Numerical Study of the Momentumless Wake of an Axisymmetric Body

Transcription

1 43rd AIAA Aerospace Scences Meetng and Exhbt Janary 2005, Reno, Nevada AIAA Nmerca Stdy of the Momentmess Wae of an Axsymmetrc Body Meng-Hang L * and Ana I. Srvente Unversty of Mchgan, Ann Arbor, MI, A second order trbence cosre s sed to cacate the trbent fow n a swrng axsymmetrc momentmess wae. The sotons are compared wth recent expermenta and nmerca data. Smatons correspondng to the component fows (.e. bare body wae and et) are aso performed wth the same trbence cosre. The comparsons wth other expermenta and nmerca data revea that the predctons of the mode for the axsymmetrc et are qte good athogh arger dscrepances exst for the Reynods stress tensor for both axsymmetrc wae and zero-momentm wae. Nomencatre (U,V,W) = mean veocty components (,v,w ) = veocty fctatons = Reynods stress tensor = Trbent netc energy Re = Reynods nmber S = Swr nmber L = body ength D = body dameter Dp = Propeer dameter ν = nematc vscosty J = Jacoban R = Jet rads (C s, C 1, C 2 ) = mode constants n the transport eqaton of the Reynods stresses (C, C 1, C 2 ) = mode constants n the -eqaton Uc = centerne veocty U 0 = free stream veocty r 1/2 = trbent ength scae r b = mean veocty ength scae U c1 = veocty defct I. Introdcton He wae of a body s characterzed by a momentm defct and many past stdes have characterzed the Tdeveopment and decay of the mean and trbence veocty and ength scaes representatve of sch fows. The wae of a three-dmensona body drven by a propeer or a et, wth appendages and contros n operaton, s easy among the most compex of shear fows. There are very few detaed anaytca or nmerca stdes, or expermenta measrements, n the swrng wae of sef-propeed bodes becase they present a chaenge to modeng as we as nstrmentaton. * Gradate Research Assstant, Nava Archtectre and Marne Engneerng Department. Assstant Professor, Nava Archtectre and Marne Engneerng Department, asrv@engn.mch.ed, AIAA Senor Member. 1 Copyrght 2005 by the, Inc. A rghts reserved.

