2.1. Definition and shock theory

The type of jets for type IV shock interaction is here studied using flow properties in the vicinity of triple points $A$ and $B$, as shown in figure 1. In this section, the flow is considered uniform in regions $(3)$ and $(5)$ and we use subscript $k$ to denote the flow parameters in region $(k)$. For instance, the Mach number in region $(3)$ is denoted $M_{3}$.

The jet is said to be supersonic if $M_{3}>1$ and $M_{5}>1$, transonic if $M_{3}>1$ and $M_{5}<1$ (if the jet exists), and subsonic if $M_{3}<1$ (in this case, shock $BA_{1}$ and the downstream wave fans do not exist). However, the transonic jet case is much more complex than this. As will be shown in § 2.3, for the case with $M_{3}>1$ and $M_{5}<1$, there are conditions where the second triple point $B$ does not exist, which means that there is no jet in the usual sense. In § 3, we will see that compression waves developed inside the triangle $ABA_{1}$ due to the pressure gradient above the slipline $AA_{1}$ will make the flow non-uniform inside the triangle and alter the transition conditions. Furthermore, we will see in §§ 3 and 4 that, in the case of a transonic jet, the region $(5)$ will be hybrid supersonic–subsonic, instead of being purely subsonic.

Here, in this section, the transition condition from a subsonic to a transonic jet is determined by the sonic condition $M_{3}=1$ and the transition condition from a transonic to a supersonic jet is determined by the sonic condition $M_{5}=1$. These two conditions, if they exist, enclose the domain for the existence of a transonic jet.

Although we will display results with a high enough Mach number, in this paper, we do not consider real gas effect. Thus, we will use the classical shock relations for a perfect gas in the analysis. Now, let us recall some classical shock expressions needed in the present paper.

If we use subscripts $u$ and $d$ to denote the flow status in regions upstream and downstream of a shock wave, then the shock angle $\beta _{ud}$ (shock slope angle with respect to the upstream flow direction) is related to the flow deflection angle $\theta _{ud}$ (angle change of flow direction across the shock) by the shock angle relation $\tan \theta _{ud}=\,f_{\theta }(M_{u} ,\beta _{ud})$ and the Mach number and pressure satisfies $M_{d}=\,f_{M} (M_{u},\beta _{d})$ and $p_{d}=p_{u}\,f_{p}(M_{u},\beta _{ud})$, with

(2.1) \begin{equation} \left\{\begin{array}{@{}l} f_{\theta}({M,\beta})=\dfrac{2\left( {M}^{2}{\sin}^{2}{\beta}-1\right) }{\left( {M}^{2}\left( \gamma+\cos{2\beta}\right) +2\right) \tan{\beta}}\\ f_{M}(M,{\beta})=\sqrt{\dfrac{\left( \gamma-1\right) {M}^{2}+2}{2\gamma {M}^{2}{\sin}^{2}{\beta}-\left( \gamma-1\right) }+\dfrac{2{M}^{2}{\cos} ^{2}{\beta}}{\left( \gamma-1\right) {M}^{2}{\sin}^{2}{\beta}+2}}\\ f_{p}(M,{\beta})=1+\dfrac{2\gamma}{\gamma+1}\left( {M}^{2}{\sin}^{2}{\beta }-1\right), \end{array}\right. \end{equation}

where $\gamma$ is the ratio of specific heats.

Let ${\beta }_{m}$ be the shock angle at the detachment condition (shock slope angle that corresponds to maximum flow deflection angle at given Mach number), i.e. at ${\beta }$ with ${{\rm d}\,f_{\theta }}/{{\rm d}{\beta }}=0$ (which occurs at the detachment angle). Using the expression in (2.1) for $f_{\theta }(M,{\beta })$, one has

(2.2)\begin{equation} \sin^{2}{\beta}_{m}=\frac{1}{\gamma M^{2}}\left( \frac{\gamma+1}{4} M^{2}-1+\sqrt{(\gamma+1)\left(1+\frac{\gamma-1}{2}M^{2}+\frac{\gamma+1}{16}M^{4} \right)}\right). \end{equation}

A downstream sonic flow occurs at $\beta =\beta _{M_{d}=1}$ where $\beta _{M_{d}=1}$ is the shock angle at which $f_{M}(M,{\beta }_{M_{d}=1} )=1$. Using the expression in (2.1) for $f_{M}(M,{\beta }_{M_{d}=1}{)}$, one has

