Thirteenth International Conference on Domain Decomposition Methods
Editors:N.Debit,M.Garbey,R.Hoppe,J.P´eriaux,D.Keyes,Y.Kuznetsov c
2001 DDM.org
46 Analysis of a defect correction method for computational
aeroacoustics
G.S.Djambazov,C.H.Lai
1
,K.A.Pericleous and Z.K.Wang
2
Introduction
Many problems of fundamental and practical importance are of multiscale nature.As a
typical example,the velocity ﬁeld in turbulent transport problems ﬂuctuates randomly and
contains many scales depending on the Reynolds number of the ﬂow.In another typical
example,which is the main concern of this paper,sound waves are several orders of magnitude
smaller than the pressure variations in the ﬂow ﬁeld that account for ﬂow acceleration.These
sound waves are manifested as pressure ﬂuctuations which propagate at the speed of sound
in the medium,not as a transported ﬂuid quantity.As a result,numerical solutions of the
NavierStokes equations which describe ﬂuid motion do not resolve the small scale pressure
ﬂuctuations.The direct numerical simulation to include the above multiple scale problems is
still an expensive tool for sound analysis [1].
In essence,there are at least three different scales embedded in the ﬂow variables,namely (i)
the mean ﬂow,(ii) ﬂow perturbations or aerodynamic sources of sound,and (iii) the acoustic
perturbation.While ﬂow perturbation or aerodynamic sources of sound may be easier to re
cover,it is not true for the acoustic perturbation because of its comparatively small magnitude.
¿Froman engineering perspective,much of the larger scales behaviour may be resolved with
the stateoftheart CFD packages which implement various numerical methods of solving
NavierStokes equations.This paper examines,in more detail,a defect correction method,
ﬁrst proposed in [2],for the recovery of smaller scales that have been left behind.The authors
have demonstrated the accurate computation of (i) and (ii) in [3][4][5].In the present study,
a twoscale decomposition of ﬂow variables is considered,i.e.the ﬂow variable
is written
as
,where
denotes the mean ﬂow and part of aerodynamic sources of sound and
denotes the remaining part of the aerodynamic sources of sound and the acoustic perturbation.
The concept of defect correction [6] has been used in various contexts since the early days.
A typical example of defect correction is the computation of a reﬁned approximation to the
approximate solution
of the nonlinear equation
.Since
is an approximate
solution,the defect may be computed as
.The idea of a defect correction method
is to use a modiﬁed/derived version of the original problem such as the one deﬁned by
.If one replaced
as
,then
is the correction
computed by solving
and a reﬁned approximation can be evaluated by using
.More details in expanding the concept to discretised problems and multigrid
methods can be found in [6].Here,the authors would like to concentrate on using the defect
correction concept at the level of the physical problemrather than the discretised problem.For
a given mathematical problemand a given approximate solution,the residue or defect may be
treated as a quantity to measure howwell the problemhas been solved.Such information may
1
C.H.Lai@gre.ac.uk
2
Same address for all authors:School of Computing and Mathematical Sciences,University of Greenwich,30
Park Row,Greenwich,London SE10 9LS,UK
446 DJAMBAZOV,LAI,PERICLEOUS,WANG
then be used in a modiﬁed/derived version of the original mathematical problem to provide
an appropriate correction quantity.The correction can then be applied to correct the approx
imate solution in order to obtain a reﬁned approximate solution to the original mathematical
problem.
This paper follows the basic principle of the defect correction as discussed above and applies
it to the recovery of the propagating acoustic perturbation.The method relies on the use of a
lower order partial differential equation deﬁned on the same computational domain where a
residue exists such that the acoustic perturbation may be retrieved through a properly deﬁned
coarse mesh.
This paper is organised as follows.First,the derivation of a lower order partial differential
equation resulting from the NavierStokes equations is given.Truncation errors due to the
model reduction are examined.Second,accurate representation of residue on a coarse mesh
is discussed.The coarse mesh is designed in such a way as to allow various frequencies of
noise to be studied.Suitable interpolation operators are studied for the two different meshes.
Third,numerical tests are performed for different mesh parameters to illustrate the concept.
Finally,future work is discussed.
The defect correction method
The aimhere is to solve the nonlinear equation
(1)
where
is a nonlinear operator depending on
.For simplicity,
is considered to have
two different scales of magnitudes as
.Here
is the mean ﬂow and
is the acoustic
perturbation as described in Section 46.Note that
and that
with
much larger than any signiﬁcant period of the perturbation velocity.The problemhere
is thus purely related to the scale of magnitude.In the case of sound generated by the motion
of ﬂuid,it is natural to imagine
as the NavierStokes operator.For a 2D problem,
where
is the density of ﬂuid and
and
are the velocity components along the two spatial
axes.Using the summation notation of subscripts,the 2DNavierStokes problem
is written as
where
is the pressure and
is the viscous force along
th axis.
