Chapter 1 Introduction of boundary layer phenomena

Transcription

1 Chaper 1 Inrodcon of bondary layer phenomena T-S Le Jan. 13, 018 Man Topcs Hsory of Fld Mechancs Developmen Idea of Bondary Layer Bondary Layer Eqaons 1

2 Fld Mechancs Developmen Hsory Ideal fld: Invscd flow (zero drag) Lagrangan/Eleran Poenal flow (Invscd+Irroaonal) Lagrange (1736~1813) Laplace Jean (1749~187) D'Alember Isaac Newon 1738 Leonhard 1687 Danel Eler Bernoll Real fld : Vscosy Vscos flow Lamnar/ Trblen flow (187~1845) Naver Cachy Posson S. Venan Soes Naver- Soes Eqaon Osborne Reynolds 184~191 Bondary layer Ldwg Prandl ~1919 Lord Raylegh Hydrodynamc nsably: Taylor Raylegh Kelvn Helmholz Benard cells EFD: Ho wre Ho flm LDV MEMS PIV 000s 1990s MD 1980s DNS 1960~70s CFD Panel Trblence mehod modelng Rchard Feynman has descrbed rblence as he mos mporan nsolved problem of classcal physcs. Bernoll Eqaon

3 Bernoll eqaon (I) BERNOULLI S EQUATION (II) & negraon BERNOULLI S EQUATION AERODYNAMICS (W-1-) 3

4 BERNOULLI S EQUATION FOR AN IRROTATION FLOW ncompressbl e Seady Irroaonal No gravy AERODYNAMICS (W-1-.1) Unseady Bernoll Eqaon Ths s no a very sefl resl n general snce vs/ can change dramacally from one pon o anoher; o se hs n pracce we need o be able o draw sreamlne shapes a each nsan n me. I wors especally for smple cases sch as mplsvely sared confned flows where sreamlnes have he same shape a each nsan and we are neresed n me reqred o sar he flow. 4

5 Eercse Flow o of a long ppe conneced o a large reservor, (1)fnd he seady sae velocy v n he ppe afer he he ransen sage ()fnd he ransen velocy v n he ppe changng wh me drng he ransen sage D'Alember's parado In fld dynamcs, d'alember's parado s a conradcon reached n 175 by French mahemacan Jean le Rond d'alember. D'Alember proved ha for ncompressble and nvscd poenal flow hedrag force s zero on a body movng wh consan velocy relave o he fld. Zero drag s n drec conradcon o he observaon of sbsanal drag on bodes movng relave o flds, sch as ar and waer; especally a hgh veloces correspondng wh hgh Reynolds nmbers. Jean le Rond d'alember ( ) hps://en.wpeda.org/w/d%7alember%7s_parado 5

6 Lagrangan/Eleran Descrpon n Fld Mechancs Assgnmen: Wach he vdeo abo Lagrangan/Eleran Descrpon n Fld Mechancs hps:// (MP4) METHODS OF DESCRIPTION Lagrangan descrpon => Sysem Eleran descrpon => Conrol volme Ch 1-6

7 Lagrangan Descrpon Aenon s focsed on a maeral volme (MV) and follow ndvdal fld parcle as move. The fld parcle s colored, agged or denfed. Deermnng how he fld properes assocaed wh he parcle change as a fncon of me. Eample: one aaches he emperare-measrng devce o a parclar fld parcle A and record ha parcle s emperare as moves abo. T A = T A ()=T ( o,y o,z o, ) where parcle A passed hrogh coordnae ( o,y o,z o ) a o The se of may sch measrng devces movng wh varos fld parcles wold provde he emperare of hese fld parcles as a fncon of me. Ch 1- Eleran Descrpon Aenon s focsed on he fld passng hrogh a conrol volme (CV) fed n he space. Obanng nformaon abo he flow n erms of wha happens a he fed pons n space as he fld flows pas hose pons. The fld moon s gven by compleely prescrbng he necessary properes as a fncons of space and me. Eample: one aaches he emperare-measrng devce o a parclar pon (,y,z) and record he emperare a ha pon as a fncon of me. T = T (, y, z, ) => feld concep. The ndependen varables are he spaal coordnaes (, y, z) and me Ch 1-7

