Compressible Flow

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Overview

This first chapter gives an overview of the field of compressible flow. In the first sections important milestones in the compressible flow history are presented. Concepts and parameters relevant for compressible flows are introduced and discussed. A short review of thermodynamics is given at the end of the chapter.

Online Flow Calculator

Sections of the online flow calculator for compressible flow problems CFLOW related to this chapter:

• Section 1: Gas Relations

Chapter Roadmap

Sections

1.2 Definition of Compressible Flow

This chapter discusses the difference between compressible and incompressible flow. The concept compressibility is defined and discussed.

Compressibility is defined as the relative change in specific volume related to a change in pressure

$$\tau=-\dfrac{1}{\nu}\dfrac{\partial \nu}{\partial p}$$

This definition must be complemented with information about the process by which the pressure is changed. Two processes are introduced, namely the constant-temperature process and the adiabatic (no heat exchange) process.

Flows where changes in density (or specific volume) are too large to be neglected should be treated as compressible. Typically this means that the relative change in density is greater than 5% (\(\Delta \rho/\rho > 0.05\)).

1.3 Flow Regimes

The free stream and local Mach numbers are defined as \(M_\infty=V_\infty/a_\infty\) and \(M=V/a\), respectively. The Mach number is used to define different regimes of compressible flows (subsonic, transonics, supersonic, hypersonic). The concept inviscid flows is introduced.

1.4 A Brief Review of Thermodynamics

This chapter gives, as the title hints, a short review of thermodynamics. Concepts important for compressible flows are introduced:

• perfect gas
• equation of state (ideal gas law) \(p=\rho RT\)
• internal energy \(e\)
• enthalpy \(h=e+p/\rho\)
• thermally perfect gas
• callorically perfect gas
• specific heat

$$C_p=\left(\frac{\partial h}{\partial T}\right)_p$$
$$C_v=\left(\frac{\partial e}{\partial T}\right)_v$$
$$C_p-C_v=R$$
$$\gamma=\frac{C_p}{C_v}$$
$$C_p=\frac{\gamma R}{\gamma-1}$$
$$C_v=\frac{R}{\gamma-1}$$

In a thermally perfect gas energy and enthalpy are functions of temperature. A calorically perfect gas is a subset of thermally perfect gases where energy and enthalpy are linear functions of temperature (the specific heats are constants). The calorically perfect gas assumption applies for temperatures in the range 100 K to 600 K while the thermally perfect gas assumption applies up to about 2500 K.

The first law of thermodynamics is introduced and described.

$$\delta q + \delta w =de$$

where \(de\) is the change in internal energy of a system (a identifiable set of molecules or a fixed mass), \(\delta q\) is heat added to the system from its surroundings, and \(\delta w\) is work done by the system on its surroundings. The heat addition and fluid work are process dependent.

Three type of processes are introduced:

• adiabatic (no heat exchange)
• reverisible (no irreversible losses introduced)
• isentropic (adiabatic and reversible)

The concept of entropy is introduced and discussed in the framework of the second law of thermodynamics. The first and second laws of thermodynamics combined gives

$$ds= C_p\dfrac{dT}{T}-R\dfrac{dp}{p}$$

Integrated between two stations 1 and 2 the relation above becomes

$$s_2-s_1=\int_1^2C_p\dfrac{dT}{T}-R\ln\left(\dfrac{p_2}{p_1}\right)$$

If the gas is calorically perfect (constant specific heats) we get

$$s_2-s_1=C_p\ln\left(\dfrac{T_2}{T_1}\right)-R\ln\left(\dfrac{p_2}{p_1}\right)$$

Finally, for an isentropic process (\(s_2=s_1\)), this leads to the isentropic relations for calorically perfect gases, which will prove to be very useful throughout the rest of the book.

$$\frac{p_2}{p_1}=\left(\frac{\rho_2}{\rho_1}\right)^{\gamma}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}$$

Aerodynamic Forces on a Body

This section describes how forces on a body immersed in a fluid flow can be calculated. As an example, a formula for the evaluation of aerodynamic forces on a wing is derived and in conjunction with that the force components lift and drag are introduced.

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Study Guide

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