DEFECTCORRECTIONMETHODFORCOMPUTATIONALAEROACOUSTICS 447
Suppose (1) may be split and rewritten as
(2)
where
and
are operators depending on the knowledge of
and
is a
functional depending on the knowledge of both
and
.Following the concept of defect
correction,
may be considered as an approximate solution to (1).Hence one can evaluate
the residue of (1) as
which may then be substituted into (2) to give
(3)
In many cases,
is small and can then be neglected.In those cases,the problem in
(3) is a linear problem and may be solved more easily to obtain the acoustics ﬂuctuation
.
A nonlinear iterative solver is required in order to obtain
for cases when
is not
negligible.Finally,to obtain the approximate solution
,one only needs to solve
.
Expanding
for
being the NavierStokes operator and rearranging we
obtain
and
(4)
It can be seen that (4) may be written in the formof (3) where
(5)
(6)
(7)
¿Fromthe knowledge of physics of ﬂuids,the acoustic perturbations
and
are of very small
magnitude (this is not true for their derivatives),therefore,
may be considered negligible
due to the reason that any feedback fromthe propagatingwaves to the ﬂowmay be completely
ignored,except in some cases of acoustic resonance,which we are not concerned with here.
448 DJAMBAZOV,LAI,PERICLEOUS,WANG
Hence the equation
,with
given by (5),which is known as the linearised Euler
equation,can be solved in an easier way.The numerics and the techniques involved here are
often referred to as Computational AeroAcoustics (CAA) methods.
The remaining question is to obtain the approximate solution
to the original problem (2).
It is well known that CFD analysis packages provide excellent methods for the solution of
.Therefore one requires to use a Reynolds averaged NavierStokes package
supplemented with turbulence models such as [7,8] to provide a solution of
.One requires
to be as accurate as possible to capture all the physics of interest,such as ﬂow turbulence
and the presence of vortices.
The use of a CFD analysis package effectively solves
instead of
.Following the concept of truncation error in a ﬁnite difference method,the truncation
error due to the removal of the perturbation part of the ﬂow variable may be deﬁned by
(8)
Using the relation
,the truncation error in the present context
is thus given by
(9)
Note that this truncation error is not related to the discretisation of continuous model.
A twolevel multigrid method
In order to simulate accurately the approximate solution,
,to the original problem,
,
the QUICK differencing scheme [9] is used which produces sufﬁciently accurate results of
for the purpose of evaluating the residue as deﬁned in (7).A sufﬁciently ﬁne mesh has to
be used in order to preserve vorticity motion.However,much coarser mesh may be used for
the numerical solutions of linearised Euler equations [3,4,5].It certainly has to obey the
Courant limit and also to account for the fact that the acoustic wavelength may be larger than
a typical ﬂow feature which needs to be resolved,e.g.a travelling vortex [10].The present
defect correction method requires to calculate the residue on the CFD mesh and to transfer
these residuals onto the acoustic mesh.Physically,the residue is effectively the sound source
that would have disappeared without the proper retrieval technique as discussed in this paper.
Let
denote the mesh to be used in the Reynolds averaged NavierStokes solver.Instead of
evaluating
,one would solve the discretised approximation
to obtain
.The
residue on the ﬁne mesh
can be computed as
by means of a higher order approxima
tion [5].Let
denote the mesh for the linearised Euler equations solver.Again instead of
evaluating
,one would solve the discretised approximation
to obtain
.Here
is the projection of
onto the mesh
.Let
be a restriction operator
to restrict the residue computed on the ﬁne mesh
to the coarser mesh
.The restricted
residue can then be used in the numerical solutions of linearised Euler equations.Therefore
the twolevel numerical scheme is (for nonresonance problems):
Solve
Solve
DEFECTCORRECTIONMETHODFORCOMPUTATIONALAEROACOUSTICS 449
Here
denotes the discretised approximation of the resultant solution on mesh
.Note
that
cannot be computed as
because
is a nonlinear operator.
In the actual implementation,a pressuredensity relation which also deﬁnes the speed of sound
in air is used:
(10)
and the ﬁrst component of the linearised Euler equations in (5) becomes
(11)
The purpose of this substitution is to make sure that the new ﬂuctuations
and
do not
contain a hydrodynamic component,and hence can be resolved on regular Cartesian meshes
[4] which is essential for the accurate representation of the acoustic waves or the ﬂuctuation
quantity
.On the other hand,an unstructured mesh may be used to obtain
.The two differ
ent meshes overlap one another on the computational domain.The computational domain for
the linearised Euler equations is not necessarily the same as the one for the CFD solutions.It
must be large enough to contain at least the longest wavelength of a particular problemunder
consideration or a number of wavelengths where propagation is of interest.The numerical ex
ample as shown in Section 46 does not contain any complicating solid objects,the restriction
operator
may then be chosen as an arithmetic averaging process [10].
Numerical experiments with various grid parameters
The propagation of the following onedimensional pulse is considered:an initial pressure
distribution with a peak in the origin generates two opposite acoustic waves in both directions.
The exact solution of this problem(12) can be veriﬁed by substitution in the linearised Euler
equations.
(12)
Here
is the amplitude and
is the wavelength of the two sound waves that start from the
origin (
) at
.The example was reported in [2].This paper provides a detailed
numerical study on various aspects of the grid parameters being used in the twolevel method.
