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A PREDICTION METHOD FOR AERODYNAMIC SOUND PRODUCED BY CLOSELY SPACED ELEMENTS IN AIR DUCTS
Journal of Sound and
A PREDICTION METHOD FOR AERODYNAMIC SOUND PRODUCED BY CLOSELY SPACED ELEMENTS IN AIR DUCTS C. M. MAK AND J. YANG Department of Building Services Engineering, ¹he Hong Kong Polytechnic ;niversity, Hung Hom, Kowloon, Hong Kong, People1s Republic of China (Received 31 March 1999, and in ,nal form 17 June 1999)
1. INTRODUCTION
The accurate prediction of aerodynamic sound production in air ducts at the design stage is of engineering importance since it is almost impossible to remedy the #ow-generated noise problems after the installation of the ventilation systems. Over the years, a number of investigators [1}6] have tried to devise a prediction technique for aerodynamic sound in air ducts. Nevertheless, their predictive technique can only be applied to an isolated element in an air duct. Current design methods such as CIBSE guide [7] and ASHRARE handbook [8] are based upon the work of several investigators on aerodynamic noise produced by an isolated element in low speed #ow ducts. However, some in-duct elements or duct discontinuities are very close to each other and the noise generated by them is very di!erent from that of an isolated element. In practice, some ventilation designers use the current design methods directly to predict the #ow-generated noise produced by closely spaced elements though they are designed for isolated elements in air ducts. The speci"c problem that motivated this study is the inaccurate noise prediction due to the interaction between closely spaced in-duct elements. The aim of the present investigation is therefore to predict sound level and spectral content of the noise radiated by the closely spaced spoilers in low speed air #ow ducts.
2. SOUND POWER RADIATED BY TWO CLOSELY SPACED SPOILERS IN AIR DUCT
2.1. INTERACTION OF TWO NOISE SOURCES It is straightforward to obtain the acoustic power spectral densities =(u) from reference [9] as follows: n o c (u)D2, =(u)"+ 0 mn lim DQ 4A T?= ¹ mn,u mn 0022-460X/00/030743#11 $35.00/0
(1) ( 2000 Academic Press
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LETTERS TO THE EDITOR
where A is the cross-section area of the duct, o is the ambient density of air, 0 c is the mode axial phase speed, and the source volume integral Q (u) is mn mn given by 1 Q (u)" mn c mn
PP
s
f (x@, y@)t* (x@, y@)eikmnz* dx@ dy@, 3 mn
(2)
where t is the normalized mode cross-section function, k is the mode axial wave mn mn number and f (x@, y@) is the force per unit density acting per unit area over the 3 cross-section z@"zA occupied by the thin rectangular spoiler, the force arising from the di!erence in #uctuating pressure acting on the two sides of the spoiler. These #uctuations in pressure are caused mainly by the formation and convection of eddies as the oncoming #ow separates at the edge of the spoiler. (x@, y@, z@) is the co-ordinate of the spoiler. The sound power radiated by two sources in the duct can be written as
PP
=(z, t)"
s
PP
IM (x, y) dx dy"
s
1 ReMp u* N dx dy 2 1`2 1`2
(3)
or, in dimensionless form,
G
H
n1 (z, t)k1* (z, t)#n1 (z, t)k2* (z, t) =(z, t) mn mn mn " + mn , #n2 (z, t)k1* (z, t)#n2 (z, t)k2* (z, t) Ao c3 mn mn mn mn 0 0 m,n
(4)
where n and k are the dimensional modal pressure ratio and the modal axial mn mn particle velocity ratio respectively. When the "rst and the second sources are located at z"0 and z"d, respectively, the power spectral density of the sound "eld in duct can therefore be written as 1 = M (u)"+ o c MS11 (u)#[S21 (u)#S12 (u)] cos(k d)#S22 (u)N, mn mn mn mn 4A 0 mn mn mn
(5)
where all quantities S (u) represent the power spectral density of the mn corresponding source volume integral Q (u), as de"ned by mn n S11 (u)" lim DQ1 (u)D2, mn mn T?= ¹
(6)
n S12 (u)"S21* (u)" lim (Q1 Q2* ), mn mn mn mn T?= ¹
(7)
n S22 (u)" lim DQ2 (u)D2. mn mn T?= ¹
(8)
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LETTERS TO THE EDITOR
It is noted that DS12 (u)D2"S11 (u)S22 (u), which is the condition of coherence, i.e., mn mn mn c2 "DS D2/S S "1. It can be veri"ed by the summation of the powers radiated by iij ij ii jj the primary and second sources.
2.2.
