On he helical behavio of ubulence in he ship wake E. Golbaikh, A. Eidelman 2, A. Soloviev 3 Physics Depamen, Ben-Guion nivesiy of he Negev, Isael 2 Mechanical Engineeing Depamen, Ben-Guion nivesiy of he Negev, Isael 3 Oceanogaphic Cene, NOVA SouhEasen nivesiy, SA Absac Tubulen ship wake consevaion a a long disance is one of unsolved poblems a pesen. I is well known ha wakes have a oaional sucue and slowly expand wih disance. Neveheless, expeimenal daa on hei sucue and popeies ae no sufficien. On he ohe hand, hese expeimenal daa show ha he divegence of wakes does no change accoding o he law /5, as pediced by he heoy. In ou wok we sudy he effec of heliciy on he paamees of a ubulen ship wake. Taking ino accoun he helical naue of he wake, we can claify he diffeence beween ubulence inside and ouside of he wake on he one hand, and slow is expansion wih ime. Inoducion A wake obseved behind a ship on he ocean suface unde elaively quie wind condiions possesses some univesal popeies. In fac, he main wake sucue includes, pobably, wo ineacing zones, namely, Kelvin's wake and a ubulen wake eained ove lage disances wih a weak angula divegence. Expeimenally obseved univesaliy of he wake sucue indicaes o he univesaliy and maybe selfsimilaiy of he pocesses in i. A sufficien amoun of numeical models have been suggesed o descibe he flow aound a moving ship and in is viciniy, which saisfacoily eflec, o he fis appoximaion, he pocesses in his zone. Regading he wake, some of is feaues (e.g., Kelvin waves) have been sudied exensively enough (see, fo example, Reed and Milgam, 993a, b; 22; Zilman e al., 24 and
efeences heein). On he ohe hand, he cenal ubulen zone of he wake, wellobseved (see, fo example, Reed e al., 22; Munk e al., 987 and efeences heein) wihin sufficienly lage disances in some field expeimens, emains pooly sudied as ye, and he descipion of is popeies is fa fom being complee. Dak-seak images, which can be seveal kilomees long unde low wind condiions (2.5 o 7.5 m/s), ae shown in Fig.. Measuemens of sho waves suppession in ship wakes wee conduced and compaed wih SAR images of simila wakes in sevaal sudies (Munk e al., 987; Milgam e al., 993a, b; Reed e al., 22; Shen e al., 22 ). They have shown ha he phenomenon can be explained by he suppession of sho sea waves in he wake, said waves being esponsible fo ada backscaeing associaed wih a bighe backgound. Fig. Ship wake. One of he ams of a naow V-wake; 2 ubulen wake; 3, 4 boundaies of he Kelvin wake (afe Zilman e al., 24). I emains unclea unil now wha physical pocesses occuing in he wake affec is popeies, how iniial (oiginaed in he nea zone) voical disubances ae developed, how a sysem of lage-scale voices aises, how enegy is supplied heein, ec. One of he main physical poblems acively discussed a pesen is ha of moion souces in fa wakes leading o hei long-living chaace. In he pesen pape, we do
no discuss enegy souces mainaining ubulen wake fa away fom a ship. Noe only ha hey can compise diffeen waves geneaed on he wae suface (fo example, Kelvin's o wind waves). We dwell on ubulen pocesses leading o wake consevaion o, in ohe wods, on he pocesses deceasing is ubulen dissipaion. Specal chaaceisics of ubulen wakes Expeimenal daa on ubulen chaaceisics of suface ship wakes ae limied. Only in few laboaoy expeimens (Pael, Sada, 99, Hoeksa, Ligelijn, 99, Benilov e al., 2, Shen, 22) ceain ubulen paamees of he wake, including specal chaaceisics, wee measued. The ship-wake ubulence is sill well deecable even in he fa wake, whee he Kolmogoov ineial ange can be idenified (Benilov e al., 2). A sufficienly inense ubulence wih Reynolds numbes coesponding o a developed ubulence is obseved in he fa wake. Benilov e al. (2) explained he long-living wake sucue by he pesence of inense ubulence agains he backgound of a shea flow suppoed by he enegy of dissipaed wind waves. Fom he sandpoin of ubulen appoach, i is geneally assumed ha ubulence is an enegy sink fo lage-scale moions. Wake velociy speca obained in he expeimens of Benilov e al. (2) a a disance x/l = 8.25 af he model a he velociy Vs=26 cm/s ae shown in Fig. 2, whee ou analyses of he speca evealed a specal slope close o 7/3. The specal slope 7/3 is a chaaceisic slope of ubulence wih a nonzeo mean heliciy (Bissaud e al., 973). Mean heliciy geneaion occus in diffeen ubulen flows wih violaed mio symmey, alhough ubulence is unifom and isoopic (e.g. Banove e al., 999).
