5.1. Stagnation point flows

Regarding the general three-dimensional stagnation point flow, the velocity at the boundary layer edge in the $x$ direction can be written as

(5.1)\begin{equation} u_e=\left(\frac{\partial u_e}{\partial x}\right)_s x, \end{equation}

where $({\partial u_e}/{\partial x})_s$ is the velocity gradient at the stagnation point external to the boundary layer. The subscript $s$ represents the stagnation point. Substituting (5.1) into (4.1a), we obtain

(5.2)\begin{equation} \xi=\int_0^x \mu_e \rho_e{\left(\frac{\partial u_e}{\partial x}\right)_s} x\, {{\rm d}\kern0.7pt x}=\frac{1}{2} \mu_e \rho_e{\left(\frac{\partial u_e}{\partial x}\right)_s} x^2. \end{equation}

Consider a self-similar solution to the governing boundary layer equations for the stagnation point case, $f$, $e$ and $g$ are assumed to be functions of $\eta$ only; hence, ${\partial f}/{\partial \xi }={\partial f^{\prime }}/{\partial \xi }={\partial e}/{\partial \xi }={\partial e^{\prime }}/{\partial \xi }={\partial g}/{\partial \xi }=0$ in (4.10). Substituting (5.1) and (5.2) into (4.10), and neglecting the terms with higher orders of $x$, the final equations for the general three-dimensional stagnation point flow are written as follows: for $\xi$-directional momentum,

(5.3a)\begin{equation} (C f^{\prime \prime})^{\prime}+\left(f+\left(\frac{{\dfrac{\partial w_e}{\partial z}}}{\dfrac{\partial u_e}{\partial x}}\right)_s e\right) f^{\prime \prime}=f^{\prime 2}-\frac{\rho_e}{\rho}; \end{equation}

for $z$-directional momentum,

(5.3b)\begin{equation} (C e^{\prime \prime})^{\prime}+\left(f+\left(\frac{{\dfrac{\partial w_e}{\partial z}}}{\dfrac{\partial u_e}{\partial x}}\right)_s e\right) e^{\prime \prime}=\left(\frac{{\dfrac{\partial w_e}{\partial z}}}{\dfrac{\partial u_e}{\partial x}}\right)_s\left(f^{\prime 2}-\frac{\rho_e}{\rho}\right); \end{equation}

for energy,

(5.3c)\begin{equation} \left(\frac{C}{P r} g^{\prime}\right)^{\prime}+\left(f+\left(\frac{{\dfrac{\partial w_e}{\partial z}}}{\dfrac{\partial u_e}{\partial x}}\right)_s e\right) g^{\prime}=0. \end{equation}

Equations (5.3) are totally identical to the general three-dimensional stagnation point flow derived by other authors (Libby Reference Libby1967). It can also be used directly as a starting point for solving the boundary layer equations over the whole body with a difference-differential method.

Consider special cases: if $(({{\partial w_e}/{\partial z}})/({{\partial u_e}/{\partial x}}))_s=0,$ (5.3) reduce directly to represent the planar stagnation point flow, where the equations are as follows: for $\xi$-directional momentum,

(5.4a)\begin{equation} (C f^{\prime \prime})^{\prime}+f f^{\prime \prime}=f^{\prime 2}-\frac{\rho_e}{\rho}; \end{equation}

for energy,

(5.4b)\begin{equation} \left(\frac{C}{P r} g^{\prime}\right)^{\prime}+f g^{\prime}=0. \end{equation}

Since the flows in the z direction are uninteresting in this case, (5.3b) is inessential and is dropped here directly. The stagnation flow for the two-dimensional cylinder is the most special case. Equations (5.4) for the two-dimensional cylinder stagnation flow are totally the same as those deduced by other authors (Anderson Reference Anderson2006).

If $(({{\partial w_e}/{\partial z}})/({{\partial u_e}/{\partial x}}))_s=1$, (5.3) reduce to represent the axisymmetric stagnation flow, with a sphere stagnation point flow as the most special case. Then, the equations are given as follows: for $\xi$-directional momentum,

(5.5a)\begin{equation} (C f^{\prime \prime})^{\prime}+(f+e) f^{\prime \prime}=f^{\prime 2}-\frac{\rho_e}{\rho};\end{equation}

for $z$-directional momentum,

(5.5b)\begin{equation} (C e^{\prime \prime})^{\prime}+(f+e) e^{\prime \prime}=f^{\prime 2}-\frac{\rho_e}{\rho}; \end{equation}

for energy,

(5.5c)\begin{equation} \left(\frac{C}{P r} g^{\prime}\right)^{\prime}+(f+e) g^{\prime}=0. \end{equation}

Further discussions regarding (5.5) will be conducted in the next section since it is also a special case of axisymmetric flow.

5.2. Axisymmetric flows

The aforementioned cases of two-dimensional and axisymmetric stagnation point flow can be regarded as specific cases of more generalized symmetry plane flow. In the case of axisymmetric flow where the incoming flow aligns with its axis, the situation becomes slightly more complicated. In the commonly used axisymmetric coordinate system applied for axisymmetric boundary layers, as detailed by Schlichting & Gersten (Reference Schlichting and Gersten2016), for example, the velocity and its gradients in the circumference direction are zero. Consequently, the boundary layer equations over an axisymmetric body are similar to those for two-dimensional flows. However, although $w_e$ is also zero in our proposed coordinates, the extra variable ${\partial w_e}/{\partial z}$, which is the gradient of the velocity at the outer edge of the boundary layer along the z direction, is not necessarily zero and is required in advance for solving (4.10). In principle, deriving this variable entails solving (3.10b) with respect to the pressure distribution on the surface. However, for an axisymmetric flow, the situation can be simplified, and this variable can be directly deduced from the kinematic property of the axisymmetric flow, with no requirement of the pressure distribution, as illustrated in figure 3.

Figure 3. Schematic geometry of the axisymmetric boundary layers.

At the stagnation point, the velocity gradients can be calculated from Newtonian theory. Consider an arbitrary point D on the surface of the axisymmetric body, apart from the stagnation point. The velocity at the boundary layer edge of this point is $u_{e, \textrm {D}}$. Imagine a line drawn tangent to the surface at point D and it intersects with the axis of symmetry at point A, making an angle $\theta$ between them, as shown in figure 3. Here CD represents the principal radius of curvature of the surface at point D, i.e. $r$ in the present study. Due to the axisymmetric property, AC also coincides with the symmetry line of the physical body.