8 Feld Represenaon of flow A a gven nsan n me, any fld propery ( sch as densy, pressre, velocy, and acceleraon) can be descrbed as a fncons of he fld s locaon. Ths represenaon of fld parameers as fncons of he spaal coordnaes s ermed a feld represenaon of flow. The specfc feld represenaon may be dfferen a dfferen mes, so ha o descrbe a fld flow we ms deermne he varos parameer no only as fncons of he spaal coordnaes b also as a fncon of me. EXAMPLE: Temperare feld T = T (, y, z, ) EXAMPLE: Velocy feld V (, y, z, ) v(, y, z, ) w(, y, z, ) Ch 1- Nare and Transformaon of Lagrangan and Eleran Descrpon I s more nare o apply conservaon laws by sng Lagrangan descrpon (e. Maeral Volme). However, he Eleran descrpon (e. Conrol Volme) s preferred for solvng mos of problem n fld mechancs. The wo descrpons are relaed and here s a ransformaon formla called maeral, oal or sbsanal dervave beween Lagrangan and Eleran descrpons. Ch 1-8

9 9 Maeral Dervave (I) Le (,y,z,) be any feld varable, e.g., ρ, T, V=(,v,w), ec. (Eleran descrpon) Observe a fld parcle for a me perod as flows (Langrangan descrpon) Drng he me perod, he poson of he fld parcle wll change by amons, y, z, whle s vale of wll change by an amon As one follow he fld parcle, So whch s called he maeral, oal, or sbsanal dervave. Ch 1-3 z z y y ),, ( ),, ( w v z y z w y v z z y y D D 0 lm Maeral Dervave (II) Use he noaon D/D o emphasze ha he maeral dervave s he rae of change seen by an observer followng he fld. The maeral dervaeve epress a Langrangan dervave n erms of Eleran dervaves. In vecor form, May also se he nde noaon and Ensen s smmaon convenon (.e, smmng over repeaed ndces) o wre Where ( 1,, 3 ) (,y,z) and ( 1,, 3 ) (,v,w) Noe: The repeaed nde ha s smmed over s called a dmmy nde ; one ha s no smmed s called a free nde. ( ) V z w y v D D D D Ch 1-3

10 Poenal Flow Theory Invscd & Irroaonal flow GOVERNING EQUATION FOR IRROTATIONAL, INCOMPRESSIBLE FLOW: LAPLACE S EQUATION Conny eqaon Incompressble: consan For ncompressble flow: here ess a sreamfncon For rroaonal flow: here ess a velocy poenal y= 0 For rroaonal, ncompressble flow: y Laplace s eqaon Laplace s eqaon For rroaonal, ncompressble flow, here are velocy poenal and sreamfncon ha boh sasfy Laplace s eqaon. 0 y 0 y AERODYNAMICS (W_1_6) 10

11 Poenal Flow Theory Governng eqaon for Poenal flow s Laplace eqaon 0 y Laplace s eqaon s a second-order lnear paral dfferenal eqaon. If 1,, 3,, n represen n separae solons of Laplace s eqaon, hen = n s also a solon of Laplace s eqaon. Comple poenal wh conformal mappng Bondary Condon for LAPLACE S EQUATION Bondary Condons: Infne bondary condons: Wall bondary condons: or or 11

12 1 Fld Flow Governng Eqaons Mass conservaon => Conny eqaon Momenm eqaon F=ma=d(mV) /d => Naver Soes Eqaon 1 s Thermaldynamc law (Conservaon of Energy) => Energy eqaon Fld Flow Governng Eqaons Conny eqaon: Momenm (N-S) eqaon: Energy eqaon: 0 ) ( D D p f ) ( ] [ ) ( ) ( T p e e ) (

13 13 p f ) ( ] [ Naver-Soes Eqaons p f ] [ For ncompressble flows, 0 V For ncompressble flows wh consan vscosy, p p f f ] [ For ncompressble,nvscd flds =0, p f ] [ The Eler eqaons 3 Fll N-S eqaon ) ( ) ( T p e e ) ( T C e v ) ( v T p T T C for consan C v & ) ( v T T T C for ncompressble flow wh consan C v & for nvscd flow wh consan C v & ) ( v T T T C ) ( v T p T T C for ncompressble nvscd flow wh consan C v &

14 where where 14

15 Moleclar and Sascal Approaches Flds conss of molecles whose moon s governng by he law of dynamcs. The macroscopc phenomena are assme o arse from he moleclar moon of he molecles. The heory aemps o predc he macroscopc behavor of he fld from he laws of mechancs and probably (or sascal) heory. Sascs => he predced macro fld behavor near an eqlbrm sae For a fld sae no far from eqlbrm, he moleclar and sascal approaches yeld he ranspor coeffcens (sch as he vscosy coeffcen and he hermal condcvy), and he eqaons of mass, momenm and energy conservaon. The heory s well developed for lgh gases, b s ncomplee for polyaomc gas molecles and for lqds. Crre Ch 1-1 The dea of Bondary Layer The occrrence of he parado s de o he negleced effecs of vscosy. In conncon wh scenfc epermens, here were hge advances n he heory of vscos fld frcon drng he 19h cenry. Wh respec o he parado, hs clmnaed n he dscovery and descrpon of hn bondary layers by Ldwg Prandl n (see he arcle: Ldwg Prandl s bondary layer, Physcs Today, 005, 58, no.1, 4-48). Ch

