* * * is the adiabatic sound speed I. INTRODUCTION II. GENERAL EQUATIONS. 76, Hamburg D-22301, Germany.

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1 Influence of grzing flow nd dissiption effects on the coustic boundry conditions t lined wll Yves Aurégn, Rudolf Strobinski, ) nd Vincent Pgneux Lbortoire d Acoustique de l Université du Mine (I. A. M.), UMR CNRS 6613, Av. O Messien, 785 Le Mns Cedex 9, Frnce Received 6 July 1999; revised 1 October ; ccepted 14 October The problem of sound propgtion ner lined wll tking into ccount men sher flow effects nd viscous nd therml dissiption is investigted. The method of composite expnsion is used to seprte the inviscid prt, in the core of the flow, from the boundry lyer prt, ner the wll. Two diffusion equtions for the sher stress nd the het flux re obtined in the boundry lyer. The mtching of the solutions of these equtions with the inviscid prt leds to modified specific coustic dmittnce in the core flow. Depending on the rtio of the coustic nd sttionry boundry lyer thicknesses, the kinemtic wll condition chnges grdully from continuity of norml coustic displcement to continuity of norml coustic mss velocity. This wll condition cn be pplied in dissiptive silencers nd in ircrft engine-duct systems. 1 Acousticl Society of Americ. DOI: 1.111/ PACS numbers: 43..Mv, 43.8.Py LCS I. INTRODUCTION In this pper, the problem of coustic propgtion in duct with prllel sher flow nd trnsverse temperture grdient is investigted. The im is to tke into ccount viscotherml effects in the boundry lyer ner lined wll. These effects re ccounted for by modifying the boundry condition t the duct wll. The effects of temperture nd velocity grdients s well s thoses cused by dissiption re concentrted in thin lyer ner the wll. In the core of the flow, the fluid is considered to be n idel gs nd the velocity nd the temperture vry only slowly. Finlly, new boundry condition on the wll is obtined for the sound in the core flow. Severl uthors 1 8 hve ddressed the problem of propgtion in lined flow ducts for dibtic, inviscid sound propgtion. In these cses, they hve ssumed continuity of displcement t the wll since it seems to be the more pproprite. 9 Nyfeh 1 hs studied how viscotherml effects ffect the impednce for the cse the coustic boundry lyer is much thinner thn the men flow boundry lyer. In this pper, by tking into ccount both viscotherml effects ner the wll nd the effects of lrge sttionry velocity nd temperture grdients, it is shown tht the effective boundry condition cn be continuity of norml coustic displcement, or continuity of norml coustic velocity, or mixed condition depending on the different length scles coustic nd men flow boundry lyer thicknesses. This wll condition cn be pplied in lined flow ducts such s dissiptive silencers nd ircrft engine-duct systems. The equtions of lossy fluid mechnics linerized bout men stte re presented in Sec. II. These equtions re simplified by mking clssicl boundry lyer ssumptions for flow ner plne wll nd scled to obtin dimensionless equtions. An symptotic representtion of these equtions is then given in Sec. III. In the core of the flow, the Present ddress: Silencers: Consulting nd Engineering, Dorotheenstr. 76, Hmburg D-31, Germny. problem reduces to solving the equtions of lossless coustics with men prllel sher flow nd men trnsverse temperture grdient. The effects of the boundry lyer cn be tken into ccount by modified dmittnce of the wll. This modified dmittnce is found in Sec. IV by solving two diffusion equtions for the sher stress nd the het flux in the boundry lyer. In Sec. V the nlysis is extended to the cse of rough lined wlls. II. GENERAL EQUATIONS The sitution for the sound propgtion being investigted is shown in Fig. 1, the lined wll lies in the plne y of coordinte system (x,y,z). The generl equtions governing the liner oscilltions of gs with men flow re t vv v v, 1 v t v vv v 1 pv v B, s t v sv s Q T, 1b 1c pc h s, 1d the terms without subscript refer to fluctuting components nd the subscript refers to men vlues v is the velocity, p is the pressure, is the density, s is the entropy. T is the temperture, c p is the specific het of the gs t constnt pressure, c is the dibtic sound speed c y p, s 59 J. Acoust. Soc. Am. 19 (1), Jnury /1/19(1)/59/6/$18. 1 Acousticl Society of Americ 59

