1. Advanced Numerical Techniques Mccormack Technique CFD Dr. Ugur GUVEN

2. McCormack Technique The techniques of Finite Differences we have learned is used to solve a single equation for a single variable. You can use the Mccormack Technique to solve for a whole set of dynamic equations. Mccormack Technique is easy to program, so it is preferred by most engineers. Mccormack Technique is suitable to a set of fluid dynamics equations

3. Application of Mccormack Technique to Fluid Flow Equations You take a set of Fluid flow equations You transform them to algebraic equations by the use of the Finite Differences Method. You solve for the variable that you want You input that variable back to the equations to resolve for t+dt You combine the two results and you take its average as the best result for accuracy.

4. Application of Mccormack Technique to Fluid Flow Equations Lets solve the set of equations below for unsteady, two dimensional inviscid flow:

5. Grid for the System

6. Solution Using Mccormack Technique We assume that all the points (values of flow such as speed, density, temperature, pressure) in the grid are known for time t. We will calculate at the same grid points for t+dt for the same values (velocity, density etc) An approach called Predictor – Corrector is used.

7. Solution Using Mccormack Technique Lets consider density, velocity and energy from the above equations and apply the Taylor series:

8. Solution Using Mccormack Technique: Predictor Step Take the First equation of continuity and rewrite it in algebraic form by forward differences.

9. Predictor Step Do the same by using the set of equations provided in the beginning for each variable to find partial derivatives of u,v, and e. Write the predictor by using these algebraic partial derivatives to find for each value in the flow at t+dt

10. Predictor Step So for density as an example the predictor is obtained as:

11. Corrector Step

12. Final Solution Now you calculate the average derivative as discussed in the beginning of the slides. You rewrite this average derivative for all flow values such as speed and density in the system. PREDICTOR STEPCORRECTOR STEP

13. SOLUTION

14. Summary of Mccormack Technique 1)Write the fluid dynamics equations with respect to time 2)Write the partial derivatives as forward differences 3)Solve for the flow value (such as density) by substituting the partial derivative in the Taylors series. 4)Write the predicted step for t+dt using step 3 5)Write the corrected step with rearward difference for t+dt by using the values found in Step 4 6)Take the average of derivative found in step 2 and 5 7)Write the final result of Predictor – Corrector value for the flow variable.