Numerical Methods for Acoustic Problems with Complex Geometries

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

Overview • Computational Aero. Acoustics • Spatial discretization • Time integration • Cut-Cell method • Results and proposals

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

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

Computational Aero. Acoustics (Dispersion relation) • A relation between angular frequency and wavenumber. • Easily determined by Fourier transforms

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

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

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

Spatial discretization (DRP) 3. Optimizing

Spatial discretization (DRP) Dispersive properties:

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

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

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

Spatial discretization (OPC) • Dispersive properties:

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

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

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

Time Integration (LDDRK) Dissipative and dispersive properties:

Cut-Cell Method • Cartesian grid • Boundary implementation

Cut-Cell Method • fn and fw with boundary stencils fn fe fw fsw • fint with boundary condition • fsw and fe with interpolation polynomials fint

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

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

Questions?