A High-Order Hybrid Numerical Scheme for Hypersonic Flow Over A Blunt Body

Abstract

A hybrid scheme is developed for direct numerical simulations of hypersonic flows over a blunt body. The scheme switches to the first-order AUSMPW+ scheme near the bow shock to provide sufficient dissipation to handle the carbuncle phenomenon. In the smooth part of the computational domain, a sixth-order central scheme with an eighth-order low-pass filter is adopted to provide high spatial accuracy to resolve turbulence. The hybrid scheme is shown to be able to obtain smooth and accurate predictions of laminar hypersonic flows over a blunt body. Using the hybrid scheme, a direct numerical simulation of a Mach 6 hypersonic flow over a circular cylinder is conducted. The result shows the turbulent structures in the near-wall region are well resolved by the hybrid scheme, and the bow shock is also captured without introducing any numerical oscillations. With the boundary layer transition on the cylinder’s surface, the simulation indicates that the heat flux peak shifts from the stagnation point to the transitional zone and its peak value is increased by 50$\%$.

Appendices

Appendix A: The Steger-Warming Flux Splitting Scheme

For the Steger-Warming scheme, the Euler flux vector, $\widehat{{\mathbf {E}}}$, is split as,

$$\begin{aligned} \widehat{{\mathbf {E}}}=\widehat{\mathbf{E}}^{+}+\widehat{{\mathbf {E}}}^{-}, \end{aligned}$$

(A1)

where $\widehat{{\mathbf {E}}}^{+}$ and $\widehat{{\mathbf {E}}}^{-}$ are respectively the forward and backward flux vectors, and determined by the local Mach number, ${\tilde{M}}$, which is calculated as,

$$\begin{aligned} {\widetilde{M}}=\frac{U}{c_a}, \end{aligned}$$

(A2)

where $U=u_i \xi _{x_i}$, $c_a=\frac{\sqrt{T} \nabla \xi }{M}$, and $\nabla \xi =\sqrt{\xi ^2_x+\xi ^2_y+\xi ^2_z}$. The positive and negative flux vector, $\widehat{{\mathbf {E}}}^{\pm }$ is given as,

$$\begin{aligned} \left\{ \begin{array}{ll} \widehat{{\mathbf {E}}}^+=\widehat{{\mathbf {E}}};\widehat{{\mathbf {E}}}^- =0 &{} ,\mathrm{for} \quad {\widetilde{M}}\ge 1 \\ \widehat{{\mathbf {E}}}^+ =0;\widehat{{\mathbf {E}}}^- =\widehat{{\mathbf {E}}} &{} ,\mathrm{for} \quad {\widetilde{M}}\le -1\\ \widehat{{\mathbf {E}}}^{\pm } = J\dfrac{\rho }{2\gamma }\left[ \begin{array}{l} 2(\gamma -1){\tilde{\lambda }}^{\pm }_1+{\tilde{\lambda }}^{\pm }_4+{\tilde{\lambda }}^{\pm }_5 \\ 2(\gamma -1){\tilde{\lambda }}^{\pm }_1u+{\tilde{\lambda }}^{\pm }_4(u+c{\tilde{\xi }}_x)+{\tilde{\lambda }}^{\pm }_5(u-c{\tilde{\xi }}_x)\\ 2(\gamma -1){\tilde{\lambda }}^{\pm }_1v+{\tilde{\lambda }}^{\pm }_4(v+c{\tilde{\xi }}_y)+{\tilde{\lambda }}^{\pm }_5(v-c{\tilde{\xi }}_y)\\ 2(\gamma -1){\tilde{\lambda }}^{\pm }_1w+{\tilde{\lambda }}^{\pm }_4(w+c{\tilde{\xi }}_z)+{\tilde{\lambda }}^{\pm }_5(w-c{\tilde{\xi }}_z)\\ (\gamma -1){\tilde{\lambda }}^{\pm }_1(u^2+v^2+w^2)+ \\ \tfrac{1}{2}{\tilde{\lambda }}^{\pm }_4\left[ \big (u+c{\tilde{\xi }}_x \big )^2+\big (v+c{\tilde{\xi }}_y \big )^2+\big (w+c{\tilde{\xi }}_z \big )^2 \right] +\\ \tfrac{1}{2}{\tilde{\lambda }}^{\pm }_5\left[ \big (u-c{\tilde{\xi }}_x \big )^2+\big (v-c{\tilde{\xi }}_y \big )^2+\big (w-c{\tilde{\xi }}_z \big )^2 \right] +W+P\\ \end{array} \right] &{} ,\mathrm{for} \\ \quad -1< {\widetilde{M}} < 1 \end{array} \right. \end{aligned}$$

