1. Finite-Volumes I Sauro Succi

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

3. Given a Volume V, enclosed in a piecewise smooth boundary (surface) S, characterized by the normal n in each point; The flux of a is defined as IF S is regular and a is continuously differentiable, we have the following: Gauss theorem

4. Gauss theorem: conservation laws

5. Volume vs Surface average

6. Gauss theorem: ADE

7. Gauss theorem: Control Volume

8. Conservativeness

9. Colocated; Control Volume Simple, but not good for surfint > geos VP uncoupled, hourglass N E W S e n s w ne

10. Centers to Edges: Interpolate E e n s w ne N

11. Staggered Laborious, no interpolation > simple geos No hourglass, VP coupled

12. Discretized Gauss: Continuity Discretized Fluxes: Interpolation

13. Discretized Fluxes: General Discretized Fluxes: Interpolation

14. Finite Volumes

15. Topological issues

16. FD<FV<FE Cartesian Non-Cartesian, Structured Unstructured

17. 1d and 2d example: advection-diffusion

18. FV: AD d=2 Discretized Fluxes

19. One Dimension

20. Linear Interp: Upwind:

21. One Dimension

23. 1d: example Give explicit expression of L_PQ and show that it reduces to standard FD for square finite volumes. Again, we don’t care about non-uniformity because the unknowns are cell averages (more physical)

24. Two-dimensions

25. Going to 2D: same principles more labour! Cartesian (Orthogonal) Trapezoidal (non-orthogonal) Unstructured (similar to FEM)

26. d d d d Cartesian d=2

27. Structured cartesian: Advection fluxes

28. Structured cartesian: diffusive flux Structured matrix: 5 nonzero entries

29. Structured cartesian: Advection fluxes

30. 2d cartesian: diffusive fluxes

31. Structured cartesian: cieff’s Structured matrix: 5 nonzero entries

32. Structured non-cartesian

33. Structured Non-Cartesian Geometrical data

35. 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

37. Co/Contravariant/Cartesian

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

41. Navier-Stokes (Compressible) Staggered FV

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

43. 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

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

45. Discretized Convective Fluxes

46. Discretized Gauss: Momentum_x Convective and Dissipative Fluxes

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

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

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

50. Unstructured FV~FEM

51. Cell vs Vertex Centered

52. Vertex control elements

55. 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 No-singularity (1/r for sherical coordinates) Commercially dominant (STAR-CD, FLUENT…)