Available online at www.sciencedirect.com
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Procedia Engineering 00 (2014) 000–000
www.elsevier.com/locate/procedia
“APISAT2014”, 2014 Asia-Pacific International Symposium on Aerospace Technology,
APISAT2014
A Frequency Domain Immersed Boundary method and its
Application to 2-dimensional Acoustic Problems
A. Liang*
AVIC Commercial Aircraft Engine co. ltd., 3998 south Lianhua Rd., Shanghai 201108, PR China
Abstract
A frequency domain Immersed Boundary (IB) method was developed and validated in the present paper using 2-dimenstional
acoustical radiation and scattering cases. The IB method was incorporated with Linearized Euler Equations (LEE) in the
frequency domain in the present work. The governing equations were spatially discretisized using the DRP scheme. A pseudo
time dependant term was added to the frequency domain equations, allowing the use of a conventional time-marching algorithm
to converge the solutions in the pseudo-time domain. Perfectly Matched Layers (PML) were placed at boundaries of
computational domain where non-reflective conditions were expected. PML technique was also implemented inside the rigid
body to stabilize the computation. The impermeable boundary condition on the surface of the geometry is guaranteed by finding
the inverse of an influence matrix, which establishes the relationship between boundary forces and induced velocity. Numerical
computations were performed for 2-dimensional acoustic radiation and scattering problems. Computational results were
compared with exact solution and yielded good agreement, providing a solid validation of the current method. The method is
expected to extend to higher dimension and applied to more complex problem like wake/airfoil interaction simulations in
turbomachinery.
© 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
Keywords: Immersed boundary method; Frequency domain; Linearized Euler equation; Computational Aeroacoustics; Acoustic scattering
* Corresponding author. Tel.: +86-021-33367194; fax: +86-021-33366688.
E-mail address: lianga@acae.com.cn
1877-7058 © 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
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A. Liang/ Procedia Engineering 00 (2014) 000–000
Nomenclature
A
B
F
M
U
Û
  x