(A3)

where $W=\frac{(3-\gamma )({\tilde{\lambda }}^{\pm }_4+{\tilde{\lambda }}^{\pm }_5)c^2}{2 (\gamma -1)}$, $P=2p(\gamma -1){\tilde{\lambda }}^{\pm }_1 {\tilde{\xi }}_x( {\tilde{\xi }}_y w- {\tilde{\xi }}_z v)$, ${\tilde{\xi }}_{x_i}= \frac{\xi _{x_i}}{\nabla \xi }$, the local speed of sound is $c=\frac{\sqrt{T}}{M}$, and ${\tilde{\lambda }}^{\pm }_i$, is calculated by,

$$\begin{aligned} {\tilde{\lambda }}^{\pm }_i=\frac{\lambda _i\pm \sqrt{\lambda _i^2+\epsilon ^2} }{2}, \end{aligned}$$

(A4)

with $\epsilon =0.04$. The eigenvalues of the Jacobian matrix of the Euler flux vector, $\lambda _i$, is given as,

$$\begin{aligned} \lambda _1 = \lambda _2 = \lambda _3 = U, \lambda _4 = U+c_a, \lambda _5 = U-c_a. \end{aligned}$$

(A5)

After the split of the Euler flux vector at each grid’s node, the positive and negative fluxes at the interface location between the $i^{th}$ and $i+1^{th}$ nodes, $\widehat{{\mathbf {E}}}_{i+1/2}^+$ and $\widehat{{\mathbf {E}}}_{i+1/2}^-$, can be reconstructed by using a upwind-biased scheme using values from nodes within its stencil. The total flux vector, $\widehat{{\mathbf {E}}}_{i+1/2}$, is then obtained through Eq.(A1). Finally, the convectional term in Eq. (1) at the $i^{th}$ can be calculated as,

$$\begin{aligned} \frac{\partial \widehat{{\mathbf {E}}}}{\partial \xi } _i= \widehat{{\mathbf {E}}}_{i+1/2} - \widehat{{\mathbf {E}}}_{i-1/2}. \end{aligned}$$

(A6)

Appendix B: The AUSMPW+ Flux Splitting Scheme

For the AUSMPW+, the flow variables at left and right sides of a node’s interface are first reconstructed using upwind-biased and downwind-biased schemes, respectively, and the final flux vector is then calculated using the AUSMPW+ scheme by using the left and right variables as,

$$ \widehat{{\mathbf{E}}} = c_{{1/2}} (\overline{{\text{M}}} _{L}^{ + } {\mathbf{\Phi }}_{L} + \overline{{\text{M}}} _{R}^{ - } {\mathbf{\Phi }}_{R} ) + (P^{ + } {\mathbf{P}}_{L} + P^{ - } {\mathbf{P}}_{R} ), $$

(B7)

where $c_{1/2}$ is the speed of sound at the node’s interface, and ${\mathbf \Phi }_{L,R}$ and the pressure vectors ${{\mathbf {P}}}_{L,R}$ are expressed as,

$$\begin{aligned} {\mathbf \Phi }_{L,R} = \left| \nabla \xi \right| _{L,R} \left[ \begin{array}{c} \rho \\ \rho u\\ \rho v\\ \rho w\\ \rho H\\ \end{array}\right] \!\!_{L,R} , \qquad {{\mathbf {P}}}_{L,R}= J_{L,R} \left[ \begin{array}{c} 0\\ \xi _x p\\ \xi _y p\\ \xi _z p\\ 0\\ \end{array} \right] _{L,R}. \end{aligned}$$

