Wave Generation by Oscillating Wall in Static Media

Transcription

1 We Genetion by Oscillting Wll in Sttic Medi Hongbin Ju Deptment of Mthemtics Floid Stte Uniesity, Tllhssee, FL Plese send comments to: Sound, oticity we nd entopy we e the thee noml modes of line Eule equtions in unifom idel flows (cht1.doc: Wes in Unifom Flow on Hlf Plne). In this chpte we will futhe conside iscous effects on the we mode seption. It will be shown tht, when medium is sttic (no men flow) nd botopic (density depends only on pessue), nd petubtion is smll so tht lineity pplies, coustic nd oticl motions cn be unmbiguously septed. They e only coupled t boundies. If the boundy conditions e lso sepble, the two wes cn then be totlly decoupled. The continuity nd Nie-Stokes equtions with isentopic pocess e: u i + u u i j - D Dt = u, (1) = - 1 p + n u i + 1 x i 3 n x i = - 1 p È + n 4 ( u ) - Í ( w ) x i 3 x i, Î i (b) p = g Constnt. (3) u ( ) () Viscous effects only ppe in momentum equtions. Two foms of momentum equtions, () nd (b), e gien. In the second fom, the net she iscous foce cting on fluid element is expessed s the sum of gdient of diltion nd cul of ottion. Het poduced by iscous stess is neglected so tht isentopic eltion Eq.(3) insted of the full enegy eqution is needed. No entopy we is ssumed in this chpte. The medium is botopic, since fo the isentopic pocess density depends only on pessue. The petubtion is smll. The lineized isentopic eqution (3) is: p / = 0. (4) 1

2 p nd e petubed pessue nd density. 0 = gp0 / 0 is the sound speed in the undistubed medium. Fom now on ibles with subscipt '0' e fo the sttic medium without petubtion. Vibles without subscipt '0' e petubtion ibles. The medium hs no men flow, u 0 = 0. Lineizing equtions (1), () nd (b), one obtins: - p = u j 0 0, (5) u i = - 1 p + n u i x i 3 n x i = - 1 p È + n 4 Í u 0 x i Î 3 x i ( ) - ( w ) i u ( ) (6) in continuity eqution (5) is substituted by p becuse of Eq.(4). Eq.(5) shows tht pessue chnging te is only elted to diltion of the fluid, which is tue fo ll botopic fluids. This tuns out to be citicl fo the we mode seption.. (6b) Seption of Wes Seption of coustic nd oticl wes cn be mde by using the noml mode method (cht1.doc: Wes in Unifom Flow on Hlf Plne). Hee we ttempt to septe the wes bsed on thei physicl oigins. As we know, coustic wes e geneted by oscilltions of the fluid element unde the blnce of ineti nd elstic estoing foces. On the othe hnd, she stesses ct tngentilly t the sufce of fluid element, if they e unblnced, will genete oticl wes. In momentum equtions (6)&(6b), thee e two foces exeted on the fluid element sufce: pessue nd iscous she stesses. Pessue is the elstic foce since it chnges s the esult of contction/expnsion of the fluid element [Eq.(5)]. In botopic fluid, pessue doesn't genete ny ottion (Thomson Theoem). Fo fluid element hs the sme cente of mss nd geomety, nd pessue foce cts though this cente geneting no ottion (Pnton1996, p.39). Theefoe pessue sees solely s the elstic foce fo coustic wes. The net she iscous stess is elted to ottion nd diltion (Eq.6b). Diltion elted iscous stess cts s fiction to coustic wes; ottion elted iscous stess is the estoing foce fo oticl wes. It is cucil to septe the two type of wes bsed on the diltion nd ottion fields of the flow. A fluid element undegoes thee diffeent type of motions when foces exet on its sufce: diltion (isotopic expnsion/contction) with olume chnge te u, igid-body ottion with oticity u, nd pue stin without olume chnge nd ottion. Accodingly, elocity t ny point in the flow cn be decomposed into two pts: u = u + u. (7)