(2.3)\begin{equation} \frac{\left( \gamma-1\right) {M}^{2}+2}{2\gamma{M}^{2}{\sin}^{2}{\beta }_{M_{d}=1}-\left( \gamma-1\right) }+\frac{2{M}^{2}{\cos}^{2}{\beta} _{M_{d}=1}}{\left( \gamma-1\right) {M}^{2}{\sin}^{2}{\beta}_{M_{d}=1}+2}=1 . \end{equation}

From the classical theory for oblique shock waves (cf. Anderson Reference Anderson2001, p. 315), we know that: (i) the shock solution admits ${\beta <\beta }_{m}$ (for which the shock is said to belong to the weak solution) and ${\beta >\beta }_{m}$ (for which the shock is said to belong to the strong solution) and (ii) for given $\theta$ and upstream $M$, one has ${\beta }_{M_{d}=1}<\beta _{m}$, (iii) for ${\beta <\beta }_{M_{d}=1}$, the downstream flow of the shock is supersonic (the shock is then called a weak shock wave) and for ${\beta >\beta }_{M_{d}=1}$ the downstream flow is subsonic (the shock is now called a strong shock).

The analysis for finding the critical conditions relies on the use of the well-known triple point theory due to von Neumann (Reference von Neumann1943, Reference von Neumann1945), which describes the relationship between flow quantities in various regions in the vicinity of a triple point (typical structure of type IV shock interaction).

Consider triple point theory applied to point $A$. The flow parameters in region $(0)$ are given, the shock angle ${{\beta _{0{1}}}}$ or the flow deflection ${\theta _{01}}$ connected to ${{\beta _{0{1}}}}$ by $\tan {\theta _{01}}={\,f_{\theta }}( {{M_{0}},{\beta _{0{1}}}})$, is also given. First, the Mach number and pressure in region $(1)$ are obtained from $M_{1}=\,f_{M}(M_{0},\beta _{01})$ and $p_{1}=p_{0}\,f_{p}(M_{0},\beta _{01})$. Then the pressure $p_{2}$ is obtained from $p_{2}=p_{0}\,f_{p}(M_{0},\beta _{02})$ for shock $AD$ and the pressure $p_{3}$ is obtained from $p_{3}=p_{1}\,f_{p} (M_{1},\beta _{13})$ for shock $AB$. The shock angle $\beta _{13}$ is solved from $\tan {\theta _{13}}=\,{f_{\theta }}( {{M_{1}},{\beta _{{13}}}})$ for the weak solution and $\beta _{02}$ is solved from $\tan {\theta _{02} }=\,{f_{\theta }}( {{M_{0}},{\beta _{{02}}}})$ for the strong solution. Here, the flow deflection angles ${\theta _{13}}$ and ${\theta _{02}}$ are obtained from the flow parallel conditions $\theta _{02}=\theta _{AA_{1}}$ and $\theta _{13}=\theta _{01}-\theta _{AA_{1}}$, where $\theta _{AA_{1}}$ is the slipline angle of $AA_{1}$ and is obtained by using the pressure balance condition $p_{2}=p_{3}$.

The triple point theory applied to point $B$ is similar to that for triple point $A$.

2.2. Conditions for a transonic jet

Here, we provide two conditions: one condition, denoted $\theta _{01} =\theta _{sub}^{tran}(M_{0})$, corresponds to $M_{3}=1$, and the other, denoted $\theta _{01}=\theta _{sup}^{tran}(M_{0})$, corresponds to $M_{5}=1$.

The condition $\theta _{01}=\theta _{sub}^{tran}(M_{0})$ is the necessary critical condition for transition from a subsonic jet to a transonic jet. The reason it is a necessary condition is that, when this condition is met, one may still have no transonic jet, as discussed in § 2.3, where we provide another critical condition. The condition $\theta _{01}=\theta _{sup }^{tran}(M_{0})$ is the critical condition for transition from a supersonic jet to a transonic jet. The detailed expressions to obtain these critical conditions will be given in Appendix A. Below, we just summarize the main steps to obtain them, before giving the critical conditions in the plane $M_{0}\text {--}{\theta _{01}}$.

The necessary condition for transition from a subsonic to a transonic jet corresponds to the sonic condition $M_{3}=1$. First, we solve the triple point theory for the triple point $A$ (see § 2.1) to get the shock angle $\beta _{13}$ for a given ${M}_{0}$ and ${\theta _{01}}$. The Mach number $M_{3}$ in region $(3)$ is then obtained by $M_{3}=\,f_{M}(M_{1},\beta _{13})$. For a given inflow Mach $M_{0}$, we search a flow deflection angle $\theta _{01}=\theta _{sub}^{tran}(M_{0})$ for which the sonic condition $M_{3}=1$ holds.

