2.1. Physical problem and flow scale

In this paper, two-dimensional steady flows are mainly considered, and the extension to three-dimensional flows is straightforward. Figure 1 shows a typical schematic of a hypersonic boundary layer involving both the continuous and rarefied regimes over a flat plate, where $M_\infty$ is the free-stream Mach number, $L$ is the global length scale and $\delta$ is the boundary-layer scale. In the continuum regime (downstream), there exist two distinct regions, i.e. the boundary layer and the shock wave, which are separated by an inviscid layer. As the flow becomes rarefied, both the boundary layer and the shock wave thicken and merge with each other, especially near the leading edge. The flow is assumed to satisfy the following requirements. First, the free-stream Mach number is large and the boundary-layer thickness is small compared with the global length, i.e. ${M_\infty } \gg 1$ , $\delta \ll L$ . Second, the local flow in the boundary layer (especially near the wall) is highly non-equilibrium, such that the magnitudes of $\left | {{\sigma _{\textit{ij}}}/p} \right |$ and $\left | {{q_i}/\textit{pa}} \right |$ are not small but can reach order unity, i.e. $\left | {{\sigma _{\textit{ij}}}/p} \right | \sim O(1)$ , $\left | {{q_i}/pa} \right | \sim O(1)$ , where $\sigma _{\textit{ij}}$ and $q_i$ are the viscous stress tensor and heat flux vector, respectively, $p$ is the hydrostatic pressure and $a$ is the sound speed. Herein, the main goal is to give the leading transport relations in the boundary layer, which also describe the momentum and heat transfer between the gas and the wall.

Figure 1.

Schematic of hypersonic rarefied flow over a flat plate.

2.2. Nonlinear transport analysis

For two-dimensional steady flow ( $\partial /\partial t=0$ ), the GHE can be written as (Myong Reference Myong1999)

(2.1) \begin{align} \begin{aligned} & \left ( {u\frac {{\partial {\sigma _{\textit{xy}}}}}{{\partial x}} + v\frac {{\partial {\sigma _{\textit{xy}}}}}{{\partial y}}} \right ) + {\sigma _{\textit{xy}}}\left ( {\frac {{\partial u}}{{\partial x}} + \frac {{\partial v}}{{\partial y}}} \right ) + p\left ( {\underline {\frac {{\partial u}}{{\partial y}}} + \frac {{\partial v}}{{\partial x}}} \right )\\ & + \left ( {{\sigma _{\textit{xx}}}\frac {{\partial v}}{{\partial x}} + {\sigma _{\textit{xy}}}\frac {{\partial v}}{{\partial y}} + {\sigma _{\textit{xy}}}\frac {{\partial u}}{{\partial x}} + \underline {{\sigma _{\textit{yy}}}\frac {{\partial u}}{{\partial y}}} } \right ) = \underline { - \frac {p}{\mu }{\sigma _{\textit{xy}}}\hat Q ( {\hat K})}, \end{aligned} \end{align}

(2.2) \begin{align} \begin{aligned} & \left ( {u\frac {{\partial {\sigma _{\textit{yy}}}}}{{\partial x}} + v\frac {{\partial {\sigma _{\textit{yy}}}}}{{\partial y}}} \right ) + {\sigma _{\textit{yy}}}\left ( {\frac {{\partial u}}{{\partial x}} + \frac {{\partial v}}{{\partial y}}} \right ) + p\left ( {\frac {4}{3}\frac {{\partial v}}{{\partial y}} - \frac {2}{3}\frac {{\partial u}}{{\partial x}}} \right )\\ & + \left ( {\frac {4}{3}{\sigma _{\textit{xy}}}\frac {{\partial v}}{{\partial x}} + \frac {4}{3}{\sigma _{\textit{yy}}}\frac {{\partial v}}{{\partial y}} - \frac {2}{3}{\sigma _{\textit{xx}}}\frac {{\partial u}}{{\partial x}}\underline { - \frac {2}{3}{\sigma _{\textit{xy}}}\frac {{\partial u}}{{\partial y}}} } \right ) = \underline { - \frac {p}{\mu }{\sigma _{\textit{yy}}}\hat Q({\hat K})}, \end{aligned} \end{align}

(2.3) \begin{align} \begin{aligned} & \left ( {u\frac {{\partial {q_y}}}{{\partial x}} + v\frac {{\partial {q_y}}}{{\partial y}}} \right ) + {q_y}\left ( {\frac {{\partial u}}{{\partial x}} + \frac {{\partial v}}{{\partial y}}} \right ) + \left [ {{\sigma _{\textit{xy}}}\left ( {u\frac {{\partial u}}{{\partial x}} + v\frac {{\partial u}}{{\partial y}}} \right ) + {\sigma _{\textit{yy}}}\left ( {u\frac {{\partial v}}{{\partial x}} + v\frac {{\partial v}}{{\partial y}}} \right )} \right ]\\ & + {\sigma _{\textit{xy}}}\frac {{\partial \left ( {{c_p}T} \right )}}{{\partial x}} + \underline {{\sigma _{\textit{yy}}}\frac {{\partial \left ( {{c_p}T} \right )}}{{\partial y}}} + \underline {p\frac {{\partial \left ( {{c_p}T} \right )}}{{\partial y}}} + \left ( {{q_x}\frac {{\partial v}}{{\partial x}} + {q_y}\frac {{\partial v}}{{\partial y}}} \right ) = \underline { - \frac {{p{c_p}}}{\kappa }{q_y}\hat Q({\hat K})} .\end{aligned} \end{align}

