SHEAR STRESS

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SHEAR STRESS
A. M. Artoli1 , D. Kandhai2, H.G. Hoefsloot3,
A.G. Hoekstra1 and P.M.A. Sloot1
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IN LATTICE BOLTZMANN
Motivation

• Shear stress plays a dominant role in
  biomechanical deseases related to blood flow
  problems.

Aorta with a bypass
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SIMULATIONS

• Conventionally, the shear stress is calculated
  from the computed gradients of velocity profiles
  obtained from experimental or simulation models.

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Why LBM?

• Recently, the Lattice Boltzmann Method (LBM) has
  attracted much attention in simulations of
  complex fluid flow problems for its simple
  implementation and inherent parallelism.
• The LBM can be used to calculate the local
  components of the stress tensor in fluid flows
  WITHOUT a need to estimate velocity gradients.
  This has two benefits over conventional CFD
  methods Increasing accuracy and decreasing
  computational cost.

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Definition

• The stress tensor is defined as1

1 L.Landau and E. Lifshitz, Fluid mechanics,
Pergamon Press (1959).
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The LBM

• The LBM is a first order finite difference
  discretization of the Boltzmann Equation
• that describes the dynamics of continuous
  particle distribution function which
  is the probability of finding a particle with
  microscopic velocity
• 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 Equation is
• The collision term is simplified to the linear
  case via the single time relaxation Approximation
  (STRA)

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Theory

• The equilibrium distribution is given by
• where
• wi 4/9 for the rest particle, 1/9 for
  particles moving in x and y directions and 1/36
• for diagonal ones. Also,
• Conservation laws are satisfied
• mass
• momentum

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Theory , cont.

• The LB equation is then discretized in space and
  time to yield
• Using the multi-scale Chapman expansion of the
  kinetic moments of the distribution functions,
  the macroscopic NS equation can be derived in the
  limit of low Mach number (u ltlt Cs the speed of
  sound).
• Where is the pressure.
• The kinematic viscosity and the equation of state
  are given by

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The stress tensor in LBM
The stress tensor for a 9 particles 2D LBM model
is given by 2 where is the
dissipative part of the momentum tensor , which
can be obtained during the collision operation,
without a need to take the derivatives
2 S. Hou, S. Chen, g. Doolen and A. Cogley.
J.Comp.Phys. 118, 329 (1995).
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How the stress tensor is computed?

• Select a model (e.gD2Q9)
• For All nodes
• compute the density and velocity from the fis
• initialize sab with 0
• for all directions
• . sab cka ckb Df (1-1/(2t)) -r cs2
  dab.
• Collide fab k fabk-(fabk- fab
  eqk)/t

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BENCHMARK-1

• Plane Poiseulle steady flow
• Analytic solutions

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BENCHMARK-2
Couette Flow with injection upper wall moves with
hrizontal velocity Un Lower veocity is
fixed Vertical injection with speed U0 Analytic
solutions
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BENCHMARK-3
Symmetric Bifurcation
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Simulations
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Simulations-2
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Ongoing Research

• 2D oscillatory Poiseuille flow
• a 3.07
• error 10-2 for integer time steps and 10-15
  for half-time steps.
• shown Full-period analytic solutions (lines) and
  simulation results (points)

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Womersley solution

• 3D Preliminary Results
• error for integer time steps and for
  half-time steps.
• shown Full-period Analytic solutions (lines) and
  simulation results (points)

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Conclusions

• With LBM, the shear stress can be obtained from
  the distribution functions without a need to
  compute derivatives of velocity profiles.
• LBM is second order accurate in space and time.
• Pulsatile shear stress can still yield accurate
  results.

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Benchmarks Analytic Solutions
2D
3D
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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.

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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 and 20 x100 for a
• 3D 50 x 50 x 100

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Results, Continued

• a
• error 10-2 for integer time steps and 10-15
  for half-time steps.
• shown Full-period Analytic solutions (lines) and
  simulation results (points)