2 Waes wth zero momentm excess are characterstc of sef-propeed movement, where the body drag s canceed by the thrst provded by the means of propson, generay a et or a propeer. The stdy of sch waes s hndered by the mted expermenta data avaabe. Ths s especay the case, as mentoned by a nmber of expermenta and theoretca stdes, when notng that these fows are hghy nfenced by varos condtons of the experments, sch as trbence eve at the net or possbe sma devatons from zero momentm. The pane momentmess wae (.e. Cmbaa and Par 1, Par and Cmbaa 2, etc.) has however receved more attenton n the past than the axsymmetrc momentmess wae. The stdes of Rdanovc 3, Wang 4, Nadascher 5, n the wae of a crcar ds wth a et at ts center concded that the far wae resembes decayng homogeneos trbence. The stdy of Hgch and Kbota 6 was concerned wth the wae of a thn crcar tbe nteractng wth a et adsted to prodce a zero net axa momentm. The stdes of Schetz and Jabows 7, Ferry and Pqet 8, Fare 9, Hyn and Pate 10 and Srvente and Pate 11,12 are among the few expermenta stdes on axsymmetrc zero momentm waes of streamned axsymmetrc bodes sef-propeed wth ets or propeers. Other stdes (sch as the one by Kostomaha and Lesnova 13 ) were condcted to expermentay measre the trbent swrng wae behnd a sphere wth compete and parta drag compensaton. A few expermenta stdes concernng momentmess waes n stratfed fows exst n the teratre, sch as Schooey and Stewart 14, Greath and Brandt 15 and Chernyh et a. 16 among others. Smarty stdes have been condcted for axsymmetrc momentmess waes n the past The rests from them seem to depend strongy on the trbence mode empoyed n the anayss. A smmary of the rests of the varos stdes can be fond n Srvente and Pate 11. Few attempts at nmercay modeng ths type of fows can be fond n the teratre sch as the stdes of Chernyh et a. 22 of a swrng momentmess trbent wae, Chernyh and Voropayeva 23 where the evoton of a zero momentm wae n a neary stratfed fd was carred ot by sng a herarchy of the second order mathematca mode. It was concded that a satsfactory agreement wth expermenta data can be obtaned when dfferenta transport eqatons for norma and one of the tangenta Reynods stresses s sed aong wth non-eqbrm agebrac reatons for other tangenta stresses. Ths paper reports on the nmerca smaton of the wae of a body of revoton at zero ncdence, wth a propeer n operaton. It docments the deveopment of the momentmess wae correspondng to the expermenta stdy of Hyn and Pate 10. The nmerca method empoyed soves the Reynods-Averaged Naver-Stoes (RANS) eqatons for an ncompressbe fd n generazed crvnear coordnates and a second order trbence cosre s sed. The performance of the trbence mode s evaated n ths stdy by frst comparng ts rests to those reported by Chernyh et a. for a swrng momentmess wae compted wth a second order cosre that combned dfferenta transport eqatons wth agebrac expressons for the norma and Reynods stresses. Ther comparsons to expermenta data ndcated very good agreement. Secondy comparsons of the nmerca smaton correspondng to the database of Hyn and Pate are performed and the rests compared wth some nmerca smatons of the same fow performed by Stern et a. 24 sng a κ mode. The sef-propeed wae can be vewed n a very smpstc manner as the combnaton of two dstnct shear ayers (.e. bondary ayer and propeer or et fow). The performance of the second order cosre s assessed by nmercay smatng the wae correspondng to the same body sed n the swrng axsymmetrc zero momentm wae of Hyn and Pate bt wthot the propeer. The expermenta data correspondng to the bare body condton s sed for comparsons. Fnay smatons of a fy sbmerged crcar et are aso condced and the rests compared to past expermenta data. The prpose of ths stdy s two fod. Frst to nmercay nvestgate the mometnmess axsymmetrc wae of a propeer-drven body wth a second order cosre. Secondy ths stdy ams at assessng the rests obtaned from the smatons of varos canonca fows (.e. axsymmetrc momentmess wae, axsymmetrc wae and et) by sng the same second order trbent cosre n a cases. II. Nmerca Methods The method sed soves the Reynods-averaged Naver-Stoes (RANS) eqatons n generazed crvnear coordnates. The veocty components are however expressed n Cartesan coordnates, foowng a parta transformaton approach. The governng eqatons are dscretzed n space on a non-staggered grd sng second order approxmatons. The convectve terms n the momentm eqatons are dscretzed by sng second-order accrate pwnd dfferencng, whe the pressre gradent and vscos terms n the momentm eqatons and the dvergence operator n the contnty eqaton mae se of second order accrate centra fnte dfferencng. A dscrete pressre-posson eqaton s sed to enforce the ncompressbty constrant. The formaton of Sotropoos and Abdaah 25, desgned to avod the odd-even decopng mpct on non-staggered grds, was sed. The dscrete momentm eqatons are ntegrated n tme to steady state by sng a for-stage expct Rnge- Ktta scheme. The pressre-veocty eqaton s soved sng the aternate-drecton-mpct (ADI) approxmate 2

