Accuracy of pulsatile 2d flow in the lattice boltzmann bgk model
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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

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Accuracy of pulsatile 2d flow in the lattice boltzmann bgk model

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


Overview
Overview model

  • Motivation

  • The Lattice Boltzmann method

  • Benchmark

    • 2D Oscillatory Channel Flow

  • Simulation Results

  • Coclusions


Motivation

Cardiovascular diseases are the main cause of human death. model

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


The lattice boltzmann lbgk
The Lattice Boltzmann LBGK model

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



Simulations
Simulations model

  • 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


Results
Results model

  • 2D oscillatory Poiseuille flow

    • a =3.07

    • shown: Full-period analytic solutions (lines) and simulation results (points)


Flow characteristics
Flow characteristics model

  • 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.


2d cont
2D, cont. model


Conclusions
Conclusions model

  • 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.


End model


Boltzmann equation
Boltzmann Equation model

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:


Bgk approximation
BGK Approximation model

  • BGK

  • LBGK


Simplified be
Simplified BE model

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



What is lost
What is lost? model

  • 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.


Boundary conditions
Boundary conditions model

  • Walls

    • Bounce-Back

    • Non slip

    • Curved

  • Inlet and Outlets

    • Velocity

    • Pressure

    • No flux


Enhancing the lbgk
Enhancing the LBGK model

  • Incompressibility

    • D2Q9 ->D2Q9i-> D2Q9ii, ...

  • Stability

    • subgrid models

  • Thermodynamic consistency

    • Non-ideal gas models



Conclusions 3d
Conclusions, 3D model

  • 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.


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