Acoustics Wave Propogation in Sheared Fluid in a Duct

8/8/2019 Acoustics Wave Propogation in Sheared Fluid in a Duct

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J. Sound Vib. (1969) 9 (l), 28-48

ACOUSTIC WAVE PROPAGATION IN A SHEARED FLUIDCONTAINED IN A DUCT

P. MUNGURAND G. M. L. GLADWELLInstitute of Sound and Vibration Research, University of Southampton,

Southampton, SO9 5NH, England(Received 12 June 1968)

In a ssessing the propagation of an acoustic w ave in a sheared fluid, first, the d erivationof the linearized w ave-equation for two-dim ensional v iscous flow is presented. Interactionswith the acoustic wave of the mean flow, the shear and the viscosity of the fluid throughshear have been included. This is followed by a numerical solution to the inviscid case,using a constant gradient and a turbulent velocity profile, for the first three symm etric modes.The plane mode is compared with results obtained by Pridmore-Brown using ananalytical approach; there is good agreem ent for the case with constant gradient profile.Methods of solution are also presented for the problems of acoustic wave propagationin a flowing medium, both with and without shear, contained in ducts with fmite walladmittance.

1. INTRODUCTIONThe stimulus for these theoretical investigations of the propagation of acoustic disturbancesin a fluid contained in a duct and having mean velocity gradients norma l to the flow direction,was the observation that high intensity sound produced in the currents of gas-cooled nuclearreactors appeared to be most strongly attenuated in those regions of the flow where highshear gradients, associated with turbulent boundary layer flow, were produced for the purposeof efiicient heat transfer. It was also interesting to determine whether mean shear effects couldproduce significantly greater attenuation of sound in gas flowing in pipes and ducts than theclassical diffusion and molecular relaxation effects associated with propagation in thequiescent fluid.Meyer, Mechel and Kurtze [I] experimented on the influence of flow on sound a ttenuationin absorbing ducts. Ing ard [2] investigated the effect of uniform flow (no shear) on the pro-pagation and attenuation in lined ducts. The effect of shear in lined ducts has been consideredby Pridmore-Brow n [3] and Tack an d Lamb ert [4]. In all the works referred to above, theeffect of viscosity has been neglected altogether.When a fluid flows past a solid boundary, the fluid immediately in contact with the wall isat rest. However, the velocity rises rapidly from zero at the wall to its value in the mainstream, the rise taking place within the thin viscous boundary layer next to the wall. In thislayer the velocity gradient is very large, so that even if the viscosity is sma ll, the tangentialstresses cannot be ignored.The paper presents a derivation of the linearized wave equation for two-dimensionalviscous flow. This is followed by a numerical solution to the inviscid case for comparisonwith Pridmore-Brow ns results. Methods of solution are also presented for the problems ofacoustic w ave propagation in a flowing fluid, both with and without shear, contained in ductswith finite wall impedance. 28


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30 P . M U N G U R A N D G . M . L . G L A D W E L LThe acoustic part of the Navier-Stokes equation can be filtered from equations (1) and (2)by subtracting the time-average of equations (1) and (2) from equations (1) and (2). If this is

done, and the products of fluctuating comp onents are neglected, one obtains

an din the x direction (3)

Pok+~&]=-$+ rloV;v+-- _+au +YE*1)30:(:; aX) axay mtheydirection. (4)The two-dimensional conservation of mass yields

a P' ,aPf ,aPfYg+ujy+vay+ p !T!Ic+~=o*( >x ay (5)As before, if the mean and fluctuating compo nents of the parameters are used, the timeaverage of equation (5) is substracted from equation (5), and the products of fluctuatingcomp onents are neglected, one obtains

aP a Pz+uax+po au+%o.( 1x aybe denoted by 4, then

+-J-($+ u$).(6)

(7)Differentiating equations (3) and (4) with respect to x and y, respectively, one obtains

zE+ ua~+2auax ay 2$+~,-&+3%!+lp+!5L~ax ay2 axayay (8)an d

a2vpoay atL--auao mu a2vay ax ~ ~_azp+90-a,~v+__+I TOa a2rlauaxay ay2 ay 3 ay2 ZSjjay (9)Differentiating equation (6) with respect to t, one obtains

