Mechanics of Fluids

5.5 A torpedo-shaped object 900 mm diameter is to move in air at
60 m · s −1 and its drag is to be estimated from tests in water
on a half-scale model. Determine the necessary speed of the
model and the drag of the full-scale object if that of the model
is 1140 N. (Fluid properties as in Problem 5.3.)
5.6 What types of force (acting on particles of fluid) would you
expect to influence the torque needed to operate the rudder of
a deeply submerged mini-submarine? To investigate the operation
of such a rudder, tests are conducted on a quarter-scale
model in a fresh-water tunnel. If, for the relevant temperatures,
the density of sea water is 2.5% greater than that of fresh
water, and the dynamic viscosity 7% greater, what velocity
should be used in the water tunnel to correspond to a velocity
of 3.5 m · s −1 for the prototype submarine? If the measured
torque on the model rudder is 20.6 N · m, what would be the
corresponding torque on the full-size rudder?
5.7 Show that, for flow governed only by gravity, inertia and pressure
forces, the ratio of volume flow rates in two dynamically
similar systems equals the 5/2 power of the length ratio.
5.8 The drag on a stationary hemispherical shell with its open, concave
side towards an oncoming airstream is to be investigated
by experiments on a half-scale model in water. For a steady
air velocity of 30 m · s −1 determine the corresponding velocity
of the water relative to the model, and the drag on the prototype
shell if that on the model is 152 N. (Fluid properties as in
Problem 5.3.)
5.9 The flow rate over a spillway is 120 m 3 · s −1 . What is the maximum
length scale factor for a dynamically similar model if a
flow rate of 0.75 m 3 · s −1 is available in the laboratory? On a
part of such a model a force of 2.8 N is measured. What is the
corresponding force on the prototype spillway? (Viscosity and
surface tension effects are here negligible.)
5.10 An aircraft is to fly at a height of 9 km (where the temperature
and pressure are −45 ◦ C and 30.2 kPa respectively) at
400 m · s −1 . A 1/20th-scale model is tested in a pressurized
wind tunnel in which the air is at 15 ◦ C. For complete dynamic
similarity what pressure and velocity should be used in the
wind-tunnel? (For air at T K, µ ∝ T 3/2 /(T + 117).)
5.11 In a 1/100th-scale model of a harbour what length of time
should correspond to the prototype tidal period of 12.4 hours?
5.12 Cavitation is expected in an overflow siphon where the
head is −7 m, the water temperature 10 ◦ C and the rate
of flow 7 m 3 · s −1 . The conditions are to be reproduced
on a 1/12th-scale model operating in a vacuum chamber
with water at 20 ◦ C. If viscous and surface tension effects
may be neglected, calculate the pressure required in the
vacuum chamber and the rate of flow in the model (Hint:
In the absence of friction the velocity in the siphon is
Problems 189