Here, only the equations of shear stress $\sigma _{\textit{xy}}$ , normal stress $\sigma _{\textit{yy}}$ and heat flux $q_y$ are given since they are related to the surface tangential momentum, normal momentum and heat transfer, respectively. In (2.1)–(2.3), $(x,y)$ are the spatial coordinates along and normal to the surface respectively, $(u,v)$ are the corresponding velocities, $T$ is the gas temperature, $c_p$ is the specific heat at constant pressure, $\mu$ is the dynamic viscosity, $\kappa$ is the thermal conductivity, $\hat Q ({\hat K})$ is the nonlinear factor connected with entropy production and $\hat K$ is a modified Rayleigh–Onsager dissipation function, written as

(2.4) \begin{align} \hat Q( {\hat K }) = \sinh ( {\hat K })/\hat K \;\;\;\;{\textrm {and}}\;\;\;\;\hat K = \frac {c}{p}{\left [ {{\boldsymbol{\sigma }}:{\boldsymbol{\sigma }} + \frac {{2\boldsymbol{q} \boldsymbol{\cdot }\boldsymbol{q}}}{{{c_p}T/\textit{Pr}}}} \right ]^{1/2}} , \end{align}

where $c = { [ { ( {2\sqrt \pi /5} ){A_2} ( \nu )\varGamma [ {4 - 2/ ( {\nu - 1} )} ]} ]^{1/2}}$ is a constant, which depends on the exponent of the inverse-power-law intermolecular force $\nu$ . Here, $\hat {K}$ and $\hat {Q}$ are both dimensionless, with $\hat {K}\gt 0$ and $\hat {Q}\gt 1$ .

In a hypersonic boundary layer (see figure 1), the relative orders of different terms in the GHE are actually very different. Considering that $\delta \ll L$ and the equation of continuity (i.e. $\partial (\rho u)/\partial x + \partial (\rho v)/\partial y = 0$ ), the flow scales in the boundary layer can be further written as

(2.5) \begin{align} \frac {x}{L} \sim \frac {u}{{{U_\infty }}} \sim O(1)\;,\;\;\;\;\frac {y}{L} \sim \frac {v}{{{U_\infty }}} \sim O(\Delta ),\;\;\;\; {\textrm {where}} \;\;\;\;\Delta =\delta /L \ll 1 , \end{align}

where $U_\infty$ is the free-stream velocity. The above equation also means that, in the hypersonic transitional boundary layer, $\partial /\partial x \ll \partial /\partial y$ and $v \ll u$ . That is, $\partial u/\partial y$ dominates all spatial derivatives of the velocity field, and $\partial T/\partial y$ dominates the derivatives of the temperature field. As for stress and heat flux, their orders are

(2.6) \begin{align} {\sigma _{\textit{xx}}} \sim {\sigma _{\textit{yy}}} \sim {\sigma _{\textit{xy}}} \sim \mu \partial u/\partial y \sim p,\;\;\;\;{q_x} \sim {q_y} \sim \kappa \partial T/\partial y \sim pa . \end{align}

With the above estimations, omitting the terms with $\partial /\partial x$ and $v$ in (2.1)–(2.3), the leading transport terms of GHE can be written as

(2.7) \begin{align} \begin{aligned} & \left ( {p + {\sigma _{\textit{yy}}}} \right )\frac {{\partial u}}{{\partial y}} + O\left ( \Delta \right ) = - \frac {p}{\mu }{\sigma _{\textit{xy}}}\hat Q( {\hat K }),\;\;\;\;\;\;\;\; - \frac {2}{3}{\sigma _{\textit{xy}}}\frac {{\partial u}}{{\partial y}} + O\left ( \Delta \right ) = - \frac {p}{\mu }{\sigma _{\textit{yy}}}\hat Q( {\hat K }),\\ & \left ( {p + {\sigma _{\textit{yy}}}} \right )\frac {{\partial \left ( {{c_p}T} \right )}}{{\partial y}} + O\left ( \Delta \right ) = - \frac {{p{c_p}}}{\kappa }{q_y}\hat Q( {\hat K }) .\end{aligned} \end{align}

