Divergence-free and curl-free wavelets in two dimensions and three dimensions: Application to turbulent flows

Journal of Turbulence

Volumn 7, Issue , 2006, Pages 1-37

  Deriaz, Erwan ^a   Perrier, Valérie ^a  

^a IMAG CNRS   (France)

Author keywords

Curl free wavelets; Divergence free wavelets; Turbulent flows

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; DIFFERENTIATION (CALCULUS); FOURIER TRANSFORMS; INCOMPRESSIBLE FLOW; INTEGRATION; INTERPOLATION; NAVIER STOKES EQUATIONS; TURBULENCE; TURBULENT FLOW; VECTORS;

TURBULENT FLOW;

EID: 31744448320     PISSN: None     EISSN: 14685248     Source Type: Journal    
DOI: 10.1080/14685240500260547     Document Type: Article

References (39)

1. Farge, M., 1992, Wavelet transforms and their applications to turbulence. Annual Review of Fluid Mechanics, 24, 395-457.
2. Meneveau, C., 1991, Analysis of turbulence in the orthonormal wavelet representation. Journal of Fluid Mechanics, 232, 469-520.
3. Lewalle, J., 1993, Wavelet transform of the Navier-Stokes equations and the generalized dimensions of turbulence. Applied Scientific Research, 51, 109-113.
4. Charton, P. and Perrier, V., 1996, A pseudo-wavelet scheme for the two-dimensional Navier-Stokes equations. Computational and Applied Mathematics, 15, 139-160.
5. Fröhlich J. and Schneider, K., 1996, Numerical simulation of decaying turbulence in an adaptive wavelet basis. Applied and Computational Harmonic Analysis, 3, 393-397.
6. Farge, M., Kevlahan, N., Perrier V. and Goirand, E., 1996, Wavelets and turbulence. Proceedings of the IEEE, 84, 639-669.
7. Schneider, K., Kevlahan, N. and Farge, M., 1997, Comparison of an adaptive wavelet method and nonlinearly filtered pseudo-spectral methods for two-dimensional turbulence. Theoretical and Computational Fluid Dynamiccs, 9, 191-206.
8. Koster, F., Griebel, M., Kevlahan, N., Farge M. and Schneider, K., 1998, Towards an adaptive wavelet-based 3D Navier-Stokes solver. In: E.H. Hirschel (Ed.), Numerical Flow Simulation I, Notes on Numerical Fluid Mechanics, vol. 66, Braunschweig: Vieweg, pp. 339-364.
9. Griebel, M. and Koster, F., 2000, Adaptive wavelet solvers for the unsteady incompressible Navier-Stokes equations, In: J. Malek and J. Necas and M. Rokyta (Eds), Advances in Mathematical Fluid Mechanics (Berlin: Springer).
10. Kevlahan, N., Vasilyev, O.V., Goldstein, D. and Jay, A., 2004, A three-dimensional adaptive wavelet method for fluid-structure interaction. In: B. J. Geurts, R. Friedrich and O. Métais (Eds) Direct and Large-Eddy Simulation V (Dordrecht: Kluwer), 8 pp.
11. Kevlahan, N.K.-R. and Vasilyev, O.V., 2005, An adaptive wavelet collocation method for fluid-structure interaction at high Reynolds numbers. SIAM Journal on Scientific Computing, 6, 1894-1915.
12. Farge, M. and Schneider, K., 2001, Coherent vortex simulation (CVS), a semi-deterministic turbulence model using wavelets. Flow, Turbulence and Combustion, 66, 393-426.
13. Goldstein, D., Vasilyev, O. and Kevlahan, N.K.-R., 2004, Adaptive LES of 3D decaying isotropic turbulence. In: P. Moin, N. Mansour and P. Bradshaw (Eds.) Studying Turbulence Using Numerical Simulation Databases - X, Proceedings of the 2004 Summer Program (Stanford, CA: Center for Turbulence Research).
14. Grossmann S. and Löhden, M., 1996, Wavelet analysis of Navier-Stokes flow. Zeitschrift für Physile B, 100, 137-147.
15. Lemarié-Rieusset, P.G., 1992, Analyses multi-résolutions non orthogonales, commutation entre projecteurs et dérivation et ondelettes vecteurs à divergence nulle. Revista Matemática Iberoamericana, 8, 221.
