2.1.1. The straight branch

The straight branch is essentially a swept cylinder with a sweep angle equivalent to the half-span angle $\beta$. When the half-span angle $\beta$ is larger than the Mach angle, a DS will be generated from the swept cylinder and gradually develop along the wall until it reaches a fully developed state. The fully developed state results in a constant standoff distance, which is denoted as $l$ in this paper. Therefore, a fully developed DS is presented as an oblique shock with a shock angle equal to $\beta$ on the $x$–$z$ symmetry plane and the flow component perpendicular to it can be treated as a supersonic flow around a cylinder with a radius $r$. Previous researchers have conducted extensive theoretical work on this flow pattern (Moeckel Reference Moeckel1949; Hida Reference Hida1953; Lighthill Reference Lighthill1957). To obtain the standoff distance, Zhang et al. (Reference Zhang, Li and Yang2021) followed the inviscid theoretical approximation proposed by Sinclair & Cui (Reference Sinclair and Cui2017). This theoretical approach demonstrates excellent and consistent agreement with the computational solutions and experimental results across a wide range of Mach numbers from 1.35 to 6. However, due to the complexity involved in the derivation, this method faces difficulties in being extended to three-dimensional flows. In this paper, an approximate method proposed by Moeckel (Reference Moeckel1949) for predicting detached shock waves ahead of blunt bodies is applied. This approach is based on a simplified form of the continuity relation and can be easily developed into three-dimensional flows, which will be discussed in § 2.1.2.

A schematic diagram of the simplified continuity method is depicted in figure 3. The black solid line represents the surface of the blunt body and the red solid line is the detached shock DS ahead of it. The vertex of the blunt body is located at the origin and is denoted as O, while the foremost point of the DS is referred to as D. The distance between two points is identified as the standoff distance $l$. Here, B and S represent the sonic points on the body surface and on the shock wave, respectively; $\delta _B$ and $\delta _S$ are the flow deflection angles at the two points. The blue dashed line connecting points B and S is assumed to be the sonic line. The free stream is oriented parallel to the $x$-axis. The DS are normal to the free stream at their foremost point D and tend to approach the free stream Mach waves at large distances from the point. Hence, the DS in figure 3 is asymptotic to the free stream Mach line, i.e. the red dashed line, which inclines at the Mach angle $\alpha$ and intersects the $x$-axis at point A. Assuming that the shock wave is a simple curve that exhibits these properties, i.e. a hyperbola, which can be expressed mathematically as

(2.1)\begin{equation} y = k\sqrt {{{( {x - {x_{{A}}}} )}^2} - {{( {{x_{{D}}} - {x_{{A}}}} )}^2}}, \end{equation}

where $k$ is the tangent of the free stream Mach angle $\alpha$. Using this equation of DS, the shock angle $\theta$, which is the angle between the stream direction and the tangent to the shock, can be determined at any point from

(2.2)\begin{equation} \tan \theta = \frac{{k(x - {x_{{A}}})}}{{\sqrt {{{( {x - {x_{{A}}}} )}^2} - {{( {{x_{{D}}} - {x_{{A}}}} )}^2}} }} = \frac{{k\sqrt {{k^2}{{( {{x_{{D}}} - {x_{{A}}}} )}^2} + {y^2}} }}{y}. \end{equation}

Figure 3.

Schematic illustration of the simplified continuity method for predicting the detached shock waves ahead of blunt bodies.

To calculate the standoff distance $l$ using the continuity method, the position of the shock wave relative to the body surface should be determined by using the geometric relation between B and S. The coordinates of the point S are

