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?
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