C&U- a2P a4axat+PO~=O- (10)If all the terms in equations (8) and (9) are brought to the left and the two equations addedand equated to equation (10) one obtains

g+ugt a+ 2 a211 u=v~P+pouax+2podt~-~770v~~-~~-2~y~.ax ay (11)4 may be eliminated from the above expression by using equation (7) and the derived relation

W=-; &(v:P)+U$$i7:p)+2ayaxay+~~~[ au a2p 1 (12)


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S O U N D P R O P A G A T IO N I N A S H E A R E D F L U I D 31Substituting equations (12) and (7) in equation (1 l), one obtains

a2P av au 4r10- =v:P-2Ua~~- UZ$$+2po__ +-- aat2 [ _(v:p)+2aua 2P a2uapax+ 3po at ay axay +ay2ax -arla2u--__ax ay2 2a211au 4710 a,,,,,,+g+g:P). (13)

To obtain a convenient wave equation, p, 7 and v mu st be expressed in terms of P inequation (13). In the simpler case of an inviscid fluid with no thermal conductivity, propaga-tion may be considered to take place adiabatically and isentropically, and the fluctuatingdensity and pressure to be related by

(14)

where c is the sound velocity. However, in viscous fluids, where the isentropic condition is nolonger v alid due to energy dissipation by viscosity, the chang e in entropy mu st be taken intoaccount. Thus, via the mechanism of viscosity, the sound wav e gives rise to an entropy wave.Through the equation of state of the fluid, the pressure wave is accom panied by a temperaturewave which in turn leads to a viscous wave.In Appen dix 2, it has been show n that w hen entropy is not conserved due to viscousdissipation, the fluctuating pressure and density are related thus :

where

pJp+@- C 2 [compare w ith equation (14)]s=-(U--_-)X jW(1 _MK)

(15)(16)

and E is a function of viscosity, thermal conductivity, shear, etc., o = angular frequency,u = ratio of the p rincipal specific heats (cP/cy). M = U/c, and K is defined in equation (26)below.In Appendix 1, it has been shown thaty=qO P_P( )O PO (17)

where a is the index re lating viscosity and temperature, and is about 0.72 to 0.75 for gases.If p is replaced by (P + S)/c2, equation (17) becomes

Substituting equations (15) and (18 ) in equation (13), and denoting (4yo)/(3poc2) by T, theviscous relaxation time, and (aqo)/(po c) by TV,one obtains


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32 P . MUNGUR AND G. M. L. GLADWELLFor harmon ic variation with time, a/at = jw an d, in this case, denoting U/c by A4, the Machnumber, one finds

2~ a y p + 6 )- - a x a t f T C M$V:(P+6)+2-[d h f a q P +6 ) +a wa ( P+q- ~ -c d y a x a y a y a x I-~2c~+r-l)%gLrq-

-27,eg [ (u- 1)a q p + 8 ) a26

a x a y -CT------.a x a y1 (20)This is the wave equation governing the propagation of sound in a viscous flowing fluidin the presence of shear. Terms involving aM /ay represent the interaction of shear with the

acoustic wave and those containing T and 7 2 are due to viscosity. Terms in M representinteraction of the mean flow with the acoustic wave and those involving the products of Tor 72 and aM /ay or a2M /ay2 represent interaction with viscosity through shear. The aboveequation simplifies to the one used by Pridmore-Brow n [3] if the terms containing T and r2are suppressed . Further, if the terms containing aM/ay and a2M/ay2 are suppressed, theequation reduces to a wave equation obtained by Ingard [2] for the case of uniform flow in aninviscid fluid.

It mu st be stressed tha t the solution to equation (20) is subject to the following limitation.Although the effect of shear has been taken into account, it has been assum ed that variationswith x of PO , U, v. and p . are zero; this assump tion is valid only under steady-state conditionsand constant cross-section. In Appendix 2, while deriving the effect of the fluctuating entropyon the fluctuating density, it has been assum ed that there is no mean temperature gradient.Equation (20) cannot be solved by the norma l separation of variable m ethod because ofthe presence of coupled terms. Substitution for 6 transforms the equation from second orderto one of fourth order. Of course, in an inviscid fluid, 6 3 0, the equation retains its second-order wave equation characteristic. It may be noted that shear affects the propagation evenin the absence of viscosity [see fourth term on the right of equation (20)]. To find the contribu-tion of viscosity to the attenua tion, the effect of shea r in the fluid with no viscosity must firstbe determined. The following section therefore deals with the solution of equation (20) forthe inviscid case. The general solution taking viscosity into account will follow in anotherpaper.

