Appendix
1.1 Explanation of incompressible approximation
Figure 6 represents the dimensional analysis on a hypersonic stagnation point flow. The typical particle diameter is around d_{p }~ 10^{−5 }m, and the computational domain character length is around ~ 10d_{p }~ 10^{−4 }m ~ δ.
The detached shock wave distance from the wall δ_{h}, and its relation with δ in terms of order are: δ_{h} ≈ 120 × 10^{−4 }m, δ ≈ 2 × 10^{−4 }m, δ_{h}/δ = 60.
According to the theoretical solution of Hiemenz flow, far from the bottom wall, larger than δ, fluid parcel vertical velocity can be approximately estimated as inviscid strain flow V = − By. The value of interest domain velocity is V_{δ} at δ, and that of velocity behind the detached shock is \({V}_{\delta_h}\approx 1000\,\mathrm{m}/\mathrm{s}\) at δ_{h}.
The linear relation of the theoretical solution of Hiemenz flow can be used to estimate the order of velocity in the computational domain:
$$\frac{V_{\delta }}{\delta}\approx \frac{V_{\delta_h}}{\delta_h},{V}_{\delta}\approx \frac{\delta }{\delta_h}{V}_{\delta_h}=\frac{1}{60}1000\approx 16\,\mathrm{m}/\mathrm{s}$$
In the stagnation point regime, the temperature can be estimated: T_{δ} > 10^{4 }K, a_{δ} > 2 × 10^{3 }m/s, V_{δ} ≈ 16 m/s and the local sonic speed is around a_{δ} ≈ 2000 m/s, thus \({M}_{\delta }=\frac{V_{\delta }}{a_{\delta }}\).
According to the definition,
$${M}_{\delta }=\frac{16\,\mathrm{m}/\mathrm{s}}{a_{\delta }>2000\,\mathrm{m}/\mathrm{s}}\approx 0.01,$$
in this respect, the blue interest domain can be treated as incompressible. In detail, starting from complete compressible fluid dynamic equations, and introducing with prior knowledge M_{δ} ≈ 0.01, the incompressible approximation can be proved.
Firstly, some important dimensional and dimensionless parameters will be defined as follows:
$${\displaystyle \begin{array}{l}{Re}=\frac{\rho_{\infty }{U}_{\infty }{L}_{\infty }}{\mu_{\infty }},Pr =\frac{C_{\mathrm{p}}{\mu}_{\infty }}{k_{\infty }}, Ma=\frac{U_{\infty }}{a_{\infty }},\\ {}{a}_{\infty }=\sqrt{\gamma R{T}_{\infty }},{p}_{\infty }={\rho}_{\infty }{U}_{\infty}^2\\ {}{C}_{\mathrm{v}}=\frac{R}{\gamma 1},{C}_{\mathrm{v}}^{\ast }=\frac{1}{\gamma \left(\gamma 1\right){Ma}^2}, Str=\frac{L_{\infty }}{U_{\infty }{t}_{\infty }}=1,\\ {}\phi =\nabla \cdot \left(\overrightarrow{u}\cdot \mu \nabla \overrightarrow{u}+\overrightarrow{u}\cdot \mu \nabla {\overrightarrow{u}}^T\overrightarrow{u}\cdot \overline{\overline{I}}\frac{2\mu }{3}\nabla \cdot \overrightarrow{u}\right)\\ {}f={f}^{\ast }{f}_{\infty}\left( ex:\nabla =\frac{\nabla^{\ast }}{L_{\infty }},\partial {t}^{\ast}\cdot {t}_{\infty }=\partial t,\overrightarrow{u^{\ast }}{U}_{\infty }=\overrightarrow{u}\dots etc\right)\end{array}}$$
∞ represents the reference physical variable, and it can be the value at infinite far field, or certain specific regime that we are interested in. For example, for wallbounded turbulence, the reference velocity is U_{∞} = U_{τ}, while for homogeneous turbulence, it is U_{∞} = U_{λ}.
