1. Finite-Volumes II: Non Cartesian Sauro Succi

2. Finite Volumes Real-life geometries: coordinate-free Courtesy of Prof. M. Porfiri, NYU Poly

3. Structured Non-Cartesian Geometrical data

5. Co/Contravariant/Cartesian

8. CEV = Centers/Edges/Vertices Non-cartesian: structured S W N Ε NΕ SΕ NW SW ne se C ew n s Non-orthogonal Still structured

9. Non-structured: diffusive flux Non-orthogonality: S W N Ε NΕ SΕ NW SW ne se C ew n s

10. CEV = Centers/Edges/Vertices Staggered S W N Ε NΕ SΕ NW SW ne se C ew n s Non-orthogonal

12. Navier-Stokes (Compressible) Staggered FV

13. NW NΕ SΕ SW n e w s P E N W S Vertex-centered staggered

14. Discretized Gauss: Continuity Discretized Convective Fluxes Same for north,west, south … Non-orthogonality issues (!) S W N Ε NΕ SΕ NW SW ne se C ew n s

15. Discretized Gauss: Continuity Discretized midpoint (2 nd order 8 neigh) Discretized Simpson (4 th order, 8 neigh)

16. Discretized Convective Fluxes

17. Discretized Gauss: Momentum_x Convective and Dissipative Fluxes

18. Non-Linear (outer) iteration Nonlinear (outer) iteration, k=0,1…

19. Real-life geometries Courtesy of Prof. M. Porfiri, NYU Poly

20. Example: Global: Cylindrical, Spherical, Local: Oblique

21. Unstructured FV~FEM

22. Reconstruction: Cell Centered

23. Mean square residual Minimize error functional:

24. Mean square residual

25. Exercise: Construct gradient on Regular cells Trapezoidal cells

26. Vertex control elements

27. Gradient computation: Gauss-Green

28. P E

32. Finite Volumes: summary Intuitive and physically sound Round-off Conservative (fluxin=-fluxout) Geo-topological ahead, laborious Interpolation to be decided (unlike FEM) Structured: Finite-Difference with non-smooth coordinates Unstructured: Close to FEM No-singularity (1/r for sherical coordinates) Commercially dominant (STAR-CD, FLUENT…)