Reformulated Dissipation for the Free-Stream Preserving of the Conservative Finite Difference Schemes on Curvilinear Grids

Abstract

In this paper, we develop a new free-stream preserving (FP) method for high-order upwind conservative finite-difference (FD) schemes on curvilinear grids. This FP method is constructed by subtracting a reference cell-face flow state from each cell-center value in the local stencil of the original upwind schemes, which effectively leads to a reformulated dissipation. It is convenient to implement this method, as it only approximates the cell-center fluxes and conservative variables before reconstructions rather than performs the FP techniques for the central and dissipation parts individually, which avoids introducing considerable complexities to the original reconstruction procedures. In addition, the proposed method removes the constraint in the traditional FP conservative FD schemes that require a consistent scheme for the metrics discretization and the central part of fluxes discretization. With this, the proposed method is more flexible in simulating the engineering problems which usually require a low-order scheme for their low-quality mesh, while the high-order schemes can be applied to approximate the flow states to improve the resolution. After demonstrating the strict FP property and the order of accuracy by two simple test cases, we consider various validation cases, including the supersonic flow around the cylinder, the subsonic flow past the three-element airfoil, and the transonic flow around the ONERA M6 wing, etc., to show that the method is suitable for a wide range of fluid dynamic problems containing complex geometries.

Appendix

The cell-face numerical fluxes $\varvec{ \tilde{{\mathcal {F}}} }_{i+1/2}^{O}$ reconstructed by the original WENO5 scheme can be expressed by the summation of the central and dissipation part [23, 35]. Substituting Eq. [SPSeqref7eqrefSPS] into Eq. (25), and then into Eq. (20) gives

$$\begin{aligned} \begin{aligned} \varvec{ \tilde{{\mathcal {F}}} }_{i+1/2}^{O}=&{\varvec{R}}\left( \varvec{ \tilde{{\mathcal {F}}} }_{i+1/2}^{+}+\varvec{ \tilde{{\mathcal {F}}} }_{i+1/2}^{-}\right) \\ =&\dfrac{1}{60} \left( \varvec{\tilde{F}}_{i-2} -8 \varvec{\tilde{F}}_{i-1}+37 \varvec{\tilde{F}}_{i} +37\varvec{\tilde{F}}_{i+1}-8 \varvec{\tilde{F}}_{i+2}+\varvec{\tilde{F}}_{i+3} \right) \\&-\dfrac{1}{60} \sum \limits _s {\varvec{R}}^s \left[ \left( 20\omega _0^+-1 \right) {\hat{f}}_{i,1}^{s,+}-\left( 10\omega _0^+ +10\omega _1^+ -5 \right) {\hat{f}}_{i,2}^{s,+} +{\hat{f}}_{i,3}^{s,+}\right] \\&+\dfrac{1}{60} \sum \limits _s {\varvec{R}}^s \left[ \left( 20\omega _0^- -1 \right) {\hat{f}}_{i,1}^{s,-}-\left( 10\omega _0^- +10\omega _1^- -5 \right) {\hat{f}}_{i,2}^{s,-} +{\hat{f}}_{i,3}^{s,-}\right] , \end{aligned} \end{aligned}$$

(66)

where

$$\begin{aligned} \begin{aligned} {\hat{f}}_{i,r+1}^{s,+} =&{\widetilde{f}}_{i+r+1}^{s,+}-3{\widetilde{f}}_{i+r}^{s,+}+3{\widetilde{f}}_{i+r-1}^{s,+}-{\widetilde{f}}_{i+r-2}^{s,+}, \qquad r=0,1,2\\ =&\dfrac{1}{2} {\varvec{L}}^s \left( \varvec{\tilde{F}}_{i+r+1}-3\varvec{\tilde{F}}_{i+r}+3\varvec{\tilde{F}}_{i+r-1}-\varvec{\tilde{F}}_{i+r-2} \right) \\&+\dfrac{1}{2}\lambda ^s {\varvec{L}}^s\left( \varvec{\tilde{Q}}_{i+r+1}-3\varvec{\tilde{Q}}_{i+r}+3\varvec{\tilde{Q}}_{i+r-1}-\varvec{\tilde{Q}}_{i+r-2} \right) . \end{aligned} \end{aligned}$$

