm d}x=\tan \left ( -\theta +\mu \right )\!, & \text{(}C_{+}\text{)} \end{array} \end{equation}

where

(2.7) \begin{equation} \varPhi (p,M)=\frac {\sqrt {M^{2}-1}}{\gamma M^{2}p}. \end{equation}

Consider now the interactive part of slipline $\textit{BG}$ (figure 2), for an arbitrary point $c$ on $\textit{BG}$ , as shown in figure 4(a).

1. (i) The flow parameters along the slipline $\textit{TBc}$ (which belongs to $C_{0}$ ) satisfy the compatibility relations (2.6b ).

2. (ii) The flow parameters along characteristic line $\textit{fc}$ (which belongs to $C_{-}$ ) satisfy the compatibility relations (2.6a ).

3. (iii) The flow in the duct below the slipline is a quasi-one-dimensional flow, satisfying (2.5).

4. (iv) The pressure above and below the slipline is balanced.

5. (v) The slipline angle $\theta _{c}$ is related to $y_{c}$ by ${{\rm d}y_{c}}/{{\rm d}x_{c}}=-\tan \theta _{c}$ .

The characteristic line $\textit{fc}$ is approximated by the second-order approximate model of Li & Ben-Dor (Reference Li and Ben-Dor1997):

(2.8) \begin{equation} y_{c}-y_{\!f}=(x_{c}-x_{\!f})\arctan \frac {2\tan \delta _{\!f}^{(\textit{fc})}+\tan (\delta _{c}^{(\textit{fc})}-\delta _{\!f}^{(\textit{fc})})}{2-\tan \delta _{\!f}^{(\textit{fc})}\tan (\delta _{c}^{(\textit{fc})}-\delta _{\!f}^{(\textit{fc})})}, \end{equation}

where $\delta _{\!f}^{(\textit{fc})}=-\theta _{\!f}^{(d)}-\mu _{\!f}^{(d)}$ and $\delta _{c}^{(\textit{fc})}=-\theta _{c}-\mu _{c}$ . The compatibility relations are approximated by first-order differential forms. Then the above models can be solved to give the unknowns $M_{c}$ , $p_{c}$ , $\theta _{c}$ , $M_{s}$ , $p_{s}$ , $ x_{c}$ , $y_{c}$ . See Bai (Reference Bai2023) for details.

2.2. Comparison of the updated model and numerical results

Figure 5.

Analytical solution superimposed on the Mach number contour obtained by for $M_{0}=4$ , $ \theta _{w}=30^{\circ }$ and $g=0.6$ .

We present a comparative analysis between the predictions of the updated model and the results obtained from computational fluid dynamics (CFD). The CFD settings are shown in Appendix A.

Figure 5 illustrates a comparison of the flow field structure predicted by the updated model (solid lines) and the Mach number contour from numerical simulations (dashed lines) under specific conditions $M_0=4,\ \theta _{w}=30^{\circ }$ and $g=0.6$ (normalised wedge trailing edge height defined as $H_R/H_A$ ). Although some discrepancies remain in the shape of the Mach stem due to the absence of this sub-model in the updated framework, the updated model demonstrates good agreement with the numerical results, accurately capturing both the shape of the reflected shock wave and the height of the Mach stem.

Table 1 presents a comparison of Mach stem heights under different inflow and geometric conditions, including results from numerical simulations, predictions from the updated model, and predictions from several classical analytical models (Li & Ben-Dor Reference Li and Ben-Dor1997; Mouton & Hornung Reference Mouton and Hornung2007; Bai & Wu Reference Bai and Wu2017). It shows that in most cases, the updated model provides predictions that more closely match the numerical simulation results.

Table 1.

The Mach stem height obtained from theoretical models and CFD under different inflow and geometrical conditions.

Figure 6 illustrates the variation of normalised Mach stem height (= $H_T/H_A$ ) with respect to $g$ under $M_{0}=4$ , $\theta _{w}=25^{\circ }$ (figure 6 a) and $M_{0}=4$ , $ \theta _{w}=30^{\circ }$ (figure 6 b). The scattered points represent the results from CFD, while the solid line indicates the predictions of the updated model. The dash-dot-dotted, dashed and dash-dotted lines correspond to the models of Li & Ben-Dor (Reference Li and Ben-Dor1997), Bai & Wu (Reference Bai and Wu2017) and Bai & Wu (Reference Bai and Wu2021), respectively. Based on figure 6, the following two key observations can be summarised.