2 i ikui d ik du i d, iui du ikp iv 1 dp w B y, isi 1 d w w B x, Q T, 5 5b 5c 5d nd, for n idel gs h y p c s c. b p The sources of forces B nd of het Q include ll the dissiption terms, nd expressions for these re given in the following. The x xis is chosen to be ligned with the men velocity. Grdients of v,, s, nd p in the x nd z directions re considered negligible in comprison with their grdients in the y direction norml to the wll ssuming fully developed sttionry flow. Furthermore, the sttionry pressure is ssumed constnt in the y direction (dp /). This lst ssumption leds to ds c p d c p dt T, 3 nd to d( /c )/. For simplicity the sound is ssumed to propgte only in the x direction n extension to propgtion in both x nd z directions is strightforwrd. Then, by introducing dimensionless quntities, the vribles my be written s follows: xxc /, yyc /, tt/, c c c y, u c u y, uc uye, v, vc vye, p c p, p c pye, T T T y, TT TyE, y, s cs p y, FIG. 1. Schemtic description of the geometry. ye, sc p sye, the subscript refers to dimensionl properties in the core of the flow for instnce in the midline, is the frequency, c is the sound speed (c c p (1)T ), u nd v re the velocity components in the x nd y directions, nd Eexp(itikx)exp(itikx) kkc / is the wve number. Using expressions 4 with the bove ssumptions, Eqs. 1 re trnsformed to 4 pc s, 5e 1ku nd v/i is the cousticl displcement in the y direction, ( / w c ) 1/ is the dimensionless coustic boundry lyer thickness, w w is the density t the wll, nd is the nmic viscosity of the fluid. For simplicity the nmic viscosity nd the therml conductivity re ssumed to be constnt nd the bulk viscosity is ssumed to be equl to zero; then, the dissiption terms B x, B y nd Q re given by B x d u 1 dv ik k u, 6 B y 4 3 Q 1 d v 1 du ik 3 k v, d T k T 1 du du ikv, 6b 6c c p / is the Prndtl number; is the therml conductivity. By retining only the two vribles nd p, Eqs. 5 led to d 1 c k p w i Q T k B x, 7 dp c B y. 7b The propgtion equtions, Eqs. 7, must be pplied with boundry conditions t the wll. The most pproprite choice to express the boundry conditions would be to use the complince of the wll which links pressure nd norml displcement. But, the liner chrcteristics re more usully given in terms of the wll dmittnce. Thus, the boundry condition t the wll is written Y c v v p p, 8 Y is the specific coustic dmittnce of the wll. III. ASYMPTOTIC REPRESENTATION For simplicity, it is helpful to seprte the problem into two regions: 1 the core of the flow, the dissiption effects cn be neglected; thin lyer ner the wll, within which the viscous nd therml dissiption effects re confined. 6 J. Acoust. Soc. Am., Vol. 19, No. 1, Jnury 1 Aurégn et l.: Boundry conditions with flow 6

3 Thus, the problem is menble to symptotic nlysis. With the method of composite expnsions, 11 the solution for ny quntity q(y), qu,v,,p,t, is expressed s qq c (y)q b () with y/. The second term with subscript b representing boundry lyer terms ner the wll y tends to zero s. The grdients of men velocity nd temperture re ssumed to be non-negligible in the boundry lyer. Then, it is convenient to express them, respectively, s u M c (y)m b () nd T o c (y) b (), the terms with subscript b ccount for the significnt grdients ner the wll. The first terms inviscid terms in the core of the flow cn be obtined from Eqs. 5 without dissiption nd without lrge grdients ner the wll. In the core of the flow, Eqs. 7 led to the convected wve eqution for the outer pressure p c d c c c dp c 1 c ck c p c, 9 c 1kM c nd c c c. This eqution my lso be written 1 d p c 1 d c k dm c c c dp c c c k p c, 1 which is the clssicl Pridmore-Brown 1 eqution with temperture grdient. An effective dmittnce could be defined for the outer region by Y c v c