1. Numerical Methods for Acoustic Problems with Complex Geometries Based on Cartesian Grids D.N. Vedder 1103784

2. Overview Computational AeroAcoustics Spatial discretization Time integration Cut-Cell method Results and proposals

3. Computational AeroAcoustics (AeroAcoustics) CFD vs AeroAcoustics AeroAcoustics: Sound generation and propagation in association with fluid dynamics.  Lighthill’s and Ffowcs Williams’ Acoustic Analogies

4. Computational AeroAcoustics (Acoustics) Sound modelled as an inviscid fluid phenomena  Euler equations Acoustic waves are small disturbances  Linearized Euler equations:

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

6. Spatial discretization (DRP) Dispersion-Relation-Preserving scheme How to determine the coefficients?

7. Spatial discretization (DRP) 1.Fourier transform  a j = -a -j

8. Spatial discretization (DRP) 2.Taylor series Matching coefficients up to order 2(N – 1) th  Leaves one free parameter, say a k

9. Spatial discretization (DRP) 3. Optimizing

10. Spatial discretization (DRP) Dispersive properties:

11. Spatial discretization (OPC) Optimized-Prefactored-Compact scheme 1.Compact scheme  Fourier transforms and Taylor series

12. Spatial discretization (OPC) 2. Prefactored compact scheme Determined by

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

14. Spatial discretization (OPC) Dispersive properties:

15. Spatial discretization (Summary) Two optimized schemes –Explicit DRP scheme –Implicit OPC scheme (Dis)Advantages –OPC: higher accuracy and smaller stencil –OPC: easier boundary implementation –OPC: solving systems Finite difference versus finite volume approach

16. Time Integration (LDDRK) Low Dissipation and Dispersion Runge- Kutta scheme

17. Time Integration (LDDRK) Taylor series Fourier transforms Optimization Alternating schemes

18. Time Integration (LDDRK) Dissipative and dispersive properties:

19. Cut-Cell Method Cartesian grid Boundary implementation

20. Cut-Cell Method f n and f w with boundary stencils f int with boundary condition f sw and f e with interpolation polynomials fnfn fwfw f sw f int fefe

21. Test case Reflection on a solid wall 6/4 OPC and 4-6-LDDRK Outflow boundary conditions

22. Proposals Resulting order of accuracy Impact of cut-cell procedure on it Richardson/least square extrapolation –Improvement of solution

23. Questions?