[10 points] The smaller Ludwig tube, high-speed wind tunnel at the Aerospace
Sciences Lab is designed to study flows at Mach 4. If the air in the Ludwig tube starts at
standard temperature and pressure, the conditions
...
[10 points] The smaller Ludwig tube, high-speed wind tunnel at the Aerospace
Sciences Lab is designed to study flows at Mach 4. If the air in the Ludwig tube starts at
standard temperature and pressure, the conditions in the test section will be the following:
M
∞=4, p∞=667.5 Pa, ρ∞=0.0339 kg/m3, T∞=68.6 K, a∞ =166 m/s, and ρ∞ =0.4625E-5
kg/(m s). Assume the largest model that will fit in the test section is 15 cm long. You
want to simulate in the wind tunnel an object flying at Mach 4 at an altitude of 30 km
(see attached standard atmosphere table).
a) How large can the full scale object be if you maintain dynamic similarity?
b) If the drag measured on the model is D=5.7 N, what will the drag be on the full scale
object (assuming it is the maximum size you computed in part (a))?
c) If the real full scale object is 3 m long, how many times larger would the wind tunnel
need to be (assuming the same flow conditions) to have dynamic similarity?
d) Assume the cost of the wind tunnel scales like the volume of the test section (or
volume of the model). How many times more would it cost to build a new wind tunnel
that could simulate the flow with full dynamic similarity? If the original wind tunnel cost
$500,000, how much would the new facility cost?
2. [5 points] Consider two solid spheres that are dropped in air at standard atmospheric
conditions at sea level. The first sphere is 2.00 mm in diameter and is made of a material
with a density 1=9 g/cm3. The second sphere is 3.00 mm in diameter and is made of a
lighter material, such that its mass is the same as the first sphere. Find the terminal
velocity for each sphere by balancing the gravitational force and the drag force. For
gravity assume g=9.81 m/s2. To compute the drag use the graph on page 11 in Reference
Figures, Tables and Equations (tables.pdf on piazza/Resources/Lecture Notes), which
gives the drag coefficient as a function of the Reynolds number. Discuss your results.
3. [5 points]. The Orion Multi-Purpose Crew Vehicle (MPCV) will replace the Space
Shuttle as America's manned spacecraft. Orion MPCV is being designed to make water
landings in case of inflight aborts. The Orion crew module (CM), therefore, needs to be
able to survive the impact with the ocean. The upward force F on the CM upon water
impact can be represented as a function of water entry velocity V, the CM nose diameter
D, water density ρ, viscosity μ and acceleration of gravity g:
a) Use the Buckingham Pi theorem to show that the force coefficient of
water impact,
C F= F
ρV 2 D2
depends on two non-dimensional parameters: the Froude number,
F= f (V , D ,ρ ,μ , g)
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Fr= V
√D g
and the Reynolds number, Re=ρ DμV
b) A ¼ scale model of the Orion CM is being tested at a water impact testing facility.
It has been found that the force coefficient is mainly determined by the Froude number
and has little sensitivity to the Reynolds number. If the maximum design velocity of CM
water landing is 30 ft/s what should be the maximum velocity at which the ¼ scale model
is tested?
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