2.1.1. Basic properties of Taylor–Maccoll equations

The T–M equations are usually formulated on the polar coordinates, where the independent variable is the polar angle ($\theta$). The dependent variables are the radial and circumferential components of the flow velocity ($u_r,u_\theta$) (Mölder Reference Mölder1967), either non-dimensionalized by the theoretically maximum speed (Maccoll & Taylor Reference Maccoll and Taylor1936) or the characteristic sonic speed (Grozdovskii Reference Grozdovskii1959). In this work, the square of the characteristic Mach number ($\lambda ^2$) and the flow inclination ($\psi$) are introduced as the dependent variables to simplify the expression of critical surfaces and facilitate the analysis. As shown in figure 2, $(x,y)$ are Cartesian coordinates, while $(r,\theta )$ are polar coordinates. Also, $\varphi$ and $\psi$ are the flow inclination with respect to the polar line and the axis, respectively, or the relative and absolute flow inclinations in short. The characteristic Mach number $\lambda$ can be related to the Mach number $M$ with

(2.1a,b)\begin{equation} \lambda^2 = \frac{\gamma+1}{2/M^2 + (\gamma-1)}\in(0,\lambda^2_{lim}),\quad \lambda^2_{lim}=\lim_{M^2\to\infty}\lambda^2=\frac{\gamma+1}{\gamma-1}, \end{equation}

where $\gamma$ is the specific heat ratio of the gas. By formulating the T–M equations in ($\lambda ^2,\psi,\theta$) space, we obtain (cf. Appendix A for details)

(2.2a)$$\begin{gather} \frac{1}{2\lambda^2} \frac{\mathrm{d}\lambda^2}{\mathrm{d}\theta} =- \frac{\sin\psi}{\sin\theta} \frac{\sin\varphi}{1-M^2\sin^2\varphi}, \end{gather}$$

(2.2b)$$\begin{gather}\frac{\mathrm{d}\psi}{\mathrm{d}\theta}=- \frac{\sin\psi}{\sin\theta} \frac{\cos\varphi}{1-M^2\sin^2\varphi}, \end{gather}$$

whose solution with initial point $(\lambda ^2_0,\psi _0,\theta _0)$ can be expressed as $\boldsymbol {F}(\theta ;\lambda ^2_0,\psi _0,\theta _0)$. The streamline and its curvature are evaluated by solving the following equations:

(2.3)$$\begin{gather} \frac{\mathrm{d}\ln r}{\mathrm{d}\theta} = \cot\varphi, \end{gather}$$

(2.4)$$\begin{gather}\kappa = \frac{\sin\varphi}{r}\frac{\mathrm{d}\psi}{\mathrm{d}\theta}. \end{gather}$$

Figure 2.

The coordinates for the T–M equations and T–M flow.

The solution to the T–M equations (2.2) and the streamline equation (2.3) with initial value $\boldsymbol {\hat {x}}(\theta _0)=(\lambda ^2_0,\psi _0,\ln r_0)$ is denoted as $\boldsymbol {\hat {F}}(\theta ; \boldsymbol {\hat {x}}(\theta _0), \theta _0)$, where three intuitive properties, namely, the properties of evolution, reversal and symmetry, can be deduced.

Property 2.3 (Symmetry)

The solutions with initial values symmetric to the lines $(\varphi,\theta )=(k{\rm \pi} /2,{\rm \pi} /2)$ are of reflection symmetry

(2.5)\begin{equation} \left.\begin{gathered} \lambda^2({\rm \pi}-\theta; \boldsymbol{\hat{x}}_{sym}, {\rm \pi}-\theta_0) = \lambda^2(\theta; \boldsymbol{\hat{x}}_0, \theta_0)\\ k{\rm \pi}-\psi({\rm \pi}-\theta; \boldsymbol{\hat{x}}_{sym}, {\rm \pi}-\theta_0) = \psi(\theta; \boldsymbol{\hat{x}}_0, \theta_0)\\ r({\rm \pi}-\theta; \boldsymbol{\hat{x}}_{sym}, {\rm \pi}-\theta_0) = r(\theta; \boldsymbol{\hat{x}}_0, \theta_0). \end{gathered}\right\},\quad \boldsymbol{\hat{x}}_{sym} =(\lambda^2_0, k{\rm \pi}-\psi_0, \ln r_0). \end{equation}

Critical surfaces can be directly deduced from (2.2)–(2.4). Taking the derivatives in (2.2) to be zero, the equilibria ($\sin \psi =0$) are yielded, which represent the uniform flows with $\psi =0$ or ${\rm \pi}$. The equilibria here are a collection of trivial solutions and denoted as $S_{equ} = \{(\lambda ^2,\varphi,\theta )\,|\,\sin \psi =0\}$. Taking the denominator in either equation (2.2), (2.3) to be zero, the infinite discontinuities for the specific equations are yielded. The integration of the T–M equations (2.2) will be terminated by the infinite discontinuity $M^2\sin ^2\varphi =1$, which is termed Mach discontinuity in this work and denoted as $S_{Ma} = \{(\lambda ^2,\varphi,\theta )\,|\,M^2\sin ^2\varphi =1\}$. Similarly, the integration of the streamline equation (2.3) can be terminated by the infinite discontinuity $\sin \varphi =0$, which is termed slip discontinuity and denoted as $S_{slip} = \{(\lambda ^2,\varphi,\theta )\,|\,\sin \varphi =0\}$. In addition, the surface $\cos \varphi =0$ consists of zero-curvature points ($\kappa =0$) for the streamlines of non-trivial solutions ($\sin \psi \neq 0$) and is denoted as $S_{\kappa =0} = \{(\lambda ^2,\varphi,\theta )\,|\,\cos \varphi =0\}$.

2.1.2. Mach discontinuity as a bound of the elementary solution

The critical surfaces and typical solutions are plotted both in the $(\lambda ^2,\varphi,\theta )$ space and on the $(\lambda ^2,\varphi )$ plane for clarity (cf. figure 3). The slice of $S_{equ}$ varies linearly along the $\varphi$-axis as $\theta$ varies, while the slices of $S_{Ma}$ and $S_{slip}$ are independent of $\theta$. The slices of critical surfaces on the plane $(\lambda ^2,\varphi )$ will be utilized consistently in this work as they are more concise for illustration, as compared with those on the phase plane $(\lambda ^2,\psi )$.

