716 J. Acoust. Soc. Am. 111 (2), February /2002/111(2)/716/13/$ Acoustical Society of America

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1 Refracted arrival wave in a zone of ilence from a finite thickne miing layer Takao Suzuki a) and Sanjiva K. Lele Department of Aeronautic and Atronautic, Stanford Univerity, Stanford, California Received 5 July 000; accepted for publication 3 October 001 Refracted arrival wave which propagate in the zone of ilence of a finite thickne miing layer are analyzed uing geometrical acoutic in two dimenion. Here, two implifying aumption are made: i the mean flow field i tranverely heared, and ii the mean velocity and temperature profile approach the free-tream condition eponentially. Under thee aumption, ray trajectorie are analytically olved, and a formula for acoutic preure amplitude in the far field i derived in the high-frequency limit. Thi formula i compared with the eiting theory baed on a vorte heet correponding to the low-frequency limit. The analyi cover the dependence on the Mach number a well a on the temperature ratio. The reult how that both limit have ome qualitative imilaritie, but the amplitude in the zone of ilence at high frequencie i proportional to 1/, while that at low frequencie i proportional to 3/, being the angular frequency of the ource. 00 Acoutical Society of America. DOI: / PACS number: 43.8.Py, 43.8.Ra MSH I. INTRODUCTION Suppoe an acoutic ource i located in lower medium, but adjacent to a fater medium. The lower or fater medium refer to the medium whoe propagation peed i lower or fater than for the other ee Fig. 1. In uch a cae, there may eit a path arriving at the oberver located in the lower medium which take horter time than direct wave; namely, once the ray arrive at the urface of the fater medium, propagate along it, and depart from it toward the oberver. A the Fermat principle indicate, under uch a condition, actual wave propagate along thi ray path, referred to a refracted arrival wave, 1 3 or ometime a head wave, 4 or lateral wave, 5 etc. The formula of refracted arrival wave can be derived uing a contour integral when the interface between the two media can be treated a dicontinuou. Thi cae i conidered to be a low-frequency limit in ene that the acoutic wavelength i much longer than the thickne of the interface. However, when the acoutic wavelength become much horter than the thickne of the interface, the low-frequency formula tend to underpredict the amplitude of refracted arrival wave. Intead, one hould rather analyze thee wave baed on geometrical acoutic, namely, the high-frequency limit. Such ditinction could be important in jet-noie problem ee Fig. a decribed in thi paper. When a noie ource i located right below or even inide a miing layer, there eit a region in which direct wave from the ource cannot reach, referred to a the zone of ilence. Intead of direct wave, econdary wave occupy thi region. In two dimenion, thee wave are particularly epreed in the form of general plane wave. Thee wave are generated by diturbance of direct wave on the other ide of the miing layer. The formula of uch wave wa derived by Gottlieb 1 uing a contour integral auming a vorte heet; however, the thickne of the miing layer, in many realitic cae, can be equivalent to or longer than the acoutic wavelength of the ound radiated from jet. 6 In other word, the high-frequency ound in the zone of ilence hould not be etimated uing a dicontinuou interface model. The purpoe of thi paper i to clarify the difference between the low- and high-frequency limit of refracted arrival wave from a tranverely heared miing layer. By auming that the thickne of the miing layer i finite and the velocity and temperature profile approach the uniform free-tream condition eponentially, ray trajectorie are aymptotically olved. Furthermore, a formula for preure amplitude can be eplicitly derived a a far-field aymptote uing the Blokhintzev invariant 7 in the high-frequency limit. Unlike direct wave, refracted arrival wave cannot be derived uing tationary-phae-type method. 8 Note that at high frequencie, intability wave inherent in a miing layer do not directly influence the noie radiation. In fact, dominant high-frequency noie ource eit relatively cloe to the jet eit in which the vortical diturbance level i relatively low. 9 Hence, the olution of refracted arrival wave from a finite thickne miing layer hould contain the baic mechanim of the high-frequency ound in the zone of ilence for jet-noie problem. Thi tudy alo indicate that the amplitude i fairly enitive to the preading rate of the miing layer in reality. The comparion between the low- and high-frequency limit demontrate that a the frequency increae, the amplitude of refracted arrival wave tend to become larger than the prediction baed on a vorte heet. The key difference i that a the frequency varie, the low-frequency formula behave a () 3/, while the high-frequency formula behave a 1/ 1 3/. Here, denote the angular frequency of the ource, denote the eponential factor of either the velocity or temperature profile, which only dea Current addre: Diviion of Engineering and Applied Science, California Intitute of Technology, Paadena, California J. Acout. Soc. Am. 111 (), February /00/111()/716/13/$ Acoutical Society of America Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

