Biomedical Engineering Reference
In-Depth Information
rest particles. However, in three dimensions, one typically does not need all 26
velocity components linking nearest neighbor lattice nodes, i.e., D3Q27. Instead,
the D3Q19 lattice scheme is often preferred.
From the discretized distribution function
f
i
(
,where
i
indexes the
c
i
velocity
components, one can determine the macroscopic density and velocity as
x
,
t
)
ρ
f
=
∑
i
f
i
(1)
1
ρ
f
∑
i
=
u
f
i
c
i
(2)
Importantly, the ideal gas equation of state in LBM gives the fluid pressure as
=
/
P
and hence the two quantities are often referred to interchangeably as
pressure/density.
Determining the distribution function at time t is a two-part process consisting of
a
streaming step
, which propagates particles between lattice nodes, followed by a
collision step
, which updates the momentum of the underlying particles as a result
of particle-particle interactions. A key development in the application of LBM to
systems involving fluid flow is the single-relaxation time Bhatnagar-Gross-Krook
(BGK) approximation [
79
]:
1
3
ρ
1
τ
(
f
eq
f
i
x
+
c
i
δ
t
,
t
+
δ
t
=
f
i
(
x
,
t
)
−
f
i
(
x
,
t
)
−
(
x
,
t
))
(3)
where
f
eq
(
x
,
t
)
is the equilibrium distribution function defined as
w
i
ρ
f
1
u
9
2
(
3
2
u
f
eq
i
2
(
x
,
t
)=
+
3
c
i
·
u
+
c
i
·
u
)
−
·
(4)
where
w
i
are lattice weights. The collision operator is defined as a relaxation towards
local equilibrium. The parameter
controls the rate of relaxation and is directly
related to the kinematic viscosity of the fluid under study:
τ
1
3
1
2
ν
=
τ
−
(5)
5.1
Boundary Conditions
Complex and irregular boundaries are simple to handle using LBM, making it
ideal for modeling vascular networks. There are essentially two important types
of boundary conditions: (a) those that involve fluid/solid interfaces at internal lattice
nodes and (b) those that involve the domain boundary. To handle the first type, a
zero-slip boundary condition is enforced on all fluid/wall interfaces. A particularly
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