High-Resolution Viscous Terms Discretization and ILW Solid Wall Boundary Treatment for the Navier–Stokes Equations

Abstract

Robust numerical methods for CFD applications, such as WENO schemes, quickly evolved in the past few decades. Together with the Inverse Lax–Wendroff (ILW) procedure, WENO ideas were also applied in the boundary treatment. Those methods are known for their high-resolution property, i.e., good representation of nonlinear phenomena, which is an important property in solving challenging engineering problems. In light of that, the objective of this work is to present a review of well-established high-resolution numerical methods to solve the Euler equations and adapt the Navier–Stokes viscous terms discretization and boundary treatment. To test the modifications, we employed the positivity-preserving Lax–Friedrichs splitting, multi-resolution WENO scheme, third-order strong stability preserving Runge–Kutta time discretization, and ILW boundary treatment. The first problems were simple flows with analytical solutions for accuracy tests. We also tested the accuracy with nontrivial phenomena in the vortex flow. Oblique shock and complicated flow structures were captured in the Rayleigh–Taylor instability and flow past a cylinder. We showed the discretization and boundary treatment can handle non-constant viscosity, are high-order, high-resolution, and behave similarly to the well-established numerical methods. Furthermore, the methods discussed here can preserve symmetry and no approximations regarding the boundary layer were made. Therefore, the discretization and boundary treatment can be considered when solving direct numerical simulations.

Appendix. Matrices and Vectors for the Rewritten Navier–Stokes Equations

To rewrite the Navier–Stokes equations, we start expanding $\varvec{S_1}$ and $\varvec{S_2}$

$$\begin{aligned}&\begin{aligned} & ({S_1}_x)_1=0,\\ &({S_1}_x)_2=\mu _x\left( \frac{4}{3}u_x-\frac{2}{3}v_y\right) +\mu \left( \frac{4}{3}u_{xx}-\frac{2}{3}v_{xy}\right) ,\\ & ({S_1}_x)_3=\mu _x\left( u_y+v_x\right) +\mu \left( u_{xy}+v_{xx}\right) ,\\ & ({S_1}_x)_4=u_x\mu \left( \frac{4}{3}u_x-\frac{2}{3}v_y\right) +v_x\mu \left( u_y+v_x\right) \\ &+\left( \frac{\mu \gamma }{Pr(\gamma -1)}\right) _x\left( \frac{p}{\rho }\right) _x+ u\mu _x\left( \frac{4}{3}u_x-\frac{2}{3}v_y\right) \\ &+ u\mu \left( \frac{4}{3}u_{xx}-\frac{2}{3}v_{xy}\right) +v\mu _x\left( u_y+v_x\right) +v\mu \left( u_{xy}+v_{xx}\right) \\ & +\frac{\mu \gamma }{Pr(\gamma -1)}\left( \frac{p_{xx}}{\rho }+\frac{2p\rho _x^2}{\rho ^3}-\frac{2p_x\rho _x}{\rho ^2}-\frac{p\rho _{xx}}{\rho ^2}\right) , \\ \end{aligned} \end{aligned}$$

(A.1)

$$\begin{aligned}& \begin{aligned} & ({S_2}_y)_1=0,\\ &({S_2}_y)_2=\mu _y\left( u_y+v_x\right) +\mu \left( u_{yy}+v_{xy}\right) ,\\ & ({S_2}_y)_3=\mu _y\left( \frac{4}{3}v_y-\frac{2}{3}u_x\right) +\mu \left( \frac{4}{3}v_{yy}-\frac{2}{3}u_{xy}\right) ,\\ & ({S_2}_y)_4=u_y\mu \left( u_y+v_x\right) +v_y\mu \left( \frac{4}{3}v_y-\frac{2}{3}u_x\right) \\ & + \left( \frac{\mu \gamma }{Pr(\gamma -1)}\right) _y\left( \frac{p}{\rho }\right) _y+u\mu _y\left( u_y+v_x\right) \\ &+ u\mu \left( u_{yy}+v_{xy}\right) +v\mu _y\left( \frac{4}{3}v_y-\frac{2}{3}u_x\right) +v\mu \left( \frac{4}{3}v_{yy}-\frac{2}{3}u_{xy}\right) \\ &+ \frac{\mu \gamma }{Pr(\gamma -1)}\left( \frac{p_{yy}}{\rho }+\frac{2p\rho _y^2}{\rho ^3}-\frac{2p_y\rho _y}{\rho ^2}-\frac{p\rho _{yy}}{\rho ^2}\right) . \end{aligned} \end{aligned}$$

(A.2)

One should notice that we did not consider $\mu$ and $Pr$ as constants nor remove any terms. We now group terms containing first and second derivatives to the primitive variables, and nonlinear terms separately

$$\begin{aligned} \varvec{S_1}_x= & {} \varvec{\psi _1}{\varvec{W}}_{xy} +\varvec{\psi _2}{\varvec{W}}_{xx}+\varvec{N_{w1}},\end{aligned}$$