The transition condition from a supersonic to a transonic jet corresponds to the sonic condition $M_{5}=1$ (for $M_{3}>1$). First, we solve the triple point theory for point $A$. Now, the shock $AB$, with its shock angle $\beta _{13}$ and its upstream Mach number $M_{1}$, already known from the triple point solution of point $A$, is considered as the incident shock wave for triple point $B$. The triple point solution of point $B$, if it exists, then gives the solution $M_{5}$. For a given inflow Mach number $M_{0}$, we search for a flow deflection angle $\theta _{01}=\theta _{sup}^{tran}(M_{0})$ for which the sonic condition $M_{5}=1$ holds.

The critical conditions ${\theta _{01}=\theta }_{sub}^{tran}(M_{0})$ and $\theta _{01}=\theta _{sup}^{tran}(M_{0})$ in the plane $M_{0}\text {--}{\theta _{01}}$ thus obtained are shown in figure 2(a) for $\gamma =1.4$ and in figure 2(b) for $\gamma =1.1$. For cold air we have $\gamma =1.4$. The reason we display the results for $\gamma =1.1$ is to show that the transition conditions also depend on the ratio of specific heats, although the result with a smaller value $\gamma =1.1$ may partly account for the high temperature effect of high Mach number flow since such an effect equivalently reduces $\gamma$.

Figure 2.

Transition criteria in the $M_{0}-\theta _{01}$ plan; (a) $\gamma =1.4$, (b) $\gamma =1.1$.

In figure 2, we also display the condition $\theta _{01} =\theta _{max}(M_{0})$ for the incident shock wave to detach, the detachment condition $\theta _{01}=\theta _{D}(M_{0})$ for Mach reflection of the triple point $A$ and the von Neumann condition $\theta _{01}=\theta _{N}(M_{0})$ for Mach reflection of triple point $A$.

For $\gamma =1.4$, the two curves $\theta _{01}={\theta }_{sub}^{tran}(M_{0} )$ and $\theta _{01}=\theta _{sup}^{tran}(M_{0})$ enclose the region inside which the jet is transonic if the second triple point exists, i.e. the first part of the jet (triangle $ABA_{1}$) is supersonic while the rest (downstream of $BA_{1}$) is (theoretically) subsonic. The critical curve $\theta _{01}=\theta _{sup}^{tran}(M_{0})$ encloses a region where the jet is purely supersonic. Outside of the critical curve $\theta _{01}=\theta _{sub}^{tran}(M_{0})$ the jet is purely subsonic (if a jet exists in this case).

We observe that both $\theta _{01}={\theta }_{sub}^{tran}(M_{0})$ and $\theta _{01}=\theta _{sup}^{tran}(M_{0})$ have a (near) C-shape in the $M_{0}\text {--}\theta _{01}$ plane. The left-most point $O$ of $\theta _{01}={\theta }_{sub}^{tran}(M_{0})$ lies at $M_{0}\approx 2.4$ and the left-most point $Q$ of $\theta _{01}=\theta _{sup}^{tran}(M_{0})$ lies at $M_{0} \approx 3.4$.

The transition curves for $\gamma =1.1$ are similar to that for $\gamma =1.4$, apart from the fact that the two curves $\theta _{01}={\theta }_{sub}^{tran}(M_{0})$ and $\theta _{01}=\theta _{sup}^{tran}(M_{0})$ move some distance to the left. In the rest of the paper we only consider air with $\gamma =1.4$.

The curve $\theta _{01}={\theta }_{nojet}^{tran}(M_{0})$ shown in figure 2 is the critical condition to have no jet and will be discussed in § 2.3.

Now, we display the shock polars for two conditions: $M_{0}=2.426$, $\theta _{01}=13^{\circ }$ and $M_{0}=3.435,\theta _{01}=13^{\circ }$, which lie on the critical transition lines $\theta _{01}={\theta }_{sub}^{tran}(M_{0})$ and $\theta _{01}={\theta }_{sup}^{tran}(M_{0})$, according to figure 2. Following Kawamura & Saito (Reference Kawamura and Saito1956), we use $\varGamma (k)$ to denote the shock polar along which is the locus of all the flow states that could be obtained from any shock wave with a fixed upstream state $(k)$. The sonic point will be marked with an open circle. The shock polars are displayed in figure 3. When $M_{0}=2.426$, $\theta _{01}=13^{\circ }$, there is only one triple point and the polar $\varGamma (0)$ intersects the polar $\varGamma (1)$ exactly at its sonic point (figure 3a). In this case, we have $M_{3}=1$, i.e. $\theta _{01}={\theta }_{sub}^{tran}(M_{0})$. When $M_{0}=3.435$, $\theta _{01}=13^{\circ }$, the polar $\varGamma (1)$ intersects the polar $\varGamma (3)$ exactly at its sonic point (figure 3b). In this case, we have $M_{5}=1$, i.e. $\theta _{01}={\theta }_{sup}^{tran}(M _{0})$.

