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2021 | OriginalPaper | Buchkapitel

No-Slip and Free-Slip Divergence-Free Wavelets for the Simulation of Incompressible Viscous Flows

verfasst von : Souleymane Kadri Harouna, Valérie Perrier

Erschienen in: Cartesian CFD Methods for Complex Applications

Verlag: Springer International Publishing

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This work concerns divergence-free wavelet-based methods for the numerical resolution of Navier–Stokes equations. It generalizes to higher dimension the approach of Kadri-Harouna and Perrier (Multiscale Model. Simul. 13:399–422; 2015) that reformulates the projection method using the Helmholtz–Hodge decomposition in wavelet domain. The solution is searched in a finite dimensional free-slip divergence-free wavelet space, with time-dependent wavelet coefficients. We prove and verify the convergence of a first-order time numerical scheme for the Helmholtz–Hodge-based projection method. Numerical simulations on the 3D lid-driven cavity flow show the accuracy and efficiency of the method.

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Vorheriges Kapitel Fluid–Structure Interaction Using Volume Penalization and Mass-Spring Models with Application to Flapping Bumblebee Flight

Nächstes Kapitel An Immersed Boundary Method on Cartesian Adaptive Grids for the Simulation of Compressible Flows

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Beylkin, G.: On the representation of operator in bases of compactly supported wavelets. SIAM J. Numer. Anal. 6, 1716–1740 (1992)MathSciNetCrossRef

Canuto, C., Masson, R.: Stabilized wavelet approximations of the Stokes problem. Math. Comp. 70, 1397–1416 (2001)MathSciNetCrossRef

Charton, P., Perrier, V.: A pseudo-wavelet scheme for the two-dimensional Navier-Stokes equations. Comp. Appl. Math. 15, 137–157 (1996)MathSciNetMATH

Chorin, A.J.: Numerical simulation of the Navier-Stokes equation. Math. Comp. 22, 745–762 (1968)MathSciNetCrossRef

Cohen, A., Daubechies, I., Vial, P.: Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1, 54–81 (1993)MathSciNetCrossRef

Cohen, A., Masson, R.: Wavelet methods for second order elliptic problems – preconditioning and adaptivity. SIAM J. Sci. Comp. 21, 1006–1026 (1999)MathSciNetCrossRef

Dahmen, W., Urban, K., Vorloeper, J.: Adaptive wavelet methods-basic concepts and applications to the Stokes problem. In: Zhou, D.-X. (ed.) Wavelet Analysis–Twenty Years Developments, pp. 39–80. World Scientific, New Jersey, (2002)CrossRef

Deriaz, E., Perrier, V.: Divergence-free and curl-free wavelets in 2D and 3D, application to turbulent flows. J. Turbu. 7, 1–37 (2006)CrossRef

Deriaz, E., Perrier, V.: Direct numerical simulation of turbulence using divergence-free wavelets. SIAM Multis. Model. Simul. 7, 1101–1129 (2008)MathSciNetCrossRef

10.

Deriaz, E., Perrier, V.: Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets. Appl. Comput. Harmon. Anal. 26, 249–269 (2009)MathSciNetCrossRef

11.

Ding, H., Shu, C., Yeo, K.S., Xub, D.: Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method. Comput. Methods Appl. Mech. Eng. 195, 516–533 (2006)CrossRef

12.

Elsasser, W.M.: The hydromagnetic equations. Phy. Rev. 79, 183–183 (1950)CrossRef

13.

Farge, M.: Wavelet transforms and their applications to turbulence. Ann. Rev. Fluid Mech. 24, 395–457 (1992)MathSciNetCrossRef

14.

Farge, M., Kevlahan, N., Perrier, V., Goirand, E.: Wavelets and turbulence. Proc. IEEE 84, 639–669 (1996)CrossRef

15.

Frohlich, J., Schneider, K.: Numerical simulation of decaying turbulence in an adaptive wavelet basis. Appl. Comput. Harm. Anal. 3, 393–397 (1996)CrossRef

16.

Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations. Springer, Berlin (1986)CrossRef

17.

Grivet-Talocia, S., Tabacco, A.: Wavelets on the interval with optimal localization. Math. Models. Meth. Appl. Sci. 10, 441–462 (2000)MathSciNetCrossRef

18.

Henry, M., Maitre, E., Perrier, V.: Optimal Transport Using HelmHoltz-Hodge Decomposition and First Order Primal-Dual Algorithm. IEEE ICIP, Piscataway, pp. 4748–4752 (2015)

19.