(A.3)

$$\begin{aligned} \varvec{S_2}_y= & {} \varvec{\psi _3}{\varvec{W}}_{xy} +\varvec{\psi _4}{\varvec{W}}_{yy}+\varvec{N_{w2}}, \end{aligned}$$

(A.4)

where

$$\begin{aligned} \varvec{\psi _1}= & {} \begin{bmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} -2\mu /3 &{} 0\\ 0 &{} \mu &{} 0 &{} 0\\ 0 &{} \mu v &{} -2\mu u/3 &{} 0\\ \end{bmatrix},\end{aligned}$$

(A.5)

$$\begin{aligned} \varvec{\psi _2}= & {} \begin{bmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 4\mu /3 &{} 0 &{} 0\\ 0 &{} 0 &{} \mu &{} 0\\ -\mu a^2/[\rho Pr(\gamma -1)] &{} 4\mu u/3 &{} \mu v &{} \mu \gamma /[\rho Pr(\gamma -1)] \end{bmatrix},\end{aligned}$$

(A.6)

$$\begin{aligned} \varvec{\psi _3}= & {} \begin{bmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} \mu &{} 0\\ 0 &{} -2\mu /3 &{} 0 &{} 0\\ 0 &{} -2\mu v/3 &{} \mu u &{} 0\\ \end{bmatrix},\end{aligned}$$

(A.7)

$$\begin{aligned} \varvec{\psi _4}= & {} \begin{bmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} \mu &{} 0 &{} 0\\ 0 &{} 0 &{} 4\mu /3 &{} 0\\ -\mu a^2/[\rho Pr(\gamma -1)] &{} \mu u &{} 4\mu v/3 &{} \mu \gamma /[\rho Pr(\gamma -1)] \end{bmatrix},\end{aligned}$$

(A.8)

$$\begin{aligned}&\begin{aligned} &(N_{w1})_1=0,\\ &(N_{w1})_2=\mu _x\left( \frac{4}{3}u_x-\frac{2}{3}v_y\right) ,\\ &(N_{w1})_3=\mu _x\left( u_y+v_x\right) ,\\ & (N_{w1})_4=u_x\mu \left( \frac{4}{3}u_x-\frac{2}{3}v_y\right) +v_x\mu \left( u_y+v_x\right) \\ &+\left( \frac{\mu \gamma }{Pr(\gamma -1)}\right) _x\left( \frac{p}{\rho }\right) _x+u\mu _x\left( \frac{4}{3}u_x-\frac{2}{3}v_y\right) \\ & + v\mu _x\left( u_y+v_x\right) +\frac{\mu \gamma }{Pr(\gamma -1)}\left( \frac{2p\rho _x^2}{\rho ^3}-\frac{2p_x\rho _x}{\rho ^2}\right) ,\\ \end{aligned}\end{aligned}$$

(A.9)

$$\begin{aligned}&\begin{aligned} & (N_{w2})_1=0,\\ &(N_{w2})_2=\mu _y\left( u_y+v_x\right) ,\\ &(N_{w2})_3=\mu _y\left( \frac{4}{3}v_y-\frac{2}{3}u_x\right) ,\\ &(N_{w2})_4=u_y\mu \left( u_y+v_x\right) +v_y\mu \left( \frac{4}{3}v_y-\frac{2}{3}u_x\right) \\ &+\left( \frac{\mu \gamma }{Pr(\gamma -1)}\right) _y\left( \frac{p}{\rho }\right) _y+u\mu _y\left( u_y+v_x\right) \\& + v\mu _y\left( \frac{4}{3}v_y-\frac{2}{3}u_x\right) +\frac{\mu \gamma }{Pr(\gamma -1)}\left( \frac{2p\rho _y^2}{\rho ^3}-\frac{2p_y\rho _y}{\rho ^2}\right) . \end{aligned} \end{aligned}$$

(A.10)

The boundary treatment is based on conservative variables, we then transform to the latter with

$$\begin{aligned}&\begin{aligned} \varvec{S_1}_x & =\varvec{\psi _1}\left[ {\varvec{M}}{\varvec{U}}_y+\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}{\varvec{U}}_{xy}\right] \\ & \quad +\varvec{\psi _2}\left[ {\varvec{M}}{\varvec{U}}_x+\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}{\varvec{U}}_{xx}\right] +\varvec{N_{w1}}, \end{aligned}\end{aligned}$$

(A.11)

$$\begin{aligned}&\begin{aligned} \varvec{S_2}_y & =\varvec{\psi _3}\left[ {\varvec{O}}{\varvec{U}}_x+\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}{\varvec{U}}_{xy}\right] \\ &\quad +\varvec{\psi _4}\left[ {\varvec{O}}{\varvec{U}}_y+\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}{\varvec{U}}_{yy}\right] +\varvec{N_{w2}}, \end{aligned}\end{aligned}$$