2 FIG. 1. Schematic of the path of a refracted wave and a direct wave. pend on the flow geometry, and denote the ditance from the ource in the flow direction. In addition, the dependence on the Mach number and the temperature ratio a well a the ource location i numerically invetigated baed on geometrical acoutic, and it i compared with the analytical epreion. It i oberved that the analytical epreion approimate the amplitude fairly well even if the ource i located inide the miing layer. Through thi tudy, the ound radiation in the zone of ilence at high frequencie can be undertood in the contet of jet-noie problem. The outline of thi paper i a follow: After the Introduction, the formula of refracted arrival wave in the highfrequency limit are derived, and thoe in the low-frequency limit are alo reviited. In Sec. III, numerical procedure of geometrical acoutic are decribed. Net, the analytical formula and the numerical reult are compared in Sec. IV; the concluion are preented in Sec. V. II. DERIVATION Conider a two-dimenional, tranverely heared miing layer. Take to be the flow direction and y to be the vertical direction, and et M and a to be the free-tream Mach number and the peed of ound on the lower ide, and M and a to be thoe on the upper ide, repectively ee Fig. 3 for chematic. In thi paper, the ubcript denote the lower ide, and the upper ide. Aume that the Mach number and temperature profile do not change in the direction, and the mean preure i contant everywhere note that the effect of miing layer preading are dicued at the FIG. 3. Schematic of the coordinate ytem of a two-dimenional miing layer. end of Sec. IV. Set a monopole ingle-frequency ource at (,y)(0,) without lo of generality. To olve acoutic field in tranverely heared flow of thi type, the thirdorder convective wave equation, called Lilley equation, 10 i adequate. The homogeneou equation can be epreed a follow: D Dt D Dt j a j u k j k a j 0, where D/Dt(/t)u 1 (/ 1 ) and 1 log(p/p ), p being the contant mean preure, and the pecific heat ratio. Furthermore, aume that 1 i nondimenionalized by taking the vorticity thickne to the length cale and the peed of ound at y, a, to be the velocity cale; therefore, u i denote the local Mach number time the local peed of ound, and a denote the local temperature. Baed on 1, the amplitude of diffracted wave in the zone of ilence, referred to a refracted arrival wave, i analytically formulated in both low-and high-frequency limit in thi ection. A. High-frequency limit finite thickne model When the acoutic wavelength i much horter than the characteritic length cale of the medium, in the preent cae the vorticity thickne, one can aume the high-frequency limit and apply geometrical acoutic. 4,7 Aume the acoutic preure fluctuation to be the following form: 1 t,e it Pepi. Subtitute into 1, and aymptotically epand it with repect to. By taking the leading term of, one can obtain the eikonal equation 1u j j a j 0, 3 where i / i, which correpond to the local wavenumber vector. By uing the method of characteritic, 4 one can reduce 3 to the following O.D.E. ytem: FIG.. Schematic of the noie from a jet. d i dt a 1u k i u i, k 4 J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave 717 Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

3 d i dt u k k 1u k k a, i a i d dt Notice that the phae ha the ame unit actually ame cale a time. Likewie, by taking the econd-highet term of, one can derive the firt-order tranport equation u j P j 1u k k a P j j a P j k 1u l l P 3 u j u k jk j u k u j k a jj P j a j u k j j u k a a j u k jk u l l k 1u l l Here, again P i P/ i. To implify 7, ue the following relation obtained by differentiating 3 by D/Dt: u j u j u k jk j u k k a j k u k 1u l l j a j j j u k 1u l l a k a4 j k jk 1u k k 0. Subtituting 8 into 7, uing 4, and auming the mean preure i contant everywhere hence, (1/)(D/Dt) (1/a )(Da /Dt)0), 7 can be implified a follow: P j d j 1u k k dt 0. 9 Hence, the quantity called the Blokhintzev invariant 7 i conerved along ray tube P S 1u k k d dtcont., 8 10 where S denote the cro ection of the ray tube normal to the ray direction. In the denominator, the mean preure, which i aumed to be contant, diappear compared with the general epreion of the Blokhintzev invariant. Thi epreion will be ued later to calculate the amplitude of refracted arrival wave. Now, when the mean velocity and temperature profile are purely tranverely heared, the O.D.E. ytem 4 6 can be implified: d /dt0 in5. In other word, i contant along the ray. Accordingly, they can be rewritten a follow: d dt dy dt d 0, dt a y 1My My, a y 1My y, FIG. 4. Ray trajectorie from a point ource above a miing layer. The ource i located at (,y)(0,), the temperature i contant everywhere, and the velocity profile i given by M(y)0.81tanh(y)/ hown on the left-hand ide. The dahed line are drawn every 3 in all direction, and the olid line are drawn by the interval of 0.3 near the limiting ray. d y dm dt dy 1My da a y dy, 14 d dt Again, M(y) denote the velocity profile not the Mach number whoe reference peed i a a(). For convenience, conider the cae in which 0, and d y /dt0 along the ray in 14, uch a ray propagating downtream above a hot jet. Among thee ray, if the initial grazing angle i lower than a certain threhold value (, where will be defined later, thi ray propagate into the lower ide and never appear on the upper ide, called a tranmitted wave in thi paper ee Fig. 4. In contrat, if, thi ray propagate on the upper ide. In particular, when the ray whoe grazing angle are only lightly higher than thi threhold, they appear a refracted arrival wave departing from the miing layer to the upper ide at nearly