3 3 factorzaton method to acceerate ts convergence, and the Thomas agorthm. Conseqenty the pressre-posson eqaton s transformed nto a dffson-e evoton eqaton. Convergence acceeraton technqes, sch as oca tme steppng and mpct resda smoothng, are sed to enhance the error dampng propertes of the tme marchng procedre. Ths nmerca methodoogy has been tested for a wde range of amnar fows and trbent fows wth varos trbence cosres The comparsons of the compted sotons wth other nmerca rests and expermenta data demonstrated ts abty to smate those fows wth great accracy. In ths stdy a second order trbence cosre s sed. The Cartesan components of the Reynods stress tensor,, are soved at the same nodes where the veocty vector s soved at. The transport eqatons for the Reynods stresses and the -eqaton can be wrtten n generazed crvnear coordnates, where J s the Jacoban of the geometrc transformaton, as foows: (2) C P C C J J V t (1) P 3 2 P C 3 2 C 3 2 J C J J g J U U V t 2 1 m n n m x x 2 1 m n m n m n x m x s x x n n = = νδ δ δ δ ν where V are the contravarant components of the mean veocty, whe U are the mean Cartesan veocty components and ν s the nematc vscosty. The frst and second terms n the rght hand sde of eqaton (1) represent the prodcton by mean shear P and the vscos dffson respectvey and can be compted exacty wthot modeng. The thrd term n the rght hand sde of (1) represents the Reynods stress trbent dffson and s modeed sng Hanac and Lander 29 approach. The ast term n the rght hand sde of eqaton (1) represents the pressre-stran correaton term and s modeed accordng to the Lander and Shma 30 proposa. In eqaton (1) C s s a mode constant, C 1 and C 2 are fnctons of the second and thrd ansotropy tensor nvarants, δ s Kronecher's deta, P s the prodcton of trbence netc energy,, s the dsspaton rate of trbence energy and the oca sotropy assmpton s foowed to compte the rate of dsspaton tensor. C, C 1 and C 2 are mode constants n eqaton (2). The atter eqatons (1) and (2) are dscretzed n space and ntegrated n tme foowng a smar procedre to that ndcated earer for the momentm eqatons. That s a second-order pwnd fnte dfferencng scheme s sed for the convectve terms and second-order accrate centra fnte dfferences are sed for the rest of the terms. A for-stage Rnge-Ktta method s aso sed to advance the dscrete eqatons n tme. In ths stdy the mode constants of eqatons (1) and (2) were C 1 =1.8, C 2 =0.6, Cs=0.22, C =0.18, C 1 =1.45, and C 2 =1.90. A. Comptatona Grd and Bondary Condtons In ths stdy the comptatona doman correspondng to the axsymmetrc bare body wae and the axsymmetrc momentmess waes extends 9.38 body engths downstream and 1.34 body enghts n the transverse drecton. The nmerca grd conssts of grd nodes n the axa, rada and crcmferenta drectons respectvey. The grd nes are stretched from the net downstream and from the centerne toward the oter bondary by sng a hyperboc tangent stretchng fncton. The comptatona doman correspondng to the crcar et extends 150 et dameters downstream and 100 et dameters n the transverse drecton. The nmerca grd aso conssts of grd nodes n the axa, rada and crcmferenta drectons respectvey. A smar stretchng to that mentoned earer for the wae grds was aso sed. A grd ndependence stdy was performed. The cross-secton vew of the grd s shown n Fgre 1.

4 Fgre 1. Cross-secton vew of the comptatona grd The bondary condtons were specfed as foows. The net bondary condton for the axsymmetrc wae and momentmess wae was ocated at x/l=1.00 and x/l=1.02 respectvey, and the data taen from the expermenta rests of Hyn and Pate 10. Inet condtons are specfed at x/d=3 for the crcar et and the data from the expermenta stdy of Sam et a. 31 sed. On the symmetry bondares mrror-mage refecton for the grd and the fow varabes s sed n a cases. The ext bondary condton s mposed n a cases by assmng zero streamwse dffson and the oter bondary condtons assme free-stream recovery. The smatons presented n ths stdy correspond to Reynods nmbers based on the body ength and the free stream veocty of Re= for the axsymmetrc waes. The Reynods nmber based on the et dameter and the centerne veocty at the ext s Re= for the crcar et. The CFL nmber sed n these comptatons s 1.2 and 1.5 for the Reynods stress eqatons. In a cases the resdas, defned by the smmaton of dfferences between the crrent and the prevos teratons, were redced by at east for orders of magntde. III. Rests and Dscsson Ths secton s sbdvded n three man parts. The frst part corresponds to the smatons of an axsymmetrc swrng momentmess wae and the correspondng comparsons wth past smatons and expermenta data. In order to assess the performance of the Reynods stress cosre sed n ths stdy, the ast two parts of ths secton present smatons correspondng to an axsymmetrc wae and a et. Those fows can be vewed as component fows of the correspondng momentmess wae and hence the abty of the second order to predct the deveopment of each of them seems crtca. A. Swrng Momentmess Wae A sef-propeed body s characterzed by a baance between the thrst generated by the propson system (e.g. propeer, et, etc.) and the drag generated by the body. The expermenta rests 13 correspondng to the axsymmetrc zero-momentm trbent wae behnd a sphere are sed frst. A swrng et ssng at the bac of the sphere compensates the sphere s drag. These experments were carred ot n a ow-trbence cosed wnd tnne. The test secton was 4m ong and 0.4m 0.4m n cross secton where a 25mm sphere wth a speca 6mm dameter nozze bt nto the sphere to form a swrng et was paced. The measrements were taen wth snge and crossed hot wre probes to measre the three veocty components as we as the sx components of the Reynods stress tensor. The experments were carred ot at a Reynods nmber, Re, based on the sphere dameter and the free-stream veocty of and varos swr nmbers, S, defned as the rato of tangenta to axa momentm at the ext: M S = M x θx ( R D) (3) 4