Fig.2 Wake ubulence unde specal waves condiions a a disance L/D=8.25 fom he model, wih he speed 26 cm/s, wave elevaion vaiance 2.54 cm, specal peak fequency Hz (afe Benilov e al., 2). Flow oaion obseved in a wake leads, among ohes facos, o mio symmey violaion of ubulence. In his case, even a unifom and isoopic ubulence acquies addiional popeies, so ha a nonzeo Euleian inegal of moion mean heliciy He u' culu' appeas in i. Hee u ' is a ubulen componen of he velociy field, and < > denoes aveaging ove he ensemble. Mean heliciy, along wih enegy, is an essenial invaian chaaceisic of a ubulen flow. Howeve, a mean heliciy value is ofen oo small o affec consideably he flow behavio. One can hadly expec ha a moion suppoed by pessue gadien only, in he absence of boundaies, would possess helical popeies. Mean heliciy geneaion in he pesence of oaion is a highly pobable and impoan naual eason fo he fomaion and consevaion of he lage-scale longiudinal voical sucues of ship wakes. The possibiliy of mean heliciy oiginaion in ubulen oaing flows was discussed in deail by Kause and Rädle (98). As shown by he auhos of his wok, in he pesence of diffeenial oaion and/o
exenal foces, heliciy eaches appeciable values and can affec flow chaaceisics. As noed by Benilov e al. (2), he flow sucue in he ubulen wake can be aed as an odinay shea, and heefoe, we can speak wih ceainy abou a elaively high heliciy in a ubulen wake (Chkheiani e al, 994). We examine some peculiaiies of speca in he pesence of heliciy. Specal chaaceisics of helical ubulence wee fis sudied by Bissaud e al. (973). Hee 3 7 / 3 he helical specum of he specal densiy of ubulen enegy ( ) 2 / E k k (whee is he heliciy flux and k is he wave veco) appeas along wih he Kolmogoov's 3 5 / 3 specum ( ) 2 / E k k (whee is he enegy flux). Accoding o he appoach of Bissaud e al. (973), helical specum should exis only fo. Howeve, as follows fom expeimenal daa (Manson and King, 985; Nasom e al., 987; Lilly and Peesen, 983; see also Banove e al., 999 and efeences heein), he helical specum is no a all exoic, being obseved in vaious flows, and in hese flows he enegy flux is also nonzeo. Boh slopes ae also obseved in ohe speca pesened by Benilov e al. (2), e.g. shown in Fig. 3 fo he wake coss-secion a he same disance and a a velociy Vs = 22 cm/s. A lage-scale ange of speca is close o 7/3 slope and an adjacen small- scale ange o 5/3 slope. Genealizing peviously obained esuls, ubulen enegy densiy specum has been analyzed using Kolmogoov's appoach fo he ineial ineval wihin he famewok of he asympoic model in he case whee and ae govening paamees no only in he ineial ineval (Golbaikh and Eidelman, 27). They have shown ha he popeies of he sucue funcion in he ineial ineval ae affeced by he behavio of he deemining paamees and in adjacen egions, boh lage-scale and dissipaive ones. In such a case, he speca become complicaed, and hei slopes defined by muual influence of he egions can aise (see Fig. 3). Theefoe, 7/3 slope of speca poins o a noiceable nonzeo mean heliciy. Basic popeies of helical ubulence having a nonzeo mean heliciy ae an enhancemen of lage-scale helical voical sucues and a decease in ubulen