The CFD domain is of 12 wavelengths and the CAA domain is of 14 wavelengths.
The effects of the following parameters on the solution accuracyare studied.These parameters
are (a) the ratio H:h,(b) number of points per wavelength,and (c) the restriction operator for
residual transfer fromﬁne grid to coarse grid.In all cases,the norm
is compared.
Here
is the approximation obtained on the coarse mesh (CAA) after correction and
is
the exact solution of the pressure variable.
Let
and
be the step lengths in the temporal axis for the CFDmesh and the CAAmesh
respectively.Figure 1 shows the effect on the accuracy for Case (a).Here
and
are
450 DJAMBAZOV,LAI,PERICLEOUS,WANG
0
2
4
6
8
10
12
14
16
18
20
0
1
2
3
4
5
6
7
8
9
10
11
P_H  P_infty
Propagation distance (wavelengths)
h = 0.05, dt_H = 0.00005875, dt_h = 0.000235
H / h = 1
H / h = 2
H / h = 4
H / h = 8
0
2
4
6
8
10
12
14
0
1
2
3
4
5
6
7
8
9
10
11
P_H  P_infty
Propagation distance (wavelengths)
h = 0.025, dt_H = 0.00005875, dt_h = 0.000235
H / h = 1
H / h = 2
H / h = 4
H / h = 8
Figure 1:The effect of mesh ratio
:
on the accuracy.
0
5
10
15
20
25
0
1
2
3
4
5
6
7
8
9
10
11
P_H  P_infty
Propagation distance (wavelengths)
H = h = 0.00005875, dt_h = 0.000235
5 points
8 points
12 points
16 points
20 points
Figure 2:The effect of number of grid points per wavelength on the accuracy.
chosen to be 0.000235 and 0.00005875 respectively.Two different mesh sizes for the CFD
are chosen and they are 0.05 and 0.025.It can be seen that when
is not ﬁne enough,say
= 0.05,to resolve some of the physics,it is still possible to use the mesh
or
to recover the small scale signal.If a ﬁner mesh was used,say
,it is possible to
use
.This property essentially links with the Courant number of the coarse mesh for
CAA [5],i.e.H,and is also conﬁrmed in the test performed for Case (b).
Figure 2 shows the effect on the accuracy for Case (b).The most accurate solution may be
achieved with more than 12 grid points per wavelength,e.g.16 or more grid points.This
conﬁrms the theoretical study based on Courant limits as discussed in [5].For number of grid
points per wavelength less than 12,the accuracy deteriorates very fast.
Figure 3 shows the effect on the accuracy for Case (c).The restriction operators being used
in this test to transfer the function
onto the coarse mesh
includes
3 point formula:
5 point formula:
7 point formula:
9 point formula:
DEFECTCORRECTIONMETHODFORCOMPUTATIONALAEROACOUSTICS 451
1.7
1.75
1.8
1.85
1.9
1.95
2
2.05
2.1
0
1
2
3
4
5
6
7
8
9
10
11
P_H  P_infty
Propagation distance (wavelengths)
H / h = 4, h = 0.015625, dt_H = 0.00005875, dt_h = 0.000235
3 point restriction
5 point restriction
7 point restriction
9 point restriction
1.75
1.8
1.85
1.9
1.95
2
2.05
2.1
2.15
0
1
2
3
4
5
6
7
8
9
10
11
P_H  P_infty
Propagation distance (wavelengths)
H / h = 8, h = 0.0078125, dt_H = 0.00005875, dt_h = 0.000235
3 point restriction
5 point restriction
7 point restriction
9 point restriction
Figure 3:The effect of restriction operators on the accuracy.
For very ﬁne CFD mesh,one can retrieve the small scale signal even on a relatively coarse
mesh.In the present study,with
one can use
while still maintain
ing the accuracy.The accuracy exhibited by using the coarse mesh
is
compatible with the result for Case (a) as depicted in Figure 1.
Conclusions
This paper provides a numerical method for the retrieval of sound signals using the defect
correction method.A detailed numerical experiments to examine various grid parameters are
provided.Truncation error of solving
instead of
is derived.
The authors are currently applying the present method to sound propagation in vortexvortex
interactions.
References
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[2] G.S.Djambazov,C.H.Lai,and K.A.Pericleous.A defect correction method for the
retrieval of acoustic waves.In Abstract:12th Domain Decomposition Conference,Chiba,
Japan,October 25  29,1999,page 93.1999.
[3] G.S.Djambazov,C.H.Lai,and K.A.Pericleous.Development of a domain decomposition
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[4] G.S.Djambazov,C.H.Lai,and K.A.Pericleous.Efﬁcient computation of aerodynamic
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[5] G.S.Djambazov.Numerical Techniques for Computational Aeroacoustics.PhD thesis,
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[6] K.B¨ohmer and H.J.Stetter.Defect Correction Methods:Theory and Applications.
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[7] N.Croft,K.Pericleous,and M.Cross.PHYSICA:A multiphysics environment for com
plex ﬂow processes.In C.Taylor et al.,editors,Num.Meth.Laminar & Turbulent Flow
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