Figure 3.

(a) Mach discontinuity, equilibria and typical solutions in $(\lambda ^2,\varphi,\theta )$ space; (b) slice of the surfaces and projection of the solutions on the $(\lambda ^2,\varphi )$ plane at $\theta =15^\circ$.

Apart from the trivial solutions on $S_{equ}$, the integrations of the T–M equations along $\theta$ directed toward 0 and ${\rm \pi}$ are both terminated by $S_{Ma}$, at the corresponding terminal points marked with $\bigcirc$ and $\triangle$, respectively, in figure 3. Thus, the elementary solution is a T–M solution defined on an interval that cannot be extended across $S_{Ma}$. For clarity, the solution branches and terminal points are defined as follows.

The leading and trailing characteristic curves of the elementary solution are tangent to the singular line at the terminal points since the relative flow inclination there is equal to the Mach angle, i.e. $|\varphi _{l,u}|=\mu$. The Mach discontinuity can be split into two families of segments based on whether the characteristics are left or right running, (cf. figure 3b).

To illustrate the influence of the left- and right-running characteristics on the solution, numerical results of four cases are obtained by solving the axisymmetric Euler equations and presented in figure 4. Four cases are divided into two groups corresponding to two elementary solutions. Two cases of the same group share the same wall surface (the bold streamlines) and free-stream Mach number, which are provided by one elementary solution. The group in the left column (cf. figure 4a,b) is based on an elementary solution called degenerate conical flow (DCF, later introduced in § 2.3.2). Alternatively, the reference T–M solution for the group in the right column (cf. figure 4c,d) is a Busemann flow. The leading characteristics are left running for the DCF solution, indicating that the disturbance induced by the wall should propagate away from the axis. Thus, the external flow (figure 4a) behaves as a conical expansion flow. In contrast, the internal flow (figure 4b) is not conical, because it is affected by the right-running characteristics, which is not expected in the DCF solution. Alternatively, since the leading characteristics are right running for the Busemann solution, the internal flow (figure 4d) behaves as the Busemann flow, while the external flow (figure 4c) does not satisfy the T–M equations. The characteristic curves at the singular line determine whether a T–M solution is internal or external.

Figure 4.

(a) External and (b) internal flow for a DCF solution. (c) External and (d) internal flow for a Busemann solution. Solid lines are contour lines of Mach number. Dashed lines are polar lines. The dot-dash-dot line is the axis.

Since the elementary solution is bounded by $S_{Ma}$, $S_{Ma}$ naturally divides the solution domain into the pre-/post-shock subdomains characterized by the Mach number corresponding to the $\theta$-component of the velocity, $M\sin \varphi$. Symbolically, these domains can be specified as

(2.6a,b)\begin{equation} D_{pre}=\{(\lambda^2,\varphi,\theta)\,|\,M^2\sin^2\varphi>1\},\quad D_{post}=\{(\lambda^2,\varphi,\theta)\,|\,M^2\sin^2\varphi<1\}. \end{equation}

Note that the shock here refers to the shock encountered in a T–M solution, which is in coincidence with the polar line. Secondary shock can emerge if the post-shock flow is supersonic. However, the secondary shock will not be located on a polar line, and the flow downstream does not satisfy the T–M equations. A typical example is the supersonic flow over a biconic body. The elementary solutions in the pre-/post-shock domains are presented in the following subsections, while discussions on shock solutions are reserved for § 3.

2.2.1. Symmetric subdomains

The pre-shock domain is cut by the Mach discontinuity into periodically spanned unconnected subdomains, in which the $k$th subdomain can be denoted as

(2.7)\begin{equation} D_{{pre},k} = D_{pre}\cap \{(\lambda^2,\varphi,\theta)\,|\,\varphi\in((k-1){\rm \pi},k{\rm \pi}),\ k\in\mathbb{Z}\}. \end{equation}

Due to the reversible nature of the T–M equations, the elementary solutions in $D_{{pre},k}$ are the same as those in $D_{{pre},k+1}$, except that the reversal in the flow direction. In addition, the equilibria ($\psi =k{\rm \pi}$) are those elementary solutions that pass through the lines $(\varphi,\theta )=((2k-1){\rm \pi} /2,{\rm \pi} /2)$, respectively. The symmetry property enables $S_{equ}$ to subdivide $D_{{pre},k}$ into two symmetric subdomains for each $k$, which are named pre-shock domains I and II, which can be specified symbolically as

(2.8a)$$\begin{gather} D_{pre,I} = D_{pre}\cap \{(\lambda^2,\varphi,\theta)\,|\, \varphi\in((k-1){\rm \pi},k{\rm \pi}-\theta),\ k\in\mathbb{Z}\}, \end{gather}$$

(2.8b)$$\begin{gather}D_{pre,II} = D_{pre}\cap \{(\lambda^2,\varphi,\theta)\,|\, \varphi\in(k{\rm \pi}-\theta,k{\rm \pi}), k\in\mathbb{Z}\}. \end{gather}$$

Note that the portions of pre-shock solutions in $D_{pre,I}$ for $\theta \in ({\rm \pi} /2,{\rm \pi} )$ and those in $D_{pre,II}$ for $\theta \in (0,{\rm \pi} /2)$ are mutually correspondent. The pre-shock solutions with initial points spanned on $\theta \in (0,{\rm \pi} /2]$ can represent the solutions in the entire pre-shock domain.

