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ME 521: Dynamics and Thermodynamics of Compressible Flows
In all problems assume that there are no chemical reactions, radiation, viscosity, or other effects leading to addition of subtraction of heat. The flows are time-independent, unless it is stated otherwise. Where necessary, assume the model of a calorically perfect gas and use y = 1.4. The universal gas constant is R=8.314J/mol K, and the molecular weight of air is 29 kg/kmol.

Briefly explain all your answers.

Problem 1
Consider a convergent-divergent nozzle with the exit-to-throat area ratio 3.85 and reservoir at pO =8 atm and TO=300 K.
a) What should be the exit pressure pe1 to realize an isentropic subsonic-supersonic flow regime?
b) Consider the situation, in which the exit pressure is gradually reduced from p = pO to p = pe1. Describe the states of the flow that appear. Calculate the values of the exit pressure, at which:
a. Flow becomes chocked
b. Flow becomes supersonic within the nozzle
c. Shock waves appear within the nozzle.

Problem 2
Give a definition of velocity potential. For the following flows, determine whether or not they can be described using the velocity potential. Justify your answers.
a) Subsonic flow past a sphere moving in air
b) Two-dimensional flow of air past a wedge of total angle 60° moving at zero attack angle (so that the deflection on each side is by 30°) at M=4.0
c) The same flow as in (b), but at M=2.0.

Problem 3
Consider a primary expansion wave (before any reflections occur) propagating in a shock tube. The driver gas before the diaphragm rupture had p =8 atm and T=2400 K. The primary shock propagating in the opposite direction creates the following gas conditions: pressure 3 atm, temperature 800 K, and gas velocity 700 m/s.
a) Find the speeds uh and ut of propagation of the head and tail of the wave.
b) Calculate pressure and temperature at the point moving with velocity (uh + ut)/2. Do they vary with time? Explain your answer.

Problem 4
Consider the flow of air past a slender body. The velocity field deviates only slightly from the uniform far-field velocity, so the flow can be considered irrotational. Consider three cases: (a) M=0.7, (b) M=0.92, and (c) M=2.5. Write the simplest possible equations describing, as accurate approximations, each flow.

Problem 5
Write the full system of differential equations for a three-dimensional unsteady flow of a compressible gas. Assume that viscous and heat conduction effects are negligible, but there is a non-negligible body force f(x, t). Write equations in component form using the Cartesian coordinates. Use the conservation version of the equations. Identify the physical principle expressed by each equation.

Extra Problem
You are asked to determine the lift force on a nearly two-dimensional (long and uniform in the direction perpendicular to the flow) wing profile. Flow of air at atmospheric conditions in the range of velocities between 100 and 200 m/s is considered. Can this be done experimentally, using the same wing and flow of water? If yes, explain the strategy and principles, on which it is based.

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