(A.12)

$$\begin{aligned}&\begin{aligned} {\varvec{M}}=\begin{bmatrix} 0 &{} 0 &{} 0 &{} 0\\ \displaystyle \frac{\rho _x u-\rho u_x}{\rho ^2} &{} \displaystyle -\frac{\rho _x}{\rho ^2} &{} 0 &{} 0 \\ \displaystyle \frac{\rho _x v-\rho v_x}{\rho ^2} &{} 0 &{} \displaystyle -\frac{\rho _x}{\rho ^2} &{} 0 \\ \left( uu_x+vv_x\right) (\gamma -1) &{} -u_x(\gamma -1) &{} -v_x(\gamma -1) &{} 0 \\ \end{bmatrix},\\ {\varvec{O}}=\begin{bmatrix} 0 &{} 0 &{} 0 &{} 0\\ \frac{\rho _y u-\rho u_y}{\rho ^2} &{} \displaystyle -\frac{\rho _y}{\rho ^2} &{} 0 &{} 0 \\ \frac{\rho _y v-\rho v_y}{\rho ^2} &{} 0 &{} \displaystyle -\frac{\rho _y}{\rho ^2} &{} 0 \\ \left( uu_y+vv_y\right) (\gamma -1) &{} -u_y(\gamma -1) &{} -v_y(\gamma -1) &{} 0 \\ \end{bmatrix}. \end{aligned} \end{aligned}$$

(A.13)

We finally write the viscous terms as

$$\begin{aligned} \varvec{S_1}_x= & \, \varvec{\psi _1}\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}{\varvec{U}}_{xy} +\varvec{\psi _2}\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}{\varvec{U}}_{xx}+\varvec{N_{1}}, \end{aligned}$$

(A.14)

$$\begin{aligned} \varvec{S_2}_y= & \, \varvec{\psi _3}\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}{\varvec{U}}_{xy} +\varvec{\psi _4}\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}{\varvec{U}}_{yy}+\varvec{N_{2}}, \end{aligned}$$

(A.15)

with

$$\begin{aligned} \varvec{N_1}= & \, \varvec{\psi _1}{\varvec{M}}{\varvec{U}}_y+\varvec{\psi _2}{\varvec{M}}{\varvec{U}}_x+\varvec{N_{w1}}, \nonumber \\ \varvec{N_2}= & {} \varvec{\psi _3}{\varvec{O}}{\varvec{U}}_x+\varvec{\psi _4}{\varvec{O}}{\varvec{U}}_y+\varvec{N_{w2}} . \end{aligned}$$

(A.16)

Introducing four new terms, we write

$$\begin{aligned} \varvec{S_1}_x+\varvec{S_2}_y= \, \varvec{\Psi _1}{\varvec{U}}_{xx} +\varvec{\Psi _2}{\varvec{U}}_{yy}+\varvec{\Psi _3}{\varvec{U}}_{xy} +{\varvec{N}} \end{aligned}$$

(A.17)

with

$$\begin{aligned} \varvec{\Psi _1}= & \, \varvec{\psi _2}\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}, \end{aligned}$$

(A.18)

$$\begin{aligned} \varvec{\Psi _2}= & \, \varvec{\psi _4}\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}, \end{aligned}$$

(A.19)

$$\begin{aligned} \varvec{\Psi _3}= & \, \varvec{\psi _1}\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}} +\varvec{\psi _3}\frac{\partial {\varvec{W}}}{\partial {\varvec{U}}}, \end{aligned}$$

(A.20)

$$\begin{aligned} {\varvec{N}}= & \, \varvec{N_1}+\varvec{N_2}. \end{aligned}$$

(A.21)

To apply the wall boundary treatment, we need to diagonalize the matrix $\varvec{\Psi _2}$. We choose the scaling factors in a way that the resulting eigenvectors are similar to those employed in [2], i.e.,

$$\begin{aligned} {\varvec{L}}_d= & {} \begin{bmatrix} \dfrac{1}{2\gamma } &{} 0 &{} 0 &{} 0 \\ -u &{} 1 &{} 0 &{} 0 \\ \dfrac{v}{2a} &{} 0 &{} -\dfrac{1}{2a} &{} 0 \\ \dfrac{q(\gamma -1)}{2a^2}-\dfrac{1}{2\gamma } &{} -\dfrac{u\left( \gamma -1\right) }{2a^2} &{} -\dfrac{v\left( \gamma -1\right) }{2a^2} &{} \dfrac{\gamma -1}{2a^2} \\ \end{bmatrix},\end{aligned}$$

(A.22)

$$\begin{aligned} {\varvec{R}}_d= & {} \begin{bmatrix} 2\gamma &{} 0 &{} 0 &{} 0\\ 2u\gamma &{} 1 &{} 0 &{} 0\\ 2v\gamma &{} 0 &{} -2a &{} 0\\ 2q\gamma +\dfrac{2a^2}{\gamma -1} &{} u &{} -2av &{} \dfrac{2a^2}{\gamma -1}\\ \end{bmatrix}, \end{aligned}$$

(A.23)

with $q=(u^2+v^2)/2$.