x
y
coefficient matrix of linearized Euler equations
coefficient matrix of linearized Euler equations
source term added to linearized Euler equations
influence matrix
state variable vector of linearized Euler equations
state variable vector of linearized Euler equations in frequency domain
Dirac delta function
angular frequency
pseudo time variable
PML coefficient
PML coefficient
1. Introduction
Aircraft noise has always been a great issue accompanying the rapid development of civil aviation. The need of
noise prediction and reduction has been impelling the development of aeroacoustics discipline, in both aspect of
theory and application. In the early ages with limited computational ability, aircraft noise analyses generally rely on
empirical models and analytical models. ANOPP (Aircraft NOise Prediction Program), which was developed under
the organization of NASA, integrates various empirical models for main noise sources of modern aircraft like fan
noise models[1], jet noise models[2-4] and so forth. TFaNS[5] and BFaNS[6, 7] are good examples for analytical
models, which predicts fan tone noise and broadband noise respectively.
In the recent decades, CFD has gradually become a common tool routinely used in aircraft and its engine design.
Tempted by the success of CFD, people were considering the application of numerical simulations for acoustic
problems. After a short period of attempt to use CFD schemes directly for acoustical simulation, people started to
realize that some important issues, including the reflection of computational boundaries, numerical dispersion and
dissipation, must be taken good care of before accurate result can be acquired. Research efforts on those aspects then
led to the emergence of a new discipline named Computational Aeroacoustics, or CAA. Lele[8] proposed a compact
differential scheme for spatial discretization while Tam and Webb[9] developed an explicit DRP scheme. For time
marching schemes, modified Adams-Bashforth[9] and modified Runge-Kutta[10] schemes were developed to
minimize the dispersion and dissipation in temporal dimension. Various versions of characteristic [11, 12],
asymptotic [9] and absorbing[13] boundary conditions were introduced to prevent sound, vorticity and entropy
waves from reflecting back into computational domain.
Complex and/or moving geometries are great challenges in both CFD and CAA. In order to simplify the grid
generation procedure and improve grid quality, computational techniques based on simple grids are of great research
interests. Besides the traditional body-fitted grid, several other options for complex geometries are available. One
possibility is the Chimera grid (also called overset grid) method[14-16], which generates a simple (e.g. Cartesian)
grid as background and a body-fitted grid for each geometry. Another concept is to treat the boundaries inside the
computational domain as force distributions and include the force term to the momentum conservation equation.
That idea was first proposed by Peskin et al.[17, 18] in 1970s and named the Immersed Boundary (IB) method. The
IB method has evolved much since its emergence. LeVeque et al.[19] described the boundaries as discontinuities of
function value and/or it derivatives and modified spatial discretization schemes to enforce the jump condition.
LeVeque’s method promoted the IB method from 1st order accuracy across boundaries to higher order. To make a
distinction, it was titled the Immersed Interface Method (IIM). Some researchers use ghost cell technique[20-22]
instead of singular force to treat complex geometries in simple background grid and sometime these methods are
also classified as IB methods.
The IB methods, including its varieties, has been applied to broad categories of fluid flow problems like flow
around heart valves[17, 18, 23], aquatic animal locomotion[24, 25], multi-phase flow[26-28], fluid-structure
interaction in turbomachinery[29] but very little attention was paid to inviscid acoustic applications[21, 22, 30].
A. Liang/ Procedia Engineering 00 (2014) 000–000
3
Moreover, most realizations of the IB concept were carried out in time domain instead of frequency domain.
However, studying acoustic problems in frequency will bring great convenience in treating boundary conditions like
impedance condition, phase-shifted condition, which is very important in turbomachinery and non-reflective
condition. Another benefit from frequency domain methods is the avoidance of recording the time history data.
In the present paper, a frequency domain IB method was developed and validated using 2 dimensional acoustic
problems. The paper is organized as follows: numerical formulations of the method will be introduced in Section 2;
numerical simulations of a series of acoustic cases will be described in Section 3, as well as the numerical results
and discussion; conclusion will be made at last.
2. Numerical formulation
2.1. Governing equations
The present paper serves as a proof of concept and the background flows are quiescent in all the computational
cases, so we start with the 2-dimensional linearized Euler equations with zero mean flow velocity. The time domain
linearized Euler equations in matrix form read as follows:
U
U
U
A
B
0
t
x
y
(1)
where

0
 

u
0

U
A
v
0
 

 p
0

0
0
c2 
0 0 
0


0 1/  
0
B
0
0 0 


0 0 
0
0 
0 0
0 0
0 

0 
1/  

2
0 c  0 
(2)
The state variables in vector U are disturbance quantities and the overbar symbols appearing in the coefficient
matrices A and B denote time averaged quantities.
The spirit of the IB method is to replace the boundaries by the effect of a force field. Thus a source term F is
added to the homogenous equations (1):
U
U
U
A
B
F
t
x
y
(3)
Physically, F represents the surface force applied to the fluid and has the following form
0




 S f n nx  x  X  ds 
F

 S f n n y  x  X  ds 


0


(4)
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A. Liang/ Procedia Engineering 00 (2014) 000–000
The quantity fn in equation (4) is the Lagrangian force on unit length of boundary curve. In the framework of inviscid
flow, the boundary forces exerted to the fluids are in the direction normal to the boundary, so the x- and ycomponent are fnnx and fnny respectively. The vector x=(x,y)T denotes the coordinate of point in fluid domain and X
the coordinate of boundary point. The integration is to be performed along the boundary curves.
The equations are then transformed to frequency domain by assuming a time harmonic variation
ˆ  x, y  eit
U  x, y , t   U
(5)
F  x, y, t   Fˆ  x, y  eit
where i  1 . Substituting equation (5) into (1) and we reach the frequency domain linearized Euler equations
ˆ
ˆ
ˆ  A U  B U  Fˆ
i U
x
y
Matrices A and B retain the same definition as equation (2) but the vectors
(6)
Û and F̂ are now complex.
2.2. Spatial discretization
The 7-point DRP scheme by Tam et al.[9] is used for the present computational cases. The DRP scheme for the
1st order derivative can be expressed as
f  x   1