THE DISCUSSION OF SOLUTION FOR SOUND PRODUCED BY TWO CLOSELY SPACED SPOILERS IN AN INFINITE HARD WALLED DUCT
In reference [5], the modal quantity S (u) is related to the power spectrum mn S (u) of the total drag force on the thin rectangular spoiler, and the resulting F expression for the spectral density S (u) is mn 1 1 S (u)" S (u) mn o2 Dc D2 F A 0 mn s
PP
Dt (x )D2 ds(x ). mn k k
1 1 S (u) S11 (u)" 1 mn A o2 Dc D2 F s1 0 mn
PP
Dt (x )D2 ds(x ), mn k1 k1
(10)
1 1 S (u) S22 (u)" 2 F mn A o2 Dc D2 s2 0 mn
PP
Dt (x )D2 ds(x ), mn k2 k2
(11)
As
(9)
Then,
As1
As2
DS12 (u)D2"DS21 (u)D2"S11 S22 . mn mn mn mn
(12)
It is de"ned that S12,DS12De*((u) and equation (5) can therefore be written as mn mn 1 1 S (u) + f (S11)"= (u), mn 11 Dc DA 4Ao F1 0 mn mn s1
PP
Dt (x )D2 ds(x ), mn k1 k1
(13)
1 1 S (u) + f (S22)"= (u), mn 22 Dc DA 4Ao F2 0 mn mn s2
PP
Dt (x )D2 ds(x ), mn k2 k2
(14)
As1
As2
1 1 f (S12, S21)"= (u), JS S + mn mn 12 F1 F2 Dc DJA A 4Ao 0 mn mn s1 s2
SPP
As2
Dt (x )D2 ds(x ) 2cos[/(u)]cos(k d). mn k2 k2 mn
SPP
As1
Dt (x )D2 ds(x ) mn k1 k1
(15)
The summation of (1/c ) and the summation of [1/c cos(k d)] for all mn mn mn propagating modes need to be solved so that the above equations can be applied in
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LETTERS TO THE EDITOR
this present investigation for practical engineering need. It can be noted that the ratios (c /c ) for a given mode can be expressed in terms of the integers m and n, 0 mn i.e., c /c "J1!(mn/ka)2!(nn/kb)2. 0 mn
(16)
If m and n are now regarded as continuous variables, the ratio (c /c ) can be 0 mn thought of as another continuous variable, which is a function of m and n, i.e. (c /c )+f (m, n). It can be estimated that 0 mn
P P
N 1 1 N c 1 kb@n ka@n + " + 0+ f (m, n) dm dn, c c c c 0 m,n mn 0 0 m,n mn 0
(17)
where (ka/n) and (kb/n) are the maximum values of the continuous variables m and n for which modes propagate at the frequency in question. According to the concept in reference [5], the following summation is k2ab k N c # (a#b). + 0" 6n 8 c m,n mn Referring to Appendix A, the sum of follows:
(18)
+N
(c /c ) cos(k d) can be obtained as m,n 0 mn mn
PP S A B A B C S A B A B D
A B
N c + 0 cos(k d)+ mn c m,n mn s
1!
mn 2 nn 2 ! cos (kd) ) ka kb
C
D
mn 2 nn 2 ! dm dn ka kb
1!
C
D
k2ab sin e 2cos e 2sin e k(a#b) J (e) " # ! # J (e)! 1 , 0 2n e2 e3 4 e e
(19)
where e"kd, and J and J are the Bessel functions. 0 1 In addition, 1 A s1
PP
C
A B A
mnd 2mnaN a 1 cos 1 Dt (x )D2 ds(x )" 1# sin mn k1 k1 a a mnd 1 As1
C
A B A
BD
b mnh 2mnbM 1 cos 1 ] 1# sin mnh b b 1
BD
+1,
(20)
where (aN , bM ) is the spoiler 1 central co-ordinate, d , h are its length and width 1 1 1 1 respectively. Similarly, 1 A s2
PP
As2
Dt (x )D2 ds(x )+1. mn k2 k2
(21)
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LETTERS TO THE EDITOR
Equations (13)}(15) can then be rewritten as
C C C
D D D
u2 3nc (a#b) = (u)" 1# 0 S (u), 11 F1 24no c3 4u A 0 0 3nc (a#b) u2 1# 0 S (u), = (u)" F2 22 A 4u 24no c3 0 0 u2 3nc (a#b) = (u)" 1# 0 Q2cos[/(u)]JS (u)S (u), 12 F1 F2 24no c3 4u A 0 0
(22a, b, c)
where Q is given by k2ab/4n2[sin e/e#2cos e/e2!2sin e/e3]#k(a#b)/82[J (e)!J (e)/e] 0 1 Q" [k2ab/4n#k(a#b)/8] and e"kd.
(23)
The total power spectral density in the duct for the case of f'f is 0 =(u)"= #= #= 11 22 12
C
D
u2 3nc (a#b) " 1# 0 MS #S #2Q cos[/(u)]JS S N. F1 F2 F1 F2 24no c3 4u A 0 0
(24)
For f(f . Equations (13)}(15) can be reduced to a simple expression for plane 0 wave propagation de"ned by m, n"0. The total power spectral density is 1 MS #S #cos(kd)2cos[/(u)]JS S N. =(u)" F1 F2 F1 F2 4Ao c 0 0
(25)
Thus, two very simple results have been derived for the source power radiated by two assumed form of source distribution, under the conditions of plane wave and multi-modal sound propagation in an in"nite duct.
3. SCALING A LAW FOR PRACTICAL ENGINEERING
3.1. THE RELATIONSHIP BETWEEN FLUCTUATING AND STEADY STATE DRAG FORCES In the development of the theory, it is assumed that the root mean square (r.m.s.) #uctuating drag force acting on the spoiler is directly proportional to the steady state drag force FM . This assumption was also used by Gordon [2, 3], Nelson and z Morfey [5] and is supported by the experiment of Heller and Widnall [4]. If the proportional frequency band de"ned by the limits ( f /a, f a) where f is the centre c c c frequency, is considered, the ratio of the r.m.s. #uctuating drag force to the steady
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LETTERS TO THE EDITOR
state drag force is dependent only on the Strouhal number only. This can be expressed as (F ) "K(St)FM z r.m.s. z
(26)
(band f /a to f a) where the numerical factor K(St) depends on the choice of a, the c c Strouhal number St is given by St"f r/; , where ; is the #ow velocity in the c c c constriction provided by the spoiler (; is de"ned by the volume #ow rate q and the c area of the duct constriction A , such that ; "q/A ) and r is a characteristic c c c dimension, and the steady state drag force acting on the spoiler can be written in terms of a drag coe$cient C and is expressed as D FM ,C (1 o ;2 )A "C (1 o ;2)p2(1!p)A, z D2 0 = s D2 0 c
(27)
where p is the open area ratio, A is the area of duct cross-section, A is the face area s of the #at spoiler and ; is the duct velocity q/A; and it is assumed that steady state = drag forces acting on the two spoilers are the same, i.e., FM "FM . z1 z2 3.2.