2.2.2. Divergence bound

Figure 5 depicts the elementary solutions in $D_{pre,I}$ for initial values $\lambda ^2_0=4,\theta _0=15^\circ$ with varying $\varphi _0$ in $(\lambda ^2,\varphi,\theta )$ space and on the $(\lambda ^2,\varphi )$ plane. Projections of the corresponding convergent, critical and divergent solutions on the $(\varphi,\theta )$ and $(\lambda ^2,\theta )$ planes are shown in figure 6. All three types of solutions persist in a similar trend, with an essential difference stemming from the upper terminal point. For the usual convergent solution (here, $\varphi _0=-30^\circ$), $\lambda ^2_u<\lambda ^2_{lim},\psi _u=0$, the upper terminal point lies on the equilibria, indicating a uniform inflow at the singular line. For the critical solution ($\varphi _0=-48.61^\circ$), $\lambda ^2_u=\lambda ^2_{lim},\psi _u=0$, the upper terminal point lies on the edge of the equilibria, which coincides with the limiting characteristic Mach number, implying a uniform inflow with the maximum speed. For the divergent solution ($\varphi _0=-60^\circ$), $\lambda ^2_u=\lambda ^2_{lim},\psi _u<0$, the upper terminal point retains the maximum speed, but no longer lies on the equilibria, nor does it pertain to a uniform inflow.

Figure 5.

(a) Divergence bound and typical solutions in $(\lambda ^2,\varphi,\theta )$ space; (b) slice of the surfaces and projection of the solutions on the $(\lambda ^2,\varphi )$ plane at $\theta =15^\circ$.

Figure 6.

(a) Values of $\varphi$ and (b) $\lambda ^2$ varying with $\theta$ for convergent, critical and divergent solutions.

The collection of all possible critical solutions mentioned above can be defined as the divergence bound, which divides the pre-shock domain into convergent/divergent domains.

Definition 2.3 (Divergence bound)

The divergence bound I (or II) is the collection of all possible elementary solutions in $D_{pre,I}$ (or $D_{pre,II}$), whose upper (or lower) terminal points lie on $S_{equ}$ and reach exactly the limit of characteristic Mach number $\lambda _{lim}$. Symbolically, the divergence bounds can be expressed as follows:

(2.9a)$$\begin{gather} S_{div,I} = \{(\lambda^2,\varphi,\theta)\,|\, \boldsymbol{F}({\rm \pi}; \lambda^2,\varphi,\theta) = (\lambda^2_{lim},(k-1){\rm \pi}),\ k\in\mathbb{Z}\}, \end{gather}$$

(2.9b)$$\begin{gather}S_{div,II} = \{(\lambda^2,\varphi,\theta)\,|\, \boldsymbol{F}(0; \lambda^2,\varphi,\theta) = (\lambda^2_{lim},k{\rm \pi}),\ k\in\mathbb{Z}\}. \end{gather}$$

The domain enclosed by $S_{Ma}$ and $S_{div}$ is termed convergent domain and denoted as $D_{conv}$, which is also divided into symmetric subdomains $D_{conv,I}$ and $D_{conv,II}$ by the equilibria. Similarly, the complementary domain of $D_{conv,I/II}$ in $D_{pre,I/II}$ is termed divergent domain I/II and denoted as $D_{div,I/II}$, respectively.

2.2.3. Inflection bound in pre-shock domain

A three-dimensional plot of the elementary solutions for initial values $\lambda ^2_0=5.0,\theta _0=15^\circ$ with varying $\varphi _0$, together with the projection on the $(\lambda ^2,\varphi )$-plane are shown in figure 7. The trend of the solutions can be observed from the projections on the $(\varphi,\theta )$- and $(\lambda ^2,\theta )$-planes as depicted in figure 8. Three types of solutions can be seen in this test case, namely, inflectional, critical and non-inflectional. The streamline of the inflectional solution (here $\varphi _0=167.1^\circ$) always consists of an inflection point, while those of the critical solution ($\varphi _0=167.135^\circ$) and the non-inflectional solution ($\varphi _0=167.2^\circ$) do not. These three types of solutions are close to each other for $\theta \le \theta _0$, and then the inflectional and non-inflectional ones bend away from the opposite sides of the critical solution as $\theta$ increases (cf. figure 8). The corresponding flow fields are illustrated in figure 9, where the usual inflectional solution (cf. figure 9a) is similar to the classic Busemann flow. The lower/upper branches of the inflectional solution terminate, respectively, at $S_{Ma}^\pm$ with $\varphi _u<{\rm \pi} /2<\varphi _l$, indicating the transition from internal compression to external compression. The critical solution terminates at exactly the sonic point $\lambda ^2_u=1,\varphi _u={\rm \pi} /2$, representing the flow with maximum isentropic compression that can be given by the T–M equations theoretically. The upper and lower branches of the non-inflectional solution both terminate at $S_{Ma}^+$, where ${\rm \pi} /2<\varphi _u<\varphi _l$.

Figure 7.

(a) Inflection bound and typical solutions in $(\lambda ^2,\varphi,\theta )$ space; (b) slice of the surfaces and projection of the solutions on the $(\lambda ^2,\varphi )$ plane at $\theta =15^\circ$.

Figure 8.

(a) Values of $\varphi$ and (b) $\lambda ^2$ varying with $\theta$ for inflectional, critical and non-inflectional solutions.

Figure 9.

(a) Inflectional, (b) critical and (c) non-inflectional solutions with $\lambda ^2_0=5.0,\theta _0=15^\circ$ and varying $\varphi _0$.

The derivative $\mathrm {d}\varphi /\mathrm {d}\lambda ^2$ at $\cos \varphi =0$ in the domain $D_{pre,II}$

(2.10)\begin{equation} \left.\frac{\mathrm{d}\varphi}{\mathrm{d}\lambda^2}\right|_{\cos\varphi=0} = \left.-\frac{\mathrm{d}\theta}{\mathrm{d}\lambda^2}\right|_{\cos\varphi=0}>0, \end{equation}

which indicates that the inflectional solution goes across $S_{\kappa =0}$ only once, rather than passing through it repeatedly or touching it tangentially. As a result, the terminal points of the inflectional solution will lie on different segments of $S_{Ma}$, while the non-inflectional solution is supported by the same one, either $S_{Ma}^+$ or $S_{Ma}^-$. For the critical solution, one of its terminal points is exactly the split point of $S_{Ma}^\pm$. Thus, the inflectional and non-inflectional solutions can be determined according to the location of the terminal point.

