Biomedical Engineering Reference
In-Depth Information
When turbulence is developed in the flow field, we would have to consider the turbu-
lent Reynolds stress, which includes time-dependent fluctuations that are difficult to solve
by hand. Recall that statistical methods were required for these formulations. The solution
to the flow profile does not only depend on the Navier-Stokes equations and the
Conservation of Mass, but also the turbulent kinetic energy ( k ), which is defined as
1
2 u i u i
k5
ð
13
:
8
Þ
where u i is the velocity of the fluid. The other variable required in turbulent flows is the
viscous dissipation rate of the turbulent kinetic energy, denoted as
ε. The dissipation rate
is defined as
t 1Δt
ð
1
Δt
ε5u i;j u i;j
u i;j u i;j dt
ð
13
:
9
Þ
t
where u i,j
is the velocity and
ν
is the kinematic viscosity. The turbulent viscosity
μ t
is
directly proportional to the turbulent quantities k and
ε
as
μ t ~ ρ 0 k 2
ε
ð
13
:
10
Þ
In the macrocirculation, blood vessels are relatively large and the flow velocity is relatively
high, and therefore, the standard k-ε
turbulence model is commonly used to solve the flow
fields (such as in the aorta). Using this model assumes an isotropic turbulence throughout
the entire flow cycle with a high Reynolds number. However, when using the k-ε
model, it
could become challenging to estimate the velocity and the stress distribution in the near-
wall regions, where the turbulent viscosity has a large effect and varies very sharply. This
requires a large amount of computational steps and a large processing time. Also, the
Reynolds number in the sub-layer of the turbulent field close to the vessel wall is usually
low because the viscosity is large, making the standard k-ε
model inadequate. Therefore, a
low-Reynolds number Wilcox turbulent k -
model is often used instead, which can more
accurately predict the velocity and stress distribution at the near-wall regions when the
viscosity is large. The k-ω
ω
ω
, rather than turbulent vis-
cous dissipation rate to characterize turbulence. In this model, the turbulent velocity u t
model uses the turbulent frequency,
is
proportional to k by means of
p
u t ~
ð
13
:
11
Þ
The turbulent frequency
ω
is also related to k and
ε
by
ε5 kω
ð
13
:
12
Þ
and the turbulent viscosity is obtained from the proportionality
μ t ~ ρ k
ω
ð
13
:
13
Þ
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