Compact Schemes for Multiscale Flows with Cell-Centered Finite Difference Method

Abstract

High-order compact interpolation schemes appropriate for multiscale flows are studied within a cell-centered finite difference method (CCFDM) framework where the robustness of high-order schemes on curvilinear grids can be greatly enhanced due to the satisfaction of geometric conservation law. Two types of compact interpolations are mainly developed in this paper for shock-free flows and shock-embedded flows respectively. The present compact schemes are verified to be superior over the explicit counterparts with same orders in terms of the spectral characteristics. Regarding the shock-free flows, low-dissipation low-dispersion properties are achieved by the spectral optimization. Three optimized compact schemes (Opt4, Opt6 and Opt8) are further validated to be attractive for shock-free problems by carrying out benchmarks from computational aeroacoustics workshops and two typical turbulence cases: Tayler–Green vortex and decaying isotropic turbulence. Regarding high-speed flows in the presence of shock waves, the shock-capturing capability is realized by extending the weighting technique to the compact interpolations. The criteria to choose optimally compact nonlinear sub-stencils on a most general compact global stencil are presented. Interestingly, the explicit WENO-type schemes can be reverted within the proposed compact framework. Three nonlinear compact schemes (UI5, CI6 and CI8) on two practical stencils are analyzed and further compared with their explicit counterparts by a series of numerical experiments. The compact ones are superior to explicit ones in resolving rich flow structures as well as discontinuities.

Appendices

Appendix A: Compact Interpolations

See Tables 11, 12, 13, 14 and 15.

Table 11 Tridiagonal (K = 1) compact central schemes

Table 12 Pentadiagonal (K = 2) compact central schemes

Table 13 Tridiagonal (K = 1) compact upwind schemes

Table 14 Pentadiagonal (K = 2) compact upwind schemes

Table 15 Heptadiagonal (K = 3) compact upwind schemes

Appendix B: Proof of Eq. (23)

Applying Taylor series analysis on Eq. (20) to achieve (2K + 2L)th-order accuracy, the following (2K + 2L) constraints are required:

$$ \begin{aligned} & \underbrace {{ 0^{q} }}_{C} + \underbrace {{\sum\limits_{n = 1}^{K} {n^{q} \left( {\beta_{n} { + }({ - 1})^{q} \beta_{{{ - }n}} } \right)} }}_{A \cdot X} = \underbrace {{\sum\limits_{m = 1}^{L + 1} {\left( {\frac{2m - 1}{ 2}} \right)^{q} \left( {c_{m} { + }({ - 1})^{q} c_{1 - m} } \right)} }}_{B \cdot Y},\\ & \quad q = 0, 1, 2 ,\ldots , 2\left( {K + L} \right) - 1, \end{aligned} $$

(49)

where the column vectors C, X, Y and matrices A, B are

Using block multiplication of matrices, Eq. (49) is written in the following form:

$$ \underbrace {{C_{1} }}_{(K + L) \times 1} + \underbrace {{A_{11} }}_{(K + L) \times K} \, \underbrace {{X_{1} }}_{K \times 1} = \underbrace {{B_{11} }}_{(K + L) \times (L + 1)} \, \underbrace {{Y_{1} }}_{(L + 1) \times 1}. $$

(50)

$$ \underbrace {{C_{2} }}_{(K + L) \times 1} + \underbrace {{A_{22} }}_{(K + L) \times K} \, \underbrace {{X_{2} }}_{K \times 1} = \underbrace {{B_{22} }}_{(K + L) \times (L + 1)} \, \underbrace {{Y_{2} }}_{(L + 1) \times 1}. $$

(51)

There are (K + L) equations but (K + L + 1) unknowns in each equation of Eqs. (50) and (51), which means both Eqs. (50) and (51) are underdetermined. In Eq. (50), the first unknown element in $X_{1}$ is $\beta_{1} { + }\beta_{ - 1}$. Assuming $\beta_{1} { + }\beta_{ - 1} = \eta$, all the entries in $X_{1}$ and $Y_{1}$ are dependent variables of $\eta$. Similarly, in Eq. (51), the first unknown element in $X_{2}$ is $\beta_{1} - \beta_{ - 1}$. Assuming $\beta_{1} - \beta_{ - 1} = \xi$, the other entries in $X_{2}$ and $Y_{2}$ are dependent variables of $\xi$. These yield:

$$ \left\{ {\begin{array}{*{20}l} {c_{1 - m} + c_{m} = c_{m}^{ + } \left( \eta \right),} \hfill & {m = 1, \ldots L + 1,} \hfill \\ {\beta_{ - n} { + }\beta_{n} = \beta_{n}^{ + } \left( \eta \right),} \hfill & {n = 1, \ldots K.} \hfill \\ \end{array} } \right. $$

(52)

$$ \left\{ {\begin{array}{*{20}l} {c_{m} - c_{1 - m} = c_{m}^{ - } \left( \xi \right),} \hfill & {m = 1, \ldots L + 1,} \hfill \\ {\beta_{n} - \beta_{ - n} = \beta_{n}^{ - } \left( \xi \right),} \hfill & {n = 1, \ldots K.} \hfill \\ \end{array} } \right. $$

(53)