5 where M θ x s the axa fx of tangenta momentm, M x s the axa fx of axa momentm, D s the body s dameter and R s the nozze or et rads. The case chosen for ths smaton corresponds to a Swr nmber of 2.2. Chernyh et a. 22 performed nmerca smatons to vadate the expermenta data of Kostomaha and Lesnova 13. The nmerca mode they sed was a RANS sover wth varos (fve dfferent modes) second order trbence cosres. The most compete mode (mode 1, as t was referred to by the athors) was based on the Reynods stress cosre of Lander and Morse 32. Mode 2 corresponds to a smpfed verson of mode 1. Mode 3 s based on the se of the agebrac representatons of the tangenta Reynods stresses. Mode 4 maes se of the agebrac representatons of the norma and one of the tangenta stresses and mode 5 based on the dfferenta transport eqatons for the norma stresses and the correspondng agebrac expressons for the tangenta stresses. The best agreement wth the expermenta data was fond from the se of Mode 5. Some of those nmerca rests are sed for comparson wth the nmerca smatons performed n ths stdy to vadate the expermenta data of Kostomaha and Lesnova (1995), where the expermenta data correspondng to x/d=20 was sed as net profe. In a cases the sphere dameter and the free stream veocty are sed as correspondng mean axa ength and veocty scaes. The rests correspondng to the veocty defect (.e. maxmm dfference between the free-stream veocty and the axa mean veocty) and the tangenta veocty components are shown n Fgre 2. In that fgre the expermenta data of Kostomaha and Lesnova (1995) at x/d=50 and the nmerca rests from Chernyh et a. (1998) correspondng to ther best trbence mode for x/d=50 and 100 are compared wth the present smatons. It can be observed that the predcton of the veocty n the oter regon of the shear ayer s consderaby better than that n the nner regon. The centerne veocty s greaty over predcted by the present method. In contrast the trbence approach empoyed by Chernyh et a. does a mch better ob a across the fow, bt more specfcay n the neghborhood of the centerne. The comparsons correspondng to the trbent fctaton ntenstes of the veocty components are presented n Fgre 3. The dstrbtons of the tangenta stresses v and vw are shown n Fgre 4. The predcton of the streamwse and rada norma stresses foows a smar trend to that seen for the veocty decay n the oter regon of the fow where the recovery of freestream condtons taes pace. That s the oter regon s better predcted than the centra porton of the fow argey domnated by strong veocty gradents. The expermenta rests as we as the nmerca smatons of Chernyh et a. ndcate that by x/d=50 the trbence s qte sotropc wth smar eves of trbent fctaton ntenstes for a the veocty components. The predcton from the trbence cosre empoyed n ths stdy however shows arger vaes for the rada veocty fctatons than those correspondng to the streamwse and tangenta veocty fctatons respectvey, showng an mportant degree of ansotropy. The tangenta stresses shown n Fgre 4 dspay a fary good agreement wth the expermenta data for vw whe the dfferences are mportant for v. Those dfferences are consderaby more sgnfcant for the present mode than that empoyed by Chernyh et a. a) b) Fgre 2. Comparson of a) the cacated dmensoness profes of the mean veocty defect and b) the tangenta veocty wth the expermenta data of Kostomaha et a. (1995) and the nmerca rests of Chernyh et a. (1998). 5