To derive ${\partial w_e}/{\partial z}$, the velocity near point D at the $z$ coordinate is required. Assume a nearby point $\textrm {D}^{\prime }$ along the $z$ coordinate, and the angle between lines AD and $\textrm {A} \textrm {D}^{\prime }$ is $\omega$. As $\textrm {D}^{\prime }$ infinitely approaches D, i.e. $\omega \rightarrow$ 0; and from the property of the axisymmetric flow, it can be obtained that the magnitude of the velocity at a nearby point $\textrm {D}^{\prime }$ along the $z$ coordinate is approximately equal to that at point $\textrm {D}, u_{e, \textrm {D}^{\prime }} = u_{e, \textrm {D}}$, which is also consistent with (3.6a). Accordingly, the velocity component in the $z$ direction at point $\textrm {D}^{\prime }$ can be expressed as

(5.6)\begin{equation} w_{e, {\rm D}^{\prime}}={u_{e, {\rm D}^{\prime}} \tan \omega} =u_{e, {\rm D}} \tan \omega . \end{equation}

In addition, the coordinate of the nearby point $\textrm {D}^{\prime }$ is

(5.7)\begin{equation} z=L_{\rm A D} \tan \omega, \end{equation}

where $L_{\textrm {AD}}$ represents the length between points A and D, and $L_{\textrm {AD}}={r}/{\tan \theta }$. Then, substituting (5.7) into (5.6) yields

(5.8a)\begin{equation} w_{e, {\rm D}^{\prime}} = u_{e, {\rm D}} \tan \omega=z \frac{u_{e, {\rm D}} \tan \theta}{r} \end{equation}

or

(5.8b)\begin{equation} \left({\frac{\partial w_e}{\partial z}}\right)_{\rm D}=\frac{u_{e, {\rm D}} \tan \theta}{r}. \end{equation}

With this relation, there is no substantial difficulty in solving the boundary layer equations for axisymmetric flows. Since point D is an arbitrary point, the subscript D is dropped in the following discussions.

Moreover, at the wall, $w={\partial w}/{\partial z}=u=0$, i.e. (5.8) hold both on the boundary layer edge and on the wall at any point. Since the boundary layer thickness is assumed to be much less than the radii of the surface, (5.8) should be valid throughout the boundary layer, that is,

(5.9)\begin{equation} \frac{\partial w(y)}{\partial z}=\frac{u(y) \tan \theta}{r}. \end{equation}

This equation indicates that ${\partial w(y)}/{\partial z}$ is directly dependent on $u(y)$ and the geometric factors throughout the boundary layer. And ${\partial w(y)}/{\partial z}$ has the same profile as $u(y)$, that is,

(5.10a)\begin{equation} e^{\prime}=f^{\prime} \end{equation}

and

(5.10b)\begin{equation} e=f. \end{equation}

By substituting (5.10) into (4.10), we have, for $\xi$-directional momentum,

(5.11a)\begin{align} (C f^{\prime \prime})^{\prime}+\left(1+\frac{2 \xi}{\mu_e \rho_e u_e} \frac{\tan \theta}{r}\right) f f^{\prime \prime}=\frac{2 \xi}{u_e} \frac{\partial u_e}{\partial \xi}\left(f^{\prime 2}-\frac{\rho_e}{\rho}\right)+2 \xi f^{\prime} \frac{\partial f^{\prime}}{\partial \xi}-2 \xi \frac{\partial f}{\partial \xi} f^{\prime \prime}; \end{align}

for energy,

(5.11b)\begin{align} \left(\frac{C}{P r} g^{\prime}\right)^{\prime}+\left(1+\frac{2 \xi}{\mu_e \rho_e u_e} \frac{\tan \theta}{r}\right) f g^{\prime}=2 \xi\left(f^{\prime} \frac{\partial g}{\partial \xi}-g^{\prime} \frac{\partial f}{\partial \xi}+\frac{f^{\prime} g}{h_e} \frac{\partial h_e}{\partial \xi}+\frac{\rho_e u_e}{\rho h_e} f^{\prime} \frac{\partial u_e}{\partial \xi}\right)-C \frac{u_e^2}{h_e}\left(f^{\prime \prime}\right)^2. \end{align}

These equations are the expressions of the axisymmetric boundary layer flow in the proposed coordinate system.

Consider special cases: for a flat plate case or a sharp cone, $u_e=$ const, $T_e=$ const, $p_e=$ const, $f$ and $g$ are assumed to be functions of $\eta$ only.

For a flat plate, we have $\tan \theta =0$ or $r \rightarrow \infty$, (5.11) reduce to

(5.12a)$$\begin{gather} (C f^{\prime \prime})^{\prime}+f f^{\prime \prime}=0, \end{gather}$$

(5.12b)$$\begin{gather}\left(\frac{C}{P r} g^{\prime}\right)^{\prime}+f g^{\prime}+C \frac{u_e^2}{h_e}\left(f^{\prime \prime}\right)^2=0. \end{gather}$$

The above equations for a flat plate are the same as those deduced by Anderson (Reference Anderson2006).

For a sharp cone, we have

(5.13)\begin{equation} \xi=\int_0^x \mu_e \rho_e u_e\, {{\rm d}\kern0.7pt x}=\mu_e \rho_e u_e x \end{equation}

and

(5.14)\begin{equation} \tan \theta=\frac{r}{x}. \end{equation}

Under these conditions, (5.11) for a sharp cone reduce to

(5.15a)$$\begin{gather} (C f^{\prime \prime})^{\prime}+3 f f^{\prime \prime}=0, \end{gather}$$

(5.15b)$$\begin{gather}\left(\frac{C}{P r} g^{\prime}\right)^{\prime}+3 f g^{\prime}+C \frac{u_e^2}{h_e}\left(f^{\prime \prime}\right)^2=0. \end{gather}$$

Moreover, if the transformation (4.1b) is replaced by

(5.16)\begin{equation} \eta=\sqrt{3} \frac{u_e}{\sqrt{2 \xi}} \int_0^y \rho\, {{\rm d} y} \end{equation}

then the boundary layer equations for a cone will be totally the same as that for a flat plate, i.e. (5.12). The derivation process is basically the same as that provided in § 4 and will not be repeated in this section. Hence, it is apparent that if the parameters at the boundary layer edge and at the wall are the same for a flat plate and a sharp cone, the variations in flow properties throughout the boundary layer will exhibit the same profiles in their transformed plane. The only difference is a factor of $\sqrt {3}$ between (4.1b) and (5.16), which indicates that the boundary layer thickness of a flat plate is $\sqrt {3}$ times greater than that of a cone. Subsequently, the relationship of heat flux for a cone and a flat plate can be easily obtained as

(5.17)\begin{equation} q_{cone}=\sqrt{3} q_{{flat\ plate }}. \end{equation}

Then, let us return to further simplify (5.5) for the axisymmetric stagnation flow, where (5.10) also hold. Thus, (5.5) can be further reduced to the following: for $\xi$-directional momentum,

(5.18a)\begin{equation} (C f^{\prime \prime})^{\prime}+2 f f^{\prime \prime}=f^{\prime 2}-\frac{\rho_e}{\rho}; \end{equation}

for energy,

(5.18b)\begin{equation} \left(\frac{C}{P r} g^{\prime}\right)^{\prime}+2 f g^{\prime}=0. \end{equation}

In summary, (5.4) for a cylinder stagnation point, (5.12) for a flat plate, (5.15) for a sharp cone and (5.18) for a sphere stagnation point are ordinary differential equations (ODEs) and are self-similar. Besides, they are all deduced in the same coordinate.

