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

Overview

• Computational AeroAcoustics• Spatial discretization• Time integration• Cut-Cell method• Testcase

• Richardson extrapolation• Interpolation• Results• Conclusions

Computational AeroAcousticsAcoustics

• Sound modelled as an inviscid fluid phenomena

Euler equations

• Acoustic waves are small disturbances

Linearized Euler equations:

Computational AeroAcousticsDispersion relation

• A relation between angular frequency and wavenumber.

• Easily determined by Fourier transforms

Spatial discretization OPC

• Optimized-Prefactored-Compact scheme

1. Compact scheme

Fourier transforms and Taylor series

xj-2 xj-1 xj xj+1 xj+2

Spatial discretization OPC

• Taylor series

Fourth order gives two equations,

this leaves one free parameter.

Spatial discretization OPC

• Fourier transforms

Theorems:

Spatial discretization OPC

Spatial discretization OPC

Optimization over free parameter:

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:

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:– Cut cells can be merged– Cut cells can be independent

Cut-Cell Method

• 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)

fn

fw

fsw fint

fe

TestcaseReflection on a solid wall

• Linearized Euler

equations

• Outflow boundary

conditions

• 6/4 OPC and

4-6-LDDRK

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?

Richardson extrapolation

Determining the order of accuracy

Assumption:

Richardson extrapolation

Three numerical solutions needed

Pointwise approach interpolation to a common grid needed

InterpolationInterpolation polynomial:

Fifth degree in x and y 36 points

1. Lagrange interpolation in interior– 6x6 squares

2. Matrix interpolation near wall– Row Scaling– Shifting interpolation procedure– Using wall condition

6th order interpolation method, tested by analytical testcase

ResultsSolution at t = 4.2 Order of accuracy at t = 4.2

Results (cont)Impact of boundary condition and filter

• Boundary condition

• Filter for removing high frequency noise

Results (cont)Order of accuracy

t = 4.2 t = 8.4

Results (cont)Impact of outflow condition

• Outflow boundary condition

• Replace by solid wall

Results (cont)Impact of cut-cell method

Order of accuracy

t = 8.4 t = 12.6

Solid wall

Results (cont)Impact of cut-cell method

• Interpolation method used for

and

• Tested by analytical testcase

• Results obtained with three norms– Order of accuracy about 0!!

fn

fw

fsw fint

fe

fsw fe

Results (cont)Richardson extrapolation

Results (cont)Richardson extrapolation

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

Questions?