Cartesian Schemes Combined with a Cut-Cell Method, Evaluated with Richardson Extrapolation

1. Cartesian Schemes Combined with a Cut-Cell Method, Evaluated with Richardson Extrapolation D.N. Vedder Prof. Dr. Ir. P. Wesseling Dr. Ir. C.Vuik Prof. W. Shyy

2. Overview • Computational AeroAcoustics • Spatial discretization • Time integration • Cut-Cell method • Testcase • Richardson extrapolation • Interpolation • Results • Conclusions

3. Computational AeroAcousticsAcoustics • Sound modelled as an inviscid fluid phenomena  Euler equations • Acoustic waves are small disturbances  Linearized Euler equations:

4. Computational AeroAcousticsDispersion relation • A relation between angular frequency and wavenumber. • Easily determined by Fourier transforms

5. Spatial discretization OPC • Optimized-Prefactored-Compact scheme • Compact scheme  Fourier transforms and Taylor series xj-2 xj-1 xj xj+1 xj+2

6. Spatial discretization OPC • Taylor series Fourth order gives two equations, this leaves one free parameter.

7. Spatial discretization OPC • Fourier transforms Theorems:

8. Spatial discretization OPC

9. Spatial discretization OPC Optimization over free parameter:

10. Spatial discretization OPC 2. Prefactored compact scheme Determined by

11. Spatial discretization OPC 3. Equivalent with compact scheme Advantages: 1. Tridiagonal system  two bidiagonal systems (upper and lower triangular) 2. Stencil needs less points

12. Spatial discretization OPC • Dispersive properties:

13. Time Integration LDDRK • Low-Dissipation-and-Dispersion Runge-Kutta scheme

14. Time Integration LDDRK • Taylor series • Fourier transforms • Optimization • Alternating schemes

15. Time Integration LDDRK Dissipative and dispersive properties:

16. Cut-Cell Method • Cartesian grid • Boundary implementation • Cut-cell method: • Cut cells can be merged • Cut cells can be independent

17. Cut-Cell Method fn fe fw • fn and fw with boundary stencils • fint with boundary condition • fsw and fe with interpolation polynomials which preserve 4th order of accuracy. (Using neighboring points) fint fsw

18. Testcase Reflection on a solid wall • Linearized Euler equations • Outflow boundary conditions • 6/4 OPC and 4-6-LDDRK

19. Results Pressure contours The derived order of accuracy is 4. What is the order of accuracy in practice? What is the impact of the cut-cell method?

20. Richardson extrapolation Determining the order of accuracy Assumption:

21. Richardson extrapolation Three numerical solutions needed Pointwise approach  interpolation to a common grid needed

22. Interpolation Interpolation polynomial: Fifth degree in x and y  36 points • Lagrange interpolation in interior • 6x6 squares • Matrix interpolation near wall • Row Scaling • Shifting interpolation procedure • Using wall condition 6th order interpolation method, tested by analytical testcase

23. Results Solution at t = 4.2 Order of accuracy at t = 4.2

24. Results (cont)Impact of boundary condition and filter • Boundary condition • Filter for removing high frequency noise

25. Results (cont) Order of accuracy t = 8.4 t = 4.2

26. Results (cont)Impact of outflow condition • Outflow boundary condition • Replace by solid wall

27. Results (cont)Impact of cut-cell method Order of accuracy t = 8.4 t = 12.6 Solid wall

28. Results (cont)Impact of cut-cell method fn fe fw • Interpolation method used for and • Tested by analytical testcase • Results obtained with three norms • Order of accuracy about 0!! fsw fe fint fsw

29. Results (cont)Richardson extrapolation

30. Results (cont)Richardson extrapolation

31. Conclusions • Interpolation to common grid • 6th order to preserve accuracy of numerical solution • Impact of discontinuities and filter • Negative impact on order of accuracy • Impact of outflow boundary conditions • Can handle waves from only one direction • Impact of cut-cell method • Lower order of accuracy due to interpolation • Richardson extrapolation • Only for “smooth” problems

32. Questions?