fluid_mechanics

7.9 Some Typical Model Studies 367
Clearly the same fluid with c c m and n n m cannot be used in model and prototype unless
the length scale is unity 1which means that we are running tests on the prototype2. In high-speed
aerodynamics the prototype fluid is usually air, and it is difficult to satisfy Eq. 7.23 for reasonable
length scales. Thus, models involving high-speed flows are often distorted with respect to
Reynolds number similarity, but Mach number similarity is maintained.
Froude number
similarity is usually
required for models
involving freesurface
flows.
V7.17 River flow
model
7.9.3 Flow with a Free Surface
Flows in canals, rivers, spillways, and stilling basins, as well as flow around ships, are all examples
of flow phenomena involving a free surface. For this class of problems, both gravitational and
inertial forces are important and, therefore, the Froude number becomes an important similarity
parameter. Also, since there is a free surface with a liquid–air interface, forces due to surface tension
may be significant, and the Weber number becomes another similarity parameter that needs
to be considered along with the Reynolds number. Geometric variables will obviously still be important.
Thus a general formulation for problems involving flow with a free surface can be expressed
as
Dependent pi term f a / i
(7.24)
/ , e
/ , rV/
m , V
2g/ , rV 2 /
s
b
As discussed previously, / is some characteristic length of the system, / i represents other pertinent
lengths, and e/ is the relative roughness of the various surfaces. Since gravity is the driving force
in these problems, Froude number similarity is definitely required so that
V m
2g m / m
V
2g/
The model and prototype are expected to operate in the same gravitational field 1g m g2, and
therefore it follows that
V m
V / m
B / 1l /
(7.25)
Thus, when models are designed on the basis of Froude number similarity, the velocity scale is determined
by the square root of the length scale. As is discussed in Section 7.8.3, to simultaneously
have Reynolds and Froude number similarity it is necessary that the kinematic viscosity scale be
related to the length scale as
n m
n 1l /2 3 2
(7.26)
The working fluid for the prototype is normally either freshwater or seawater and the length scale
is small. Under these circumstances it is virtually impossible to satisfy Eq. 7.26, so models involving
free-surface flows are usually distorted. The problem is further complicated if an attempt is made
to model surface tension effects, since this requires the equality of Weber numbers, which leads to
the condition
V7.18 Boat model
s mr m
sr 1l /2 2
(7.27)
for the kinematic surface tension 1sr2. It is again evident that the same fluid cannot be used in
model and prototype if we are to have similitude with respect to surface tension effects for l / 1.
Fortunately, in many problems involving free-surface flows, both surface tension and viscous
effects are small and consequently strict adherence to Weber and Reynolds number similarity is
not required. Certainly, surface tension is not important in large hydraulic structures and rivers.
Our only concern would be if in a model the depths were reduced to the point where surface tension
becomes an important factor, whereas it is not in the prototype. This is of particular importance
in the design of river models, since the length scales are typically small 1so that the width of
the model is reasonable2, but with a small length scale the required model depth may be very small.
To overcome this problem, different horizontal and vertical length scales are often used for river