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

This study presents an innovative approach to computational aeroacoustics by combining Cartesian schemes with a cut-cell method, evaluated using Richardson extrapolation. We discuss spatial discretization, time integration, and test cases involving linearized Euler equations, focusing on the sound modeled as inviscid fluid phenomena. Key findings on order of accuracy and the impact of boundary conditions are highlighted. The effectiveness and optimization of these computational methods provide insights into their application for modeling acoustic waves and their disturbances in various scenarios.

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

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  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?

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