Fig. 3 Tubulen speca in he wake coss-secion a he disance L = 3 m fom he model. Model speed Vs=22 cm/s (afe Benilov, 2) diffusiviy (see Moiseev e al., 983a, b; Belyan e al., 993; Belyan e al., 998; Banove e al., 999 and efeences heein). Noe ha helical ubulence geneaes and/o enhances coheen sucues in ubulen flows. When speaking abou geneaional popeies of ubulence, we imply ha a ubulen cascade diffes fom Kolmogoov's one, whee he enegy supplied fom he ouside ino lage-scale flucuaions is diecly ansfeed ino small scales and dissipaes heein. I can lead o a change in he specal behavio of ubulen flucuaions conneced wih he mean heliciy level of he ubulen field. Heliciy level in a fa wake ( 2km af a ship) can be esimaed using field expeimenal daa of Pelze e al. (992), whee he chaaceisic voiciy scale L 2m. I is assumed ha wo voices coexis in he fa wake and ha ubulence is mainly concenaed wihin.5-.6 of he voex adius (Pael and Sada, 99). Main voiciy flucuaions poduced wih flucuaions of laeal velociy ae locaed a a ceain disance fom he suface (Shen e al., 22) and amoun o
.5./ 8 7 3 L, s. The esimaed heliciy flux eads 3.3 m / s, and 6 5 2 3 he esimaed ubulen enegy flux amouns o 6 5 m / s. On he ohe hand, unde expeimenal condiions (Pelze e al., 992) wih he wind speed no exceeding 5 m/s, he value of fo backgound ubulence should have amouned o 2 s 6 2 3 3 m / (Soloviev and Lukas, 26). Hence, ubulence in he wake was songe han beyond i. Popeies of flows possessing heliciy can be explained by he fac ha he lae effecively educes he acion of nonlinea pocesses esponsible fo enegy ansfe and edisibuion beween vaious scales. Indeed, wiing Navie-Sokes equaion fo a voex culv (in incompessible case) (v) and a elaion fo Lamb s veco v v 2 v 2 2 v 2, (2) we can see ha wih gowing heliciy, he nonlinea em in Eq. () deceases and becomes zeo fo Belami flow, whee () v ( being a ceain consan). The same easoning can be applied o he ubulen pa of he velociy field, subsiuing mean velociies in Eq. (2) wih flucuaional ones and aveaging ove he ensemble. Howeve, as known (aking ubulence ino accoun), he nonlinea pa of Eq. () is conneced wih ubulen viscosiy, so ha he helical em leads o is decease. Vaious sudies of he popeies of helical ubulence (see, fo example, Belian e al., 994, 998; Banove e al., 999 and efeences heein) demonsae ha non-zeo heliciy leads o a decease in he enegy flux fom lage o small scales. Tubulen enegy edisibuion beween small and lage scales a nonzeo heliciy depends on he aio of heliciy modulus o ubulence enegy (Moiseev e al., 983a, b) and, as shown in his wok, helical ubulence can lead o he geneaion (o sabilizaion) of lage-scale voical sucues. Thus, he main popeies of helical ubulence ae non-kolmogoov's specal dependence of enegy densiy E (k) (whee k is wave veco); ubulen viscosiy decease; enegy ansfe fom small- o lage-scale moion invese enegy ansfe,
and a coesponding decease in he ubulen enegy ansfe fom lage o small scales. Regading he case of helical ubulence epesening a basically 3D-phenomenon (alhough is anisoopy can be high), we emphasize ha he invese enegy ansfe is is impoan popey. The invese enegy ansfe ofen acquies he chaace of an exponenially fas geneaion of a lage-scale sucue due o he moion insabiliy. One of he mos impoan esuls of he developmen of he helical ubulence model is he undesanding of he fac ha he 3D-chaace of moion deemines he opology and, finally, ininsic popeies of ubulence. Thus, we poceed o discussing ubulen wake popeies conneced wih he pesence of