(2.3)\begin{equation} \left.\begin{array}{c@{}} \displaystyle {x_{{S}}} = \dfrac{{( {{x_{{D}}} - {x_{{A}}}} )}}{{\sqrt {1 - {k^2}{{\cot }^2}{\theta _{{S}}}} }} + {x_{{A}}}\\ \displaystyle {y_{{S}}} = \dfrac{{{k^2}( {{x_{{D}}} - {x_{{A}}}} )\cot {\theta _{{S}}}}}{{\sqrt {1 - {k^2}{{\cot }^2}{\theta _{{S}}}} }} \end{array}\right\}, \end{equation}

where $\theta _S$ is the shock angle at point S and can be calculated from the oblique shock relations. For bodies with sharp or clearly defined shoulders, research has found that the sonic point B is located at the shoulder (Ladenburg, VanVoorhis & Winckler Reference Ladenburg, VanVoorhis and Winckler1946). For more gradually curved bodies, such as ogives, Busemann (Reference Busemann1949) suggested that the shoulder is located at the point where the contour of the body is inclined at the wedge angle or cone angle corresponding to shock detachment. For the cylinder with a radius $r$, by using $\delta _d$ to denote the shock wave detachment angle, the coordinates of the sonic point B can be expressed as

(2.4)\begin{equation} \left.\begin{array}{c@{}} {x_{{B}}} = r( {1 - \sin {\delta _d}} )\\ {y_{{B}}} = r\cos {\delta _d} \end{array}\right\}. \end{equation}

From figure 3, the geometric relationship between sonic points S and B can be described as follows:

(2.5)\begin{equation} {x_{{S}}} = {x_{{B}}} + ( {{y_{{B}}} - {y_{{S}}}} )\tan \eta, \end{equation}

where $\eta$ is the inclination of the sonic line and can be estimated by the arithmetic mean of the inclinations at the two extremities,

(2.6)\begin{equation} \eta = \frac{{{\delta _d} + {\delta _s}}}{2}. \end{equation}

By substituting the coordinates (2.3) of point S into (2.5), we obtain

(2.7)\begin{equation} \frac{{( {{x_{{D}}} - {x_{{A}}}} )}}{{\sqrt {1 - {k^2}{{\cot }^2}{\theta_{{S}}}} }} + {x_{{A}}} = {x_{{B}}} + \left[ {{y_{{B}}} - \frac{{{k^2}( {{x_{{D}}} - {x_{{A}}}})\cot {\theta _{{S}}}}}{{\sqrt {1 - {k^2}{{\cot }^2}{\theta _{{S}}}} }}} \right]\tan \eta. \end{equation}

This equation establishes the relationship between $x_{{D}}$ and $x_{{A}}$, from which we can solve for $x_{{A}}$:

(2.8)\begin{equation} {x_{{A}}} = \frac{{( {1 + {k^2}\cot {\theta _{{S}}}\tan \eta } ){x_{{D}}} - \sqrt {1 - {k^2}{{\cot }^2}{\theta _{{S}}}} ( {{x_{{B}}} + {y_{{B}}}\tan \eta } )}}{{1 + {k^2}\cot {\theta _{{S}}}\tan \eta - \sqrt {1 - {k^2}{{\cot }^2}{\theta _{{S}}}} }}. \end{equation}

Through the above derivation of the geometric relationship, only the quantity $x_{{D}}$ remains to be determined for the prediction of the detached shock DS. To determine this value, the continuity relation should be applied to the control volume surrounded by the body contour, the detached shock, the sonic line and the symmetry plane (or the $x$-axis), i.e. the subsonic flow through the shock segment SD is isentropically accelerated to the speed of sound and across the sonic line. As the distribution of the flow variables along the sonic line is unknown, the stagnation pressure immediately behind the shock is applied for the calculation. For planar flow, the average value of this quantity can be approximated by using the stagnation pressure $P_C$ behind the shock at the $y_{{C}}=y_{{S}}/2$. By denoting the stagnation pressure of the free stream and the contraction ratio required to decelerate the free stream to sonic velocity isentropically as $P_0$ and $\sigma$, respectively, the simplified continuity equation may be expressed as

(2.9)\begin{equation} {A_{{S}}} = \sigma {A_0}\frac{{{P_0}}}{{{P_{{C}}}}}, \end{equation}

where $A_0$ is the projected height of the shock segment SD on the free stream direction and AS is the length of the sonic line BS, i.e.