Definition 2.4 (Inflection bound in the pre-shock domain)

The collection of the elementary solutions in $D_{pre,I}$, whose lower branches terminate at exactly the zero-curvature point, is termed inflection bound I and denoted as $S_{inflect,I}$.

Similarly, the collection of the elementary solutions in $D_{pre,II}$, whose upper terminal points are sonic, is termed inflection bound II and denoted as $S_{inflect,II}$. Symbolically

(2.11a)$$\begin{gather} S_{inflect,I} = D_{pre}\cap \{(\lambda^2,\varphi,\theta)\,|\, \boldsymbol{F}(\theta_l; \lambda^2,\varphi,\theta) = (1.0,(2k-1){\rm \pi}/2),\ k\in\mathbb{Z}\}, \end{gather}$$

(2.11b)$$\begin{gather}S_{inflect,II} = D_{pre}\cap \{(\lambda^2,\varphi,\theta)\,|\, \boldsymbol{F}(\theta_u; \lambda^2,\varphi,\theta) = (1.0,(2k-1){\rm \pi}/2),\ k\in\mathbb{Z}\}. \end{gather}$$

The domain enclosed by $S_{Ma}$, $S_{equ}$, $S_{inflect,I}$ (or $S_{inflect,II}$) and the surface $\lambda ^2=\lambda ^2_{lim}$ is termed inflection domain I (or II) and denoted as $D_{inflect,I}$ (or $D_{inflect,II}$). The complementary domain of $D_{inflect,I/II}$ in $D_{pre,I/II}$ is termed non-inflection domain I/II and denoted as $D_{non\textrm {-}inflect,I/II}$.

The intersection of $S_{div}$ and $S_{inflect}$ divides $S_{pre}$ into four subdomains: the convergent inflectional domain (Busemann domain, $D_{Bu}$), the convergent non-inflectional domain ($D_{non\textrm {-}inflect,Bu}$), the divergent inflectional domain ($D_{inflect,div}$) and the divergent non-inflectional domain ($D_{non\textrm {-}inflect,div}$).

2.3.1. Symmetric subdomains

The post-shock domain is the connected domain across the entire $\varphi$-axis with $\lambda ^2<\lambda ^2_{Ma}$. To facilitate the analysis, the post-shock domain is firstly divided into subdomains according to the property of symmetry. The elementary solutions passing through the lines $(\varphi,\theta )=((2k-1){\rm \pi} /2,{\rm \pi} /2)$ are exactly the equilibria, $\psi =k{\rm \pi}$, which divide the post-shock domain into periodical subdomains as

(2.12)\begin{equation} D_{{post},k} = D_{post}\cap \{(\lambda^2,\varphi,\theta)\,|\,\psi\in((k-1){\rm \pi},k{\rm \pi}),\ k\in\mathbb{Z}\}. \end{equation}

The elementary solutions passing through the lines $(\varphi,\theta )=(k{\rm \pi},{\rm \pi} /2)$ are termed symmetry bound and denoted as $S_{sym}$ (cf. figure 10)

(2.13)\begin{equation} S_{sym} = \{(\lambda^2,\varphi,\theta)\,|\,(\lambda^2,\varphi) = \boldsymbol{F}(\theta; \lambda^2_0,k{\rm \pi},{\rm \pi}/2),\lambda^2_0\in(0,\lambda^2_{lim}),\ k\in\mathbb{Z}\}. \end{equation}

The symmetry bound in each $D_{{post},k}$ can further subdivide it into two symmetric portions, the post-shock domains I and II, which can be specified symbolically as follows:

(2.14a)$$\begin{gather} D_{post,I} = D_{post}\cap \{(\lambda^2,\varphi,\theta)\,|\, \varphi\in(k{\rm \pi}-\theta,\varphi_{sym}),\ k\in\mathbb{Z}\}, \end{gather}$$

(2.14b)$$\begin{gather}D_{post,II} = D_{post}\cap \{(\lambda^2,\varphi,\theta)\,|\, \varphi\in(\varphi_{sym},(k+1){\rm \pi}-\theta),\ k\in\mathbb{Z}\}. \end{gather}$$

As a result, the portions of post-shock solutions $D_{post,I}$ for $\theta \in ({\rm \pi} /2,{\rm \pi} )$ are equivalent to those in $D_{post,II}$ for $\theta \in (0,{\rm \pi} /2)$. The analysis performed in $D_{post}$ for $\theta \in (0,{\rm \pi} /2]$ would be representative for the entire post-shock domain.

Figure 10.

(a) Critical surfaces and typical solutions in $(\lambda ^2,\varphi,\theta )$ space; (b) slice of the surfaces and projection of the solutions on the $(\lambda ^2,\varphi )$ plane at $\theta =15^\circ$.

The elementary solutions on $D_{{sym}}$ are literally symmetric, as shown in figure 11(b). Also, the solutions above and below the symmetry bound (cf. figure 11a,c) are symmetric to each other. The flow above the slipstream resembles the classic ECF solution, while the flow below differs entirely from the nature of those four classic solutions. Thus, the corresponding solution branches can be abbreviated as the ECF and the inner flow of ECF (IECF), respectively.

Figure 11.

The ECF–IECF solutions (a) with $\theta _0<\theta _{sym}$, (b) on $S_{sym}$ and (c) with $\theta _0>\theta _{sym}$.

2.3.2. Degeneration bound

The elementary solutions for $\varphi _0=150^\circ,\theta _0=15^\circ$ by varying $\lambda ^2_0$ show a similar trend as depicted in figures 12 and 13, where the essential difference appears at the upper terminal points. The elementary solution that passes through the slip discontinuity is split into the ECF and IECF branches (cf. figure 14c). However, unlike the classic ECF solution, the ECF branch here contains an inflection point. As $\lambda ^2_0$ increases, the slipstream moves toward the axis, and the IECF branch shrinks accordingly. When $\lambda ^2_0$ reaches a critical value, the slipstream coincides with the axis while the IECF branch disappears (cf. figure 14b).

Figure 12.

(a) Critical surfaces and typical solutions in $(\lambda ^2,\varphi,\theta )$ space; (b) slice of the surfaces and projection of the solutions on the $(\lambda ^2,\varphi )$ plane at $\theta =15^\circ$.