nonzeo mean heliciy. Wake expansion in field expeimens To evaluae popeies of ubulen viscosiy in a ubulen wake, we examine he dependence of is widh on he disance o he ship. An asympoic elaion fo he expansion of he widh of he ubulen wake af a self-popelled ship wih zeo axial ne momenum was pediced in 957 by Bikhoff and Zaanonello, bu only ecenly i has been veified fo full-scale vessels in field expeimens (Pelze e al., 992, Milgam e al., 993a). Noe wo ses of pefomed expeimens: () an iniial se of measuemens of waves and a few suface ension measuemens in and nea he wakes of commecial vessels, and (2) moe compehensive measuemens in and nea he wakes of seveal.s. Navy vessels. In addiion, ada and backgound meeoological measuemens wee also caied ou. Waves wee measued wih specially designed esisance ansduces having a high accuacy and high signal-o-noise aio fo he ange of fequencies ha included waves associaed wih he Bagg backscaeing of adas (Milgam e al, 993a). Suface ension disibuions calibaed in a wide ange fom 44.5 o 73 mn/m wee measued by speading oils (Pelze e al., 992). Exensive daa on sufacans disibuions acoss ship wakes wee obained in 989 Field Expeimen (Pelze e al., 992). Wake widhs W wee
obained a 3 o 4 disances x af of wo Navy ships, a desoye and a figae, a he velociies fom 6.2 m/s (2 knos) o 2.9 m/s (25 knos) coesponding o he ange of he Foude numbes F = V s /(gl) /2 =.56.366, whee V s and L ae ship velociy and lengh, g gaviaional acceleaion. The suface ension disibuion and SAR aicaf image inensiy acoss a wake of 5 min age wee obained appoximaely a he same disance 3577 m af he ship (a desoye a 2.9 m/s on 28//89) (see Milgam e al., 993a, Fig. ). The widhs of he wake deeced in hese expeimens using sufacans disibuion and SAR images (especially in C-band) aveaged ove m along he wake and 2 m acoss he wake ae evidenly ahe close. Milgam e al. (993a, b) have concluded he following:. Reduced ada eun leading o a dak ceneline wake image is undoubedly associaed wih educed sho-wave enegy in he ceneline ship wake. 2. Boh ubulence and concenaion of sufacans in longiudinal bands caused by he passage of a ship play a ceain ole in aenuaing sho waves. The dependence W / B vs. x/l obained by Milgam e al. (993a) eads W B W B x L n (3) whee W is a paamee chaaceizing he widh of a self-simila fa wake a a disance educed o x = L af a ship, B and L ae he widh and lengh of a ship. The dependence was appoximaed in log-log coodinaes by a slope n = /5 fo eigh wakes (Milgam e al., 993a). The slopes n decease fo boh he desoye and he figae, and chaaceisic widhs of he wake W /B incease wih gowing F. The widhs of he wakes diffe significanly (by 5%) in he expeimens whee F numbes vaied wo- fold fo boh he desoye and he figae. Fuhemoe, he figae wakes W nomalized o B ae appoximaely 3% lage han he desoye wakes fo
evey velociy used in he expeimens. This also poins o a dependence on F numbes, since F values fo he figae ae less han hose fo he desoye a he same velociy. We have pocessed he same daa independenly fo each wake. The slopes fo seven wakes of eigh ae evidenly less han /5, which is obained fo only one wake. The mean value of n fo he desoye eads.53 and.33 fo he figae, ha is, he slope is close o /7. We include he F numbe ino he equaion in ode o eveal is effec on he wake widh W W 2 F B B x L n Tha esuls in eviden