Geodesic
High order finite volume schemes for numerical solution of 2D and 3D transonic flows
Kybernetika, Tome 45 (2009) no. 4, pp. 567-579
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The aim of this article is a qualitative analysis of two modern finite volume (FVM) schemes. First one is the so called Modified Causon's scheme, which is based on the classical MacCormack FVM scheme in total variation diminishing (TVD) form, but is simplified in such a way that the demands on computational power are much smaller without loss of accuracy. Second one is implicit WLSQR (Weighted Least Square Reconstruction) scheme combined with various types of numerical fluxes (AUSMPW+ and HLLC). Two different test cases were chosen for the comparison $-1$) two-dimensional transonic inviscid nonstationary flow over an oscillating NACA 0012 profile and 2) three-dimensional transonic inviscid stationary flow around the Onera M6 wing. Nonstationary effects were simulated with the use of Arbitrary Lagrangian–Eulerian Method (ALE). Experimental results for these regimes of flow are easily available and so the numerical results are compared both in-between and with experimental data. The obtained numerical results in all considered cases (2D and 3D) are in a good agreement with experimental data.
The aim of this article is a qualitative analysis of two modern finite volume (FVM) schemes. First one is the so called Modified Causon's scheme, which is based on the classical MacCormack FVM scheme in total variation diminishing (TVD) form, but is simplified in such a way that the demands on computational power are much smaller without loss of accuracy. Second one is implicit WLSQR (Weighted Least Square Reconstruction) scheme combined with various types of numerical fluxes (AUSMPW+ and HLLC). Two different test cases were chosen for the comparison $-1$) two-dimensional transonic inviscid nonstationary flow over an oscillating NACA 0012 profile and 2) three-dimensional transonic inviscid stationary flow around the Onera M6 wing. Nonstationary effects were simulated with the use of Arbitrary Lagrangian–Eulerian Method (ALE). Experimental results for these regimes of flow are easily available and so the numerical results are compared both in-between and with experimental data. The obtained numerical results in all considered cases (2D and 3D) are in a good agreement with experimental data.
Classification : 76H05, 76M12, 76N99
Keywords: ALE method; AUSMPW+; finite volume method; HLLC; nonstationary flow; transonic flow; TVD 
@article{KYB_2009_45_4_a1,
     author = {F\"urst, Ji\v{r}{\'\i} and Kozel, Karel and Furm\'anek, Petr},
     title = {High order finite volume schemes for numerical solution of {2D} and {3D} transonic flows},
     journal = {Kybernetika},
     pages = {567--579},
     year = {2009},
     volume = {45},
     number = {4},
     mrnumber = {2588623},
     zbl = {1251.76027},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2009_45_4_a1/}
}
TY  - JOUR
AU  - Fürst, Jiří
AU  - Kozel, Karel
AU  - Furmánek, Petr
TI  - High order finite volume schemes for numerical solution of 2D and 3D transonic flows
JO  - Kybernetika
PY  - 2009
SP  - 567
EP  - 579
VL  - 45
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/KYB_2009_45_4_a1/
LA  - en
ID  - KYB_2009_45_4_a1
ER  - 
%0 Journal Article
%A Fürst, Jiří
%A Kozel, Karel
%A Furmánek, Petr
%T High order finite volume schemes for numerical solution of 2D and 3D transonic flows
%J Kybernetika
%D 2009
%P 567-579
%V 45
%N 4
%U http://geodesic.mathdoc.fr/item/KYB_2009_45_4_a1/
%G en
%F KYB_2009_45_4_a1
Fürst, Jiří; Kozel, Karel; Furmánek, Petr. High order finite volume schemes for numerical solution of 2D and 3D transonic flows. Kybernetika, Tome 45 (2009) no. 4, pp. 567-579. http://geodesic.mathdoc.fr/item/KYB_2009_45_4_a1/

[1] P. Batten, M. A. Leschziner, and U. C. Goldberg: Average-state Jacobians and implicit methods for compressible viscous and turbulent flows. J. Comput. Phys. 137 (1997), 38–78. | MR

[2] D. M. Causon: High resolution fnite volume schemes and computational aerodynamics. In: Nonlinear Hyperbolic Equations – Theory, Computation Methods and Applications (Notes on Numerical Fluid Mechanics volume 24, J. Ballmann and R. Jeltsch, eds.), Vieweg, Braunschweig 1989, pp. 63–74. | MR

[3] Compendium of Unsteady Aerodynamic Measurements. AGARD Advisory Report No. 702, 1982.

[4] J. Donea: An arbitrary Lagrangian–Eulerian finite element method for transient fluid- structur interactions. Comput. Methods Appl. Mech. Engrg. 33 (1982), 689–723.

[5] M. Feistauer, J. Felcman, and I. Straškraba: Mathematical and Computational Methods for Compressible Flow. (Numerical Mathematics and Scientific Computation.) Oxford University Press, Oxford 2003. | MR

[6] J. Fürst: Numerical Solution of Transonic Flow Using Modern Schemes of Finite volume Method and Finite Differences. Ph.D. Thesis (in Czech), ČVUT, Praha 2001.

[7] J. Fürst: A weighted least square scheme for compressible flows. Submitted to Flow, Turbulence and Combustion 2005.

[8] J. Fürst, M. Janda, and K. Kozel: Finite volume solution of 2D and 3D Euler and Navier–Stokes equations. Math. Fluid Mechanics (J. Neustupa and P. Penel, eds.), Birkhäuser Verlag, Basel 2001. | MR

[9] J. Fürst and K. Kozel: Application of second order TVD and ENO schemes in internal aerodynamics. J. Sci. Comput. 17 (2002), 1–4, 263–272. | MR

[10] J. Fürst and K. Kozel: Second and third order weighted ENO scheme on unstructured meshes. In: Proc. Finite Volumes for Complex Applications III (D. Herbin and D. Kröner, eds.), Hermes Penton Science, pp. 737–744.

[11] Kyu Hong Kim, Chongam Kim, and Oh-Hyun Rho: Methods for the accurate computations of hypersonic flows. AUSMPW+ scheme. J. Comput. Physics 174 (2001), 38–80. | MR

[12] M. Lesoinne and C. Farhat: Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Comput. Methods Appl. Mech. Engrg. 134 (1996), 71–90.

[13] V. Schmitt and F. Charpin: Pressure Distributions on the ONERA-M6-Wing at Transonic Mach Numbers. Experimental Data Base for Computer Program Assessment. Report of the Fluid Dynamics Panel Working Group 04, AGARD AR 138, 1979.

[14] H. C. Yee: A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods. Technical Memorandum 101088, NASA, Moffett Field, California 1989.