C.R.Acad.Sci.Paris,t.334,Serie II b,p.1?6,2004  PXPP????.TEX 
Rubrique/Heading
(Sousrubrique/SubHeading)
Computational AeroAcoustics using Bspline
Collocation Method
Ronny Widjaja
a
,Andrew Ooi
b
,Li Chen
c
,Richard Manasseh
d
a
Department of Mechanical Engineering,University of Melbourne,Parkville,3010,Australia
Email:ronnyw@mame.mu.oz.au,Ph:(613) 8344 7723,Mb:(61) 421 573 051
b
Department of Mechanical Engineering,University of Melbourne,Parkville,3010,Australia
Email:a.ooi@unimelb.edu.au
c
Maritime Platforms Division,Defence Science Technology Organization,Fishermans Bend,3027,Australia
Email:Li.Chen@dsto.defence.gov.au
d
Energy and Thermouids Engineering,Commonwealth Scientic and Industrial Research Organization,
Highett,3190,Australia
Email:Richard.Manasseh@csiro.au
(Rec¸u le jour mois ann´ee,accept´e apres r´evision le jour mois ann´ee)
Abstract.One of the major problems in computational aeroacoustics is the disparity in length scales
between the?ow?eld and the acoustic?eld.As a result,a mapping function is normally
used to achieve a nonuniformgrid distribution.In this paper,a Bspline collocation method
with an arbitrary grid placement capability is proposed.This capability not only allows an
optimum grid distribution but also avoids the numerical complexities associated with the
mapping function.The Bspline collocation method is applied to the case of spinning co
rotating vortices.The result agrees well with the matched asymptotic solution.c 2004
Academie des sciences/
Editions scienti?ques et medicales Elsevier SAS
Computational aeroacoustics/Bspline collocation method
Collocation par Bspline appliquee aux Simulations Numeriques en Aeroacoustique
R´esum´e.Un probl?eme rencontre dans les simulations numeriques en aeroacoustique est la disparite
des echelles de longueurs sur lesquelles sont resolus le champ de vitesse de l’ecoulement
?uide et le champ de pression acoustique.Habituellement une fonction de transformation
est utilisee pour generer un maillage nonuniforme.Dans cet article une methode de col
location par Bspline est proposee.Cette methode permet un maillage optimum du do
maine et evite les complexites numeriques associees avec les fonctions de transformation.
Le champ acoustique g
en
er
e par une paire de tourbillons corotatifs est simul
e en utilisant
cette methode.Les resultats de cette simulation numerique sont en accord avec la solution
asymptotique associee.c 2004 Academie des sciences/
Editions scienti?ques et medicales
Elsevier SAS
Simulation num´erique en a´eroacoustique/M´ethode de collocation par Bspline
Note pr´esent´ee par First name NAME
S16207742(01)0?????/FLA
c
2004 Academie des sciences/
Editions scienti?ques et medicales Elsevier SAS.Tous droits reserves.1
R.Widjaja,A.Ooi,L.Chen,R.Manasseh
1.Introduction
Computational aeroacoustics (CAA) emerges from the success of computational uid dynamics (CFD)
in solving many physical problems.Nevertheless,C.K.W.Tam [1] pointed out that there are some issues
that are unique to CAA.These issues include the longpropagation distance and life of acoustic waves;and
the disparity in length scales between the oweld and acoustic eld.The former provides enough time for
any dissipation and dispersion errors to grow and contaminate the acoustic eld.The latter requires both a
dense mesh and a large computational domain,causing a uniformmesh to be impractical in CAA.
To overcome the rst issue,many studies have been conducted to improve the numerical schemes com
monly used in CFD.C.K.W.Tam and J.C.Webb [2] developed a DispersionRelationPreserving (DRP)
scheme.This DRP scheme is an optimised nite difference scheme where the order of accuracy of the
numerical scheme has been sacriced for a much better resolution at high wave number.This considerably
reduces the dispersion error.Another type of optimised scheme is the compact difference scheme with
spectrallike resolution developed by S.K.Lele [3].This scheme was further enhanced by J.W.Kim and
D.J.Lee [4] using different optimisation constraints to ensure a minimum dispersion error over a certain
range of wave number.
