fluid_mechanics

7.9 Some Typical Model Studies 369
SOLUTION
The width, w m , of the model spillway is obtained from the length
scale, l / , so that
Therefore,
(Ans)
Of course, all other geometric features 1including surface roughness2
of the spillway must be scaled in accordance with the same
length scale.
With the neglect of surface tension and viscosity, Eq. 7.24 indicates
that dynamic similarity will be achieved if the Froude
numbers are equal between model and prototype. Thus,
and for g m g
Since the flowrate is given by Q VA, where A is an appropriate
cross-sectional area, it follows that
where we have made use of the relationship A mA 1/ m/2 2 . For
l and Q 125 m 3 / 1
s
15
Q m
Q V m
VA
w m
w l / 1
15
w m 20 m
15 1.33 m
V m
2g m / m
V m
V / m
B /
A m
1l / 2 5 2
/
2
m
B / a/ m
/ b
Q m 1 1
152 5 2 1125 m 3 s2 0.143 m 3 s
(Ans)
The time scale can be obtained from the velocity scale, since
the velocity is distance divided by time 1V /t2, and therefore
V
/ t m
V m t / m
V
2g/
or
t m
t
V / m
V m / / m
B / 2l /
This result indicates that time intervals in the model will be
smaller than the corresponding intervals in the prototype if
l For l / 1 / 6 1.
15 and a prototype time interval of 24 hr
t m 2 1 15 124 hr2 6.20 hr (Ans)
COMMENT As indicated by the above analysis, the time
scale varies directly as the square root of the length scale. Thus,
as shown in Fig. E7.8b, the model time interval, t m , corresponding
to a 24-hr prototype time interval can be varied by changing
the length scale, l / . The ability to scale times may be very useful,
since it is possible to “speed up” events in the model which
may occur over a relatively long time in the prototype. There is
of course a practical limit to how small the length scale (and the
corresponding time scale) can become. For example, if the
length scale is too small then surface tension effects may
become important in the model whereas they are not in the prototype.
In such a case the present model design, based simply
on Froude number similarity, would not be adequate.
t m , hr
20
16
12
8
4
(1/15, 6.20 hr)
0
0 0.1 0.2
m ___
0.3 0.4 0.5
F I G U R E E7.8b
V7.20 Testing of
large yacht mode
There are, unfortunately, problems involving flow with a free surface in which viscous, inertial,
and gravitational forces are all important. The drag on a ship as it moves through water is
due to the viscous shearing stresses that develop along its hull, as well as a pressure-induced component
of drag caused by both the shape of the hull and wave action. The shear drag is a function
of the Reynolds number, whereas the pressure drag is a function of the Froude number. Since both
Reynolds number and Froude number similarity cannot be simultaneously achieved by using water
as the model fluid 1which is the only practical fluid for ship models2, some technique other than
a straightforward model test must be employed. One common approach is to measure the total drag
on a small, geometrically similar model as it is towed through a model basin at Froude numbers
matching those of the prototype. The shear drag on the model is calculated using analytical techniques
of the type described in Chapter 9. This calculated value is then subtracted from the total
drag to obtain pressure drag, and using Froude number scaling the pressure drag on the prototype
can then be predicted. The experimentally determined value can then be combined with a calculated
value of the shear drag 1again using analytical techniques2 to provide the desired total drag