# Introduction to numerical methods

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[[Fluid]]s could be regarded as substances those on application of external force deform or show very little resistance to deformation. This deformation in the cases of liquid and gases is substantial. Both obey common lows of motion and both for most practical computational purposes can be regarded as continuum. Forces can be considered as body forces, example: forces due to gravity or they can be considered as surface forces example: surface tension.

+ Numerical methods are at the heart of the CFD process. Researchers dedicate their attention to two fundamental aspects in CFD; i.e. physical modeling and numerics. -

+ - Based on speed and nature of flow, flows could be categorized into:
+ - *[[Laminar flow]]
+ - *[[Turbulent flow]]
+ - Or it could be categorized into:
+ - *[[Subsonic flow]]
+ - *[[Supersonic flow]]
+ - Based upon their [[Mach number]]. This Mach number could be used to determine whether the flow shall be considered as [[Incompressible flow ]] or [[Compressible flow]] for computational purposes. For the flows of Mach number smaller than 0.3, we can safely assume them as incompressible for all computational purposes.

+ -

+ - Based upon the degree of effect upon the viscosity on the flow, the flows could be classified as [[Viscous flow]] or [[Inviscid flow]] in nature. Fluids obeying [[Newton’s law]] are called [[Newtonian fluid | Newtonian fluids]]. + - Further based upon the phases of flow, the flows can be considered as [[multiphase]] or single phase flows.

+ - == Control Volume Approach == + In physical modeling, we seek a set of equations or mathematical relations that allow us to close the governing equations. In turbulence modeling for example, one is interested in devising new equations for the extra unknowns that resulted from the averaging process. -

+ - Conservation laws can be applied to given quantity of fluid. However, it is more convenient to apply them on a small spatial region, called Control volume. For this purposes the geometry under consideration is subdivided into smaller sub-regions. Together they are called mesh or grid. And the approach used is called Control volume approach for solving the flow problem.
+ - The figure 1.1 shows one such computational domain.
+ - [[Image:Img_nm_intro_01.jpg]]
+ On the other hand, the focus in numerics is to devise efficient, robust, and reliable algorithms for the solution of PDEs. PDEs are a combination of differential terms (rates of change) that describe a conservation principle. Without loss in generality, all physical processes can be described by PDEs. Now, the CFD process requires the discretization of the governing PDEs, i.e. the derivation of equivalent algebraic relations that should faithfully represent the original PDEs. This is done by transforming each differential term into an approximate algebraic relation (see [[Generic scalar transport equation]]). - Figure 1.1
+ -

+ - == Conservation Principles == + - == Conservation of scalar quantities == + - == Simple Flows == + - === [[Incompressible flow]] === + - === [[Inviscid flow | Euler or inviscid flow]] === + - === [[Potential flow]] === + - === [[Stokes flow | Stokes or creeping flow]] === + - ---- + Deriving an original numerical algorithm is not only a mathematical challenge. The investigator should also bear in mind the physics behind the term that is being discretized. For example, there are various discretization schemes for the convection term (upwind, QUICK, SOU etc...) because of the special behavior of the convection process. Similarly, the diffusion, convection, and source terms have very specialized treatments. - Return to [[Numerical methods | Numerical Methods]] +

## Latest revision as of 21:02, 24 May 2007

Numerical methods are at the heart of the CFD process. Researchers dedicate their attention to two fundamental aspects in CFD; i.e. physical modeling and numerics.

In physical modeling, we seek a set of equations or mathematical relations that allow us to close the governing equations. In turbulence modeling for example, one is interested in devising new equations for the extra unknowns that resulted from the averaging process.

On the other hand, the focus in numerics is to devise efficient, robust, and reliable algorithms for the solution of PDEs. PDEs are a combination of differential terms (rates of change) that describe a conservation principle. Without loss in generality, all physical processes can be described by PDEs. Now, the CFD process requires the discretization of the governing PDEs, i.e. the derivation of equivalent algebraic relations that should faithfully represent the original PDEs. This is done by transforming each differential term into an approximate algebraic relation (see Generic scalar transport equation).

Deriving an original numerical algorithm is not only a mathematical challenge. The investigator should also bear in mind the physics behind the term that is being discretized. For example, there are various discretization schemes for the convection term (upwind, QUICK, SOU etc...) because of the special behavior of the convection process. Similarly, the diffusion, convection, and source terms have very specialized treatments.