Note that, in analysing the equation of $q_y$ , $ \textit{Pr}=\mu c_p/\kappa \sim O(1)$ and $q_i \sim pa \sim \sigma _{\textit{xy}}u$ are also considered. Detailed analyses show that all omitted terms are of the order of $O(\Delta )$ or higher. Major contributions to $\sigma _{\textit{ij}}$ and $q_i$ only come from the underlined terms in (2.1)–(2.3). From (2.7), more explicit expressions for $\sigma _{\textit{xy}}$ , $q_y$ and $\sigma _{\textit{yy}}$ can be given as

(2.8) \begin{align} {\sigma _{\textit{xy}}} = F\left ( { - \mu \frac {{\partial u}}{{\partial y}}} \right )\!,\;\;\;\;\;\;{q_y} = F\left ( { - \kappa \frac {{\partial T}}{{\partial y}}} \right )\!,\;\;\;\;\;\;{\sigma _{\textit{yy}}} = \big( {F\hat Q - 1} \big)p, \end{align}

where $F$ is a nonlinear function depending on $\hat Q ( {\hat K } )$ and $K_\sigma$ , as

(2.9) \begin{align} F = \frac {1}{{\hat Q({\hat K}) + 2 K_\sigma ^2/ \big(3\hat Q(\hat K) \big)}},\;\;\;\; {\textrm {where}} \;\;\;\; K_\sigma =\left | \frac {\mu }{p}\frac {{\partial u}}{{\partial y}} \right | , \end{align}

where $\hat Q ( {\hat K } )$ is defined in (2.4), which increases monotonically with $\hat {K}$ , and $K_\sigma$ is a local non-equilibrium parameter characterising the shear non-equilibrium degree. Equations (2.8) and (2.9) imply that the nonlinear transport in a hypersonic transitional boundary layer is influenced by the shear non-equilibrium effect (indicated by $K_\sigma$ ) and dissipation rate (donated by $\hat {K}$ ). Since the above derivations follow a general boundary-layer concept, the transport relations (2.8) and (2.9) are in fact applicable to general boundary-layer flows, and they are applicable in both continuous and transitional regimes. In the continuum regime ( ${K_\sigma } \to 0, \hat {K} \to 0$ and ${F} \to 1$ ), (2.8) recovers to the NSF linear constitutive relations. As flow becomes rarefied ( ${K_\sigma } \gt 0, \hat {K} \gt 0$ and $F \lt 1$ ), the shear stress $\sigma _{\textit{xy}}$ and heat flux $q_y$ deviate from the linear constitutive values, and the normal stress $\sigma _{\textit{yy}}$ becomes comparable to the hydrostatic pressure $p$ .

Equations (2.8) and (2.9) are still coupled since $\hat K$ is related to all components of the stress tensor and heat flux vector. Here, we consider two typical scenarios. First, in most of the boundary-layer region (especially downstream), the flow is dominated by velocity shear. Under such circumstances, the nonlinear factor $\hat Q$ does not play significant role and can be assumed to be 1 for simplicity. Hence, (2.8) and (2.9) can be written as

(2.10) \begin{align} \begin{aligned} & {\sigma _{\textit{xy}}} = {F_\sigma }\sigma _{\textit{xy}}^{({\textit{NS}})},\;\;\;\;\;\;{q_y} = {F_\sigma }q_y^{({{NS}})},\;\;\;\;\;\;{P_{\textit{yy}}} = {F_\sigma }p,\\ {\textrm {where}}\;\;\;\; & {F_\sigma } = \frac {1}{{1 + 2 K_\sigma ^2 /3}}\;\;\;\;{\textrm {and}}\;\;\;\;{K_\sigma } = \left | {\sigma _{\textit{xy}}^{({\textit{NS}})}/p} \right | \! .\end{aligned} \end{align}

In writing the above relations, ${\sigma _{\textit{xy}}^{(\textit{NS})}} = - \mu \partial u/\partial y$ and ${q_{y}^{(\textit{NS})}} = - \kappa \partial T/\partial y$ are considered in the boundary layer, and $P_{\textit{yy}} = \sigma _{\textit{yy}} + p$ is the normal pressure. In (2.10), $F_\sigma$ is always smaller than 1, indicating that the actual $\sigma _{\textit{xy}}$ and $q_y$ are smaller than the corresponding NSF linear constitutive values (so-called shear thinning phenomenon), and the normal pressure $P_{\textit{yy}}$ is smaller than the hydrostatic pressure $p$ due to finite negative $\sigma _{\textit{yy}}$ . Moreover, the nonlinear stress and heat flux are only correlated with $K_\sigma$ and can be regarded as the NSF constitutive values multiplied by the same nonlinear function $F_\sigma$ .