Figure 13.

(a) Values of $\varphi$ and (b) $\lambda ^2$ varying with $\theta$ for degenerate, critical and non-degenerate solutions.

Figure 14.

(a) The DCF, (b) critical and (c) ECF–IECF solutions with $\varphi _0=150^\circ,\theta _0=15^\circ$ and varying $\lambda ^2_0$.

The elementary solution that consists of ECF and IECF branches and passes through the slip discontinuity can be stated as the non-degenerate solution in contrast to the solution for the DCF, whose IECF branch disappears. In general, the upper terminal points of the degenerate and critical solutions are located exactly on the equilibria (cf. figure 13a), implying uniform flow (cf. figure 14a,b). In particular, all the critical solutions approach tangentially to the singular points $(1,(2k+1){\rm \pi} /2,{\rm \pi} /2)$. The collection of the above critical solutions can be defined as the degeneration bound, which separates the degenerate solutions from the non-degenerate ones.

Definition 2.5 (Degeneration bound)

The collection of the elementary solutions in $D_{post}$, which are terminated by and tangent to $S_{Ma}$ at the singular points $(1,(2k+1){\rm \pi} /2,{\rm \pi} /2)$, is termed the degeneration bound and denoted as $S_{degen}$

(2.15)\begin{equation} S_{degen} = D_{post}\cap\{(\lambda^2,\varphi,\theta)\,|\, \boldsymbol{F}({\rm \pi}/2; \lambda^2,\varphi,\theta)= (1.0,(2k+1){\rm \pi}/2)\}. \end{equation}

The domain enclosed by $S_{Ma}$, $S_{equ}$ and $S_{degen}$ can be defined as the degenerate domain and denoted as $D_{DCF}$. The associated complementary domain in $D_{post}$ is termed non-degenerate domain and denoted as $D_{CF}$.

Accordingly, the degeneration bound divides the post-shock domain into non-degenerate and degenerate domains consisting of elementary solutions with and without the IECF branch, respectively. The non-degenerate domain can be subdivided further into subdomains corresponding to the ECF and IECF branches, considering that the streamline of ECF–IECF solutions is not continuous at $S_{slip}$. The IECF domain is enclosed by $S_{Ma}$, $S_{slip}$ and $S_{sym}$ and denoted as $D_{IECF}$, while the ECF domain is the complementary domain of $D_{IECF}$ in $D_{CF}$ and denoted as $D_{ECF}$. The associated subsets of $D_{ECF}$ in $D_{post,I/II}$ are denoted as $D_{ECF,I/II}$.

2.3.3. Sonic bound

The local minimum of $\lambda ^2$ for the elementary solution in $D_{post}$ is located at $\mathrm {d}\lambda ^2/\mathrm {d}\theta =0$, or $\sin \varphi =0$. Thus, elementary solutions passing through the sonic points on the slip discontinuity are supersonic elsewhere. Accordingly, the collection of the above solutions is termed a sonic bound and denoted as $S_{sonic}$. Symbolically,

(2.16)\begin{equation} S_{sonic} = \{(\lambda^2,\varphi,\theta)\,|\,(\lambda^2,\varphi) = \boldsymbol{F}(\theta; 1.0,k{\rm \pi},\theta_0),\theta_0\in(0,{\rm \pi}),\ k\in\mathbb{Z}\}. \end{equation}

The elementary solutions on $S_{degen}$ terminate exactly at the sonic point $(1,(2k+1){\rm \pi} /2,{\rm \pi} /2)$, among which the sonic equilibrium has the minimum speed. Thus, the solutions in $D_{DCF}$ are supersonic. As illustrated in figure 15, $S_{sonic}$ and $S_{degen}$ divide $D_{post}$ into the transonic domain $D_{trans}$ and the supersonic domain $D_{super}$. Since the post-shock solutions with inflection points (cf. figure 31c) must intersect with $S_{\kappa =0}$, where the intersection points are subsonic, all the elementary solutions in $D_{super}$ are non-inflectional.

Figure 15.

(a) Critical surfaces in $(\lambda ^2,\varphi,\theta )$ space and (b) their slices at $\theta =15^\circ$.

3.1.1. Classic solutions with uniform outflow

The collection of the elementary solutions tangent to $S'_{equ}$ (cf. figure 16a) is termed the classic Busemann bound and denoted $S_{cBu}$. The elementary solutions in $D_{pre}$ outside $S_{cBu}$ do not intersect with $S'_{equ}$, which implies that the classic solutions with uniform outflows, i.e. the classic Busemann and ICFB solutions, may not exist for the present case. In addition, each elementary solution on $S_{cBu}$ is tangent to $S'_{equ}$, corresponding to one uniform outflow. Every elementary solution interior to $S_{cBu}$ intersects with $S'_{equ}$ at two points, which produces two types of shock-connected solutions with strong and weak shocks, respectively. The numerical results reveal that $S_{cBu}$ does not intersect with $S_{inflect}$, and the domain consisting of the classic pre-shock solutions is contained in $D_{inflect}$. Hence, there exists an inflection point upstream of the shock in the classic Busemann flow, which is consistent with the conclusion given by Mölder (Reference Mölder2019).

Figure 16.

(a) Classic Busemann bound and elementary solutions in $(\lambda ^2,\varphi,\theta )$ space. Slices of critical surfaces and projection of the solutions on the $(\lambda ^2,\varphi )$ plane at (b) $\theta =15^\circ$, (c) $\theta =78.56^\circ$.

The combinations of the upper pre-shock branches and the lower post-shock branches in figure 16(b,c) are illustrated in figure 17(a,b). It is seen that the same pre-shock solution may be connected to different uniform post-shock flows with shocks of different intensities and locations. Here, the pre-shock solution is the same in the sense that the pre-shock branches that split at different shock locations share the same elementary solution.

Figure 17.

Classic Busemann solutions with a (a) weak and (b) strong shock based on the same pre-shock solution.

