A conservative FE-discretisation of the Navier-Stokes equation

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

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

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. Navier-Stokes-Equation The Navier-Stokes-Equation writes: Equation of momentum Continuity equation with Velocityfield Pressurefield Density Dynamic viscosity

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

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

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

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

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

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

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

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

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

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

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

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. Conservative FE-Elements 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 • We use a partially staggered grid for our discretisation. We write:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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