Biomedical Engineering Reference
In-Depth Information
6.9 CHARACTERISTICS OF TWO-PHASE FLOW
In Chapter 2, we briefly described the nature of two-phase flows. In typical engineering
applications, the two phases are either different fluids or the same fluid that exists in two
phases (i.e., gas and liquid). Blood is a two-phase flow because the plasma component has
different physical properties as compared with the cellular components. Under most con-
ditions, it is not necessary to use two-phase flow principles within the macrocirculation.
This is because the cellular component has little to no effect on the overall flow conditions.
However, as discussed in this chapter, the cellular component can greatly affect the flow
conditions within the microcirculation. For the remainder of this section, we will discuss
some of the approaches available to solve two-phase flow problems. For this type of analy-
sis, it is important to know the exact location of each particle within the fluid. Therefore,
the Lagrangian model would be used to solve the two-phase flow problem. However, if
average values are more important, then the Eulerian approach should be used.
Using the Lagrangian approach, our goal would be to solve Newton's Second Law of
Motion for each particle within the flow field. In this instant, Newton's law takes the
form of
dv
-
dt
F
-
ð
t
Þ
5
m
p
ð
6
:
17
Þ
where F
-
is the summation of all forces acting on one particular particle. To solve this
equation, information regarding the exact instantaneous velocity as well as the position of
each particle would be required. If at time t, the velocity and the position of a particle is
known, then the particle will move to a new position at time
ð
t
Þ
δ
t, according to
ð
1
δ
Þ
5
ð
Þ
1
δ
ð
:
Þ
x
t
t
x
t
v
p
t
6
18
The velocity of the particle is developed using a known mean velocity component and a
fluctuating component, which arises from turbulence, variations in the particles, or varia-
tions in the flow field. This is similar to turbulence modeling discussed briefly in
Section 5.10. To predict the variation, a number of approaches have been developed which
are all related to a Lagrangian correlation function. Most of these approaches use statistical
distributions to predict the fluctuations in velocity. The general Lagrangian correlation
function is defined as the time average of the velocity fluctuations (u
0
)as
u
0
ð
t
Þ
u
0
ð
t
1
τÞ
q
u
0
2
q
u
0
2
R
ð
t
Þ
5
ð
t
Þ
ð
t
1
τÞ
In most two-phase flow approaches, this correlation function is linearized based on
some fluid parameters (such as the time of fluctuations or the length scale of fluctuations).
One of the most common and simplest approaches to linearize the correlation function is
to use the eddy decay lifetime in the form of
t
2t
D
R
ð
t
Þ
5
1
ð
6
:
19
Þ
2
where t
D
is the average eddy decomposition time.
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