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A Numerical Strategy for Freestream Preservation of the High Order Weighted Essentially Non-oscillatory Schemes on Stationary Curvilinear Grids

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Abstract

The weighted essentially non-oscillatory (WENO) schemes have been extensively employed for the simulation of complex flow fields due to their high order accuracy and good shock-capturing properties. However, the standard finite difference WENO scheme cannot hold freestream automatically in general curvilinear coordinates. Numerical errors from non-preserved freestream can hide small scales such as turbulent flow structures; aero-acoustic waves which can make the results inaccurate or even cause the simulation failure. To address this issue, a new numerical strategy to ensure freestream preservation properties of the WENO schemes on stationary curvilinear grids is proposed in this paper. The essential idea of this approach is to offset the geometrically induced errors by proper discretization of the metric invariants. It includes the following procedures: (1) the metric invariants are retained in the governing equations and the full forms of the transformed equations on the general curvilinear coordinates are solved; (2) the symmetrical, conservative form of the metrics instead of the original ones are used; (3) the WENO schemes which are applied for the inviscid fluxes of the governing equations are employed to compute the outer-level partial derivatives of the metric invariants. In other words, the outer-level derivative operators for the metric invariants are kept the same with those for the corresponding inviscid fluxes. It is verified theoretically in this paper that by using this approach, the WENO schemes hold the freestream preservation properties naturally and thus work well in the generalized coordinate systems. For some well-known WENO schemes, the derivative operators for the metric invariants are explicitly expressed and thus this approach can be straightforwardly employed. The effectiveness of this strategy is validated by several benchmark test cases.

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References

  1. Shu, C.W.: High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD. Int. J. Comput. Fluid Dyn. 17(2), 107–118 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Deng, X.G., Zhang, H.X.: Developing high-order weighted compact nonlinear schemes. J. Comput. Phys. 165(1), 22–44 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Thomas, P.D., Lombard, C.K.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17(10), 1030–1037 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Vinokur, M.: An analysis of finite-difference and finite-volume formulations of conservation laws. J. Comput. Phys. 81, 1–52 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Visbal, R.M., Gaitonde, D.V.: On the use of higher-order finite-difference schemes on curvilinear and deforming meshes. J. Comput. Phys. 181, 155–185 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Nonomura, T., Iizuka, N., Fujii, K.: Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids. Comput. Fluids 39, 197–214 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Étienne, S., Garon, A., Pelletier, D.: Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow. J. Comput. Phys. 228, 2313–2333 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Trulio, J.G., Trigger, K.R.: Numerical solution of the one-dimensional hydrodynamic equations in an arbitrary time-dependent coordinate system. Technical Report UCLR-6522, University of California Lawrence Radiation laboratory (1961)

  11. Pulliam, T.H., Steger, J.L.: On implicit finite-difference simulations of three-dimensional flow. AIAA Paper 78–10 (1978)

  12. Zhang, H., Reggio, M., Trépanier, J.Y., et al.: Discrete form of the GCL for moving meshes and its implementation in CFD schemes. Comput. Fluids 22(1), 9–23 (1993)

    Article  MATH  Google Scholar 

  13. Farhat, C., Geuzaine, P., Grandmont, C.: The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids. J. Comput. Phys. 174, 669–694 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mavriplis, D.J., Yang, Z.: Achieving higher-order time accuracy for dynamic unstructured mesh fluid flow simulations: role of the GCL. AIAA Paper 2005–5114 (2005)

  15. Sjögreen, B., Yee, H.C., Vinokur, M.: On high order finite-difference metric discretizations satisfying GCL on moving and deforming grids. J. Comput. Phys. 265, 211–220 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deng, X.G., Mao, M.L., Tu, G.H., et al.: Geometric conservation law and applications to high-order finite difference schemes with stationary grids. J. Comput. Phys. 230, 1100–1115 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Deng, X.G., Min, Y.B., Mao, M.L., et al.: Further studies on geometric conservation law and applications to high-order finite difference schemes with stationary grids. J. Comput. Phys. 239, 90–111 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Abe, Y., Iizuka, N., Nonomura, T., et al.: Conservative metric evaluation for high-order finite difference schemes with the GCL identities on moving and deforming grids. J. Comput. Phys. 232, 14–21 (2013)

    Article  Google Scholar 

  19. Abe, Y., Nonomura, T., Iizuka, N., et al.: Geometric interpretations and spatial symmetry property of metrics in the conservative form for high-order finite difference schemes on moving and deforming grids. J. Comput. Phys. 260, 163–203 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liao, F., Zhang, Z.Y., Zhang, L.X.: Extending geometric conservation law to cell-centered finite difference methods on stationary grids. J. Comput. Phys. 284, 419–433 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Deng, X.D., Jiang, Y., Mao, M.L., et al.: A family of hybrid cell-edge and cell-node dissipative compact schemes satisfying geometric conservation law. Comput. Fluids 116, 29–45 (2015)

    Article  MathSciNet  Google Scholar 

  22. Cai, X., Ladeinde, F.: Performance of WENO scheme in generalized curvilinear coordinate systems. AIAA Paper 2008–36 (2008)

  23. Jiang, Y., Shu, C.W., Zhang, M.: An alternative formulation of finite difference weighted ENO schemes with Lax–Wendroff time discretization for conservation laws. SIAM J. Sci. Comput. 35(2), 1137–1160 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jiang, Y., Shu, C.W., Zhang, M.: Free-stream preserving finite difference schemes on curvilinear meshes. Methods Appl. Anal. v21, 1–30 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Asahara, M., Nonomura, T., Fujii, K. et al.: Comparison of resolution and robustness with TSFDWENO schemes. In: Proceedings of 27th Computational Fluid Dynamics Symposium, C03-4, (2013) (in Japanese)

  26. Nonomura, T., Terakado, D., Abe, Y., et al.: A new technique for freestream preservation of finite-difference WENO on curvilinear grid. Comput. Fluids 107, 242–255 (2015)

    Article  MathSciNet  Google Scholar 

  27. Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This study is supported by the project of Aeronautical Science Foundation of China (Grant No. 201514U8005), National Natural Science Foundation of China (Grant No. 11302250), Natural Science Foundation of Shaanxi Province (Grant No. 2015JQ1008) and National Science Foundation for post-doctoral Scientist of China (Grant No. 2015M570084).

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Authors and Affiliations

  1. Xi’an Research Institute of High-Tech, Xi’an, 710025, Shaanxi, China

    Yujie Zhu, Zhensheng Sun, Yu Hu & Shiying Zhang

  2. Department of Mechanics, Tsinghua Universty, Beijing, 100084, China

    Yuxin Ren

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  1. Yujie Zhu
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  2. Zhensheng Sun
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  3. Yuxin Ren
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  4. Yu Hu
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  5. Shiying Zhang
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Correspondence to Zhensheng Sun.

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Zhu, Y., Sun, Z., Ren, Y. et al. A Numerical Strategy for Freestream Preservation of the High Order Weighted Essentially Non-oscillatory Schemes on Stationary Curvilinear Grids. J Sci Comput 72, 1021–1048 (2017). https://doi.org/10.1007/s10915-017-0387-x

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  • DOI: https://doi.org/10.1007/s10915-017-0387-x

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