Journal of Sound and Vibration (1988) 122(2),389-392

LEITERS TO THE EDITOR

PLANE ACOUSTIC WAVE PROPAGATION IN HOT GAS FLOWS

The propagation of sound in a hot gas flowing through a uniform pipe, with heat lossthrough the walls, is a common practical problem. It arises, for example in the analysisof the acoustic performance of piston engine exhaust systems. Flow and gas conditionswill both vary as the flow temperature decreases with axial distance x along the duct.Thus, associated with a flow temperature T(x), there wiII be a corresponding axial velocityu(x, t), density p(x, t), pressure p(x, t), and entropy s(x, t), with corresponding changesin gas properties.

Relations expressing conservation of mass, momentum and energy, for a general case,are, respectively,

p = pes, p),

(l,2)

(3,4)

where k r is the thermal conductivity of the gas and eij is the viscous stress tensor. If oneassumes small viscosity and neglects diffusion, conservation of energy simplifies to

p(Ds/Dt) =0,

with a corresponding change to equation (2).With one-dimensional flow in x, the relevant equations, with the notation D.JDt =

c1/at + u a/ax, are

1 Dxp au--+-=0p Ot ax '

Dxu 1 ap-+--=0Dt p ax '

O~sp Ot =O. (l(a), 2(a), 3(a»

Combining equation (3a) with equation (4) gives, alternatively, for equation (la),

(l/yp) Dxp/Dt+iJu/iJx =0, (4a)

which may be preferred to equation (la) when the ratio of specific heats 'Y variessignificantly with gas temperature. The new one·dimensional flow equations can beinterpreted to include the mean viscous losses with the heat losses, but not the fluctuatingones.

The axial temperature gradient along the duct will depend on the convective heat flowto the walls, whose temperature will also depend on the ambient conditions outside thepipe. Values of gas physical properties such as viscosity and thermal conductivity increasesignificantly with gas temperature, while that of 'Y decreases slowly. Thus it seems morerealistic to rely on practical observation to establish the gas temperature and its gradientsrather than on any strictly analytical approach.

Typical exhaust systems for vehicles consist of sections of uniform pipe, normally lessthan some two meters long, separated by expansion boxes: Observations indicate that,within such limited pipe lengths, taking the local temperature gradieOnt as constant is inclose agreement with reality. In such pipes, the axial temperature distribution will thenbe described by

T(x) = To(l - ax),389

0022-460X/88/080389 +O-t S03.00/0

ax<O'l, (5)

© 1988 Academic Press Limited

390 LETTERS "TO THE EDITOR

where To is the gas temperature at the upstream end of each pipe section and a =(I/To)(dT/dx). Observations indicate that O<a<O'l, while for most cases a<0·05.

The cyclically varying total exhaust gas flow will have a time averaged part and afluctuating component. Thus the gas :velocity u(x, 1):= ii(x)+ I/'(x, I) density p(x, I) =p(x)+p'(x, I) and pressure p(x, I) =p(x)+p'(x, I), where the overbar indicates the timeaveraged value. The energy equation (3a) for the time averaged flow, to first order in thefluctuations, is

(3b)

ii:::=110(I- ax),or

dT* d [ P ii2

]~ --=- c T+-+-'P dx dx v P 2 '

where the stagnation temperature T* is related to T by T* = T(I +!(Y-t)M 2), M is the

corresponding Mach number ii/c, and c2 = yRT.Upon assuming" y is locally independent of T, then p and p may be eliminated from

equation (3b) with appropriate substitutions from equations (Ia) and (2a) to give

2 1 dii -a(I-yM )-:--=--,

II dx I-ax

where Uo is the time averaged velocity, at the upstream end of the pipe. With a similarapproximation M 2 =M o(1- ax), the time averaged. flow conditions are described by

ii/Uo =(1- aX)[l-axyM~], PIPo =(1- ax)-'[l +axyM~], (6,7)

Plpo=l+axyM~, (8)

(9a, 10)ilp' [illI' all' , au]-+p -+ii-+II - =0.ilx ill ilx ilx

where Po, or Po, and Co are the values at the upstream end of the pipe. Since the factorsyM~ and ax are normally both less than 0,1, their product is less than second ordercompared to unity and may be neglected to that order. Furthermore, observations indicatethat the ~alue of a normally decreases as M increases. Similarly, equations (6) and (7)yield pu=Poiio=constant to within better than fourth order; note that ri(pII)/ ilx =0, hutiI(pii)/ilx:;= 0 necessarily, as has found to be the case with the approximations adopted.