1. (i) In most cases, the updated model yields more accurate predictions of Mach stem height, exhibiting closer agreement with numerical results, although slightly larger errors are observed in a few specific cases.

2. (ii) The curve predicted by the updated model (solid line) is essentially parallel to scattered points obtained from numerical simulations. This indicates that the updated model is particularly effective in capturing the correct slope of the linear variation between $H_T/H_A$ and $g$ . The minor discrepancies observed in specific cases are mainly due to systematic deviations.

Figure 6.

Comparison of different models and the CFD result: (a) $M_{0}=4$ , $ \theta _{w}^{\circ }=25^{\circ }$ ; (b) $M_{0}=4$ , $ \theta _{w}=30^{\circ }$ .

In addition, a comparison among various analytical models (Azevedo & Liu Reference Azevedo and Liu1993; Li & Ben-Dor Reference Li and Ben-Dor1997; Mouton & Hornung Reference Mouton and Hornung2007; Bai & Wu Reference Bai and Wu2017; and the updated model), CFD simulations (Vuillon et al. Reference Vuillon, Zeitoun and Ben-Dor1995; Mouton & Hornung Reference Mouton and Hornung2007; and current CFD results) and experimental results (Hornung & Robinson Reference Hornung and Robinson1982; Mouton & Hornung Reference Mouton and Hornung2008) is presented in figure 7, where the four cases were given by Hornung & Robinson (Reference Hornung and Robinson1982).

The experimental study of Hornung & Robinson (Reference Hornung and Robinson1982) alters the shock angle by rotation of the wedge, so when the shock angle of the wedge changes, the ratio $ H_{\!R}/w$ also varies. The updated model and the present CFD results are based on a fixed $H_{\!R}/w$ . Some previous models also assumed $H_{\!R}/w$ to be fixed. It is seen that the updated model compares with the present CFD results well in all four cases.

Figure 7.

Comparison of the updated Mach stem height ( $H_T/w$ ) calculations against previous works for $H_R/w=0.4$ and $M_0=2.84, 3.49, 3.98, 4.96$ .

3.1. The influence of the free part of the slipline sub-model

There are two primary approaches for modelling the free part of the slipline $ TB$ . The first approach assumes $\textit{TB}$ to be a straight line, a simplification commonly adopted in early Mach reflection theories. The second approach treats $\textit{TB}$ as a curved line; it was modelled by Gao & Wu (Reference Gao and Wu2010) first, and later applied in several following studies, such as Bai & Wu (Reference Bai and Wu2017). The special formulations for each approach are presented below.

(i) Straight (e.g. Li & Ben-Dor Reference Li and Ben-Dor1997; Mouton & Hornung Reference Mouton and Hornung2007). In this approach, the coordinates $(x,y)$ along the slipline satisfy

(3.1) \begin{equation} y-y_{T}=-(x-x_{T})\tan (\theta _{03}^T). \end{equation}

(ii) Curved (e.g. Gao & Wu Reference Gao and Wu2010; Bai & Wu Reference Bai and Wu2017; the updated model). The details of this model, including the assumed conditions and the expressions of the relevant flow parameters, are presented in § 2.1.2.

Figure 8 compares the predicted Mach stem heights obtained via different methods of the free part of the slipline, with CFD results under $M_{0}=4$ , $\theta _{w}=25^{\circ }$ and $M_{0}=4$ , $ \theta _{w}=30^{\circ }$ . The results indicate that assuming $\textit{TB}$ to be a straight line will reduce the predictive accuracy of the model and lead to irregular errors. It can be observed that the underestimation becomes more pronounced when $g$ is smaller, whereas the predicted Mach stem height exceeds the CFD result when $g$ is larger. The curved-line model shows more consistent agreement with the numerical data.

The results demonstrate that the use of the curved sub-model for the free part of the slipline yields better agreement with numerical results, suggesting that the curved assumption more accurately reflects the physical behaviour of the flow.

