## Chapter 9

Compressible Flow

## The Time-Marching Technique

### Overview

Study Guide

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

- How is the Mach number defined?
- 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* - What is the upper Mach number often used as the limit for incompressible flow?
- Name the different Mach number regimes in compressible flow and specify the corresponding Mach number ranges
- What is an adiabatic process?
- What is required for a process to be isentropic?
- Write down the relation between internal energy and temperature and between enthalpy and temperature for
- an ideal gas with constant specific heats (calorically perfect gas)
- an ideal gas with specific heats that are functions of temperature (thermally perfect gas)

- 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?
- 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 $$\dfrac{p_2}{p_1}=\left(\dfrac{\rho_2}{\rho_1}\right)^\gamma$$
- The adiabatic energy equation
- 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.
- Rewrite the energy equation from the previous question but now expressed in terms of temperature
- Now, derive the following relation: $$\dfrac{T_o}{T}=1+\dfrac{\gamma-1}{2}M^2$$

- 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?
- What is a critical property (such as for example the critical temperature \(T^\ast\)?
- 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
- Explain the concept of choking
- Normal shocks:
- 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?
- 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)?
- How is the Hugoniot relation derived?
- Derive a relation for the pressure ratio over a normal shock
- 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?

- 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?
- convergent nozzle
- convergent-divergent nozzle

- Make a schematic sketch showing how pressure waves expands around an object moving at
- subsonic speed
- sonic speed
- supersonic speed

- Oblique shocks:
- 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, \(\theta\).
- Show schematically how the oblique shock formed ahead of a wedge traveling at supersonic speed if
- \(\theta<\theta_{max}\)
- \(\theta>\theta_{max}\)

- Can the relations for total quantities for a normal shock be used for an oblique shock? Explain why/why not.

- Prandtl-Meyer expansions:
- What is a Prandtl-Meyer expansion. Show with a figure.
- 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 |