1. Finite Difference Methods
Dr. Ugur GUVEN
Aerospace Engineer (P.hD)
Nuclear Science and Technology Engineer (M.Sc)

2. 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 Methods

3. 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 value

4. Taylor Series

5. 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 equation

6. 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 difference

7. Rearward Differences
Lets now write the Taylor series for the point u(i-1,j) in a 2d flow

8. 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.

9. 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 equations

10. Central Differences
Lets subtract the following equations to try to increase the order of accuracy

11. 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:

12. 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)

13. 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 differentiation

14. Partial Derivatives Using Finite Differences

15. 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 differences

16. 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)

17. Second Order Partial Derivatives
Summation of the above equations gives:
As a result, the second order partial derivative is written in central differences as:

18. Second Order Partial Derivatives
Hence, the second order partial derivatives written in the central difference notation would be:

19. Mixed Partial Derivatives
By writing a mixed partial derivative , we will be able to write second order equations as well.

20. Second Order Partial Derivatives

21. Transformation of Partial Differential Equations
Transform the following partial differential equation into an algebraic equation by using Finite Differences

22. Homework

23. Engineering Example 1
Calculate the shear stress and the heat transfer at the wall with the following data by using finite differences

24. Example 1
The shear stress and the heat transfer at the wall is given by the following equations in first dimensional problems.

25. Example 1
First Order Difference Solution, solve for the other points

26. 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.

27. Example of Time Marching Solution
Lets assume the unsteady, one dimensional heat conduction equation with constant thermal diffusivity

28. Example of Time Marching Solution

29. 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 Solution

30. 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.

31. 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.

32. THANK YOU
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Dr. Ugur GUVEN