3.1.2. Classic solutions with uniform inflow

Post-shock elementary solutions cannot approach tangentially to $S'_{equ}$. They rather intersect with $S'_{equ}$ at a single point or emerge without intersection. Then, the boundary of $S'_{equ}$ ($\partial S'_{equ}$), determines the existence of the classic post-shock solution. Accordingly, the collection of the elementary solutions integrated from the points on the boundary is termed the classic ECF bound and denoted as $S_{cECF}$. In addition, the elementary solutions integrated from the points on $S'_{equ}$ which satisfy the shock detachment condition (B6) turn out to be exactly the detached solution of classic ECF type. Likewise, the collection of the above elementary solutions can be termed the detachment bound and denoted as $S_{detach}$.

The bounds $S_{cECF}$ and $S_{detach}$ are illustrated in figure 18. The elementary solutions outside $S_{cECF}$ or on the right of $S_{degen}$ do not intersect with $S'_{equ}$, indicating that the classic solutions with uniform inflow, i.e. classic ECF and ICFA solutions, do not exist. The domain enclosed by $S_{cECF}$ can be subdivided further by $S_{detach}$ into subdomains for classic solutions with strong (left side) and weak (right side) shocks. Figure 19 depicts two types of ECF branches integrated downstream of strong and weak shocks, respectively, with the same uniform pre-shock inflow.

Figure 18.

(a) Classic ECF bound, detachment bound and elementary solutions in $(\lambda ^2,\varphi,\theta )$ space. Slices of critical surfaces and projection of the solutions on the $(\lambda ^2,\varphi )$ plane at (b) $\theta =15^\circ$, (c) $\theta =42.5^\circ$, (d) $\theta =87.14^\circ$.

Figure 19.

Classic ECF solutions with a (a) weak and (b) strong shock under the same inflow velocity and cone half-angle.

3.1.3. Classic solutions as solution branches

A classic solution is a combination of two solution branches sharing a common shock as their initial stations. One branch shows the features of classic solutions, while the other is part of the equilibria. As illustrated in figure 20, with a pre-shock elementary solution in $D_{pre}$ and its corresponding trivial solution in $D_{post}$, two pairs of different branches can be obtained to form two classic solutions. Specifically, the lower branch of the pre-shock solution can be combined with the upper branch of the post-shock equilibria to form the classic ICFB solution (cf. figure 20b), while the upper pre-shock branch and the lower post-shock branch form the classic Busemann solution (cf. figure 20c). Similarly, for the shock-connecting solution with a post-shock solution in $D_{post}$ and its corresponding pre-shock trivial solution (cf. figure 21), the upper branch of the post-shock solution is part of the classic ICFA solution (figure 21b), and the lower one is part of the classic ECF solution (figure 21c).

Figure 20.

(a) Branches of the elementary solutions. Classic (b) ICFB solution (dashed lines in (a)) and (c) Busemann solution (solid lines in (a)).

Figure 21.

(a) Branches of the elementary solutions. Classic (b) ICFA solution (dashed lines in (a)) and (c) ECF solution (solid lines in (a)).

In summary, the four classic solutions are combinations of branches extracted from two pairs of elementary solutions. The classic Busemann and ICFB solutions belong to one, while the classic ECF and ICFA solutions belong to the other. Further, following notations of the four classic solutions, the branch of solutions in $D_{Bu}$ with uniform flows can be denoted as the Busemann branch, and the remaining branch the ICFB branch. Meanwhile, the branch of solutions that approaches $S_{slip}$ in $D_{ECF}$ can be denoted as the ECF branch, and the opposite branch is denoted as the ICFA branch.

3.2.1. Critical points for supersonic shock-connected solutions

Analogously to the shock opposite of equilibria, the mapping of the shock relations (3.1) and (3.2) can bring the critical surfaces defined in § 2 to parametric curvilinear surfaces, called the shock-opposite surfaces and denoted as $S'$. The shock-opposite domains can be defined similarly and denoted as $D'$. Note that the shock-opposite surfaces are no longer collections of elementary solutions. Instead, they serve only as the references where the pair of pre- and post-shock points are located on $S_{\theta =\theta _{s}}$. In other words, the topology of the critical surfaces together with their associated shock opposites determine which pairs of pre- and post-shock subdomains $D_{1,2}$ can be connected for a shock located at $\theta =\theta _s$, or symbolically, $D'_1\cap D_2\cap S_{\theta =\theta _{s}}\ne \varnothing$.

To avoid a lengthy discussion, here, efforts will be focused on the combination with elementary solutions in $D_{super}$, and the symbols introduced before will be used extensively (cf. Appendix C for the nomenclature). Figure 22 depicts the slices of most critical surfaces and their associated shock opposites at critical shock locations. The topologies on different slices are significantly distinct from each other. Thus, the permissible shock-connected solution varies correspondingly with respect to the variation of $\theta _{s}$.

Figure 22.

Critical surfaces (solid lines) and their shock opposite (dotted lines) on $S_{\theta =\theta _{s}}$. The relevant subdomains on the critical slice are filled with different colours. The dashed arrows indicate the direction the intersection curves move as $\theta$ increases. Panels show (a) $\theta _s=23.41^{\circ }$, (b) $\theta _s=27.8^{\circ }$, (c) $\theta _s=55.35^{\circ }$, (d) $\theta _s=58.92^{\circ }$, (e) $\theta _s=63.77^{\circ }$, (f) $\theta _s=66.85^{\circ }$.

As illustrated in figure 22, there are three relevant subdomains in $D_{post}$ for $\theta \in (0,90^\circ ]$, i.e. $D_{super,ECF,I}$, $D_{super,IECF}$ and $D_{DCF}$; $D'_{super,ECF,I}$ behaves as a simply connected region contained in $D_{inflect,I}\cup D_{pre,II}$ on $S_{\theta =\theta _s}$ until $\theta _{s}$ reaches $58.92^\circ$. On $S_{\theta =58.92^\circ }$, $S_{sonic}$ touches $S_{Ma}$ and $D_{super,ECF,I}$ is split into two separate portions (cf. figure 22d), among which the smaller portion near the sonic equilibrium shrinks quickly as $\theta _s$ increases. When $\theta _{s}>66.85^\circ$, $D'_{super,ECF,I}$ retreats from $D_{inflect,div,I}$ (cf. figure 22f). In addition, the larger portion of $D'_{super,ECF,I}$ also retreats from $D_{Bu,I}$ and becomes contained in $D_{pre,II}$, while the smaller portion still intersects with $D_{Bu,I}$.