3. THE INVISCID CASEWhen the viscosity is zero, T, 72 and 6 are also zero, and equation (20) becomes

;!$41 -M3$+~+2poc;$-~$~. (21)As the problem is now one involving the effect of shear on the propagation, one may assu mea solution of the form p = F(y) e-ax elw-kxx) (22)and U= G(y) e-CcXJ(wt-kxX)_ (23)Here F(y) and G(y) are the amp litudes of the pressure wave and the transverse particlevelocity, respectively, and a, k, are respectively the attenuation and the propagation constants


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SOUND PROPAGATION IN A SHEARED FLUID 33in the x direction. As there is no viscosity, the attenuation would be due to finite wa ll admit-tance. For rigid walls CL ould tend to zero.

Substituting equations (22) and (23) in (21), one obtains-$E=(l -M3(~ +j/~,)~F+~~-2p~c~~ (~+j~JG+j+~.+j~~)F. (24)

From equation (4), in absence of viscosity,

or[jw-(u+jk,)U]G=-;z

an dG=_ 1 aFjccJpo( - MK) ay (25)

whereK = (g + jk,)jw/c (26)

the normalized complex wave vector in the x direction and k = w/c.If one substitutes the expression in equation (25) for G, equation (24) becomes

d2F@ +

&K dr;$+k2[1 -K2(1- M2) - 2MK ]F=O. (27)This equation is identical to that o btained by Pridmore-Brow n if Kis taken to be wholly real(that is if the wa lls are rigid). Once the flow profile M w hich is a function of y has been chosen,

equation (27) may be solved to yield the pressure profile across the duct. In order to obtainan explicit solution, the boundary conditions at the wa lls or at one wall and the centre of theduct must be specified. This leads to specific eigenvalues of K, the imaginary part of whichyields the attenuation.

4. SOLUTION FOR RIGID WALLSPridmore-Brow n obtained an approximate solution to equation (27) by a method proposed

by Langer [6]. For a constant gradient boundary layer, he obtainedF= d~~q-~~f (H) (28)

wheree=K-I-M,q=fP-1 ,s = f q112 e = +{8(e2 - 1)2 - cash- B},1

H = (3clK~/2)~,d = kl(dMdy),

andf(H) is the general solution of Airys differential equation, namely,f(H) + Hf(H) = 0.


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34 P. MUNGU R AND G. M. L. GLADWELLIn the case of a turbulent boundary layer, taking M = M ,,(J&)~ /~, and using M as theindependent variable, one obtains

d2 F--dM 2 dg+49(kL)2M,14M12{(1 -MIQ2-K2}F=0. (29)A solution to the above equation, obtained by Pridmore-Brow n, is

F= M3 t9s6q-4f (H)where q = M12(e2 - l),

A = 7kLMG7,(30)

and the rema ining symbols are defined as above.Due to the singularity in equation (29) at M = 0, the solution given by equation (30) breaksdown at the wall. However, the solution is valid asymptotically for (1 --f w in the neighbour-hood of y = 0. Values of F across the flow profile w ere obtained using Laurent series andinvolved tedious calculations and approximations.

5. NUME RICAL SOLUTIONEquation (27) has been solved numerically for a number of different velocity protjles M(y).The method used was the fourth-order Rung e-Kutta, a good description of which is given by

Hildebrand [7]. Equation (17) may be written in the formd2 Y@ +fi(X)dr;+f2(X) Y= 0 (31)

where Y = F, X E y/L.Equation (31) may be rewritten as a pair of equations

dYdx=Z and g=-fi(X)Z-f2(X) Y=f(X, Y,Z)for which the Rung e-Kutta method gives the approximate solution

Y(+) = Y() + hZ + &?, + rnz + m3) + O(V)an d

(32)

(33)

