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

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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
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Overview

• Computational AeroAcoustics
• Spatial discretization
• Time integration
• Cut-Cell method
• Testcase
• Richardson extrapolation
• Interpolation
• Results
• Conclusions

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Computational AeroAcousticsAcoustics

• Sound modelled as an inviscid fluid phenomena
• ? Euler equations
• Acoustic waves are small disturbances
• ? Linearized Euler equations

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Computational AeroAcousticsDispersion relation

• A relation between angular frequency and
  wavenumber.
• Easily determined by Fourier transforms

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Spatial discretization OPC

• Optimized-Prefactored-Compact scheme
• Compact scheme
• ? Fourier transforms and Taylor series

xj-2
xj-1
xj
xj1
xj2
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Spatial discretization OPC

• Taylor series
• Fourth order gives two equations,
• this leaves one free parameter.

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Spatial discretization OPC

• Fourier transforms
• Theorems

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Spatial discretization OPC

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Spatial discretization OPC

• Optimization over free parameter

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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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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
• Cut-cell method
• Cut cells can be merged
• Cut cells can be independent

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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
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Testcase

• Reflection on a solid wall
• Linearized Euler
• equations
• Outflow boundary
• conditions
• 6/4 OPC and
• 4-6-LDDRK

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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?
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Richardson extrapolation

• Determining the order of accuracy

Assumption
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Richardson extrapolation

• Three numerical solutions needed
• Pointwise approach ? interpolation to a
  common grid needed

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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

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Results
Solution at t 4.2
Order of accuracy at t 4.2
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Results (cont)Impact of boundary condition and
filter

• Boundary condition
• Filter for removing high frequency noise

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Results (cont)
Order of accuracy
t 8.4
t 4.2
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Results (cont)Impact of outflow condition

• Outflow boundary condition
• Replace by solid wall

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Results (cont)Impact of cut-cell method
Order of accuracy
t 8.4
t 12.6
Solid wall
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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
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Results (cont)Richardson extrapolation
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Results (cont)Richardson extrapolation
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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

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Questions?