Figure 8.

The influence of the free part of the slipline on the accuracy of model: (a) $M_{0}=4$ , $ \theta _{w}=25^{\circ }$ ; (b) $M_{0}=4$ , $ \theta _{w}=30^{\circ }$ .

3.2. The influence of the reflected shock sub-model

There are two primary approaches for modelling the reflected shock. The special formulations for each approach are presented as follows.

1. (i) Identical parameter sub-model (Bai & Wu Reference Bai and Wu2017). In this approach, the reflected shock is represented as a curved line with analytical expressions. Most of the applied flow relations are consistent with those in the updated model described in § 2.1.1, including the Prandtl–Meyer relations, characteristic theory and oblique shock relations. However, in this sub-model, the flow parameters in regions (7) and (8) (shown in figure 3 b) are assumed to be equal.

2. (ii) Distinct parameter sub-model (the updated model). The detailed conditions and the corresponding expressions of the relevant flow parameters are provided in § 2.1.1. Compared with the identical parameter sub-model, the flow parameters in regions (7) and (8) (shown in figure 3 b) are inconsistent. The parameters in region (7) are used to solve the transmitted Mach wave and interactive part of the slipline, while the parameters in region (8) are used to advance the calculation of the reflected shock wave.

Figure 9 compares the predicted Mach stem heights from the two methods against numerical simulations under $M_{0}=4$ , $ \theta _{w}=25^{\circ }$ and $M_{0}=4$ , $\theta _{w}=30^{\circ }$ . The results show that the identical parameter sub-model reduces predictive accuracy and consistently leads to an overestimation of Mach stem height compared to the distinct parameter sub-model, particularly at higher wedge angles. In contrast, the distinct parameter sub-model shows closer agreement with the numerical data.

Figure 9.

Comparison of Mach stem height predictions from different sub-models for reflected shock against CFD for $M_{0}=4$ : (a) $ \theta _{w}=25^{\circ }$ , (b) $ \theta _{w}=30^{\circ }$ .

Figure 10.

Comparison of the results obtained from the distinct parameter sub-model and the numerical method for $M_{0}=4$ , $ \theta _{w}=26^{\circ }$ , $g=0.6$ . The parameters are defined along the interactive part of the reflected shock ( $\textit{FE}$ ), located just downstream of the interactive part of the reflected shock.

Furthermore, figure 10 displays a comparison of flow parameters just downstream of the reflected shock $\textit{FE}$ (see figure 2) between the distinct parameter sub-model and CFD. Figure 10(a) displays the comparison for the local slope $ \delta _{\!f}^{(\textit{fc})}$ , and figure 10(b) displays the comparison for the pressure ratio $p_f^{(d)}/p_0$ . Their abscissae are the same, i.e. the dimensionless distance $\bar X_{EF}$ ( $ =(x-x_{F})/(x_{E}-x_{F})$ ). Overall, the distinct parameter sub-model yields good agreement with numerical simulations.

In general, the distinct parameter sub-model provides a more physically accurate representation of the flow field.

3.3. The influence of the transmitted Mach wave sub-model

Though there are three primary approaches for the sub-model of the transmitted Mach wave, the sub-models of Li & Ben-Dor (Reference Li and Ben-Dor1997) and Bai & Wu (Reference Bai and Wu2017) have a constant solution on pressure $p$ and flow angle $\theta$ , therefore they are categorised here as straight models. Further details about each approach are summarised as follows.

1. (i) Straight (e.g. Li & Ben-Dor Reference Li and Ben-Dor1997 and Bai & Wu Reference Bai and Wu2017). This approach treats the characteristic line $\textit{fc}$ as straight, and the flow parameters ( $p$ and $\theta$ ) along $\textit{fc}$ are constant.

2. (ii) Curved (the updated model). In this approach, the characteristic line $ fc$ is treated as a curved line with small curvature (see (2.8)). The flow parameters ( $p$ and $\theta$ ) satisfy the $C\_$ compatibility relation (2.6a ).

