Accuracy of Pulsatile 2D flow in the Lattice Boltzmann BGK model

1. Accuracy of Pulsatile 2D flow in the Lattice Boltzmann BGK model A. M. Artoli, A. G. Hoekstra and P. M. A. Sloot Section Computational Science Institute Informatics Faculty of Science University of Amsterdam http://www.science.uva.nl/research/scs Emails: [artoli, alfons, sloot]@science.uva.nl

2. Overview • Motivation • The Lattice Boltzmann method • Benchmark • 2D Oscillatory Channel Flow • Simulation Results • Coclusions

3. Cardiovascular diseases are the main cause of human death. Atherogenesis grows at locations of low and oscillating shear stress. Shear stress can be computed easily and up to the same accuracy as the flow fields in LBM. Motivation Aorta with a bypass

4. The Lattice Boltzmann LBGK The LBM is a first order finite difference discretization of the Boltzmann Equation that describes the dynamics of continuous particle distribution function. The velocity is descritized into a set of vectors ei The inter-particle interactions are contained in the collision term W The resulting Lattice Boltzmann involves two steps: streaming and Collision

5. Benchmarks Analytic Solutions 2D 3D

6. Simulations • Flow is driven by a time dependent body force P = A sin(w t) in the x-direction. A= initial Magnitude of P, w= angular frequency, t= simulation time . • Boundary conditions • inlet and outlet: Periodic boundaries. • Walls: bounce-Back • Parameters • a ranges from 1-15 • t = 1 • Grid size : • 2D : 10 x50 for a = 3.07 and 20 x100 for a= • 3D : 50 x 50 x 100

7. Results • 2D oscillatory Poiseuille flow • a =3.07 • shown: Full-period analytic solutions (lines) and simulation results (points)

8. Flow characteristics • There is a Phase lag between the pressure and the fluid motion. • At low a, steady Poiseuille flow is obtained. • At high a, we have the annular effect: • Profiles are flattened • The phase lag increases toward the center. • The shear stress is very low near the center and reasonably high at the walls.

9. 2D, cont.

10. Conclusions • Obtaining accurate reproduction of 2D oscillatory flow is possible with incompressible LBGK. • Simulation results are more accurate if they are compared to analytic solutions at half time steps. • Pulsatile shear stress, directly computed from the distribution functions, yield accurate results.

11. End

12. Boltzmann Equation Nonlinear integrodifferential equation in Kinetic theory of dilute monatomic gases. Describes the temporal evolution of the one-particle distribution function in a gas of particles with binary collisions:

13. BGK Approximation • BGK • LBGK

14. Simplified BE Equation is solvable near equilibrium ->Molecules are assumed Maxwellians -> gI(c) and not the energy. Linearize the collision term and put fn = feqhn -> Linearized BE

15. Introduction

16. What is lost? • Thermodynamic consistency between pressure and forcing • BGL? • Mass, radius and V ->0, N and r2 are finite => Perfect gas! -> Compressibility • But: • A gas that is not too rarefied (mfp->0) ~Fluid • NOT valid initially, near the boundaries and around shocks.

17. Boundary conditions • Walls • Bounce-Back • Non slip • Curved • Inlet and Outlets • Velocity • Pressure • No flux

18. Enhancing the LBGK • Incompressibility • D2Q9 ->D2Q9i-> D2Q9ii, ... • Stability • subgrid models • Thermodynamic consistency • Non-ideal gas models

19. Results, 3D

20. Conclusions, 3D • LBGK with the simple bounce back produces an error > 12% for steady and unsteady flows. • Enhanced LBGK or LBE model reduces the error . • Curved boundary conditions should be used.