16 Prandl made he hypohess ha he vscos effecs are mporan n hn layers called bondary layers adacen o sold bondares, and ha vscosy has no role of mporance osde. The bondary-layer hcness becomes smaller when he vscosy redces. The fll problem of vscos flow, descrbed by he nonlnear Naver Soes eqaons. p F Usng hs hypohess (and baced p by epermens) Prandl was able o derve an appromae model for he flow nsde he bondary layer, called bondary-layer heory; whle he flow osde he bondary layer cold be reaed sng nvscd flow heory. The prncpal concep of he bondary orgnally sprngs from he parclar form of he fld connm eqaons n whch he dsspaon erms nvolve hgher order dervaves han he neral, advecve erms, e.g. for he Naver Soes eqaons for a non roang fld: p F For flds le ar or waer he coeffcen of vscosy s ofen sffcenly small, n a non-dmensonal sense o be clarfed more formally below, sch ha he physcal effecs of frcon wold seem o be neglgble allowng he neglec of he las erm on he rgh hand sde of he eqaon. 16

17 17

18 Pressre dsrbon of flow over a crclar cylnder Pressre dsrbon for he flow arond a crclar cylnder. The dashed ble lne s he pressre dsrbon accordng o poenal flow heory, reslng n d'alember's parado. The sold ble lne s he mean pressre dsrbon as fond n epermens a hgh Reynolds nmbers. The pressre s he radal dsance from he cylnder srface; a posve pressre (overpressre) s nsde he cylnder, owards he cenre, whle a negave pressre (nderpressre) s drawn osde he cylnder. hps://en.wpeda.org/w/d%7alember%7s_parado 18

19 19

20 0

21 Reynolds epermen sng waer n a ppe o sdy ranson from lamnar o rblence Lamnar ppe flow Re < 100 Transen ppe flow 100<Re<4000. Trblen ppe flow Re>4000. Bondary Layer descrbed by sng vorcy sorce and vorcy sn pon of vew 1

22 Soes Theorem Consder a D nform flow passng hrogh a fla plae, bondary = layer s developng along drecon near he srface. All areas for = regon = abcd are 1 wh he lengh (a) Please calclae he crclaon along he pah abcd and he oal vorcy nsde regon abcd a hree locaons 1, and 3? Compare he oal vorces nsde he regon abcd a locaon 1, and 3. (4%) (b) We now all vorccy s generaed on he bondary of srface. Is any vorcy generaed on he bondary of srface from locaon 1 o locaon? If yor answer s no, please descrbe he reasonng. If yor answer s yes, please descrbe he locaon where he vorcy comes from. (4%) (c) Is any vorcy generaed on he bondary of srface from locaon o locaon 3? Why or why no? (4%) U U U b a c d X b Y 1 3 a c d nda A b a c d A n da Bondary Layer The German physcs Ldwg Prandl sggesed n 1904 ha he effecs of a hn vscos bondary layer possbly cold be he sorce of sbsanal drag. Prandl p forward he dea ha, a hgh veloces and hgh Reynolds nmbers, a noslp bondary condon cases a srong varaon of he flow speeds over a hn layer near he wall of he body. Ths leads o he generaon of vorcy and vscos dsspaon of nec energy n he bondary layer. Bondary-layer heory s amenable o he mehod of mached asympoc epansons for dervng appromae solons. In he smples case of a fla plae parallel o he ncomng flow, bondary-layer heory resls n (frcon) drag, whereas all nvscd flow heores wll predc zero drag. Imporanly for aeronacs, Prandl's heory can be appled drecly o sreamlned bodes le arfols where, n addon o srface-frcon drag, here s also form drag. Form drag s de o he effec of he bondary layer and hn wae on he pressre dsrbon arond he arfol.

