Purdue University AAE 333 Fluid Mechanics Homework #2
1. [40 pts] A large water tank empties slowly through a small hole under the force of
gravity. The flow is steady, and the volumetric flow rate ??̇ (in m3/s) depends on the exit velocity
V, the gravitational acceleration
g, the depth of the water
h, and the diameter of the hole
D. the
viscosity
µ, and the surface tension σ (force/unit length).
(a) Express the nondimensional volumetric flow rate ??̇ in terms of its dependence on the
other nondimensional groups.
(b) A test is to be made on a 1/4th scale model. What is the ratio of the model volumetric
flow rate to the prototype volumetric flow rate that would be needed to obtain dynamical
similarity?
(c) (i) | If you now decide that, instead of the volumetric flow rate, you are interested in
the mass flow rate ??̇ , and the density ?? must therefore be included in the
dimensional analysis, will the number of nondimensional groups change?
Without going through the full derivation again, use the results of part (a) to
(ii) express the nondimensional mass flow rate ??̇ in terms of its dependence on the
other nondimensional groups.
(d) If you later discover that the mass flow rate ??̇ depends on the viscosity ?? and the surface
tension ?? (dimensions of force per unit length), in addition to ??, ??, ??, ℎ, and ??, find the
mass flow rate ??̇ in terms of its dependence on the other nondimensional groups. Hint:
you may avoid the full Buckingham Pi derivation by using the nondimensional groups
from part (c) and compiling additional groups with “new” variables and “old” repeating
variables simply by inspection.
2. [30 pts] Tests on a model propeller in a wind tunnel at sea level (air density ρ= 1
.2
kg/m3) gave the following results for the thrust at given forward velocities:
Velocity
(m/s) | 0 | 10 | 20 | 30
Thrust (N) | 300 | 278 | 211 | 100The propeller diameter was 0
.8 m and it was spun at 2000 rpm.
(a) Using dimensional analysis, find the nondimensional parameters which govern this observed
behavior.
(b) Using the experimental data given in the table, find the thrust generated by a
geometrically similar propeller of diameter 3 m, spinning at 1500 rpm at a forward velocity
of 45 m/s, while operating at an altitude where the density is half that at sea level. You may
need to interpolate from tabulated values.
3. [30 pts] The lift force (
F) produced by a fly depends on the wing beat frequency (
f), its
forward velocity (
V), the density (ρ) and viscosity (
µ) of the fluid, the length or “span” (
S), and
width or “chord” (
c) of the wings, and the Young’s modulus of the wing material (
E, dimensions
of stress).
(a) Express the non-dimensional lift force as a function of the other non-dimensional groups.
(b) A fly is observed to travel at 1 m/s when it beats its wings at 120 Hz. If a dynamically
similar robotic fly was built 100 times larger than a real fly, what would the forward
velocity and wing beat frequency be if it was tested in silicon oil with a density equal to
that of water, but with a viscosity 50 times that of water? (Assume µwater = 1×10-3 Pa·s)