Second, at the stagnation point, the velocity shear is nearly zero ( $K_\sigma =0$ ), while the thermal gradient is especially large, which causes prominent non-Fourier heat conduction. In fact, only the wall-normal heat flux $q_{y}$ is non-zero at the stagnation point, and $\hat {K}$ can be written as $\hat K = c{[2 \textit{Pr}q_y^2/({c_p}T)]^{1/2}}/p$ . Hence, (2.8) and (2.9) for the stagnation-point heat flux can be explicitly written as

(2.11) \begin{align} {q_y} = {F_q}q_y^{(\textit{NS})},\;\;\;\;{\textrm {where}}\;\;\;\;{F_q} = \ln \left ( {{K_q} + \sqrt {K_q^2 + 1} } \right )/{K_q}\;\;{\textrm {and}}\;\;{K_q} = \tilde c\left | {\frac {{q_y^{({\textit{NS}})}}}{{pa}}} \right | , \end{align}

where $\tilde {c} = c\sqrt {2 ( {\gamma - 1} )Pr}$ is a constant, $a={\sqrt {\gamma RT}}$ is the sound speed, with $\gamma$ being the ratio of specific heats and $R$ being the gas constant, and $K_q$ is a local non-equilibrium parameter characterising the thermal-gradient non-equilibrium degree. In fact, $c$ has a value between 1.0138 and 1.2232 and is close to 1 for a usual gas viscosity. For instance, $c=1.0179$ for $\omega =0.75$ and $c=1.0138$ for $\omega =1$ , where $\omega$ is the temperature exponent for the power-law viscosity. Hence, herein, $c$ is assumed to be 1 for simplicity, which gives $\tilde {c} = \sqrt {2 ( {\gamma - 1} )Pr}$ . From (2.11), $F_q$ is always smaller than 1, meaning that the actual stagnation-point heat flux $q_y$ is smaller than the corresponding Fourier (NSF) heat flux.

On the whole, (2.7) and (2.8) give the leading transport terms in a hypersonic transitional boundary layer based on the GHE. Equation (2.10) describes the nonlinear transport of shear dominated flow in most boundary-layer regions, and (2.11) describes the non-Fourier heat conduction in the stagnation region. In fact, the nonlinear transport caused by the shear non-equilibrium effect (i.e. (2.10)) can be observed in different shear flows by different methods. For instance, theoretical analysis of the Boltzmann equation in pure shear flow (Garzó & Santos Reference Garzó and Santos2003) and DSMC simulations of high-speed rarefied Couette flow (Ou & Chen Reference Ou and Chen2020b ) also illustrate that the nonlinear shear stress and heat flux depend only on a non-equilibrium parameter like $K_\sigma$ , and the nonlinear transport function is similar to the present $F_\sigma$ . Meanwhile, (2.10) can not only be derived from the GHE, but also can be derived from the classical Grad’s 13 moment equations (G13). For two-dimensional steady boundary layer, with similar analysis based on the G13, i.e. omitting the terms of the order of $\Delta$ and higher, the leading transport terms of the G13 can be written as

(2.12) \begin{align} & \left ( {p + {\sigma _{\textit{yy}}}} \right )\frac {{\partial u}}{{\partial y}} + 2\varepsilon \frac {{\partial {q_x}}}{{\partial y}} + O\left ( \Delta \right ) = - \frac {p}{\mu }{\sigma _{\textit{xy}}},\;\;\;\; - \frac {2}{3}{\sigma _{\textit{xy}}}\frac {{\partial u}}{{\partial y}} + \frac {8}{3}\varepsilon \frac {{\partial {q_y}}}{{\partial y}} + O\left ( \Delta \right ) = - \frac {p}{\mu }{\sigma _{\textit{yy}}}, \nonumber \\& \left ( {p + {\sigma _{\textit{yy}}}} \right )\frac {{\partial \left ( {{c_p}T} \right )}}{{\partial y}} + 2\varepsilon \left [ {{q_x}\frac {{\partial u}}{{\partial y}} + \frac {{\partial \left ( {{c_p}T{\sigma _{\textit{yy}}}} \right )}}{{\partial y}}} \right ] + O\left ( \Delta \right ) = - \frac {{p{c_p}}}{\kappa }{q_y}, \end{align}

where $\varepsilon =(\gamma -1)/(2\gamma )$ . Usually, $\varepsilon$ can be assumed small (i.e. $\varepsilon \ll 1$ ) in a thin viscous shock-layer analysis (Cheng et al. Reference Cheng, Lee, Wong and Yang1989). Omitting the terms gauged by $\varepsilon$ , we can obtain the same relations as (2.10). As for the (2.11), it can only be derived from the GHE (not from the G13) since it comes from the nonlinear factor $\hat {Q}(\hat {K})$ . Equation (2.11) shows that the actual stagnation-point heat flux in a hypersonic transitional flow is smaller than the corresponding Fourier heat flux, which is consistent with previous theoretical observations based on the Burnett/super-Burnett equations and DSMC simulations (Singh & Schwartzentruber Reference Singh and Schwartzentruber2016).

