Fluids: Liquids and Gases - Computer Animation: Algorithms and Techniques - page 272

FIGURE 8.19
Hybrid method.

direction of flow

flow out

differential element

flow in

FIGURE 8.20
Differential element used in Navier-Stokes.

The full CFD equations consist of equations that state, in a closed system, the following:

￿ Mass is conserved.
￿ Momentum is conserved.
￿ Energy is conserved.

When expressed as equations, these are the Navier-Stokes (NS) equations. The NS equations are
nonlinear differential equations. There are various simplifications that can be made in order to make
the NS equations more tractable. First of all, for most all graphics applications, energy conservation
is ignored. Another common simplification used in graphics is to assume the fluid is nonviscous.
These assumptions result in what are called Euler equations [ 12 ]. By ignoring energy conservation
and viscosity, the Euler equations describing the conservation of mass and conservation of momen-
tum can be readily constructed for a given flow field. When dealing with liquids (as opposed to a
gas), another useful assumption is that the fluid is incompressible, which means that its density does
not change.

Conservation of mass
Conservation of mass is expressed by the continuity equation [ 4 ]. The underlying assumption of the
conservation of mass is that mass is neither created nor destroyed inside of the fluid. The fluid flows
from location to location and, if compressible, can become more dense or less dense at various loca-
tions. To determine how the total mass inside of an element is changing, all of the sources and sinks in
the element have to be added up. The divergence of a flow field describes the strength of the source or