the ame angle. The ray whoe initial grazing angle i eactly i called a limiting ray. A Fig. 4 how, the turning point of the refracted arrival ray, at which the ray become parallel to the miing layer, are fairly cloe to the lower free-tream region when the ray are propagating far downtream. Accordingly, thee ray propagate horizontally jut beneath the miing layer for long ditance. To olve thee ray trajectorie, aume that the velocity and temperature profile approach the lower free-tream condition eponentially. In other word, the velocity and the temperature profile near the turning point can be approimated by MyM Me 1 y, 16 a y1a e y, 17 a y, where 1, 0, and M and a are ome contant determined from the flow field. In many real phyical flow, 1 and can be common near the free-tream 718 J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

4 region e.g., the Crocco Buemann relation: Tu 1 /C p C 1 u 1 C, where C 1 and C are contant. Hence, et 1. If 1, jut retain the term whoe i maller. Thi model hould cover realitic flow; however, if the profile do not follow the formula 16 and 17, etenion of the preent method i required, uch a curve fitting. Nonethele, the proportionality of the frequency and the ditance from the ource hould how imilar feature a dicued at the end of thi ection. Subtitute 16 and 17 into the O.D.E. ytem 11 15, and take the leading-order term auming Me y and a e y to be mall. Conequently, one can implify them a follow: d dt 1 1M M, dy dt 1 1M y, d 0, dt d y dt d dt M 1M a e y, 1 Notice that at the leading order, d/dt become contant from 18. Differentiating 19 with repect to t, and ubtituting 1 into it, yield d y dt 1 d y 1M dt M 1M a e y Ae y, 3 where A M/(1M )(a /), which i contant and aumed to be non-negative along the ray. Redefining z e y and ubtituting it into 3, one can obtain the following O.D.E.: z d z dt dz dt Az To reduce 4 to an integrable form, convert the variable by etting 0 z and 1 ż. After calculating d 1 /d 0, 4 yield dż ż dz z Az. From 5, the general olution can be obtained a 5 ż Az z. 6 Here, i an arbitrary contant defined to be 0,) here for convenience alway atifying z 0. When one take the plu ign in 6, it correpond to a ray of a tranmitted wave. In contrat, with the minu ign, z at a certain point, correponding to a ray of a refracted arrival wave. Thi point i actually the turning point, which can be epreed a y* log. 7 In thi paper, the upercript * denote the quantity at the turning point. Note that if 0 in6, it correpond to the limiting ray. Firt, to olve the ray trajectorie of refracted arrival wave, take the minu ign in 6, convert the variable by etting (z ) tan (0/), and integrate it. After ome calculation, it yield A tt*. 8 Here, t* denote the time when the ray pae through the turning point. Equation 8 indicate that the trajectory i ymmetric about the turning point. Rewriting 18 and 8 in the phyical domain, one can obtain the ray trajectory near the turning point a follow: *Btt*, 9 y log. 30 co A/tt* where BM (1M ) /(1M ). Combining 9 and 30, it can be rewritten by y log coc*, 31 where C A/B. Thi equation will be ued later to derive the amplitude of refracted arrival wave. Second, pecial olution, the limiting ray, can be obtained by etting 0 in6. By directly integrating 6, one can obtain y log 1 A/tt 0 log 1 C 0. 3 Here, 0 denote a certain reference point. Finally, to olve the ray of tranmitted wave, take the plu ign in 6, and et z tan (0/). After integrating the equation, one can obtain y 1 log 1co co log inhc Net, conider the trajectory of the turning point for refracted arrival wave propagating far downtream ee Fig. 5. The initial grazing angle of thee ray are lightly higher than the angle of the limiting ray; hence, the location where thee ray enter the miing layer are approimately the ame. Here, thee location are called the incident point, denoted by in in thi paper. On the other hand, due to the light difference of the initial angle, the ditance from the incident point to the turning point are quite different; accordingly, the location at which the ray depart from the miing layer are alo different. Thee location are called the departing point, denoted by out. Recalling the ray trajectorie are ymmetric about the turning point from 8, the J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave 719 Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

5 B 1M A dc * 4C 3 d in d FIG. 5. Eample of ray trajectorie of refracted arrival wave. The velocity profile i depicted on the left-hand ide (M(y)0.81tanh(y)/), and the temperature i contant everywhere. The ource i located at (,y) (0,) ame a Fig. 4. The initial angle of the ray are with the interval of Solid dot denote the turning point. ditance from the incident point to the departing point i twice that to the turning point. Now, for the ray propagating far downtream, typically y* become a relatively large value; hence, e y*/ tend to be a fairly mall value. For eample, when y*5, On the other hand, near the incident point, y in 0; accordingly, /coc(* in )1 from 31. Therefore, ince 1, coc(* in )1 mut be atified; namely C* in. 34 Hence, uing 7 the trajectory of the turning point can be approimated by y* log C* in. 35 Now, the preure amplitude of refracted arrival wave i derived uing the Blokhintzev invariant 10. By calculating the departing point of adjacent ray, the amplitude can be approimately olved. Firt, uing 35, calculate the turning point of adjacent ray. Knowing that C i a function of, differentiate 35 a follow: * in dy*d*e y*/ C dc d d. 36 On the other hand, the relation between y* and can be obtained from the eikonal equation 3. Knowing that y 0 give the turning point, differentiate 3 and implify it a follow: 1M Ae y* dy*bd 0. Furthermore, from 3, for the limiting ray i given by 1 1M Here, the upercript denote the quantity of the limiting ray. Subtituting 35 and 37 into 36 yield 4C * in 3 d*0. 