(2.10)$$\begin{gather} {A_0} = {y_{{S}}}, \end{gather}$$

(2.11)$$\begin{gather}{A_{{S}}} = \frac{{{y_{{S}}} - {y_{{B}}}}}{{\cos \eta }}. \end{gather}$$

Figure 4 provides a comparison of the non-dimensional shock standoff distance $l/r$ estimated by the continuity method with the theoretical method of Sinclair & Cui (Reference Sinclair and Cui2017), the experimental results of Alperin (Reference Alperin1950), Kaattari (Reference Kaattari1961) and Kim (Reference Kim1956), as well as the inviscid computational fluid dynamics (CFD) results of this work. While both theoretical methods have been found to exhibit good agreement with the experimental and simulation results for low free stream Mach numbers, it has been noted that the method of Sinclair & Cui (Reference Sinclair and Cui2017) shows some deviation at higher Mach numbers. At the same time, the continuity method can present excellent agreement across the entire range of Mach numbers from 1 to 10. This result indicates that the continuity method is a reliable and robust approach for modelling the supersonic flow around a cylinder, especially at higher Mach numbers. Moreover, this method also applies to bodies with a variety of contour shapes, such as ellipses, wedges, etc., with minor modifications of the shoulder.

Figure 4.

The non-dimensional shock standoff distance $l/r$ for a cylinder as a function of the free stream Mach number $M_0$ obtained by different methods.

Figure 5 further compares the non-dimensional shock standoff distance $l/r$ for swept cylinders between the theoretical results and viscous CFD results. The sweep angle $\beta$ is fixed at $24^\circ$ in figure 5(a) and the free stream Mach number $M_0$ is fixed at 6 in figure 5(b). It is observed that the theoretical estimations closely agree with the numerical results for relatively large sweep angles ($\beta >24^\circ$) and free stream Mach numbers ($M_0>6$), although viscosity effects have not been considered in theoretical prediction. This indicates that the shock standoff distance is rarely affected by the boundary layer on the straight branch in the present conditions. The difference between the theoretical and CFD results for small values of $\beta$ and $M_0$ is mainly caused by the limited length of the straight branch (Zhang et al. Reference Zhang, Li and Yang2021). Generally, it is harder for a DS to achieve the fully developed state due to the weaker vertical compression when the flow component $M_0\sin {\beta }$ is small. Therefore, this study focuses on cases in which the vertical Mach number component is relatively large. Also, the method is not applicable when $\beta$ is large enough to make the flow behind the DS subsonic, where disturbances generated by the crotch of the VBLE may affect the upstream flow and the shape of the DS. This limitation, however, does not impact the current investigation, as the MR structure and the CS at the crotch cannot be generated under such a condition.

Figure 5.

The non-dimensional shock standoff distance $l/r$ for swept cylinders obtained by theoretical method and numerical simulations as the function of (a) the free stream Mach number at a fixed sweep angle of $24^\circ$ and (b) the sweep angle at a fixed free stream Mach number of 6.

2.1.2. The crotch

Apart from solving the DS of the straight branch, the continuity method can also be utilized for the three-dimensional interaction structure of the primary MR at the crotch with reasonable simplification and assumption of the flow behaviour. Figure 6 presents the primary MR configuration and the secondary interactions of the TS at the crotch in the $x$–$z$ symmetry plane. In figure 6(a), it can be observed that a pair of DSs with standoff distance $l$ and shock angle $\beta$ from opposite families meet in front of the crotch, producing a Mach stem MS and two triple points T. The transmitted shock TS and the shear layer SL emanate from point T. The vertical distance between two triple points is defined as the MS height $H_m$. The horizontal distance between the triple point T and the stagnation point of the crotch is denoted as the triple point position $d$. By positioning the origin at the stagnation point O, the geometry equation of the shock waves and wall surfaces in the plane can be easily established. To begin with, we can describe the contour of the wall by employing the radius $R$ of the crotch and the half-span angle $\beta$,

(2.12)$$\begin{gather} \text{the crotch:}\qquad z ={\pm} [ { - \sqrt {{R^2} - {{( {x + R} )}^2}} }], \end{gather}$$

(2.13)$$\begin{gather}\text{the straight branch:}\qquad z ={\pm} \tan \beta \left( {x + R - \frac{R}{{\sin \beta }}} \right), \end{gather}$$

where the signs $-$ and $+$ represent the contour at the upper and lower branches, respectively. According to the analysis in § 2.1.1, DS is an oblique shock parallel to the straight branch with a standoff distance $l$. Thus, the geometry of DS can be expressed as follows:

(2.14)\begin{equation} z ={\pm} \tan \beta \left( {x + R + \frac{{l - R}}{{\sin \beta }}} \right). \end{equation}

Figure 6.

