This process starts with converting unsolvable governing
equations (Navier-Stokes equations) to a solvable
set of algebraic equations for a finite number of
points (usually around one million of cells) within
the space under consideration. By visiting and solving
the equations cell by cell as well as an iteration
technique, all detailed information for velocity,
pressure, temperature, and chemical species within
that space are acquired as a whole.
Theoretically, to analyze the fluid
flow, the basic conservation equations have to be
solved. The equations that govern the flow include
those for the conservation of momentum (Navier-Stokes
equations), the conservation of mass (continuity
equation) and the conservation of energy (energy
equation). All of these equations are in a form
of a partial differential, non-linear equation that
rarely issues exact solutions for most of the cases.
The principal approach of CFD is
to represent those equations as well as flow domain
in discretized form by using one of “finite
differencing”, “finite element”,
or “finite volume methods”. Each discretization
scheme differs in the assumption of profile within
a small volume considered and the way space is discretized.
Once discretized, it leaves meshes that cover the
whole domain and a set of algebraic equations for
that small volume (control volume). Whenever the
linearization procedure is necessary, an iterative
calculation procedure must be adopted, whereby the
equations are successively re-linearized and solved
until the solution to the original numerical form
of the equations is attained.
There are two ways to solve the
set of algebraic equations obtained by discretization:
direct method (i.e., those requiring no iteration)
and iterative method. One of the direct methods
is called the Tri-diagonal Matrix Algorithm (TDMA),
which is very efficient but applicable only in 1-D
applications because the direct method is usually
involved with matrix inversion that may cause very
expensive calculations in 2-D or 3-D problems. Alternatively,
the iterative method is widely accepted since it
is stable and applies to 2-D and 3-D situations.
It starts with guessed variables and uses the algebraic
equations to get improved variables. It goes on
until the difference between new values and the
previous values is minor. Then the converged solution
is acquired.
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