3 u is the elocity ssocited with the diltion field of the flow: u = u, u = 0. (8) u is the elocity ssocited with the oticity distibution of the flow: u = u, u = 0. (9) Any elocity ssocited with pue stin cn be dded to u nd/o u since it hs no diltion nd ottion. Theefoe u is the iottionl (oticity fee) elocity, nd u is the solenoidl (diltion fee) elocity. At this point we ssume pessue cn be decomposed ccodingly: p is ssocited with diltion field u, nd p = p + p. (10) p ssocited with oticity field u. Plugging Eq.(7) nd (10) into Eqs.(5)&(6) nd equting tems with subscipt nd tems with subscipt, one obtins two sets of equtions. Acoustic we The equtions bout the iottionl field e: u i Ê u w i = e k ijk Á Ë - u j x k ˆ = 0, (11) - p u = 0 i 0, x i (1) = - 1 p + n 4 ( u ). 0 x i 3 x i (13) The second fom of the momentum eqution Eq.(6b) is used hee fo conenience. In Eq.(13), the ineti is mostly blnced by pessue, the estoing foce fo coustic wes. Just s we sid befoe, u epesents the oticity fee field. Its pue stin nd the ssocited iscous stess e not necessily zeo. The iscous tem in (13) is the net iscous stess due to diltion, which cts s fiction to the coustic we. Voticl We The equtions bout the solenoidl field e: 3

4 u j = 0, (14) u i p = 0, (15) = p x i + n u i. (16) Momentum eqution Eq.(6) is conenient to use fo the oticl mode. Integting Eq.(15) gies p ( x,t) = p ( x ). By tking diegence of (16), one obtins Lplce eqution p / x i x i = 0. Thee e no nonsingul solutions fo Lplce s eqution except p = constnt. Genelly otexes e compct in spce with p = 0 in the f field. Theefoe the solution to Lplce s eqution is p = 0. The oticl we equtions become: p = 0, (17) u i w i = n u i, o, (18) = n w i, (19) Ê u whee w i = e k ijk Á Ë - u j x k ˆ. Fo idel gs, iscous tems e dopped, then u i / = 0, u i = 0, which mens no oticl motion in idel sttic medium. Acoustic nd oticl wes popgte independently in the sttic medium. This is tue due to the lineity nd unifom of men flow. Nonlineity o non-unifom men flow will ineitbly couple the two we modes. Acoustic wes nd oticl wes only couple t boundies whee the totl elocity u = u + u nd totl pessue p = p + p stisfy boundy conditions. In some simple situtions, boundy conditions e lso sepble so tht the two types of motions e totlly decoupled in the whole field. In the following sections we will he fou exmples. In the fist exmple pue oticl we is geneted by plne wll oscillting on its own plne. In the second exmple pue coustic we is excited when the plne wll oscilltes in its noml diection. In the thid exmple, the wll oscilltes in n oblique diection; the elocities of the wll in its tngentil nd noml diections die oticl we nd sound we espectiely. In the lst exmple, the oscillting body hs n bity shpe; when oscillting fequency is in cetin nge, the boundy conditions cn be ppoximtely decoupled. 4

5 Stokes Lye The simplest but most typicl exmple of oticl we is the we geneted by plne wll oscillting in iscous medium. The medium is bounded by n infinitie igid wll on (x,z) plne in the thee dimensionl Ctesin system (x,y,z) shown in Fig.1. The wll oscilltes on its own plne. The iscous we is geneted by the wll nd popgtes into the fluid. Fig.1, Wll oscillting on its own plne. Anlyticl solution of the full incompessible N.S. equtions fo this poblem ws found by Stokes (G.G.Stokes1851). The iscous we is clled Stokes we. Detils of the solution is in the book of Lndu&Lifshitz1959. Hee we will sole this poblem using the method of we seption. The wll stts to oscillte t t = 0 in x diection. Asymptoticlly the wll ibtes hmoniclly t nnul fequency w, i.e., s t Æ : u = Ae -iwt, = 0, w = 0, t y = 0; (0) u = 0, = 0, w = 0, t y =. We will ssume hmonic solutions while still consideing initil field sttic. Complex ibles my be used in the nlysis. As long s the equtions e line, the finl esult cn be obtined by tking el pts of complex quntities. Acoustic We We fist show tht the oscilltion of the wll on its own plne will not die sound wes. Boundy condition (0) hs no dependence on x nd z. Fom physics intuition of symmety, ll ibles will only depend on y nd t. Momentum equtions in x nd z diections fom Eq.(13) e: u = 0, w = 0. (1) Initilly thee is no sound in the field. This immeditely gies: u = 0, w = 0. () 5