The primary MR configuration at the crotch of VBLEs, (b) the secondary interaction between the TS and the SS, (c) the secondary MR and (d) the secondary RR between the TS and the CS.

With (2.14), it is possible to derive an expression that relates the coordinates of the point T to the MS height $H_m$:

(2.15)\begin{equation} \left.\begin{array}{c@{}} {x_{{T}}} =- \dfrac{{{H_m}}}{{2\tan \beta }} - R - \dfrac{{l - R}}{{\sin \beta }},\\ {z_{{T}}} ={\pm} \left(-\dfrac{{{H_m}}}{2}\right). \end{array}\right\} \end{equation}

Downstream of the triple point T, the TS will form a secondary RR or secondary MR interaction with the separation shock SS or the curved shock CS generated on the converging wall. The different secondary interaction types in the black box are enlarged in figure 6(b–d). Behind the MS, the shear layer-bounded supersonic jets travel downstream along the wall and collide near the stagnation point, forming a large counter-rotating vortex pair (CVP). Due to the impact of the CVP, the MS sometimes appears as an arch shape with a raised middle. Given the diversity and complexity of these secondary flow structures, this study focuses only on the primary shock structures, and the interactions between the TS and the SS or CS are ignored. To clarify the analysis, the MS and the TS are both assumed to be straight and do not experience any deformation. The TS is extended linearly to point $\textrm {O}_2$ on the body contour. Then the geometric equation of the TS can be written as

(2.16)\begin{equation} z ={\pm} [-\tan {\theta _{{TS}}}( {x - {x_{{T}}}} ) + {z_{{T}}}], \end{equation}

where $\theta _{{TS}}$ is the angle between the TS and the free stream direction, which can be expressed as the function of the free stream Mach number $M_0$ and the DS angle $\beta$ through the use of the shock relations and the three-shock theory (von Neumann Reference von Neumann1943, Reference von Neumann1945). At the same time, the triple point T is projected to point $\textrm {O}_1$ on the wall along the free stream direction. The coordinates of the points $\textrm {O}_1$ and $\textrm {O}_2$ can be solved by using the geometric equations (2.12)–(2.13) and (2.16).

With these simplifications, the shock structure of the primary MR configuration can be described graphically in the three-dimensional space, as shown in figure 7. In the illustration, the grey surface is the body contour (BC) of the VBLE. The green surface, red surface and purple surface positioned around the BC represent the detached shock wave DS, the Mach stem MS and the transmitted shock wave TS, respectively. The orange surface, referred to as the side surface, is parallel to the $x$–$y$ symmetry plane and the triple point T is located on it. The line DT is the intersection of the MS and the $x$–$z$ symmetry plane and $\textrm {OO}_1$ is the line where the BC intersects the plane. The $\textrm {SS}_1$ and $\textrm {BB}_1$ represent sonic lines on the MS surface and the BC, respectively, and the blue area enclosed by them is defined as the sonic throat (ST). The colour of the surfaces and the labels of the points in the $x$–$z$ symmetry plane are all consistent with those in figure 6. The origin of the coordinate is still positioned at the stagnation point O.

Figure 7.

The simplified three-dimensional shock interaction configuration of the primary MR at the crotch.

