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Numerical Methods for Acoustic Problems with Complex Geometries Based on Cartesian Grids

Numerical Methods for Acoustic Problems with Complex Geometries Based on Cartesian Grids. D.N. Vedder 1103784. Overview. Computational AeroAcoustics Spatial discretization Time integration Cut-Cell method Results and proposals. CFD vs AeroAcoustics

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Numerical Methods for Acoustic Problems with Complex Geometries Based on Cartesian Grids

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

  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) • Fourier transform  aj = -a-j

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

  9. Spatial discretization (DRP) 3. Optimizing

  10. Spatial discretization (DRP) Dispersive properties:

  11. Spatial discretization (OPC) • Optimized-Prefactored-Compact scheme • 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 fn fe fw • fn and fw with boundary stencils • fint with boundary condition • fsw and fe with interpolation polynomials fint fsw

  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?

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