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Tu bule ce

F om Wikipedia, he f ee e cyclopedia Jump o aviga io Jump o sea ch Mo io cha ac e ized by chao ic cha ges i p essu e a d flow veloci y .mw-pa se -ou pu .ha o e{fo -s yle:i alic}.mw-pa se -ou pu div.ha o e{paddi g-lef :1.6em;ma gi -bo om:0.5em}.mw-pa se -ou pu .ha o e i{fo -s yle: o mal}.mw-pa se -ou pu .ha o e+li k+.ha o e{ma gi - op:-0.5em}Fo he u bule ce fel o a ai pla e, see Clea -ai u bule ce. Fo o he uses, see Tu bule ce (disambigua io ). I fluid dy amics, u bule ce o u bule flow is fluid mo io cha ac e ized by chao ic cha ges i p essu e a d flow veloci y. I is i co as o a lami a flow, which occu s whe a fluid flows i pa allel laye s, wi h o dis up io be wee hose laye s.[1] Tu bule ce is commo ly obse ved i eve yday phe ome a such as su f, fas flowi g ive s, billowi g s o m clouds, o smoke f om a chim ey, a d mos fluid flows occu i g i a u e o c ea ed i e gi ee i g applica io s a e u bule .[2][3]: 2  Tu bule ce is caused by excessive ki e ic e e gy i pa s of a fluid flow, which ove comes he dampi g effec of he fluid's viscosi y. Fo his easo u bule ce is commo ly ealized i low viscosi y fluids. I ge e al e ms, i u bule flow, u s eady vo ices appea of ma y sizes which i e ac wi h each o he , co seque ly d ag due o f ic io effec s i c eases. This i c eases he e e gy eeded o pump fluid h ough a pipe. The o se of u bule ce ca be p edic ed by he dime sio less Rey olds umbe , he a io of ki e ic e e gy o viscous dampi g i a fluid flow. Howeve , u bule ce has lo g esis ed de ailed physical a alysis, a d he i e ac io s wi hi u bule ce c ea e a ve y complex phe ome o . Richa d Fey ma has desc ibed u bule ce as he mos impo a u solved p oblem i classical physics.[4] The u bule ce i e si y affec s ma y fields, fo examples fish ecology,[5] ai pollu io [6] a d p ecipi a io .[7] Co e s 1 Examples of u bule ce 2 Fea u es 3 O se of u bule ce 4 Hea a d mome um a sfe 5 Kolmogo ov's heo y of 1941 6 See also 7 Refe e ces a d o es 8 Fu he eadi g 8.1 Ge e al 8.2 O igi al scie ific esea ch pape s a d classic mo og aphs 9 Ex e al li ks Examples of u bule ce[edi ] Lami a a d u bule wa e flow ove he hull of a subma i e. As he ela ive veloci y of he wa e i c eases u bule ce occu s. Tu bule ce i he ip vo ex f om a ai pla e wi g passi g h ough colou ed smoke Smoke isi g f om a ciga e e. Fo he fi s few ce ime e s, he smoke is lami a . The smoke plume becomes u bule as i s Rey olds umbe i c eases wi h i c eases i flow veloci y a d cha ac e is ic le g h scale. Flow ove a golf ball. (This ca be bes u de s ood by co side i g he golf ball o be s a io a y, wi h ai flowi g ove i .) If he golf ball we e smoo h, he bou da y laye flow ove he f o of he sphe e would be lami a a ypical co di io s. Howeve , he bou da y laye would sepa a e ea ly, as he p essu e g adie swi ched f om favo able (p essu e dec easi g i he flow di ec io ) o u favo able (p essu e i c easi g i he flow di ec io ), c ea i g a la ge egio of low p essu e behi d he ball ha c ea es high fo m d ag. To p eve his, he su face is dimpled o pe u b he bou da y laye a d p omo e u bule ce. This esul s i highe ski f ic io , bu i moves he poi of bou da y laye sepa a io fu he alo g, esul i g i lowe d ag. Clea -ai u bule ce expe ie ced du i g ai pla e fligh , as well as poo as o omical seei g ( he blu i g of images see h ough he a mosphe e). Mos of he e es ial a mosphe ic ci cula io . The ocea ic a d a mosphe ic mixed laye s a d i e se ocea ic cu e s. The flow co di io s i ma y i dus ial equipme (such as pipes, duc s, p ecipi a o s, gas sc ubbe s, dy amic sc aped su face hea excha ge s, e c.) a d machi es (fo i s a ce, i e al combus io e gi es a d gas u bi es). The ex e al flow ove all ki ds of vehicles such as ca s, ai pla es, ships, a d subma i es. The mo io s of ma e i s ella a mosphe es. A je exhaus i g f om a ozzle i o a quiesce fluid. As he flow eme ges i o his ex e al fluid, shea laye s o igi a i g a he lips of he ozzle a e c ea ed. These laye s sepa a e he fas movi g je f om he ex e al fluid, a d a a ce ai c i ical Rey olds umbe hey become u s able a d b eak dow o u bule ce. Biologically ge e a ed u bule ce esul i g f om swimmi g a imals affec s ocea mixi g.