6 Eq.(1) nd momentum eqution (13) in y diection become, - p = 0 0 y, (3) = p y n y. (4) By eliminting p in Eqs.(3) nd (4), the sound we eqution bout is: Ê = Á Ë 0 y 4 + n 3 y ˆ t. (5) It will be shown in the oticl we nlysis next Eq.(9), = 0. Then boundy condition (0) mens: The solution of Eq.(5) with boundy (6) is = 0, t y = 0 nd y Æ. (6) = 0, p = 0. (7) Viscous stess fom the oscillting wll doesn t squeeze fluid nd thus genetes no sound wes. Voticl We The oticl elocity is diltion fee [Eq.(14)], theefoe: y = 0. (8) Fom momentum eqution (18) nd Eq.(8): = n = 0. (9) y Initilly thee is no otex, theefoe oticl elocity in y diection is lwys zeo no mtte wht boundy conditions e: Momentum eqution (18) in z diection is: = 0. (30) 6

7 w = n w y. (31) Its solution fo w = 0 t y = 0 nd y Æ is: Momentum eqution in x diection: w = 0. (3) u = n u y. (33) We he ledy shown in coustic we solution Eq.(), u = 0. Fom (0), the boundy condition fo u becomes: u = Ae -iwt, t y = 0; (34) u = 0, t y Æ. Eq.(33) with boundy conditions (34) cn be soled by using the noml mode method. Assuming this fom of solution: u Ae i( ky-wt) =. (35) Substituting it into (33), we obtin the dispesion eltion: Suppose w is el, then, iw = nk. (36) k = ±(1+ i) /d, d = n /w. (37) k with negtie sign should be emoed since it gies solution with exponentil gowth s y Æ. Then the solution is: u - y / i( y / d wt) = Ae d e -. (38) This descibes tnsese we in the iscous fluid with the popgtion diection pependicul to the oscilltion diection. d is penettion depth of the we popgting into the medium. A thin boundy lye, clled Stokes lye, is fomed ne the wll. Within this boundy lye, sufficient esolution must be wnted if numeicl methods e used to sole this poblem. 7

8 Eqs.(17), (30), (3), nd (38) fom the full set of oticl solutions to the Stokes lye poblem. The oscilltion of the unifom iscous stess on the medium by the wll doesn't squeeze the fluid nd thus genetes no sound. When the iscous stess is non-unifom, sound cn be geneted. This is consistent with Lighthill s Anlogy Theoy in which the sound souce is the double diegence of iscous stesses: s ij / x i. The equtions we soled e lineized continuity eqution (5) nd N.S. equtions (6). Howee, this set of solutions lso stisfy the full incompessible N.S. equtions (Lndu&Lifshitz1959) nd the full compessible N.S. equtions with isentopic eltionship, Eqs.(1)~(3). The eson is tht fo these solutions, nonline tem u u / x in the N.S. equtions is zeo. Theefoe the full N.S. eqution is line no mtte if the oscilltion is wek o stong. Simil solutions cn be found fo n oscillting cylinde ound its xis, o n oscillting sphee ound its cente. j i j Plne Wll Oscilltion in Noml Diection Fig., Plne wll oscillting in noml diection. Now let s discuss the we geneted by plne wll oscillting in its noml diection s in Fig.. The boundy conditions e: -iwt = Be, u = 0, w = 0, t y = 0; (39) = 0, u = 0, w = 0, s y Æ. The boundy conditions he no dependence on x nd z. The solutions e only functions of y nd t. Fo the sme eson s in the lst section, coustic elocities in x nd z diections e: u = 0, w = 0. (40) Eq.(5) is the we eqution fo. Eq.(30) still holds, then the boundy condition bout bsed on (39) is: 8

