## The Time-Marching Technique

### Study Guide

The questions below are intended as a "study guide" and may be helpful when reading the text book.

1. How is the Mach number defined?
2. Show by estimation of the density variation in a fluid flow that the criteria for incompressible flow is $$M\ll1$$
hint: see Eqn 4.13 - 4.17 in White
3. What is the upper Mach number often used as the limit for incompressible flow?
4. Name the different Mach number regimes in compressible flow and specify the corresponding Mach number ranges
5. What is an adiabatic process?
6. What is required for a process to be isentropic?
7. Write down the relation between internal energy and temperature and between enthalpy and temperature for
1. an ideal gas with constant specific heats (calorically perfect gas)
2. an ideal gas with specific heats that are functions of temperature (thermally perfect gas)
8. Derive an expression for the speed of sound for a generic fluid. In the derivation, an assumption regarding the pressure derivative is made -- what is the assumption?
9. The speed of sound in a generic fluid is given by $$a=\sqrt{\left(\dfrac{\partial p}{\partial \rho}\right)_s}$$ From the relation above, derive an expression for the speed of sound in a calorically perfect gas (perfect gas with constant specific heats) as a function of temperature. Use the isentropic relation $$\dfrac{p_2}{p_1}=\left(\dfrac{\rho_2}{\rho_1}\right)^\gamma$$
1. Write down the energy equation for steady-state adiabatic compressible flow without potential energy changes and no viscous work. The equation should be written out using enthalpy.
2. Rewrite the energy equation from the previous question but now expressed in terms of temperature
3. Now, derive the following relation: $$\dfrac{T_o}{T}=1+\dfrac{\gamma-1}{2}M^2$$
11. Which of the properties $$h_o$$, $$T_o$$, $$a_o$$, $$p_o$$, and $$\rho_o$$ are constants in a flow if the flow is adiabatic and isentropic, respectively?
12. What is a critical property (such as for example the critical temperature $$T^\ast$$?
13. Using the continuity equation and energy equation on differential form together with the definition of speed of sound, the following relation can be derived $$\dfrac{dV}{V}=\dfrac{dA}{A}\dfrac{1}{M^2-1}=-\dfrac{dp}{\rho V^2}$$ Show, using the relation given above, how the velocity and pressure changes in a flow through a divergent or convergent duct for initially subsonic flow or initially supersonic flow
14. Explain the concept of choking
15. Normal shocks:
1. Explain what a normal shock is. What happens with the velocity, pressure and total temperature over a normal shock? How is the critical area, $$A^\ast$$ effected?
2. The normal shock equation system has two solutions. How do we know which solution that is the correct one? %What is the implication of this for very weak waves (such as for example acoustic/sound waves)?
3. How is the Hugoniot relation derived?
4. Derive a relation for the pressure ratio over a normal shock
5. How does pressure ($$p$$), temperature ($$T$$), density ($$\rho$$), Mach number ($$M$$), total pressure ($$p_o$$), and total temperature ($$T_o$$) change over a normal shock?
16. Explain, using a schematic figure, how pressure and velocity varies in a nozzle (the two types of nozzles specified below) as the downstream pressures (back pressures) changes. The upstream end of the nozzle is connected to a large reservoir (tank) with the pressure $$p_o$$ and temperature $$T_o$$. What happens with the massflow when the downstream pressure changes?
1. convergent nozzle
2. convergent-divergent nozzle
17. Make a schematic sketch showing how pressure waves expands around an object moving at
1. subsonic speed
2. sonic speed
3. supersonic speed
Show the location of object at the time $$t$$ and the time $$t+\Delta t$$. In your sketch, show features such as Mach waves and the Mach angle
18. Oblique shocks:
1. Show schematically how the velocity (normal velocity component, tangential velocity component, and the total velocity) changes over an oblique shock. Indicate the shock angle, $$\beta$$, and the deflection angle, $$\theta$$.
2. Show schematically how the oblique shock formed ahead of a wedge traveling at supersonic speed if
1. $$\theta<\theta_{max}$$
2. $$\theta>\theta_{max}$$
3. Can the relations for total quantities for a normal shock be used for an oblique shock? Explain why/why not.
19. Prandtl-Meyer expansions:
1. What is a Prandtl-Meyer expansion. Show with a figure.
2. How does pressure ($$p$$), temperature ($$T$$), density ($$\rho$$), Mach number ($$M$$), total pressure ($$p_o$$), and total temperature ($$T_o$$) change over an expansion region?

 Document Archive MTF053_C09.pdf lecture notes chapter 9 MTF053_Compressible-Flow-Hugoniot-Equation.pdf Complementary material - Derivation of the Hugoniot equation - a relation of thermodynamic properties over a normal shock MTF053_Formulas-Tables-and-Graphs.pdf A collection of formulas, tables, and graphs