6.1. Approximation of the crossflow velocity gradient at the boundary layer edge for hypersonic flows

Newtonian theory for prediction of the pressure distribution at a point depends only on the knowledge of its local surface inclination relative to the free stream. However, the second-order pressure gradient in the $z$ direction in (3.10b) will be affected by its surrounding flow field or the shape geometry. An equivalent cone is formed to consider the effect of surface geometry on the crossflow velocity. Point D is considered to be on the symmetry plane. Due to the symmetry property, the surface near the symmetry plane is approximated as an equivalent cone here, as shown in figure 4. Moreover, the semiangle of the cone is defined as

(6.1)\begin{equation} \theta=\arctan \frac{{\rm d} r}{{\rm d}\kern0.7pt x} \quad\text{ or }\quad \tan \theta=\frac{{\rm d} r}{{\rm d}\kern0.7pt x}, \end{equation}

where AC is the axis and the length of CD represents the principal radius of curvature of the surface at point D. Line AD is tangent to the surface at point D. Unlike the tangent-cone method commonly used to estimate the local surface pressure, where $\theta$ represents the local deflection angle of the surface, the parameter $\theta$ in this study is defined by (6.1). This definition allows for the consideration of the influence of surface geometry variation on the second-order pressure gradient or the crossflow velocity gradient. For axisymmetric flows, a sharp cone, for example, AC coincides with the axis line of the actual body. For most cases, AC will continuously vary along $x$ as the surface geometry changes, and it is specifically determined by (6.1).

Figure 4. Illustration of the tangent-cone method for the approximation of the crossflow velocity gradient.

We assume a nearby point $\textrm {D}^{\prime \prime }$ near the symmetry plane, which also comes from point D rotated by a small angle $\varphi$ around the axis of the cone, i.e. $\varphi$ represents the angle between line BD and $\textrm {BD}^{\prime \prime }$. The direction of the normal vector to the surface at point $\textrm {D}^{\prime \prime }$ is consistent with the direction of the vector $\boldsymbol {\textrm {CD}}^{\prime \prime }$. Thus, in Cartesian coordinates $(X, Y, Z)$, the unit normal vectors to the surface at points D and $\textrm {D}^{\prime \prime }$ are expressed as

(6.2a,b) \begin{equation} \boldsymbol{n}_{\rm D}=\left(\begin{array}{@{}c@{}} -\sin \theta \\ \cos \theta \\ 0 \end{array}\right), \quad \boldsymbol{n}_{{\rm D}^{\prime \prime}}=\left(\begin{array}{@{}c@{}} -\sin \theta \\ \cos \theta \cos \varphi \\ \cos \theta \sin \varphi \end{array}\right). \end{equation}

To calculate the vector dot product of the free-stream velocity and the normal vector to the surface, it is necessary to transform both to the same coordinate system. Here, we transform both to the coordinate system $(x, y, z)$. Transformation between the coordinates $(x, y, z)$ and the coordinates $(X, Y, Z)$ is expressed as

(6.3) \begin{equation} \left(\begin{array}{@{}c@{}} x \\ y \\ z \end{array}\right)=\left(\begin{array}{@{}ccc@{}} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{@{}c@{}} X \\ Y \\ Z \end{array}\right). \end{equation}

Thus, the unit normal vector to the surface at point $\textrm {D}^{\prime \prime }$ in the coordinates $(x, y, z)$ is expressed as

(6.4) \begin{align} \boldsymbol{n}_{{\rm D}^{\prime \prime}}=\left(\begin{array}{@{}ccc@{}} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{@{}c@{}} -\sin \theta \\ \cos \theta \cos \varphi \\ \cos \theta \sin \varphi \end{array}\right)=\left(\begin{array}{@{}c@{}} -\cos \theta \sin \theta+\sin \theta \cos \theta \cos \varphi \\ \sin ^2 \theta+\cos ^2 \theta \cos \varphi \\ \cos \theta \sin \varphi \end{array}\right). \end{align}

Consider the free-stream velocity as a vector and $\beta$ is the angle between the tangent to the surface and the free stream. Then, the unit velocity vector in the coordinates $(x, y, z)$ is expressed as

(6.5) \begin{equation} \boldsymbol{n}_{V_{\infty}}=\left(\begin{array}{@{}c@{}} \cos \beta \\ -\sin \beta \\ 0 \end{array}\right). \end{equation}

From Newtonian theory, the pressure coefficient $C_p$ at point $\textrm {D}^{\prime \prime }$ is given by

(6.6)\begin{equation} C_p=\frac{p-p_{\infty}}{\dfrac{1}{2} \rho_{\infty} V_{\infty}^2}=2(\boldsymbol{n}_{{\rm D}^{\prime \prime}} \boldsymbol{\cdot} \boldsymbol{n}_{V_{\infty}})^2, \end{equation}

where the subscript $\infty$ represents the free-stream flow parameter. Substitution of (6.4) and (6.5) into (6.6) yields

(6.7)\begin{align} C_p=2(-\cos \beta \cos \theta \sin \theta+\cos \beta \sin \theta \cos \theta \cos \varphi-\sin \beta \sin ^2 \theta-\sin \beta \cos ^2 \theta \cos \varphi)^2. \end{align}

Differentiating the above equation with respect to $\varphi$ yields

(6.8)\begin{equation} \left.\begin{array}{c} \dfrac{\partial C_p}{\partial \varphi}=4(-\cos \beta \cos \theta \sin \theta+\cos \beta \sin \theta \cos \theta \cos \varphi-\sin \beta \sin ^2 \theta-\sin \beta \cos ^2 \theta \cos \varphi) \\ (-\cos \beta \sin \theta \cos \theta \sin \varphi+\sin \beta \cos ^2 \theta \sin \varphi). \end{array}\right\} \end{equation}

As $\textrm {D}^{\prime \prime }$ infinitely approaches $\textrm {D}, \varphi \rightarrow 0$ and $\omega \rightarrow 0$. Thus, $\cos \varphi \approx 1, \cos \omega \approx 1$ and $\sin \varphi \approx \varphi$. From figure 4 we can obtain the following expression:

(6.9)\begin{equation} \varphi \approx \frac{L_{{\rm D D}^{\prime \prime}}}{L_{{\rm B D}}} = \frac{L_{{\rm A D}} \omega}{r \cos \theta}. \end{equation}