3.1. Formulation of the correlations

From (2.10) and (2.11), in a hypersonic transitional boundary layer, the actual shear stress $\sigma _{\textit{xy}}$ , heat flux $q_y$ and normal pressure $P_{\textit{yy}}$ can be regarded as the corresponding linear stress and heat flux multiplied by a nonlinear function. For simplified theoretical modelling, we might take continuum-based solutions as an approximation for the linear stress and heat flux instead of solving the GHE equations. An estimation shows that the error is of the order of $O(K_\sigma ^2)$ . This enables us to construct local aerothermodynamic correlations for near-continuum flows with only the NSF solutions.

In this paper, the surface pressure, shear stress and heat flux coefficients are defined as

(3.1) \begin{align} {C_{\kern-1pt p}} = \frac {{{P_w}}}{{0.5{\rho _\infty }{U_\infty ^2}}},\;\;\;\;{C_{\kern-1pt f}} = \frac {{ {\sigma _w}}}{{0.5{\rho _\infty }{U_\infty ^2 }}},\;\;\;\;{C_{\kern-1pt h}} = \frac {{ {q_w}}}{{0.5{\rho _\infty }{U_\infty ^3 }}} , \end{align}

where ${P_w} = {\left . {{P_{\textit{yy}}}} \right |_w}$ is the surface normal pressure and $\sigma _w$ and $q_w$ are the surface shear stress and total heat flux, respectively. Considering that the hypersonic boundary layer is dominated by the shear non-equilibrium effect, we first give a set of correlations based on (2.10) as

(3.2) \begin{align} {C_{\kern-1pt p}} = {F_\sigma }\left ( {{K_\sigma }} \right ){C_{\kern-1pt p}^{(\textit{NS})}},\;\;\;\;{C_{\kern-1pt f}} = {F_\sigma }\left ( {{K_\sigma }} \right ){C_{f}^{(\textit{NS})}},\;\;\;\;{C_{\kern-1pt h}} = {F_\sigma }\left ( {{K_\sigma }} \right ){C_{\kern-1pt h}^{(\textit{NS})}} , \end{align}

where $C_{\kern-1pt p}^{(\textit{NS})}$ , $C_{f}^{(\textit{NS})}$ and $C_{\kern-1pt h}^{(\textit{NS})}$ are the continuum-based solutions, which can be either given by analytical formulae or NSF simulations. For blunt-body flows, the above correlations for $C_{\kern-1pt p}$ and $C_{\kern-1pt f}$ can keep the same form, but the heat flux correlation $C_{\kern-1pt h}$ becomes ineffective in the stagnation region due to the significant thermal-gradient non-equilibrium effect. Here, a modified heat flux correlation is given by combining (2.10) and (2.11) as

(3.3) \begin{align} {C_{\kern-1pt h}} = {F}\left ( {{K_\sigma },{K_q}} \right ){C_{\kern-1pt h}^{(\textit{NS})}},\;\;\;\;{\textrm {where}}\;\;\;\;{F}\left ( {{K_\sigma },{K_q}} \right ) = F_\sigma F_q . \end{align}

From (2.10) and (2.11), the $K_\sigma$ and $K_q$ values at the wall can be written as

(3.4) \begin{align} {K_\sigma } = \left | {C_{\kern-1pt f}^{({{NS}})}/C_{\kern-1pt p}^{({{NS}})}} \right |,\;\;\;\;\;\;{K_q} = \left | {C_{\kern-1pt h}^{({\textit{NS}})}/C_{\kern-1pt p}^{({{NS}})}} \right | \tilde c {M_\infty }/{\sqrt {\hat {T}} } , \end{align}

where $\hat {T}={\left . T \right |_{y=0}}/T_\infty$ is the dimensionless gas temperature at the wall. In the following, the accuracy of the above correlations is carefully verified by comparing with DSMC simulations and experimental data. The correlation formulae are first used to correct the analytical formulae of a rarefied flat-plate flow and then applied to correct the NSF numerical solutions of various hypersonic rarefied flows over different geometries.

3.2. Theoretical modelling of hypersonic rarefied flat-plate flow

This section presents theoretical modelling of the surface aerothermodynamics for a hypersonic rarefied flow over a flat plate, which is a typical case that has been widely studied. Within the NSF framework, various analytical correlations have been developed to predict the surface pressure, shear stress and heat flux for a hypersonic flat-plate flow, such as the strong interaction (SI) theory (Hayes & Probstein Reference Hayes and Probstein1959) and empirical formulae (EF) by fitting numerical solutions (Shorenstein & Probstein Reference Shorenstein and Probstein1968). Usually, the surface quantities are correlated with the viscous interaction parameter $\bar \chi = {M_\infty ^3 }\sqrt {C/R{e_x}}$ , where $C = ({\mu _w}{T_\infty })/({{\mu _\infty }{T_w}})$ and $ \textit{Re}_{x} = {\rho _\infty }{U_\infty }x/{\mu _\infty }$ . As for the surface coefficients $C_{\kern-1pt p}$ , $C_{\kern-1pt f}$ and $C_{\kern-1pt h}$ , it is more appropriate to correlate them with a parameter $\bar V = {M_\infty }\sqrt {C/R{e_x}}$ considering ${C_{\kern-1pt p}} \propto \bar V$ from the SI theory (Anderson Reference Anderson2006). Here, $\bar V$ can be also regarded as a local rarefaction parameter. The classical SI and EF formulae predict the surface coefficients in terms of $\bar V$ for hypersonic flat-plate flow as (Shorenstein & Probstein Reference Shorenstein and Probstein1968)