Jouini, A., Lemarié-Rieusset, P.G.: Analyse multirésolution biorthogonale sur l’intervalle et applications. Ann. Inst. Henri Poincaré Sect. C 10, 453–476 (1993)CrossRef

20.

Kadri-Harouna, S., Perrier, V.: Helmholtz-Hodge decomposition on [0, 1]^d by divergence-free and curl-free wavelets. In: Boissonnat, J.-D., Chenin, P., Cohen, A., Gout, C., Lyche, T., Mazure, M.-L., Schumaker, L. (eds.) Curves and Surfaces, Proceedings of the 7th International Conference, Avignon, June 24–30, 2010. Lecture Notes in Computer Science Series, vol. 6920, pp. 311–329. Springer, Berlin (2012)MATH

21.

Kadri-Harouna, S., Perrier, V.: Effective construction of divergence-free wavelets on the square. J. Comput. Appl. Math. 240, 74–86 (2013)MathSciNetCrossRef

22.

Kadri-Harouna, S., Perrier, V.: Divergence-free wavelet projection method for incompressible viscous flow on the square. Multiscale Model. Simul. 13, 399–422 (2015)MathSciNetCrossRef

23.

Kadri-Harouna, S., Perrier, V.: Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and related applications. Preprint hal-01568431v2 (2018)

24.

Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comp. Phys. 59, 308–323 (1985)MathSciNetCrossRef

25.

Lemarié-Rieusset, P.G.: Analyses multi-résolutions non orthogonales, commutation entre projecteurs et dérivation et ondelettes vecteurs à divergence nulle. Rev. Mat. Iberoamericana 8, 221–236 (1992)MathSciNetCrossRef

26.

Liu, J.-G., Liu, J., Pego, R.: Stable and accurate pressure approximation for unsteady incompressible viscous flow. J. Comput. Phys. 229, 3428–3453 (2010)MathSciNetCrossRef

27.

Masson, R.: Biorthogonal spline wavelets on the interval for the resolution of boundary problems. Math. Models Methods Appl. Sci. 6, 749–791 (1996)MathSciNetCrossRef

28.

Monasse, P., Perrier, V.: Orthogonal wavelet bases adapted for partial differential equations with boundary conditions. SIAM J. Math. Anal. 29, 1040–1065 (1998)MathSciNetCrossRef

29.

Sass-Hansen, M., Larsen, R., Christensen, N.-V.: Curl-gradient image warping-introducing deformation potentials for medical image registration using Helmholtz decomposition. In: International Conference on Computer Vision Theory and Applications, vol. 1, pp. 79–185 (2009)

30.

Schneider, K., Vasilyev, O.: Wavelet methods in computational fluid dynamics. Ann. Rev. Fluid Mech. 42, 473–503 (2010)MathSciNetCrossRef

31.

Stevenson, R.: Divergence-free wavelet bases on the hypercube: free-slip boundary conditions, and applications for solving the instationary Stokes equations. Math. Comp. 80, 1499–1523 (2011)MathSciNetCrossRef

32.

Stevenson, R.: Divergence-free wavelets on the hypercube: general boundary conditions. Const. Approx. 44, 233–267 (2016)MathSciNetCrossRef

33.

Temam, R.: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II. Arch. Rational Mech. Anal. 33, 377–385 (1969)MathSciNetCrossRef

34.

Temam, R.: Navier Stokes Equations. North Holland, New York (1977)MATH

35.

Urban, K.: Using divergence-free wavelets for the numerical solution of the Stokes problem. In: AMLI’96: Proceedings of the Conference on Algebraic Multilevel Iteration Methods with Applications University of Nijmegen, vol. 2, pp. 261–277 (1996)

36.

Urban, K.: Wavelet bases in H(div) and H(curl). Math. Comput. 70, 739–766 (2000)MathSciNetCrossRef

37.

Wang, C., Liu, J-G.: Convergence of Gauge method for incompressible flow. Math. Comput. 69, 1385–1407 (2000)MathSciNetCrossRef

38.

Weinan, E., Guo-Liu , J.: Projection method I: convergence and numerical boundary layers. SIAM J. Numer. Anal. 32, 1017–1057 (1995)MathSciNetCrossRef

Titel: No-Slip and Free-Slip Divergence-Free Wavelets for the Simulation of Incompressible Viscous Flows
verfasst von: Souleymane Kadri Harouna
Valérie Perrier
Verlag: Springer International Publishing
Buch: Cartesian CFD Methods for Complex Applications
Print ISBN: 978-3-030-61760-8

Electronic ISBN: 978-3-030-61761-5

Copyright-Jahr: 2021
DOI: https://doi.org/10.1007/978-3-030-61761-5_3

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