x
 x
m
a
j  n
nm
j

f  x  jx    O  x 4 

(7)
where n is the number of points to the left and m to the right. The coefficients ajnm are given by Table 1. Central
difference, i.e. n=3, is used wherever possible and forward or backward difference is used near the boundaries of
computational domain. An artificial selective damping[31] is added in the computation to selectively damp out the
short waves and avoid affecting on the long waves.
Table 1. Coefficients of 7-point DRP scheme
j=-n
j=-n+1
j=-n+2
j=-n+3
j=-n+4
j=-n+5
j=-n+6
n=0
-2.192280339
4.748611401
-5.108851915
4.461567104
-2.833498741
1.128328861
-0.203876371
n=1
-0.209337622
-1.084875676
2.147776050
-1.388928322
0.768949766
-0.281814650
0.048230454
n=2
0.049041958
-0.468840357
-0.474760914
1.273274737
-0.518484526
0.166138533
-0.026369431
n=3
-0.0208431428
0.166705904
-0.770882381
0
0.770882381
-0.166705904
0.0208431428
n=4
0.026369431
-0.166138533
0.518484526
-1.273274737
0.474760914
0.468840357
-0.049041958
n=5
-0.048230454
0.281814650
-0.768949766
1.388928322
-2.147776050
1.084875676
0.209337622
n=6
0.203876371
-1.128328861
2.833498741
-4.461567104
5.108851915
-4.748611401
2.192280339
A. Liang/ Procedia Engineering 00 (2014) 000–000
5
2.3. Pseudo time marching
In order to find the solution of the frequency domain equations (6), we follow the approach of Verdon[32] and
add a pseudo time dependant term:
ˆ
ˆ
ˆ
U
ˆ  A U  B U  Fˆ
 i U

x
y
(8)
where  is the pseudo time variable. This treatment allows the use of a conventional time-marching algorithm to
converge the solutions to constant values. In the present paper, the LDDKR scheme proposed by Hu et al. [10] is
applied as the time-marching algorithm. Although this may not be the fastest converging scheme, it does not seem to
bring any other disadvantages. The pseudo time marching procedure consists of 2 steps. In the first step, equation (8)
is advanced without the inhomogenous source term F̂ , and then the state vector is updated in the second step by
considering the source term by
ˆ  Fˆ 
U
(9)
The method for source term computation will be introduced in Section 2.5.
2.4. Non-reflective boundary condition
Perfectly Matched Layer[13] (PML) can be easily realized and give a satisfactory performance when the
background flow velocity is absent and is thus selected in the present paper for the treatment of non-reflective
boundary condition. As stated by Hu[33], the key step of PML is a complex change of variable in frequency domain
as follows

1


x 1  i x x



y
1
(10)
1

i y y

Substituting equation (10) into (8) and we get the PML equation in frequency domain.
ˆ
ˆ
U
ˆ  1 A U 
 i U
i

1  x x

1
ˆ
1
U
B
 Fˆ
i y y
(11)

For the present usage, the deduction may end up here but if the calculation is performed in time domain, an auxiliary
variable vector must be introduced[33]. In the PML zones, the absorbing coefficient in x- direction is given as
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A. Liang/ Procedia Engineering 00 (2014) 000–000
x 
 0 x  x0
x
2
(12)
D
where x=x0 is the interface between PML and regular zone, D is the width of PML and  0 is a constant typically
selected to be equal to 4. Configuration of PML coefficients in y- direction is similar.
2.5. Impermeable boundary condition
Impermeable condition should be satisfied on the boundaries of solid body inside the inviscid fluid. In the present
work, the solid boundary is described by a series of discrete points immersed in the fluid background grid as
illustrated in Fig. 1 and the impermeable boundary condition is guaranteed by introducing a force field normal to the
boundary curves as indicated in equation (3) and (4).
It can be noted in equation (4) that the force exerted on the fluid is singular with the existence of a Dirac delta
function. This problem is handled by approximate the Dirac delta by a smooth function[34]
  x  h  x 
1
 x   y 
    
xy  x   y 
(13)
where



1
2
 8 3  2 r  1  4 r  4r

1
  r    5  2 r  7  12 r  4r 2
8
0


The plot of
 r 
in equation (14) is illustrated in Fig. 2.
Fig. 1. Discrete boundary points.
, 0  r 1

