1. A conservative FE-discretisation of the Navier-Stokes equation JASS 2005, St. Petersburg Thomas Satzger

2. 2 Overview Navier-Stokes-Equation –Interpretation –Laws of conservation Basic Ideas of FD, FE, FV Conservative FE-discretisation of Navier-Stokes- Equation

3. 3 Navier-Stokes-Equation The Navier-Stokes-Equation is mostly used for the numerical simulation of fluids. Some examples are -Flow in pipes -Flow in rivers -Aerodynamics -Hydrodynamics

4. 4 Navier-Stokes-Equation The Navier-Stokes-Equation writes: Equation of momentum Continuity equation withVelocity field Pressure field Density Dynamic viscosity

5. 5 Navier-Stokes-Equation The interpretation of these terms are: Derivative of velocity field Convection Pressure gradient Diffusion Outer Forces

6. 6 Navier-Stokes-Equation The corresponding for the components is: for the momentum equation, and for the continuity equation.

7. 7 Navier-Stokes-Equation With the Einstein summation and the abbreviationwe get: for the momentum equation, and for the continuity equation.

8. 8 Navier-Stokes-Equation Now take a short look to the dimensions:

9. 9 Navier-Stokes-Equation - Interpretation We see that the momentum equations handles with accelerations. If we rewrite the equation, we get: This means: Total acceleration is the sum of the partial accelerations.

10. 10 Navier-Stokes-Equation - Interpretation Interpretation of the Convection fluid particle Transport of kinetic energy by moving the fluid particle

11. 11 Navier-Stokes-Equation - Interpretation Interpretation of the pressure Gradient fluid particle Acceleration of the fluid particle by pressure forces

12. 12 Navier-Stokes-Equation - Interpretation Interpretation of the Diffusion fluid particle Distributing of kinetic Energy by friction

13. 13 Navier-Stokes-Equation - Interpretation Interpretation of the continuity equation Conservation of mass in arbitrary domain this means: influx = out flux forwe get

14. 14 Navier-Stokes-Equation - Laws of conservation Conservation of kinetic energy: We must know that the kinetic energy doesn't increase, this means: Proof:

15. 15 Navier-Stokes-Equation - Laws of conservation With the momentum equation it holds Using the relations(proof with the continuity equation) and

16. 16 Navier-Stokes-Equation - Laws of conservation Additionally it holds Therefore we get Due to Greens identity we have

17. 17 Navier-Stokes-Equation - Laws of conservation This means in total We have also seen that the continuity equation is very important for energy conservation.

18. 18 Basic Ideas of FD, FE, FV We can solve the Navier-Stokes-Equations only numerically. Therefore we must discretise our domain. This means, we regard our Problem only at finite many points. There are several methods to do it: Finite Difference (FD) One replace the differential operator with the difference operator, this mean you approximate by or an similar expression.

19. 19 Basic Ideas of FD, FE, FV Finite Volume (FV) -You divide the domain in disjoint subdomains -Rewrite the PDE by Gauß theorem -Couple the subdomains by the flux over the boundary Finite Elements (FE) -You divide the domain in disjoint subdomains -Rewrite the PDE in an equivalent variational problem -The solution of the PDE is the solution of the variational problem

20. 20 Basic Ideas of FD, FE, FV Comparison of FD, FE and FV Finite Difference Finite Element Finite Volume

21. 21 Basic Ideas of FD, FE, FV Advantages and Disadvantages Finite Difference: + easy to programme - no local mesh refinement - only for simple geometries Finite Volume: + local mesh refinement + also suitable for difficult geometries Finite Element: + local mesh refinement + good for all geometries BUT: Conservation laws aren't always complied by the discretisation. This can lead to problems in stability of the solution.

22. 22 Conservative FE-Elements We use a partially staggered grid for our discretisation. We write: for the number of grid points for the horizontal velocity in the i-th grid point for the vertical velocity in the i-th grid point

23. 23 Conservative FE-Elements The FE-approximation is an element of an finite-dimensional function space with the basis The approximation has the representation whereby

24. 24 Conservative FE-Elements If we use a Nodal basis, this means we can rewrite the approximation and

25. 25 Conservative FE-Elements Every approximation should have the following properties: continuous conservative In the continuous case the continuity equation was very important for the conservation of mass and energy. If the approximation complies the continuity pointwise in the whole area, e.g., then the approximation preserves energy.

26. 26 Conservative FE-Elements Now we search for a conservative interpolation for the velocities in a box. We also assume that the velocities complies the discrete continuity equation.

27. 27 Conservative FE-Elements Now we search for a conservative interpolation for the velocities in a box. We also assume that the velocities complies the discrete continuity equation.

28. 28 Conservative FE-Elements Now we search for a conservative interpolation for the velocities in a box. We also assume that the velocities complies the discrete continuity equation.

