Application of High-Order Optimized Upwind Schemes for 1-D Computational Aeroacoustics Simulation

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22 Φεβ 2014 (πριν από 3 χρόνια και 5 μήνες)

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Application of High-Order Optimized
Upwind Schemes for 1-D
Computational Aeroacoustics
Simulation
AKHILESH MIMANI
School of Mechanical Engineering,
Presentation to the Flow and Noise Group on 10/07/2012
Computational AeroAcoustics(CAA)
1. The use of numerical techniques to simulate noise from unsteady flows
2. Unlike, the classical frequency domain approach, CAA is inherently
time-domain analysis because of the presence of sources which vary non-
periodically with time
3. Numerical techniques used include Finite-Difference Schemes for spatial
as well as temporal discretization of the Non-Linear/Linear Euler
equations –These are briefly reviewed as follows.
Literature Review

Dispersion-Relation-Preserving Finite Difference Schemes for Computational
Aeroacoustics –Tam and Webb, JCP, 107, 1993

Computational Aeroacoustics –Issues and Methods –Tam, AIAA, 33, 1995

Wall Boundary Conditions for High-Order Finite Difference Schemes in
Computational Aeroacoustics –Tam and Dong, Theoretical and Computational Fluid
Dynamics, 6, 1994

Application of higher-order optimized Upwind Schemes for Computational
Aeroacoustics –Zhuangand Chen, AIAA, 40 , 2002

Pseudo-characteristic formulation and dynamic boundary conditions for computational
aeroacoustics–Lu and Sagaut, International Journal for Numerical methods in Fluids,
53, 2007

High-Accuracy Algorithms for Computational Aeroacoustics –Lockardet al., AIAA,
33, 1995

A coupled time-reversal/complex differentiation method for aeroacousticsensitivity
analysis: towards a source detection procedure –Deneuveet al. JFM, 642, 2010.
Problem Proposal

Implementation of higher order optimized schemes in a pseudo-characteristic
formulation by appropriate use of the upwindingdirection for each of the
antagonistic fluxes. In essence, the number of points in the upwind and
downwind sides of a finite-difference stencil used to compute the derivative
at a particular node in the discretized computational domain is decided by the
direction of flux.

Validate the numerical solution with exact analytical solution for the case of
linearized Euler equations incorporating the convective effects of mean flow.

Compare different spatial discretization schemes.

Present an improved TR formulation in this work, wherein it is shown that a
minimum of two nodal points (adjacent or non-adjacent) at each side of the
computational domain is needed to accurately determine or recover the
pressure and velocity distribution at the initial time-instant.

An interpolation technique is also proposed and shows promising results. It
provides impetus for the future experimental work on the optimallocation of
microphones.
Pseudo-Characteristic Formulation…1
The 1-D continuity equation and inviscid momentum equationare shown
hereunder.


(1) ,0





z
u
t


)2(
z
p
z
u
u
t
u

















)3(
2




ZZ
c
t
p



)ba,4(
1
,
1

























z
u
z
p
c
cuZ
z
u
z
p
c
cuZ



)5(
2
1




ZZ
t
u
Pseudo-Characteristic Formulation…2
Linearized Euler Equation with Super-Imposed Mean flow
where
uUu
~
0


ppp
~
0


ρρρ
~
0




)6( ,
2
~
linearlinear
00




ZZ

t
p




),7( ,
~~
1
1
~~
1

,
~
~
1
1
~
~
1
linear
00
0
00
00
linear
00
0
00
00
baZ
z
u
z
p
c
Mc
z
u
z
p
c
cUZ
Z
z
u
z
p
c
Mc
z
u
z
p
c
cUZ






















































)8( .1
00



cUM


)9( .
2
1
~
linearlinear




ZZ
t
u
Analytical Solution for Linearized Euler
Equations…1
,0
D
~
D1
~
2
2
2
0
2
2



t
p
cz
p
)10( .
D
D
0
z
U
tt





where


)11( .0
~
1
1
1
1
0

0



























p
tMcztMcz








)12( ,00,
~
,0,
~




tzuztzp



)13( ,11
2
1
,
~
00
tcMztcMztzp



)14( 11
2
1
,
~
00
00
tcMztcMz

tzu

Analytical Solution for Linearized Euler
Equations…2


)15( ,
2
1
,
~
00
tcztcztzp



)16(
2
1
,
~
00
00
tcztcz

tzu


,0
~
1
1
~
0







t
p
Mcz
p

)ba,17( .0
~
1
1
~
0







t
p
Mcz
p

,1
~
0tcMzp








)ba,18( .1
~
0tcMzp




Higher Order Optimized Stencil: Spatial
Discretization Scheme…1
Higher Order Optimized Stencil: Spatial
Discretization Scheme…2
Higher Order Optimized Stencil: Spatial
Discretization Scheme
(Upwind Scheme)…3
Zhuangand Chen
[4]