Figure 11 shows a comparison of the results from the curved model, the straight model and CFD under $M_{0}=4$ , $\theta _{w}=25^{\circ }$ and $M_{0}=4$ , $\theta _{w}=30^{\circ }$ . It can be seen that the results obtained from the straight model and the curved model differ only slightly, and are nearly identical under specific conditions. However, the trend of Mach stem height variation given by the curved model agrees more accurately with the CFD results than those of the straight model.

Figure 11.

Comparison of different methods on the transmitted Mach waves: (a) $M_{0}=4$ , $ \theta _{w}=25^{\circ }$ ; (b) $M_{0}=4$ , $ \theta _{w}=30^{\circ }$ .

Figure 12.

Comparison of the results along the characteristic line $\textit{CE}$ from the curved model and CFD under different conditions. Here, $\bar X_{\textit{CE}}$ is the dimensionless distance and $p/p_0$ is the pressure ratio.

Figure 13.

Schematics of the transmitted Mach wave sub-model: (a) wave reflection; (b) compatibility based.

Furthermore, a specific transmitted characteristic line $\textit{CE}$ (shown in figure 2) is selected for analysis. Figure 12 presents a comparison of the results from the curved model and CFD under different conditions. The variation of the pressure ratio $p/p_0$ with the dimensionless distance $\bar X_{\textit{CE}}$ ( $=(x-x_{E})/(x_{C}-x_{E})$ ) along the specified characteristic line is illustrated. It is obvious that the pressure along the transmitted characteristic line is different and has significant changes.

In general, either the straight model or the curved model for the transmitted Mach wave has little effect on the predicted accuracy.

3.4. The influence of the interactive part of the slipline sub-model

The approaches for modelling the interactive part of the slipline are various and have different assumptions.

Three approaches are selected to analyse the influence of the interactive part of the slipline sub-model, as follows.

1. (i) Geometric based (Li & Ben-Dor Reference Li and Ben-Dor1997). This approach decribes the interactive part of the slipline with a first-order approximation, where the slipline angle $\theta _{c}$ at the sonic throat equals zero. The physical relations are not involved.

2. (ii) Wave reflection (Bai & Wu Reference Bai and Wu2017). As shown in figure 13(a), the transmitted Mach waves are reflected at the slipline and form the reflected Mach waves. The pressure decrease due to the transmitted Mach wave plus the pressure increase due to the reflected Mach wave is consistent with the pressure in the duct below the slipline.

3. (iii) Compatibility based (the updated model). As shown in figure 13(b), the flow parameters just above the slipline satisfy two compatibility relations: (2.6a ) ( $C_-$ ) along $\textit{fc}$ , and (2.6b ) ( $C_0$ ) along $Tc$ .

Figure 14 compares the predicted Mach stem heights from different sub-models for the interactive part of the slipline against numerical simulations under $M_{0}=4$ , $\theta _{w}=25^{\circ }$ and $M_{0}=4$ , $\theta _{w}=30^{\circ }$ . The results show that assuming the interactive part of the slipline as a quadratic line (geometric based model) overestimates the Mach stem height and reduces the model’s predictive accuracy for given conditions. The wave reflection model also reduces the accuracy but leads to irregular errors. In contrast, the compatibility based model agrees more consistently with the numerical results and captures the variation of the Mach stem height slope with $g$ greatly. This also indicates that the use of a compatibility relation, rather than a wave reflection over the slipline, allows the updated model to capture the slope accurately.

Figure 15 shows the flow field structure under the conditions $M_0=4$ , $\theta _w=30^{\circ }$ and $g=0.62$ . The results obtained from CFD are plotted with dotted lines, while the predictions from the compatibility based model and the wave reflection model are shown with solid lines and dashed lines, respectively. It can be observed that the compatibility based model provides a more accurate representation, matching the CFD results closely, whereas the slipline shape predicted by wave reflection model exhibits deviations.

Figure 14.

Comparison of different methods on the interactive part of the slipline: (a) $M_{0}=4$ , $ \theta _{w}=25^{\circ }$ ; (b) $M_{0}=4$ , $ \theta _{w}=30$ .

Figure 15.

The flow field structure obtained by CFD and different models under the conditions $M_0=4$ , $\theta _w=30^{\circ }$ and $g=0.62$ .

Figure 16.

The variation between the dimensionless distance $\bar X_{\textit{CT}}$ and the slipline angle ( $ \theta _{\textit{slipline}}$ ) along slipline $\textit{TC}$ for different conditions.