Substitution of (5.7) into (6.9) yields

(6.10)\begin{equation} \varphi \approx \frac{z}{r \cos \theta}. \end{equation}

Thus, (6.8) can be simplified as

(6.11a)\begin{equation} \frac{\partial C_p}{\partial z} r \cos \theta=4(-\sin \beta)(-\cos \beta \sin \theta \cos \theta+\sin \beta \cos ^2 \theta) \frac{z}{r \cos \theta} \end{equation}

or

(6.11b)\begin{equation} \frac{\partial C_p}{\partial z}=4 \sin ^2 \beta\left(\frac{\tan \theta}{\tan \beta}-1\right) \frac{z}{r^2}. \end{equation}

Differentiating the above equation with respect to $z$ yields

(6.12a)\begin{equation} \frac{\partial^2 C_p}{\partial z^2}=4 \sin ^2 \beta\left(\frac{\tan \theta}{\tan \beta}-1\right) \frac{1}{r^2} \end{equation}

or

(6.12b)\begin{equation} \frac{\partial^2 p}{\partial z^2}=2 \rho_{\infty} V_{\infty}^2 \sin ^2 \beta\left(\frac{\tan \theta}{\tan \beta}-1\right) \frac{1}{r^2}. \end{equation}

In the symmetry plane, i.e. $\varphi =0$, the pressure coefficient is given by

(6.13)\begin{equation} C_p|_{\varphi=0}=2 \sin ^2 \beta=\frac{p_{\varphi=0}-p_{\infty}}{\dfrac{1}{2} \rho_{\infty} V_{\infty}^2}. \end{equation}

Substitution of this equation into (6.12) yields

(6.14)\begin{equation} \left.\frac{\partial^2 p}{\partial z^2}\right|_{\varphi=0}=\frac{2(p_{\varphi=0} -p_{\infty})}{r^2}\left(\frac{\dfrac{{\rm d} r}{{\rm d}\kern0.7pt x}}{\tan \beta}-1\right). \end{equation}

Given our focus on the symmetry plane flow, the subscript $\varphi =0$ is dropped in the following discussions. Equation (6.14) indicates that the contribution to ${\partial ^2 p}/{\partial z^2}$ can be divided into two parts. The first part is the contribution of the kinetic factor, which is characterized by a constant, i.e. 1 in (6.14). This part can also be seen as the contribution of an equivalent cylinder with a radius of $r$ to ${\partial ^2 p}/{\partial z^2}$. The second part is the contribution of the geometric factor, which is characterized by ${\textrm {d} r}/{\tan \beta }$, and it can also be seen as the influence of the local geometric factor or the deviation of the geometry from an equivalent cylinder with a radius of $r$ on ${\partial ^2 p}/{\partial z^2}$. Furthermore, ${\textrm {d} r}/{\textrm {d}{\kern0.8pt}x}$ represents the bluntness effects of the blunt body on ${\partial ^2 p}/{\partial z^2}$, where $\tan \beta$ represents the equivalent axisymmetric effects or axisymmetric effects in simple terms.

Consider special cases: for a hemisphere or a cylinder, ${\textrm {d} r}/{\textrm {d}{\kern0.8pt}x}=0$ or $({{\textrm {d} r}/{\textrm {d}{\kern0.8pt}x}})/{\tan \beta }=0,$ (6.14) degenerates to a formula similar to that at the stagnation point, i.e. ${\partial ^2 p}/{\partial z^2}=-{2(p_e-p_{\infty })}/{r^2}$; for a sharp cone with zero angle of attack, $\theta =\beta$ or $({{\textrm {d} r}/{\textrm {d}{\kern0.8pt}x}})/{\tan \beta }=1$, namely, the contributions of geometric and kinetic factors offset each other out. As a result, ${\partial ^2 p}/{\partial z^2}=0$.

Associating (6.14) with (3.10b) yields

(6.15)\begin{equation} {-}2(p_e-p_{\infty})\left(\frac{\dfrac{{\rm d} r}{{\rm d}\kern0.7pt x}}{\tan \beta}-1\right) \frac{1}{r^2}=\rho_e\left[u_e \frac{\partial}{\partial x}\left({\frac{\partial w_e}{\partial z}}\right)+\left({\frac{\partial w_e}{\partial z}}\right)^2+\frac{u_e^2}{R r}\right]. \end{equation}

Similarly, the kinetic factor and the geometric factor in (6.15) will affect ${{\partial w_e}/{\partial z}}$, just as they affect ${\partial ^2 p}/{\partial z^2}$.

In this study, Newtonian theory is used to obtain the pressure distribution, which in turn facilitates the calculation of the crossflow velocity gradient ${\partial w_e}/{\partial z}$. However, if more accurate pressure distributions are obtained through alternative methods, they can be applied directly to (3.10b), resulting in a more precise crossflow velocity gradient. Notably, the range of applicability of (6.14) and (6.15) is consistent with Newtonian theory, which is applicable only under hypersonic conditions. Anderson (Reference Anderson2006) extensively discussed the historical background, various improved forms and the scope of application of Newtonian theory in his book. In the Newtonian model of fluid flow, the particles in the free stream impact only on the frontal area of the body; they cannot curl around the body and impact on the back surface. Hence, the application of Newtonian theory appears most suitable for bodies turning convex surface parts toward hypersonic oncoming gas. Moreover, it is worth noting that, similar to various methods for solving wall pressure, the precision and range of applicability of these methods are not entirely definitive. Nevertheless, Newtonian theory is straightforward and easy to apply in hypersonic flows.

6.2. Boundary layer edge parameters of blunt bodies under hypersonic conditions

Until now, this paper has articulated the new contributions made in the realm of existing boundary layer theories. We have selected several common blunt bodies encountered in hypersonic flight for developing our solving program and validating our theory. Another reason for choosing blunt bodies is the existence of efficient approximate methods for solving the boundary layer edge parameters for such shapes. Notably, for non-blunt bodies, considerations must be given to the applicability of Newtonian theory and the selection of appropriate methods for solving the boundary layer edge parameters.

