Numerical Methods for Acoustic Problems with Complex Geometries Based on Cartesian Grids
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Numerical Methods for Acoustic Problems with Complex
Geometries Based on Cartesian Grids
D.N. Vedder1103784
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Overview
• Computational AeroAcoustics
• Spatial discretization
• Time integration
• Cut-Cell method
• Results and proposals
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Computational AeroAcoustics(AeroAcoustics)
• CFD vs AeroAcoustics
AeroAcoustics: Sound generation and propagation in association with fluid dynamics.
Lighthill’s and Ffowcs Williams’ Acoustic Analogies
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Computational AeroAcoustics(Acoustics)
• Sound modelled as an inviscid fluid phenomena Euler equations
• Acoustic waves are small disturbances Linearized Euler equations:
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Computational AeroAcoustics(Dispersion relation)
• A relation between angular frequency and wavenumber.
• Easily determined by Fourier transforms
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Spatial discretization (DRP)
• Dispersion-Relation-Preserving scheme
• How to determine the coefficients?
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Spatial discretization (DRP)
1. Fourier transform
aj = -a-j
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Spatial discretization (DRP)
2. Taylor seriesMatching coefficients up to order 2(N – 1)th
Leaves one free parameter, say ak
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Spatial discretization (DRP)
3. Optimizing
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Spatial discretization (DRP)
Dispersive properties:
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Spatial discretization (OPC)
• Optimized-Prefactored-Compact scheme
1. Compact scheme
Fourier transforms and Taylor series
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Spatial discretization (OPC)
2. Prefactored compact scheme
Determined by
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Spatial discretization (OPC)
3. Equivalent with compact scheme
Advantages:1. Tridiagonal system two bidiagonal systems (upper and lower triangular)2. Stencil needs less points
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Spatial discretization (OPC)
• Dispersive properties:
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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
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Time Integration (LDDRK)
• Low Dissipation and Dispersion Runge-Kutta scheme
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Time Integration (LDDRK)
• Taylor series• Fourier transforms• Optimization
• Alternating schemes
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Time Integration (LDDRK)
Dissipative and dispersive properties:
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Cut-Cell Method
• Cartesian grid• Boundary implementation
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Cut-Cell Method
• fn and fw with boundary
stencils
• fint with boundary condition
• fsw and fe with interpolation polynomials
fn
fw
fsw fint
fe
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Test case
Reflection on a solid wall• 6/4 OPC and 4-6-LDDRK• Outflow boundary conditions
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Proposals
• Resulting order of accuracy
• Impact of cut-cell procedure on it
• Richardson/least square extrapolation– Improvement of solution
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Questions?