gouping of expeimenal daa shown in Fig. 4. Noe ha he definiion of W is changed in his case, and i becomes a paamee 2 x chaaceizing he widh of a self-simila fa wake if F af a ship. L The values of n and W /B in Equaion (4) fo eigh diffeen wakes ae pesened in Table, whee W /B values fo each ship ae close and he values fo he figae (FF) ae lage han fo he desoye (DDG) by abou 5- %. Possible easons of his sligh scaeing ae diffeen paamees of he hulls o/and wind condiions. (4) Table. Chaaceisics of desoye and figae wakes Run Velociy m/s F DDG n DDG W /B DDG F FF n FF W /B FF W B / W / B DDG FF 6.2.8.2.83.6.6.92.9 2 2.9.37.7.86.32..92.93 3 9.3.27.2.85.23.3.94.9 4 2.9.37.5.82.32.5.86.95
...9 FF() 2k FF(2) 25k FF(3) 8k FF(4) 25k DDG() 2k DDG(2) 25k DDG(3) 8k DDG(4) 25k Linea (DDG() 2k) Linea (FF() 2k) Linea (DDG(2) 25k) Linea (FF(2) Linea (DDG(3) 8k) Linea (DDG(4) 25k) Linea (FF(3) 8k) Linea (FF(4) lg W /B y =.49x +.859 y =.95x +.924 y =.24x +.945 desoye & figae.8 y =.62x +.96 y =.26x +.86 y =.526x +.856 y =.87x +.8472.7 y =.2x +.8337 lg (F^2 x/l).6 -. -.5..5..5 2. Fig.4. Dependence of he wake widh Foude numbe. W / B on he disance af a ship x/l and he Discussion of effecive ubulen viscosiy in he wake The value of n = /5 in Eq. () suggesed fo expeimenal daa by Milgam e al. (993a) coesponds o he heoeical esul obained fo he model of a wake of a self-popelled axisymmeic submeged body (Tennekes, Lumley, 99; Bikhoff, Zaanello, 957). Accoding o Tennekes and Lumley (99), he value n = /5 should be obained fo wakes of axisymmeic submeged bodies in he appoximaion of a consan ubulen viscosiy ove he wake cosssecion. In ode o impove he descipion of a swiled wake, whee he ansvese componen of he velociy field should be aken ino accoun besides he longiudinal and adial ones, we can use a univesal ansvese pofile fo he enie wake. Because of his,
u equaion v (whee and u ae mean and ubulen longiudinal x velociy componens, v - ubulen componen of he adial velociy; see also (4..9) in (Tennekes & Lumley, 99)) is no valid any moe. A longiudinal deivaive of pessue appeas in he igh-hand pa of his equaion (Loysansky, 97) because of he adial componen of he pessue gadien. This deivaive balances cenifugal foces aising due o he wake swil in he saionay case. We assume ha he longiudinal pessue gadien componen weakly affecs he univesal chaace of he pofile in ceain limied secions of he wake, i.e., we inoduce a piecewise univesaliy of he wake. Hee he oal momenum flux hough he wake coss-secion is conseved (Loysansky, 97): ( p 2 ) d cons J and compises he local pessue. In his secion, he dependences of he chaaceisic n n widh l and velociy defec ) on he disance x, x and x, especively, ( obained by Tennekes & Lumley (99) also hold ue. On he ohe hand, he consevaion of he pincipal momen of momenum ansfe hough he je cosssecion in case of a swiling flow (Loysansky, 97) aises side by side wih he consevaion of he oal momenum flux: 2 Wd cons L (5) whee W is he ansvese velociy componen. The momen L is consan along he wake and seves as a measue of he wake swiling. In he case of solid oaion wih W, 2 L cons ( ) d, which leads o he value n afe de-dimensionalizaion. Howeve, a wake of a ship is fa fom 5 solid oaion. We assume ha, o he fis appoximaion, W ( ) cul! ( ) ( ) (whee ", V, W# is he mean velociy field in he wake; he pseudo-scala 2 W W cul!
