Adaptive Wavelet Method for Incompressible Flows in Complex Domains

An adaptive wavelet-based method provides an alternative means to refine grids according to local demands of the physical solution. One of the prominent challenges of such a method is the application to problems defined on complex domains. In the case of incompressible flow, the application to problems with complicated domains is made possible by the use of the Navier-Stokes-Brinkman equations. These equations take into account solid obstacles by adding a penalized velocity term in the momentum equation. An adaptive wavelet collocation method, based on interpolating wavelets, is first applied to a benchmark problem defined on a simple domain to demonstrate the accuracy and efficiency of the method. Then the penalty technique is used to simulate flows over obstacles

[1]  Jörg M. Hahn.  Geometric continuous patch complexes , 1989, Comput. Aided Geom. Des..

[2]  Yevgenii Anatolyevich Rastigejev.  Multiscale computations with a wavelet-adaptive algorithm , 2002 .

[3]  A. Devaney,et al.  Inverse scattering using the Heitler equation , 1989 .

[4]  Naoki Saito,et al.  Multiresolution representations using the autocorrelation functions of compactly supported wavelets , 1993, IEEE Trans. Signal Process..

[5]  M. M. Zdravkovich,et al.  REVIEW—Review of Flow Interference Between Two Circular Cylinders in Various Arrangements , 1977 .

[6]  O. Vasilyev,et al.  A Fast Adaptive Wavelet Collocation Algorithm for Multidimensional PDEs , 1997 .

[7]  O. Botella,et al.  BENCHMARK SPECTRAL RESULTS ON THE LID-DRIVEN CAVITY FLOW , 1998 .

[8]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[9]  R. LeVeque,et al.  Analysis of a one-dimensional model for the immersed boundary method , 1992 .

[10]  Jaroslav Mackerle,et al.  Error estimates and adaptive finite element methods , 2001 .

[11]  Philip M. Gresho,et al.  On the theory of semi‐implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory , 1990 .

[12]  Jochen Fröhlich,et al.  An Adaptive Wavelet Galerkin Algorithm for one and two Dimensional Flame Computations , 1993 .

[13]  J. Ferrari,et al.  NUMERICAL SIMULATION OF FLOW INTERFERENCE BETWEEN TWO CIRCULAR CYLINDERS IN TANDEM AND SIDE-BY-SIDE ARRANGEMENTS , 2001 .

[14]  Leland Jameson,et al.  A Wavelet-Optimized, Very High Order Adaptive Grid and Order Numerical Method , 1998, SIAM J. Sci. Comput..

[15]  O. Vasilyev,et al.  A Dynamically Adaptive Multilevel Wavelet Collocation Method for Solving Partial Differential Equations in a Finite Domain , 1996 .

[16]  Jacques Liandrat,et al.  Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation , 1990 .

[17]  Philippe Angot,et al.  A penalization method to take into account obstacles in incompressible viscous flows , 1999, Numerische Mathematik.

[18]  Khodor Khadra,et al.  Fictitious domain approach for numerical modelling of Navier–Stokes equations , 2000 .

[19]  M. Olshanskii,et al.  On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid , 2000 .

[20]  M. Braza,et al.  A nonreflecting outlet boundary condition for incompressible unsteady Navier-Stokes calculations , 1993 .

[21]  Frans N. van de Vosse,et al.  An approximate projec-tion scheme for incompressible ow using spectral elements , 1996 .

[22]  R. Sani,et al.  Résumé and remarks on the open boundary condition minisymposium , 1994 .

[23]  Silvia Bertoluzza.  Adaptive wavelet collocation method for the solution of Burgers equation , 1996 .

[24]  Silvia Bertoluzza.  Adaptive wavelet collocation for the solution of steady-state equations , 1995, Defense, Security, and Sensing.

[25]  J. P. Beyer.  A computational model of the cochlea using the immersed boundary method , 1992 .

[26]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[27]  Gilles Deslauriers,et al.  Symmetric iterative interpolation processes , 1989 .

[28]  Mats Holmström,et al.  Solving Hyperbolic PDEs Using Interpolating Wavelets , 1999, SIAM J. Sci. Comput..

[29]  L. Sirovich,et al.  Modeling a no-slip flow boundary with an external force field , 1993 .

[30]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[31]  Jie Shen,et al.  Hopf bifurcation of the unsteady regularized driven cavity flow , 1991 .

[32]  Michael Griebel,et al.  Numerical Simulation in Fluid Dynamics: A Practical Introduction , 1997 .

[33]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[34]  Samuel Paolucci,et al.  Wavelet‐based adaptive multiresolution computation of viscous reactive flows , 2006 .

[35]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[36]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[37]  R. Glowinski,et al.  A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow , 2001 .