ICSV22, Florence (Italy) 12-16 July 2015 1
INVESTIGATION ON EFFECT OF ROTOR BLADES ON THE
SOUND PROPAGATION
Xiaoyu Wang, Di Zhou and Xiaofeng Sun
School of Jet Propulsion, Beijing University of Aeronautics & Astronautics, 37 Xueyuan Road,
Beijing, 100191, China.
e-mail: bhwxy@buaa.edu.cn
In this paper, a new model including the influence of rotor blades on sound propagation will
be set up to calculate the rotor and stator interaction noise. Firstly, three-dimensional lifting
surface theories are used to describe the sound and vortex propagation through a single blade
row. Secondly, the basic model in finite element about sound generation and propagation
will be set up. Finally, the relation of acoustic wave and vortex wave between rotor blades
row and stator row is studied by using TEM (Transfer Element Method), in which we set up
a matching surface in order to determine the sound field and unsteady loading on rotor
blades and stator vanes. The prediction of rotor blades and stator vanes interaction will be
given by the TEM model. Various examples are presented here to illustrate the effects of
number of stator vanes. It is found that rotor blades have a good “shield” effect on the up-
stream sound. Other results will show that the interaction between rotor and stator has a great
impact on the unsteady loading of blades.
1. Introduction
From the view of noise characteristics in modern high bypass ratio turbofan engine, the discrete
frequency noise generated by fan rotor and OGV is still one of the major noises. There are many
works which focus on the prediction and control of this kind of noise. The basic physics is that the
impact of periodic rotor wake on the stator vanes results in the unsteady responding which is regard
as a dipole source. Therefore a lot of works investigate the wake-stator interaction. In fact the
acoustic wave traveling form stator will couple with vortical wave on rotor blades and induce sound
transmission and reflection. Sometimes the rotor blades are the good shield for sound propagation.
Therefore it is important to include the effect of rotor blades on the prediction of noise.
In 1970’s, Kaji and Okazaki [1] gave the propagation model of sound waves through a blade row
by using acceleration potential method. They found the effect of blade spacing on sound propaga-
tion through a cascade is not considerable. In 1997, Hanson [2] investigated the unsteady acoustic
and vortical coupling between adjacent blade rows. In order to explain the sound transmission and
reflection phenomenon, he introduced the special condition where wavefronts are parallel or verti-
cal to blade chords. However the 2D flat plate unsteady cascade model cannot consider the radial
interaction of unsteady loading on the blade surface, although the more realistic flow and geometry
has been included. For 3D problems Namba [3] and Schulten [4] developed the lifting surface theo-
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ry to investigate the sound generation by wake and blades interaction, respectively. Schulten inves-
tigated the sound propagation through 3D rotor cascade. In these two methods the cascade has been
considered as isolation in the fan channel.
Recently Wang and Sun [5] developed a new method called “Transfer Element Method” to in-
vestigate the effect of stator on the liner design. In the work, the solutions of sound propagation in
finite domain with liner and stator vanes are set up. However the effect of rotor blades on the acous-
tic scattering is absent. In the present paper the basic solution of rotor blades in finite domain will
be established firstly. Then combining the element of stator and rotor, it is possible to predict the
noise generation by rotor and stator interaction.
2. Transfer Element Method for rotor cascade
2.1 Basic Solution
In this paper, the sound generation and propagation of rotor cascade will be calculated in a hard
wall duct, subject to the following simplifying assumptions: The system is three-dimensional with a
compressible, inviscid, isentropic and mean flow; The mean angle of incidence is zero and the
mainstream flow pass through the cascade undeflected; The blade are flat plates of negligible thick-
ness; All perturbations are small relative to uniform mean flow, and thus the flow equation may be
linearized and the principle of superposition can be applied; The flow is subsonic; The unsteady
blade loading at the trailing edge is finite.
Basing on above assumptions, the sound generated by unsteady loading on the rotor [6] can be
considered as
(1)
()
( , ) ( )
T
i
s
i
T
G
p x t p ds y d
y
.
where
()
()
s
ds y
is the blade surface integral.
i
p
represents the pressure difference on the blade
surface and
(2)
( tan ) ( )
iz
i
r
p p p p p
y r z r z r U z
.
