# with a Cut-Cell Method,

Mécanique

22 févr. 2014 (il y a 4 années et 4 mois)

92 vue(s)

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

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

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?