Biomedical Engineering Reference
In-Depth Information
investigate the turbulent characteristic of stenotic flows using different turbulence
models (Birchall et al. 2006; Lee et al. 2008; Younis and Berger 2004).
The pulsatile blood flow is clearly unsteady with an oscillatory motion. In order
to assess the importance of the unsteadiness on the mean or average flow character-
istics obtained through a steady solution, we make use of the Womersley number,
α
and the Strouhal number,
S
. The Womersley number is a ratio of unsteady forces
to viscous forces named after John R. Womersley (1907-1958) and is defined as
0.5
=
D
ω
(7.4)
α
,
2
T
g
where
D
is the characteristic length which was taken as the inlet hydraulic diameter
of the carotid artery equal to approximately 5 mm. Then,
ν
is the kinematic viscos-
ity of blood (3.50 × 10
−3
Pa
ʘ
ଉ
s), ω is the blood pumping frequency equal to
ωπ
−
1
= =
and
u
ave
is the average velocity through the artery passage un-
der the flow rate of approximately 10 ml/s or 10
4
mm
3
/s, which is equal to 510 mm/s.
When α is small (1 or less), the oscillatory effects are sufficiently low that the inlet
conditions such as a parabolic velocity profile has time to develop during each cy-
cle, and the flow will be very nearly in phase with the pressure gradient. When α is
large (10 or more), the oscillation effects are sufficiently large that the velocity
profile does not develop in time and the mean flow characteristics lags the pressure
gradient by about 90 degrees, (Womersley 1955).
The Strouhal number is a ratio of the unsteady forces to the inertial forces named
after Vincenc Strouhal (1850-1922) and is defined as
2
f
6.28
s
ω
D
S
=
,
(7.5)
u
ave
where
u
ave
is the mean airflow velocity. For large
S
(> 1), the oscillations become
important. For low
S
(<< 1), the contribution of the velocity dominates the oscilla-
tions. The calculated
α
and
S
numbers for the artery in this case study are 0.1059 and
0.0616 respectively. When
α
is smaller than 1, then the frequency is sufficiently low
that a parabolic velocity profile has time to develop during each cycle. On the other
hand, the low value for
S
suggests that the flow may be assumed to be quasi-steady.
Although the assumption of Newtonian behavior of blood is acceptable for high
shear flows and also has been validated by several numerical studies, which indi-
cated that the influence of shear thinning properties of blood are not significant
(Cho and Kensey 1991; Perktold et al. 1991), it is not valid in low shear rate regions
where the value is less than 100 s
−1
(Chien 1982; Chua et al. 2005). The shear
thinning behavior of the blood flow was incorporated by Carreau-Yasuda model
(Bird et al. 1987) to predicate the flow pattern in the low shear rate region more
accurately, and the correlation between blood viscosity and the shear rate variation
is governed by
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