(3.5) \begin{align} \begin{aligned} & {\textrm{SI}}:\;\;\;\;{C_{p,\textit{SI}}} = \frac {{2{a_0}}}{\gamma }\bar V,\;\;\;\;{C_{f,\textit{SI}}} = {b_0}{{\bar V}^{\frac {3}{2}}},\;\;\;\;{C_{h,\textit{SI}}} = {c_0}{{\bar V}^{\frac {3}{2}}}\\& {\textrm {EF}}:\;\;\left \{ \begin{aligned} & {C_{p,\textit{EF}}}/{C_{p,\textit{SI}}} = 0.5\left [ {1 - \tanh \left ( {0.89{{\log }_{10}}\beta + 1.12} \right )} \right ],\;\;\\ & {C_{f,\textit{EF}}}/{C_{f,\textit{SI}}} = {C_{h,\textit{EF}}}/{C_{h,\textit{SI}}} = 0.5\left [ {1 - \tanh \left ( {0.91{{\log }_{10}}\beta + 1.10} \right )} \right ], \end{aligned} \right . \end{aligned} \end{align}

where the coefficients $a_0$ , $b_0$ and $c_0$ depend on the wall temperature ratio, written as ${a_0} = 0.554{T_w}/{T_0} + 0.0973$ , ${b_0} = 2 ( {0.368{T_w}/{T_0} + 0.0684} )$ and ${c_0} = {b_0} ( {1 - {T_w}/{T_0}} )/2$ , with $T_0$ the free-stream stagnation temperature, the correlation parameter $\beta = { ( {{T_w}/{T_0}} )^{1/2}}{{\bar V}^2}$ . For the present correlation, (3.2) is adopted, and the SI and EF formulae are chosen as the continuum-based solutions. That is, $C_{\kern-1pt p}^{(\textit{NS})}$ , $C_{f}^{(\textit{NS})}$ and $C_{\kern-1pt h}^{(\textit{NS})}$ in (3.2) are replaced by the $C_{p,\textit{SI}}$ , $C_{f,\textit{SI}}$ and $C_{h,\textit{SI}}$ or $C_{p,\textit{EF}}$ , $C_{f,\textit{EF}}$ and $C_{h,\textit{EF}}$ in (3.5), and the parameter $K_\sigma$ can be written in terms of $\bar V$ as ${K_\sigma } = (\gamma {b_0}/2{a_0}){{\bar V}^{1/2}}$ based on the SI formula.

Figure 2 shows the variations of surface coefficients ( $C_{\kern-1pt f}$ , $C_{\kern-1pt h}$ and $C_{\kern-1pt p}$ ) with $\bar V$ predicted by different theoretical models, DSMC simulations and experiments at around $M_\infty =20$ . From the distributions of $C_{\kern-1pt f}$ and $C_{\kern-1pt h}$ , different analytical correlations compare well with the DSMC data and experiments for $\bar V \lt 0.2$ . As $\bar V$ further increases, the SI formula predicts that the shear stress and heat flux coefficients increase monotonically, showing large discrepancies with the DSMC and experimental data. With considering the shear non-equilibrium effect, the present correlations largely improve the accuracy of the SI formula. Combined with a more accurate empirical formula, the present correlations agree well with the DSMC and experiments up for $\bar V \sim 1$ , and can give accurate peaks of shear stress and heat flux. From the pressure distribution ( $C_{\kern-1pt p}$ ), the SI formula and the empirical formula deviate from the DSMC data (indicated by $P_w$ -DSMC) and experiments for $\bar V \gt 0.2$ . Again, the present correlations can largely improve the accuracy of these two continuum-based formulae. Moreover, the hydrostatic pressure adjacent to the wall (indicated by $p_w$ -DSMC) is also plotted. Surprisingly, it is observed that the SI formula coincides with the DSMC data in hydrostatic pressure ( $p_w$ ) even when $\bar V \sim 1$ . However, the hydrostatic pressure ( $p_w$ ) differs greatly from the normal pressure $P_w$ (i.e. actual gas-surface normal momentum transfer) under rarefied conditions (i.e. large $\bar V$ ). The present correlations can well capture such effects.

