fluid_mechanics

a minor role and can be neglected. However, in other problems 1such as turbulent flow through
pipes2 roughness can be very important.
It follows from this discussion that for flow in closed conduits at low Mach numbers, any
dependent pi term 1the one that contains the particular variable of interest, such as pressure drop2
can be expressed as
Dependent pi term f a / i
/ , e
/ , rV/
m b
7.9 Some Typical Model Studies 361
(7.16)
This is a general formulation for this type of problem. The first two pi terms of the right side of
Eq. 7.16 lead to the requirement of geometric similarity so that
or
/ im
/ i
/ m /
e m
/ m
e /
Accurate predictions
of flow behavior
require the correct
scaling of
velocities.
/ im
/ i
e m
e / m
/ l /
This result indicates that the investigator is free to choose a length scale, l / , but once this scale is
selected, all other pertinent lengths must be scaled in the same ratio.
The additional similarity requirement arises from the equality of Reynolds numbers
r m V m / m
m m
rV/
m
From this condition the velocity scale is established so that
V m
(7.17)
V m m r /
m r m / m
and the actual value of the velocity scale depends on the viscosity and density scales, as well as
the length scale. Different fluids can be used in model and prototype. However, if the same fluid
is used 1with m m m and r m r2, then
V / / m
Thus, V m Vl / , which indicates that the fluid velocity in the model will be larger than that in the
prototype for any length scale less than 1. Since length scales are typically much less than unity,
Reynolds number similarity may be difficult to achieve because of the large model velocities required.
With these similarity requirements satisfied, it follows that the dependent pi term will be equal
in model and prototype. For example, if the dependent variable of interest is the pressure differential,
3 ¢p, between two points along a closed conduit, then the dependent pi term could be
expressed as
ß 1 ¢p
rV 2
The prototype pressure drop would then be obtained from the relationship
V m
¢p r a V 2
b ¢p
r m V m
m
so that from a measured pressure differential in the model, ¢p m , the corresponding pressure differential
for the prototype could be predicted. Note that in general ¢p ¢p m .
3 In some previous examples the pressure differential per unit length, ¢p / , was used. This is appropriate for flow in long pipes or conduits
in which the pressure would vary linearly with distance. However, in the more general situation the pressure may not vary linearly with position
so that it is necessary to consider the pressure differential, ¢p, as the dependent variable. In this case the distance between pressure
taps is an additional variable 1as well as the distance of one of the taps measured from some reference point within the flow system2.