Reference mass flux: \(\frac{\rho_{\infty }{U}_{\infty }}{L_{\infty }}=\left(\frac{\rho_{\infty }{U}_{\infty }}{L_{\infty}^3}\right){L}_{\infty}^2\)
Reference momentum flux: \(\frac{\rho_{\infty }{U}_{\infty}^2}{L_{\infty }}=\left(\frac{\rho_{\infty }{U}_{\infty}^2}{L_{\infty}^3}\right){L}_{\infty}^2\)
Reference energy flux: \(\frac{\rho_{\infty }{U}_{\infty}^3}{L_{\infty }}=\left(\frac{\rho_{\infty }{U}_{\infty}^3}{L_{\infty}^3}\right){L}_{\infty}^2\)
Three basic control equations:
mass conservation:
$$\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty }}\left(\frac{\partial \rho }{\partial t}+\nabla \cdot \overrightarrow{u}=0\right)$$
(23)
momentum conservation:
$$\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^2}\left\{\frac{\partial \rho \overrightarrow{u}}{\partial t}+\nabla \cdot \left(\rho \overrightarrow{u}\otimes \overrightarrow{u}+\overline{\overline{I}}p\right)\right\} =\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^2}\left\{\nabla \cdot \left(\mu \nabla \overrightarrow{u}+\mu \nabla {\overrightarrow{u}}^T\overline{\overline{I}}\frac{2\mu }{3}\nabla \cdot \overrightarrow{u}\right)\right\}$$
(24)
energy conservation:
$${\displaystyle \begin{array}{c}\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\Big\{\frac{\partial \rho {C}_{\mathrm{v}}T}{\partial t}+\frac{\partial \rho \frac{{\leftV\right}^2}{2}}{\partial t}+\nabla \cdot \left(\rho {C}_{\mathrm{v}}T\overrightarrow{u}\right)+\nabla \cdot \left(\rho \overrightarrow{u}\frac{{\leftV\right}^2}{2}\right)+\nabla \cdot \left(\overline{\overline{I}}\cdot \overrightarrow{u}p\right)\Big\}\\ {}=\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\Big\{\nabla \cdot \left(k\nabla T\right)+\nabla \cdot \left(\overrightarrow{u}\cdot \mu \nabla \overrightarrow{u}+\overrightarrow{u}\cdot \mu \nabla {\overrightarrow{u}}^T\overrightarrow{u}\cdot \overline{\overline{I}}\frac{2\mu }{3}\nabla \cdot \overrightarrow{u}\right)\Big\}\end{array}}$$
(25)
dimensionless mass conservation:
$$Str\cdot \frac{\partial {\rho}^{\ast }}{\partial {t}^{\ast }}+{\nabla}^{\ast}\cdot \overrightarrow{u^{\ast }}=0$$
(26)
dimensionless momentum conservation:
$${\displaystyle \begin{array}{c} Str\cdot \frac{\partial {\rho}^{\ast}\overrightarrow{u^{\ast }}}{\partial {t}^{\ast }}+{\nabla}^{\ast}\cdot \left({\rho}^{\ast}\overrightarrow{u^{\ast }}\otimes \overrightarrow{u^{\ast }}+\overline{\overline{I}}{p}^{\ast}\right)\\ {}=\frac{1}{{Re}}{\nabla}^{\ast}\cdot \left({\mu}^{\ast }{\nabla}^{\ast}\overrightarrow{u^{\ast }}+{\mu}^{\ast }{\nabla}^{\ast }{\overrightarrow{u^{\ast}}}^T\overline{\overline{I}}\frac{2{\mu}^{\ast }}{3}\nabla \cdot \overrightarrow{u^{\ast }}\right)\end{array}}$$
(27)
dimensionless energy conservation:
$${\displaystyle \begin{array}{c}\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\Big\{\frac{\rho_{\infty}\frac{R}{\gamma 1}{T}_{\infty }}{t_{\infty }}\frac{\partial {\rho}^{\ast }{T}^{\ast }}{\partial {t}^{\ast }}+\frac{\rho_{\infty }{U}_{\infty}^2}{t_{\infty }}\frac{\partial {\rho}^{\ast}\frac{{\left{V}^{\ast}\right}^2}{2}}{\partial {t}^{\ast }}+\frac{\rho_{\infty}\frac{R}{\gamma 1}{T}_{\infty }{U}_{\infty }}{L_{\infty }}{\nabla}^{\ast}\cdot \left({\rho}^{\ast }{T}^{\ast}\overrightarrow{u^{\ast }}\right) +\\ {} \frac{\rho_{\infty }{U}_{\infty}^3}{L_{\infty }}{\nabla}^{\ast}\cdot \left({\rho}^{\ast}\overrightarrow{u^{\ast }}\frac{{\left{V}^{\ast}\right}^2}{2}\right)+\frac{\rho_{\infty }{U}_{\infty}^3}{L_{\infty }}{\nabla}^{\ast}\cdot \left(\overline{\overline{I}}\cdot \overrightarrow{u^{\ast }}{p}^{\ast}\right)\Big\} \\ {}=\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\Big\{\frac{k_{\infty }{T}_{\infty }}{L_{\infty}^2}{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right)+\frac{\mu_{\infty }{U}_{\infty}^2}{L_{\infty}^2}{\phi}^{\ast}\Big\}\end{array}}$$
(28)
Due to the complexity of energy equation, the terms are developed one by one in details as follows:

1st term: unsteadiness of internal and pressure energy of fluid parcel
$${\displaystyle \begin{array}{c}\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\frac{\rho_{\infty}\frac{R}{\gamma 1}{T}_{\infty }}{t_{\infty }}\frac{\partial {\rho}^{\ast }{T}^{\ast }}{\partial {t}^{\ast }}=\frac{L_{\infty }}{U_{\infty }{t}_{\infty }}\frac{R{T}_{\infty }}{U_{\infty}^2\left(\gamma 1\right)}\frac{\partial {\rho}^{\ast }{T}^{\ast }}{\partial {t}^{\ast }}\\ {}= Str\cdot \frac{1}{M{a}^2\gamma \left(\gamma 1\right)}\frac{\partial {\rho}^{\ast }{T}^{\ast }}{\partial {t}^{\ast }}=\frac{\partial {\rho}^{\ast }{C_{\mathrm{v}}}^{\ast }{T}^{\ast }}{\partial {t}^{\ast }}\end{array}}$$

2nd term: unsteadiness of kinetic energy of fluid parcel
$$\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\frac{\rho_{\infty }{U}_{\infty}^2}{t_{\infty }}\frac{\partial {\rho}^{\ast}\frac{{\left{V}^{\ast}\right}^2}{2}}{\partial {t}^{\ast }}={Str}\cdot \frac{\partial {\rho}^{\ast}\frac{{\left{V}^{\ast}\right}^2}{2}}{\partial {t}^{\ast }}=\frac{\partial {\rho}^{\ast}\frac{{\left{V}^{\ast}\right}^2}{2}}{\partial {t}^{\ast }}$$

3rd term: convection of internal and pressure energy of fluid parcel
$$\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\frac{\rho_{\infty}\frac{R}{\gamma 1}{T}_{\infty }{U}_{\infty }}{L_{\infty }}{\nabla}^{\ast}\cdot \left({\rho}^{\ast }{T}^{\ast}\overrightarrow{u^{\ast }}\right)={\nabla}^{\ast}\cdot \left({\rho}^{\ast }{C_{\mathrm{v}}}^{\ast }{T}^{\ast}\overrightarrow{u^{\ast }}\right)$$

4th term: convection of kinematic energy of fluid parcel