Unfortunately,all the above numerical schemes were developed by assuming uniformmesh.A mapping
function is commonly used to extend the schemes for nonuniform mesh.The use of a mapping function
usually results in more grid points than necessary and in some cases may lead to numerical instabilities.In
this paper,an alternative approach using a Bspline collocation method is proposed.The Bspline colloca
tion method is a collocation method using Bsplines as the trial functions.Due to the exibility of Bsplines
in the local representation,the Bspline collocation method allows the mesh points to be placed arbitrarily.
This capability not only allows an optimumgrid distribution but also avoids the numerical complexities as
sociated with the use of a mapping function.Furthermore,a uniformC
k1
continuity throughout a Bspline
element of order k gives the Bspline collocation method a highresolution property.
2.Numerical formulations
The properties of the Bspline collocation method depend very much on the trial functions.A Bspline
of order k is made up of a polynomial of order k and has a compact support consisting of k + 1 knot
points.Knot points are a set of points on which Bsplines are dened.The distribution of these knot points
determines the shape and distribution of the Bsplines and consequently the resolution of the mesh.
Following the formulation by Y.Morinishi,S.Tamano and K.Nakabayashi [5],the knot points
[t
k
;t
k+1
;:::;t
N1
;t
N
] are related to the mesh points [x
1
;x
2
;:::;x
N1
;x
N
] by
t
j
=
8
>
>
<
>
>
:
x
1
for k j 0
1
2
x
j+k/2
+x
j+k/2+1
for 0 < j < (N k);even k
x
j+(k+1)/2
for 0 < j < (N k);odd k
x
N
for (N k) j N
The corresponding N numbers of Bsplines of order k can be computed using the following recursive
formula
B
k
j
(x) =
x t
j
t
j+k
t
j
B
k1
j
(x) +
t
j+k+1
x
t
j+k+1
t
j+1
B
k1
j+1
(x) (1)
where the zeroth order Bspline is dened by
B
0
j
(x) =
1 for t
j
x t
j+1
0 otherwise
(2)
The expressions for the rst and second derivatives (i.e.
d
dx
B
k
j
(x) and
d
2
dx
2
B
k
j
(x) ) can be obtained by
differentiating Eq.1 and 2 with respect to x.
2
CAA using Bspline Collocation Method
0.00
0.25
0.50
0.75
1.00
0.0
0.5
1.0
0.00
0.25
0.50
0.75
1.00
0.0
0.5
1.0
PSfrag replacements
(a)
(b)
B(x)
B(x)
x
Figure 1:Distributions of Bsplines (
);knot points (2);and mesh points (3) for (a) uniform mesh and
(b) nonuniformmesh with local stretching factor of 0.1.
To show a typical distribution of Bsplines,consider a domain x 2 [0;1] discretized into 15 intervals
with uniform and nonuniform grid spacings.The nonuniform mesh is constructed using a constant local
stretching factor of lsf = 0:1 where the local stretching factor is dened as lsf =
x
i+2
x
i+1
x
i+1
x
i
1.The
proles of fourth order Bsplines,B
4
j
(x),for both meshes are plotted in Fig.1.Squares and diamonds
represent the knot and mesh points respectively.The distribution of Bsplines is clearly seen to follow the
distribution of the knot points and the grid stretching is found to alter the shapes of the Bsplines.
In solving differential equations,the computational variable (e.g.(x)) and its derivatives are repre
sented by a linear combination of Bspline trial functions as
(x) =
N
X
j=1
j
B
k
j
(x);
d
dx
(x) =
N
X
j=1
j
d
dx
B
k
j
(x);and
d
2
dx
2
(x) =
N
X
j=1
j
d
2
dx
2
B
k
j
(x):
In matrix form,these equations can be written as
fg = [M] fg;
d
dx
= [D] fg and
d
2
dx
2
= [V ] fg
where M
ij
= B
k
j
(x
i
),D
ij
=
d
dx
B
k
j
(x
i
) and V
ij
=
d
2
dx
2
B
k
j
(x
i
).These [M],[D] and [V ] matrices are in
fact banddiagonal matrices where the elements of the matrices are nonzero only along the few diagonal
lines adjacent to the main diagonal.As a result,conventional fast algorithms for banddiagonal matrices
can be utilized to minimize the computational time.