p c, 11 while the reltionship between the norml velocity nd the norml pressure grdient given by Eq. 5c cn be simplified s i c v c dp c /. In the boundry lyer, tking the limit, Eqs. 7 reduce to d b d w i Q b k T B b, 1 dp b d, 1b B b d u b d, 13 nd Q b 1 d T b d 1 dm b du b d d. 13b The solution of Eq. 1b which tends to zero s is p b. Thus, the pressure is constnt cross the boundry lyer to the first order in. Integrtion of Eq. 1 leds to b w i b b Q b c b kb b c b d. 14 Without ny dissiption, the kinemtic condition in the cse of vnishing sttionry boundry thickness is continuity of coustic norml displcement. 9 Using the bove nottion, this mens tht b (). Eqution 14 shows tht this condition does not hold when there is dissiption, nd tht n dded norml displcement b () is introduced by the viscotherml effects. 13 The expression for the dded displcement my be simplified if w c () nd w c () b () is the wll temperture nd M M c () re smll compred to 1. To the first order in the prmeters M nd, the dded displcement my be written b i n 1 w dt b d k du b d. 15 The dded displcement b (), which needs to be included in the boundry conditions, is defined only in terms of the het flux qdt b /d nd the sher stress du b /d t the wll. How these re determined is shown in the next section. IV. DETERMINATION OF THE ADDED DISPLACEMENT The diffusion of momentum nd het hs to be determined in the boundry lyer to find the kinemtic condition which cn be pplied. The momentum eqution in the x direction Eq. 5b becomes, to the first order in M nd d u b d iu b iu c i dm b d cik p c. w 16 This eqution my be trnsformed into n expression for the sher stress du b /d which is involved in the dded displcement d d i df d, 17 f i dm b d cik p c. 18 w The boundry conditions ssocited with Eq. 17 re when nd u()u b ()u c (), which cn be trnsformed using Eq. 16 into d/d()f (). The sher wves described by Eq. 17 is excited in two wys. In the first wy, sher wve is excited by the coustics in the core of flow i.e., the second term in the definition of f (). This wve is the clssicl one found in nmic boundry lyers ner rigid wll. The second wy which corresponds to the first term in the definition of f () is very wek when the wll is rigid (). It corresponds to wves induced by chnges to the sttionry velocity nd by the norml displcement ner the wll. Eqution 17, subject to the boundry conditions, leds to 1 d 1 1 f d d, J. Acoust. Soc. Am., Vol. 19, No. 1, Jnury 1 Aurégn et l.: Boundry conditions with flow 61

4 1 exp(1i) is the solution of the homogeneous prt of Eq. 17 vnishing t infinity. Then, the sher stress t the wll is 1ik p c i M w eff c, dm M eff b exp1id b d is the effective men velocity involved in the dded displcement. This effective velocity my be seen 14 s n verge of the men velocity over the boundry lyer weighted by 1 nd cn be written M eff v M, M M c () with v 1. In the sme wy, the conservtion of energy Eq. 5d leds to vlue for the het flux qdt b /d t the wll 1 q 1i 1 p c i w eff c, 1 eff d b d exp1id 1b is the effective difference between men temperture nd wll temperture involved in the dded displcement nd cn be written eff t, w with t 1. The dded displcement my be written b 1i w km eff eff w 1 w k p c c, which leds to reltion giving the modified dmittnce 11 v km w t Y c Y 1i w k 1. 3 It cn be seen from Eq. 3 tht the effect of the clssicl sher nd therml wves induced by the coustics in the core of the flow which leds to the second term on the right-hnd side of 3 is wek of the order. Since Y is much lrger thn for typicl lined wll, this term is only importnt for hrd wll nd will be neglected in front of Y in wht follows. When v nd t, Eq. 3 is equivlent to the continuity of coustic norml displcement cross the boundry lyer: b () ory c Y /(1kM ). When v nd t 1, Eq. 3 is trnsformed in condition of conservtion of the norml mss velocity cross the boundry lyer: c ()v c () w v() or Y c w Y. This behvior is illustrted here for three simplified men velocity profiles with constnt temperture. The outer men velocity is tken s constnt, i.e., M c (y)m. The slope t the origin is the sme for ll three profiles: du /()M / is the sttionry boundry FIG.. Vrition of the effective velocity divided by the core velocity, v, s function of the rtio of the coustic nd sttionry boundry lyer thicknesses,, for three men velocity profiles. Solid line: liner; dshed line: qudrtic, nd dsh-dot line: exponentil. b Men velocity profiles. lyer thickness see Fig. b. The rtio of the coustic over the sttionry boundry lyer thickness is clled /. 