Figure 3.

Shock polars for $\theta _{01}=13^{\circ }$. (a) On the critical line $\theta _{sub}^{tran}$ with $M_{0}=2.426$. (b) On the critical line $\theta _{sup}^{tran}$ with $M_{0}=3.435$.

2.3. Condition of no jet and region with a transonic jet

As mentioned in § 2.1, for the case with $M_{3}>1$ and $M_{5}<1$, there are conditions where the second triple point $B$ does not exist, which means that there is no jet in the usual sense. This occurs when the polar $\varGamma (3)$ lies completely inside of $\varGamma (1)$, as displayed in figure 4(a) for $M_{0}=2.5$, $\theta _{01}=13^{\circ }$. Appendix A.3 used the theory of Chernyshov, Kapralova & Savelova (Reference Chernyshov, Kapralova and Savelova2021) to derive the condition $\theta _{01}={\theta }_{nojet}^{tran}(M_{0})$, beyond which the second triple point $B$ and the jet do not exist. The critical condition $\theta _{01}={\theta }_{nojet}^{tran}(M_{0})$ corresponds to the polar $\varGamma (3)$ tangent to $\varGamma (1)$, as shown in figure 4(b) for $M_{0}=2.548,$ $\theta _{01}=13^{\circ }$.

Figure 4.

Shock polars for $\theta _{01}=13^{\circ }$. (a) Between the critical lines $\theta _{sub}^{tran}$ and $\theta _{nojet}^{tran}$ ($M_{0}=2.5$). (b) On the critical line $\theta _{nojet}^{tran}$ ($M_{0}=2.548$).

The curve $\theta _{01}={\theta }_{nojet}^{tran}(M_{0}{)}$ is also displayed in figure 2(a,b). For $\gamma =1.4$, its left-most point $P$ has $M_{0}\approx 2.5422$. The curve $\theta _{01}={\theta }_{nojet}^{tran} (M_{0}{)}$ lies between $\theta _{01}={\theta }_{sub}^{tran}(M_{0}{)}$ (necessary condition) and $\theta _{01}={\theta }_{sup}^{tran}(M_{0}{)}$. In the region between $\theta _{01}={\theta }_{sub}^{tran}(M_{0}{)}$ and $\theta _{01}={\theta }_{nojet}^{tran}(M_{0}{)}$, the flow downstream of $AB$ is still supersonic but the second triple point $B$ does not exist.

In conclusion, a transonic jet occurs in the region bounded by the critical conditions $\theta _{01}={\theta }_{nojet}^{tran}(M_{0}{)}$ and $\theta _{01}=\theta _{sup}^{tran}(M_{0}{)}$, which are shown in figure 2(a,b).

In the present paper, we focus on the case with a transonic jet, so in the following we only consider the region enclosed by $\theta _{01}=\theta _{nojet}^{tran}(M_{0})$ and $\theta _{01}={\theta }_{sup}^{tran}(M _{0})$.

2.4. Numerical evidence of a transonic jet and observation of a two-branch transonic jet

CFD is used to obtain numerical solutions. The inviscid compressible Euler equations for a perfect gas with a ratio of specific heat $\gamma =1.4$ are solved numerically using the second-order Roe scheme (Roe Reference Roe1986) on a structured grid with $500\times 800$ points. The grid is refined in the jet and there are approximately $50$ grid points in the direction perpendicular to the jet. The grid has been chosen such that the flow structure no longer changes and the accuracy is checked using the Edney (Reference Edney1968b) experimental measurement for $M_{0}=4.6$ and $\theta _{01}=10^{\circ }$. According to the transition criterion as shown in figure 2(a), the jet should be supersonic. The computed Mach contours shown in figure 5(a) agree well with figure 13 of Edney (Reference Edney1968b).

Figure 5.

Mach contours at two instants for two set of conditions. (a) Supersonic jet at $\bar {t}=115$ with $M_{0}=4.6$, $\theta _{01}=10^{\circ }$, (b) transonic jet at $\bar {t}=52$ with $M_{0}=3$ and $\theta _{01}=13^{\circ }$, (c) supersonic jet at another instant $\bar {t}=120$ and (d) transonic jet at $\bar {t}=62$. The time $\bar {t}$ is normalized by the reference time (radius of the blunt body divided by the free-stream velocity).