THE RELATIONSHIP BETWEEN SOUND POWER AND FLOW PARAMETERS
Using previously derived equations (24) and (25) for the sound power transmitted along the duct under multi-mode and plane wave radiation conditions, the sound power radiated in a given bandwidth can be expressed as follows, for frequencies above and below the cut-on frequency of the lowest transverse duct mode. The mean square value of the #uctuating force in a given band is given by
P
(FM 2)D " z F
u2
u1
S (u) du, F
(28)
where S (u) is the power spectrum of the total drag force on the spoiler. F Using equations (26) and (27), the sound power can be expressed in terms of drag coe$cient. After some algebraic manipulation, it can be shown that when the two spoilers have the same shape, f (f , c 0 =D +(1/4Ao c )M(FM 2 )D #(FM 2 )D #2cos(kd)cos[/(u)] J(FM 2 )D (FM 2 )D N F 0 0 z1 F z2 F z1 F z1 F "(o /16c )AK2(St)[p2(1!p)]2C2 ;4 M2#2cos(kd)cos[/(u)]N, 0 0 D c f 'f c 0 =D +(u2/24no c3)[1#(3nc /4u )(a#b)/A]M(FM 2 )D #(FM 2 )D F c 0 0 0 c z1 F z2 F #2Q cos[/(u)] J(FM 2 )D (FM 2 )D N z1 F z1 F
(29)
LETTERS TO THE EDITOR
749
"(o n/24c3)[1#3nc /4u )(a#b)/A](A/r)2(St)2K2(St) 0 0 0 c ][p2(1!p)]2C2 ;6 M2#2Q cos[/(u)]N, D c
(30)
where a, b are the duct cross-section dimensions and d is the distance between two spoilers. Since the sound power measurements are made in proportional frequency bands, the above scaling laws may be used as they stand to normalize an experimental data. The inferred in"nite-duct values of radiated sound power level SWL in the frequency band can thus be normalized by evaluating D f (f c 0 120#20 log K(St)"SWL !10 log Mo A[p2(1!p)]2C2 ;4/16c N 10 D 10 0 D c 0 !10 log [2#2cos(kd)], 10
(31)
f 'f c 0 120#20 log K(St)"SWL !10 log Mo nA2(St)2[p2(1!p)]2C2 ;6/24c3r2N 10 D 10 0 D c 0 !10 log [1#(3nc /4u )(a#b)/A]!10 log M2#2QN. 10 0 c 10 (32) It is assumed in the above equations that S "S , i.e., /(u)"0. It should be F1 F2 noted that when the two sources are located at the same position, the total sound power should be the same as that of a single source. It can be seen that these predictive equations (equations (31) and (32)) are similar to those derived by Nelson and Morfey in reference [5] and can be expressed in terms of easily measurable engineering parameters, together with the single Strouhal number-dependent constant K(St). The additional terms in equations (31) and (32) as they are compared to equations of Nelson and Morfey represent the interaction between two aerodynamic noise sources. Nelson and Morfey measured the sound power levels radiated by di!erent #at plate #ow spoilers in a rectangular air duct at various velocities and "nally they produced a normalized spectrum [5]. It should be noted that 6 dB corrections need to be subtracted from the values of K2(St) in their normalized spectrum [10]. Together with the corrected normalized spectrum K2(St) of Nelson and Morfey, the inferred-duct values of sound power level in 1 octave 3 bands radiated by two closely spaced spoilers can be obtained by equations (31) and (32). The predictive equations (31) and (32) can therefore form a basis of a generalized prediction method for aerodynamic sound generated by closely spaced induct elements. However, the validity of these predictive equations should be checked against experimental data. This will certainly require further work.
4. CONCLUSIONS
The predictive equations developed here which permit the determination of sound power radiated by two closely spaced spoilers only require simple #ow
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LETTERS TO THE EDITOR
parameters and corrected normalized spectrum K2(St) derived by Nelson and Morfey. It needs to be seen whether this predictive technique based on their corrected normalized spectrum K2(St) can be applied to a wider range of #ow duct discontinuities. The ultimate objective of the present study is therefore to extend this method to predict the sound level and spectral distribution of the additional acoustic energy produced by the combination of any given duct discontinuities in any given duct air #ow velocities. This study certainly provides a basis for further investigation.
ACKNOWLEDGMENTS
The authors would like to thank the Hong Kong Polytechnic University Research Committee (Project account code: A-PA37) for the "nancial support during the course of this work.