Parameters at the boundary layer edge, i.e. the values of ${u_e}$, ${\rho _e}$, ${h_e}$, ${\mu _e}$ and ${\partial w_e}/{\partial z}$, must be determined to solve (4.10). While ${\partial w_e}/{\partial z}$ is deduced from (6.15), other boundary layer edge parameters are obtained using the simplest and, therefore, the most commonly employed method (Griffith et al. Reference Griffith, Adams and Majors1981). From the exact normal shock wave theory, ${p_s}$ behind a normal shock is then determined as

(6.16)\begin{equation} {\frac{p_{s}}{p_{\infty}}=\left[\frac{(\gamma+1)^{2}{M_{\infty}}^{2}}{4 \gamma{M_{\infty}}^{2}-2(\gamma-1)}\right]^{\gamma/(\gamma-1)} \left[\frac{1-\gamma+2\gamma{M_{\infty}}^{2}}{\gamma+1}\right],} \end{equation}

where $\gamma$ is the ratio of specific heats and $p_s$ is the total pressure behind a normal shock wave. Subsequently, $p_e$ over the body of interest is obtained from the modified Newtonian method firstly as

(6.17) \begin{equation} {C_p=\frac{p_e-p_\infty}{\dfrac12\rho_\infty v_\infty^2}=\frac{p_s-p_\infty}{\dfrac12\rho_\infty v_\infty^2}\cos^2\theta ,} \end{equation}

where $C_p$ is the pressure coefficient. The subscript $\infty$ denotes the free-stream flow parameter. Subsequently, the local flow conditions along the boundary layer edge, i.e. $u_e$, $\rho _e$, $T_e$, are deduced by isentropically expanding from these stagnation conditions to the known local pressure $p_e$, thus treating the boundary layer edge as an isentropic surface. The isentropic relations are given as

(6.18a,b)\begin{align} {\left(\frac{\rho_{e}}{\rho_{\infty}}\right)^{\gamma}= \frac{p_{e}}{p_{\infty}}=\left(\frac{1+\dfrac{\gamma-1}{2} M_{\infty}^{2}}{1+\dfrac{\gamma-1}{2}M_{e}^{2}} \right)^{{\gamma}/({\gamma-1})}=\left(\frac{T_{e}}{T_{\infty}} \right)^{{\gamma}/({\gamma-1})},\quad u_{e}=M_{e}\sqrt{\gamma\frac{p_{e}}{\rho_{e}}}.} \end{align}

The viscosity $\mu$ and $\mu _e$ are calculated according to Sutherland's law in this study. And for air at standard conditions, ${Pr}$ equals 0.71 in this study.

To complete the system, the following two thermodynamic state equations are required:

(6.19a)\begin{equation} T=T(p,\rho),\quad h=h(p,\rho). \end{equation}

Here $T$ is the temperature. A calorically perfect gas is applied in the present study, then

(6.19b)\begin{equation} p=\rho RT,\quad h=c_pT, \end{equation}

where $R$ is the specific gas constant of the gas and $c_p$ is the specific heat capacity at constant pressure; $h_e$ is calculated from (6.19b). At this point, all the necessary outer boundary layer parameters required for solving (4.10) have been obtained.

Notably, the above method does not consider the effect of the entropy layer on the boundary layer edge parameters. This method is satisfactory for the forward partition of a blunt body. Nevertheless, alternative existing methods for taking the outer edge properties can replace the methods outlined in this section, such as extracting the boundary layer edge quantities from the computational fluid dynamic solutions (Hamilton & Greene Reference Hamilton and Greene1994; Liechty et al. Reference Liechty, Berry, Hollis and Horvath2003). Solving the Navier–Stokes equations is more time consuming. It can yield more accurate parameters such as $u_e$, $\rho _e$, $h_e$ and $\mu _e$. However, even modern CFD poses significant uncertainty in deducing ${\partial w_e}/{\partial z}$, as already described earlier. Thus, we mainly opt for ((6.15)–(6.19)) to provide a rapid estimation of the boundary layer conditions. Anderson (Reference Anderson2006) also discussed the entropy effects on aerodynamic heating under hypersonic flows in Chapter 6 and provided some methods to mitigate their influence.

6.3. Numerical solution of the boundary layer equations

Compared with the planar boundary layer equations (Anderson Reference Anderson2006), exact solutions of (4.10) do not add inherent difficulty if the parameters at the boundary layer edge are known. Commonly used methods for solving these general, non-similar planar boundary layer equations, such as local similarity, difference-differential approach and finite-difference solutions (Anderson Reference Anderson2006), are all applicable for solving (4.10). Furthermore, the existing literature in this field is quite expansive. Finite-difference solutions discretize the equations into finite grids and solve them numerically, requiring appropriate grid resolution and numerical schemes similar to those used for solving the Navier–Stokes equations. Instead, we choose the difference-differential approach, which can simplify (4.10) to ODEs for solution in some cases and is also an exact solution for general non-similar boundary layers. The Crank–Nicolson type differential method is similar to that used by other authors (Blottner & Ellis Reference Blottner and Ellis1973; Wang Reference Wang1975; Popinski & Baker Reference Popinski and Baker1976) is adopted here.

Schematic for finite-difference solution of the boundary layer is shown in figure 5. Assume that we wish to calculate the boundary layer profiles at the station denoted by $i$. The derivatives in the direction along the surface are replaced with finite-difference relations. The first-order derivatives with respect to $\xi$ in (4.10) are replaced by the two-point backward finite-difference quotients as

(6.20)\begin{equation} {\frac{\partial f}{\partial\xi}=\frac{f_i-f_{i-1}}{\Delta \xi_i} \text{ at a given } j.} \end{equation}

Figure 5. Schematic for finite-difference solution of the boundary layer.

Identical expressions are used for other parameters as those used for the derivative with respect to $\xi$, such as ${\partial f'}/{\partial \xi }$, ${\partial e'}/{\partial \xi }$ and ${\partial g}/{\partial \xi }$. At the stagnation point, i.e. $i = 1$ in figure 5, the velocity gradients at the stagnation point in the $x$ and $z$ directions are calculated from Newtonian theory, that are

(6.21a,b)\begin{equation} {\left(\frac{{\rm d}u_e}{{\rm d}\kern0.7pt x}=\frac{1}{R}\sqrt{\frac{2(p_e-p_\infty)}{p_e}}\right)_s, \quad \left(\frac{{\rm d}w_e}{{\rm d}z}= \frac{1}{r}\sqrt{\frac{2(p_e-p_\infty)}{p_e}}\right)_s,} \end{equation}

where the subscript $s$ represents the stagnation point. Afterward, we proceed to solve the ordinary differential boundary layer equations (5.3) to obtain the boundary layer profiles at $i = 1$. Choosing the stagnation point as the starting point, the difference-differential method is then implemented to move to the next downstream location. In other words, $f_{i-1}$ in (6.20) are known numbers. The only unknown number in (6.20) is $f_i$. Recall that ${\textrm {d}u_e}/{\textrm {d}\xi }$ and ${\partial w_e}/{\partial z}$ are also known numbers, obtained from the known velocity variation at the boundary layer edge. Thus, when the difference expressions such as (6.20) are substituted into (4.10), the only derivatives that appear are $\eta$. Hence, (4.10) are ODEs that can be integrated across the boundary layer at station $i$. A more detailed description of the solution method in this section can be found in the book by Anderson (Reference Anderson2006).