Now we select the typical point $M_{0}=3$ and $\theta _{01}=13^{\circ }$ inside the transonic jet region of figure 2(a). According to the numerical result shown in figure 5(b), the jet downstream of the inner shock indeed has a subsonic region. However, the jet is not purely subsonic but displays a two-branch structure: the upper one is supersonic while the lower one is subsonic. This unusual phenomenon will be considered in § 4. Note that the present numerical result is for inviscid flow and, due to the sharp gradients in the jet, viscous effects in real flow could very well make the noted two-branch structure more difficult to obtain or detect. In future work the viscous effect on the two-branch structure may be considered. We also note that the entropy is continuous across the interface between the two branches.

We have also computed the flow for $M_{0}=2$ and $\theta _{01}=13^{\circ }$, which should have a subsonic jet according to the transition criterion, as shown in figure 2(a). According to our numerical results, not shown here, there is only one triple point and one slipline.

The type IV interaction is known to display unsteadiness (cf. Holden et al. Reference Holden, Wieting, Moselle and Glass1988; Gaitonde & Shang Reference Gaitonde and Shang1995; Chu & Lu Reference Chu and Lu2012; Windisch et al. Reference Windisch, Reinartz and Müller2016; Khatt & Jagadeesh Reference Khatt and Jagadeesh2018). The numerical method used has a second-order time accuracy and the flow is unsteady. The Mach contours displayed in figures 5(a) and 5(b) correspond to some instant after a long time of iteration. As shown in figures 5(c) and 5(d), the local flow parameters at other instants have minor changes compared with those shown in figures 5(a) and 5(b), showing unsteadiness. However, the observed overall flow configuration, of interest in this paper, does not change with time.

Note that Lu et al. (Reference Lu, Yue, Xiao and Zhang2013) studied the interaction between isentropic compression waves and a bow shock and noted for some condition a transonic jet in their numerical results. The exact flow pattern for their transonic jet and the transition conditions to have their transonic jet may deserve a separate study.

3.1. Compression waves in the triangle $ABA_{1}$

Figure 6(a) displays, for the transonic jet case with $M_{0}=3$ and $\theta _{01}=13^{\circ }$, the pressure contours in terms of the pressure coefficient. It is clear that, above the slipline $AA_{1}$, there is a positive pressure gradient. The pressure just above the slipline $AA_{1}$ will be denoted by $p_{slip}$. Positive pressure gradient here means ${\rm d}p_{slip}/{\rm d}s>0$, where $s$ is the distance measured along the slipline $AA_{1}$.

Figure 6.

Flow pattern for $M_{0}=3$ and $\theta _{01}=13^{\circ }$. (a) Pressure contours showing positive pressure gradient above $AA_{1}$. (b) Illustration of compression waves $C_{1},C_{2},\ldots$. The lines $Ab$ and $Ba_{1}$ are straight lines tangent to the shocks $AB$ and $BA_{1}$ at point $A$ and $B$, respectively.

Now we provide a simplified model of compression waves. Since the pressure is continuous across the slipline $AA_{1}$, the flow inside the triangle $ABA_{1}$ also has a positive pressure gradient and this pressure gradient generates a series of compression waves $C_{1},C_{2},\ldots$ that propagate in the way illustrated in figure 6(b). To account for the influence of these compression waves, we simply assume the flow is isentropic and irrotational inside the triangle $ABA_{1}$, so we represent the compression waves by straight characteristic lines of the second family satisfying ${{\rm d} y}/{{\rm d} x}=\tan (\theta +\mu )$, where $\mu$ is the Mach angle. Along each characteristic line the pressure is assumed to be constant, i.e. $p=p_{slip}$ (this appears to be so according to figure 6a). The possible reflected waves of $C_{1},C_{2},\ldots$ from $AB$ is not considered.

Under the above simplification, the variation ${\rm d}\theta$ of the flow deflection angle across the compression waves can be related to the pressure variation ${\rm d}p={\rm d}p_{slip}$ and the Mach number variation ${\rm d}M$ through the Mach wave relations

(3.1a,b)$$\begin{gather} {\rm d}p_{slip}=\varPhi(M,p)\,{\rm d}\theta,\quad \varPhi(M,p)=\frac{\gamma Mp}{\sqrt{M^{2}-1}}, \end{gather}$$

(3.2a,b)$$\begin{gather}{\rm d}M=\varPsi(M,p)\,{\rm d}\theta,\quad \varPsi(M,p)={-}\frac{M}{\sqrt{M^{2}-1}}\left( 1+\frac{\gamma-1}{2}M^{2}\right) . \end{gather}$$