REFERENCES 1. C. G. GORDON and G. MAIDANIK 1966 Bolt Beranek and Newman Inc. Report No. 1426. In#uence of upstream #ow discontinuities on the acoustic power radiated by a model air jet. 2. C. G. GORDON 1968 Journal of the Acoustical Society of America 43, 1041}1048. Spoiler-generated #ow noise. I: the experiment. 3. C. G. GORDON 1969 Journal of the Acoustical Society of America 45, 214}223. Spoiler-generated #ow noise. II: results. 4. H. H. HELLER and S. E. WIDNALL 1970 Journal of the Acoustical Society of America 47, 924}936. Sound radiation from rigid #ow spoilers correlated with #uctuating forces. 5. P. A. NELSON and C. L. MORFEY 1981 Journal of Sound and
APPENDIX A: CALCULATION OF THE SUMMATION IN EQUATION (19)
According to the concept of reference [5], the double integration expressed in equation (19) evaluates the volume enclosed by the surface (c /c )cos k d for all 0 mn mn propagating modes.
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LETTERS TO THE EDITOR
Let x"n/ka m, y"n/kb n. Then equation (19) can be written as
PP 1
I"
J
1~y2
A
0 0
A BA BP GP ka n
"
kb n
BA BA B
J1!x2!y2 cos kdJ1!x2!y2 1
J
0
0
ka n
kb dx dy n
H
1~y2
J1!x2!y2 cos(kdJ1!x2!y2) dx dy, (A1)
let x "J1!x2!y2/1!y2, e"kd; then dx"x J1!y2/1!x2 . Equation 1 1 1 (A1) can be simpli"ed as
A BA BP ka n
I"
kb n
1
(1!y2)
0
GP
H
1
x2 1 cos (eJ1!y2 x ) dx dy. 1 1 J1!x2 0 1
(A2)
On the right of equation (A2), the bracket of the integration term can be divided into the following forms:
P
1
P
1 cos(cx ) dx ! J1!x2 cos(cx ) dx , 1 1 1 1 1 0 J1!x21 0 1
where c"eJ1!y2. Considering the integration formulation in reference [11]
P
a
0
AB
Jn 2a b~1@2 (a2!x2)b~1 cos(bx) dx" C(b)J (ab) b~1@2 2 b a, Reb'0, Darg bD(n
(A3)
where C(x) and J (x) represent the Gamma function and the sth order Bessel's s function respectively. Applying equation (A3) to equation (A2),
A BA BP
I"
ka n
kb n
1
(1!y2)
0
G
AB
AB H
Jn 2 Jn 1 3 C J (c)! C J (c) dy, 2 0 2 1 2 2 c
where C (3)"1 C(1)"1 Jn. 2 2 2 2 It is therefore obtained as
A BA B
I"
ka n
kb n [I !I ], 2 n 2 1
(A4)
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LETTERS TO THE EDITOR
where
P
I " 1
1
0
P
1 1 I " J(1!y2J (eJ1!y2 dy. 2 e 1 0
(1!y2)J (eJ1!y2) dy, 0
For integration I and I , let y "J1!y2; then dy "!(J1!y2/y ) dy, 1 2 1 1 1 1
P
I " 1
1
0
P
y3 (1!y2)~1@2J (ey ) dy , 1 1 0 1 1
I " 2
1
0
y2 (1!y2)~1@2J (ey ) dy. 1 1 1 1 (A5, A6)
Considering the integration formulation in reference [12]:
P
a
0
AB
1 2 n`b x0`2n`1(a2!x2)b~1Jt (cx) dx" (t#1) C(b)a2n`2b`t n 2 ac
A BA B
ac k (ac) J k`n`b`t 2
n (!1)k n + (t#1) k k k/0
(a, Re b'0, Re t'!n!1).
(A7)
Application of equation (A7) to equation (A5) gives
S C
I " 1
D
e 2n J (e)! J (e) . 3@2 2 5@2 e3
(A8)
Considering another integration formulation in reference [12]:
P
a
0
2b~1ab`t C(b)J t (ac) (a, Re b'0, Re t'!1) xt`1(a2!x2)b~1Jt (cx) dx" b` cb (A9)
Application of equation (A9) to equation (A6) gives
S
n J (e). I " 2 2e 3@2
(A10)
Equation (A4) can therefore be rewritten as
A BA B S C
I"
ka n
kb n n 2
D
n 1 J (e)!J (e) , 5@2 2e e 3@2
(A11)
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LETTERS TO THE EDITOR
where
S A
S CA
J (e)" 5@2
B
2e sin e cos e ! , e n e2
J (e)" 3@2
B
D
1 3 3 ! sin e! !cos e . e2 e3 e
2e n
Substitution of the above formulations into equation (A11) gives the following:
C
D
k2ab sin e 2cos e 2sin e 3 # ! . I" e2 e3 e 6n
(A12)
It is noted that when eP0, ID "(k2ab/6n)3[1!2/3]"k2ab/6n, it is the same e?0 as the volume of the ellipsoidal segment given in reference [5]. In addition, the volume of the two &&slices'' should be considered in the summation of (c /c )cosk d. First, the area of the two &&slice'', can be obtained as 0 mn mn
P S A B
S" x
ka@n
1!
