CFD Introduction :: Continued

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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