29. 29 Conservative FE-Elements Now we search for a conservative interpolation for the velocities in a box. We also assume that the velocities complies the discrete continuity equation: (1)

30. 30 Conservative FE-Elements The bilinear interpolation isn't conservative

31. 31 Conservative FE-Elements The bilinear interpolation isn't conservative It is easy to show that

32. 32 Conservative FE-Elements The bilinear interpolation isn't conservative Basis on the box

33. 33 Conservative FE-Elements These basis function for the bilinear interpolation are called Pagoden. The picture shows the function on the whole support.

34. 34 Conservative FE-Elements Now we are searching a interpolation of the velocities which complies the continuity equation on the box. How can we construct such an interpolation?

35. 35 Conservative FE-Elements Now we are searching a interpolation of the velocities which complies the continuity equation on the box. How can we construct such an interpolation? Divide the box in four triangles.

36. 36 Conservative FE-Elements Now we are searching a interpolation of the velocities which complies the continuity equation on the box. How can we construct such an interpolation? Divide the box in four triangles. Make on every triangle an linear interpolation.

37. 37 Conservative FE-Elements What's the right velocity in the middle?

38. 38 Conservative FE-Elements What's the right velocity in the middle? We must have at every point in the box the following relations:

39. 39 Conservative FE-Elements What's the right velocity in the middle? We must have at every point in the box the following relations:

40. 40 Conservative FE-Elements What's the right velocity in the middle? We must have at every point in the box the following relations:

41. 41 Conservative FE-Elements What's the right velocity in the middle? We must have at every point in the box the following relations:

42. 42 Conservative FE-Elements What's the right velocity in the middle? We must have at every point in the box the following relations:

43. 43 Conservative FE-Elements What's the right velocity in the middle? We must have at every point in the box the following relations:

44. 44 Conservative FE-Elements What's the right velocity in the middle? We must have at every point in the box the following relations:

45. 45 Conservative FE-Elements What's the right velocity in the middle?

46. 46 Conservative FE-Elements What's the right velocity in the middle?

47. 47 Conservative FE-Elements What's the right velocity in the middle?

48. 48 Conservative FE-Elements What's the right velocity in the middle?

49. 49 Conservative FE-Elements What's the right velocity in the middle?

50. 50 Conservative FE-Elements What's the right velocity in the middle?

51. 51 Conservative FE-Elements What's the right velocity in the middle?

52. 52 Conservative FE-Elements What's the right velocity in the middle?

53. 53 Conservative FE-Elements What's the right velocity in the middle?

54. 54 Conservative FE-Elements What's the right velocity in the middle?

55. 55 Conservative FE-Elements What's the right velocity in the middle?

56. 56 Conservative FE-Elements Till now we have:

57. 57 Conservative FE-Elements Till now we have: With the discrete continuity equation we get

58. 58 Conservative FE-Elements Till now we have: With the discrete continuity equation we get Therefore we choose

59. 59 Conservative FE-Elements What's the right velocity in the middle?

60. 60 Conservative FE-Elements Now we calculate the basis.

61. 61 Conservative FE-Elements Now we calculate the basis.

62. 62 Conservative FE-Elements Now we calculate the basis.

63. 63 Conservative FE-Elements Now we calculate the basis.

64. 64 Conservative FE-Elements Now we calculate the basis.

65. 65 Conservative FE-Elements Now we calculate the basis.

66. 66 Conservative FE-Elements Now we calculate the basis.

67. 67 Conservative FE-Elements Now we calculate the basis.

68. 68 Conservative FE-Elements Now we calculate the basis.

69. 69 Conservative FE-Elements Now we calculate the basis.

70. 70 Conservative FE-Elements Now we calculate the basis.

71. 71 Conservative FE-Elements Now we calculate the basis.

72. 72 Conservative FE-Elements Now we calculate the basis.

73. 73 Conservative FE-Elements Now we calculate the basis.

74. 74 Conservative FE-Elements Now we calculate the basis.

75. 75 Conservative FE-Elements Now we calculate the basis.

76. 76 Conservative FE-Elements Now we calculate the basis.

77. 77 Conservative FE-Elements Now we calculate the basis.

78. 78 Conservative FE-Elements Now we calculate the basis.

79. 79 Conservative FE-Elements Linear interpolation provides the basis.

80. 80 Conservative FE-Elements View on conservative elements in 3D

81. 81 Conservative FE-Elements View on conservative elements in 3D Partially staggered grid in 3D

82. 82 Conservative FE-Elements We also search for a conservative interpolation of the velocities.

83. 83 Conservative FE-Elements We also search for a conservative interpolation of the velocities. Divide every box into 24 tetrahedrons, on which you make a linear interpolation