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 AeroAcoustics Acoustics Sound modelled as an inviscid fluid phenomena  Euler equations Acoustic waves are small disturbances  Linearized Euler equations:

4. Computational AeroAcoustics Dispersion relation A relation between angular frequency and wavenumber. Easily determined by Fourier transforms

5. Spatial discretization OPC Optimized-Prefactored-Compact scheme 1.Compact scheme  Fourier transforms and Taylor series x j-2 x j-1 xjxj x j+1 x j+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. 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 f n and f w with boundary stencils f int with boundary condition f sw and f e with interpolation polynomials which preserve 4 th order of accuracy. (Using neighboring points) fnfn fwfw f sw f int fefe

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 1.Lagrange interpolation in interior –6x6 squares 2.Matrix interpolation near wall –Row Scaling –Shifting interpolation procedure –Using wall condition 6 th order interpolation method, tested by analytical testcase

23. Results Solution at t = 4.2Order 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 = 4.2 t = 8.4

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 = 8.4t = 12.6 Solid wall

28. Results (cont) Impact of cut-cell method Interpolation method used for and Tested by analytical testcase Results obtained with three norms –Order of accuracy about 0!! fnfn fwfw f sw f int fefe f sw fefe

29. Results (cont) Richardson extrapolation

31. Conclusions Interpolation to common grid –6 th 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?