The shock-opposite subdomain $D'_{super,IECF}$ is contained in $D_{non\textrm {-}inflect,div,I}$ initially. As $\theta _{s}$ increases, $D'_{super,IECF}$ begins to intersect with $D_{inflect,div,I}$, $D_{non\textrm {-}inflect,Bu,I}$ and $D_{Bu,I}$ at $\theta _{s}=23.41^\circ$, $55.35^\circ$, $63.77^\circ$, respectively (cf. figure 22a,c,e). The shock-opposite subdomain $D'_{DCF}$ intersects with $D_{inflect,I}$ initially. As $\theta _{s}$ increases, $D'_{DCF}$ retreats from $D_{inflect,div,I}$ and becomes contained in $D_{Bu,I}$ at $\theta _{s}=27.87^\circ$ (cf. figure 22b). As illustrated also in figure 22(b), $D_{conv,I}$ and $D_{non\textrm {-}inflect,I}$ begin to intersect with each other and form $D_{non\textrm {-}inflect,Bu,I}$ on $S_{\theta =\theta _s}$ after $\theta _{s}$ reaches $27.75^\circ$.

The critical slices related to $S_{sym}$, the dividing surface between $D_{ECF,II}$ and $D_{IECF}$ for $\theta \in (0,{\rm \pi} /2]$, are illustrated in figure 23. After $\theta _{s}$ reaches $69.69^\circ$, $D'_{super,ECF,II}$ emerges and intersects with $D_{non\textrm {-}inflect,Bu,I}$ on $S_{\theta =\theta _s}$ (cf. figure 23a). As $\theta _{s}$ increases further, $D'_{super,ECF,II}$ intersects with the other three subdomains in $D_{pre,I}$, $D_{Bu,I}$, i.e. $D_{inflect,div,I}$ and $D_{non\textrm {-}inflect,div,I}$, subsequently at $\theta _{s}=74.25^\circ$, $77.60^\circ$, $77.86^\circ$ (cf. figure 23b,c). In contrast, $D'_{super,IECF}$ intersects with the four pre-shock subdomains until $\theta _{s}$ reaches approximately $80^\circ$ (cf. figure 23d). Specifically, $D'_{super,IECF}$ retreats from $D_{non\textrm {-}inflect,Bu,I}$ at $\theta _{s}=79.98^\circ$ and $D_{Bu,I}$ at $\theta _{s}=80.10^\circ$.

Figure 23.

Critical surfaces related to the symmetry bound (solid lines) and their shock opposite (dotted lines) on $S_{\theta =\theta _{s}}$. The relevant subdomains on the critical slice are filled with different colours. The dashed arrows indicate the direction in which the intersection curves move as $\theta$ increases. Panels show (a) $\theta _s=66.69^{\circ }$, (b) $\theta _s=74.25^{\circ }$, (c) $\theta _s=77.7^{\circ }$, (d) $\theta _s=80.0^{\circ }$.

3.2.2. Classification of supersonic shock-connected solutions

The critical cases analysed above are summarized in table 1, together with the associated polar angles $\theta _{s}$. Those critical cases, as manifesting topological changes on the corresponding slices, are utilized to clarify the $\theta _{s}$ ranges compatible with the shock-connected solutions. In other words, the pre-shock subdomains can be connected to the post-shock subdomains in the specific $\theta _{s}$ ranges shown in figure 24. Leaving out the transonic solutions, there are 16 types of supersonic shock-connected solutions identified for $\theta _{s}\in (0,90^\circ )$. The 4 subdomains contained in $D_{pre,II}$ ($D_{non\textrm {-}inflect/inflect,II}\cap D_{conv/div,II}$) intersect with only $D'_{ECF,I}$ for $\theta _{s}\in (0,90^\circ )$, while the 4 subdomains in $D_{pre,I}$ may be connected to $D_{ECF,II}$ and $D_{IECF}$ with a total of 8 solutions under different $\theta _{s}$ ranges. Here, $D_{Bu,I}$ and $D_{inflect,div,I}$ can also be connected to $D_{ECF,I}$ and $D_{DCF}$ with 4 solutions. Note that $D_{ECF,I}$ and $D_{ECF,II}$ are shown on the same track. The dashed line near $66.85^\circ$ indicates that the larger portion of $D_{super,ECF,I}$ cannot be connected to $D_{Bu,I}$ for $\theta _{s}\ge 66.85^\circ$, while the smaller portion can.

Table 1.

Critical $\theta _{s}$ values for $\gamma =1.4$, where the topology on the slice changes. Here, $+$/$-$ indicate that something appears or disappears, respectively, as $\theta$ increases.

Figure 24.

The $\theta _{s}$ ranges for typical supersonic shock-connected solutions.

The previous discussions focus on the topological relations between pre- and post-shock subdomains, or in other words, the connection of pre- and post-shock elementary solutions. For a pair of pre- and post-shock elementary solutions, the upper/lower branches of the pre-shock solution can be connected to the lower/upper branches of the post-shock one, respectively, and thus form two different shock-connected solutions. Recall that, in § 3.1.3, the elementary solutions in $D_{Bu}$ can be defined as a combination of the Busemann branch and the ICFB branch, while those in $D_{ECF}$ are composed of the ECF branch and the ICFA branch. Similarly, the elementary solutions in $D_{non\textrm {-}inflect,Bu}$ can also be split into the Busemann branch and the ICFB branch, while the divergent solutions in $D_{inflect,div}$, $D_{non\textrm {-}inflect,div}$ only have the ICFB branches. Further, the branch of the elementary solution in $D_{DCF}$ with uniform inflow is a conical expansion flow and can be denoted as the CEF branch, while the opposite one is a conical compression flow denoted as the CCF branch.

