s
Lattice Boltzmann-Based Immersed Boundary Calculations
L INDSAY M. C ROWL AND A ARON L. F OGELSON
Department of Mathematics, University of Utah
Convergence in Time
Immersed Boundary Method
We want to include a fluid-structure interaction, the
Convergence in Time: Fluid Velocity
particles can
F(q; t) = s
dt 2
−X ||
dt/2
0.6
|| /||X
dt 2
dt/2 2
u : dx=1/32
0.4
0.3
dt/4
y
u : dx=1/64
x
u : dx=1/64
0.5
−X
x
−u
dt/2 2
u : dx=1/32
y
0.2
0.2
u : dx=1/128
x
0.1
0.1
u : dx=1/128
@2
X(q; t)
@q
0.2
0.4
0.6
0.8
1
0
0
1.2
dt
0.2
0.4
0.6
0.8
1
1.2
dt
−3
x 10
−3
x 10
For our lattice Boltzmann-Immersed Boundary simulations, convergence in space
2
and time is first order, most likely due to the thinness of the elastic membrane.
Coupling (boundary moves with fluid velocity)
@
tribution functions f (x; t) at each node. This eliminates noise from the model but
keeps the simple evolution equations.
0.5
0
0
– move to neighboring nodes
– collide with each other.
Instead of keeping track of discrete particles, the LBM keeps track of particle dis-
0.7
X : dx=1/32
Y : dx=1/32
X : dx=1/64
Y : dx=1/64
X : dx=1/128
Y : dx=1/128
y
Simple elastic force rule:
The LBM originates from discrete kinetic theory lattice-gas-automata where gas
0.8
0.3
Lagrangian frame.
Lattice Boltzmann: Origin
0.8
0.4
dt/4
X(q; t) denotes the location of a point q at time t in the
0.9
0.6
|| /||u
assume a thin massless membrane that resists
stretching. This membrane lives in a Lagrangian
frame and is coupled to an Eulerian fluid velocity
field.
0.9
0.7
−u ||
We
1
dt/2
goal of which is being able to model biological fluid
dynamics.
Convergence in Time: Location of IB points
1
||u
The lattice Boltzmann Method (LBM) is a relatively new computational approach to
solving fluid flow problems that uses a mesoscopic particle-based technique. Its advantage over traditional computational fluid dynamics is that it bypasses the need to solve
for the pressure gradient and so all calculations become local ones. This poster presents
a lattice Boltzmann algorithm that incorporates the Immersed Boundary Method (IBM).
This method is being developed with the goal of simulating fluid filled deformable particles, such as platelets or red blood cells. The accuracy and speed of this new method
will be compared to the original Navier Stokes Immersed Boundary Method.
||X
Abstract
X(q; t) = u (X(q; t); t)
@t
Comparison to Navier Stokes
Coupling (boundary force affects fluid for Incompressible Navier Stokes)
(ut + u ru) = rp + r u + F
Navier Stokes Immersed Boundary:
t = 0.2, max vel = 1.117
2
Navier Stokes Immersed Boundary:
t = 0.2 max vel = 1.117
1
1
0.8
0.8
Force Term
0.6
In order to couple the LBM to an Immersed Boundary, we need to add a time
dependent, spatially varying force term into the lattice Boltzmann equations.
Since the Navier Stokes equations can be recovered from a Chapman-Enskog ex The LBM can also be seen as a discretization of the continuous Boltzmann equation [3] if we assume a simplification of the collision operator to a single relaxation
time towards equilibrium:
@f (x; e; t)
1
g(x; e; t));
+ e rf (x; e; t) = C (f ) = (f (x; e; t)
@t
where is the relaxation time and g (x; e; t) is the Maxwell Boltzmann distribution
about the macroscopic velocity of the fluid, u.
pansion [1] of the governing equations, we alter our equations in order to get the
correct macroscopic approximation to the Navier Stokes equations with an external force [2].
fi(x + eit; t + t) fi(x; t)
1
=
fi(x; t)
w
+ i t
F e
t F
u + 2
T
C
:
(
ee
i
+
cs
2
We need to redefine the momentum density v = u +
eq
fi
c
2 4s
:::
2
c I)
s
:
F to include the Immersed
t
2
Boundary force, since it has large spatial variations [4].
When we recover the Navier Stokes equations from the lattice Boltzmann equa-
Lattice Boltzmann: Governing Equations
tions, we find that:
During a given time step, the particle distributions fi advect to corresponding
neighboring nodes (in the direction of ei ) and relax toward equilibrium:
f|i(x + tei; t {z
+ t )
fi(x; t}) =
Advection
where = t
1
|
f
( i(
x; t)
eq
fi (
{z
x; t)) ;
}
Relaxation/Collision
and i’s denote the velocity direction at any particular node.
