Summary
In this paper we apply the finite difference method on adaptive sparse grids [9] to the simulation of turbulent flows. This method combines the flexibility and efficiency of finite difference schemes with the advantages of an adaptive approximation by tensor product multiscale bases. We shortly discuss the method. Then, we present numerical results for a simple linear convection problem for a validation of our scheme. Finally, results for three-dimensional turbulent shear layers are shown.
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References
Brown G.L., Roshko A. (1974) On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64.
Cohen A., Daubechies I. et al. (1992) Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45, 485–560
Dahmen W., Schneider R. et al. (1998) Nonlinear Functionals of Wavelet Expansions. IGPM, RWTH Aachen, to appear in Numerische Mathematik.
Daubechies I. (1988) Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41, 909–996
Deslaurier G., Dubuc S. (1989) Symmetric iterative interpolation processes. Constr. Appr. 5, 49–68
DeVore R. (1999) Nonlinear Approximation. Acta Numerica 8.
Donoho D. (1992) Interpolating wavelet transform. Preprint Stanford University.
Faber G. (1909) Über stetige Funktionen. Math. Annalen 66, 81–94.
Griebel M. (1998) Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences. Computing 61, 151–180.
Griebel M., Koster F. (2000) Adaptive Wavelet Solvers for the Unsteady Incompressible Navier-Stokes Equations. in Malek J., Rokyta M. (eds.), Advanced Mathematical Theories in Fluid Mechanics.
Harten A. (1995) Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. 48, 1305–1342.
Ho C.-M., Huerre P. (1984) Perturbated free shear layers. Ann. Rev. Fluid Mech. 16, 365–424.
Hochmuth R. (1999) Wavelet Bases in Numerical Analysis and Restricted Nonlinear Approximation. Habilitationsschrift, Freie Universität Berlin.
Koster F. (2000) A Proof of the Consistency of the Finite Difference Technique on Sparse Grids. Computing 65, 247–261.
Koster F., Schneider K., Griebel M., Farge M. (2000) Adaptive Wavelet Methods for the Navier-Stokes equations. in E.H. Hirschel (ed.) Notes on Numerical Fluid Mechanics.
Koster F. (2001) Multiskalen-basierte Finite Differenzen Verfahren auf adaptiven Düinnen Gittern. PhD thesis, Universität Bonn.
Koster E (2001) Preconditioners for Sparse Grid Discretizations. Preprint SFB-256 Universität Bonn.
MaIlat S. (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Patt. Anal. and Mach. Intell. 7, 674–693
Rogers M.M., Moser R.D. (1993) Direct simulation of a self-similar turbulent mixing layer. Phys. Fluids 6, 903–923.
Schiekofer T. (1998) Die Methode der finiten Differenzen auf dünnen Gittern zur Lösung elliptischer und parabolischer PDEs. PhD thesis, Universität Bonn.
Temlyakov V. (1993) Approximation of Periodic Functions. Nova Science Publishers.
Townsend A.A. (1976) Structure of Turbulent Shear Flow. Cambridge University Press.
Triebe! H. (1992) Theory of function spaces II. Birkhäuser Verlag.
Wesseling P. (2001) Principles of Computational Fluid Dynamics. Springer Verlag.
Yserentant H.(1986) On the multilevel splitting of finite element spaces. Num. Math. 49, 379–412.
Zenger C. (1991) Sparse Grids. in Hackbusch W. (ed.) Notes on Numerical Fluid Mechanics 31.
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Griebel, M., Koster, F. (2003). Multiscale Methods for the Simulation of Turbulent Flows. In: Hirschel, E.H. (eds) Numerical Flow Simulation III. Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45693-3_13
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DOI: https://doi.org/10.1007/978-3-540-45693-3_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-53653-3
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