Figure 16 shows the variation of the slipline angle $ \theta _{\textit{slipline}}$ obtained from CFD against the dimensionless horizontal distance $\bar X_{\textit{CT}}$ ( $=(x-x_{T})/(x_{C}-x_{T})$ ) along $\textit{TC}$ . It can be seen that the trend of the curve’s change indicates that the slipline angle changes gradually rather than suddenly, while in the wave reflection model there is a sudden change in slipline angle, which decreases the accuracy.

Figure 17 displays a comparison of the results from the compatibility based model and CFD under $M_{0}=4$ , $\theta _{w}=26^{\circ }$ , $g=0.6$ . Figure 17(a) shows the variation of the local slope $\delta _c^{(\textit{fc})}$ against the dimensionless horizontal distance $\bar X_{CB}$ ( $=(x-x_{B})/(x_{C}-x_{B})$ ), and figure 17(b) shows the variation of the pressure ratio $p_{c}/p_{0}$ against $\bar X_{CB}$ . It can be seen that the pressure prediction along the slipline given by the compatibility based model is accurate, while the slipline angle prediction along the slipline has a certain degree of error, and this may be the reason for the slight systematic deviation between the Mach stem height predicted by the compatibility based model and that from CFD.

3.5. Summary

A summary of the sensitivity analysis for the sub-models is presented in table 2. Sub-models of the free part of the slipline, the interactive part of the slipline and the reflected shock have a great influence on the overall accuracy of the Mach reflection model. Specifically, the curved model, compatibility based model and distinct parameter model are the preferred choices for the free part of the slipline, interactive part of the slipline, and reflected shock sub-models, respectively. For the transmitted Mach waves, choice of the straight model or the curved model has little effect on the predicted accuracy.

Table 2.

Sensitivity analysis summary.

Figure 17.

The variation of (a) the angle of the characteristic line and (b) the pressure ratio above the slipline for $M_{0}=4$ , $ \theta _{w}=26^{\circ }$ , $g=0.6$ .

4.1. The linear variation law of Mach stem height and wedge trailing edge height

The linear variation of Mach stem height with wedge trailing edge height has been noted in earlier studies, such as Schotz et al. (Reference Schotz, Levy, Ben-Dor and Igra1997). Bai & Wu (Reference Bai and Wu2021) and Qin et al. (Reference Qin, Shi, Dowell, Pei and Huang2022) validated this relationship numerically under limited conditions, leaving it unclear whether the observed trend represents a specific case or a universal characteristic of Mach reflection.

To address this uncertainty, we conducted a series of systematic numerical simulations of Mach reflection across a broader range of flow and geometric conditions. The variation of Mach stem height with wedge trailing edge height was analysed.

Representative CFD results are presented in figure 18, alongside comparative predictions from the updated theoretical model. The data reveal that across all tested conditions, Mach stem height exhibits a clear linear dependence on wedge trailing edge height. This confirms that the trend is not condition-specific but rather a general feature of shock wave reflection behaviour. Furthermore, the updated model demonstrates excellent agreement with CFD results in capturing this linear trend, including the slope of variation. This confirms that the proposed model offers a reliable predictive capability for the linear dependence of Mach stem height on wedge trailing edge height in Mach reflection configurations.

Although the updated model predicts the slope of the linear variation, it tends to slightly overestimate the absolute value of the Mach stem height. Therefore, we next introduce a calibrated formula based on extensive numerical simulations and the underlying physical trends governing Mach stem height variation.

Figure 18.

Comparison of the updated model and CFD results under different conditions: (a) $M_{0}=3$ , (b) $M_{0}=4$ , (c) $M_{0}=5$ .

4.2. Approximate functional forms of the slope $k$ and intercept $m$

Extensive CFD results have demonstrated a clear linear relationship between Mach stem height and wedge trailing edge height. To characterise this behaviour, a linear expression is postulated as

(4.1) \begin{equation} \frac {H_{T}}{H_{A}}=k(M_{0},\theta _{w})\,g +m(M_{0},\theta _{w}), \end{equation}

where the slope $k$ and intercept $m$ depend solely on the inflow Mach number $ M_{0}$ and the wedge angle $\theta _{w}$ .