The equilibrium distribution is an expansion in u of the Maxwell Boltzmann distribution function up to second order:
u
eq
fi (; ) = wi
e
(ei u)
i u
1+
+
2
c
uu
2c
2
2c
4
2
:
We typically discretize velocity into eight non-zero directions since we want a
0.6
0.4
0.4
0.2
0
0.2
0
0.2
0.4
– the macroscopic viscosity is = c2s 1
2
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
wi
:
w 2i
1
0
Lattice Boltzmann Immersed Boundary
Density Profile: ρ(x,y,t=0.2)
1.06
50
1.04
40
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
1.02
30
60
1.0008
50
1.0006
40
1.0004
30
1.0002
20
1
1
0.9998
10
0.9996
20
30
40
50
60
10
20
30
40
50
60
=
1
Space discretization: dx 64
,
1
Time discretizationdt 5120
.
=
=
=
1
Space discretization: dx 64
,
1
Time discretizationdt 640
.
LBM is compressible, but as the Mach number goes to zero (c = dxdt
Lattice Boltzmann-Immersed Boundary Algorithm
1
Lattice Boltzmann Immersed Boundary
Density Profile: ρ(x,y,t=0.2)
60
10
fluid approaches an incompressible limit.
>>
kuk), the
LBM fluid velocity appears slightly slower than Navier Stokes fluid velocity.
1. Calculate force on IB (Fib(X;t) ).
2. Spread force onto Eulerian grid (FLB (x; t)).
3. Advect fi ’s to neighboring nodes.
Future Work
4. Sum over i at each node to determine (x; t) and u(x; t).
5. Calculate fieq (; u).
6. Collide (relax to equilibrium) by 1= .
Compare LBM and Navier Stokes for IB problem in more de-
7. Move IB with fluid velocity.
tail.
We test convergence for the ellipse problem with periodic boundary conditions
where = 0:01, = 1, Re 10 , and s = 1. Initially the ellipse is set to rx = 0:40,
1=9
0.8
0.96
though we do require a small enough time step to ensure incompressibility.
i = 1; 3; 5; 7
= 1=36 i = 2; 4; 6; 8:
=
0.6
0.2
Lattice Boltzmann Immersed Boundary:
t = 0.2 max velocity = 0.9805
0.8
10
– the pressure is given by an equation of state: p = c2s .
0
0.8
0.98
t
0
1
20
For smooth external forces, convergence is second order in space and time, al4=9
1
1
Test speed of current LBM.
Parallelize LBM for simulating particles in flow.
Simulate flow of platelets in blood vessels.
Convergence in Space
=
0.8
Lattice Boltzmann Immersed Boundary:
t = 0.2 max velocity = 0.9948
Cartesian geometry and the more simple square velocity model has non-physical
macroscopic properties. We choose the appropriate weighting by distance for each
velocity:
8
< w0
0.6
Courtesy of Dr. Elisabeth Maurer-Spurej.
2
ry = 0:15.
References
Convergence of Fluid Velocity in Space
"X" − velocity in x direction, "O" − velocity in y direction
1
[1] J. M. B UICK AND C. A. G REATED, Gravity in a lattice Boltzmann model, Physical Review E, 61(5) (2000), pp. 5307–5320.
0.9
X
X
i
x
fi ( ; t) =
x ei =
fi ( ; t)
i
X
i
X
eq
x
fi ( ; t) = (x; t)
x; t)ei
eq
fi (
=
x ux
( ; t) ( ; t):
i
We recover these macroscopic properties of the fluid by summing over i at each
node at each time step.
ing at size of the fraction
kudx=
kudx=
4
2
for the fluid velocity.
k
udx k
udx=2
2
2
:
0.8
||u128−u64||2/||u64−u32||2
The governing equations conserve macroscopic density and momentum:
We check the convergence rate by look-
0.7
[2] Z. G. F ENG AND E. E. M ICHAELIDES, The immersed boundary-lattice Boltzmann
method for solving fluid-particles interaction problems, J. Comp. Phys., 195 (2004),
pp. 602–628.
0.6
0.5
0.4
[3] X. H E AND L. S. L UO, A priori derivation of the lattice Boltzmann equation, Phys. Rev.
E., 55(6) (1997), pp. R6333–6.
0.3
0.2
0.1
0
0
0.5
1
1.5
2
dt
2.5
3
3.5
−4
x 10
[4] A. J. C. L ADD AND R. V ERBERG, Lattice-Boltzmann Simulations of Particle-Fluid Suspensions, J. Stat. Phys., 104(5/6) (2001), pp. 1191–1251.