Figure 2.

Variations of $C_{\kern-1pt f}$ , $C_{\kern-1pt h}$ and $C_{\kern-1pt p}$ with $\bar V$ for hypersonic rarefied flat-plate flow, and comparison among different theoretical models, DSMC results by Chen et al. (Reference Chen, Wang and Yu2015) and present simulations and experiments (Vidal & Bartz Reference Vidal and Bartz1969). The experiments are conducted at ${M_\infty } = 19.0 \sim 21.8$ , ${T_w}/{T_0} = 0.059 \sim 0.074$ , $Re_\infty = {\rho _\infty }{U_\infty }/{\mu _\infty }=$ $9.45 \times {10^3} \sim 3.55 \times {10^5}\;{{\textrm {m}}^{-1}}$ for air, and in the DSMC and theoretical model, ${M_\infty } = 20$ , ${T_w}/{T_0} = 0.06$ , $\omega = 0.75$ for nitrogen.

3.3. Numerical modelling of hypersonic rarefied flows over different geometries

For complex flows, there exist no general analytical formulae for the surface aerothermodynamics of continuum-based solutions. Here, the NSF equations are numerically solved along with the no-slip and Maxwell–Smoluchowski slip boundary conditions (Ou & Chen Reference Ou and Chen2020a ). Various hypersonic flows over different geometries are considered, including flat-plate, sharp-wedge, cylinder and blunt-cone flows, as shown in figure 3. In the NSF simulation, the surface pressure, shear stress and heat flux are computed as

(3.6) \begin{align} {p_{w}^{(\textit{NS})}} = {\left . p \right |_0},\;\;\;\;\;\;{\sigma _{w}^{(\textit{NS})}} = \mu\! {\left . \partial u/\partial y \right |_0},\;\;\;\;\;\;{q_{w}^{(\textit{NS})}} = \kappa\! {\left . \partial T/\partial y \right |_0} + {u_s}{\sigma _{w}^{(\textit{NS})}} , \end{align}

where $y$ represents the wall-normal direction. Note that, for surface heat flux, $\kappa \partial T/\partial y$ represents the Fourier heat conduction and ${u_s}{\sigma _{w}^{(\textit{NS})}}$ donates the shear work done by sliding friction, where $u_s$ is the slip velocity. With no-slip boundary conditions, $u_s$ is zero and only the Fourier heat conduction exists. In this section, the correlations (3.2) and (3.3) are adopted, and the numerical solutions of the NSF equations (see (3.6)) are used to give the $C_{\kern-1pt p}^{(\textit{NS})}$ , $C_{f}^{(\textit{NS})}$ and $C_{\kern-1pt h}^{(\textit{NS})}$ in (3.2)–(3.4). When applying the present correlations for slip flows, the total heat flux contains two terms: the Fourier term is corrected by $F_\sigma$ or $F_\sigma F_q$ according to the $C_{\kern-1pt h}$ correlation of (3.2) or (3.3), and the sliding-friction term is corrected by $F_\sigma$ according to the $C_{\kern-1pt f}$ correlation of (3.2). For all cases, argon is chosen as the working gas. The hard sphere or variable hard sphere model is used for intermolecular collisions and the fully diffuse reflection is adopted at the wall.

Figure 3.

Schematics of computational geometries.

First, the hypersonic rarefied flows past a flat plate and sharp wedge are analysed. The continuum-based solutions are given by NSF slip solutions. The distributions of surface coefficients of pressure ( $C_{\kern-1pt p}$ ), shear stress ( $C_{\kern-1pt f}$ ) and heat flux ( $C_{\kern-1pt h}$ ) by different models are shown in figure 4. As can be seen, for the Mach-10 flat-plate flow, the NSF solutions show large discrepancies with the DSMC data in the upstream ( $x/{\lambda _\infty } \lt 20$ ), while the present correlations (3.2) significantly improve the NSF solutions and agree well with DSMC for all surface coefficients up to $x/{\lambda _\infty } = 2$ ( $K_\sigma \approx 1.5$ ). The modified heat flux correlation (3.3) exhibits little difference from the correlation (3.2). This is because the boundary layer over the entire surface is dominated by velocity shear in such sharp leading flow, such that the correction on $C_{\kern-1pt h}$ is mainly induced by $K_\sigma$ (i.e. shear non-equilibrium) and the effect of $K_q$ (i.e. thermal-gradient non-equilibrium) is very limited (also see panel $a_{\textit{iii}}$ ). For the Mach-15 sharp-wedge flow at an incident angle, the rarefaction effects in the leeward are much stronger than that in the windward, causing the deviations of the NSF solutions and the DSMC data to be more significant in the leeward. Again, the present correlations largely improve the NSF solutions and agree well with DSMC up to $x/\lambda _\infty = 3$ ( $K_\sigma \sim 1$ ) in both the leeward and windward. We also notice that some differences exist at the leading edge ( $x/\lambda _\infty \lt 2$ ) between the present correlations and DSMC, which is because the local flow is too rarefied to be predicted by such a simple model.

