A time-reversal lattice Boltzmann method

In this paper we address the time-reversed simulation of viscous flows by the lattice Boltzmann method (LB). The theoretical derivation of the reversed LB from the Boltzmann equation is detailed, and the method implemented for weakly compressible flows using the D2Q9 scheme. The implementation of boundary conditions is also discussed. The accuracy and stability are illustrated by four test cases, namely the propagation of an acoustic wave in a medium at rest and in an uniform mean flow, the Taylor-Green vortex decay and the vortex pair-wall collision.

[1]  Mickael Tanter,et al.  Time-reversed acoustics , 2000 .

[2]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[3]  Cyrus K. Aidun,et al.  Lattice-Boltzmann Method for Complex Flows , 2010 .

[4]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[5]  Pierre Sagaut,et al.  A coupled time-reversal/complex differentiation method for aeroacoustic sensitivity analysis: towards a source detection procedure , 2009, Journal of Fluid Mechanics.

[6]  Orestis Malaspinas,et al.  Straight velocity boundaries in the lattice Boltzmann method. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases , 1939 .

[8]  Pierre Sagaut,et al.  Comparison between lattice Boltzmann method and Navier-Stokes high order schemes for computational aeroacoustics , 2009, J. Comput. Phys..

[9]  Peter M. A. Sloot,et al.  Lattice-Boltzmann hydrodynamics on parallel systems , 1998 .

[10]  M. Fink,et al.  Time-reversal of ultrasonic fields. III. Theory of the closed time-reversal cavity , 1992, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[11]  H. Grad.  Note on N‐dimensional hermite polynomials , 1949 .

[12]  L. Luo,et al.  Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model , 1997 .

[13]  X. Yuan,et al.  Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation , 2006, Journal of Fluid Mechanics.

[14]  Bastien Chopard,et al.  a Benchmark Case for Lattice Boltzmann:. Turbulent Dipole-Wall Collision , 2007 .

[16]  Michel Deville,et al.  Lattice Boltzmann method for the simulation of viscoelastic fluid flows , 2010 .

[17]  Jan Westerholm,et al.  An efficient swap algorithm for the lattice Boltzmann method , 2007, Comput. Phys. Commun..

[18]  Hjh Herman Clercx,et al.  Benchmark computations of the normal and oblique collision of a dipole with a no-slip boundary , 2006 .

[19]  C. Bogey,et al.  An analysis of the correlations between the turbulent flow and the sound pressure fields of subsonic jets , 2007, Journal of Fluid Mechanics.

[20]  Peter V. Coveney,et al.  HemeLB: A high performance parallel lattice-Boltzmann code for large scale fluid flow in complex geometries , 2008, Comput. Phys. Commun..

[21]  H. Grad.  On the kinetic theory of rarefied gases , 1949 .

[22]  Skordos,et al.  Initial and boundary conditions for the lattice Boltzmann method. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  P. Dellar.  Bulk and shear viscosities in lattice Boltzmann equations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Dawn Walker,et al.  A Complex Automata approach for in-stent restenosis: Two-dimensional multiscale modelling and simulations , 2011, J. Comput. Sci..

[25]  J. Boon.  The Lattice Boltzmann Equation for Fluid Dynamics and Beyond , 2003 .

[26]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[27]  Comparison of viscous flows backward in time with small data , 1989 .

[28]  Timothy J. Ulrich,et al.  Investigation of the robustness of time reversal acoustics in solid media through the reconstruction of temporally symmetric sources , 2008 .

[29]  S. Turek,et al.  Benchmark computations based on Lattice-Boltzmann, Finite Element and Finite Volume Methods for laminar Flows , 2006 .

[30]  L. Payne.  On Stabilizing Ill—Posed Cauchy Problems for the Navier—Stokes Equations , 1993 .

[31]  Lawrence E. Payne,et al.  On the stability of solutions of the Navier-Stokes equations backward in time , 1968 .

[32]  Pierre Sagaut,et al.  Lattice Boltzmann method with selective viscosity filter , 2009, J. Comput. Phys..

[33]  Sauro Succi,et al.  Massively Parallel Lattice-Boltzmann Simulation of Turbulent Channel Flow , 1997 .

[34]  C. Airiau,et al.  Adjoint analysis for noise control in a two-dimensional compressible mixing layer , 2008 .

[35]  Massimo Bernaschi,et al.  MUPHY: A parallel MUlti PHYsics/scale code for high performance bio-fluidic simulations , 2009, Comput. Phys. Commun..