, 1 r  2
,
r 2
(14)
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Fig. 2. Plot of function   r  .
The force in fluid domain is calculated via a discrete form of equation (4)
Fx  i, j     f n nx  k   h  xi , j  X k  w 
k
Fy  i, j     f n n y  k   h  xi , j  X k  w
(15)
k
In equation (15), Fx, Fy and fn are complex while the other variables and functions are real. The remaining
problem is the determination of the boundary force fn. This is handled by constructing and solving an influence
matrix as used in literature[30, 35]. As introduced in section 2.3, the first step of time matching is to advance
equation (8) is advanced to the next pseudo time step without the boundary force term. The resulting flow field does
not satisfy the impermeable boundary conditions in general and the normal component of fluid velocity at kth
boundary point can be calculated via an interpolation formula
un  k   nx  u  i, j   h  xi , j  X k  xy   ny  v  i, j   h  xi , j  X k  xy 
i, j
(16)
i, j
Since equation (15) and (16) are linear, an influence matrix M can be constructed[30] such that
Mfn 
U n

(17)
where fn is a vector whose kth component represents the Lagrangian force density on kth boundary point and kth
component U n is the corresponding variation of normal velocity. The influence matrix M is sometimes singular
and the SVD algorithm is applied to solve equation (17) for fn.
Another problem arising together with impermeable boundary condition is the flow field inside the geometries.
For viscid flow with non-slip boundary condition, the IB methods treat those domains as regular fluid domains and
govern them by the same differential equations as outside. However, for the cases of inviscid flow where viscous
dissipation is absent, flow field inside geometries may cause severe stability problems. One possible solution is to
place a PML domain inside each geometry and set the absorbing coefficient according to the distance to the
boundary[30].
Finally, we summarize the steps for the computation as follows:
(1) Advance equation (8) without force term from pseudo time step  n to  n+1;
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A. Liang/ Procedia Engineering 00 (2014) 000–000
(2) Calculate normal velocity on each boundary point using equation(16), and then calculate the error of normal
velocity;
(3) Solve equation (17) to determine the force term;
(4) Spread the boundary force to the fluid domain using equation (9) and (15).
3. Cases and results
3.1. Radiation from pulsating cylinder
Consider a pulsating cylinder in free space illustrated in Fig. 3(a). Denote the radius of the cylinder by r0 and
i t
express the boundary normal velocity as vn e . The radiated sound is a function only of the distance between
observing point and the center of the cylinder, r. The theoretical solution of the radiated sound field is given in [36]
as
ivn H 0(1)  kr  it
p
e
kH1(1)  kr0 
(18)
where k is the wave number and Hm(1) is the Hankel function of mth order. The parameters in equation (18) and
computational configuration are listed in Table 2. It should be noted that this is actually a moving boundary problem.
However, in the present simulation we assume the displacement of the cylinder is small and only consider the
velocity boundary condition.
The resulting radiation sound field is shown in Fig. 4 where the real part of sound pressure is plotted. It can be
witnessed from the figure that the pressure field shows perfect axial symmetry and the PML zones both at the
computational boundaries and inside the cylinder give good performance. A more detailed comparison is given in
Fig. 5 where pressure distribution along line y=0 is compared with theoretical solution. It is seen from the figure that
the computational data matches well with theoretical data (the region x<0.5 and x>1.1 are within PML zones).
Table 2. Parameters for pulsating cylinder case and vibrating cylinder case
Parameter
Pulsating Cylinder
Vibrating Cylinder
Sound speed
1
1
Mean density
1
1
Angular frequency
8
8
Diameter of the cylinder
1
1
Amplitude
vn=1e-4
ua=1e-4
Computational domain
[-1.4, 1.4]×[-1.4, 1.4]
[-1.4, 1.4]×[-1.4, 1.4]
Width of PML zones
0.3
0.3
Grid number (Nx×Ny)
281×281
281×281
Number of boundary points
315
315
Fig. 3. (a) pulsating cylinder; (b) vibrating cylinder.
A. Liang/ Procedia Engineering 00 (2014) 000–000
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Fig. 4. Computed radiation sound field of a pulsating cylinder.
Fig. 5. Comparison of pressure distribution for pulsating cylinder case. (a) real part; (b) imaginary part.
3.2. Radiation from vibrating cylinder
The case of vibrating cylinder, shown in Fig. 3(b), is similar to the pulsating cylinder, except that the cylinder
i t
moves as a rigid body in the x direction with velocity u a e . The normal velocity of a boundary point is easily
i t
deduced as u a e cos  . The theoretical solution is again given in [36] as
p
2  ua i cos  H1(1)  kr 
k  H 0(1)  kr0   H 2(1)  kr0  
eit
(19)
Parameters necessary for case setup and numerical computation are given in Table 2. The computed sound field is
shown in Fig. 6. It can be clearly observed that the radiation field is equivalent to that from a dipole. The
comparison with theoretical solution along line y=0 is given in Fig. 7. Also, good comparisons are observed.
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A. Liang/ Procedia Engineering 00 (2014) 000–000
Fig. 6. Computed radiation sound field of a vibrating cylinder.
Fig. 7. Comparison of pressure distribution for vibrating cylinder case. (a) real part; (b) imaginary part.
3.3. Scattering by a cylinder
We use a benchmark case in the second CAA workshop[37] as further validation of present method. As illustrated
in Fig. 8, sound generated by a Gaussian spatially distributed source located at (4, 0) is scattered by a cylinder with
diameter equal to 1 and its center located at (0, 0). Sound pressures are to be compared at observation points along
the arc r=7. The sound source is described by a source term added to the linearized Euler equation:
u p

