A time-reversal Lattice Boltzmann Method E. Vergnault a,˚ , O. Malaspinas a , P. Sagaut a a Institut Jean Le Rond d’Alembert, UMR 7190, Université Pierre et Marie Curie - Paris 6 4, place Jussieu, case 162, F-75252 Paris cedex 5, France Abstract 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 Boltz- mann 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. Keywords: Lattice Boltzmann Equation, Time-reversal 1. Introduction Noise source identification is of major interest in the transports industry. The sound generated by an aerodynamic source is radiated in the flow in a one-way process : the source defines the sound field in the flow, but it is very difficult to identify the location of emission from the sound field. Noise source identification has been addressed by a large variety of methods. Among the three major families of methods, namely, those based on aeroacoustic analogies (see [1]), on statistical definition by correlation (see [2]) and on an inverse problem. Here, we focus on the last one. The methods in this family solve the inverse propagation problem. After running a simulation for some time, the time is reversed and the simulation is run backwards. The study of inverse problems was at first used for antenna problems, where the reverse problem corresponds to a very simple wave-propagation model. The progresses in computational fluid dynamics allow more complex physical models to be solved (in our case the weakly compressible Navier– Stokes equations), and hence more accurate solutions of the inverse problem. The noise source detection is then performed using a sensitivity analysis, arguing that the higher the sensitivity of the acoustic field to a hydrodynamic event is, the more likely it is to be its source. The sensitivity analysis can be done with an adjoint problem [3] or with complex differentiation [4], this specific topic is left for a future work. Here, we focus on the resolution of the reverse hydrodynamic problem. In the past two decades, the Lattice Boltzmann Method (see, e.g. the book by Succi [5], Benzi et al. [6] or Aidun and Clausen [7]) has gained fame amongst the computational fluid dynamics community. Among others it has been used for the simulation of weakly ˚ Corresponding author Email addresses: vergnault@lmm.jussieu.fr (E. Vergnault), malaspinas@lmm.jussieu.fr (O. Malaspinas), pierre.sagaut@upmc.fr (P. Sagaut) Preprint submitted to Journal of Computational Physics October 25, 2013