16. Albukrek, C.-M., Urban, K., Dahmen, W., Rempfer, D. and Lumley, J.-L., 2002, Divergence-free wavelet analysis of Turbulent Flows. Journal of Scientific Computing, 17, 49-66.
17. Ko, J., Kurdila, J.A. and Rediniotis, O.K., 2000, Divergence-free bases and multiresolution methods for reduced-order flow modeling. AIAA Journal, 38, 2219-2232.
18. Urban, K., 2002, Wavelets in Numerical Simulation (Berlin: Springer).
19. Urban, K., 1994, A wavelet-galerkin algorithm for the driven-cavity-Stokes problem in two space dimensions. In: M. Feistauer, R. Rannacher, K. Kozel (Eds), Numerical Modelling in Continuum Mechanics, Charles: University Prague, pp. 278-289.
20. Urban, K., 1996, Using divergence-free wavelets for the numerical solution of the Stokes problem. In: Proceedings of the Conference on Algebraic Multilevel Iteration Methods with Applications, Vol. 2, University of Nijmegen, The Netherlands, pp. 261-277.
21. Dahmen, W., Kunoth, A. and Urban, K., 1996, A wavelet-Galerkin method for the Stokes problem. Computing, 56, 259-302.
22. Urban, K., 2000, Wavelet Bases in H(div) and H(curl). Mathematics of Computation, 70, 739-766.
23. Daubechies, I., 1992, Ten Lectures on Wavelets, (Philadelphia, Pennsylvania: SIAM).
24. Kahane, J.-P. and Lemarié-Rieusset, P.G., 1995, Fourier Series and Wavelets, London: Gordon and Breach.
25. Mallat, S., 1998, A Wavelet Tour of Signal Processing. New York: Academic Press.
26. Cohen, A., 2000, Wavelet methods in numerical analysis. Handbook of Numerical Analysis, Vol. VII. In: P.G. Ciarlet and J.L.Lions (Eds) Amsterdam: Elsevier.
27. Girault, V. and Raviart, P.A., 1986, Finite Element Methods for Navieer-Stokes Equations (Berlin: Springer).
28. Chorin, A.J. and Marsden, J.E., 1993, A Mathematical Introduction to Fluid Mechanics, 3rd edition (Berlin: Springer).
29. Bernardi, C. and Maday, Y., 1992, Approximations spectrales de problèmes aux limites elliptiques. Mathématiques & Applications, Vol. 10 (Paris: Springer).
30. De Boor, C., 2001, A Practical Guide to Splines (New York: Springer).
31. Hua, B.L. and Klein, P, 1998, An exact criterion for the stirring properties of nearly two-dimensional turbulence, Physica D, 113, 98-110.
32. Lapeyre, G., 2000, Topologie de mélange dans un fluide turbulent géophysique. Thèse de doctorat, L'Université Paris VI.
33. Cottet, G.-H., Michaux, B., Ossia, S. and Vanderlinden, G., 2002, A comparison of spectral and vortex methods in three-dimensional incompressible flows. Journal of Computational Physics, 175, 1-11.
34. Chiavassa, G. and Liandrat, J., 1997, On the effective construction of compactly supported wavelets satisfying boundary conditions on the interval. Applied and Computational Harmonic Analysis, 4, 62-73.
35. Monasse, P. and Perrier, V., 1998, Orthonormal wavelet bases adapted for partial differential equations with boundary conditions. SIAM Journal on Mathematical Analysis, 29, 1040-1065.
36. Cohen, A., Daubechies, I. and Feauveau, J.C., 1992, Biorthogonal bases of compactly supported wavelets. Communications in Pure and Applied Mathematics, 45, 485-560.
37. Schneider, K. and Farge, M., 2000, Numerical simulation of a mixing layer in an adaptive wavelet basis, Comptes Rendus de l'Academie des Sciences, Série II, 263-269.
38. Lemarié-Rieusset, P.G., 1994, Un théorème d'inexistance pour les ondelettes vecteurs à divergence nulle. Comptes Reaches de l'Academie des Sciences, Série 1, 319, 811-813.
39. Vasilyev, O.V. and Kevlahan, N.K.-R., 2005, An adaptive multilevel wavelet collocation method for elliptic problems. Journal Computational Physics (to be published).