Due to compression waves, the flow is non-uniform in regions $(3)$ and $(5)$. Thus, $M_{3}^{(B)}$, which denotes the Mach number in region $(3)$ and in the vicinity of point $B$, is different to $M_{3}^{(A)}$, which denotes the Mach number in region $(3)$ but in the vicinity of point $A$. To further simplify the analysis, we assume the compression waves are weak enough so that the variation of parameters inside the triangle $ABA_{1}$ is small. Under this assumption, the coefficients $\varPhi (M,p)$ and $\varPsi (M,p)$ in (3.1a,b) and (3.2a,b) and those in similar formulas given below will be evaluated using the uniform solutions $M_{3}$, $p_{3}$, ${\beta }_{13}$ and ${\beta } _{35}$ given in § 2. For instance, we will simply put $\varPhi (M,p)=\varPhi (M_{3},p_{3})$ and $\varPsi (M,p)=\varPsi (M_{3},p_{3})$. The expressions (3.1a,b) and (3.2a,b) can then be used to give a relation between $dM$ and $dp$ (across the compression waves)

(3.3)\begin{equation} \frac{{\rm d}M}{{\rm d}p_{slip}}={-}\left( 1+\frac{\gamma-1}{2}M_{3}^{2}\right) \frac {1}{\gamma p_{3}} . \end{equation}

Below, we will use the Mach wave relations (3.1a,b), (3.2a,b) and (3.3) to account for the influence of compression waves on the deflection of transmitted shock $AB$ from a straight line (similar effect by compression waves on the inner shock $BA_{1}$ will be considered in § 4). Note that, the interaction between Mach waves and shock waves has been considered for various other purposes (Moeckel Reference Moeckel1952; Rosciszewski Reference Rosciszewski1960; Hasimoto Reference Hasimoto1964; Li & Ben-Dor Reference Li and Ben-Dor1995; Meshkov, Omelchenko & Uskov Reference Meshkov, Omelchenko and Uskov2002; Silnikov, Chernyshov & Uskov Reference Silnikov, Chernyshov and Uskov2014; Uskov & Chernyshov Reference Uskov and Chernyshov2014).

3.2. Advance of transition from a supersonic jet to a transonic jet

Now we consider the role of compression waves $C_{1},C_{2},\ldots,C_{i}$ in the shape of transmitted shock $AB$ as well as on the advance of transition from a supersonic jet to a transonic jet. For this purpose, we rewrite the shock angle relation as $\theta =F(M,{\beta })$ with $F(M,{\beta })=\arctan \,f_{\theta }({M,\beta })$, where $f_{\theta }({M,\beta })$ is defined in (2.1).

The Mach number $M_{1}$ upstream of shock $AB$ is constant so that differentiating $\theta =F(M,{\beta })$ gives

(3.4a,b)\begin{equation} \frac{{\rm d}{\beta}}{{\rm d}\theta}=\frac{1}{G(M_{1},{\beta}_{13})},\quad G(M,{\beta })=\frac{\partial F(M,{\beta})}{\partial{\beta}}.\end{equation}

Combining shock angle relation (3.4a,b) and the Mach wave relation (3.1a,b) then gives the following relation between the shock angle of $AB$ and the pressure gradient ${\rm d}p_{slip}$ along the slipline $Aa$:

(3.5)\begin{equation} \frac{{\rm d}{\beta}}{{\rm d}p_{slip}}=\frac{\sqrt{M_{3}^{2}-1}}{\gamma M_{3}p_{3} G(M_{1},{\beta}_{13})}. \end{equation}

One can check that, for a weak shock $AB$, we have ${{\rm d}{\beta }}/{{\rm d}p_{slip} }>0$, thus the shock $AB$ deflects from a straight line toward the lower side as shown in figure 6(b) and, at the triple point $B$, the shock angle ${\beta }_{13}^{(B)}$ is larger than ${\beta }_{13}$ (shock angle in case of no compression waves). Integrating (3.5) from point $A$ to point $B$ gives

(3.6)\begin{equation} {\beta}_{13}^{(B)}={\beta}_{13}+\frac{\sqrt{M_{3}^{2}-1}}{\gamma M_{3} p_{3}G(M_{1},{\beta}_{13})}(p_{slip}^{(a)}-p_{3}) , \end{equation}

where $p_{slip}^{(a)}$ is the pressure at point $a$ (this point is connected to $B$ by a characteristic line). There is no simple theory to provide the pressure $p_{slip}^{(a)}$ in this paper. Below, we simply use CFD data for $p_{slip}^{(a)}$ once it is needed in analysis. After ${\beta }_{13}^{(B)}$ is thus obtained, the shock angle relation $\theta _{13}^{(B)}=F(M_{1},{\beta }_{13}^{(B)})$ is used to evaluate $\theta _{13}^{(B)}$.

