计算流体力学(CFD)文档——2._Governing_equations.pdf

2. FLUID-FLOW EQUATIONS SPRING 2011

Control-volume approach
Conservative integral and differential equations
Non-conservative differential equations
Non-dimensionalisation
Summary
Examples


Fluid dynamics is governed by conservation equations for:
• mass;
• momentum;
• energy;
•(for a non-homogenous fluid) individual constituents.

The continuum equations for these can be expressed mathematically in many different ways.
In this section we shall show that they can be written as equivalent:
• integral (. control-volume ) equations;
• differential equations;
In addition, the differential equations may be either conservative or non-conservative .

The focus will be the integral equations because they are physically more fundamental and
form the basis of the finite-volume method. However, the equivalent differential equations
are often easier to write down, manipulate and, in some cases, solve analytically.

Although there are different fluid variables, most of them satisfy the same generic equation,
which can therefore be putationally by the same subroutine. This is called the
scalar-transport or advection-diffusion equation.


Control-Volume Approach

The rate of change of some quantity within an arbitrary control volume is determined by:
• rate of transport across the bounding surface (“ flux ”);
• rate of production within that control volume (“ source ”).
RATE OF CHANGE  FLUX  SOURCE 
+ = (1)
 inside V  out of boundar y  inside V  V

The flux across the bounding surface can be divided into:
• advection : transport with the flow;
• diffusion : net transport by molecular or turbulent fluctuations.
(Some authors – but not this one – prefer the term convection to advection .)

RATE OF CHANGE  ADVECTION + DIFFUSION  SOURCE 
+ = (2)
 inside V  through boundary  inside V 

The finite-volume method is a natural disc