with a Cut-Cell Method,

mustardarchaeologistMechanics

Feb 22, 2014 (3 years and 4 months ago)

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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

Overview


Computational AeroAcoustics


Spatial discretization


Time integration


Cut
-
Cell method


Testcase



Richardson extrapolation


Interpolation


Results


Conclusions

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

OPC


Optimized
-
Prefactored
-
Compact scheme


1.
Compact scheme








Fourier transforms and Taylor series

x
j
-
2

x
j
-
1

x
j

x
j+1

x
j+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


f
n

and
f
w

with boundary


stencils



f
int

with boundary condition



f
sw

and
f
e

with interpolation polynomials which
preserve 4
th

order of accuracy. (Using neighboring
points)

f
n

f
w

f
sw

f
int

f
e

Testcase

Reflection 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


Interpolation


Interpolation 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


6
th

order interpolation method, tested by analytical testcase

Results

Solution 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!!

f
n

f
w

f
sw

f
int

f
e

f
sw

f
e

Results (cont)

Richardson extrapolation

Results (cont)

Richardson extrapolation

Conclusions


Interpolation to common grid


6
th

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?