After substituting the derivatives with respect to $\xi$ into (4.10) they are reformulated into a system of linear algebraic equations in terms of the single independent variable $\eta$. The successive over-relaxation iterative method is then adopted to solve these equations. To accelerate convergence, a relaxation factor is applied in the iterative progress. Once (4.10) has been solved for all grid points, the transformed streamfunction values at discretized points within the boundary layer are known. These values can be used to calculate detailed parameters such as velocity, enthalpy and temperature at each grid. Consequently, they can be employed to compute other boundary layer characteristics of interest, such as skin friction and heat transfer.

7.1. Comparison with experimental results

7.1.1. Stagnation point heat flux

Considerable attention is often given to the stagnation point of blunt bodies, where the local heat flux is usually the greatest. Additionally, the subsequent calculations for the blunt bodies in this paper also choose the stagnation point as the starting point, requiring the boundary layer information at this initial point. Thus, we firstly compare the stagnation point heat flux of spheres obtained by solving (4.10) with both the experimental data, and the results estimated using the analytical approach proposed by Fay & Riddell (Reference Fay and Riddell1958). Because of its simplicity, the Fay and Riddell approach has been used extensively for re-entry flight design as well as ground test facility applications, where the expression is shown in (7.1) as below:

(7.1)\begin{equation} {q_s=0.763Pr^{{-}0.6}(\rho_e\mu_e)^{0.5}\sqrt{\frac{{\rm d}u_e}{{\rm d}\kern0.7pt x}}(h_0-h_w).} \end{equation}

Equation (7.1) is its simplified form and is valid for equilibrium and a non-reacting gas.

The heat flux values corresponding to the ‘present method’ column in table 1 are obtained by solving (5.18). Equation (5.18) are ODEs and are self-similar. The boundary layer edge parameters are obtained based on ((6.15)–(6.19)). It can be observed from table 1 that the stagnation points heat flux obtained by the present method aligns very well with both experimental values and the results from the commonly used Fay and Riddell approach.

Table 1. Heat flux at the stagnation point of spheres in air between the present theoretical results and the experimental data and the Fay–Riddell results. Experimental data of case 1 are from Irimpan & Menezes (Reference Irimpan and Menezes2023), cases 2 and 3 are from Irimpan et al. (Reference Irimpan, Mannil, Arya and Menezes2015), case 4 are from Kaattari (Reference Kaattari1978).

7.1.2. Heat flux distribution on the sphere

The second validation case involves a sphere, and the results from the present method by solving (4.10) are also compared with the experimental data. Two sets of experimental data are considered. The experiments were conducted in the Langley 31-inch Mach tunnel (Hamilton, Millman & Greendyke Reference Hamilton, Millman and Greendyke1992). This study primarily employs ((6.15)–(6.19)) to obtain the boundary layer edge parameters. As a comparison, we also obtain the boundary layer edge parameters of these cases by solving the Navier–Stokes equations, which are solved by an in-house code named the advanced research tool in science and technology CFD (ARTIST-CFD). The ARTIST-CFD is a parallel, finite volume code that solves two-dimensional, axisymmetric and three-dimensional Navier–Stokes equations based on multi-block structured meshes. Roe upwind scheme with second-order MUSCL reconstruction and minmod limiter is applied for inviscid fluxes, and viscous fluxes are discretized by second-order central difference scheme. For details about the ARTIST-CFD, we refer the reader to Wang et al. (Reference Wang, Guo, Hong and Li2023); Guo, Wang & Li (Reference Guo, Wang and Li2024). The boundary layer edge in the computations is defined where the local stagnation enthalpy is equal to 99.5 % of the free-stream stagnation enthalpy. However, due to the difficulty in solving ${\partial w_e}/{\partial z}$, it is still obtained through (6.15).

Figure 6(a) first presents a comparison of the boundary layer edge parameters, i.e. $T_e$ and $u_e$, derived from ((6.16)–(6.19)) with those obtained from CFD. In the front portion of the model, the differences between the two methods are minor, while the main discrepancies appear in the rear part of the model. As mentioned in § 6.2, ((6.16)–(6.19)) provide a quick and straightforward method to solve for the boundary layer edge parameters but do not account for the entropy effect. Computational fluid dynamics methods, although inevitably introducing additional complexity, can provide more reliable boundary layer edge parameters, except for ${\partial w_e}/{\partial z}$. After obtaining the boundary layer edge parameters using the two methods described above, we used them as inputs to solve the boundary layer equation (4.10), resulting in the respective boundary layer parameter profiles. These were then compared with the boundary layer parameter distributions obtained from CFD, as shown in figures 6(b) and 6(c), which respectively present the temperature and velocity profiles. When the boundary layer edge parameters are consistent, the theoretical solutions from this study match quite well with the CFD-derived boundary layer parameter distributions at both $\theta =0^{\circ }$ and $\theta =41.7^{\circ }$ positions. Here $\theta$ is the angle between the radius vector and the flight direction, as illustrated in figure 6(b). Since the velocity at the edge of the boundary layer at the stagnation line is zero, as also illustrated in figure 6(a), only the velocity profiles at $\theta =41.7^{\circ }$ are provided.

Figure 6. (a) Temperature and velocity profiles at the boundary layer edge. (b) Temperature profiles in the boundary layer. (c) Velocity profiles in the boundary layer. Two methods are used to obtain the boundary layer edge parameters; one involves deriving them through ((6.16)–(6.19)) (dash dot), while the other involves solving the Navier–Stokes equations (solid). The results are obtained under the conditions of $R = r = 50.8$ mm, $T_w = 300$ K, $\gamma = 1.4$, $M_{\infty } = 9.74$, $p_{\infty } = 65.2$ Pa, $T_{\infty } = 53.17$ K and zero angle of attack.

Furthermore, the heat flux distribution on the sphere between the present theoretical results and the experimental data are shown in figure 7, where two boundary layer edge parameters are utilized. One involves deriving them through ((6.16)–(6.19)) (dash dot), while the other involves using CFD (solid). As shown in figure 7, in the forward region of the model, particularly where $\theta$ is smaller than $70^{\circ }$, the results by the present method exhibit a close agreement with experimental data (i.e. within $\pm 10\,\%$) for the two boundary layer parameters. However, as $\theta$ further increases, disparities between theoretical and experimental results become more pronounced while using ((6.16)–(6.19)) to deduce the boundary layer edge parameters. This discrepancy primarily arises because the simplified method employed in ((6.16)–(6.19)) neglects the influence of the entropy layer. This issue becomes prominent at the rear of blunt bodies, where the boundary layer thickness increases and interacts with the entropy layer, resulting in vorticity interaction. Consequently, the entropy layer poses analytical challenges when calculating the above boundary layer parameters on the surface. This is also evident in figure 6(a), where the discrepancy between the boundary layer edge parameters derived from ((6.16)–(6.19)) and those obtained via CFD increases progressively towards the rear part of the model. Certainly, while choosing alternative methods, such as extracting the boundary layer edge quantities from CFD, these discrepancies in the rear part of the sphere are subject to improvement (solid line in figure 7).