(67)

Similarly, the cell-face numerical fluxes $\varvec{ \tilde{{\mathcal {F}}} }_{i+1/2}^{FP}$ obtained with the proposed FP WENO5 scheme can be given by applying the approximations suggested in Eqs. (51) and (52), i.e.

$$\begin{aligned} \begin{aligned}&\varvec{ \tilde{{\mathcal {F}}} }_{i+1/2}^{FP}={\varvec{R}}\left( \varvec{ \tilde{{\mathcal {F}}} }_{i+1/2}^{FP,+}+\varvec{ \tilde{{\mathcal {F}}} }_{i+1/2}^{FP,-}\right) \\&\quad =\dfrac{1}{60} \left( \varvec{\tilde{F}}_{i-2} -8 \varvec{\tilde{F}}_{i-1}+37 \varvec{\tilde{F}}_{i} +37\varvec{\tilde{F}}_{i+1}-8 \varvec{\tilde{F}}_{i+2}+\varvec{\tilde{F}}_{i+3} \right) +\Delta \varvec{\tilde{F}}^*\\&\qquad -\dfrac{1}{60} \sum \limits _s {\varvec{R}}_{i+1/2}^s \left[ \left( 20\omega _0^+-1 \right) {\hat{f}}_{i,1}^{s,+}-\left( 10\omega _0^+ +10\omega _1^+ -5 \right) {\hat{f}}_{i,2}^{s,+} +{\hat{f}}_{i,3}^{s,+}\right] \\&\qquad +\dfrac{1}{60} \sum \limits _s {\varvec{R}}_{i+1/2}^s \left[ \left( 20\omega _0^- -1 \right) {\hat{f}}_{i,1}^{s,-}-\left( 10\omega _0^- +10\omega _1^- -5 \right) {\hat{f}}_{i,2}^{s,-} +{\hat{f}}_{i,3}^{s,-}\right] , \end{aligned} \end{aligned}$$

(68)

where

$$\begin{aligned} \begin{aligned} {\hat{f}}_{i,r+1}^{s,+} =&{\widetilde{f}}_{i+r+1}^{s,+}-3{\widetilde{f}}_{i+r}^{s,+}+3{\widetilde{f}}_{i+r-1}^{s,+}-{\widetilde{f}}_{i+r-2}^{s,+}, \qquad r=0,1,2\\ =&\dfrac{1}{2} {\varvec{L}}_{i+1/2}^s \left( \varvec{\tilde{F}}_{i+r+1}-3\varvec{\tilde{F}}_{i+r}+3\varvec{\tilde{F}}_{i+r-1}-\varvec{\tilde{F}}_{i+r-2} \right) \\&+\dfrac{1}{2}\lambda ^s {\varvec{L}}_{i+1/2}^s\left( \varvec{\tilde{Q}}_{i+r+1}-3\varvec{\tilde{Q}}_{i+r}+3\varvec{\tilde{Q}}_{i+r-1}-\varvec{\tilde{Q}}_{i+r-2} \right) \\&-\dfrac{1}{2} {\varvec{L}}_{i+1/2}^s \left( \varvec{\tilde{F}}_{i+r+1}^{*}-3\varvec{\tilde{F}}_{i+r}^{*}+3\varvec{\tilde{F}}_{i+r-1}^{*}-\varvec{\tilde{F}}_{i+r-2}^{*} \right) \\&-\dfrac{1}{2}\lambda ^s {\varvec{L}}_{i+1/2}^s\left( \varvec{\tilde{Q}}_{i+r+1}^{*}-3\varvec{\tilde{Q}}_{i+r}^{*}+3\varvec{\tilde{Q}}_{i+r-1}^{*}-\varvec{\tilde{Q}}_{i+r-2}^{*} \right) , \end{aligned} \end{aligned}$$

(69)

and $\Delta \varvec{\tilde{F}}^*$ is given by Eq. (60).