Page 77 - Mechanical Engineers' Handbook (Volume 4)

66 Fluid Mechanics

turbines, and so forth are made on this basis. Many practical problems exist, however, and
it is not always possible to achieve complete dynamic similarity. When viscous forces govern
the flow, the Reynolds number should be the same for model and prototype, the length in
the Reynolds number being some characteristic length. When gravity forces govern the flow,
the Froude number should be the same. When surface tension forces are significant, the
Weber number is used. For compressible gas flow, the Mach number is used; different gases
2
may be used for the model and prototype. The pressure coefficient C
p/( V /2), the
p
drag coefficient C drag/( V /2)A, and the lift coefficient C lift/( V /2)A will be the
2
2
L
D
same for model and prototype when the appropriate Reynolds, Froude, or Mach number is
2
the same. A cavitation number is used in cavitation studies, (p p )/( V /2) if vapor
v
v
pressure p is the reference pressure or (p p )/( V /2) if a cavity pressure is the
2
c
c
v
reference pressure.
Modeling ratios for conducting tests are listed in Table 6. Distorted models are often
used for rivers in which the vertical scale ratio might be 1/40 and the horizontal scale ratio
1/100, for example, to avoid surface tension effects and laminar flow in models too shallow.
Incomplete similarity often exists in Froude–Reynolds models since both contain a
length parameter. Ship models are tested with the Froude number parameter, and viscous
effects are calculated for both model and prototype.
Table 6 Modeling Ratios a
Modeling Parameter
Froude Froude Mach
Number, Number, Mach Number,
Reynolds Undistorted Distorted Number, Different
Ratio Number Model b Model b Same Gas d Gas d
1/2 1/2 1/2 1/2
Velocity
V m L m L m L m
m kR
m
m
p
m
p
m
m
p
V p L p L p L p V
p kR
p
p
Angular
velocity
2 1/2 — c 1/2 1/2
m
m
m L p m L p
m L p kR
m L p
p
m p p
p L m p L m
p L m kR
p L m
Volumetric
flow rate
5/2 — c — c
3/2
p
Q m L m L m L m L m
m
m
p
Q p L p L p L p V L p H
Time
2 1 /2 1 /2 1/2 1/2
1/2
1/2
p
m
t m L m p L m g p L m L p g p
p L m kR
p L m
p
t p L p m L p g m L p H L m V g m
m L p kR
m L p
p
m
m
Force
2 3 2 2 2
F m m p L m m m L m L m
m L m K m L m
m
p
F p p m L p p p L p L p
p L p K p L p
H V
a Subscript m indicates model, subscript p indicates prototype.
b For the same value of gravitational acceleration for model and prototype.
c Of little importance.
d Here
refers to temperature.