,
Δ
1

0
2 5
nodes














Mk
Nk
NM
k
c
Ni
kia
zz



.
Δ
1

0
4 3
nodes














Mk
Nk
NM
k
c
Nj
kja
zz







,2
20
1
1
2
1
3
1
12
4
1
3
30
1
Δ
1
0
4

















iiiiii
zz
c
i







.2
20
1
1
2
1
3
1
12
4
1
3
30
1
Δ
1
0
3-

















jjjjjj
zz
c
Nj





,1
3
1
2
1
12
6
1
Δ
1
0
3

















iiii
zz
c
i





.1
3
1
2
1
12
6
1
Δ
1
0
2-

















jjjj
zz
c
Nj


Higher Order Optimized Stencil: Spatial
Discretization Scheme
(Upwind Scheme)…7


,
Δ
1
,1
Δ
1
6
0
nodes
60
0

6
0
06
0
1
nodes



























k
k
k
c
Ni
k
k
k
c
i
kNb
zz
kb
zz






,1
Δ
1
,2
Δ
1
5
1
nodes
51
0
1
5
1
15
0
2
nodes



























k
k
k
c
Nj
k
k
k
c
i
kNb
zz
kb
zz




Temporal Discretization
Third order Total Variation Diminishing (TVD) Runge-
Kutta Scheme




.Δ,...,,
3
2
3
2
3
1
,Δ,...,,
4
1
4
1
4
3
,Δ,...,,
2,2,
2
2,
1
2,
11
Δ
1
1,1,
2
1,
1
1,
11
2,
1
211
1,
1
tL
tL
tL
nnnnnn
nnnnnn
nnnnn
t
N
tttttt
t
N
ttttt
t
N
tttt







Validation of the Finite Difference
Scheme…1
Anechoic or absorbing boundaryconditions at both the ends of computational domain
are implementedsetting the out fluxes to be zero at the either of the boundaries. This is
realized by the following conditions in the computer programme.
51,2,3,..., 0,
linear


i ZZ
.,...,3,4 0,
nodesnodesnodeslinear
NNNiZZ

.3,...,-4,- ,0
1

1,2,...,5, ,0
1
nodesnodesnodes
NNNi
z
u
z
p
c
i
z
u
z
p
c


























Validation of the Finite Difference
Scheme…2
Validation of the Finite Difference
Scheme…3
Validation of the Finite Difference
Scheme…4
Validation of the Finite Difference Scheme…5
Duct with one of the ends having rigid termination and other end
having pressure release condition.




.0 , ,0 ,0




tLzutzp
Initial Conditions:




.00 , ,0 ,




tzuzftzp





,
2
12
cos
2
12
sin ,
0
3 ,2 ,1


















L
tcm
L
zm
Ctzp
m
m






,
2
12
sin
2
12
cos
1
,
0
3 ,2 ,1
00


















L
tcm
L
zm
C

tzu
m
m

Validation of the Finite Difference
Scheme…6
Comparison of the Different Spatial
Discretization Schemes…1
Comparison of the Different Spatial
Discretization Schemes…2
Time-reversal Simulations:
Backward or Retarded Time-Evolution
The invariance of the 1-D Euler equations given by the above
mentioned transformations may be verified, the details are
avoided here.



.,,
,,,
,,,
,
tzutzu
tzptzp
tzρtzρ
tt





Future Work

Use of higher order upwind biased Scheme to solve the
forward Problem in 2-D Cartesian Co-ordinates having multi-
pole sources.

Develop methods to deal with singularities in circular polar
and spherical co-ordinates.

Develop effective absorbing boundary conditions for the 2-D
problem.

Time-reverse the 2-D problem to detect acoustic sources