39 Here, the econd term in become negligible far downtream a * in ). Thu, the intenity i proportional to I d d* 1M A BC 3 * in * in On the other hand, near the ource the cro-ection area can be calculated from the difference of the initial grazing angle. From 11 and 1, calculate the change of the ray path with repect to near the ource d dt 1, n dy dt n n, 41 4 where the quantitie with the ubcript are evaluated at the ource point, and n (1M )/a. A een later, n behave a a refraction inde. Now, to apply the ray tube theory uing 10, it i convenient to calculate the following quantity: d/dt ds 1My d d dt, dy dt d dt, dy dtdt 1M dt n 3 n. 43 The ditance from the ource dr and the time dt i related a dr dt d dt dy a dt 1 M M 1/. n 44 Likewie, calculate the ame quantity at the departing point d/dt d ds dt, dy dt d out d,0 1My 1M d out n d out. 45 d n Here, 45 i evaluated in the uniform region right above the miing layer. Remember that in the upper free-tream region, the ray are almot parallel, and refracted arrival wave propagate in the form of general plane wave. Now, the olution cloe to a monopole ource can be written a 70 J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

6 M co 1 epi M r 1 1 M /a 1/ r 1/ 3/ 1 M in in r 3 4 1/4, 46 where r co and yr in. In particular, the initial grazing angle for refracted arrival wave i given by dy tan dt d dt n 1 M M, 47 where i approimated by 38. The epreion 46 can be obtained from A11 hown later by auming that the flow field near the ource i uniform, and taking the limit of a far-field aymptote, r. Combining 40 and 43 46, the amplitude of refracted arrival wave from a finite thickne miing layer i approimated by d/dt ds 1M,,y dr 1 d/dt 1M d d* 1/ dr dt 1/ M 1/ 3/ 1 1/ n d ds d out in 1/4 1/ n 3/ 4 n 4 n n n 3/ in 4 n 1 M 1 M n, 48 1/4 X 3/ where X( n / )y. Here, ince the ray trajectory i ymmetric about the turning point, it i aumed that * in out in. Note when the ource i at a large ditance from the miing layer (1), the correction for the ditance from the ource to the incident point need to be included ee Eq. 59 hown later. On the other hand, when the ource approache the lower free-tream region, e.g., 1, the correponding incident point are no longer identical for the ray of refracted arrival wave, and the approimation fail. Thi epreion 48 will be compared with the epreion baed on a vorte heet a well a the numerical reult. B. Low-frequency limit vorte heet model When the acoutic wavelength i much longer than the vorticity thickne, a vorte heet can be ued, which correpond to the low frequency limit. Refracted arrival wave of thi type have been reported in everal tudie. 1 3 In thi ection, the reultant formula of refracted arrival wave in the low-frequency limit are hown in two cae the ource i located above and below the vorte heet. For their derivation, pleae refer to the Appendi. When the ource i located above the miing layer ( 0), the abolute value of preure amplitude yield 1 n,,y 3/ a n n X. 49 3/ Here the notation i the ame a 48, and thi epreion i valid only in the zone of ilence on the upper ide. Likewie, when the ource i located below the miing layer (0) 1 4 n,,y 3/ 4 n n X. 3/ 50 Note that the epreion 50 give larger amplitude than the epreion 49, a hown later. Here, one can ee that the decay rate of 49 or 50 for a vorte heet and that of 48 for a finite thickne miing layer are common (X 3/ (n )y/ 3/ ). However, their coefficient are different. It i important to notice that a the frequency varie, 49 and 50 are proportional to 3/, while 48 i proportional to 1/ with a fied. In other word, a the frequency increae with the flow geometry fied, the amplitude i guaranteed to eceed the prediction baed on the vorte heet model. Thi proportionality i till valid for the finite thickne miing layer with the velocity and temperature profile other than e y. Remember that the ray trajectorie are independent of the ource frequency o long a the frequency i conidered high enough; hence, the only part in which the frequency dependence appear i the amplitude epreion near the ource 46. Thee theoretical epreion 48, 49, and 50, are compared with the numerical reult in Sec. IV later. J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave 71 Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

7 III. NUMERICAL SIMULATION To compare the analytical formula with more accurate olution, preure amplitude of refracted arrival wave i numerically olved baed on geometrical acoutic. The procedure are to imply integrate the eikonal equation and to apply the ray-tube theory, which are decribed in thi ection. To olve ray trajectorie, the O.D.E. ytem of the eikonal equation, 11 15, wa numerically integrated uing the tandard fourth-order Runge Kuttcheme. The initial condition are a follow: 00, 51 y0, 5 co i 0, M co i in i y 0, M co i 00, where the initial grazing angle i given by tan in i /(M co i ). For implicity, the velocity profile wa et to be My M 1tanhy M Thi formula provide M(y) M M e 4y a y, which i conitent with 16 (M 0., MM, and 4.). In addition, thi velocity profile yield the vorticity thickne of M/(dM/dy) ma 1. Similarly, the temperature profile wa et to be a y 1a 1tanhya. 