Chapter 1

Introduction to Fluid Mechanics

table of content

Compressible Flow


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.

Sections

1.2 Definition of Compressible Flow

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

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.

Flow regimes
Figure 1: Compressible flows can be divided into flow regimes based on the local flow Mach number and the free stream Mach number

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

The first law of thermodynamics is introduced and described.

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

Three type of processes are introduced:

  • adiabatic
  • reverisible
  • isentropic

The concept of entropy is introduced and discussed in the framework of the second law of thermodynamics. Calculation of entrpoy finally leads us to the isentropic relations, 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.




Study Guide

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


  1. Explain how a fluid differs from a solid. How does a fluid element and solid element react under the presence of shear forces?
  2. The continuum concept is very central in fluid mechanics - explain this concept.
  3. Primary and secondary dimensions:
    1. What does primary dimension mean?
    2. Which are the primary dimensions used in fluid mechanics?
    3. Give examples of secondary dimensions.
    4. What does a dimensional homogenous equation imply?
  4. Explain the difference between Eulerian and Lagrangian frame of reference
  5. Show that if the shear stress is proportional to the fluid element strain rate \(\delta \theta /\delta t\), it is also proportional to the velocity gradient \(du/dy\)
  6. What is the viscosity of a fluid?
  7. For a viscous flow, what is the fluid velocity at a wall? What is this boundary condition called?
  8. What does it mean that a fluid is Newtonian?
  9. How is the Reynolds number defined? Explain in words what the Reynolds number is.
  10. Explain the following concepts:
    1. Steady-state flow
    2. Unsteady flow
    3. Incompressible flow
    4. Inviscid flow
    5. Turbulent flow
  11. What is the vapor pressure of a fluid? Explain why cavitation may occur if the pressure becomes low enough in a fluid flow
  12. Explain the difference between streamline, pathline, and streakline. Under what circumstances do these three line types coincide in a fluid flow?
  13. The energy per unit mass \(e\) can be expressed as follows: $$e=\hat{u}+\dfrac{1}{2}V^2+gz$$ The three components on the right-hand side represents different forms of energy what does each of the terms represent physically?
  14. How does the fluid viscosity vary with temperature in liquids and gases, respectively.

Document Archive
MTF053_C01.pdf Lecture notes chapter 1
MTF053_Formulas-Tables-and-Graphs.pdf A collection of formulas, tables, and graphs



Videos


Grad - Div - Curl

Viscosity

Eulerian and Lagrangian Description in Fluid Mechanics

Flow Visualization

Laminar Flow