Figure 7. Heat flux distribution on the sphere in air between the present theoretical results and the experimental data under the conditions of $R = r = 50.8$ mm, $T_w = 300$ K, $\gamma = 1.4$ and zero angle of attack. Two methods are used to obtain the boundary layer edge parameters, one involves deriving them through ((6.16)–(6.19)) (dash dot), while the other involves solving the Navier–Stokes equations (solid). Other conditions are as follows: Case 1: $M_{\infty } = 9.74$, $p_{\infty } = 65.2$ Pa, $T_{\infty } = 53.17$ K; Case 2: $M_{\infty } = 9.55$, $p_{\infty } = 29.8$ Pa, $T_{\infty } = 54$ K.

7.1.3. Heat flux distribution on the spherical blunt cone

The third comparison case involves a $15^{\circ }$ semiapex spherical blunt cone. The experimental tests were conducted in air at the Ames 3.5-foot hypersonic wind tunnel (Cleary Reference Cleary1969). The tests were made at a Mach number of 10.6 and a total temperature of 1111 K. The authors conducted tests at different Reynolds numbers, angles of attack and nose radius. We selected a specific set of conditions with a nose radius of $R = 9.5$ mm, a total model length of $L = 541.53$ mm and an incoming unit Reynolds number of $3.9 \times 10^6\textrm{ m}^{-1}$ for comparison. The heat flux results at three different angles of attack were compared. The boundary layer edge parameters were obtained both through ((6.16)–(6.19)) and CFD, similar to the previous sphere case. Notably, ${\partial w_e}/{\partial z}$ was still obtained through (6.15).

The comparative results are depicted in figure 8. Notably, the $X$ in figure 8 is based on values in the Cartesian coordinate system, with the coordinate origin at the stagnation point of the model. For the heat flux results (solid) obtained by solving (4.10) with boundary layer edge parameters derived from CFD, there is relatively good agreement with experimental data across all three conditions as shown in figure 8. When using boundary layer edge parameters quickly obtained from ((6.16)–(6.19)), excellent agreement is observed between the theoretical (dash dot) and experimental heat flux values for the model's forward region at $0^{\circ }$ angle of attack, as well as for the entire model at $15^{\circ }$ and $20^{\circ }$ angles of attack. However, at the rear section of the model with a $0^{\circ }$ angle of attack, disparities between theoretical (dash dot) and experimental results become more pronounced. This discrepancy is attributed to the same reason for the increasing disparity in heat flux observed at the rear section of the model in figure 7. At the rear section of the model with a $0^{\circ }$ angle of attack, the boundary layer is thicker, making it more susceptible to the entropy effects, thereby affecting the accuracy of solving ((6.16)–(6.19)) to obtain boundary layer edge parameters. Similar to the cases shown in figure 7, CFD simulations provide more accurate boundary layer edge parameters, thereby enhancing the accuracy of heat flux calculation at the rear portion of the model under $0^{\circ }$ angle of attack.

Figure 8. Heat flux distribution along the windward meridian of the $15^{\circ }$ semiapex spherical blunt cone in air, comparing the present theoretical results with the experimental data under three different angles of attack. The test conditions are as follows: $R = r = 9.5$ mm, $T_w = 294.4$ K, $\gamma = 1.4$. $M_{\infty } = 10.6$, $p_{\infty } = 132.1$ Pa, $T_{\infty } = 47.33$ K, $L = 541.53$ mm. Two methods are used to obtain the boundary layer edge parameters, one involves deriving them through ((6.16)–(6.19)) (dash dot), while the other involves solving the Navier–Stokes equations (solid).

Based on the above analysis, it can be concluded that solution accuracy of the boundary layer equations (4.10) primarily relies on the precision of the boundary layer edge parameters. With reliable boundary layer edge parameters, solving (4.10) can yield reliable results. Besides, the impact of entropy on the boundary layer edge parameters is itself an important topic, with dedicated research being conducted in this area. This paper does not delve into the discussion of how entropy affects the accuracy of solving ((6.16)–(6.19)) to obtain boundary layer edge parameters. Additionally, for most regions of the blunt body considered in this study, ((6.16)–(6.19)) can still produce satisfactory results. Thus, ((6.16)–(6.19)) are still employed in the following examples.

7.2. Leading edge of a saucer-shaped body

One example to be examined here has a very simple shape, namely, a semi-circular disk with a semi-circular edge. Similar shapes can be found in many hypersonic vehicles, such as X51 and some other wave riders. In these shapes, the most severe aerodynamic heating occurs at the leading edge, and existing methods can only predict it approximately. Thus, this paper aims at providing more precise results using the proposed method. The schematic geometry of the semi-circular disk is shown in figure 9, where $R$ and $r$ remain constant along the whole symmetry plane. In the case where $R / r \rightarrow \infty$, the body can be treated as a cylinder with a radius of $r$ and if $R / r=1$, the body is a sphere.

Figure 9. Schematic geometry of a semi-circular disk body.

Figure 10 shows the theoretical heat flux at the stagnation point and at the symmetry plane using the non-dimensional form of $q / q_{{s}, {sphere}}$, where $q_{s}$ represents the heat transfer rate at the stagnation point and the subscript ‘sphere’ denotes the case under $R / r=1$. As shown in figure 10(a), the heat flux rate at the stagnation point $(\theta =0)$ decreases rapidly as $R / r$ increases, and the final result converges to that of a cylinder where $q_{{s}, {cylinder}} / q_{{s}, {sphere}}=0.7274$. This value can be interpreted as the difference between the leading coefficient in the approximate expression for the stagnation point heat transfer rate, i.e. 0.57 for a cylinder and 0.763 for a sphere, as shown by Anderson (Reference Anderson2006).

Figure 10. Theoretical heat flux under the conditions of $M_{\infty }=7$, $p_{\infty }=1197$ Pa, $T_{\infty }=226.5$ K, $T_w=293$ K, $\gamma =1.4$, $r=5$ mm and zero angle of attack. (a) At the stagnation point. (b) At the symmetry plane. The boundary layer edge parameters are obtained based on ((6.16)–(6.19)).

Figure 10(b) shows $q / q_{{s}, {sphere}}$ as a function of $\theta$ at the symmetry plane, including the results from Lees’ approximation (Lees Reference Lees1956) under $R / r=1$. Lees’ approximation is a method used to estimate aerodynamic heating in hypersonic flows. It involves an analytical approximation developed by Lees (Reference Lees1956), which provides a simplified approach to calculate heat flux at the surface of a body moving at hypersonic speeds. Despite its simplicity, Lees’ approximation has been widely used in the field of hypersonic aerodynamics due to its ease of implementation and reasonable accuracy for certain flow conditions. At $\theta$ greater than $60^{\circ }$, the change in $q / q_{{s}, {sphere}}$ with $R / r$ is relatively small. At $\theta$ smaller than $60^{\circ }, q / q_{{s}, {sphere}}$ decreases with an increasing $R / r$ and converges to the heat flux distribution in the symmetry plane of a cylinder. At $R / r=10$, the maximum deviation of $q / q_{{s},{sphere}}$ from that for the cylinder heat flux distribution is within $4\,\%$. Besides, the results presented in this paper also align well with Lees’ approximation under the condition $R / r=1$, which is acceptable for aerodynamic heating calculations.