The relation of unsteady force direction is illustrated in Fig. 1. Because the attack angle and flow
deflection are not considered in this model, the shape of rotor blade depends on the velocity triangle,
as shown in Fig. 1(B).
Figure 1. The direction of unsteady force
For the infinite straight annular duct the corresponding Green`s function is
(3)
( ) ( )
2 2 2 2 2
1
00
( ) ( )
1
42
im im
i t i z z
m mn m mn
mn
mn mn
k r k r e e
e e d d
G
Ma k k k
p
p
U
W
r
z
p
z
rotor blade
tip
b
mid
hub
()A
()B
z
()C
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where
()
m mn
kr
and
mn
k
are the eigenfunction and eigenvalue in the annular duct.
22
1 Ma
,
and
Ma
expresses axial Mach number,
is angular frequency and
00
kc
(
0
c
is the local speed
of sound). For the rotor cascade, it is supposed that the frequency
of the unsteady loading on eve-
ry blade is equivalent. The circumferential unsteady force of the k-th blade can be written as
(4)
2
( 1)
, 1,2, , ,
B
i i k
p p e k B
denotes the inter-blade phase angle,
B
is the number of rotor blades. Substituting Eqs. (2)-(4)
into Eq. (1) and considering the rotational coordinate system
( , , )r r z z
, where,
is the rotating speed, the pressure field generated by unsteady loading can be expressed as
(5)
2
( 1)
2
()
11
()
2 2 2 2
00
()
( , ) ( )
4
( ) ( )
2
im
B
i m k
im
m
B
m
s
m n k
m
T
i z z
m
it
T
i
re
i
p x t p r e e
r m e
e e d d d ds y
U r Mk k k
By utilizing the fact in the generalized function theory and the principle of geometric progres-
sion, it is easy to find out only when
m
and
m sB
are satisfied the right side of the
preceding equation is not zero. Letting
2
mn mn mn
and integrating with
, the acoustic
pressure generated by unsteady loading on the rotor blades can be obtained that
(6)
12
()
1
( ) ( )
12
()
( , ) ( )
4
( )( ) ( )( ) ( )
im
it
im
mn mn
q mn mn
s
sn
nm
i z z i z z
k r e
Be
p x t p k r e
m r m r
H z z e H z z e ds y
r U r U
where,
1
and
2
are the wave number for downstream and upstream, respectively.
2.2 Unsteady loading on blades
The key of noise prediction generated by rotor cascades is solving the unsteady loading on the
blades. We will obtain an equation by letting the sum of
-direction velocity
v
and upwash ve-
locity
w
become zero for satisfying the boundary condition in the normal direction on the blades
surface. The equation is that
(7)
,
0
i
vw
where,
,i
w
denotes the wake model of stator or the velocity of potential flow. The relationship of
rotating frame and convection frame can be found in Fig. 1(c).
2.2.1 The normal velocity of scattering wave on the blades
In the convection frame the momentum equation is
(8)
1
vv
p
W
t z r
Basing on this equation the velocity in the normal direction of blades can be expressed as
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(9)
00
1
3
( ) ( )
1
()
()
2
11
, , 0
()
()
2
2 2 2
0,
( ) ( ) ( )
2
( ) ( )( )
2 ( )
(
it
im im
s mn mn mn mn
ss
sn
im z z
i z z
U
n m n m
im z z
i z z
U
mn
Be
v p K ds y k r e p k r e
W
Me e m r m r
H z z
M k r U r U
M e e m
k M k r
2
33
()
()
2
22
, 0 ,
)( )
( ) ( )( ) ( )
2 ( )
im z z
i z z
U
n m n m
r m r
U r U
Me e m r m r
H z z ds y
k M r U r U
where
W
is convection velocity.
1
and
2
represent the same items discussed above.
3
is the
wave number of vortex wave propagating downstream. The first and last terms in Eq. (9) represent
the response velocity of acoustic wave in the normal direction of blades. The second one describes
the vortex wave and it is necessary because the Kutta condition is implemented at the tailing edge
with uniform flow.