3.Modi?ed wave number analysis
The modied wave number analysis is commonly used to determine the resolution property of a nu
merical scheme.For the Bspline collocation method,the modied wave number for the rst and second
derivatives,
0
and
00
,can be expressed analytically as (see A.G.Kravchenko and P.Moin [6] for deriva
tions)
0
() =
I
P
i=1
2D
ij
sin(i)
M
0j
+
I
P
i=1
2M
ij
cos (i)
and (
00
())
2
=
I
P
i=1
2V
ij
(1 cos (i))
M
0j
+
I
P
i=1
2M
ij
cos (i)
3
R.Widjaja,A.Ooi,L.Chen,R.Manasseh
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
PSfrag replacements
(a) (b)
0
00
Figure 2:Modied wave numbers for (a) the rst derivative and (b) the second derivative for spectral (
);
FD4 (  );FD6 ();CD4 ( );CD6 (
);BSC4 ();and BSC6 (
) methods.
where I is the half bandwidth size of the [M],[D] and [V ] matrices.The subscript j is chosen such that the
mesh point x
i
= 0 is located at the peak of j
th
Bspline.
The modied wave numbers of 4
th
and 6
th
order Bspline collocation methods (BSC4,BSC6) are com
pared to those of 4
th
and 6
th
order nite difference (FD4,FD6) and compact difference (CD4,CD6)
schemes in Fig.2.The Bspline collocation method is shown to be capable of correctly representing the
waves up to a higher wave number than the other schemes of the same order.The unique convergence of
00
of the Bspline collocation method at higher wave numbers demonstrates its superiority in resolution for
high wave number waves.This reduces the number of grid points required in the computation.
4.Acoustic?eld froma spinning corotating vortex pair
PSfrag replacements
R
Figure 3:Schematic diagramof ow conguration.
To demonstrate the application of the Bspline collocation method,the acoustic eld froma spinning co
rotating vortex pair is simulated.As shown in Fig.3,the vortices whose strengths are = are separated
at a distance of 2R = 10 between the cores.They rotate about their mid point with a rotation rate of
=
4R
2
= 0:01 and a rotating Mach number of M
r
=
4Rc
0
= 0:05 where c
0
is the speed of sound.
The proles of the vortices are modelled using Gaussian vortex which vorticity distribution is given by
!
z
=
exp
r
2
:
The corresponding velocity eld can be obtained by solving the streamfunction Poisson equation,
r
2
= !
z
;
where ~u = r
^
k
,and
^
k is the unit vector normal to the plane of rotation.
4
CAA using Bspline Collocation Method
0 10 20 30 40 50
0.0
0.5
1.0
1.5
2.0
0 200 400 600 800 1000 1200
0
2
4
6
8
10
12
14
16
PSfrag replacements
r
r
r
r
Figure 4:Radial grid spacings using Bsplines (
);and mapping function (
).
In the acoustic calculation,Powell's acoustic analogy is used to simulate the production and radiation of
the acoustic waves.The governing equation is
1
c
2
0
@
2
p
@t
2
r
2
p =
0
r:(~!~u) (3)
where p is the acoustic pressure uctuation,~!is the vorticity and
0
is the uid density.The right hand
side of Eq.3 is called the acoustic source and is computed using the incompressible ow parameters.The
acoustic eld is simulated using a polar coordinate system.At the computational boundary (r
boundary
=
1241:5),a radiation boundary condition based on C.K.WTam [7] is applied to convect the acoustic waves
out of the domain.The spatial derivatives are calculated using the 6
th
order Bspline collocation method
and a Fourier Galerkin method.
The domain discretization uses 192 64 grid points in the radial and azimuthal directions respectively.
The radial mesh is nonuniform while the azimuthal mesh is uniform.The radial discretization involves
three regions,a uniformne mesh (r
near
= 0:2) in the near eld,a uniformcoarser mesh (r
far
= 15)
in the far eld and a stretched mesh (lsf = 0:05) connecting the two meshes.A plot of grid spacing
at different radial position is given in Fig.4.The discontinuity in the slope at the intersections of the
regions,which is a problem when using a mapping function,does not deteriorate the accuracy of the B
spline collocation method.This exibility allows the grids to be distributed optimally.For a comparison,a
continuous hyperbolic tangent mapping function is also plotted in Fig.4.The mapping function results in
242 grid points,which is 26%more than that using Bsplines.