(B8)

The enthalpy, H, is defined as,

$$\begin{aligned} H=\frac{E+p}{\rho }. \end{aligned}$$

(B9)

The subscripts, L, R, stand for variables at the left and right sides of a node’s interface, calculated by the upwind-biased and downwind-biased schemes, respectively. It is noted that the coordinate transformation matrix, $\xi _i$, and its Jacobian, J, are also reconstructed onto the node’s interface to preserve the geometric conservation.

For $m_{1/2}=M^+ + M^- \ge 0$,

$$ \begin{aligned} \overline{{\text{M}}} _{L}^{ + } & = M^{ + } + M^{ + } [(1 - w)(1 + f_{R} ) - f_{L} ] \\ \overline{{\text{M}}} _{R}^{ - } & = M^{ - } w(1 + f_{R} ), \\ \end{aligned} $$

for $m_{1/2} < 0$,

$$ \begin{aligned} \overline{{\text{M}}} _{L}^{ + } & = M^{ + } w(1 + f_{L} ) \\ \overline{{\text{M}}} _{R}^{ - } & = M^{ - } + M^{ - } [(1 - w)(1 + f_{L} ) - f_{R} ], \\ \end{aligned} $$

with,

$$\begin{aligned} w(p_L,p_R)=1-\mathrm{min}\left( \frac{p_L}{p_R}, \frac{p_R}{p_L}\right) ^3, \end{aligned}$$

(B10)

and

$$\begin{aligned} f_{L,R}=\left\{ \begin{array}{ll} \left( \frac{p_{L,R}}{p_s} - 1 \right) \mathrm{min}\left( 1, \frac{\mathrm{min}(p_{1L},p_{1R},p_{2L},p_{2R})}{\mathrm{min}(p_L,p_R)} \right) ^2 &{},\quad p_s \ne 0\\ 0 &{}, \mathrm{elsewhere} \end{array} \right. \end{aligned}$$

(B11)

where $p_s=P^+ p_L+P^- p_R$, $p_1$ and $p_2$ are the pressure from neighboring nodes in the $\eta$ direction.

The Mach number and pressure splitting functions are

$$\begin{aligned} M^{\pm }=\left\{ \begin{array}{ll} \pm \tfrac{1}{4}(M\pm 1)^2 &{},\vert M\vert \le 1,\\ \tfrac{1}{2}(M\pm \vert M\vert )^2 &{}, \vert M\vert > 1, \end{array} \right. \end{aligned}$$

(B12)

where M is the local Mach number.

$$\begin{aligned} P^{\pm }\vert _\alpha =\left\{ \begin{array}{ll} \pm \tfrac{1}{4}(M\pm 1)^2 (2\mp M)\pm \alpha M(M^2-1)^2 &{},\vert M \vert \le 1,\\ \tfrac{1}{2}(1\pm \mathrm{sign}(M)) &{}, \vert M\vert > 1. \end{array} \right. \end{aligned}$$

(B13)

In the present study, $\alpha$ is set to 0, and the average speed of sound $c_{1/2}$ is calculated by,

$$\begin{aligned} c_{1/2}=\left\{ \begin{array}{cc} c^2_s/\mathrm{max}(\vert U_L\vert ,c_s), &{} \mathrm{for} \quad \tfrac{1}{2}(U_L+U_R)>0,\\ c^2_s/\mathrm{max}(\vert U_R\vert ,c_s), &{} \mathrm{for} \quad \tfrac{1}{2}(U_L+U_R)<0, \end{array} \right. \end{aligned}$$

(B14)

where,

$$\begin{aligned} U_{L,R} = (u\xi _x + v\xi _y + w\xi _z)_{L,R}/{\left| \nabla \xi \right| }_{L,R}, \\ V_{L,R} = \sqrt{(u^2+v^2+w^2)_{L,R}-U^2_{L,R}},\\ c_s = \sqrt{\tfrac{1}{2}(H_L - \tfrac{1}{2}V^2_L + H_R - \tfrac{1}{2}V^2_R)}.\\ \end{aligned}$$