6 a) b) c) Fgre 3. Comparson of the cacated dmensoness profes of the fctaton ntenstes of the veocty components wth the expermenta data of Kostomaha et a. (1995) and the nmerca rests of Chernyh et a. (1998). a) b) Fgre 4. Comparson of the cacated dmensoness profes of the tangenta Reynods stresses a) v and b) vw wth the expermenta data of Kostomaha et a. (1995) and the nmerca rests of Chernyh et a. (1998). 6

7 To evaate the abty of the sed second order cosre to smate the evoton of the momentmess swrng wae, the decay of the axa and tangenta veocty scaes was compted and s shown n Fgre 5. Comparsons of the rests obtaned from the expermenta stdy of Kostomaha et a. (1995) and two dfferent nmerca sotons (correspondng to modes 2 and 5; refer to secton II) of Chernyh et a. (1998), as we as wth other expermenta rests (Fare, 1995 and Schetz and Jabows, 1975) correspondng to axsymmetrc momentmess waes wth dfferent nta condtons, as we as Re and S nmbers, are aso shown n Fgre 5. a) b) Fgre 5. Evoton of wae veocty scaes a) centerne veocty and b) maxmm tangenta veocty, correspondng to axsymmetrc momentmess waes From Fgre 5a t can be observed that wthn 10D regardess of the dfferences n the nta bondary condtons correspondng to each of the fows represented, the centerne veocty becomes wthn 3% of the free-stream veocty. The sma-defect assmpton that s conventonay made n cassca smarty theory of waes s met qte eary by ths canonca fow. The agreement of the nmerca predctons among themseves and wth the array of expermenta rests presented n Fgre 5a s rather good. On the other hand the decay of the tangenta veocty taes pace at a comparatvey mch sower pace than that observed for the streamwse veocty scae. Frthermore the predctons correspondng to the nmerca smatons of the mode 2 of Chernyh et a. (1998) and the present smatons seem to have a smar sope, however they try mss the expermenta trend as we as the rests obtaned by the mode 5 of Chernyh et a. The atter s n good agreement wth the expermenta data. The evoton correspondng to the ntensty and ength scae of s potted n Fgre 6, where the evoton of the maxma correspondng to the streamwse veocty fctatons s presented. As the maxmm vae of occrs at the centerne, the correspondng ength scae, r 1/2, s taen as the rada dstance where the vae of the axa trbence ntensty fas to haf of ts maxmm. The max taes pace at the centerne p to x=1d. From approxmatey x=10d the decay of the streamwse veocty fctatons seem to tae pace at a faster pace emphaszed by the sope shown by the expermenta data, as we as the nmerca predctons. The evoton of the trbence ength scae seems to be we predcted by any of the smatons. a) b) Fgre 6. Evoton of the ntensty scae of and correspondng trbent ength scae for axsymmetrc momentmess waes 7

