Higher accuracy scheme based on edge-oriented quasi-1В reconstruction of variables for solving aerodynamics and aeroacoustics problems on unstructured meshes

I. V. Abalakin ; T. K. Kozubskaya

Matematičeskoe modelirovanie, Tome 25 (2013) no. 8, pp. 109-136.

Résumé

The paper presents a description in details of EBR scheme for solving aerodynamics and aeroacoustics problems on unstructured triangular and tetrahedral meshes. The EBR scheme is vertex-centered and thereby presupposes the construction of control volumes. Its higher accuracy is provided at the expense of edge-based quasi-1D reconstruction of variables used for the calculation of numerical fluxes through the cell faces. The results of analytical study of EBR scheme accuracy for “Cartesian” meshes and experimental study by solving the test problem on propagation of acoustic Gaussian pulse on meshes of different quality are given.

Détail

Keywords: higher-accuracy schemes, cells, unstructured meshes, aeroacoustics.
Mots-clés : reconstruction of variables, Euler equations

Bibliographie

[1] Stolarski T., Nakasone Y., Yoshimoto S., Engineering Analysis with ANSYS Software, Elsevier, Butterworth–Heinemann, 2006

[2] Fluent. User's Guide Fluent 6.3.26, Fluent Incorporated, Lebanon, NH, 2006

[3] Li B. Q., Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Springer, London, 2006 | MR

[4] Zhang Y.-T., Shu C.-W., “Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes”, Commun. Comput. Phys., 5:2–4 (2009), 836–848 | MR

[5] Dumbser M., Kaser M., “Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems”, J. Comput. Phys., 221:2 (2007), 693–723 | DOI | MR | Zbl

[6] Tsoutsanis P., Titarev V. A., Drikakis D., “WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions”, J. Comput. Phys., 230:2 (2011), 1585–1601 | DOI | MR | Zbl

[7] Wolf W. R., Azevedo J. L. F., “High-order ENO and WENO schemes for unstructured grids”, Int. J. Numer. Meth. Fluids, 55:10 (2007), 917–943 | DOI | MR | Zbl

[8] Debiez S., Approximations decentrees a faible dissipation pour des problemes hyperboliques, INRIA Report 2811, Février, 1996

[9] Abalakin I. V., Kozubskaya T. K., Dervieux A., “High Accuracy Finite Volume Method for Solving Nonlinear Aeroacoustics Problems on Unstructured Meshes”, Chinese Journal of Aeronautics, 19:2 (2006), 97–104 | DOI | MR

[10] Abalakin I. V., Kozubskaya T. K., “Mnogoparametricheskoe semeistvo skhem povyshennoi tochnosti dlya lineinogo uravneniya perenosa”, Matematich. modelirov., 19:7 (2007), 56–66 | MR | Zbl

[11] Shokin Yu. I., Yanenko H. H., Metod differentsialnogo priblizheniya. Primenenie k gazovoi dinamike, Nauka, Sib. otd., Novosibirsk, 1985, 364 pp. | MR | Zbl

[12] Oran E., Boris Dzh., Chislennoe modelirovanie reagiruyuschikh potokov, Mir, M., 1990, 660 pp. | MR

[13] Chakravarthy S. R., Osher S., A new class of high accuracy TVD schemes for hyperbolic conservation laws, AIAA Paper 85-0363, New York, 1985

[14] Roe P. L., “Approximate Riemann Solvers, Parameter Vectors and Difference Schemes”, J. Comput. Phys., 43:2 (1981), 357–372 | DOI | MR | Zbl

[15] Wu H., Wang L., “Non-existence of third order accurate semi-discrete MUSCL-type schemes for nonlinear conservation laws and unified construction of high accurate ENO schemes”, Proceedings of the 6th International Symposium on CFD (Lake Tahoe, USA, September 4–8, 1995)

[16] Huang L. C., “Pseudo-Unsteady Difference Schemes for Discontinuous Solution of Steady-State, One-Dimensional Fluid Dynamics Problems”, J. Comput. Phys., 42:1 (1981), 195–211 | DOI | MR | Zbl

[17] Gourvitch N., Rogé G., Abalakin I., Dervieux A., Kozubskaya T., A tetrahedral-based superconvergent scheme for aeroacoustics, INRIA Report 5212, May, 2004

[18] Jameson A., Numerical Solution of the Euler Equations for Compressible inviscid Fluids, Numerical. Methods for the Euler Equations of Fluid Dynamics, SIAM, Philadelphia, 1985, 199 pp. | MR | Zbl

[19] Proceedings of ICASE/LaRC. Workshop on Benchmark Problems in Computational Aeroacoustics (CAA) (May 1995), NASA Conference Publication, 3300, eds. J. C. Hardin, J. R. Ristorcelli, C. K. W. Tam

[20] Tam C. K. W., Webb J. C., “Dispersion-relation-preserving finite-difference schemes for computational acoustics”, J. Comput. Phys., 107:2 (1993), 262–281 | DOI | MR | Zbl

[21] Kozubskaya T., Abalakin I., Dervieux A., Ouvrard H., “Accuracy Improvement for Finite-Volume Vertex-Centered Schemes Solving Aeroacoustics Problems on Unstructured Meshes”, 16th AIAA/CEAS Aeroacoustics Conference (Stockholm, Sweden, June 2010); AIAA paper 2010-3933, 2010

[22] Abalakin I. V., Derve A., Kozubskaya T. K., Uvrar Kh., “Metodika povysheniya tochnosti pri modelirovanii perenosa akusticheskikh vozmuschenii na nestrukturirovannykh setkakh”, Uchenye zapiski TsAGI, XLI:1 (2010), 110–125

[23] Abalakin I. V., Bakhvalov P. A., Gorobets A. V., Duben A. P., Kozubskaya T. K., “Parallelnyi programmnyi kompleks NOISETTE dlya krupnomasshtabnykh raschetov zadach aerodinamiki i aeroakustiki”, Vychislitelnye metody i programmirovanie, 13 (2012), 110–125

[24] Volkov K. N., Emelyanov V. N., Modelirovanie krupnykh vikhrei v raschetakh turbulentnykh techenii, Fizmatlit, M., 2008, 368 pp.

[25] Liu X.-D., Osher S., Chan T., “Weighted essentially non-oscillatory schemes”, J. of Comput. Phys., 115 (1994), 200–212 | DOI | MR | Zbl