9 Be -iwt =, t y = 0; (41) = 0, s y Æ. Assume the next fom of solution: Be i( ky-wt) =. (4) Substituting it into Eq.(5), we obtin the dispesion eltion: w = 0 k - i 4 3 nk w. (43) Fo el w, w k = ± Ê Á 0 - i 4 Ë 3 nw ˆ. (44) 1/ k with + sign epesents decy wes s y Æ. The bnch cut fo the sque oot is shown by Fig.3. Fig.3, Bnch cut fo w'+ 0 Acoustic pessue is obtined fom Eq.(1): 4 = plne. 3 ( ) 1/ on w' -i nw p i( ky- w t) = Bke / w. (45) 0 0 Eqs.(40), (4) nd (45) fom the coustic solution of the poblem. It cn be shown tht the wll oscillting in its noml diection does not genete oticl wes. Plne Wll Oscilltion in Oblique Diection 9

10 Fig.4, Plne wll oscillting in oblique diection. Suppose the wll oscilltes in n oblique diection s in Fig4. At the boundy, u = Ae -iwt, = Be -iwt, w = 0, t y = 0. (46) u = 0, = 0, w = 0, s y Æ. We he ledy known tht oscilltion of the plne wll in its noml diection does not die oticl wes, nd oscilltion of the wll on its own plne does not die sound wes. Theefoe, = Be -iwt t the wll dies only sound we with solutions (40), (4) nd (45). u = Ae -iwt t the wll dies only oticl we with solutions (17), (30), (3), nd (38). Oscilltion of Object with Abity Shpe Fig.5, Oscilltion of n bity object. As the oscillting object hs n bity shpe s in Fig.5, the nlyses fo plne wlls in peious sections do not pply since flow ibles e no longe only functions of y nd t. Howee, if the object sufce is smooth nd the oscilltion fequency is high, the coustic we length nd Stokes lye e smll, nd locl body sufce my be consideed s plne wll. Suppose the object oscilltes ound point o in Fig.5. Sufce ound point A oscilltes nely on its tngentil diection, geneting oticl we. Sufce ound point B oscilltes nely in its noml diection, mostly diing n 10

11 coustic we. Most of the sufce segments oscillte in oblique diections nd die both oticl nd sound wes. Voticl wes e only impotnt in the Stokes lye with thickness d [Eq(37)]. Wes outside the Stokes lye e iottionl/cousticl. Hee we discuss limiting cse when d is smll comped with body dimension l nd the sound welength l, i.e., l w >> n, w << p 0 /n. (47) This is the mid nge of fequency. Outside the Stokes lye, coustic wes cn be soled if boundy condition is set t the outside sufce of the Stokes lye (shown in i t Fig.5 by dshed line), i.e., noml elocity u e - w ˆ n t this sufce. Since d is ey smll, the boundy cn ppoximtely be set t the body sufce, i.e., uˆ ª ˆ, (48) n u wn whee û wn is the noml oscillting elocity on the object sufce. Acoustic equtions (1)&(13) with boundy condition (48) cn be soled nlyticlly o numeiclly. In the locl Ctesin coodintes ( x ', y' ) (Fig.5), the oticl we eqution is: u ' = n u ' y'. (49) Suppose the tngentil elocity fom the coustic solution t the object sufce is û t. -iwt The tngentil oscillting elocity t the body sufce is uˆ wt e. Then the boundy conditions fo u ˆ ' is: uˆ ' = uˆ - uˆ, t y = 0; (50) wt t u ˆ ' = 0, t y =. The solution is: u ( ˆ ˆ - y'/ i( y'/ d wt) ' = uwt - ut ) e d e -. (51) In summy, the soling pocedues fo the coustic nd oticl wes geneted by n oscillting object with bity shpe in the fequency nge (47) e: (1)Acoustic solution with noml elocity of the body sufce s the boundy condition; ()Anlyticl iscous solution in the inne field, Eq.(51), with tngentil elocity (50) t the body sufce. 11