23 -Growh of bondary layer de o he vscos effec (from he vewpon of vorcy dynamcs) Non-slp bondary condon: no slp on a wall srface de o vscos effec (Prandl, 1904). How does a bondary layer develop on a wall? Consder flow over he leadng edge of an arfol, how he bondary layer s naed and developed? (Lghhll, 1963) In he neghborhood of he sagnaon pon, he eernal flow velocy U rses from s vale zero a he frs pon of aachmen o he mamm posve vale,, herefore he vorcy-flow ncreases. Downsream of,, he vorcy-flow decreases, rn no he possbly of flow separaon. (Lghhll, 1963) acceleraon deceleraon 46 3

24 Chec a case The concep of vorcy fl 48 4

25 -See he vorcy sorce and sn n he momenm eqaons. Adverse pressre graden; wall as a vorcy sn Favorable pressre graden; wall as a vorcy sorce 5

26 Bondary Layer Eqaons -Bondary layer assmpon and bondary layer eqaons ( Whe, F. M., Vscos fld flow. McGraw-Hll, Chaper 4) Characersc scales of a bondary layer In he prevos dscsson, was arged by non ha near he wall Ths relaon can be frher verfed wh he bondary layer assmpon, whch wll be nrodced n hs secon. Frs of all, le s defne he characersc scales of a bondary layer as follows. Arfol srface 5 6

27 :characersc lengh along he sreamwse drecon :bondary layer hcness :freesream velocy Bondary layer assmpon: (Nearly parallel flow assmpon) Ths assmpon mples ha he bondary layer conanng vorcy s relavely hn compared o he characersc scale of a body, on whch he bondary layer s developed. Therefore, he bondary layer s also referred o as a hn shear layer. In a broad sense, bondary layer s a erm referred o all he shear layers of whch he assmpon s vald, ncldng e, wae and mng layer. Non-dmensonalzed conservaon eqaons for wo-dmensonal, ncompressble flows Conny eqaon: 54 7

28 Momenm eqaons: 55 In a bondary layer, boh of he convecve and vscos dffson effecs shold be consdered. Therefore, Reynolds nmber 56 8

29 Conseqenly, momenm eqaon n he drecon Evalae each erm n he momenm eqaon n he y drecon l O( U V )( U 0 U )( l l V ) ~ O( )( )( U l U 0 U )( l V ) ~ O( U 0 U )( l ) Reynolds nmber Therefore, 58 Compared o he momenm eqaon n he drecon, he momenm eqaon n he y drecon can be gnored. Ths mples ha pressre varaon along he y drecon, across he bondary layer, s nsgnfcan. 9

30 The momenm eqaon n he y drecon redced o: Ths mples ha pressre osde he bondary layer of an arfol srface can be evalaed by sng poenal flow resls f separaon flow (or sall) does no occr on he arfol.. The bondary layer eqaon s referred o he momenm eqaon n he drecon. Reynolds nmber Incompressble D ( ) 0 D

31 The bondary-layer assmpon s applcable when he Reynolds nmber s large. Cases of bondary layer phenomena: 1. wall-bonded shear layer. free shear layer: e, wae and mng layer 31

32 -Trblen bondarylayer flow M. Van Dye, (ed.) An Albm of fld moon, The Parabolc Press, Sanford, Calforna, Je flow a low Reynolds nmber M. Van Dye, (ed.) An Albm of fld moon, The Parabolc Press, Sanford, Calforna,

33 -Comparson of rblen waes a hgh and low Reynolds nmbers M. Van Dye, (ed.) An Albm of fld moon, The Parabolc Press, Sanford, Calforna, Flow Separaon and Reaachmen 33

34 -Flow separaon and reaachmen Physcally speang, as flow over an obec flow deached from he srface of he obec s referred o as an occrrence of flow separaon. On he oher hand, a separaed flow aached on he srface of an obec s referred o an occrrence of flow reaachmen. Sharp-edge separaon: flow over a nfe edge; flow over a dela wng Flow separaon nsensve o Reynolds nmber Bondary-layer separaon: flow n a dffser ; flow over an arfol 67 Flow over a dela wng 34

35 Flow separaon n a dffser flow separaon over an arfol -Two-dmensonal bondary-layer separaon Pranl creron -Three-dmensonal separaon (Lghhll, 1963; Whe, 1974, p. 365, Fg. 4-40): Spral focs ; separaon lne 70 Lmng sreamlne: sreamlne of flow very near he wall or he shear sress lne 35

36 -Two-dmensonal flow reaachmen Flow over a bacward facng sep -Three-dmensonal flow reaachmen Lmng sreamlne: sreamlne of flow very near he wall or he shear sress lne 71 -Three-dmensonal bondary layer: flows over 3-D srfaces Flow over a swep-bac wng ;amospherc bondary layer Noe ha flow near he wall s domnaed by he pressre graden The velocy n he bondary layer changes drecon wh wall dsance, b s nearly parallel o he wall. 7 36

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39 Vore Breadown -Compressble bondary layer (Whe, 1974) Effec of Mach nmber Shoc-bondary layer neracon; shoc ndced separaon Aerodynamc heang 78 39