0
nm 2 dm. ka
(A13)
Let x J1!(nm/ka)2. Then 1
P
C
D
ka J (e) ka 1 x2 cos(ex ) 1 1 dx " J (e)! 1 , S" 1 0 x 2 e n J1!x2 0 1
(A14)
where equation (A3) is used. Similarly,
C
D
J (e) kb J (e)! 1 . S" 0 y e 2
(A15)
Thus, the total volume representing the summation of (c /c )cosk d can be 0 mn mn written as I "I#S /2#S /2 t x y
C
D
C
D
k2ab sin e 2cos e 2sin e k(a#b) J (e) " # ! # J (e)! 1 , 0 2n e2 e3 4 e e where 1 is the width of each &&slice'' (similar to the concept in reference [5]). 2
(A16)
A PREDICTION METHOD FOR AERODYNAMIC SOUND PRODUCED BY CLOSELY SPACED ELEMENTS IN AIR DUCTS C. M. MAK AND J. YANG Department of Building Services Engineering, ¹he Hong Kong Polytechnic ;niversity, Hung Hom, Kowloon, Hong Kong, People1s Republic of China (Received 31 March 1999, and in ,nal form 17 June 1999)
1. INTRODUCTION
The accurate prediction of aerodynamic sound production in air ducts at the design stage is of engineering importance since it is almost impossible to remedy the #ow-generated noise problems after the installation of the ventilation systems. Over the years, a number of investigators [1}6] have tried to devise a prediction technique for aerodynamic sound in air ducts. Nevertheless, their predictive technique can only be applied to an isolated element in an air duct. Current design methods such as CIBSE guide [7] and ASHRARE handbook [8] are based upon the work of several investigators on aerodynamic noise produced by an isolated element in low speed #ow ducts. However, some in-duct elements or duct discontinuities are very close to each other and the noise generated by them is very di!erent from that of an isolated element. In practice, some ventilation designers use the current design methods directly to predict the #ow-generated noise produced by closely spaced elements though they are designed for isolated elements in air ducts. The speci"c problem that motivated this study is the inaccurate noise prediction due to the interaction between closely spaced in-duct elements. The aim of the present investigation is therefore to predict sound level and spectral content of the noise radiated by the closely spaced spoilers in low speed air #ow ducts.
2. SOUND POWER RADIATED BY TWO CLOSELY SPACED SPOILERS IN AIR DUCT
2.1. INTERACTION OF TWO NOISE SOURCES It is straightforward to obtain the acoustic power spectral densities =(u) from reference [9] as follows: n o c (u)D2, =(u)"+ 0 mn lim DQ 4A T?= ¹ mn,u mn 0022-460X/00/030743#11 $35.00/0
(1) ( 2000 Academic Press
744
LETTERS TO THE EDITOR
where A is the cross-section area of the duct, o is the ambient density of air, 0 c is the mode axial phase speed, and the source volume integral Q (u) is mn mn given by 1 Q (u)" mn c mn
PP
s
f (x@, y@)t* (x@, y@)eikmnz* dx@ dy@, 3 mn
(2)
where t is the normalized mode cross-section function, k is the mode axial wave mn mn number and f (x@, y@) is the force per unit density acting per unit area over the 3 cross-section z@"zA occupied by the thin rectangular spoiler, the force arising from the di!erence in #uctuating pressure acting on the two sides of the spoiler. These #uctuations in pressure are caused mainly by the formation and convection of eddies as the oncoming #ow separates at the edge of the spoiler. (x@, y@, z@) is the co-ordinate of the spoiler. The sound power radiated by two sources in the duct can be written as
PP
=(z, t)"
s
PP
IM (x, y) dx dy"
s
1 ReMp u* N dx dy 2 1`2 1`2
(3)
or, in dimensionless form,
G
H
n1 (z, t)k1* (z, t)#n1 (z, t)k2* (z, t) =(z, t) mn mn mn " + mn , #n2 (z, t)k1* (z, t)#n2 (z, t)k2* (z, t) Ao c3 mn mn mn mn 0 0 m,n
(4)
where n and k are the dimensional modal pressure ratio and the modal axial mn mn particle velocity ratio respectively. When the "rst and the second sources are located at z"0 and z"d, respectively, the power spectral density of the sound "eld in duct can therefore be written as 1 = M (u)"+ o c MS11 (u)#[S21 (u)#S12 (u)] cos(k d)#S22 (u)N, mn mn mn mn 4A 0 mn mn mn
(5)
where all quantities S (u) represent the power spectral density of the mn corresponding source volume integral Q (u), as de"ned by mn n S11 (u)" lim DQ1 (u)D2, mn mn T?= ¹
(6)
n S12 (u)"S21* (u)" lim (Q1 Q2* ), mn mn mn mn T?= ¹
(7)
n S22 (u)" lim DQ2 (u)D2. mn mn T?= ¹
(8)
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LETTERS TO THE EDITOR
It is noted that DS12 (u)D2"S11 (u)S22 (u), which is the condition of coherence, i.e., mn mn mn c2 "DS D2/S S "1. It can be veri"ed by the summation of the powers radiated by iij ij ii jj the primary and second sources.
2.2.
