Numerical Methods for Acoustic Problems with Complex

Geometries Based on Cartesian Grids

D.N. Vedder1103784

Overview

• Computational AeroAcoustics

• Spatial discretization

• Time integration

• Cut-Cell method

• Results and proposals

Computational AeroAcoustics(AeroAcoustics)

• CFD vs AeroAcoustics

AeroAcoustics: Sound generation and propagation in association with fluid dynamics.

Lighthill’s and Ffowcs Williams’ Acoustic Analogies

Computational AeroAcoustics(Acoustics)

• Sound modelled as an inviscid fluid phenomena Euler equations

• Acoustic waves are small disturbances Linearized Euler equations:

Computational AeroAcoustics(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 seriesMatching 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

• fint with boundary condition

• fsw and fe with interpolation polynomials

fn

fw

fsw fint

fe

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?