Modifications to the SIMPLE Method for Non-Orthogonal, Non-Staggered Grids in k - E Turbulence Flow Model of an NACA 0

1. Modifications to the SIMPLE Method for Non-Orthogonal, Non-Staggered Grids in k-E Turbulence Flow Model of an NACA 0012 Airfoil

2. Research Overview Grid Generation Base Equations Transformation Discretization Pressure/Convection Correction SIMPLE Modifications Results Discussion Presentation Overview

3. Research Overview Primary Paper • 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 • 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. Grid Generation Discretization HW#3 Top & Bottom Boundaries

5. Grid Generation Solution Form • Central Difference for 2nd Derivatives • TDMA or Pt. Gauss-Seidel Solvers • Solve X, Solve Y, Reiterate to Convergence • Exponential Stretching, Discontinuity Removal

6. Base Equation General Transformation

7. Convective (Velocity) Terms Integral Form Converted to Finite Difference Approximation

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

9. Pressure Corrections

10. Pressure Correction Equation • Oscillatory P-field Produced • Due to 2nd Order Central Diff. Approx. • (cannot resolve 1DX oscillations) • Staggered mesh approach not allowed since u & v are not orthogonal to grid

11. Modifications • New Treatment of G1 and G2 • Recall G1 & G2 values obtained by linear interpolation in the physical plane • Propose to calculate G1 & G2 values directly from • Where formulated in terms of P star

12. Resulting Correction Equation • Converting the Central Difference Approximation to an Upwind Difference Approximation • Better convergence than original SIMPLE • If P-field linear, than modification is redundant • B.C. assumes derivative of P’ = 0 at boundary

13. Solution Procedure • Set preliminary pressure field • Velocity components calculated (including G1 & G2 values ) • Pressure Correction Calculated • Adjustments Made • k-E Turbulence Scalars Solved • Iterate until Convergence Obtained

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

15. Results Discussion (paper’s comments) • 77x34 Grid, Boundaries @ 8-12 chord lengths away from airfoil • 500 Iterations w/out separation 2X with • Leading Edge Errors (5% to 40%) due to Insufficent Grid Resolution • Trailing Edge Errors due to poor turbulence modeling

16. Summary • Non-orthogonal transformation is lengthy; however, straightforward • Non-staggered grid is required due to non-orthogonal relationship between u,v & grid • Pressure oscillations eliminated through partial use of upwind differencing • Satisfactory results obtained