ZCr+) = Z@ ) + *(ml + 2m2 + 2m3 + m 4) + O(h5)where Y() = Y(rh), Ztr) = Z(rh), the interval 0 G X G 1 is divided into n intervals of lengthh = l/n, and

m, = h x f (X@ ), Y(), Z@ )),m2 = h x f (X(r) + +h, Y(r) + +hZ(), Z(r) + *ml),m3 = h x f (Xc) + *h, Y() + 3hZ() + ihm,, Zcr) + Jm2),m4 = h x f (Xc) + h, Y() + hZ() + + hm2, Zcr) + m3). (34)

6. APPLICATION OF RUNG E-KUITA TO THE INVISCID CASETack a nd Lambert [4] showed that the velocity profiles M(y) in a duct can be approximatedeither by an exponential function or by a fractional power. In the present analysis a powerlaw will be assumed, namely,

M= Mo(y/L)lN = MO X1N for 0 G XG 1 (35)


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SOUND PROPAGATION IN A SHEARED FLUID 35where L is the value of y at the centre of the duct, M e is the midstream Mach number, and N isa positive number whose value depend s on the nature of the wall. For flow without shear, thevelocity profile is uniform so that N -+ co. In the case of a constant velocity gradient, N = 1,and for a fully-developed turbulent flow, N is approximately equal to 7. It follows that

dM 1 MO XN--= -dy N L (36)Equation (28) now becomes

d2 Y dZ 2Kjj,f,, X N-1dXZ=dX=- N(l -MO XIN) Z - [(1 - M ,, KX N)2 - K2] Y, (37)

= -_f&w -.lxX) K=f(X K a, (38)

where Y G F, X E y /L and Z = (d Y/dZ). Equation (37) is of the same form as equation (32),the solution of which is given by equation (33).The boundary conditions that m ust be satisfied in solving equation (38) are as follows:at the wall X = 0, Y may be set equal to unity. Z, being actually (d Y/dX ), is proportional tothe particle velocity normal to the flow and is therefore equal to zero at a rigid wall. Also atX= 1, the centre of the du ct, Z is zero for symm etric mod es and Y is zero for antisymm etricmodes. Several values of K may be found to satisfy the above boundary conditions ; the modeof propagation depends on the magnitude of K . For plane wave propagation in the absenceof flow K is unity in the lowest mode.A computer program was written to evaluate Y and Z at various steps of X between0 and 1 for various values of (wL/c), Me and N. A subroutine selects the appropriate value of Kto satisfy the boundary condition at X = 1. Various mod es of propagation may be investigatedin this way. However, the effect due to shear or viscosity is best demonstrated by comp aringthe results for the lowest mode of propagation. In the absence of shear, that is for uniformflow, the pressure is uniform across the flow. The way in which shear alters the pressureprofile is show n in Figures 2 and 3 where N has been set equal to unity, equivalent to a constantgradient flow. The results are in good agreem ent with those predicted by Pridmore-Brow nusing an analytical solution. Figure 4 show s the pressure profile for a turbulent flow; in thiscase, unlike Pridmore-Brow ns results, the present solution does not break down at the wall.Figures 6 and 7 show the effect of flow on the pressure distribution for the next two highersymm etric modes. In Appen dix 3 an expression is obtained for K , the normalized wavevector in the x direction when shear is absent, and the magnitude of K so deduced is comparedin Table 1 with those obtained when shear is present. It may be noted that for the lowest mode(that is mode zero) Kis a function of the mean flow only for the no-shear case, but the presenceof shear brings in dispersion by making K dependent on the frequency as well. Figure 5 show sthe variation of K with MO and k L .

7 . S O L U T I O N F O R W AL L S W I T H F I N I T E A D M I T T AN C EBefore the general n umerical solution of this class of problem is described, the slightly

simpler case of absence of shear will be discussed as this can be solved analytically.7.1. IN ABSENCE OF SHEAR

In this case the separated wave equation becomesf$ + (k L)2 [( 1 - M K)2 - K2] F = 0. (39)


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S O U N D P R O P A G A T IO N I N A S H E A R E D F L U I D 39The finite admittance of the wall leads to attenuation of the sound w ave and this makes Knecessarily complex. Let K = B +jA [see equation (26)] (40)where B is the normalized propagation vector along the duct = k,lk an d A =-u/k is thenormalized attenuation coefficient along the duct.Let

- k; = (kL)*[(l - MK )* - K*],= (kL)* [l - 2MB + (M* - 1) (B* - A*) +j2A(M 2 B - B - M)],= al +jbl (say). (41)

Letk, = cl +jd, ;

then al = d; - c;and br = - 2ci dl.