57 It alo yield a (y) 1(1a )e 4y a y. If a 1, the flow correpond to a hot jet, while if a 1, it correpond to a cold jet. In thi cae, M mut be reaonably large o that A i alway non-negative. See Eq. 3. Once the ray trajectorie were computed, the Blokhintzev invariant 10 wa ued to obtain preure amplitude by calculating cro ection between adjacent ray. Defining ( n,y n ) to be a certain grid point of the nth ray, the infiniteimal cro ection of the nth ray wa computed by the following midpoint rule: n1 n1,y n1 y n1 d dt, dy dtn ds n d dt dy, 58 n dtn where d/dt and dy/dt were given by 11 and 1, repectively. A total of 100 ray wa iued with the interval of i from the angle of the limiting ray. The time tep wa taken to be dt0.05 (/a ). The ratio of the infiniteimal cro ection at the grid cloet from the ource to that at the grid right above y almot uniform flow wa ued to calculate amplitude. In addition, the amplitude near the ource point wa calculated uing 46, which i FIG. 6. Turning-point trajectorie for different ource location. The lower free-tream velocity i M 0.8, and the temperature i contant everywhere. Symbol were computed by numerical integration:, ;, 1;, 0; *, 0.5; and, 1. Line were calculated uing 35 correponding to, 1, 0,0.5, and 1 from the top. conitent with the analytic epreion. Thu, preure amplitude of refracted arrival wave wa numerically calculated baed on the ray-tube theory. IV. RESULTS AND DISCUSSION A. Turning point trajectory Firt, to oberve the accuracy of the analytical epreion, turning-point trajectorie were calculated uing both analytical epreion 35 and numerical integration 11 15, and the reult are compared. Here, the incident location in 35 were approimated by the following form: in tan, 59 where /,0) i defined by 47. Figure 6 repreent the dependence of the turning-point trajectorie on the ource location. It how that a the ource location become lower cloer to the higher velocity ide, the trajectorie hift downward. When 0.5, the theoretical prediction agree with the numerical olution fairly well. But, when the ource location approache the lower free-tream (1. cae, the theoretical prediction deviate far lower than the numerical olution. Remember that the formula 35 aume the incident point of the ray to be identical; hence, when the ource approache the lower freetream region, thi epreion tend to fail. Nonethele, the analytical epreion approimate the ray trajectorie fairly well when the ource i above or cloe to the center line of the miing layer. Figure 7 repreent the dependence of the ray trajectorie on the lower free-tream velocity. Although M actually yield the Mach number of the lower free-tream, the term free-tream velocity i ued intead of free-tream Mach number to emphaize that M(y) denote the velocity normalized by a a oppoed to the local Mach number. Thi figure indicate that the analytical epreion cover a 7 J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

8 FIG. 7. Turning-point trajectorie for different lower free-tream velocitie. The ource location i, and the temperature i contant everywhere. Symbol were computed by numerical integration:, M 0.3;, M 0.8; and, M 1.5. Line were calculated uing 35 correponding to M 0.3, 0.8, and 1.5 from the top. Two cae (M 0.8 and 1.5 almot overlap. wide velocity range. Thu, one can epect that the analytic epreion provide a good approimation to the ray trajectorie a long a the ource i above or cloe to the center line of the miing layer. B. Preure amplitude ditribution To validate the analytical epreion for variou lower free-tream velocitie and ource location, preure amplitude of refracted arrival wave wa calculated uing 48, and it i compared with the numerical integration uing 10 and Figure 8a c repreent the amplitude profile in the direction for different lower free-tream velocitie and ource location. A een in Fig. 7, the theoretical prediction and the numerical reult agree very well when the ource i 0.5. Hence, one can epect that thi epreion can be ued to etimate the noie generated inide the miing layer and propagating in the zone of ilence. Since the analytical formula 48 aume a far-field aymptote, the theory and the numerical reult agree better a X increae in all cae. Each figure how that a the ource approache the lower free tream ( decreae, the amplitude increae; in particular, when the ource i below y0, the amplitude eem to be fairly enitive to the ource location. Thi erie of figure alo indicate that the noie from nearly the lower free tream tend to be trongly amplified a the velocity increae. C. Comparion between the finite thickne miing layer model and the vorte heet model FIG. 8. Comparion of preure amplitude between the finite thickne miing layer model and the numerical integration. The amplitude at y are plotted. The lower free-tream velocity i a M 0.3; b M 0.8; and c M 1.5. The temperature i contant everywhere. Symbol were computed by numerical integration:, ;, 0; and *, 0.5. Line were calculated uing 48 correponding to, 0, and 0.5 from the bottom. Net, two analytical model, the finite thickne miing layer and the vorte heet model, are compared. Remember that the finite thickne miing layer model 48 correpond to the high-frequency limit, while the vorte heet model 49 and 50, correpond to the low-frequency limit. Figure 9a c repreent the comparion of thee model at different lower free-tream velocitie. Both model how that the amplitude increae a the ource approache the lower free tream. Moreover, all thee cae how that a the frequency increae, the amplitude of the finite thickne model eceed that of both vorte heet model, a mentioned before. J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave 73 Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