THE DISCUSSION OF SOLUTION FOR SOUND PRODUCED BY TWO CLOSELY SPACED SPOILERS IN AN INFINITE HARD WALLED DUCT
In reference [5], the modal quantity S (u) is related to the power spectrum mn S (u) of the total drag force on the thin rectangular spoiler, and the resulting F expression for the spectral density S (u) is mn 1 1 S (u)" S (u) mn o2 Dc D2 F A 0 mn s
PP
Dt (x )D2 ds(x ). mn k k
1 1 S (u) S11 (u)" 1 mn A o2 Dc D2 F s1 0 mn
PP
Dt (x )D2 ds(x ), mn k1 k1
(10)
1 1 S (u) S22 (u)" 2 F mn A o2 Dc D2 s2 0 mn
PP
Dt (x )D2 ds(x ), mn k2 k2
(11)
As
(9)
Then,
As1
As2
DS12 (u)D2"DS21 (u)D2"S11 S22 . mn mn mn mn
(12)
It is de"ned that S12,DS12De*((u) and equation (5) can therefore be written as mn mn 1 1 S (u) + f (S11)"= (u), mn 11 Dc DA 4Ao F1 0 mn mn s1
PP
Dt (x )D2 ds(x ), mn k1 k1
(13)
1 1 S (u) + f (S22)"= (u), mn 22 Dc DA 4Ao F2 0 mn mn s2
PP
Dt (x )D2 ds(x ), mn k2 k2
(14)
As1
As2
1 1 f (S12, S21)"= (u), JS S + mn mn 12 F1 F2 Dc DJA A 4Ao 0 mn mn s1 s2
SPP
As2
Dt (x )D2 ds(x ) 2cos[/(u)]cos(k d). mn k2 k2 mn
SPP
As1
Dt (x )D2 ds(x ) mn k1 k1
(15)
The summation of (1/c ) and the summation of [1/c cos(k d)] for all mn mn mn propagating modes need to be solved so that the above equations can be applied in
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LETTERS TO THE EDITOR
this present investigation for practical engineering need. It can be noted that the ratios (c /c ) for a given mode can be expressed in terms of the integers m and n, 0 mn i.e., c /c "J1!(mn/ka)2!(nn/kb)2. 0 mn
(16)
If m and n are now regarded as continuous variables, the ratio (c /c ) can be 0 mn thought of as another continuous variable, which is a function of m and n, i.e. (c /c )+f (m, n). It can be estimated that 0 mn
P P
N 1 1 N c 1 kb@n ka@n + " + 0+ f (m, n) dm dn, c c c c 0 m,n mn 0 0 m,n mn 0
(17)
where (ka/n) and (kb/n) are the maximum values of the continuous variables m and n for which modes propagate at the frequency in question. According to the concept in reference [5], the following summation is k2ab k N c # (a#b). + 0" 6n 8 c m,n mn Referring to Appendix A, the sum of follows:
(18)
+N
(c /c ) cos(k d) can be obtained as m,n 0 mn mn
PP S A B A B C S A B A B D
A B
N c + 0 cos(k d)+ mn c m,n mn s
1!
mn 2 nn 2 ! cos (kd) ) ka kb
C
D
mn 2 nn 2 ! dm dn ka kb
1!
C
D
k2ab sin e 2cos e 2sin e k(a#b) J (e) " # ! # J (e)! 1 , 0 2n e2 e3 4 e e
(19)
where e"kd, and J and J are the Bessel functions. 0 1 In addition, 1 A s1
PP
C
A B A
mnd 2mnaN a 1 cos 1 Dt (x )D2 ds(x )" 1# sin mn k1 k1 a a mnd 1 As1
C
A B A
BD
b mnh 2mnbM 1 cos 1 ] 1# sin mnh b b 1
BD
+1,
(20)
where (aN , bM ) is the spoiler 1 central co-ordinate, d , h are its length and width 1 1 1 1 respectively. Similarly, 1 A s2
PP
As2
Dt (x )D2 ds(x )+1. mn k2 k2
(21)
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Equations (13)}(15) can then be rewritten as
C C C
D D D
u2 3nc (a#b) = (u)" 1# 0 S (u), 11 F1 24no c3 4u A 0 0 3nc (a#b) u2 1# 0 S (u), = (u)" F2 22 A 4u 24no c3 0 0 u2 3nc (a#b) = (u)" 1# 0 Q2cos[/(u)]JS (u)S (u), 12 F1 F2 24no c3 4u A 0 0
(22a, b, c)
where Q is given by k2ab/4n2[sin e/e#2cos e/e2!2sin e/e3]#k(a#b)/82[J (e)!J (e)/e] 0 1 Q" [k2ab/4n#k(a#b)/8] and e"kd.
(23)
The total power spectral density in the duct for the case of f'f is 0 =(u)"= #= #= 11 22 12
C
D
u2 3nc (a#b) " 1# 0 MS #S #2Q cos[/(u)]JS S N. F1 F2 F1 F2 24no c3 4u A 0 0
(24)
For f(f . Equations (13)}(15) can be reduced to a simple expression for plane 0 wave propagation de"ned by m, n"0. The total power spectral density is 1 MS #S #cos(kd)2cos[/(u)]JS S N. =(u)" F1 F2 F1 F2 4Ao c 0 0
(25)
Thus, two very simple results have been derived for the source power radiated by two assumed form of source distribution, under the conditions of plane wave and multi-modal sound propagation in an in"nite duct.
3. SCALING A LAW FOR PRACTICAL ENGINEERING
3.1. THE RELATIONSHIP BETWEEN FLUCTUATING AND STEADY STATE DRAG FORCES In the development of the theory, it is assumed that the root mean square (r.m.s.) #uctuating drag force acting on the spoiler is directly proportional to the steady state drag force FM . This assumption was also used by Gordon [2, 3], Nelson and z Morfey [5] and is supported by the experiment of Heller and Widnall [4]. If the proportional frequency band de"ned by the limits ( f /a, f a) where f is the centre c c c frequency, is considered, the ratio of the r.m.s. #uctuating drag force to the steady
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LETTERS TO THE EDITOR
state drag force is dependent only on the Strouhal number only. This can be expressed as (F ) "K(St)FM z r.m.s. z
(26)
(band f /a to f a) where the numerical factor K(St) depends on the choice of a, the c c Strouhal number St is given by St"f r/; , where ; is the #ow velocity in the c c c constriction provided by the spoiler (; is de"ned by the volume #ow rate q and the c area of the duct constriction A , such that ; "q/A ) and r is a characteristic c c c dimension, and the steady state drag force acting on the spoiler can be written in terms of a drag coe$cient C and is expressed as D FM ,C (1 o ;2 )A "C (1 o ;2)p2(1!p)A, z D2 0 = s D2 0 c
(27)
where p is the open area ratio, A is the area of duct cross-section, A is the face area s of the #at spoiler and ; is the duct velocity q/A; and it is assumed that steady state = drag forces acting on the two spoilers are the same, i.e., FM "FM . z1 z2 3.2.