8 The rests of Hyn and Pate (1991) correspondng to a detaed expermenta stdy of an axsymmetrc body sef-propeed by a three-bade propeer were compared wth the expermenta data of Srvente and Pate (2000). The atter data base correspond to the same body as that sed by Hyn and Pate bt sef-propeed wth a swrng et nstead for the same Re and S. Both cases were smated wth the present nmerca approach and comparsons are shown for the evoton of the centerne veocty and the maxmm tangenta veocty n Fgre 7. Comparsons of veocty and trbence profes for two dfferent ocatons x/d p =1.87 and 7.12, where D p represents the dameter of the propeer, are shown n Fgre 8. Both fgres 7 and 8 ncde some of the nmerca predctons of the Hyn and Pate data performed by Stern et a. (1991) wth a κ mode. Fgre 7 ndcates that the Reynods stress cosre performs better n the predcton of the evoton of the decay of the maxmm tangenta veocty than of the centerne veocty, whch seems to be better predcted by the sotropc trbence cosre. Fgre 7. Comparson of the evoton of the wae veocty scaes correspondng to the axsymmetrc momentmess wae data of Hyn and Pate (1991) and Srvente and Pate (2000) wth the present smaton The rests presented n Fgre 8 show n genera a mch better agreement of the present smaton than that correspondng to the κ mode for both veocty and trbence predctons. It shod be noted that the rada veocty fctatons are arger than the streamwse and tangenta veocty fctatons respectvey by x/d p =1.87, whe the trend has changed ( >v >w ) by 7.12D p. B. Wae of an Axsymmetrc Body The wae of an axsymmetrc body s characterzed by an axa momentm defct whch corresponds to the drag of the body. The data of Hyn and Pate (1991) aso ncdes measrements of the three veocty components and the sx Reynods stresses of the wae of the axsymmetrc body wthot the propeer, for the same Reynods nmber as that empoyed for the zero-momentm wae measrements. The abty of the second order cosre to propery predct the deveopment of the wae of ths bare h was tested as we. The comparsons of the veocty and trbence profes correspondng to x/d p =1.87 and 7.12 are presented n Fgre 9, aong wth the correspondng smatons performed by Stern et a. (1991) sng a κ mode. The rests correspondng to the decay of the centerne veocty and of r 1/2 are shown n Fgre 10. From Fgre 9 t can be observed a sgnfcant overpredcton of the norma stresses n comparson to the trbence netc energy obtaned by the κ mode empoyed by Stern et a. The atter comptatons were started over the body and extended nto the wae. The present smatons on the other hand made se of the expermenta data gathered n the near wae (x/dp=0.3675) as net bondary condton. The evoton of the centerne veocty seems to be better predcted by the κ mode, at east n the near wae. The rests obtaned from the present smaton were aso compared wth those cacated sng the comerca software Fent wth a second order cosre. The dscrepances obtaned for the norma and Reynods stresses n comparson to the expermenta data of Hyn and Pate were comparabe to those shown n Fgre 9. The mpact of the net bondary condton sed n ths comptaton was not tested. That s assessment of the rests obtaned wth the next avaabe expermenta profe was not done. The gradents n the near wae regon are mportant and conseqenty the expermenta errors assocated wth ntrsve hot-wre anemometry are arger than n the far wae. The stdy of the effect of the net bondary condton deserves frther nvestgaton. 8

9 Fgre 8. Comparsons of the veocty and trbence profes of the momentmess swrng wae for x/d p =1.87 and 7.12 wth the expermenta data of Hyn and Pate (1991) and the nmerca mode of Stern et a. (1991) C. Isoated Crcar Jet Smatons correspondng to an axsymmetrc et were aso done. The expermenta data of Sam et a. (1967) at x/d=3 was sed as net bondary condton. The Reynods nmber n ths case was Re= The prpose of ths nmerca experment was to compare the performance of the second order trbence cosre for ths canonca fow, whch can be vewed n a smpstc manner as representatve of the means of propson of a sef-propeed body. Nonetheess the condtons of the experment of Sam et a. dd not correspond to the axa momentm constrants to match the Hyn and Pate stdy. Nonetheess the smtons were taed as representatve of the genera trends shown by the trbence cosre for ths type of free shear ayers. 9

10 Fgre 9. Comparsons of the veocty and trbence profes of the wae of an axsymmetrc body for x/d p =1.87 and 7.12 wth the expermenta data of Hyn and Pate (1991) and the nmerca mode of Stern et a. (1991) The rests correspondng to the decay of the centerne veocty and of r b, defned as the rada dstance where the vae of mean axa veocty s haf of the dfference between centerne and free stream veocty are shown n Fgre 11. The trends shown by the centerne veocty are ceary consstent wth other expermenta data. On the other hand the nmerca rests correspondng to r b ntay agree wth the expermenta data of Sam et a. bt there s not enogh expermenta data n the wae to perform a deta vadaton. The nmerca data correspondng to rather wea swrng ets, aso shown n the fgre, ndcates a dfferent evoton for the non-swrng et. The comparsons of the veocty and trbence profes correspondng to varos steamwse ocatons are presented n Fgre 12, aong wth the expermenta data of Sam et a. (1967) and Hssen et a. (1994). Fgre 12 ndcates that the mean veocty becomes sef-smar qte eary by x/d=40, whe the trbence reqres a onger streamwse 10

