MULTIPHYSICS 2009

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

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. Mezrhabb, M. Bouzidic 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 UniversitéBlaise Pascal – Clermont II/ IUT/ LaMI EA3867 – FR TIMS 2856 CNRS, Av. A. Briand, F-03101 Montluçon cedex, France

3. Why use the LBM? Motivation Choice • Lattice Boltzmann Method (LBM) • An alternative to classic CFD Method • Bigprogress made over the last decade • Becomespopular • Advantages of LBM • Ease of BC implementation • Welladapted for parallel computations • No Poisson equation for pressure

4. What is the origin of this method? From a historic point view • LBM has been derivedfrom the LatticegasAutomata (LGA) • also • It canbeobtainedusing a first order explicit upwind FD discretisatin of the discrete Boltzmann equation. • Seenextslide

5. Modelling of the fluidprocess • Threelevels • Macroscopic, mesoscopic and microscopic Moleculardynamics Intermolecularpotential Microscopic LatticeGasautomata Lattice Boltzmann Equation Fluid Particule Probabilty Mesoscopic Fluiddynamics Navier-Stokes Equations ContinumFluid Macroscopic Fig.1. Threelevels of modeling

6. Particlevelocity Vector position Complexform !!!! External force (EF) If we assume: and neglect EF feq denotes the equilibrium function;  is the relaxation time

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

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

9. As computationaltool, LBE methoddiffersfrom NS equationsbasedmethod in various aspects: • NS eqs are 2dorderPDEs; LBE is 1storder PDE • NS solversneed to treat the NL convective term; the LBE avoidsthisterm; • CFD solversneed to solve the Poisson equation; the LBE methodisalways local: the pressure isobtainedfrom an equation of state; • In the LBE, the CFL numberis ~ to Δ t/Δx; • Since The LBE iskinetic-based, the physicsassociatedwith the molecularlevel interaction canbeincorporated more easily in the LBE model. Hence, the LBE model canbefruitfullyapplied to micro-scalefluid flow problems; • The couplingbetweendiscretizedvelocityspace and configuration spaceleads to regular square grids. This is a limitation of the LBE, especially for Aerodynamicswhereboth the far fieldboundary condition and the nearwallboundary layer need to becarefullyimplemented.

10. The latticeBolzmannMethod for the D2Q9 square lattice model • Instead of a complexintegro-differentialoperatorΩwecan use twodfferent approximations:  Single Relaxation Time (SRT) model  Multi Relaxation Time (MRT) model whereMf=m In thisaproach the distribution istransformedinto moment spacebefore relaxation The utilisation of severaldifferent relaxation rates for the non-conserved moments leads to an increase in stability and thus to more efficient simulations

11. is the transformation matrix such that For the D2Q9 square lattice model • the form of the transformation matrixis • the relaxation matrix S in the moment space is S = diag(0,s1,s2,0,s4,0,s6,s7,s8); sibeing the collision rates

12. The moments (9) are separated into two groups: (ρ, m3, m5) are the conserved moments which are locally conserved in the collision process; (m1, m2, m4, m6, m7, m8) are the non-conserved moments and they are calculated from the relaxation equations: where mjacis the moment after collision, mjbcis the moment before collision (the post-collision value) sjare the relaxation rates which are the diagonal elements of the matrix S and are the corresponding equilibrium moments The macroscopicfluid variables, (ρ , velocityu and pressure P, are obtainedfrom the moments of the distribution functions as follows

13. 6 2 5 e2 e5 e6 e1 e3 e0 3 1 e7 e8 e4 7 4 8 The D2Q9 model In LB flow simulations, Discrete Particle Distribution Functions (PPDF) are propagated with discrete velocities Fig. 3. A 2-D 9-velocity lattice (D2Q9) model

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 the rod (Bi-partition) h w xp xb the obstacle Fig. 4. Schematic representation of the configuration and nomenclature

16. e2 e6 e5 e3 e1 e0 e8 e4 e7 Fig. 5. Mesh structure around the control partition and square cylinder

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

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

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

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

21. The bi-partitionheighteffects (a) (b) (c) (d) 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

22. (e) (f) (g) (h) 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. Conclusion • The present LBM showed: • LBM is a reliable alternative Method to the classicalbased on the resolution of the NS-equations (at least for incompressiblflows) • LBM canbeused for research and development • BUT •  Need to convince more researchers (e.g. turbulence community) to use LBM (How ?)