THE RELATIONSHIP BETWEEN SOUND POWER AND FLOW PARAMETERS
Using previously derived equations (24) and (25) for the sound power transmitted along the duct under multi-mode and plane wave radiation conditions, the sound power radiated in a given bandwidth can be expressed as follows, for frequencies above and below the cut-on frequency of the lowest transverse duct mode. The mean square value of the #uctuating force in a given band is given by
P
(FM 2)D " z F
u2
u1
S (u) du, F
(28)
where S (u) is the power spectrum of the total drag force on the spoiler. F Using equations (26) and (27), the sound power can be expressed in terms of drag coe$cient. After some algebraic manipulation, it can be shown that when the two spoilers have the same shape, f (f , c 0 =D +(1/4Ao c )M(FM 2 )D #(FM 2 )D #2cos(kd)cos[/(u)] J(FM 2 )D (FM 2 )D N F 0 0 z1 F z2 F z1 F z1 F "(o /16c )AK2(St)[p2(1!p)]2C2 ;4 M2#2cos(kd)cos[/(u)]N, 0 0 D c f 'f c 0 =D +(u2/24no c3)[1#(3nc /4u )(a#b)/A]M(FM 2 )D #(FM 2 )D F c 0 0 0 c z1 F z2 F #2Q cos[/(u)] J(FM 2 )D (FM 2 )D N z1 F z1 F
(29)
LETTERS TO THE EDITOR
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"(o n/24c3)[1#3nc /4u )(a#b)/A](A/r)2(St)2K2(St) 0 0 0 c ][p2(1!p)]2C2 ;6 M2#2Q cos[/(u)]N, D c
(30)
where a, b are the duct cross-section dimensions and d is the distance between two spoilers. Since the sound power measurements are made in proportional frequency bands, the above scaling laws may be used as they stand to normalize an experimental data. The inferred in"nite-duct values of radiated sound power level SWL in the frequency band can thus be normalized by evaluating D f (f c 0 120#20 log K(St)"SWL !10 log Mo A[p2(1!p)]2C2 ;4/16c N 10 D 10 0 D c 0 !10 log [2#2cos(kd)], 10
(31)
f 'f c 0 120#20 log K(St)"SWL !10 log Mo nA2(St)2[p2(1!p)]2C2 ;6/24c3r2N 10 D 10 0 D c 0 !10 log [1#(3nc /4u )(a#b)/A]!10 log M2#2QN. 10 0 c 10 (32) It is assumed in the above equations that S "S , i.e., /(u)"0. It should be F1 F2 noted that when the two sources are located at the same position, the total sound power should be the same as that of a single source. It can be seen that these predictive equations (equations (31) and (32)) are similar to those derived by Nelson and Morfey in reference [5] and can be expressed in terms of easily measurable engineering parameters, together with the single Strouhal number-dependent constant K(St). The additional terms in equations (31) and (32) as they are compared to equations of Nelson and Morfey represent the interaction between two aerodynamic noise sources. Nelson and Morfey measured the sound power levels radiated by di!erent #at plate #ow spoilers in a rectangular air duct at various velocities and "nally they produced a normalized spectrum [5]. It should be noted that 6 dB corrections need to be subtracted from the values of K2(St) in their normalized spectrum [10]. Together with the corrected normalized spectrum K2(St) of Nelson and Morfey, the inferred-duct values of sound power level in 1 octave 3 bands radiated by two closely spaced spoilers can be obtained by equations (31) and (32). The predictive equations (31) and (32) can therefore form a basis of a generalized prediction method for aerodynamic sound generated by closely spaced induct elements. However, the validity of these predictive equations should be checked against experimental data. This will certainly require further work.
4. CONCLUSIONS
The predictive equations developed here which permit the determination of sound power radiated by two closely spaced spoilers only require simple #ow
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LETTERS TO THE EDITOR
parameters and corrected normalized spectrum K2(St) derived by Nelson and Morfey. It needs to be seen whether this predictive technique based on their corrected normalized spectrum K2(St) can be applied to a wider range of #ow duct discontinuities. The ultimate objective of the present study is therefore to extend this method to predict the sound level and spectral distribution of the additional acoustic energy produced by the combination of any given duct discontinuities in any given duct air #ow velocities. This study certainly provides a basis for further investigation.
ACKNOWLEDGMENTS
The authors would like to thank the Hong Kong Polytechnic University Research Committee (Project account code: A-PA37) for the "nancial support during the course of this work.