11 dstance to acheve sef-smarty, at east 75D. The predctons of the norma and trbent stresses are qte good n comparson to the expermenta data. Fgre 10. Comparson of the decay of the centerne veocty and the r b ength scae correspondng to the wae of the axsymmetrc body of Hyn and Pate (1991) Fgre 11. Comparson of the decay of the centerne veocty and the r b ength scae correspondng to an axsymmetrc et IV. Smmary and Concsons A Reynods-averaged Naver-Stoes sover n concton wth a second order Reynods stress cosre was sed to cacate the trbent fow of a swrng axsymmetrc momentmess wae. The compted rests were compared wth varos expermenta and nmerca data bases. The same nmerca mode was aso se to nmercay smate the so caed component fows of the zero-momentm wae, that s the wae of the bare body and the et e type of fow generated by the means of propson. The foowng concsons can be drawn from the present stdy: 11

12 Fgre 9. Comparsons of the veocty and trbence profes of the axsymmetrc et for varos ocatons wth the expermenta data of Hssen et a. (1994) 1. The present comptatons reprodce, wth remarabe accracy, the evoton of the mean and trbent characterstcs correspondng to the axsymmetrc et fow stded as one of the component fows of the axsymmetrc momentmess wae. 2. These nmerca experments showed that the smatons correspondng to the axsymmetrc momentmess wae and wae of the bare body tend to overpredct the norma and Reynods stresses n comparson wth expermenta data. The trends fond n both comptatons were remaraby smar. Ths seems to ndcate that the nabty of the second order cosre to accratey captre the correct evoton of the zero-momentm wae s ned to ts performance n the smaton of the bare body wae. 3. One of the modes empoyed by Chernyh et a. (1998) for the smaton of a swrng axsymmetrc momentmess wae was abe to prodce better predctons than the present mode for the norma stresses of the swrng axsymmetrc momentmess wae. 12

