Finite Difference Methods - PowerPoint Presentation

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Slide1

Finite Difference Methods

Dr.

Ugur

GUVEN

Aerospace Engineer (

P.hD

)

Nuclear Science and Technology Engineer (

M.Sc

)Slide2

Discretization of Equations

The first step in solving any CFD problem is to discretize the equations.

Usually all fluid dynamics equations are in a partial differential equation format

These differential equations must be transformed into algebraic form, so that they can become solvable by the computer

The most common Discretization technique for Partial Differential Equations is the Finite Difference MethodsSlide3

Taylor Series

The primary background of any discretization using Finite Differences depends on using the Taylor series.

In the Taylor series, you can approximate a solution to any function at (x +

dx

) as long as you know the initial valueSlide4

Taylor SeriesSlide5

Forward Differences

Lets write the X component of velocity at Point (

i,j

) for a 2d flow.

Lets solve for the first term derivative of the above equationSlide6

Forward Differences

Hence after the first term, the remaining terms can be classified as truncation error and can be discarded (if you accept the magnitude of the error)

Then, we get a first order algebraic equation that approximates the partial derivative and it is called a forward differenceSlide7

Rearward Differences

Lets now write the Taylor series for the point u(i-1,j) in a 2d flowSlide8

Rearward Differences

Thus, if we solve the above equations to get a partial derivative of u to x, then we will have a first order accurate algebraic form of the partial derivative in the rearward difference form.Slide9

Central Differences

However, the problem with both the rearward differences as well as the forward differences is the fact that they are first order algebraic representations of a partial derivative.

Hence, the accuracy is greatly decreased due to this and we will need to look for ways to increase the accuracy of the partial derivative to reduce the overall error that will be formed in the equationsSlide10

Central Differences

Lets subtract the following equations to try to increase the order of accuracySlide11

Central Differences

Hence, solving the equation would cause the following equation to be formed:

Hence, the following is a second order accurate central difference representation:Slide12

Central Differences

Hence, the following equation is a more accurate representation of a partial derivative of u over x and thus it has less errors. Moreover, it takes data from both sides of the grid over point (

I,j

)Slide13

Differences with Respect to Y

Using the methodology defined above, it will be possible to create forward, rearward or central differences with respect to y in partial differentiationSlide14

Partial Derivatives Using Finite DifferencesSlide15

Writing Partial Derivatives in Algebra

Hence, using the methods of finite differences, you can easily transform first degree partial derivatives so that you can create an algebraic equation.

For example, transform the following partial differential equation using finite differencesSlide16

Second Order Partial Derivatives

To find the second order partial derivatives with respect to x, lets add the Taylor series expansion for u(i+1,j) and u(i-1,j)Slide17

Second Order Partial Derivatives

Summation of the above equations gives:

As a result, the second order partial derivative is written in central differences as:Slide18

Second Order Partial Derivatives

Hence, the second order partial derivatives written in the central difference notation would be:Slide19

Mixed Partial Derivatives

By writing a mixed partial derivative , we will be able to write second order equations as well.Slide20

Second Order Partial DerivativesSlide21

Transformation of Partial Differential Equations

Transform the following partial differential equation into an algebraic equation by using Finite DifferencesSlide22

HomeworkSlide23

Engineering Example 1

Calculate the shear stress and the heat transfer at the wall with the following data by using finite differencesSlide24

Example 1

The shear stress and the heat transfer at the wall is given by the following equations in first dimensional problems.Slide25

Example 1

First Order Difference Solution, solve for the other pointsSlide26

Time Marching Solution

In the solution of fluid dynamics equations, it is usually customary to solve flows changing over time. At each ‘t’ step of the flow, the flow properties will change accordingly. This is called Time Marching Solution. Slide27

Example of Time Marching Solution

Lets assume the unsteady, one dimensional heat conduction equation with constant thermal diffusivitySlide28

Example of Time Marching SolutionSlide29

Example of Time Marching Solution

Now, we are going to write the equation only for T, since we are interested in finding the temperature at different time points for x distance. Hence, the solution for Temperature is solved by Time Marching SolutionSlide30

Time Marching Solution

Time marching means that T at all grid points at time level n+1 are calculated from known values at time level n. Then n+2 is calculated pretty much the same way as n+1 levels are used for calculation. Slide31

Explicit and Implicit Approach

If a difference equation contains only one unknown and all other variables are know, then that is called an explicit solution.

If more then one unknown variable exists, then you will need to solve a set of algebraic equations simultaneously. Hence, this is called the implicit approach.Slide32

THANK YOU

Please download this lecture as well as other lectures and numerical examples from my website:

www.cfdlectures.co.cc

Dr.

Ugur

GUVEN

drguven@live.com