The solution of equation (39) may be written as(42)

therefore,F = D, ekyx + Dz ebkpx (43)

dy= k,,[D, ekyX Dz emklx] (44)where D1 and D 2 are constants to be determined from the boundary conditions.

At the centre of the duct, X = 1, dF/dX = 0 for symmetric modes and F= 0 for anti-symm etric mod es. From equations (43) and (44), it follows thatDz = fD, ezky,

the positive sign referring to the symmetric mode and the negative sign to the antisymmetricmode.Equations (43) and (44) now become

F= D, e+ky[e-kH-X) f ekWO ]and (45)

_d!!d!&, Di e+kr[e-ky(i-X)7 ekyW)]_ (46).At the w all, X = 0, the ratio of i,,, the transverse wall velocity, to P, the pressure is equal

to the normal specific adm ittance of the wall, A. ThusA i_=_w_

POC p (47)In problems where there is relative motion between the fluid and the boundary it is necessary

to use the basic continuity of acoustic particle displacemen t rather than velocity because ofthe extra convective terms in the substantial derivative of the displacement. The importanceof using the basic boundary condition had been pointed out before by Miles [8], Ribner [9]and Ingard [2]. In the absence of relative m otion, continuity of particle displacem ent alsoleads to continuity of particle velocity.Let 5;, = ,Q,) e-orxeiWt-kxx) (48)represent the displacemen t in the med ium.


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40 P . MUNGU R AND G. M. L. GLADWELLThen u, the particle velocity in the medium , is given by

orv = jw(1 - MK) &.

Using continuity of acoustic particle displacem ent, one may write5;, = 5, (at the wall),

Vc-- jw( 1 - MK) For the wall,

Therefore,

Substituting equation (50) in equation (47) one obtainsA 1 v-=pot (1 - MK)P

=(l &K);.

(49)

(50)

(51)From equation (25), the amplitude of the transverse particle velocity is given by

G= j ldFkL(1 - MK)PocdX (52)Equation (51) becomesdF/dX

A=iX(l kK)2-F. (53)Substituting for dF/dX and F from equations (45) and (4 6) in equation (53), one obtains

Thus

an d

A = jkYe-b&

[ 1L(1 - M K ) z e y k y * t iy -jk,A = k L ( 1 - M K ) 2 tanb k , for symmetric modes (54)

A = kL(1 - M K ) 2 cotb k , for antisymmetric modes. (55)Equation (54) has been obtained previously by Ingard [2] and that an d equation (55) reduceto the expressions obtained by Morse [lo] when there is no flow. At first sight it seems quitelogical to replace the wall admittance by some equivalent wall admittance through thefactor (1 - M K ) 2 . Unfortunately Kcontains the attenuation coefficient A , whose computationis the whole object of the analysis. However, a numerical method can be applied. k , an d Kare related by equation (41). Let A = C +jS where C and S are the conductance and thesusceptance of the wall, respectively. Equations (54) and (55) may then be written as

C + jS = (~2 + jb2) tanb (cl+ jd,) (56)


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S O U N D P R O P A G AT I O N I N A S H E A R E D F L U I D 41or

C +jS = (uz +jbJ coth (cl + jd,)where u2, b2, cl an d dl are functions of B, A and M only.Thus for given values of C and S, B and A can be solved graphically after separating thereal and imaginary parts of equation (56).7.2. I N P R E S E N C E O F S H E A R

When there is shear b ut the flow occurs w ithin soft walls an analytical solution of theproblem is apparently impossible; however, a numerical method can be found. T he separatedwave equation to be solved is still equation (27), but now that there is wall absorption, K,the normalized wave vector, is complex.