2.2.2 The upwash velocity
Two kinds of mechanism of sound generation are considered in this paper. The first type is due to
wakes behind the adjacent vanes which is rotating relatively. In this mechanism the upwash velocity
can be described as
(10)
,
2
2
iqV z
U
iq
q
r
w W e
Ur
(in rotating frame )
The other mechanism is called potential interaction. A rigid thin airfoil placed in the periodically
fluctuating velocity field is directly equivalent to a layer of acoustic doublets in the sound field, and
in this section we emphasize the interaction between sound wave and rotor blades.
Considering the sound generated by rotor-wake and stator vanes, the acoustic field in the rotation
system can be expressed as
(11)
1
( , ) ( )
im i t i z
mn mn mn
mn
p r t p k r e e e
And the frequency and mode characteristic of the acoustic field generated by wake-stator interac-
tion are
,qV m B qV
. In this manner for the rotor cascades the frequency of the unsteady
fluctuation is
qV
. According to the three-dimensional lifting surface theory that mentioned
above, the corresponding frequency and mode of acoustic field generated by interaction between
sound wave and rotor blades are
sB
and
m s B qV
, respectively.
It can been seen that a given wave appears at blade passing frequency in the stator coordinate
system and at vane passing frequency in the rotor coordinate system. In other words interaction of
the blades with waves of the form of Eq.(11) will produce output waves at the same frequency and
corresponding mode. Besides the frequency scattering produces corresponding modal scattering
with the potential interaction in the rotor reference frame.
Substituting Eq. (11) into Eq. (8), we can obtain the normal velocity on the rotor blades
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ICSV22, Florence, Italy, 12-16 July 2015 5
(12)
()
,
1
1
()
i m z
it
U
i mn m mn
qn
mr
rU
w p k r e e
W
m
UU
Up to now, the upwash velocity has been obtained that result in the integral equation.
2.3 Transfer element method for fan rotor
In order to investigate the effect of rotor on the sound generation, it is necessary to set up an in-
dependent element, in which the pressure wave and vortex wave interact with rotor blades. As
shown in Fig. 2,
,1D
p
,
,1A
p
and
,1D
w
are the “assumed wave”, whose amplitudes and phase are un-
known. However the value of interface wave can be determined by applying the continuity and
momentum equation on the interface. Besides, the difficult point of this model is how to express the
scattering wave (
s
p
and
s
w
) as explicit function of assumed wave.
Figure 2. Transfer Element Method for rotor blades
The section between interface
aa
and
dd
is defined as a transfer element. We suppose the
assumed waves at interface can be expressed as the mode form of pressure wave and vortex wave.
In the rotor coordinate system, corresponding unsteady fluctuation of these three interface wave in
the normal direction of blade surface can be obtained by applying Eq. (8). The scattering pressure
wave
s
p
and the scattering vortex wave
s
w
are generated by the interaction between three assumed
waves and rotor blades. As mentioned above, the velocity satisfies the following condition
(13)
, , , ,1 , ,1 , ,1
( , ) ( , ) ( , ) ( , ) ( , ) 0
s s D A D
u r z w r z u r z u r z w r z
Although the coefficient, such as
1p
m
D
etc. in the
, ,1D
u
,
, ,1A
u
and
, ,1D
w
, is unknown, the
s
p
and
s
w
can be expressed as the explicit function of
1p
m
D
,
1p
m
A
and
1
0
w
D
. In Eq. (13) it is noted that
the generation of unsteady loading on the blades includes three parts: the first is caused by the up-
stream pressure disturbance
, ,1
( , )
D
u r z
, the second is done by the downstream pressure disturbance
, ,1
( , )
A
u r z
and the third is due to the upstream vortex disturbance
, ,1
( , )
D
w r z
. In terms of the lin-
ear superposition principle, the unsteady loading due to the pressure and vortex waves can be con-
sidered separately by letting
,s
u
and
,s
w
satisfy the boundary condition, respectively. Therefore,
the three integral equation can be obtained, and for example
(14)
, ,1
()
( , ) ( )
Ds
s
u r z p K ds y
which is for upstream pressure disturbance. Considering the unsteady loading satisfies the infinite
condition on the leading edge and the Kutta condition on the trailing edge, the pressure difference
can be expressed as
i
p
,0A
p
,2D
p
,2D
w
i
w
,1D
p
,1D
w
,1A
p
s
p
a
a
d
d
s
w
d
R
h
R
1
l
b
2
l
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(15)
01
12
( ) cot sin( 1)
2
JI
j j i j
ji
p r A A i
Applying point matching method, the solution of integral equation is
(16)
1
( , )
11
IJ
i j ij v ij i j
ij
Av
where,
1,2,3
and
1
, ,1
( , )
ij D
v u r z
,
2
, ,1
( , )
ij A
v u r z
,
3
, ,1
( , )
ij D
v w r z
.
