1. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Modifications to the SIMPLE Method for Non-Orthogonal, Non-Staggered Grids in k- E Turbulence Flow Model of an NACA 0012 Airfoil

2. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Presentation Overview u Research Overview u Grid Generation u Base Equations u Transformation u Discretization u Pressure/Convection Correction u SIMPLE Modifications u Results u Discussion

3. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Research Overview Primary Paper u Rhie, C.M. and Chow, W.L., “Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation”, AIAA, November 1993 Secondary Paper u Lonsdale, R.D., “An Algebraic Multi-Grid Solver for the Navier-Stokes Equations on Unstructured Meshes”, International Journal of Numerical Methods for Heat and Fluid Flow, v3 n1 February 1993 (PennState)

4. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Grid Generation HW#3 Top & Bottom Boundaries Discretization

5. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Grid Generation Solution Form u Central Difference for 2nd Derivatives u TDMA or Pt. Gauss-Seidel Solvers u Solve X, Solve Y, Reiterate to Convergence u Exponential Stretching, Discontinuity Removal

6. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Base Equation General Transformation

7. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Convective (Velocity) Terms Integral Form Converted to Finite Difference Approximation

8. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Generalized Result u Hybrid Scheme Utilized for Coefficients Source Term Terms due to cross-derivatives in diffusion terms and non-orthogonality (can be neglected for ‘near’ orthogonal grids) Rhie, C.M., “A Numerical Study of the Flow Past an Isolated Airfoil with Separation”, Ph.D. Thesis, 1981, U. of Illinois - Urbana-Champaign (for details refer to)

9. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Pressure Corrections

10. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Pressure Correction Equation u Oscillatory P-field Produced u Due to 2nd Order Central Diff. Approx. u (cannot resolve 1DX oscillations) u Staggered mesh approach not allowed since u & v are not orthogonal to grid

11. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Modifications u New Treatment of G 1 and G 2 u Recall G 1 & G 2 values obtained by linear interpolation in the physical plane u Propose to calculate G 1 & G 2 values directly from u Where formulated in terms of P star

12. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Resulting Correction Equation u Converting the Central Difference Approximation to an Upwind Difference Approximation u Better convergence than original SIMPLE u If P-field linear, than modification is redundant u B.C. assumes derivative of P’ = 0 at boundary

13. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Solution Procedure u Set preliminary pressure field u Velocity components calculated (including G 1 & G 2 values ) u Pressure Correction Calculated u Adjustments Made  k- E Turbulence Scalars Solved u Iterate until Convergence Obtained

14. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Results ( as published by Rhie & Chow ) u Code tested on known laminar flow solutions with satisfactory results u Experimental results from NACA 0012 airfoil obtained for turbulence comparisons u 0, 6 & 12 degree angles of attack examined u (2) Re examined, w&w/out separation u Good Fit with minor discrepancies

15. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Results Discussion ( paper’s comments ) u 77x34 Grid, Boundaries @ 8-12 chord lengths away from airfoil u 500 Iterations w/out separation 2X with u Leading Edge Errors (5% to 40%) due to Insufficent Grid Resolution u Trailing Edge Errors due to poor turbulence modeling

16. CFD II w/Dr. Farouk By: Travis Peyton7/18/2015 Summary u Non-orthogonal transformation is lengthy; however, straightforward u Non-staggered grid is required due to non- orthogonal relationship between u,v & grid u Pressure oscillations eliminated through partial use of upwind differencing u Satisfactory results obtained