[8] S ow fe ces wo k by i duci g u bule ce i he wi d, fo ci g i o d op much of i s s ow load ea he fe ce. B idge suppo s (pie s) i wa e . Whe ive flow is slow, wa e flows smoo hly a ou d he suppo legs. Whe he flow is fas e , a highe Rey olds umbe is associa ed wi h he flow. The flow may s a off lami a bu is quickly sepa a ed f om he leg a d becomes u bule . I ma y geophysical flows ( ive s, a mosphe ic bou da y laye ), he flow u bule ce is domi a ed by he cohe e s uc u es a d u bule eve s. A u bule eve is a se ies of u bule fluc ua io s ha co ai mo e e e gy ha he ave age flow u bule ce.[9][10] The u bule eve s a e associa ed wi h cohe e flow s uc u es such as eddies a d u bule bu s i g, a d hey play a c i ical ole i e ms of sedime scou , acc e io a d a spo i ive s as well as co ami a mixi g a d dispe sio i ive s a d es ua ies, a d i he a mosphe e. .mw-pa se -ou pu .u solved{ma gi :0 1em 1em;bo de :#ccc solid;paddi g:0.35em 0.35em 0.35em 2.2em;backg ou d-colo :#eee;backg ou d-image:u l("h ps://upload.wikimedia.o g/wikipedia/commo s/2/26/Ques io %2C_Web_Fu dame als.svg");backg ou d-posi io : op 50%lef 0.35em;backg ou d-size:1.5em;backg ou d- epea : o- epea }@media(mi -wid h:720px){.mw-pa se -ou pu .u solved{floa : igh ;max-wid h:25%}}.mw-pa se -ou pu .u solved-label{fo -weigh :bold}.mw-pa se -ou pu .u solved-body{ma gi :0.35em;fo -s yle:i alic}.mw-pa se -ou pu .u solved-mo e{fo -size:smalle } U solved p oblem i physics: Is i possible o make a heo e ical model o desc ibe he behavio of a u bule flow—i pa icula , i s i e al s uc u es? (mo e u solved p oblems i physics) I he medical field of ca diology, a s e hoscope is used o de ec hea sou ds a d b ui s, which a e due o u bule blood flow. I o mal i dividuals, hea sou ds a e a p oduc of u bule flow as hea valves close. Howeve , i some co di io s u bule flow ca be audible due o o he easo s, some of hem pa hological. Fo example, i adva ced a he oscle osis, b ui s (a d he efo e u bule flow) ca be hea d i some vessels ha have bee a owed by he disease p ocess. Rece ly, u bule ce i po ous media became a highly deba ed subjec .[11] Fea u es[edi ] Flow visualiza io of a u bule je , made by lase -i duced fluo esce ce. The je exhibi s a wide a ge of le g h scales, a impo a cha ac e is ic of u bule flows. Tu bule ce is cha ac e ized by he followi g fea u es: I egula i y Tu bule flows a e always highly i egula . Fo his easo , u bule ce p oblems a e o mally ea ed s a is ically a he ha de e mi is ically. Tu bule flow is chao ic. Howeve , o all chao ic flows a e u bule . Diffusivi y The eadily available supply of e e gy i u bule flows e ds o accele a e he homoge iza io (mixi g) of fluid mix u es. The cha ac e is ic which is espo sible fo he e ha ced mixi g a d i c eased a es of mass, mome um a d e e gy a spo s i a flow is called "diffusivi y".