To accurately derive a fitting formula for Mach stem height, it is essential to first analyse the variation patterns of the slope $k$ and intercept $m$ . Figures 19 and 20 illustrate the behaviours of $k$ and $m$ under various conditions, plotted on different planes. The observed trends can be summarised as follows.

1. (i) The slope $k$ is consistently negative, while the intercept $m$ remains positive.

2. (ii) For a fixed $M_{0}$ , the slope $k$ decreases and the intercept $m$ increases as the wedge angle $\theta _{w}$ increases. Furthermore, the rates of change for both parameters accelerate with increasing $\theta _{w}$ .

3. (iii) For a fixed wedge angle $\theta _{w}$ , the slope $k$ increases and the intercept $m$ decreases as the Mach number $M_0$ increases. Furthermore, the rates of change for both parameters gradually diminish with further increases in $M_0$ .

Figure 19.

The variation of (a) slope $k$ and (b) intercept $m$ with $M_0$ for different wedge angles obtained by CFD.

Figure 20.

The variation of (a) slope $k$ and (b) intercept $m$ with $\theta _{w}$ for different Mach numbers obtained by CFD.

The slope $k$ and the intercept $m$ play an important role in the rapid prediction of Mach stem height. To facilitate efficient estimation, approximate functional forms for $k$ and $m$ are proposed. It is well established, for a fixed specific heat ratio $\gamma =1.4$ , that both the slope $k$ and the intercept $m$ are determined by the inflow Mach number $M_{0}$ and the wedge angle $\theta _{w}$ . Accordingly, the calibrated formulas are assumed to take the form of a product of two single-variable functions, expressed as

(4.2) \begin{equation} \left \{ \begin{array}{l} k(M_{0},\theta _{w})=f_{k1}(M_{0})\times f_{k2}(\theta _{w}), \\[3pt] m(M_{0},\theta _{w})=f_{m1}(M_{0})\times f_{m2}(\theta _{w}). \end{array} \right . \end{equation}

According to figures 19 and 20, the functions $f_{k1}(M_{0})$ and $f_{m1}(M_{0})$ tend towards zero as the inflow Mach number $M_{0}$ approaches infinity for a fixed wedge angle $ \theta _{w}$ . Conversely, for a fixed Mach number $M_{0}$ , $f_{k2}(\theta _{w})$ and $f_{m2}(\theta _{w})$ tend towards infinity as $\theta _{w}$ approaches the detached angle of the incident shock $\theta _{\textit{max}}$ . These functions also vanish when $\theta _{w}=\theta ^{(N)}$ , corresponding to the von Neumann condition.

Based on the observed behaviours, $f_{k1}$ and $f_{m1}$ exhibit characteristics similar to power functions, while $f_{k2}$ and $f_{m2}$ resemble logarithmic functions. The following assumptions are adopted.

1. (i) When $M_{0}$ and $\theta _{w}$ satisfy the von Neumann criterion, both $k$ and $m$ vanish, i.e.

   (4.3) \begin{equation} \left \{ \begin{array}{l} k=0,\ \theta _{w}=\theta ^{(N)}, \\[3pt] m=0,\ \theta _{w}=\theta ^{(N)}. \end{array} \right . \end{equation}

2. (ii) When $\theta _{w}$ approaches $\theta _{\textit{max}}$ , where $\theta _{\textit{max}}$ denotes the detachment angle of the incident shock and is a function dependent on inflow Mach number, the values of $k$ and $m$ diverge:

   (4.4) \begin{equation} \left \{ \begin{array}{l} k\rightarrow -\infty ,\ \theta _{w}\rightarrow \theta _{\textit{max}}, \\[3pt] m\rightarrow +\infty ,\ \theta _{w}\rightarrow \theta _{\textit{max}}. \end{array} \right . \end{equation}

Here, infinity $\infty$ means very large.