Appendix C: Reconstruction Scheme of WENO7

For the WENO7 reconstruction,the positive flux at the interface, $f_{i+1/2}^+$, is reconstructed from sub-stencils as,

$$\begin{aligned} f_{i+1/2}^+ = \sum ^3_{k=0}\omega _k q_k, \end{aligned}$$

(C15)

in which the reconstruction from each sub-stencils, $q_k$, is give as,

$$\begin{aligned} q_0 =&\tfrac{1}{12}(-3f_{i-3} + 13f_{i-2} - 23f_{i-1} + 25f_{i} ),\\ q_1 =&\tfrac{1}{12}(f_{i-2} -5f_{i-1} +13f_{i} + 3f_{i+1} ),\\ q_2 =&\tfrac{1}{12}(-f_{i-1} +7f_{i} + 7f_{i+1} - f_{i+2} ),\\ q_3 =&\tfrac{1}{12}(3f_{i} +13f_{i+1} -5f_{i+2} + f_{i+3} ),\\ \end{aligned}$$

and the weight, $\omega _k$, is expressed as,

$$\begin{aligned} \omega _k=\frac{\alpha _k}{\alpha _0+\alpha _1+\alpha _2+\alpha _3}, \alpha _k = \frac{C_k}{(\epsilon + IS_k)^2}, k=0,1,2,3. \end{aligned}$$

(C16)

In the expression of $\alpha _k$, $C_k$ and $IS_k$, are respectively, the optimal weights and smoothness estimators to ensure the WENO scheme avoids reconstruction using a sub-stencil containing a discontinuity, and returns to the optimal linear scheme in the smooth region. In Eq. C16, $\epsilon$ is set to $10^{-10}$ to avoid a zero denominator. The smoothness estimators $IS_k$ are calculated as,

$$\begin{aligned} IS_0 = &f_{i-3}(547f_{i-3}-3882f_{i-2} + 4642f_{i-1} - 1854f_i) + f_{i-2}(7043f_{i-2}-17246f_{i-1} \\&+7042f_i)+ f_{i-1}(11003f_{i-1}-9402f_i)+2107f^2_i, \\ IS_1 = &f_{i-2}(267f_{i-2}-1642f_{i-1} + 1602 f_{i} - 494 f_{i+1}) + f_{i-1}(2843 f_{i-1}-5966 f_{i} \\&+1922 f_{i+1})+ f_{i}(3443 f_{i}-2522 f_{i+1})+ 547 f^2_{i+1}, \\ IS_2 = &f_{i-1}(547 f_{i-1} - 2522 f_{i} + 1922 f_{i+1} - 494 f_{i+2}) + f_{i}(3443 f_{i}-5966 f_{i+1} \\&+1602 f_{i+2})+ f_{i+1}(2843 f_{i+1}-1642 f_{i+2})+ 267 f^2_{i+2}, \\ IS_3 = &f_{i}(2107 f_{i} - 9402 f_{i+1} + 7042 f_{i+2} - 1854 f_{i+3}) + f_{i+1}(11003 f_{i+1}-17246 f_{i+2} \\&+4642 f_{i+3})+ f_{i+2}(7043 f_{i+2}-3882 f_{i+3})+ 547 f^2_{i+3}, \\ \end{aligned}$$

and the optimal weights, $C_k$, are give as,

$$\begin{aligned} C_0= \frac{1}{35},\quad C_1=\frac{12}{35},\quad C_2=\frac{18}{35},\quad C_3=\frac{4}{35}. \end{aligned}$$

For a reconstruction in a fully smooth region, the WENO7 scheme will have $\omega _k=C_k$, and the scheme returns to the standard seventh-order linear upwind scheme.