9 FIG. 10. Comparion between the finite thickne miing layer model and the vorte heet model in different peed of ound. The peed of ound on the upper ide i a a 0.7 and b a 1.. The lower free-tream velocity i M 0.3. The ret of the condition and notation are the ame a Fig. 9. FIG. 9. Comparion between the finite thickne miing layer model and the vorte heet model in different lower free-tream velocitie. The preure amplitude above the miing layer are plotted. The lower free-tream velocity i a M 0.3; b M 0.8, and c M 1.5. The temperature i contant, and X0. Line repreent a follow:, finite thickne model 48 with ; - - -, that with 0;, vorte heet model with the ource above the miing layer 49; and, the ource below the miing layer 50. Thi tendency i particularly triking in lower ubonic flow. Figure 9a clearly demontrate that the vorte heet model far underetimate the amplitude of refracted arrival wave in a wide range on the higher-frequency ide. Remember that a the frequency increae, the finite thickne model decay a 1/, while the vorte heet model a 3/. Figure 10a and b how the comparion of thee model at different peed of ound. Here, a repreent a hot jet and b a cold jet, and the contant temperature cae correpond to Fig. 9a. They how that in the vorte heet model, the temperature variation hardly affect the preure amplitude. In contrat, in the finite thickne model, the amplitude trongly increae a the ource approache the lower free tream in cold jet; however, it barely change in hot jet. Notice that due to the definition of the ource term (ˆ n k ˆ (y)/, refer to A1 hown in the Appendi, the amplitude may even decreae a the ource approache the core of hot jet though the ditance between the adjacent ray become narrower. Thee tendencie will be ummarized in the net figure. Finally, to oberve the dependence on the lower freetream velocity and the peed of ound, the amplitude of refracted arrival wave wa mapped onto the M and a plane. Figure 11a c repreent the preure amplitude contour calculated by the finite thickne model 48. The region where contour line are miing the left top corner indicate that refracted arrival wave do not or barely eit under uch condition: n approache zero in 48. Hence, there are no ray which initially propagate downward and get refracted upward. Thi erie of figure demontrate the feature oberved in Fig. 8a c: A the ource approache the lower free tream, the amplitude tend to increae over the whole range; particularly, thi tendency become triking when the ource i below the center line of the miing layer (0.5). Figure 11 alo reveal that the amplitude become more enitive to the lower free-tream velocity a decreae. They alo how that a the jet become hotter (a decreae, the amplitude increae in the uperonic range (M 1) in all cae. On the other hand, in low ubonic flow the amplitude become fairly large when the jet become colder (a increae. In thi region, the critical angle defined by 47 become coniderably hallow o that wide angle of the ray are captured within the miing layer, and the ditinction between direct wave and refracted arrival wave become ambiguou. For reference, Fig. 1 repreent the amplitude contour calculated uing the vorte heet model for the ource a above and b below the miing layer, repectively. Note that the direct comparion of the magnitude between Fig. 11 and 1 may not be meaningful, ince the amplitude ratio between them depend on the ratio of /. Figure 11b and 1a a well a Fig. 11c and 1b how ome qualitative imilaritie. However, when the ource i below the miing layer, the vorte heet model indicate that the amplitude ubtantially increae a the velocity increae. It i important to notice that the olution for refracted 74 J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

10 FIG. 1. Preure amplitude contour in the plane of the lower free-tream velocity and the peed of ound baed on the vorte heet model. The condition and notation are the ame a Fig. 11. The ource location i a above the miing layer 49, and b below the miing layer 50. FIG. 11. Preure amplitude contour in the plane of the lower free-tream velocity and the peed of ound baed on the finite thickne miing layer model. The preure amplitude of the finite thickne model 48 are hown. /4 and X0. The ource i et to be a ; b 0; and c 0.5. The thicker dahed line denote the ioenthalpy line. arrival wave belong to the ame family a Mach wave-type ound. 11 From a one-dimenional point of view, namely the linear analyi baed on A1, thi family atifie the boundary condition of eponential decay toward the lower ide high peed and ocillation toward the upper ide low peed, and change it nature at the turning point. In thi ene, refracted arrival wave from a finite thickne miing layer have imilar feature of Mach wave tudied in ome previou work However, it i worthwhile to oberve that uperonic phae velocity can be obtained not only by ource uperonically convected: Wave iued from an uptream ource and refracted near the lower free tream have phae velocity of u a where u denote the jet velocity; hence, they can propagate in the zone of ilence although the intenity of refracted arrival wave tend to be fairly mall, a Fig. 11 and 1 indicate. It hould alo be emphaized that the preent analyi i baed on a parallel miing layer. Of coure, when the jet i preading, 14 the turning point hift cloer to the core; a a reult, refracted arrival wave become more like direct wave and their amplitude i enhanced refer to Ref. 15 for calculation in a more realitic flow geometry. Hence, the high-frequency ound in the zone of ilence meaured in eperiment might be caued mainly by direct wave from the end of the potential core. 6 To etimate the miing layer preading effect on the ound radiation field, the preure amplitude wa numerically calculated at different preading rate. The velocity profile wa et to be M,y M y 1tanh 60 1, J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave 75 Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