Based on assumptions (4.3)–(4.4), the calibrated functional forms for $k$ and $m$ are proposed as

(4.5) \begin{equation} \left \{ \begin{array}{l} f_{k1}(M_{0})=c_{k1}M_{0}^{c_{k2}}\,{\rm e}^{c_{k3}M_{0}+c_{k4}}, \\[5pt] f_{m1}(M_{0})=c_{k1}M_{0}^{c_{k2}}\,{\rm e}^{c_{k3}M_{0}+c_{k4}}, \end{array} \right . \end{equation}

and

(4.6) \begin{equation} f_{k2}(\theta _{w})=f_{m2}(\theta _{w})=\ln\! \left(\frac {\theta _{\textit{max}}-\theta _{w} } {\theta _{\textit{max}}-\theta ^{(N)}}\right)\!. \end{equation}

Using the least squares method and CFD data for $M_{0}=2,3,5$ , the coefficients in (4.5) are determined as

(4.7) \begin{equation} \left \{ \begin{array}{l} c_{k}=0.59,0.14,-0.15,2.25, \\ c_{m}=-1.31,-0.31,-0.10,1.53. \end{array} \right . \end{equation}

Therefore, the calibrated formulas of the slope $k$ and intercept $m$ are

(4.8) \begin{equation} \left \{ \begin{array}{l} {f_{k1}}({M_0}) = 0.59M_{0}^{0.14}{\,{\rm e}^{ - 0.15{M_0} + 2.25}}, \\[5pt] {f_{m1}}({M_0}) = - 1.31M_{0}^{ - 0.31}{\,{\rm e}^{ - 0.10{M_0} + 1.53}}. \end{array} \right . \end{equation}

The normalised Mach stem height can be predicted from

(4.9) \begin{equation} H_{T}/H_{A}=\{ f_{k1}({M_0})\,g+f_{m1}({M_0})\} \ln\! \left(\frac {\theta _{\textit{max}}-\theta _{w} }{\theta _{\textit{max}}-\theta ^{(N)}}\right)\!. \end{equation}

The first factor, $f_{k1}({M_0})\,g+f_{m1}({M_0})$ , describes the influence of the geometric parameter $g$ and is negative. The second factor, $\ln (({\theta _{\textit{max}}-\theta _{w} })/({\theta _{\textit{max}}-\theta ^{(N)}}))$ , isolates the influence of the wedge angle $\theta _w$ . This form is derived from the boundary conditions, and is also negative for all valid Mach reflection configurations. This negativity arises because the wedge angle $\theta _w$ satisfies $\theta _N \lt \theta _w \lt \theta _{\textit{max}}$ for such configurations, which yields a ratio inside the logarithm that is positive but less than 1. This factorised form facilitates the independent quantification of the contributions of $M_0$ and $\theta _w$ , improving the parametric analysis capability. The valid range is $M_0=2{-}5$ .

Table 3.

Comparion of the slope $k$ and intercept $m$ obtained from the calibrated formula and CFD.

4.3. Analyses and results

The slope $k$ and intercept $m$ can be quantitatively predicted using the calibrated expressions from (4.2) and (4.6)–(4.8). Figure 21 presents a comparative analysis among three approaches: the calibrated formula, the updated model and the CFD results under $M_{0}=5$ . It can be seen that the $k$ and $m$ values given by the updated model are already quite accurate, but the results given by the calibrated model are basically consistent with the numerical results. Additional comparisons under varying conditions are provided in table 3 and figure 22, which further demonstrate that the calibrated formula closely agrees with the CFD data.

Furthermore, figure 23 shows a comparison of the CFD results and predictions from the calibrated formula under $M_{0}=2.2$ , $M_{0}=3.5$ and $M_{0}=4$ . It can be seen that the accuracy of the calibrated formula shows good agreement with the results obtained by CFD.

In general, these results confirm that the calibrated formula (4.8) provides sufficient accuracy for predicting both $k$ and $m$ , and (4.9) provides sufficient accuracy for predicting for Mach stem height.

Figure 21.

Comparison of the slope $k$ and intercept $m$ obtained from the calibrated formula, updated model and CFD for $M_{0}=5$ .

Figure 22.

Comparison of the slope $k$ and intercept $m$ obtained from the calibrated formula and CFD.

Figure 23.

The results obtained by the calibrated formula under different conditions: (a) $M_{0}=2.2$ , (b) $M_{0}=3.5$ , (c) $M_{0}=4$ .