1
( , )v ij i j
represents the
coefficient matrix produced by kernel function. Substituting Eqs. (15) and (16) into Eqs. (5) and (9),
we can obtain the scattering field
s
p
and
s
w
expressions in the independent element.
As what has discussed above, in order to obtain acoustic field generated by unsteady loading
on the blades, it is noted that the pressure and velocity is continuous on the interface plane, so we
have
(17)
,,
a a a a a a
p p u u w w
They are the matching condition for interface
aa
. Using the orthogonality of Bessel function, we
can write the linear equations as following
(18)
0 1 1 1
0
0 1 1 1
0
1
0
1 1 1 2
0
1 1 1 2
0
00
00
0 0 0 0 0
00
00
a a a a a a a a
p p p w
m m m
a a a a a a a a
p p p w
m m m
aa
w
d d d d d d d d
p p w p
m m m
d d d d d d d d
p p w p
m m m
p p p p
A D A D
u u u u
A D A D
w
D
p p p p
D A D D
u u u u
D A D D
ss ss ss ss
ss ss ss ss
ss
ss ss ss ss
ss ss ss ss
1 1 1 2
00
0
1
1
0
1
0
2
2
0
0
0
0
00
d d d d d d d d
p p w w
mm
p
p
m
m
p
u
m
m
p
w
m
w
p
m
w
w w w w
D A D D
A
I
D
I
A
I
D
D
D
ss ss ss ss
3. Results and disccusion
0 5 10 15 20 25 30 35 40 45
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
E
H
E
H
E
Q
E
Q
E
H
,E
Q
N
w
M
a
=0.35
Namba
TEM
M
a
=0.25
Namba
TEM
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
E
Q
E
Q
E
H
E
H
E
H
,E
Q
N
w
=0
o
Namba
TEM
=10
o
Namba
TEM
Figure 3. Variation of the upstream and downstream acoustic powers
In order to verify the model, we need to make some comparison with the existing results [7] from
different methods. The interaction between the external disturbance and rotor blades is considered,
with following parameters: the number of blades
40B
, blades chord
0.054978b
, the rotational
speed of rotor
294.4536
, and the hub/tip ratio
0.4
hd
RR
. The fundamental circumferential
wave number of unsteady inflow is
35
w
N
, which rotates in the same direction to rotor and the
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angular velocity is
210.347
w
. So the unsteady fluctuation frequency in the rotor coordinate sys-
tem is
( ) 2943.72
ww
N
.
Fig. 3 gives the comparison for variation of the upstream and downstream acoustic powers with
the circumferential wave number of the external disturbance. It is found that results agree with each
other very well. The differences may be due to the application of different methods.
Figure 4. Interaction between rotor and stator
As shown in Fig. 4, three independent elements are established to investigate the effect of rotor
blades on the sound generation by interaction between rotor and stator. In the numerical examples
given at present the geometrical parameters are fixed as follows:
16B
,
11,21V
,
0.15b
，
axial flow Mach number
0.245
a
M
,
2
0.5lb
,
RS
l l b
, rotor tip Mach number
0.406
T
M
,
upwash velocity
[ / ]
( , , ) 0.1
iB x U t
w r z t e
and a hub-tip ratio,
0.6
hd
RR
. Fig. 5 shows the varia-
tion of sound power level with the number of stator vanes. In order to guarantee the shield effect of
rotor the distance between rotor and stator is big enough, there is only one cut-on mode.
