Integral Forms of the Conservation Equations for Inviscid Flows

Overview

The second chapter is devoted to the derivation of the integral formulation of the governing equations for inviscid compressible flow.

Chapter Roadmap

Sections

2.2 Approach

The governing equations for inviscid flows on integral form are derived using a control volume approach.

Conservation of mass, momentum, and energy for over a fixed control volume leads to the governing flow equations on integral form.

2.3 Continuity Equation

Mass can be neither created nor destroyed
Conservation of mass is applied on a control volume which leads to the continuity equation on integral formulation.

$$\frac{d}{dt}\iiint_{\Omega}\rho d{\mathscr{V}}+\oint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=0$$

The rate of change of mass within the control volume \(\Omega\) equals the net flux of mass over the control volume surface \(\partial\Omega\).

2.4 Momentum Equation

The time rate of change of momentum of a body equals the net force exerted on it
Newtons's second law is applied on a control volume leading to the integral formulation of the momentum equation.

$$\frac{d}{dt}\iiint_{\Omega}\rho{\mathbf{v}} d{\mathscr{V}}+\oint_{\partial \Omega}\left[\rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}+p{\mathbf{n}}\right]dS=\iiint_{\Omega}\rho{\mathbf{f}} d{\mathscr{V}}$$

Note that the surface pressure force is lumped together with the net flux of momentum over the control volume surface since these two terms are both surface integrals on the same form. The Term on the right-hand-side of the equation is the volume integral of body forces (gravity, magnetic fields, Coreolis forces, etc).

2.6 Energy Equation

Energy can be neither created nor destroyed; it can only change in form.
The first law of thermodynamics is applied to a control volume leading to the integral formulation of the energy equation.

$$\frac{d}{dt}\iiint_{\Omega}\rho e_o d{\mathscr{V}}+\oint_{\partial \Omega}\left[\rho h_o {\mathbf{v}}\cdot {\mathbf{n}}\right]dS=\iiint_{\Omega}\rho{\mathbf{f}}\cdot{\mathbf{v}} d{\mathscr{V}}$$

Note that total energy \(e_o\) and total enthalpy \(h_o\) are used here. The total energy is the energy \(e\) plus the kinetic energy per unit mass \(0.5\mathbf{v}\cdot\mathbf{v}\)

Study Guide

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

  1. What is the physical interpretation of each of the terms in the continuity equation on integral form $$\frac{d}{dt}\iiint_{\Omega}\rho d{\mathscr{V}}+\oint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=0$$
  2. What is the physical interpretation of each of the terms in the momentum equation on integral form $$\frac{d}{dt}\iiint_{\Omega}\rho{\mathbf{v}} d{\mathscr{V}}+\oint_{\partial \Omega}\left[\rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}+p{\mathbf{n}}\right]dS=\iiint_{\Omega}\rho{\mathbf{f}} d{\mathscr{V}}$$
  3. What is the physical interpretation of each of the terms in the energy equation on integral form $$\frac{d}{dt}\iiint_{\Omega}\rho e_o d{\mathscr{V}}+\oint_{\partial \Omega}\left[\rho e_o ({\mathbf{v}}\cdot{\mathbf{n}}) +p {\mathbf{v}}\cdot {\mathbf{n}}\right]dS=\iiint_{\Omega}\rho{\mathbf{f}}\cdot{\mathbf{v}} d{\mathscr{V}}$$
  4. How can the control volume formulations of the governing flow equations be used?
  5. Check through the three conservation theorems (the integral forms for the conservation of mass, momentum and energy) and make sure you understand how to apply them for a specific case. As a simple example, look at the derivation on the slides of Lecture 2. There are also some examples on how to use these relations in the book.
  6. In Formulas, Tables & Graphs, check the conservation laws. Make sure you are familiar with the notation.