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Computational Fluid Dynamics

Lecture IINumerical Methods and Criteria for CFD

Dr. Ugur GUVEN

Professor of Aerospace Engineering

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Discretization

The process of converting differentialequations into algebraic equations is called

discretization.

Discretization is necessary for numerical

methods and computational programming to

be employed in solving fluid dynamics

equations

Of course, improper discretization may cause

errors in calculation

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Discretization Methods

Make sure that you choose the correct

discretization methods for solving partial

differential equations. These include:

1) Finite Difference Methods

2) Finite Volume Methods3) Finite Elements Methods

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Discretization Methods

These methods can be summarized as:

A) Finite Difference Methods are usedespecially for easy geometries and for simple

flow problem equations. B) Finite Volume Methods are more

applicable to a wide variety of flow problems

and wide variety of geometries C) Finite Elements Method is more flexible,

but it is more suitable for solid structures.

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Finite Differences Methods

Finite Differences Methods are used to turndifferential equations into algebraic equations

by transforming derivatives into limited terms

using Taylors series. For simple differential equations with first to

nth degree derivatives, you can use finite

differences to create algebraic equations of

any equation that contains partial derivatives.

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Finite Volume Methods

In the finite volume methods, you concentrate

on the geometry of the solution domain and

on the geometry of the boundary values.

Hence, the equations are solved on the

integral level by limiting the geometricdomains.

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Grid Types for Numerical Methods

Block Structured Grid

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Grid Types for Numerical Methods

Unstructured Grid

Composite (Chimera) Grid

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Meshing

Meshing is the concept of adjusting the grid to

cover as many points as possible, in order to

increase the accuracy without creating complex

calculations.

Each cell is calculated separately and then all cells

are superimposed to create one big solution.

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Properties of the Solution Method

Consistency

Stability

Convergence

Conservation

Boundedness

Realizability

Accuracy

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Consistency

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Stability

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Convergence

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Conservation

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Boundedness

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Realizability

Divergence

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Accuracy It is to be remembered that the

final solution is only anapproximation of the real

differential equations.

Hence, there may be accuracyissues due to the amount of

errors in the approximation

process

It is important for accuracy to be

within acceptable boundaries

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Errors in the Numerical Solution

Various errors can be introduced into the solutionduring the course of the development of the

solution algorithm as well as in the programming

stage or in setting up of the boundary conditions

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Error Control in Numerical Solutions

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Thank You

www.cfdlectures.co.cc

drguven@live.com

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