Fundamentals of Fluid Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 108

surface. The results show that for a given inertial force, the higher dynamic viscosity
in Case 1 produces a higher friction force or diffusion of the inertial flow property.
This increased diffusion inhibits the fluid momentum and leads to a quicker transition
of the flow to a fully developed parabolic profile at a shorter distance. The developed
profile is reached at a downstream distance of y = 1.2D. Case B has a smaller viscos-
ity, so its diffusion term is smaller and there is less frictional resistance. This leads to
a slower development of the parabolic profile where the fully developed flow occurs
at a distance of y = 6D.
In fluid dynamics, the concept of dynamic similarity is frequently encountered.
This involves normalizing the mathematical equations to yield the non-dimensional
governing equations. In the example presented in Fig. 5.7 , viscosity plays a role in
holding the fluid together by resisting any motion. The inertia or flow velocity, in
contrast to viscosity, moves the fluid with through the pipe. When the viscosity was
decreased by a factor of four from 4 10 − 5 10 − 5

s, the fluid velocity
travelled farther before it was developed. If we were to decrease the velocity by
the same factor for Case 2, then this would equalise the change in viscosity and,
consequently, the developing length would be the same.
Likewise for any combination of different inlet velocities and dynamic viscosities,
the same fluid flow effect is obtained in reference to the development of the flow.
Extending this idea we can also increase the air density by a factor of four, from
ρ 1 = 1.2 kg/m 3 to ρ 2 = 4.8 kg/m 3 , while fixing the inlet velocity and dynamic viscosity
at 0.01 m/s and μ =4

× to 1 × kg/m ·

10 − 5 kg/m

s, respectively; and the same results are obtained.
The same results occur because the increase of density contributes to the increase
of the inertia force, which has the same effect as increasing the inlet velocity. An
important non-dimensional parameter that encapsulates these variables and describes
the flow characteristics is the Reynolds number ( Re ) defined as:

× ·

Inertia Force
Friction Force =

ρu in D
μ

Re = (5.15)

where ρ is density, D h is the hydraulic diameter such as in pipes or ducts, U is the
fluid velocity, and μ is the viscosity.
For a given Re number, different combinations of the density, dynamic viscosity,
and velocity values will yield the same flow entrance behaviour. Another important
use of the Reynolds number is to indicate whether the flow is laminar or turbulent.
For internal circular pipe flows that are smooth and free of sharp curvatures, the
flow will remain laminar for Re < 2,300, turbulent for Re > 4,000, and transitional
in- between those numbers.
In fluid dynamics, the dimensionless number groups play an important role in
describing fluid flow under different scales. It reduces the need to test and evaluate the
different parameters which are correlated within the dimensionless parameter. In the
Reynolds number example described above, the parameters, ρ , u in , D , μ need not be
evaluate separately as they are neatly described by the Reynolds number. Furthermore
non-dimensionalization of the governing equations of fluid flow can be performed
which provides further physical insight into the importance of various terms in the