0
t x
v p

0
t y

  x  4 2  y 2  
p u v


 exp   ln 2 
  sin t
2


t y y
0.2



where the angular frequency  is set to 8 .
(20)
A. Liang/ Procedia Engineering 00 (2014) 000–000
11
Fig. 8. Geometry of the scattering problem.
Fig. 9. Computed sound field of the scattering case.
Fig. 10. Comparison of directivity at r=7.
The theoretical solution is given in [37] and will not be repeated here. For the present numerical study, the
computational domain [-8, 8]×[-2, 8] is discretized to a 769×481 uniform Cartesian grid and the cylinder is
described by 500 uniformly distributed points. PML zones are again used on the computational domain boundary
and inside the cylinder. The computed sound field is plotted in Fig. 9 and the comparison of directivity is shown in
Fig. 10. The good agreement further validates the present methods.
4. Conclusion
A frequency domain Immersed Boundary method for inviscid flow with impermeable boundary condition was
developed in the present paper. As a proof of concept and a preliminary validation, numerical calculations for cases
without background flow velocity were carried out and compared with exact solutions. The computed results and
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A. Liang/ Procedia Engineering 00 (2014) 000–000
exact solutions matched well, validating the present method to be suitable for acoustic problems with complex
geometries.
Some further developments of the present method are expectable. Firstly, the advantages of frequency domain
methods are the convenience in treating phase-lagged boundary conditions and impedance conditions. However,
these advantages were not demonstrated in the present work and future research effort is required. Secondly, the
frequency domain equations were solve by adding a pseudo time variable and algorithm for time marching is used.
This may not be the most efficient treatment and fast-converging solver for time domain equations may be desirable.
Finally, the frequency domain IB method may be extended to more complex flow phenomenon and more complex
geometry and applied to cases with more practical interests.
Acknowledgements
The funding of the “973 Program” (Grant No. 2012CB720201) is gratefully acknowledged.
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