REFERENCES 1. C. G. GORDON and G. MAIDANIK 1966 Bolt Beranek and Newman Inc. Report No. 1426. In#uence of upstream #ow discontinuities on the acoustic power radiated by a model air jet. 2. C. G. GORDON 1968 Journal of the Acoustical Society of America 43, 1041}1048. Spoiler-generated #ow noise. I: the experiment. 3. C. G. GORDON 1969 Journal of the Acoustical Society of America 45, 214}223. Spoiler-generated #ow noise. II: results. 4. H. H. HELLER and S. E. WIDNALL 1970 Journal of the Acoustical Society of America 47, 924}936. Sound radiation from rigid #ow spoilers correlated with #uctuating forces. 5. P. A. NELSON and C. L. MORFEY 1981 Journal of Sound and
APPENDIX A: CALCULATION OF THE SUMMATION IN EQUATION (19)
According to the concept of reference [5], the double integration expressed in equation (19) evaluates the volume enclosed by the surface (c /c )cos k d for all 0 mn mn propagating modes.
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LETTERS TO THE EDITOR
Let x"n/ka m, y"n/kb n. Then equation (19) can be written as
PP 1
I"
J
1~y2
A
0 0
A BA BP GP ka n
"
kb n
BA BA B
J1!x2!y2 cos kdJ1!x2!y2 1
J
0
0
ka n
kb dx dy n
H
1~y2
J1!x2!y2 cos(kdJ1!x2!y2) dx dy, (A1)
let x "J1!x2!y2/1!y2, e"kd; then dx"x J1!y2/1!x2 . Equation 1 1 1 (A1) can be simpli"ed as
A BA BP ka n
I"
kb n
1
(1!y2)
0
GP
H
1
x2 1 cos (eJ1!y2 x ) dx dy. 1 1 J1!x2 0 1
(A2)
On the right of equation (A2), the bracket of the integration term can be divided into the following forms:
P
1
P
1 cos(cx ) dx ! J1!x2 cos(cx ) dx , 1 1 1 1 1 0 J1!x21 0 1
where c"eJ1!y2. Considering the integration formulation in reference [11]
P
a
0
AB
Jn 2a b~1@2 (a2!x2)b~1 cos(bx) dx" C(b)J (ab) b~1@2 2 b a, Reb'0, Darg bD(n
(A3)
where C(x) and J (x) represent the Gamma function and the sth order Bessel's s function respectively. Applying equation (A3) to equation (A2),
A BA BP
I"
ka n
kb n
1
(1!y2)
0
G
AB
AB H
Jn 2 Jn 1 3 C J (c)! C J (c) dy, 2 0 2 1 2 2 c
where C (3)"1 C(1)"1 Jn. 2 2 2 2 It is therefore obtained as
A BA B
I"
ka n
kb n [I !I ], 2 n 2 1
(A4)
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where
P
I " 1
1
0
P
1 1 I " J(1!y2J (eJ1!y2 dy. 2 e 1 0
(1!y2)J (eJ1!y2) dy, 0
For integration I and I , let y "J1!y2; then dy "!(J1!y2/y ) dy, 1 2 1 1 1 1
P
I " 1
1
0
P
y3 (1!y2)~1@2J (ey ) dy , 1 1 0 1 1
I " 2
1
0
y2 (1!y2)~1@2J (ey ) dy. 1 1 1 1 (A5, A6)
Considering the integration formulation in reference [12]:
P
a
0
AB
1 2 n`b x0`2n`1(a2!x2)b~1Jt (cx) dx" (t#1) C(b)a2n`2b`t n 2 ac
A BA B
ac k (ac) J k`n`b`t 2
n (!1)k n + (t#1) k k k/0
(a, Re b'0, Re t'!n!1).
(A7)
Application of equation (A7) to equation (A5) gives
S C
I " 1
D
e 2n J (e)! J (e) . 3@2 2 5@2 e3
(A8)
Considering another integration formulation in reference [12]:
P
a
0
2b~1ab`t C(b)J t (ac) (a, Re b'0, Re t'!1) xt`1(a2!x2)b~1Jt (cx) dx" b` cb (A9)
Application of equation (A9) to equation (A6) gives
S
n J (e). I " 2 2e 3@2
(A10)
Equation (A4) can therefore be rewritten as
A BA B S C
I"
ka n
kb n n 2
D
n 1 J (e)!J (e) , 5@2 2e e 3@2
(A11)
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where
S A
S CA
J (e)" 5@2
B
2e sin e cos e ! , e n e2
J (e)" 3@2
B
D
1 3 3 ! sin e! !cos e . e2 e3 e
2e n
Substitution of the above formulations into equation (A11) gives the following:
C
D
k2ab sin e 2cos e 2sin e 3 # ! . I" e2 e3 e 6n
(A12)
It is noted that when eP0, ID "(k2ab/6n)3[1!2/3]"k2ab/6n, it is the same e?0 as the volume of the ellipsoidal segment given in reference [5]. In addition, the volume of the two &&slices'' should be considered in the summation of (c /c )cosk d. First, the area of the two &&slice'', can be obtained as 0 mn mn
P S A B
S" x
ka@n
1!
0
nm 2 dm. ka
(A13)
Let x J1!(nm/ka)2. Then 1
P
C
D
ka J (e) ka 1 x2 cos(ex ) 1 1 dx " J (e)! 1 , S" 1 0 x 2 e n J1!x2 0 1
(A14)
where equation (A3) is used. Similarly,
C
D
J (e) kb J (e)! 1 . S" 0 y e 2
(A15)
Thus, the total volume representing the summation of (c /c )cosk d can be 0 mn mn written as I "I#S /2#S /2 t x y
C
D
C
D
k2ab sin e 2cos e 2sin e k(a#b) J (e) " # ! # J (e)! 1 , 0 2n e2 e3 4 e e where 1 is the width of each &&slice'' (similar to the concept in reference [5]). 2
(A16)