1 For the first profile, the inner men velocity is liner M b M 1 M b for 1/, nd in this cse for 1/, v 1i 1exp 1i. 4 The second profile is qudrtic M b M 1/ for /, M b for /, nd in this cse v i exp 1i 1 1i. 5 3 The lst profile is exponentil, i.e., M b () M exp(), nd in this cse v /(1i). The rel nd imginry prts of v s function of re plotted in Fig. for the three profiles. When the coustic boundry lyer thickness,, is smll compred to the sttionry boundry lyer thickness i.e., 1, v goes to zero. In this cse, continuity of displcement cn be pplied cross the boundry lyer. On the other hnd, when 1, v goes to 1, which mens tht continuity of velocity is pplicble cross the boundry lyer. For given sttion- 6 J. Acoust. Soc. Am., Vol. 19, No. 1, Jnury 1 Aurégn et l.: Boundry conditions with flow 6

5 FIG. 3. Schemtic description of the rough wll geometry. ry boundry lyer thickness,, continuity of displcement pplies t high frequencies while continuity of mss velocity pplies t low frequencies. For given frequency, continuity of displcement pplies t low Mch number i.e., giving thick sttionry boundry lyer, while continuity of mss velocity pplies t high Mch number i.e., resulting in thin sttionry boundry lyer. These findings re in qulittive greement with the experimentl observtions of Ingrd nd Singhl. 15 It should be noted tht, when both coustic nd sttionry boundry lyer thicknesses re of the sme order, v nd t re complex, so they not only chnge the vlue but lso the chrcter of the dmittnce. It my be seen from Eq. tht the most importnt prt of the coustic sher stress comes from the trnsfer, by the norml fluctuting displcement, of xil momentum from the sttionry flow into the lined wll, 16 this effect being induced by viscosity. The prmeter v controls this trnsfer from v no trnsfer to v 1 full trnsfer. v cn be seen from Eq. b to be the rtio of the men velocity in the lyer the sher wve is significnt over the core men velocity. The sme resoning holds for the therml flux nd the prmeter t. 14 V. ADMITTANCE OF A ROUGH LINED WALL The men velocity profiles of turbulent flow over rough wll is schemticlly depicted in Fig. 3. Compred to smooth wll profile, the min difference is the slip velocity M 1 t the outer boundry of the equivlent roughness thickness. This slip velocity depends on the equivlent roughness nd on the core velocity. 17 Tking into ccount the bove nlysis of the viscous effects, the xil momentum linked to the slipping velocity M 1 is trnsferred into the roughness of the wll even if the coustic boundry lyer thickness is smll compred to. Then, the continuity of mss velocity must be pplied over distnce equl to the roughness of the wll. 13 This cn be written s 1 v()i(1km 1 ) 1 () w Yp(), 1 nd w re the density corresponding, respectively, to the y nd to the wll temperture. The origin of the coordintes y nd is tken t the outer boundry of the equivlent FIG. 4. Vrition of the effective velocity divided by the core velocity, v, s function of the rtio of the coustic nd sttionry boundry lyer thickness,, for rough wll with n exponentil men velocity profile. Solid line: rough wll; dshed line: smooth wll smesfig.. roughness thickness. Assuming tht the xil coustic velocity is equl to t y, Eq. 3 then becomes 11 vkm w t Y c Y, 6 v 1 M M 1 dm b d exp1id, 7 nd 1 d t 1 b d exp1id ; 7b 1 w 1 is the difference between the wll temperture nd the temperture t y. The effect of roughness is illustrted for the cse of n exponentil velocity profile with slip velocity, in the cse of constnt temperture. The sttionry velocity profile