Next, we will employ the simplified continuity method to the control volume encircled by the Mach stem MS, the body contour BC, the side surface, the sound throat ST and the two symmetry planes. It is assumed that the MS is hyperbolic both in the $x$–$y$ symmetry plane and the side surface. The geometrical equation of the MS in the $x$–$y$ symmetry plane follows (2.1), while the equation describing its geometry in the side surface is expressed as

(2.17)\begin{equation} y = k\sqrt {{{[ {( {x - {x_{{O1}}}} ) - ({x_{{A1}}} - x{_{{O1}}})} ]}^2} - {{( {{x_{{T}}} - {x_{{A1}}}} )}^2}}. \end{equation}

In the equation, $x_{{O}1}$, $x_{{A}1}$ and $x_{{T}}$ are the $x$-coordinates of the points $\textrm {O}_1$, ${A}_1$ and the triple point T, respectively (${A}_1$ is the intersection of the MS asymptote and the $x$–$z$ symmetry plane in the side surface and can be solved by applying the same method described in § 2.1.1). Based on the derivation presented above, one can determine the three-dimensional geometry of the shock waves with the MS height $H_m$ when the free stream parameters and the VBLE geometry are fixed. Similarly, to determine the height, the continuity relation is applied to the control volume. At this time, aside from the subsonic flow entering from the MS, the control volume is also receiving a supersonic jet entering via the side surface. It is assumed that both of these flows undergo isentropic acceleration or deceleration to reach the speed of sound within the control volume before exiting through the sonic throat ST. Given the unknown distributions of flow variables through both the ST and the side surface, the stagnation pressure is still used for the calculation. The average stagnation pressure of the flow through the MS is approximated by using the stagnation pressure P_C behind it at $y_{{C}}=y_{{S}}/2$. The flow parameters through the side surface are estimated by the values in the $x$–$z$ symmetry plane. By using $P_{DS}$ and $P_{TS}$ to denote the stagnation pressure behind the DS and the TS, the contraction ratio required to decelerate the flow in front of the side surface to sonic velocity can be approximated as follows:

(2.18)\begin{equation} {\sigma _{{{Side}}}} = {\sigma _{{{DS}}}}\frac{{{P_{{{DS}}}}}}{{{P_{{{TS}}}}}} \frac{{{d_{{{T1}}}}}}{l}, \end{equation}

where $\sigma _{{DS}}$ is the contraction ratio required to decelerate the flow behind the DS to sonic velocity isentropically. When calculating $P_{{TS}}$, the shock angle of the TS is averaged by the value at the triple point T and the point $\textrm {O}_2$. Then the simplified continuity equation can be written as follows:

(2.19)\begin{equation} {A_{{{ST}}}} = \sigma {A_{{{MS}}}}\frac{{{P_0}}}{{{P_C}}} + {\sigma _{{{side}}}}{A_{{{Side}}}}, \end{equation}

where $A_{{ST}}$, $A_{{MS}}$ and $A_{{Side}}$ represent the projected area of the ST, the MS and the side surface on the flow direction through them, respectively. The above process establishes the correlation between the MS height and specific free stream and geometric conditions. In the calculation, an initial value for the MS height is first assumed. Using this initial estimate, the three-dimensional interaction geometry can be calculated. Next, we assess whether the simplified continuity equation (2.19) is met. If not, the MS height value is adjusted, and the complete process outlined above is repeated until a converged value for $H_m$ is achieved.

The above process enables the determination of the triple point position. It is worth noting that while the process ignores the secondary interactions between the TS and the CS or CWs, solving for the CS is still essential when studying the transition between the primary MR and sRR. This is because the transition boundary is jointly determined by the position of the CS and the triple point. Given a linear decrease in Mach number along the stagnation line behind the DS, one can use theoretical estimates to determine the local temperature, velocity components and flow deflection angles $\theta$ from the DS to the wall. This is possible as the inviscid flow behind the DS in the $x$–$z$ symmetry plane is isentropic. With the flow variables behind the DS, the parameters associated with the CWs and the CS can be solved through an analytical, iterative method (Emanuel Reference Emanuel1982, Reference Emanuel1983). The detailed solving process has been outlined in Zhang et al. (Reference Zhang, Li and Yang2021), and the approach used in this paper is consistent with it. In the current study, the solved positions of the CS were only used for identifying the primary interaction type.