Figure 4.

Distributions of surface $C_{\kern-1pt p}$ , $C_{\kern-1pt f}$ and $C_{\kern-1pt h}$ , and comparison among NSF slip solutions (red dashed line), present correlations (3.2) (black solid line), present modified heat flux correlation (3.3) (blue dashed line) and DSMC data (symbol). The subfigures in ( $a_{\textit{iii}}$ ) and ( $b_{\textit{iii}}$ ) plot the surface $K_\sigma$ and $K_q$ of NSF slip solutions. ( $a_{i}$ – $a_{\textit{iii}}$ ) Flat-plate flow: $M_\infty = 10$ , ${T_\infty } = 200\;{\textrm {K}}$ , ${T_w} = 500\;{\textrm {K}}$ , $\omega$ = 0.5, ${\lambda _\infty } = 16.8\;{\textrm {mm}}$ ; ( $b_{i}$ – $b_{\textit{iii}}$ ) sharp-wedge flow: ${M_\infty } = 15$ , ${T_\infty } = 200\;{\textrm {K}}$ , ${T_w} = 1000\;{\textrm {K}}$ , $\omega$ = 0.81, ${\lambda _\infty } = 15.3\;{\textrm {mm}}$ , $\theta =10^{\textrm {o}}$ and $\alpha =9^{\textrm {o}}$ .

Second, the hypersonic rarefied flows past blunt bodies are simulated, including cylindrical flow ( $M_\infty =10$ ) and blunt-cone flow ( $M_\infty =20$ ) at different Knudsen numbers. The present correlations combined with NSF solutions at both no-slip and slip boundary conditions are presented. Figure 5 shows the distributions of surface $C_{\kern-1pt p}$ , $C_{\kern-1pt f}$ and $C_{\kern-1pt h}$ by different methods for the cylindrical and blunt-cone flows. For the surface pressure ( $C_{\kern-1pt p}$ ), different models agree very well with each other, even for the $ \textit{Kn}=0.25$ case. This is because the flow is strongly compressed, such that the pressure computation is not sensitive to the rarefaction effects. Similar phenomena can be also seen in the windward of the sharp-wedge case. For the shear stress ( $C_{\kern-1pt f}$ ), the NSF no-slip solutions show the largest discrepancies with DSMC in the downstream of the leading edge ( $\phi \gt 30^{\textrm {o}}$ or $x / R_n \gt 0.15$ ). The present correlations can improve both the no-slip and slip solutions, and compare well with the DSMC data by combining with the slip solutions. For the surface heat flux ( $C_{\kern-1pt h}$ ), there exist large differences between the NSF solutions and DSMC at the leading edge ( $\phi \lt 60^{\textrm {o}}$ or $x / R_n \lt 0.5$ ), especially at the stagnation point. The correlation (3.2) fails in such a region but gradually agrees with DSMC in the downstream. This is because, in the stagnation region, $K_\sigma$ is nearly zero while $K_q$ is very large (see panel $b_{vi}$ ), such that the correlation (3.2) fails. On moving downstream, the shear non-equilibrium effect (i.e. $K_\sigma$ ) quickly increases and dominates the nonlinear transport. Considering both the thermal-gradient and shear non-equilibrium effects, the modified heat flux correlation (3.3) exhibits excellent agreement with DSMC over the entire surface. On the whole, we suggest that the present correlations (3.2) combined with the NSF slip solutions are used to predict the surface aerodynamics in the hypersonic near-continuum regime and the modified heat flux correlation (3.3) is adopted to compute the non-Fourier heat conduction for blunt-body flows (especially in the stagnation region), which can largely enhance the accuracy of the NSF solutions.

Figure 5.

Distributions of surface $C_{\kern-1pt p}$ , $C_{\kern-1pt f}$ and $C_{\kern-1pt h}$ and comparison among different models. Panel ( $b_{\textit{vi}}$ ) plots the surface $K_\sigma$ and $K_q$ of NSF slip solution. ( $a_{i}$ – $a_{\textit{vi}}$ ) Cylinder flow: $M_\infty = 10$ , ${T_\infty } = 200\;{\textrm {K}}$ , ${T_w} = 500\;{\textrm {K}}$ , $\omega = 0.734$ , $ \textit{Kn} = \lambda _\infty /D =$ 0.05 and 0.25; ( $b_{i}$ – $b_{\textit{vi}}$ ) blunt-cone flow: $M_\infty = 20$ , $\omega =0.81$ , ${T_w} = 1000\;{\textrm {K}}$ , ${T_\infty } =$ 200 K and 180.7 K, $ \textit{Kn} = \lambda _\infty /R_n =$ 0.046 and 0.246, cone angle $\theta = 5^{\textrm {o}}$ .