After subtracting the time averaged qua~tities from equations (1b) and (2a) neglectingfluctuating terms of order (11')2, conservation of mass and momentum for the fluctuationsare described by "

ilp' ilp' illI'-+ ii-+yp-= 0ill ilx ax '

(9)

Substitution for ilp'/ilx from equation (10) in equation (9a) gives the result

ap' [ilU' ilu' , ilU] illI'--pu -+ii-+II - +yp-=O.ill ill ilx ax ax

The fluctuating pressure p' can now be eliminated between equations (9) and (IO) bysubtracting illill of equation (IO) from iI/ilx of equation (9). The result of doing so yields

iI 2 illI' iI[ [DxU' ,ilii]] iI [DX ,ilU]-[yp-pii]--- pu --+11 - --p -+11 - =0. (It)ilx ilx ilx DI ilx ill DI ilx

Hitherto, it has been assumed that the local value of y was independent of the flowtemperature T. In fact, its value is C1osely'represented by y ~ y(O) - 8T, over the usefulr·ange 300 K < T < 1060 K. With air y(O) =1·433 and 8 =0·93 X 10-4 K- 1, while withexhaust gas the values are typically y(O) =1·425 and 8 =1·25 X 10-4 K- t

• Thus the gradientiI(yp)/ilx, normalized by yp, will be represented by (l/y)(ily/ax)which is of order 8acompared to unity. Equivalent approximations to second order for the distribution of the

LEITERS TO THE EDITOR 391

speed of sound c and Mach number M along each section of pipe are

c2 = c~(1- ax)[1- BT], M 2 = M~(1- ax)[1 + BT-2ax'YM~]. (12,13)

These, with equations (6)-(8) describe the variation of mean flow and gas propertiesalong each section of the pipe.

With acoustic excitation, one can substitute the equivalent acoustic particle velocityU'(x) exp(iwt) for the fluctuating velocity u'(x, t) in equation (11). After doing so anddividing by "iP, equation (11) becomes

2 d2u [. 1 dii]dU [ 2 ik dii]

(1-M )dx2-2M Ik+; dx dx + k --;- dx U=O (lla)..

when the term (11 "i)(a'Y/ax) and the term in a2 alax2 are both neglected, with k;;::;.wi c.An analytical solution can be found to equation (1Ia) along lines suggested for exampleby Kamke [1]. Equation (1Ia) is of the form

(a2x+ b2)u"+ (a,x+ b.)u'+ (aox+ bo)u ;;::;. O.

This may be transformed to a confluent hypergeometric form, with the substitutionU(x);;::;.expsx1J(t), a2t;;::;.a2x+b2' where S is the root of a2s2+a.s+ao=0. Followingthis, standard forms of solution may then be found, for example, in reference [2].Alternatively, equation (1Ia) may be solved by numerical integration.

If one assumes that 'Y is constant and independent of flow temperature, this affordssome simplification to equation (IIa). After substituting "iP;;::;' "ioPo, pa ;;::;. PoliO, ii=i'ioO- ax), and p;;::;.Po(1 + ax) this becomes

(1- M~) d2

~-2Mo[iko- Moa] ddU

+(1 + ax)[k~+ikoMa] U;;::;. 0, (1Ib)dx x

where ko=wlco.Chow [3] investigated analytical and numerical solutions to both equations (IIa) and

(11 b) for a set of representative examples of the flow of a hot exhaust gas through a ductof constant cross-section and limited length I. He found that, with al< 0·15 and T - 800 K,the predicted behaviour differed only slightly among the different solutions. It was alsoin similar agreement with predictions based on the mean flow and gas properties appropri­ately averaged over I, the presence of the temperature gradient thus being neglected.

These results appear to justify, for the conditions stated, an approach adopted in theprogramme suite currently used at Southampton for predicting component acoustic waveamplitudes along flow ducts, with axial temperature gradients. Here, the duct is subdividedappropriately into axial segments with flow conditions and gas properties in each segmentrepresented by their averaged values [4]. Experience over the years has shown that thisapproximate procedure yields effective results in practice, provided the fluctuating press­ure amplitude remains small enough. That is, when the corresponding gas temperaturefluctuations remain sufficiently small to be neglected, as has been implicitly assumedhere, so that the wave propagates without change of shape. In many practical exhaustsystems this may not be the case, particularly in the pipe close to the exhaust manifold.

The writing of this letter was prompted by a query concerning practical methods ofacoustical analysis for hot flow ducts with temperature gradients. This was raised recentlyby the referee of another paper [5].

Insti/llte of Sound and Vibration Research,Unit'ersity of SOli/hampton,Highfield, Southampton S09 5NH, England.

(Received 4 December 1988)

P. O. A. L. DAVIES

392 LEITERS TO THE EDITOR

REFERE.NCES

1. F. KAMKE 1948 Differential Gleichungen Vol. I. New York: Chelsea.2. M. ABRAMOWITZ and I. A. STEGUN 1970 Handbook of Mathematical Functions, 503-514. New

York: Dover.3. C. F. CHOW 1985 M.Sc. Thesis, University ofSouthampton. Propagation of plane acoustic waves

in axial flow duct with temperature gradient.4. W. J. ADAMS 1975 ISVR Internal Report (unpublished).5. P. O. A. L. DAVIES 1988 Journal of Sound and Vibration 123, (3), (submitted for publication).

Practical flow duet acoustics.