and in saionay piecewise univesaliy case weakly depends on x ). Then i follows fom (5) ha L 2 d, (6) ( ) n and fo he dependence l x o hold ue and no o divege fom () a, he condiion d( ( ) d 2 ) should be valid in he piecewise univesaliy appoximaion. This leads o he dependence ( ), and hee we can wie afe inegaing by pas ( L cons ) d, (7) whee is he longiudinal velociy value beyond he wake. Theefoe, o obain he expeimenal value of he paamee n, we aive a he equaion by de - dimensionalizing equaion (7). n (8) 2 The magniude () is a pseudo-scala and should change is sign a subsiuion fo. Hence, he paamee should assume only he values fo which Expeimenal. n values (see Table ) coespond, accoding o (8), o values 6 8 wihin he ineval fom 4 o 6. Thus, 5 coesponds o aveage value of expeimenal n. 7 Effecive ubulen diffusiviy u u eff x exp deemined fom he expeimenal daa on he aio of he measued componen " x # of Reynolds sesses enso o he adial deivaive of he longiudinal velociy is no consan ove he wake cosssecion and deceases fom he cene o he voex peiphey (Pael, Sada, 99). Such behavio of he effecive ubulen viscosiy can be due o he fac ha in he geneal case of local homogeneous isoopic ubulence wih violaed mio symmey, he coelaion velociy enso includes, besides he symmeical pa, an x
asymmeic one poduced by mean heliciy. The symmeic pa is esponsible fo he dissipaion, i.e. can be idenified wih ubulen viscosiy (see, e.g., Kause, Redle, 98; Monin, Yaglom,996). The asymmeic pa can compensae, unde ceain condiions, he symmeic pa deceasing he effecive ubulen viscosiy (Belian e al., 994, 998). To analyze ubulen viscosiy popeies in a wake coss-secion, we make use of he equaion deived by Moiseev e al. (983) fo mean velociy (voiciy) in he pesence of nonzeo mean ubulen heliciy: whee o ( ) cul$ cul (9) cul, $ % He, % - effecive elaxaion ime, - ubulen viscosiy defined by he symmeical pa of he ubulen velociy field coelao, i.e. % E, whee E is he mean enegy of ubulence (see also Banove e al., 999, and efeences heein). If $ and ae consans, we obain he well-known equaion o ( ) $ cul () descibing voex geneaion. This equaion in a somewha expanded fom was used fo he sudy of lage-scale voical sucues geneaion in vaious condiions including planeay amosphees (see Banove e al., 999; Ivanov e al., 996 and 2 efeences heein). Acually, a posiive paamee of insabiliy ' i& $ k k (whee k is wave veco) fo modes wih $ k ( can be obained wih Fouie ansfomaion of (). A he same ime, modes wih k $ ) aenuae, bu he ae of his aenuaion deceases wih he gowh of $. On he ohe hand, k coesponds o he opeao in Eq. (9), which deemines ubulen viscosiy. Thus, he pesence of nonzeo mean heliciy leads o he educion of he effecive ubulence viscosiy (Belyan e al., 994, 998; Golbaikh e al., 998). To demonsae his asseion, we ewie he expession unde cul opeao in Eq. (9) as
( ) $ () whee in he above appoximaions in he cylindical coodinaes igh-hand side of () as x x and W. Then we can ewie he x x x $ ( ) (2) If he signs of he mean ubulen heliciy and lage-scale heliciy ae he same, he eff $ effecive ubulen viscosiy (whee he same ime, since $ ) deceases. A $ eff gows fom he cene o he peiphey, should decease fom he cene o he peiphey, which is obseved expeimenally (Pael, Sada, 99) and, in pinciple, agees wih he esuls of Hoeksa and Ligelijn (99). Now we esimae he odes of magniude of he paamees, $ and fo expeimenal daa (Pael, Sada, 99). We inoduce a adius-aveaged value R R d, whee R is he effecive voex size. The model dimensions wee as follows: L = 3.48 m, B =.35 m and D =.9 m. A a modeae disance fom he ship, we can esimae as W and $ fom he diffeence beween paamees a he levels Z / D and Z / D. 5 (size of he voex is on he ode of half-deph of he wake). Since.D and. B and eff. 6 D ( ) B eff, which poins o abou.5-fold viscosiy decease fom he voex cene o is peiphey. Thus, he geneaed helical ubulence leads o a decease in he effecive ubulen viscosiy, which is efleced in a lowe incease in he wake widh wih he disance han he pevious model pedics.