Let K = B + jA where A is associated with the attenuation [see equation (26)]. To solveequation (27) it is necessary to separate the real and imaginary parts of the coefficients ofdFjdX and F. If this is done the equation can be written in the form

(57)The solution of equation (57) will consist of a real and an imaginary part; let it be

F= Yl +jY2. (58)If equation (58) is substituted back in equation (57), one has, after separating the real andimaginary parts,

Y;+aY;-bY;+cYr-dY2=0an d (59)Y;+bY;+aY;+cY;!+dYr=Owhere dashes denote derivative with respect to X.Let

Yl = Yl,y2 = y2,r; = Y3,r; = Y4.

Substituting equation (60) in equation (59), one obtainsY;=-cY1+dY2-aYs+bY,and Y;=-dY,-cY,-by,-uY+

In the matrix form, one may write

In general,

In the vector notation,Y = Q(x)Y.

(60)

(61)

69

(63)

(64)


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42 P . MUNGUR AND G. M. L. GLADWELLEquation (64) may now be solved by a generalization of the Rung e-Kutta method. Thisstates that if

g =fi(X Yl, y2, y,, 1.. Yn) (65)then

Y(+i) = yW + ; (m(i) + 2m Z+ 2,,+3 + &4)) (66)where, as before, Y() = Y(rh) = (y@z), yz(rh), . . . y,(A)),

rn () = f (X, , Ylr)) ,mc2)= f(X, + j&z,Y() + + hml) ,mc3)= f (X, + *h , Y( ) + + hm,) ,mc4)= f ( X , + h , Y( r )+ hm , ),

and f (X, Y) = Q (X) Y [ see also equations (62), (63), (64)].As in the case w ith rigid w alls, ( Y: + Y $) can be set equal to unity at the wall. Ys and Y,, beingderivatives of Yr and Y2, respectively, and therefore proportional to the transverse particlevelocity, can be set equal to the conductance and susceptance of the wall, respectively. Conti-nuity of acoustic particle velocity is used here, b ecause the flow profile assures tha t there is norelative motion of the fluid and the wall. For sym metric mod es, the transverse particle velocityat the centre of the duct m ust be zero. Tw o subrou tines are used to select values of the realand imaginary parts of K to mak e (Y$ f Yi)1/2 equal to zero at the centre of the duct. Theprogram computes finally the values of Yr, Y 2, Ys and Y4 for various values of X between0 and 1. From the values of Yr and Y2 the mag nitude of F is given byF= (Y; + Y;)12.From the imaginary part of K, the attenuation a can be obtained.

8. CONCLUDING REMARKS

The results for the pressure profile across a constant gradient shear flow are in good agree-men t with those predicted by Pridmore-Brow n using an analytical solution. In the case of theturbulent flow profile, unlike Pridmore-Brow ns results, the present solution does not failat the wall.In this paper the results of the effect of shear on the first two modes have been presented.By changing the limits on the frequency range in the hunt for eigenvalues to satisfy theboundary conditions, the effect of shear on the other m odes can easily be obtained. However,information about the overall pressure profile cannot be obtained from the individual mod esuntil a knowledge of the energy distribution produced by the sound source in the variousmodes is specified.One basic difference between this and the analytical method is in the calculation of theattenuation due to finite admittance of the wall. In the latter m ethod it is assum ed tha t thepressure profile ac ross the duct will not change from that calculated for rigid walls, whereasin the numerical method, the eigenvalues of K can be found to match the wall admittanceand the imaginary part of K will yield the attenuation straightaway.

ACKNOWLEDGMENTS

The authors gratefully acknow ledge the financial support for this research given to theInstitute of Sound and Vibration Research by Atomic Power Constructions Ltd., Sutton,


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SOUND P ROPA GATION IN A SHEARED FLUID 43Surrey, England. They also wish to express their thanks to Mr. F. J. Fahy for suggesting theproblem and making useful comments, and to Mr. C. L. Morfey for helpful discussions.