$$\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\frac{\rho_{\infty }{U}_{\infty}^3}{L_{\infty }}{\nabla}^{\ast}\cdot \left({\rho}^{\ast}\overrightarrow{u^{\ast }}\frac{{\left{V}^{\ast}\right}^2}{2}\right)={\nabla}^{\ast}\cdot \left({\rho}^{\ast}\overrightarrow{u^{\ast }}\frac{{\left{V}^{\ast}\right}^2}{2}\right)$$

5th term: pressure gradient work \(\overrightarrow{u^{\ast }}\cdot {\nabla}^{\ast }{p}^{\ast }\) and pressure work on the compressible fluid parcel \({p}^{\ast }{\nabla}^{\ast}\cdot \overrightarrow{u^{\ast }}\)
$$\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\frac{\rho_{\infty }{U}_{\infty}^3}{L_{\infty }}{\nabla}^{\ast}\cdot \left(\overline{\overline{I}}\cdot \overrightarrow{u^{\ast }}{p}^{\ast}\right)={\nabla}^{\ast}\cdot \left(\overline{\overline{I}}\cdot \overrightarrow{u^{\ast }}{p}^{\ast}\right)$$

6th term: heat conduction
$${\displaystyle \begin{array}{c}\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\frac{k_{\infty }{T}_{\infty }}{L_{\infty}^2}{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right)=\frac{1}{\rho_{\infty }{U}_{\infty }{L}_{\infty }}\frac{\mu }{\mu}\frac{C_{\mathrm{v}}}{C_{\mathrm{v}}}\frac{k_{\infty }{T}_{\infty }}{U_{\infty}^2}{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right) \\ {}=\frac{1}{Re}\frac{1}{Pr}\frac{C_{\mathrm{v}}{T}_{\infty }}{U_{\infty}^2}{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right) =\frac{1}{Re}\frac{1}{Pr}\frac{\gamma R{T}_{\infty }}{\gamma \left(\gamma 1\right){U}_{\infty}^2}{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right) \\ {}=\frac{1}{Re}\frac{1}{Pr}\frac{1}{\left(\gamma 1\right)M{a}^2}{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right)\end{array}}$$

7th term: viscous work on the fluid parcel
$$\frac{L_{\infty }}{\rho_{\infty }{U}_{\infty}^3}\frac{\mu_{\infty }{U}_{\infty}^2}{L_{\infty}^2}{\nabla}^{\ast}\cdot \left(\overrightarrow{u^{\ast }}\cdot {\mu}^{\ast }{\nabla}^{\ast}\overrightarrow{u^{\ast }}\right)=\frac{1}{Re}{\phi}^{\ast }$$
dimensionless energy conservation:
$${\displaystyle \begin{array}{c}\frac{\partial {\rho}^{\ast }{C_{\mathrm{v}}}^{\ast }{T}^{\ast }}{\partial {t}^{\ast }}+\frac{\partial {\rho}^{\ast}\frac{{\left{V}^{\ast}\right}^2}{2}}{\partial {t}^{\ast }}+{\nabla}^{\ast}\cdot \left({\rho}^{\ast }{C_{\mathrm{v}}}^{\ast }{T}^{\ast}\overrightarrow{u^{\ast }}\right)+\\ {}{\nabla}^{\ast}\cdot \left({\rho}^{\ast}\overrightarrow{u^{\ast }}\frac{{\left{V}^{\ast}\right}^2}{2}\right)+{\nabla}^{\ast}\cdot \left(\overline{\overline{I}}\cdot \overrightarrow{u^{\ast }}{p}^{\ast}\right)\\ {}=\frac{1}{Re}\frac{\gamma }{Pr }{C}_{\mathrm{v}}^{\ast }{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right)+\frac{1}{Re}{\phi}^{\ast}\end{array}}$$
(29)
dimensionless energy conservation·γ(γ1)Ma^{2}:
