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2. THE FLUID-FLOW EQUATIONS SPRING 2005
2.1 General control-volume form of the governing equations
2.2 Fluid-flow equations
2.2.1 Mass (continuity)
2.2.2 Momentum
2.2.3 General scalar
2.2.4 Momentum components as transported scalars
2.2.5 Non-gradient diffusion
2.2.6 Moving control volumes
2.3 Other differential equations for fluid flow
2.3.1 Material derivatives and non-conservative form of the flow equations
2.3.2 Equations for derived variables
2.4 Compressible and incompressible flow
2.5 Non-dimensionalisation
2.6 Summary
Fluid dynamics is governed by conservation of:
• mass;
• momentum;
• energy;
• any additional constituents.
These basic physical principles can be expressed mathematically in many different ways.
Because they form the basis of the finite-volume method for CFD we will focus on control-
volume (i.e. integral) forms of the governing equations, but show also how these are related
to equivalent differential equations which are easier to write down and manipulate.
2.1 General Control-Volume Form of the Governing Equations
The rate of change of some quantity within an arbitrary control volume is V
determined by the net rate of transport across its bounding surface (“f lux”)
and the rate of production within that control volume (“source”):
RATE OF CHANGE FLUX SOURCE
+ = (1)
inside V out of boundary inside V
Flux (= rate of transport across a surface) can be divided into:
• advection (others prefer convection): movement with the flow;
• diff usion: net transfer across a surface due to molecular or turbulent fluctuations.
RATE OF CHANGE ADVECTION + DIFFUSION SOURCE
+
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