Biomedical Engineering Reference
In-Depth Information
state of motion—a condition of turbulence. These disturbances may originate from
the free stream of the fluid motion, or induced by the surface roughness where they
are amplified in the direction of the flow, in which case turbulence will occur. The
onset of turbulence depends on the ratio of the inertia to viscous force, which is in-
dicated by the Reynolds number, discussed in the previous section. At low Reynolds
number, inertia forces are smaller than the viscous forces. The naturally occurring
disturbances are dissipated away and the flow remains laminar. At high Reynolds
number, the inertia forces are sufficiently large to amplify the disturbances, and a
transition to turbulence occurs. The velocity and all other flow properties vary in a
random and chaotic manner.
Turbulence is associated with the existence of random fluctuations in the fluid.
If we consider flow through a blood vessel in Fig. 5.12 , then at any moment in time
its motion is random and unpredictable. If we measured its velocity at Point X over
time, then the velocity variation would exhibit random fluctuations.
We observe that over time there is an averaged value for the velocity u , with
some deviation, defined as the fluctuating component (  t ) at any moment in time.
This means that the instantaneous velocity is decomposed as the sum of the aver-
age and its instantaneous fluctuation:
ut u u t . In general, t o characterize
a turbulent flow the mean values of other flow properties ( u , v , w , p etc.) and its
fluctuating component (  , , , etc.) is used.
The random fluctuations from turbulent flow should not be confused with time de-
pendent oscillatory fluctuations. There is a common misconception that if a flow fluc-
tuates then it is turbulent. To illustrate this, we consider a transient laminar flow over
a cylinder (Fig. 5.13 ). Flow over a cylinder exhibits vortices that shed away (known
as vortex shedding) from the cylinder in an oscillatory and periodic motion. Taking
Point × in the flow field and tracking its velocity over time, we get a velocity profile
shown in Fig. 5.13d that fluctuates periodically over time. This suggests that the ve-
locity fluctuates but is orderly and is predictable—unlike turbulence which is random.
Turbulence models involve additional equations or modifications to the gov-
erning equations (continuity, momentum, and energy) to account for the turbulent
fluctuations in the flow field by finding a solution for the Reynolds stresses in
Eq. (5.23). They have been derived and improved over time by many researchers,
based on experimental measurements, boundary layer theory, wall bounded flows
and simple free shear flows. There are a number of turbulence models that range in
complexity which is summarised later in this section.
()
=+′
()
Fig. 5.12  a Schematic of instantaneous flow fluctuations in a blood vessel. b Velocity measure-
ment taken at Point X over time displaying a averaged value with random deviations from the
averaged value at any moment in time
 
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