REFERENCES1. E. MEYER, F . MECHEL an d G . KURTZE 1958 J. acoust. Sot. Am. 30, 165. Experiments on theinfluence of tlow on sound attenuation in absorbing ducts .2. U. INGARD 1959 J. acoust. Sot. Am. 31,103 5. Influence of fluid motion past a plane boundary onsound reflection, absorption and transmission.3. D. C. PRIDMORE-BROWN 1958 J. Fluid. Mech. 4, 393. Sound propagation in a fluid flowingthrough an attenuating duct.4. D. H. TACK and R. F. LAMBERT 1965J. acoust. Sot. Am. 38,655. Influence of shear flow on soundattenuation in lined ducts .5. L. HOWARTH 1964 Modern developments in fluid dynamics-high speed flow, vol . I , p. 379.London: Oxfo rd University P ress.6. R. E. LANGER 1937 Phys. Rev. 51,669 . On the connection formula and the solutions of the waveequation.7. F. B. HILDEBRAND 1956 Introduction to Numerical Analysis. New York: McGraw-Hill.8. J. W. MILES 1957 J. acoust. Sot. Am. 29, 226. On reflection of sound at an interface of relativemotion.9. H. S. RIBNER 1957 J. acoust. Sot. Am. 29, 435. Reflection, transmission and amplification ofsound by a moving medium.

10. P. M. MOR SE 1948 Vibration and Sound, p. 369. New Y ork: McGraw-Hill.11. L . H OWARTH 1964 Modern developments in fluid dynamics-high speed flow, vol. I, p. 55.London: Oxf ord University Pre ss.12. H. SCHLICHTING 1955 Boundary Layer Theory, p. 338. London: Pergamon Press.

APPENDIX 1: THE EVALUATION OF AND +/axayThese tw o terms refer to spatial variation of the fluctuating viscosity in the viscous wave

which accompanies the temperature wave. The variation of viscosity with the absolutetemperature may be expressed by [12]

r) = AT. (A.1)Substituting the static and fluctuating compo nents of 7 and T in equation (A. 1) one obtains

q. + 7)= AT:(l + T/To). (A.3Time-averaging equation (A.2), one obtains

rlo = AT,,and by neglecting the products of fluctuating quantities, one obtains

rl =a~o(Wo).From the equation of state of the gas, one may write

(PO + P) = (PO+ P) Wo + T).Subtracting the time average of equation (A.4) from equation (A.4), one obtains

P= R(poT+pTo)

(A.3)

(A.4)

orT/To = P/PO - P/ PO .

Equation (A.3) may now be written as17= WOW0 - P/ PO) .

(A.3(A.6)


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44It follows that

P . MUNGUR AND G. M. L. GLADWELL

877-=a.Q ( aP 1 ap----_i3X PO ax poax )an da27 ( a 2 p i azp- ~ - - - .--=arlo p o a x a yx ay >oaxay

(A-7)

(AJOAPPENDIX 2: THE RELATION BETWEEN P AND p

If entropy is conserved during the passage of the sound wave, p and P are relatedby the adiabatic relation P = pc 2. If there is heat dissipation through viscosity, thermalconductivity or any other agency, the entropy of the system can no longer be consideredconstant.

Let p be a function of P and S where S refers to the total entropy which consists of thesum of So and S, the static entropy and the fluctuating entropy, respectively. Then

6P = (g$SP + (igJ/S~ . (A.9Denoting 6p, 6P and 65 by p, P and S , the fluctuating parts of p, P and S, one obtains

(A. 10)(i5 P/ap)s is the adiabatic speed of sound squared, and may be denoted by c2. The expressionfor (ap/%), can be obtained as follows.

From the first law of thermodynam ics, one may write

Therefore

TdS = dE + Pd(l/p) where E is the internal energy=dE-clidp.P (A.ll)

= (-g)p(gp-;=c, _TI( 1cP P12 for a perfect gas.

Therefore=------7.

Thus equation (A.lO) now becomesp=~_!Js.P

(A.12)

(A.13)The fluctuating entropy S in equation (A.13) can be obtained by considering the energyequation in terms of entropy, which in the two-dimensional case is given by [ 1 ]

p T g = @ + div (K; grad T) (A.14)


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SOUND P ROP AGATION IN A SHEARED FLUID 45where Kj = thermal conductivity of the gas,ICp= 17 & + f [(err - e&* + e& + e:,],

= 77 & + !C [e?, + e& - err e&i3 7

Therefore

an d

The dashes denote the total magnitude of the parameter concerned.T = T ,, + T = sum of static temperature and fluctuating temp erature andKi = &, + Kt = sum of the static and fluctuating thermal conductivity.