11 FIG. 13. Comparion of preure amplitude profile of different preading rate. The lower free-tream velocity i M 0.8, the ource location i 0, and the temperature i contant everywhere. Symbol were computed by numerical integration:, 0;, 0.05; *, 0.10; and, 0.0. A olid line wa calculated uing the finite thickne miing layer model 48. where the free-tream velocity wa et to be M 0.8 and the temperature to be contant everywhere. Here, the ource wa placed at the origin. The reult are plotted in Fig. 13 and compared with the parallel miing layer model at high frequencie. Figure 13 clearly how that even 0.1 of the preading rate yield everal time a large preure amplitude a that in the parallel miing layer cae. Note that previou eperimental and numerical tudie ummarized in Ref. 16 have indicated that the preading rate can be up to 0. a the jet Mach number decreae. Therefore, to compare thee theoretical epreion with actual eperiment, one need the information about the mean flow a well a rigorou ource model and their ditribution. Furthermore, in term of the frequency range, the mot untable mode of intability wave i likely to be omewhere in between the low- and high-frequency limit. The highfrequency formula derived here focue on noie due to rather finer cale turbulence. A oberved by eperiment, 9 the high-frequency component of the jet noie i mainly generated near the nozzle lip, in which large-cale vortical diturbance have not yet ignificantly grown. In addition, a numerical tudy by Suzuki 17 how that thi high-frequency formula for refracted arrival wave i applicable when the ratio of the acoutic wavelength to the vorticity thickne become unity or le. Therefore, the analyi in thi tudy i epected to be ueful for noie generated near the jet eit. However, once intability wave have developed into largecale vortical tructure downtream, uch a at the end of the potential core, the current analyi would no longer be valid. Other iue aociated with jet noie, uch a the multipole and moving ource, are alo dicued in Ref. 17. V. CONCLUSION Through thi tudy, refracted arrival wave propagating in the zone of ilence are formulated in the high-frequency limit and compared with the formula in the low-frequency limit. Thee formula how that the amplitude at high frequencie i proportional to 1/, while that at low frequencie i proportional to 3/, being the ource angular frequency. Thi indicate that the eiting low-frequency formula,,3 namely the vorte heet model, tend to underetimate the ound-preure level in the zone of ilence a the frequency increae. It alo implie that the previou highfrequency theory 8 ignoring refracted arrival wave doe not correctly repreent the ound radiation pattern in the zone of ilence. Thi high-frequency formula of refracted arrival wave ha ignificant implication for the application to jet noie. In mot previou tudie, general plane-wave-type radiation ha been conidered only for uperonic jet flow referred to a Mach wave. However, thi tudy indicate that even in ubonic miing layer, general plane-wave-type ound can theoretically propagate in the zone of ilence. In particular, thi formula i epected to be applicable to the highfrequency noie mainly generated near the nozzle lip of the jet eit. Therefore, it hould be ued to etimate the oundpreure level in the zone of ilence at high frequencie. However, at preent the eitence of uch wave in real flow i uncertain, and a noted below, ome etenion of the preent theory are required for quantitative prediction. For eample, one may need to more rigorouly analyze ome additional effect of real flow: A the miing layer pread, more ray are trapped inide of it and the ditinction between refracted arrival wave and direct wave become ambiguou. In fact, thi tudy how that light increae in the preading rate of the miing layer dratically enhance the amplitude of refracted arrival wave. Other eample which are not tudied here are effect of unteady flow diturbance, ource model for turbulent miing noie, and o on. Nonethele, it i important to note thi tudy demontrate that ound radiation pattern in the zone of ilence i fundamentally different from the region in which direct wave propagate and the amplitude of refracted arrival wave i different everal time over between the low- and highfrequency limit. ACKNOWLEDGMENTS The author would like to thank Profeor Brian J. Cantwell and Profeor Joeph B. Keller for many ueful uggetion. We gratefully acknowledge the financial upport by NASA Ame Reearch Center Grant No. NAG APPENDIX: Derivation for the low-frequency limit To rederive the formula of refracted arrival wave in the low-frequency limit, derivative matching 4 i ued here. Firt, take a Fourier tranform of 1 in time and the flow direction, and et a delta function at y, a ˆ y y y, where k M y ˆ kma y km a k ˆ A1 76 J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