Appendix D: Reconstruction Scheme of MP7

The MP reconstruction has an explicit linear part and a nonlinear part. For the reconstruction of $f_{i+1/2}^+$, the following procedures are conducted in turn,

• (i) The original interface value $f^L_{i+1/2}$ is calculated by the standard seventh-order linear upwind scheme as,

  $$\begin{aligned} f^L_{i+1/2}= \frac{1}{420}(-3f_{i-3} + 25f_{i-2} - 101f_{i-1} + 319f_{i} + 214f_{i+1} -38f_{i+2} + 4f_{i+3}). \end{aligned} $$

  (D17)

• (ii) The smooth regions are determined by the following criterion,

  $$\begin{aligned} \big (f^L_{i+1/2}-f_i \big )\big (f^L_{i+1/2}-f^{MP} \big ) \le 10^{-10}, \end{aligned}$$

  (D18)

  where,

  $$\begin{aligned} f^{MP} = f_i + \mathrm{minmod}\left[ (f_{i+1} - f_i), 4(f_i - f_{i-1}) \right] , \end{aligned}$$

  (D19)

  and the minmod function is defined as,

  $$\begin{aligned} \mathrm{minmod}(z_1,z_2,\cdots ,z_k)=s \cdot \mathrm{min}(\vert z_1\vert ,\vert z_2\vert ,\cdots ,\vert z_k\vert ),\\ s=\tfrac{1}{2}[\mathrm{sgn}(z_1) + \mathrm{sgn}(z_2)] \cdot \vert \tfrac{1}{2}[\mathrm{sgn}(z_1) + \mathrm{sgn}(z_3)]\cdot \cdots \cdot \tfrac{1}{2}[\mathrm{sgn}(z_1) + \mathrm{sgn}(z_k)]\vert , \\ \mathrm{sgn}(z_k)=\left\{ \begin{array}{ll} 1 &{} \mathrm{if}z_k>0 \\ 0 &{} \mathrm{if}z_k=0 \\ -1&{} \mathrm{if}z_k<0 \end{array}. \right. \end{aligned}$$

  For the smooth region, the final interface’s value is equal to the value from the linear scheme as,

  $$\begin{aligned} f_{i+1/2}^+ = f^L_{i+1/2}. \end{aligned}$$

  (D20)

• (iii) The MP limiter is applied to the original interface value $f^L_{i+1/2}$, when Eq. D18 is not satisfied. The MP limiter is expressed as follows,

  $$\begin{aligned} {\left\{ \begin{array}{ll} d_i=f_{i-1}-2f_i+f_{i+1} \\ d^{M4}_{i+1/2} = \mathrm{minmod}[(4d_i-d_{i+1}),(4d_{i+1} - d_i) , d_i , d_{i+1}]\\ f^{UL} = f_i + 4(f_i - f_{i-1}) \\ f^{AV} = (f_i + f_{i+1}) / 2 \\ f^{MD} = f^{AV} - \tfrac{1}{2}d^{M4}_{i+1/2} \\ f^{LC} = f_i + \tfrac{1}{2}(f_i - f_{i-1}) + \tfrac{4}{3}d^{M4}_{i-1/2} \\ f^{\mathrm{min}} = \mathrm{max}[\mathrm{min}(f_i, f_{i+1}, f^{MD}) , \mathrm{min}(f_i, f^{UL},f^{LC}) ] \\ f^{\mathrm{max}} = \mathrm{min}[\mathrm{max}(f_i, f_{i+1}, f^{MD}), \mathrm{max}(f_i,f^{UL},f^{LC}) ] \end{array}\right. }. \end{aligned}$$

  (D21)

  The final interface’s value is obtained as,

  $$\begin{aligned} f_{i+1/2}^+ = \mathrm{median}\big (f^L_{i+1/2}, f^\mathrm{min}, f^{\mathrm{max}} \big ), \end{aligned}$$

  (D22)

  where the function $\mathrm{median}(x,y,z)$ is defined as,

  $$\begin{aligned} \mathrm{median}(x,y,z)=x+\mathrm{minmod}(y-x,z-x). \end{aligned}$$