10 12 14 16 18 20 22
100
105
110
115
120
125
130
135
140
Soudn Power Level (dB)
Number of stator vanes
SPLu-for TEM
SPLd-for TEM
SPLu-for LST
SPLd-for LST
10 12 14 16 18 20 22
0
2
4
6
8
10
12
14
Transmission Loss (dB)
Number of stator vanes
TL throuth a rotor
SPLu(LST) - SPLu(TEM)
Figure 5. The comparison results Figure 6. Transmission loss of acoustic power
The sound power level is defined as
12
, 10 ,
10log ( 10 )
u d u d
SPL E
, the subscript u and d describe
the up- and downstream acoustic wave. TEM in Fig. 5 expresses sound generated by the rotor-stator
interaction, and LST means sound generated by wake-stator interaction. It is easy to see that when
the rotor “shield” effect is considered, the upstream sound power is much less than in wake-stator
interaction model. Meanwhile, there is no obvious increase of downstream sound power. The bene-
fit closes to 15dB in this model if we carefully design the number of stator vanes in certain areas.
The shield effect depends mainly on the angle between the directions of sound wave fronts and
the normal of blades at tip of rotor [8]. As show in Fig. 6, the transmission loss of sound power is a
function of acoustic mode when the sound propagating through a rotor row.
Another reason for reduction of the upstream sound wave is the interaction between sound field
and vortex field on the blades surface. The difference of unsteady loading is obvious when the ro-
tor-stator interaction is considered, as shown in Fig. 7. The black line describes the unsteady load-
ing on the blades tip when a single acoustic mode wave propagating through the rotor row. The red
line is the unsteady loading when the rotor-stator interaction is considered. The different unsteady
Rotor
A
p
i
w
S
l
H
p
H
w
2
l
R
l
Stator
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ICSV22, Florence, Italy, 12-16 July 2015 8
lift results in the different ratio between sound field and vortex field. The many times of reflection
between rotor and stator enhance the energy conversion. Therefore the difference of upstream SPL
between wake-stator interaction and rotor-stator interaction is little higher than the transmission loss
for single acoustic scattering by rotor, as shown in Fig. 6. Obviously the interaction between rotor
and stator is important for the noise prediction.
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
Unsteady Loading
z
Real part for rotor-stator interaction
Imaginary part for rotor-stator interaction
Real part for sound propagation through a rotor
Imaginary part for sound propagation through a rotor
Figure 7. The unsteady loading on the rotor blades
4. Conclusions
A transfer element model which includes sound generation and scattering effect of rotor blades
has been established. This unique model is quite different with the typical one that only considering
the noise generated by interaction of rotor wake and stator. By applying the proposed model, rotor-
stator interaction noise has been investigated. The numerical results of noise prediction indicate the
effect of interaction depends mainly on the shied effect of rotor blade when there is only one cut-on
mode between fan rotor and OGV. However, the downstream sound power had no evident increase
when the upstream sound power reduced greatly. The difference of sound power between wake-
stator interaction and rotor-stator interaction illustrates there is multi reflections between rotor and
stator and interactions between sound wave and vortical wave on the blades that results in compara-
tively big change of unsteady loading. Obviously, the effect of rotor blades on the sound generation
is important.
5. Acknowledgement
The authors are grateful to the National Natural Science Foundation of China (Grant No. 51106005),
and acknowledges the National Basic Research Program of China (Grant No. 2012CB720201).
REFERENCES
1 Kaji, S. and Okazaki, T., Generation of sound by rotor-stator interaction, Journal of Sound and Vibration, 13(3),
281-307, (1970).
2 Hanson, D B., Acoustic reflection and transmission of rotors and stators including mode and frequency scatter-
ing, AIAA Paper, AIAA-97-1610-CP, (1997).
3 Namba, M., Three-Dimensional Flows, in: Aero-elasticity in axial flow turbomachinery aerodynamics, M. F.
Platzer and F. O. Carta, eds., AGARD-AG-298, Vol.1.
4 Schulten, J. B. H. M., Vane sweep effects on rotor/stator interaction noise, AIAA Journal, 35(6), 945-951,
(1997).
5 Wang, X. and Sun, X., On the interaction of fan stator and acoustic treatments using transfer element method,
Fluid Dynamics Research, 42(1), 103-119, (2010).
6 Goldstein, M. E., Aeroacoustics, McGraw-Hill, New York, (1976).
7 Namba, M., Three-dimensional analysis of blade force and sound generation for an annular cascade in distorted
flows, Journal of Sound and vibration, 50(4), 479-508, (1977).
8 Schulten, J. B. H. M., Sound generation by ducted fans and propellers as a lifting surface problem, Ph.D Thesis,
Netherland: Univ. of Twente, (1993).