Equation (A. 14) reduces to

(A.15)When there is no mean temperature gradient,

c_aT and aT aTa x a x -=---*a y a y

If S, u, v, T, Ki are replaced by S,, + S, U + U , v, T,, + T and Ko, + K,, respectively, andthe products of fluctuating quantities neglected, equation (A.15) becomes(PO+P)VO+T) x+z =rl ay +2r) 5 ay+z +KotDSODS) (au)2 -J(au 8,) (!?+ $). (A.16)If (p. + p)(To + T) is written as p o T o equation (A. 16) becomes

+ 2770a u a u a u--(-+-)+s($+g). (A.17)polo ay ay axFrom the time average of equation (A.17), one obtains

Uas,=~o au2--. ( )x polo ay (A.18)


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46It then follows that

P. MUNGUR AND G. M. L. GLADWELL

+ 2110au a u a 0- - ( - +- ) +f $ ( $ +g ) . (A.19)p o l o y a y a xReplacing W,/ax from equation (A.18) in the first term and substituting&[1 - M K] S for(g+ Ug),onefinds

In Appendix 1 it has been show n that

andSubstitute the above two expressions; then equation (A.20) becomes

(A.20)

or1s=-x EPOTo Ml - MN

where E is defined by equation (A.21). F rom equation (A. 13), one may now writep,P_dxIx Ec2 C, p. To jw(1 - MK)

P (a- 1) E=- c2--7-x jW(1 - MK)(P +a>- c2

where 8 = - (u - 1) x jW(1 - MK) *

(A-21)

(A.22)

Equation (A.22) expresses the relation between p and P ; 6 may be considered as a perturbedacoustic pressure due to entropy change by dissipation. It may be noted that 6 = 0 if there isno dissipation.


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S O U N D P R O P A G AT I O N I N A S H E A R E D F L U I D 47A P P E N D I X 3 : T H E E V AL U A T IO N O F K I N T H E A B S E N C E O F

S H E A R WI T H R I G I D W A L LIn the absence of shear, equation (27) becomes

$+(kL)[(lMK)2-K2]F=0.For rigid walls, K is wholly real.

T A B L E 1Variation of K with MO and kL

m = 0 = plane mode

(A.23)

Constant gradient flow TurbulentFlow at No shear , 3 flowcentre freq. indep. kL=m kL=2rr kL= 1 0 kL=20 kL=20MO) (K) (K) (K) (K) (K) (K)0.0 l.oooo - - - - -0 .1 0 .9091 0 .9526 0 .9566 0 .9633 0 .9761 0 .92220 .2 0 .8330 0 .9095 0 .9224 0 .9393 0 .9618 0 .85840 .3 0 .7692 0 .8695 0 .8932 0 .9194 0 .9497 0 .80330 .4 0 .7143 0 .8322 0 .8672 0 .9015 0 .9388 0 .75500 .5 0 .6667 0 .7970 0 .8431 0 .8849 0 .9286 0 .7121

m = 1 =jirst m o d ekL=2?r kL= 10 kL=20Flow at r ,

c e n t r e N o s h e a r c .g . f l ow N o s h e a r c .g . f l ow N o s h e a r c .g . f l ow( M O ) (K ) W I ( K ) 0 0 W I W I

0.0 0.866 - 0.949 - 0.987 -0 .1 0.775 0.816 0.850 0.894 0.895 0.9290.2 0.700 0.768 0.776 0.841 0.819 0.8860.3 0.636 0.724 0.710 0.793 0.755 0.8530.4 0,582 0.684 0.65 0.751 0.700 0.8240.5 0.535 0647 0.61 0.713 0.653 0.78

m = 2 = second modekL=20Flow at , 7centre No shear Turbulent flow(MO) (K) (K)

o-o 0 .949 -O* l 0 .851 0 .86300 .2 0 .776 0 .79280 .3 0 .710 0 .73260 .4 0 .653 0 .68040 .5 0 .616 0 .6347


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Acoustics Wave Propogation in Sheared Fluid in a Duct

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