1. 1 MULTIPHYSICS 2009 9-11 December 2009 Lille, FRANCE

2. 2 Computation and control of the near- wake flow over a square cylinder with an upstream rod using an MRT lattice Boltzmann model by H. Naji a, A. Mezrhab b, M. Bouzidi c a Université Lille 1 - Sciences et Technologies/ Polytech'Lille/ LML UMR 8107 CNRS, F-59655 Villeneuve d'Ascq cedex, France b Laboratoire de Mécanique & Energétique, Département de Physique, Faculté des sciences, Université Mohamed 1, Oujda, Maroc c Université Blaise Pascal – Clermont II/ IUT/ LaMI EA3867 – FR TIMS 2856 CNRS, Av. A. Briand, F-03101 Montluçon cedex, France

3. 3  Lattice Boltzmann Method (LBM)  A n alternative to classic CFD Method  Big progress made over the last decade  Becomes popular  Advantages of LBM  Ease of BC implementation  Well adapted for parallel computations  No Poisson equation for pressure  Choice Why use the LBM? Motivation

4. 4 What is the origin of this method?  LBM has been derived from the Lattice gas Automata (LGA) also  It can be obtained using a first order e xplicit upwind FD discretisatin of the discrete Boltzmann equation. See next slide From a historic point view

5. 5  Modelling of the fluid process  Three levels  Macroscopic, mesoscopic and microscopic Molecular dynamics Intermolecular potential Lattice Gas automata Lattice Boltzmann Equation Fluid Particule Probabilty Fluid dynamics Navier-Stokes Equations Continum Fluid Fig.1. Three levels of modeling Microscopic Mesoscopic Macroscopic

6. 6 Vector position Particle velocity External force (EF) If we assume: Complex form !!!! and neglect EF f eq denotes the equilibrium function;  is the relaxation time

7. small Knudsen number small Mach number Chapman-Enskog-Expansion Bhatnagar-Gross-Krook-Approximation (BGK) discretisation in velocity space discretisation in space and time Chapman-Enskog-Expansion small Knudsen number 7 Boltzmann equationBoltzmann equation discrete Boltzmann equation Navier Stokes equations continuity equation Boltzmann equation Lattice Boltzmann equation (LBGK) Overview of the Modelling

8. The post-collision state propagation collision 8 Splitting process Fig. 2. Collision and propagation steps

9. 9  As computational tool, LBE method differs from NS equations based method in various aspects:  NS eqs are 2 d order PDEs; LBE is 1 st order PDE  NS solvers need to treat the NL convective term; the LBE avoids this term;  CFD solvers need to solve the Poisson equation; the LBE method is always local: the pressure is obtained from an equation of state;  In the LBE, the CFL number is ~ to Δ t/Δx;  Since The LBE is kinetic-based, the physics associated with the molecular level interaction can be incorporated more easily in the LBE model. Hence, the LBE model can be fruitfully applied to micro- scale fluid flow problems;  The coupling between discretized velocity space and configuration space leads to regular square grids. This is a limitation of the LBE, especially for Aerodynamics where both the far field boundary condition and the near wall boundary layer need to be carefully implemented.

10. 10 The lattice Bolzmann Method for the D2Q9 square lattice model  Instead of a complex integro-differential operator Ω we can use two dfferent approximations:  Multi Relaxation Time (MRT) model  Single Relaxation Time (SRT) model In this aproach the distribution is transformed into moment space before relaxation The utilisation of several different relaxation rates for the non-conserved moments leads to an increase in stability and thus to more efficient simulations where Mf=m

11. 11 For the D2Q9 square lattice model  the form of the transformation matrix is is the transformation matrix such that  the relaxation matrix S in the moment space is S = diag(0,s 1,s 2,0,s 4,0,s 6,s 7,s 8 ); s i being the collision rates

12. 12 where m j ac is the moment after collision, m j bc is the moment before collision (the post- collision value) s j are the relaxation rates which are the diagonal elements of the matrix S and are the corresponding equilibrium moments The moments (9) are separated into two groups: (ρ, m 3, m 5 ) are the conserved moments which are locally conserved in the collision process; (m 1, m 2, m 4, m 6, m 7, m 8 ) are the non-conserved moments and they are calculated from the relaxation equations: The macroscopic fluid variables, (ρ, velocity u and pressure P, are obtained from the moments of the distribution functions as follows

13. 13 The D2Q9 model Fig. 3. A 2-D 9-velocity lattice (D2Q9) model In LB flow simulations, Discrete Particle Distribution Functions (PPDF) are propagated with discrete velocities e8e8 e4e4 e7e7 e3e3 e0e0 e1e1 e6e6 e2e2 e5e5 1 2 3 4 56 78

14. 14 In the discrete velocity space, the density and momentum fluxes can be evaluated as As for the pressure, it was can be computed simply by The corresponding viscosity in the NS equations is

15. Flow configuration 15 Fig. 4. Schematic representation of the configuration and nomenclature h w xpxp xbxb the rod (Bi-partition) the obstacle

16. 16 e8e8 e4e4 e7e7 e3e3 e0e0 e1e1 e6e6 e2e2 e5e5 Fig. 5. Mesh structure around the control partition and square cylinder

17. 17 (a) (b) Fig. 6. Comparison with previous work for: (a) the drag coefficient, (b) Strouhal number.

18. 18 (a )(b) Fig. 7. Comparison with previous work for: (a) the drag coefficient, (b) rms lift coefficient at w/d=1.5.

19. 19 (a) (c) Fig. 8. Streamlines at Re = 250 and h * =0.5: (a) without control, (b) w/d = 1, (c) w/d = 2; (b) The bi-partition loc ation effects

20. 20 (d) (e) Fig. 9. Streamlines at Re = 250 and h * =0.5: (d) w/d = 3, (e) w/d = 4, (f) w/d = 5. (f) Effect of the bi-partition location (continued)

21. 21 (b) (d) (a) Fig. 10. Streamlines for different bi-partition heights at w/d =5 and Re=250; (a) h*= 0.1, (b) h* = 0.3, (c) h* = 0.5, (d) h* = 0.6 (c) The bi-partition height effects

22. 22 (f) (g) (h) (e) Fig. 11. Streamlines for different bi-partition heights at w/d =5 and Re=250; (e) h* = 0.7, (f) h* = 0.8, (g) h* = 0.9, (h) h* =1.

23. 23  The present LBM showed: 1.LBM is a reliable alternative Method to the classical based on the resolution of the NS-equations (at least for incompressibl flows) 2.LBM can be used for research and development BUT  Need to convince more researchers (e.g. turbulence community) to use LBM (How ?) Conclusion