$${\displaystyle \begin{array}{c}\gamma \left(\gamma 1\right)M{a}^2\Big\{\frac{\partial {\rho}^{\ast }{C_{\mathrm{v}}}^{\ast }{T}^{\ast }}{\partial {t}^{\ast }}+\frac{\partial {\rho}^{\ast}\frac{{\left{V}^{\ast}\right}^2}{2}}{\partial {t}^{\ast }}+{\nabla}^{\ast}\cdot \left({\rho}^{\ast }{C_{\mathrm{v}}}^{\ast }{T}^{\ast}\overrightarrow{u^{\ast }}\right)+\\ {}{\nabla}^{\ast}\cdot \left({\rho}^{\ast}\overrightarrow{u^{\ast }}\frac{{\left{V}^{\ast}\right}^2}{2}\right)+{\nabla}^{\ast}\cdot \left(\overline{\overline{I}}\cdot \overrightarrow{u^{\ast }}{p}^{\ast}\right)\Big\}\\ {}=\gamma \left(\gamma 1\right)M{a}^2\Big\{\frac{1}{Re}\frac{\gamma }{Pr }{C}_{\mathrm{v}}^{\ast }{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right)+\frac{1}{Re}{\phi}^{\ast}\Big\}\\ {}\\ {}\frac{\partial {\rho}^{\ast }{T}^{\ast }}{\partial {t}^{\ast }}+\gamma \left(\gamma 1\right)M{a}^2\frac{\partial {\rho}^{\ast}\frac{{\left{V}^{\ast}\right}^2}{2}}{\partial {t}^{\ast }}+{\nabla}^{\ast}\cdot \left({\rho}^{\ast }{T}^{\ast}\overrightarrow{u^{\ast }}\right)+\\ {}\gamma \left(\gamma 1\right)M{a}^2{\nabla}^{\ast}\cdot \left({\rho}^{\ast}\overrightarrow{u^{\ast }}\frac{{\left{V}^{\ast}\right}^2}{2}\right)+\gamma \left(\gamma 1\right)M{a}^2{\nabla}^{\ast}\cdot \left(\overline{\overline{I}}\cdot \overrightarrow{u^{\ast }}{p}^{\ast}\right)\\ {}=\frac{1}{Re}\frac{\gamma }{Pr }{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right)+\gamma \left(\gamma 1\right)M{a}^2\frac{1}{Re}{\phi}^{\ast}\end{array}}$$
(30)
Thus, it is clear that, when γ(γ − 1)Ma^{2} becomes vanishingly small, certain terms can be neglected, which are the kinetic energy of fluid parcel, the pressure gradient and pressure work on the fluid parcel, and the viscous work on the fluid parcel.
$${\displaystyle \begin{array}{c}\frac{\partial {\rho}^{\ast }{T}^{\ast }}{\partial {t}^{\ast }}+\gamma \left(\gamma 1\right)M{a}^2\frac{\partial{\rho}^{\ast}\frac{{\left{V}^{\ast}\right}^2}{2}}{\partial{t}^{\ast }}+{\nabla}^{\ast}\cdot \left({\rho}^{\ast }{T}^{\ast}\overrightarrow{u^{\ast }}\right)+\gamma \left(\gamma 1\right)M{a}^2{\nabla}^{\ast}\cdot \left({\rho}^{\ast}\overrightarrow{u^{\ast }}\frac{{\left{V}^{\ast}\right}^2}{2}\right),\\ {}\gamma \left(\gamma 1\right)M{a}^2{\nabla}^{\ast}\cdot \left(\overline{\overline{I}}\cdot \overrightarrow{u^{\ast }}{p}^{\ast}\right),\\ {}\gamma \left(\gamma 1\right)M{a}^2\frac{1}{Re}{\phi}^{\ast}\end{array}}$$
Once M_{τ} = M_{δ} ≈ 0.01, the energy equation can be simplified as:
$$\frac{\partial{\rho}^{\ast }{T}^{\ast }}{\partial{t}^{\ast }}+{\nabla}^{\ast}\cdot {\left({\rho}^{\ast }{T}^{\ast }\overrightarrow{{u}^{\ast}}\right)}=\frac{1}{Re}\frac{\gamma }{Pr }{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right)$$
(31)