Now we discuss how the transition condition $\theta _{01}=\theta _{sup} ^{tran}(M_{0}{)}$ is modified by the compression waves. The expression (3.6) provides an approximate evaluation of the shock angle ${\beta }_{13}^{(B)}$ of $AB$ near the triple point $B$ when the role of compression waves $C_{1},C_{2},\ldots,C_{i}$ is accounted for. The triple point theory at point $B$ using this updated shock angle ${\beta }_{13}^{(B)}$ for the incident shock wave $AB$ gives an updated value of $M_{5}$, now denoted $M_{5}^{(B)}$. In the modified transition criterion, it is this updated $M_{5}^{(B)}$ which is used to define the jet type: if $M_{5}^{(B)}>1$, the jet is supersonic and if $M_{5}^{(B)}<1$, the jet is transonic. Since ${\beta }_{13}^{(B)} >{\beta }_{13}$, thus $M_{5}^{(B)}$ is smaller than $M_{5}$. Table 1 gives, for four conditions, the jet types predicted using the original criterion (shown in figure 2a), the modified transition criterion and CFD observation. For Case 2, while the original theory predicts a supersonic jet, both the modified criterion and CFD show a transonic jet. Case 3 is close to the modified transition criterion, i.e. for $\theta _{01}=13^{\circ }$, the modified transition occurs at $M_{0}\approx 3.7$, while the original transition criterion corresponds to $M_{0}\approx 3.42$. Thus, compression waves developed due to the pressure gradient ${\rm d}p_{slip}/{\rm d}s>0$ advance transition from a supersonic jet to a transonic jet.

Table 1.

Predicted and observed types of the jet cases for $\theta _{01}=13^{\circ }$ and various Mach numbers.

4.1. Observation of a sonic line apparently emanating from point $A_{1}$

Figure 7 shows the two-branch structure of the transonic jet, for three different conditions, which are inside the transonic jet region according to table 1 or figure 2(a). For each case we observe a sonic line emanating apparently from point $A_{1}$. Note that some parts of the sonic line in figure 7(b) touch the lower slipline, while in figures 7(a) and 7(c) the sonic line is away from the lower slipline. The solution shown in figure 7(b) is in fact near the transition condition from a transonic to a supersonic jet, so the sonic line gets closer to the lower slipline. This sonic line will be used to explain, in § 4.2, the production of the observed two-branch structure of the transonic jet.

Figure 7.

Mach contours of the transonic jet, where the dashed line represents a sonic line with $M=1$; (a) $M_{0}=3$ and $\theta _{01}=13^{\circ }$, (b) $M_{0}=3.5$ and $\theta _{01}=13^{\circ }$, (c) $M_{0}=3$ and $\theta _{01}=15^{\circ }$.

To verify that the sonic line emanates from point $A_{1}$ and not from any inner point of the reflected shock, we perform a grid refinement study using the case shown in figure 7(a). The starting point of the sonic line indeed gets closer to point $A_{1}$ if the grid is further refined, as shown in figure 8(a–d) with four different grids. The number of grids is $9\times 10^{4}$ for grid A (approximately $45$ grid points along $BA_{1}$), $3.4\times 10^{5}$ for grid B (approximately $90$ grid points along $BA_{1}$), $1.36\times 10^{6}$ for grid C (approximately $180$ grid points along $BA_{1}$) and $2.08\times 10^{6}$ for grid D (approximately $360$ grid points along $BA_{1}$).

Figure 8.

Transonic jet for $M_{0}=3$ and $\theta _{01}=13^{\circ }$. (a) Grid A. (b) Grid B. (c) Grid C. (d) Grid D.

The mechanism for the creation of the sonic line at $A_{1}$ is rather complex. According to a simplified analysis in Appendix B, it appears that the sonic line results from the weakening of $BA_{1}$ by the combined effect of upstream flow disturbance (due to the preceding compression waves) and downstream flow disturbance (due to local flow stream deflection near $A_{1}$ and this deflection follows from the pressure continuity requirement across the slipline).

4.2. Postulated mechanism by which the two-branch transonic jet structure is produced

Now we postulate the mechanism for the production of two-branch transonic jet, based on the observation of a sonic line emanating from point $A_{1}$ and using the schematic displayed in figure 9 of the wave pattern in the vicinity of the inner shock $BA_{1}$. Figure 9 is sketched following the numerical solution shown in figure 7(a).

Figure 9.

Schematic display of the wave pattern in the vicinity of the inner shock $BA_{1}$.