tkes the form M b ()(M M 1 )exp() nd in this cse v ((1i)M 1 /M )/(1i). The rel nd imginry prts of v s function of re plotted in Fig. 4 for M 1.5M. It cn be seen tht continuity of norml displcement ( v) is never ttined for rough wll solid line in Fig. 4. When the coustic boundry lyer thickness is smll compred to the sttionry boundry lyer thickness i.e., 1, v M 1 /M, nd the boundry condition is Y c Y /(1 km 1 ) insted of Y c Y /(1kM ) for the cse of smooth wll i.e., continuity of displcement. VI. CONCLUSIONS The effective coustic dmittnce of liner, tking ccount of viscotherml effects, cn be computed for the cse coustic nd sttionry boundry lyer thicknesses re smll compred to the wvelength. The min effect of viscosity is the trnsfer of xil momentum nd het flux of the sttionry flow into the lined wll. The effective dmittnce is given s function of two coefficients v nd t which minly depend on the rtio of the coustic nd sttionry 63 J. Acoust. Soc. Am., Vol. 19, No. 1, Jnury 1 Aurégn et l.: Boundry conditions with flow 63

6 boundry lyer thicknesses. When the coustic boundry lyer thickness is smll compred with the sttionry boundry lyer thickness, continuity of norml displcement pplies cross the boundry lyer. On the other hnd, when the coustic boundry lyer thickness is lrge compred with the sttionry boundry lyer thickness, it is continuity of mss velocity which pplies cross the boundry lyer. If the lined wll is rough, norml displcement continuity never pplies. In this pper, only moleculr diffusion effects described by the nmic viscosity nd the therml conductivity re tken into ccount. Further work is needed to include the turbulent diffusion effects which cn be incorported in complex effective viscosity nd which will depend both on the norml coordinte nd on frequency. 1 D. H. Tck nd R. F. Lmbert, Influence of sher flow on sound ttenution in lined duct, J. Acoust. Soc. Am. 38, W. Eversmn, Effect of boundry lyer on the trnsmission nd ttenution of sound in n cousticlly treted lined duct, J. Acoust. Soc. Am. 49, S. H. Ko, Sound ttenution in cousticlly lined circulr ducts in the presence of uniform flow nd sher flow, J. Sound Vib. 54, M. E. Goldstein nd E. Rice, Effect of sher on duct wll impednce, J. Sound Vib. 31, B. J. Tester, The propgtion nd ttenution of sound in lined ducts contining uniform or plug flow, J. Sound Vib. 8, B. J. Tester, Some spects of sound ttenution in lined ducts contining inviscid men flows with boundry lyers, J. Sound Vib. 8, R. S. Brnd nd R. T. Ngel, Reflection of sound by boundry lyer, J. Sound Vib. 851, R. T. Ngel nd R. S. Brnd, Boundry lyer effects on sound in circulr duct, J. Sound Vib. 851, A. H. Nyfeh, J. E. Kiser, nd D. P. Telionis, Acoustics of ircrft engine-duct systems, AIAA J. 13, A. H. Nyfeh, Effect of coustic boundry lyer on the wve propgtion in ducts, J. Acoust. Soc. Am. 54, A. H. Nyfeh, Perturbtion Methods Wiley, New York, 1973, Chp D. C. Pridmore-Brown, Sound propgtion in fluid flowing through n ttenuting duct, J. Fluid Mech. 4, R. Strobinski, Sound propgtion in lining duct with essentilly nonuniform distribution of velocity nd temperture, in Jet Engine Noise, Trnsctions of the Centrl Institute of Avition Engine CIAM, N 75, Moscow, 1978, pp in Russin. 14 R. Strobinski, Theory nd synthesis of the silencers for intke nd exhust systems of internl combustion engine, D. of Sc. thesis, Toglitti University, 1983 in Russin. 15 U. Ingrd nd V. Singhl, Upstrem nd downstrem sound rdition into moving fluid, J. Acoust. Soc. Am. 54, J. Rebel nd D. Ronneberger, The effect of sher stress on the propgtion nd scttering of sound in flow ducts, J. Sound Vib. 1583, H. Schlichting, Boundry Lyer Theory, 7th ed. McGrw-Hill, New York, J. Acoust. Soc. Am., Vol. 19, No. 1, Jnury 1 Aurégn et l.: Boundry conditions with flow 64

Consequently, the temperature must be the same at each point in the cross section at x. Let:

Consequently, the temperature must be the same at each point in the cross section at x. Let: HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the