2008 Compressible Flow Final Exam
December 16th 2008
15:00 — 18:00
ANSWER ALL 6 PROBLEMS; ALL PROBLEMS HAVE EQUAL VALUE; NO
NOTES OR BOOKS; TAKE γ = 1.4 IN ALL CASES; USE GAS TABLES
THAT WERE DISTRIBUTED.
Question #1
A 2-D converging-diverging nozzle is driven from a reservoir at 800 kPa and
300 K. The exit-to-throat area ratio is 2.005. Once the flow exiting the nozzle
reaches steady state, a Schlieren photograph is taken:
Determine quantitatively the range of back pressure for which the shock
configuration, illustrated in the Schlieren photograph above, will be observed.
The flow within the nozzle can be assumed essentially frictionless.
Question #2
Consider an essentially frictionless converging-diverging nozzle connected to a
pipe in which friction effects take place:
Then, what is the Mach number distribution along the pipe if the nozzle exit
2
Mach number is Me = 1.65 and the discharge tank pressure is 220 kN/m ?
Question #3
A two-dimensional intake diffuser for an air-breathing engine for supersonic
aircraft consists of a wedge forebody leading into a cowl, as shown in the
following sketch:
The purpose of the forebody is to generate an oblique shock (or shock waves) for
external compression. This is followed by a series of internal oblique shock waves
and a single normal shock wave to complete the compression to subsonic levels
compatible with engine combustor requirements. The external portion of the
compression process can be done in a one-step deflection or multi-step deflections
as shown in the (a) and (b) versions of the sketch. For M∞ = 3 , determine
quantitatively the relative merits (compression ratio, stagnation pressure loss,
Mach number reduction) of each type of configuration for the external portion of
the compression process only.
Question #4
As shown below, a cambered supersonic aerofoil is simulated by an articulated
∘
flat plate where the articulated deflection is 4 :
∘
If the angle of attack for the aerofoil is α = 2 , determine the lift and drag forces
for the aerofoil per unit span using exact shock-expansion theory.
Question #5
For the airfoil in the previous problem, calculate the lift and drag forces per unit
span using first order linearized theory. Recall that linearized theory yields a
pressure coefficient of:
CP
f,g
= ∓
2θ defl
−
−
−
−
−
−
−
2
√M∞
− 1
Question #6
After graduation, you are working at a petro-chemical plant. Your office is in a
building, a few hundred meters away but facing process equipment that handles
highly volatile materials (i.e. hydrogen, ethylene, propane, etc). During the
course of the day a pressure vessel ruptures and releases some hot volatile gases
which then mix with air to form an explosive cloud of significant size (
Rinit ∼ 5 m), which, as sometimes happens, then detonates. You are located
200 m away and just happen to be looking out the window facing the explosion
when the latter occurs. As the detonation transmits from the cloud at Rinit ≈ 5 m
into the surrounding air, the initial shock strength characterized by the
dimensionless shock over-pressure ΔP /Px = (Py − Px )/Px corresponds to a
shock Mach number Ms ≃ 7 .
What will be the damage to your window and office building caused by the
explosion?
Note that the spherical blast (shock wave) strength will decay roughly as the
inverse cube of the distance, i.e.
ΔP
1
∝
Px
3
R
and that the window and building will “feel” the reflected shock pressure.
Damage data
ΔP /Px ≈ 0.001 breaks windows
ΔP /Px ≈ 0.01 causes minor structural damage
ΔP /Px ≈ 0.1 causes major structural damage
ΔP /Px ≈ 1 is total destruction (rubble)
Answers
1. 179.5 kPa, 314.2 kPa.
2. 252.2 kPa.
3a. P2 /P∞ = 3.37 , P
∘
2
∘
/P∞ = 0.84 , M2 = 2.1 .
3b. M2 = 2.18, P2 /P∞ = 3.46 , P
∘
2
∘
/P∞ = 0.979 .
2
2
2
2
4. FL = 3.89L kN/m , FD = 0.252L kN/m .
5. FL = 3.94L kN/m , FD = 0.256L kN/m .
6. ΔP /Px = 0.00175.
π