Furthermore,the acoustic eld is time marched using a 4
th
order RungeKutta scheme with a time step
of t = 0:125.This results in a maximumCFL number of 0.625.The effect of the initial acoustic transient
is minimized by employing a ramping function and a numerical lter proposed by S.K.Lele [3].
Figure 5a shows the acoustic eld at t = 3700 whereby the acoustic eld has reached its steady periodic
state.The vortices are located close to the center of the domain.They generate a pair of positive and a pair
of negatives spikes in the near eld as they rotate.This spike pattern denotes a quadrupole source for the
acoustic radiation.The radiated acoustic waves have a wavelength of = 314.Its amplitude decays as
r
1=2
in the far eld,which is in agreement with 2D wave propagation theory.The acoustic eld can be
further validated against the matched asymptotic solution which is given by
p(r;;t) =
0
4
64
3
R
4
c
2
0
J
2
2
r
c
0
sin(2 2
t) +Y
2
2
r
c
0
cos (2 2
t)
(4)
where J
2
2
r
c
0
and Y
2
2
r
c
0
are second order Bessel functions of the rst and second kinds.
5
R.Widjaja,A.Ooi,L.Chen,R.Manasseh
0
200
400
600
800
1000
1200
4
2
0
2
4
(×10
5
)
X
Y
Z
PSfrag replacements
(a) (b)
r
p
r
1/2
Figure 5:(a) Acoustic eld fromthe spinning corotating vortices at t = 3700 and (b) comparison of radial
cut of acoustic eld at = 0
() to the matched asymptotic solution (
).
Shown in Fig.5b is the radial cut of the acoustic eld at = 0
.The far eld acoustic signal agrees
very well with Eq.4.In the near eld however,there are some discrepancies at r < 20.This is due to the
fact that the analytical solution based on matched asymptotic expansions is derived assuming point vortices
where the vorticity is concentrated at a single point at the vortex core.These discrepancies in the near eld
were also observed by S.A.Slimon,M.C.Soteriou and D.W.Davis [8].
5.Conclusion
A collocation method based on Bsplines as the trial functions is proposed.Its unique arbitrary grid
placement capability is shown to be efcient in resolving the ow and acoustic length scales with 26%
fewer grid points than that using a hyperbolic mapping function.Moreover,the resolution property of the
Bspline collocation method is found to be superior to nite difference and compact difference schemes.
Along with its robust formulation,these features make the Bspline collocation method be a suitable method
for computational aeroacoustics.
Acknowledgements.This research is funded by Defence Science and Technology Organization,Australia and sup
ported by Commonwealth Scienti?c and Industrial Research Organization,Australia.The computer resources are
supplied by Advanced Research Computing Center at University of Melbourne.
References
[1] C.K.WTam,Computational aeroacoustics:Issues and methods,AIAA Journal 33 (1995) 17881796.
[2] C.K.WTam,J.C.Webb,Dispersionrelationpreserving?nite difference schemes for computational acoustics,J.
Comput.Phys.107 (1993) 262281.
[3] S.K.Lele,Compact?nite difference schemes with spectrallike resolution,J.Comput.Phys.103 (1992) 1642.
[4] J.W.Kim,D.J.Lee,Optimized compact?nite difference schemes with maximum resolution,AIAA Journal 34
(1996) 887893.
[5] Y.Morinishi,S.Tamano,K.Nakabayashi,A DNS algorithmusing Bspline collocation method for compressible
turbulent channel?ow,Computers and Fluids Journal,32 (2003) 751776.
[6] A.G.Kravchenko,P.Moin,Bspline methods and zonal grids for numerical simulations of turbulent?ow,Ph.D.
Thesis,Stanford University,1998.
[7] C.K.W.Tam,Advances in numerical boundary conditions for computational aeroacoustics,J.Comput.Phys.6
(1998) 377402.
[8] S.A.Slimon,M.C.Soteriou,D.W.Davis,Computational aeroacoustics simulations using the expansion about
incompressible?ow approach,AIAA Journal,37 (1999) 409416.
6
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