7.3. Leading edge of an ellipsoid-cone-shaped body

Blunt shapes are of particular interest for various applications, including aerodynamic decelerators for aerobraking systems or as re-entry vehicles. Among these configurations, the spheroid has garnered attention as an orbital, ballistic-entry vehicle. Thus, a spheroid, defined as an ellipsoid of revolution about the major axis of an ellipse, is examined here. The equation of a spheroid with $X$ as the symmetry axis is given as

(7.2)\begin{equation} \frac{X^2}{a^2}+\frac{Y^2+Z^2}{c^2}=1 . \end{equation}

The semi-axis $c$ is the equatorial radius of the spheroid and $a$ is the distance from centre to pole along the symmetry axis. The model is an oblate spheroid if $c > a$ and a prolate spheroid if $c < a$. The case of $a = c$ corresponds to a sphere. The schematic geometry of the spheroid is shown in figure 11. A full mathematical description of the analytical expression for the principal curvature is given by Weatherburn (Reference Weatherburn1927).

Figure 11. Schematic geometry of a spheroid body.

In figure 12 the crossflow velocity gradient and heat flux at the symmetry plane are plotted as functions of $\theta$ for $c / a=0.6$, 0.8, 1, 1.3, 1.5 and 2. Both are displayed using the non-dimensional form and normalized by their value at the stagnation point. It is important to note that at the stagnation point for the model illustrated in figure 11, the flow diverges, indicated by $({\partial w_e}/{\partial z})_s$ being positive in the current coordinate system. Therefore, values of ${({\partial w_e}/{\partial z})}/{({\partial w_e}/{\partial z})_s}$ larger than zero indicate that the flows are diverging from the symmetry plane, and there are mass flows diverging from the symmetry plane for all cases, as evidenced in figure 12(a). Comparatively, the prolate spheroid exhibits a smaller crossflow velocity gradient than the sphere, whereas the oblate spheroid displays a larger crossflow velocity gradient. Generally, a larger crossflow velocity gradient corresponds to greater flow divergence from the symmetry plane, thinner boundary layer thickness or higher temperature gradients near the wall. This trend is also reflected in figure 12(b), where the heat flux values along the symmetry plane gradually increase from $c / a=0.6$ to $c / a=2$.

Figure 12. Crossflow velocity gradient (a) and heat flux (b) distributions at the symmetry plane under the conditions of $M_{\infty }=7, p_{\infty }=1197\ \mathrm {Pa}, T_{\infty }=226.5\ \mathrm {K}, T_w=293\ \mathrm {K}, \gamma =1.4, a=50\ \mathrm { mm}$ and zero angle of attack. The boundary layer edge parameters are obtained based on ((6.16)–(6.19)).

Furthermore, the heat flux distribution on the oblate spheroid exhibits greater complexity, where the heat flux on some aft section of the body is even larger than that at the stagnation point, as observed for $c / a=2$ in figure 12(b). On a general three-dimensional blunt body, the longitudinal and circumferential compression both contribute to heat fluxes. In the case of $c / a=2$, the crossflow velocity gradient is larger than that at the stagnation point. Consequently, the accompanying increase in circumferential compression on some portion of the boundary layers may dominate the compression effect, resulting in higher heat flux values. These findings also reflect that the impact of three dimensionality on heat fluxes is much more complicated than the heat fluxes analysed in two-dimensional cases where only longitudinal compression exists.

Another cone-shaped body examined in this study is an elliptic cone, which shares similarities with vehicles like the HIFiRE-5 (Juliano, Jewell & Kimmel Reference Juliano, Jewell and Kimmel2019) and HyTRV (Xiang et al. Reference Xiang, CHEN, Yuan, Wan, Zhuang, Zhang and Tu2022). The schematic geometry of the elliptic cone considered here is shown in figure 13, where $r = 20$ mm, $L = 100$ mm and the half-angle in the minor axis $(X\unicode{x2013} Z)$ plane is 7$\deg$. The ellipticity $c/a$ is set to 1, 1.5, 2 and 3. Notably, the case of $c/a = 1$ corresponds to a 7-deg sphere–cone configuration.

Figure 13. Schematic geometry of an elliptic cone body (not to scale).

The crossflow velocity gradient and heat flux at the $\phi =0^{\circ }$ symmetry plane exhibit similar trends to those observed for a spheroid body, as shown in figure 14. Hence, we will not discuss this in detail here. Instead, more attention is drawn to the parameters at the $\phi =90^{\circ }$ symmetry plane, as shown in figure 15. The values of ${({\partial w_e}/{\partial z})}/{({\partial w_e}/{\partial z})_s}$ under $c / a=1$ are positive, indicating that the flows are diverging from the symmetry plane. However, for $c / a$ that are larger than 1, the values of ${({\partial w_e}/{\partial z})}/{({\partial w_e}/{\partial z})_s}$ can become negative at certain sections in the aft body of the cone, indicating converging flow to the symmetry plane; and the larger the $c / a$ is, the further forward ${({\partial w_e}/{\partial z})}/{({\partial w_e}/{\partial z})_s}$ becomes negative.

Figure 14. Crossflow velocity gradient (a) and heat flux (b) distributions at the $\phi =0^{\circ }$ symmetry plane under the conditions of $M_{\infty }=7$, $p_{\infty }=1197$ Pa, $T_{\infty }=226.5$ K, $T_w=293$ K, $\gamma =1.4$, $r=20$ mm and zero angle of attack. The boundary layer edge parameters are obtained based on ((6.16)–(6.19)).

Figure 15. Crossflow velocity gradient (a) and heat flux (b) distributions at the $\phi =90^{\circ }$ symmetry plane under the conditions of $M_{\infty }=7$, $p_{\infty }=1197$ Pa, $T_{\infty }=226.5$ K, $T_w=293$ K, $\gamma =1.4$, $r=20$ mm and zero angle of attack. The boundary layer edge parameters are obtained based on ((6.16)–(6.19)).

In the case of converging flow, the shock will be pushed farther away, resulting in a thicker boundary layer thickness and reduced heat flux, as illustrated in figure 15(b). Nevertheless, the non-dimensional heat fluxes at the $\phi =90^{\circ }$ symmetry plane are relatively close to each other for all cases. Notably, we do not consider the presence of separated flows in this analysis.