Conclusions Helical ubulence has no been pacically applied o oceanic phenomena. I is paially a esul of limied expeimenal daa (as compaed, e.g., o he amosphee) on oceanic ubulen fields. We make an aemp, based on available expeimenal daa, o show ha helical ubulence geneaed in ubulen ship wakes can essenially affec dynamic chaaceisics of he wake. In paicula, such effec should be obseved in speca of ubulen velociy flucuaions in he wake, which become seepe 7 / 3 ( E ( k) k ) han Kolmogoov's speca. A he same ime, effecive ubulen viscosiy in he wake should decease, which esuls in slowing-down of he wake widening wih he disance fom he ship. Besides, i should lead o a fa wake sabilizaion, which is possible in he pesence of helical ubulence due o he pemanen invese enegy cascade fom small o lage scales, side by side wih he egula enegy flux fom lage o small scales. In he pesen sudy, we do no discuss enegy souces mainaining ubulence in he wake, bu i is noewohy ha since heliciy mixes up velociy componens, any lage-scale enegy inpu ino ubulence in he fa wake mainains ubulence on he whole. Howeve, i is noewohy ha hey can include pocesses aising agains he backgound of wind waves geneaion, changes in he suface ension coefficien in he wake as compaed wih he suoundings, ineacion of wake wih Kelvin s waves, as well as pessue flucuaions conneced wih he ship and ansmied ove lage disances, and some ohes o be discussed in conclusion. Geneaive popeies of helical ubulence have been examined in numeous sudies boh in conducive and non-conducive media. Vaious exenal impacs conneced wih oaion, shea flows, hemal convecion, ec. have been aken ino accoun in hese sudies. Fo example, Moiseev e al. (983) have shown using equaions (9) o () ha in L ) scales insabiliy aises leading o he geneaion (enhancemen) of $ coheen voical sucues of yphoon ype. We believe ha simila pocesses of geneaion (sabilizaion) also occu in he case of longiudinal voices enhancemen in a ship wake.
This follows fom he wok of Novikov (996) who has shown ha he change in he velociy ciculaion even in he pesence of a fee suface and sufacan is deemined only by he ciculaion of viscous foce * eff i' i (i vaies fom o 3). In ou wok, we have inoduced an effecive viscosiy insead of kinemaic one ino Novikov's expession fo velociy ciculaion. Theefoe, if ubulence possesses helical popeies, and he effecive ubulen viscosiy can be educed, he voiciy will be mainained long enough. We have shown ha he widening of a ubulen ship wake can be elaed o he eff heliciy magniude while he effecive viscosiy deceases fom he cene of he wake o is peiphey. Accodingly, he ae of is widening deceases while he lifeime of he ship wake inceases. Refeences Belian, A.V., Moiseev, S.S. and Chkheiani, O.G. On eddy viscosiy in helical ubulence. Physics Doklady, v. 39, No., 3-5, 994. Belian, A., Chkheiani, O., Golbaikh, E. and Moiseev, S. Helical ubulence: ubulen viscosiy and insabiliy of he second momens. Physica A, 258, No. -2, 55-68, 998. Benilov, A., Bang, G., Safay, A., and Tkachenko, I. Ship Wake Deecabiliy in he Ocean Tubulen Envionmen, in 23d Symposium on Naval Hydodynamics, Val de Reuil, Fance, -5, 2. Bikhoff G, Zaanonello EH. Jes, Wakes and Caviies. NY: Academic Pess, 957. Banove, H., A. Eidelman, E. Golbaikh and S. Moiseev. Tubulence and Sucues, Academic Pess, NY, Boson, London, 999, 265 p. Bissaud, A., Fisch,., Leoa, J., Lesieu, M. and Mazue, A. Heliciy cascades in fully developed isoopic ubulence. Phys. of Fluids, 6, No 8, 366-367, 973. Chkheiani, O.G., Moiseev, S.S., Peosyan, A.S. and Sagdeev, R.Z. The lage scale sabiliy and self-oganizaion in homogeneous ubulen shea flow. Physica Scipa, 49, 24-22, 994. Golbaikh, E., Chkheiani, O., and Moiseev, S. The ole of heliciy in ubulen MHD flows. JETP, 87, No., 95-, 998.
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