First, it is physically impossible that, for a transonic jet, the flow behind the inner shock is subsonic everywhere. In fact, the pressure continuity condition, similar to that stated in the Introduction for a supersonic jet, is violated if the flow downstream of the inner shock is subsonic everywhere. To see this, let us consider the three points $i$, $j$ and $k$ as shown in figure 9. If a pure subsonic flow is assumed in all regions downstream of $BA_{1}$, then the expansion fan between points $j$ and $k$ does not exist. In this case, $p_{k}=p_{j}>p_{i}$, which is impossible since, due to pressure continuity across a slipline, the pressures at points $i,j,k$ should be balanced by the pressure in the subsonic region $(0)$ above the slipline $AA_{1}$, and the inequality $p_{k}=p_{j}>p_{i}$ means pressure discontinuity in the subsonic region $(0)$, which is impossible since pressure discontinuity cannot be sustained by a subsonic flow.

As we have shown in § 4.1, at point $j$, the flow should be sonic, i.e. $M_{5}^{(A_{1})}=1$. A sonic flow at point $j$ is able to accelerate to supersonic flow at point $k$ through generating an expansion fan, as displayed in figure 9. Due to this expansion fan the pressure continuity condition ($p_{k}=p_{i}$) is met as in case of a pure supersonic jet. The reason that the flow is not supersonic at $j$ would be that the condition $M_{5}^{(A_{1})}=1$ corresponds to the least effort required by the compressive waves ahead of $BA_{1}$ to weaken $BA_{1}$ to allow for pressure continuity.

Finally, through generating both disturbances upstream and downstream of $BA_{1}$, the pressure gradient outside the slipline appears to play the essential role to weaken $BA_{1}$ until the downstream flow has a sonic line emanating from $A_{1}$. A last question is whether the pressure gradient is large enough to sustain such a structure. Since the flow describing the shock $BA_{1}$ depends on the pressure gradient outside of the upper slipline (see Appendix B), the Mach number $M_{5}^{(A_{1})}$ should depend on $p_{slip}^{(A_{1})}$ (pressure at point $A_{1}$) and $p_{slip}^{(a)}$ (pressure at point $a$ as shown in 6a). Let the functional form of $M_{5}^{(A_{1})}$ be $M_{5}^{(A_{1})}=\,F_{1}(M_{3}^{(B)},{\beta }_{35} ^{(B)},\bar {p}_{slip}^{(A_{1})},\bar {p}_{slip}^{(a)})$, where $\bar {p}_{slip}^{(a)}=p_{slip}^{(a)}/p_{3}$ and $\bar {p} _{slip}^{(A_{1})}=p_{slip}^{(A_{1})}/p_{3}$. The sonic line assumption $M_{5}^{(A_{1})}=1$ then gives the pressure constraint

(4.1)\begin{equation} F_{1}(M_{3}^{(B)},{\beta}_{35}^{(B)},\bar{p}_{slip}^{(A_{1})} ,\bar{p}_{slip}^{(a)})=1. \end{equation}

In other words, the condition (4.1) ensures that $M_{5} ^{(A_{1})}=1$, which has been shown to be the condition for the production of the two-branch transonic jet. The transonic jet is likely to adjust its size (notably the length of $AA_{1}$) so that the pressure $p_{slip}$ at $A_{1}$ satisfies (4.1). This should be possible if $p_{slip}^{(A_{1} )}$ determined by (4.1) is well below the maximum possible value of $p_{slip}$ that can be created over the upper slipline. The maximum possible value of $p_{slip}$ should be given by the stagnation pressure $p_{s}$. It is well known that the stagnation pressure is well above the pressure behind any shock wave, thus $p_{slip}^{(A_{1})}$ is well below $p_{s}$.

In conclusion, the possible mechanism to have a two-branch structure of a transonic jet may be summarized as follows: it is impossible to have $M_{5}^{(A_{1})}<1$ since this will violate the pressure continuity condition, the upstream disturbance (preceding compressive waves) and downstream disturbance (all may be due to the pressure gradient outside of the jet) then weaken $BA_{1}$ to achieve a sonic condition $M_{5}^{(A_{1})}=1$ at $A_{1}$, which is possible when the inner shock adjusts its shape and size in such a way that the pressure constraint (4.1) for $p_{slip}^{(A_{1})}$ is met and this sonic condition corresponds to the least effort required to maintain pressure continuity near $A_{1}$. This sonic flow at point $j$ accelerates to supersonic flow through creating an expansion fan in the upper part of the jet and, downstream of this expansion fan, a combination of compressive and expansive waves are created for the same reason as in a pure supersonic jet.