12 1 ˆ,k,y t,,ye i(tk) dt d. A When the wavelength i much longer than the vorticity thickne, the third term of A1 become much maller than the firt two term. Therefore, in the low-frequency limit A1 can be approimated by a y y ˆ kmy y y kmy. A3 Hence, the quantity a (y)/(km(y)) ˆ /y i contant acro the vorte heet. Subequently, auming ˆ /y i finite, it can be hown that ˆ i continuou acro the vorte heet. Although the preure itelf i continuou acro the vorte heet, the firt derivative ha a dicontinuity. Thu, the jump condition acro the vorte heet become ˆ y0 ˆ y0, 1 ˆ n y 1 ˆ, n y0 y y0 A4 A5 where n (km )/a, which i equivalent to n n i defined after 41 and 4. Likewie, k i equal to ued in the high-frequency limit. To derive the formula for refracted arrival wave, firt put the ource above the vorte heet (0). In the tranvere direction, incident and reflected wave propagate on the upper ide, and tranmitted wave on the lower ide. Knowing that the econd term of A1 vanihe in the uniform flow region, the form of the olution on the upper and lower ide can be epreed a follow: ˆ ein k (y) ia n k C re in k y, A6 ˆ C t e in k y. A7 Here, C r and C t (C) are the reflection and tranmiion coefficient, repectively. Note that the reonance mode 3,18 i not taken into account here for implicity. Subtituting A6 and A7 into A4 and A5, tranmitted wave can be obtained a follow: ˆ ia n e in k e in k y n k n n k n. A8 By taking an invere Fourier tranform of A8, the twodimenional formula can be derived a,r, i 1 e in k e i(k co n k in )r a n n k n n k n dk. A9 To evaluate A9, aume r and r1, and ue the tationary phae method. Defining the phae part to be (k) k co (n k )in, (k)0 give the tationary point, which become k co 1M M. A10 1M in A a reult, A9 can be approimated in the far field a follow: 1/,r, r 3/ r in k ein e im co 1M in /1M r3/4) 1M in 3/4 a n n k n n k n, A11 where k and n are evaluated at the tationary point A10. The olution for refracted arrival wave mut match with A11 acro the miing layer. To apply the derivative matching A5, differentiate A11 with repect to y and et y0,,ya n k k y e i(kn k y). A13 1/ e in y y0 3/ k e i(k3/4) a n k. A1 Similarly, differentiate A13 with repect to y and evaluate at y0 retaining the lowet order of namely, the A term i eliminated Here, the tationary point i k(0)/(1m ). On the other hand, refracted arrival wave on the upper ide hould be epreed in the form of general plane wave; hence, they can be written by in y y0 k A e ik. A14 Subtituting A1 and A14 into A5, it yield a follow: J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave 77 Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51

13 ,,y 1/ n a n n k ein k e i(kn k y/4) n k k. 3/ y A15 One can obtain the ame reult by taking a contour integral of A9. By uing the ame notation a 48, the abolute value of A15 become 49. Likewie, put the ource below the vorte heet ( 0), and follow the ame procedure a decribed above. It i noticed that there eit incident wave and reflected wave on the lower ide, but only the reflected wave contribute the derivative matching. Conequently, the abolute value of preure amplitude for refracted arrival wave become 50 in thi cae. 1 P. Gottlieb, Sound ource near a velocity dicontinuity, J. Acout. Soc. Am. 3, A. B. Friedland and A. D. Pierce, Reflection of acoutic pule from table and intable interface between moving fluid, Phy. Fluid 16, M. S. Howe, Tranmiion of an acoutic pule through a plane vorte heet, J. Fluid Mech. 43, J. B. Keller and R. M. Lewi, Aymptotic method for partial differential equation: The reduced wave equation and Mawell equation, in Survey in Applied Mathematic, edited by J. B. Keller, D. W. McLaughlin, and G. C. Papanicolaou Plenum, New York, 1995, Vol. 1, pp L. D. Landau and E. M. Lifhitz, Fluid Mechanic, Tran., nd ed. Pergamon, New York, 1987, pp H. K. Tanna, An eperimental tudy of jet noie. I. Turbulent miing noie, J. Sound Vib. 503, D. I. Blokhintzev, Acoutic of a nonhomogeneou moving medium, Tran., NACA TM M. E. Goldtein, High frequency ound emiion from point multiple ource embedded in arbitrary tranverely heared mean flow, J. Sound Vib. 80, B. R. Dougherty, Phaed array beamforming for aeroacoutic, Lecture Note in an AIAA Profeional Development Short Coure, G. M. Lilley, On the noie from jet, AGARD, CP-13, C. K. W. Tam and D. E. Burton, Sound generated by intability wave of uperonic flow. I. Two-dimenional miing layer, J. Fluid Mech. 138, J. E. Ffowc William, The noie from turbulence convected at high peed, Philo. Tran. R. Soc. London, Ser. A 55A, T. F. Bala, The far field of high frequency convected ingularitie in heared flow, with an application to jet-noie prediction, J. Fluid Mech. 74, L. K. Schubert, Numerical tudy of ound refraction by a jet flow. I. Ray acoutic, J. Acout. Soc. Am. 51, C. K. W. Tam and L. Auriault, Mean flow refraction effect on ound radiated from localized ource in a jet, J. Fluid Mech. 370, J. B. Freund and S. K. Lele, Computer imulation and prediction of jet noie, to appear in High Speed Jet Flow, edited by G. Raman, D. K. McLaughlin, and P. J. Morri Taylor & Franci, London, T. Suzuki, Acoutic wave propagation in tranverely heared flow, Ph.D. diertation, Stanford Univerity, J. W. Mile, On the reflection of ound at an interface of relative motion, J. Acout. Soc. Am. 9, J. Acout. Soc. Am., Vol. 111, No., February 00 Takao Suzuki and Sanjiva Lele: Refracted arrival wave Reditribution ubject to ASA licene or copyright; ee Download to IP: On: Tue, 03 May :33:51