dimensionless mass conservation:
$$\frac{\partial {\rho}^{\ast }}{\partial {t}^{\ast }}+{\nabla}^{\ast}\cdot \overrightarrow{u^{\ast }}=0$$
(32)
dimensionless momentum conservation:
$${\displaystyle \begin{array}{c}\frac{\partial {\rho}^{\ast}\overrightarrow{u^{\ast }}}{\partial {t}^{\ast }}+{\nabla}^{\ast}\cdot \left({\rho}^{\ast}\overrightarrow{u^{\ast }}\otimes \overrightarrow{u^{\ast }}+\overline{\overline{I}}{p}^{\ast}\right)=\\ {}\frac{1}{Re}{\nabla}^{\ast}\cdot \left({\mu}^{\ast }{\nabla}^{\ast}\overrightarrow{u^{\ast }}+{\mu}^{\ast }{\nabla}^{\ast }{\overrightarrow{u^{\ast}}}^T\overline{\overline{I}}\frac{2{\mu}^{\ast }}{3}\nabla \cdot \overrightarrow{u^{\ast }}\right)\end{array}}$$
(33)
dimensionless energy conservation:
$$\frac{\partial{\rho}^{\ast }{T}^{\ast }}{\partial{t}^{\ast }}+{\nabla}^{\ast}\cdot {\left({\rho}^{\ast }{T}^{\ast }\overrightarrow{{u}^{\ast}}\right)}=\frac{1}{Re}\frac{\gamma }{Pr }{\nabla}^{\ast}\cdot \left({k}^{\ast }{\nabla}^{\ast }{T}^{\ast}\right)$$
(34)
\(\frac{\partial \rho }{\rho }={Ma}^2\frac{\partial u}{u}=0,\partial \rho =0,\rho = cte,{\rho}^{\ast }= cte=\frac{\rho }{\rho_{\infty }}=1\), here ∞ represents the character physical variable in the near wall stagnation point regime, where M_{δ} ≈ 0.01.
dimensionless mass conservation:
$$0+{\nabla}^{\ast}\cdot \overrightarrow{u^{\ast }}=0$$
(35)
dimensionless momentum conservation:
$${\displaystyle \begin{array}{c}\frac{\partial \overrightarrow{u^{\ast }}}{\partial {t}^{\ast }}+{\nabla}^{\ast}\cdot \left(\overrightarrow{u^{\ast }}\otimes \overrightarrow{u^{\ast }}+\overline{\overline{I}}{p}^{\ast}\right)=\\ {}\frac{1}{Re}{\nabla}^{\ast}\cdot \left(\frac{\mu^{\ast }}{\rho^{\ast }}{\nabla}^{\ast}\overrightarrow{u^{\ast }}+\frac{\mu^{\ast }}{\rho^{\ast }}{\nabla}^{\ast }{\overrightarrow{u^{\ast}}}^T\overline{\overline{I}}\frac{2{\mu}^{\ast }}{3}\nabla \cdot \overrightarrow{u^{\ast }}\right)\end{array}}$$
(36)
dimensionless energy conservation:
$$\frac{\partial{T}^{\ast }}{\partial{t}^{\ast }}+{\nabla}^{\ast}\cdot {\left({T}^{\ast }\overrightarrow{{u}^{\ast}}\right)}=\frac{1}{Re}\frac{\gamma }{Pr }{\nabla}^{\ast}\cdot \left(\frac{k^{\ast }}{\rho^{\ast }}{\nabla}^{\ast }{T}^{\ast}\right)$$
(37)
Therefore, the energy equation becomes the heat convection and conduction equation, and its influence on the dynamic equation alters the dynamics viscosity \(\frac{\mu^{\ast }}{\rho^{\ast }}\), which is equivalent to the density stratification effect in incompressible fluid. Temperatureinduced density stratification effect is beyond the scope of current study. In general, in near wall stagnation point regime with extreme high temperature, M_{δ }~ 0, the complete compressible fluid dynamic equations are retrograded into incompressible equations.