13 V. References 1. Cmbaa, J.M., and Par, W.J., (1990), Expermenta Investgaton of the Trbent Strctre n a Two Dmensona Momentmess Wae, Jorna of Fd Mechancs, Vo. 213, pp Par, W.J., and Cmbaa, J.M., (1991), Effect of et Inecton Geometry on Two-Dmensona Momentmess Waes, Jorna of Fd Mechancs, Vo. 224, pp Rdanovc, M., (1963), "Wae wth Zero Change of Momentm Fx," Ph.D. Dssertaton, Dept. of Mechanca Engneerng, Unversty of Iowa, Iowa Cty, IA. 4. Wang, H., (1965), "Fow Behnd a Pont Sorce of Trbence," Ph.D. Dsseraton, Department of Mechanca Engneerng, Unversty of Iowa, Iowa Cty, IA. 5. Nadascher, E., (1965), "Fow n the Wae of Sef-Propeed Bodes and Reated Sorces of Trbence," Jorna of Fd Mechancs, Vo. 22, pp Hgch, H., and Kbota, T., (1990), "Axsymmetrc Waes Behnd a Sender Body Incdng Zero- Momentm Confgratons," Physcs of Fds A-Fd Dynamcs, Vo. 2, pp Schetz, J.A., and Jabows, A.K., (1975), Expermenta Stdes of the Trbent Wae behnd Sef- Propeed Sender Bodes, AIAA Jorna, Vo. 13(12), pp Ferry, M., and Pqet, J., (1987), "Sage Vsqex Lontan d'n Corps Sos-Marn Atopropse," SIREHNA, Rapport d'ettde 86/14R, Nantes, France. 9. Fare, T.M., (1995), Etde expermentae d sage trbent d n corps a symetre de revoton atopropse par hece. Ph.D. Thess, No 95-01, Ecoe Centrae de Lyon. 10. Hyn, B.S., and Pate, V.C., (1991), Measrements n the Fow arond a Marne Propeer at the Stern of an Axsymmetrc Body, Experments n Fds, Vo. 11, pp Srvente, A.I. and Pate, V.C. (2001), Trbence Strctre of the Wae of a Sef- Propeed Body wth and wthot Swr, AIAA Jorna, Vo. 39(12), pp Srvente, A.I. and Pate, V.C. (2000), Wae of a Sef-Propeed Body. Part I: Momentmess Wae, AIAA Jorna, Vo. 38(4), pp ; Part II: Momentmess Wae wth Swr," AIAA Jorna, Vo. 38, pp Kostomaha, V.A., and Lesnova, N.V., (1995), Trbent Swrng Wae behnd a Sphere wth Compete or Parta Drag Compensaton, Jorna of Apped Mechancs and Technca Physcs, Vo. 36(2), pp Schooey, A.H., and Stewart, R.W., (1963), Experments wth a Sef-Propeed Body Sbmerged n a Fd wth Vertca Densty Gradent, J Jorna of Fd Mechancs, Vo. 15, pp Greath, H.E., and Brandt, A., (1985), Experments on the Generaton of Interna Waves n a Strafed Fd, AIAA Jorna, Vo.23(5), pp Chernyh, G.G., Iyshn, B.B., and Voropayeva, O.F. (2003), Ansotropy Decay of Trbence n a Far Momentmess Wae n a Lneary Stratfed Medm, Rss. J. Nmer. Ana. Math. Modeng, Vo. 18, pp Tennees, H., and Lmey, J. L., (1972), A Frst Corse n Trbence, MIT Press, Cambrdge, MA. 18. Fnson, M.L., (1975), Smarty Behavor of Momentmess Trbent Waes, Jorna of Fd Mechancs, Vo. 71, pp Hassd, S., (1980), Smarty and Decay Law of Momentmess Waes, Physcs of Fds, Vo. 23, No. 2, pp Ferry, M., and Pqet, J., (1987), "Sage Vsqex Lontan d'n Corps Sos-Marn Atopropse," SIREHNA, Rapport d'ettde 86/14R, Nantes, France. 21. Srvente, A. I., (1996), Wae of an AxsymmetrcBody Propeed by a Jet wth and wthot Swr, Ph.D. Dssertaton, Dept. of Mechanca Engneerng, Unv. of Iowa, Iowa Cty, IA. 22. Chernyh, G.G., Demenov, A.G., and Kostomaha, V.A., (1998), Nmerca Mode of a Swrng Momentmess Trbent Wae, Rss. J. Nmer. Ana. Math. Modeng, Vo. 13, pp Chernyh, G.G., and Voropayeva, O.F., (1999), Nmerca Modeng of Momentmess Trbent Wae Dynamcs n a Lneary Stratfed Medm, Compters & Fds, Vo. 28, pp Stern, F., Toda, Y., and Km, H.T., (1991), Comptaton of Vscos Fow arond Propeer-Body Confgratons: Iowa Axsymmetrc Body, Jorna of Shp Research, Vo. 35, pp Sotropoos, F., and Abdaah, S., (1991), The dscrete contnty eqaton n prmtve varabe sotons of ncompressbe fow, J. of Comptatona Physcs, Vo. 95, pp Sotropoos, F. and Pate, V.C., (1992), Fow n Crved Dcts of Varyng Cross-Secton, IIHR Report no

14 27. Snha, S.K., Sotroposos, F. and Odgaard, A.J., (1998), Three-dmensona Nmerca Mode for Fows Throgh Natra Rvers, Jorna of Hydrac Engneerng, Vo. 124, pp Km, K., Srvente, A.I. and Bec, R.F., (2005), The compementary RANS eqatons for the Smaton of Vscos fows, Internatona Jorna for Nmerca Methods n Fds (n press). 29. Hanac, K., and Lander, B.E., (1976), Contrbton toward a Reynods-Stress Cosre for Low- Reynods-Nmber Trbence, Jorna of Fd Mechancs, Vo. 74, pp Lander, B.E., and Shma, N., (1989), Second-Moment Cosre for the Near-Wa Sbayer. Deveopment and Appcaton, AIAA Jorna, Vo. 27, pp Sam, S., Carmody, T., and Rose, H., (1967), Jet Dffson n the Regon of Fow Estabshment, Jorna of Fd Mechancs, Vo. 27, pp Lander, B.E., and Morse, A., (1979), Nmerca Predcton of Axsymmetrc Free Shear Fows wth a Second-Order Reynods Stress Cosre, In Trbent Shear Fows, 1, Sprnger, Bern, pp