Apart from the solutions relating to $D_{div}$, four types of shock-connected solutions are allowed for $\theta _{s}<55.35^\circ$ (cf. figure 24), i.e. solutions across the domains $D_{Bu,I}\times D_{ECF,I}$, $D_{Bu,I}\times D_{DCF}$, $D_{Bu,II}\times D_{ECF,I}$, $D_{non\textrm {-}inflect,Bu,II}\times D_{ECF,I}$. As an example, four typical shock-connected solutions at $\theta _{s}=15^\circ$ are presented in figure 25–28. Among which, the elementary solutions across $D_{Bu,I}$ and $D_{ECF,I}$ are depicted in figure 25(a). The combination of the pre-shock upper branch with the post-shock lower branch forms a solution of Busemann–ECF type, which is a compression flow from the inlet to the outlet. On the other hand, the combination of the pre-shock lower branch with the post-shock upper branch forms an ICFB–ICFA solution. Here, the branches are expansion flow while strong compression occurs across the shock due to the flow deflection between the branches.

Figure 25.

(a) Shock-connected elementary solutions across $D_{Bu,I}$ and $D_{ECF,I}$, and shock-connected solution branches (b) Bu$_{\textrm {I}}$–ECF$_{\textrm {I}}$, (c) ICFB$_{\textrm {I}}$–ICFA$_{\textrm {I}}$.

For the elementary solutions across $D_{Bu,II}$ and $D_{ECF,I}$ (cf. figure 26), the combination of the pre-shock lower branch and post-shock upper branch forms a Busemann–ICFA solution, which is a compression–expansion flow. The combination of the pre-shock upper branch and post-shock lower branch forms a solution of ICFB–ECF type, revealing an expansion–compression flow. In accordance with the symmetry property of the T–M solutions, the shock-connected solution Bu$_{\textrm {II}}$–ECF$_{\textrm {I}}$ at $\theta _{s}=15^\circ$ will be the same as the Bu$_{\textrm {I}}$–ECF$_{\textrm {II}}$ solution at $\theta _{s}=165^\circ$.

Figure 26.

(a) Shock-connected elementary solutions across $D_{Bu,II}$ and $D_{ECF,I}$, and shock-connected solution branches (b) Bu$_{\textrm {II}}$–ICFA$_{\textrm {I}}$, (c) ICFB$_{\textrm {II}}$–ECF$_{\textrm {I}}$.

The elementary solutions across $D_{non\textrm {-}inflect,Bu,II}$ and $D_{ECF,I}$ (cf. figure 27) are similar to those across $D_{Bu,II}$ and $D_{ECF,I}$. The structures of the Busemann–ICFA-type solutions, constructed by the Busemann branches in $D_{Bu,II}$ and $D_{non\textrm {-}inflect,Bu,II}$, respectively, are basically the same. However, due to the inflection point located on the ICFB branch, significant discrepancies can be observed between the inflectional and non-inflectional ICFB branches under slight changes of the initial values ($\Delta \varphi _0=0.2^\circ$). The non-inflectional ICFB branch is terminated quickly, while the inflectional ICFB branch covers a wider range of polar angles.

Figure 27.

(a) Shock-connected elementary solutions across $D_{non\textrm {-}inflectBu,II}$ and $D_{ECF,I}$, and shock-connected solution branches (b) non-inflectBu$_{\textrm {II}}$–ICFA$_{\textrm {I}}$, (c) non-inflectICFB$_{\textrm {II}}$–ECF$_{\textrm {I}}$.

For the elementary solutions across $D_{Bu,I}$ and $D_{DCF}$ (cf. figure 28), the combination of the pre-shock upper branch and post-shock lower branch forms a solution of Busemann–CEF type, which is a compression–expansion flow in nature. On the other hand, the combination of the pre-shock lower branch and post-shock upper branch forms an expansion–compression flow of ICFB–CCF type.

Figure 28.

(a) Shock-connected elementary solutions across $D_{Bu,I}$ and $D_{DCF}$, and shock-connected solution branches (b) Bu$_{\textrm {I}}$–CEF, (c) ICFB$_{\textrm {I}}$–CCF.

There are four additional types of shock-connected solutions for $\theta _{s}\ge 69.69^\circ$, i.e. the solutions across the domains $D_{Bu,I}\times D_{ECF,II}$, $D_{Bu,I}\times D_{IECF}$, $D_{non\textrm {-}inflect,Bu,I}\times D_{ECF,II}$, $D_{non\textrm {-}inflect,Bu,I}\times D_{IECF}$. The inflection point is now located on the Busemann branch and behaves similarly to the inflection point in $D_{pre,II}$. However, since the inflection point in the present case is close to the shock, the differences between the inflectional and non-inflectional branches are not as significant as those in the previous cases for small $\theta _{s}$. This implies that there is little influence of the inflection point on the behaviour of the connected solution, and it suffices to investigate simply the nature of the solutions across $D_{Bu,I}\times D_{ECF,II}$ and $D_{Bu,I}\times D_{IECF}$. The solution across $D_{Bu,I}\times D_{ECF,II}$ with the pre-shock initial value $(\lambda ^2_0,\varphi _0,\theta _0)=(3.0,-123^\circ,75^\circ )$ is illustrated in figure 29. The types of the shock-connected solution branches across $D_{Bu,I}\times D_{ECF,II}$ are the same as those across $D_{Bu,II}\times D_{ECF,I}$, i.e. the Busemann–ICFA type and the ICFB–ECF type. When the initial value becomes $(\lambda ^2_0,\varphi _0,\theta _0)=(3.1,-123^\circ,75^\circ )$, the post-shock solution moves to $D_{IECF}$. However, the location of the slipstream is close to $90^\circ$. Specifically, $\theta _{slip}=89.87^\circ$ for the IECF solution and $\theta _{slip}=90.35^\circ$ for the ECF$_{\textrm {II}}$ solution. Thus, the ECF and IECF solutions here are almost the same, and so are the shock-connected solutions.

Figure 29.

(a) Shock-connected elementary solutions across $D_{Bu,I}\times D_{ECF,II}$ and shock-connected solution branches (b) Bu$_{\textrm {I}}$–ICFA$_{\textrm {II}}$, (c) ICFB$_{\textrm {I}}$–ECF$_{\textrm {II}}$.