[12] Tu bule diffusio is usually desc ibed by a u bule diffusio coefficie . This u bule diffusio coefficie is defi ed i a phe ome ological se se, by a alogy wi h he molecula diffusivi ies, bu i does o have a ue physical mea i g, bei g depe de o he flow co di io s, a d o a p ope y of he fluid i self. I addi io , he u bule diffusivi y co cep assumes a co s i u ive ela io be wee a u bule flux a d he g adie of a mea va iable simila o he ela io be wee flux a d g adie ha exis s fo molecula a spo . I he bes case, his assump io is o ly a app oxima io . Neve heless, he u bule diffusivi y is he simples app oach fo qua i a ive a alysis of u bule flows, a d ma y models have bee pos ula ed o calcula e i . Fo i s a ce, i la ge bodies of wa e like ocea s his coefficie ca be fou d usi g Richa dso 's fou - hi d powe law a d is gove ed by he a dom walk p i ciple. I ive s a d la ge ocea cu e s, he diffusio coefficie is give by va ia io s of Elde 's fo mula. Ro a io ali y Tu bule flows have o -ze o vo ici y a d a e cha ac e ized by a s o g h ee-dime sio al vo ex ge e a io mecha ism k ow as vo ex s e chi g. I fluid dy amics, hey a e esse ially vo ices subjec ed o s e chi g associa ed wi h a co espo di g i c ease of he compo e of vo ici y i he s e chi g di ec io —due o he co se va io of a gula mome um. O he o he ha d, vo ex s e chi g is he co e mecha ism o which he u bule ce e e gy cascade elies o es ablish a d mai ai ide ifiable s uc u e fu c io .[13] I ge e al, he s e chi g mecha ism implies hi i g of he vo ices i he di ec io pe pe dicula o he s e chi g di ec io due o volume co se va io of fluid eleme s. As a esul , he adial le g h scale of he vo ices dec eases a d he la ge flow s uc u es b eak dow i o smalle s uc u es. The p ocess co i ues u il he small scale s uc u es a e small e ough ha hei ki e ic e e gy ca be a sfo med by he fluid's molecula viscosi y i o hea . Tu bule flow is always o a io al a d h ee dime sio al.[13] Fo example, a mosphe ic cyclo es a e o a io al bu hei subs a ially wo-dime sio al shapes do o allow vo ex ge e a io a d so a e o u bule . O he o he ha d, ocea ic flows a e dispe sive bu esse ially o o a io al a d he efo e a e o u bule .[13] Dissipa io To sus ai u bule flow, a pe sis e sou ce of e e gy supply is equi ed because u bule ce dissipa es apidly as he ki e ic e e gy is co ve ed i o i e al e e gy by viscous shea s ess. Tu bule ce causes he fo ma io of eddies of ma y diffe e le g h scales. Mos of he ki e ic e e gy of he u bule mo io is co ai ed i he la ge-scale s uc u es. The e e gy "cascades" f om hese la ge-scale s uc u es o smalle scale s uc u es by a i e ial a d esse ially i viscid mecha ism. This p ocess co i ues, c ea i g smalle a d smalle s uc u es which p oduces a hie a chy of eddies. Eve ually his p ocess c ea es s uc u es ha a e small e ough ha molecula diffusio becomes impo a a d viscous dissipa io of e e gy fi ally akes place. The scale a which his happe s is he Kolmogo ov le g h scale. Via his e e gy cascade, u bule flow ca be ealized as a supe posi io of a spec um of flow veloci y fluc ua io s a d eddies upo a mea flow. The eddies a e loosely defi ed as cohe e pa e s of flow veloci y, vo ici y a d p essu e. Tu bule flows may be viewed as made of a e i e hie a chy of eddies ove a wide a ge of le g h scales a d he hie a chy ca be desc ibed by he e e gy spec um ha measu es he e e gy i flow veloci y fluc ua io s fo each le g h scale (wave umbe ). The scales i he e e gy cascade a e ge e ally u co ollable a d highly o -symme ic. Neve heless, based o hese le g h scales hese eddies ca be divided i o h ee ca ego ies. I eg al ime scale The i eg al ime scale fo a Lag a gia flow ca be defi ed as: T = ( 1 ⟨ u ′ u ′ ⟩ ) ∫ 0 ∞ ⟨ u ′ u ′ ( τ ) ⟩ d τ {\displays yle T=\lef ({\f ac {1}{\la gle u'u'\ a gle }}\ igh )\i _{0}^{\i f y }\la gle u'u'(\ au )\ a gle \,d\ au } whe e u′ is he veloci y fluc ua io , a d τ {\displays yle \ au } is he ime lag be wee measu eme s.[14] I eg al le g h scales La ge eddies ob ai e e gy f om he mea flow a d also f om each o he . Thus, hese a e he e e gy p oduc io eddies which co ai mos of he e e gy. They have he la ge flow veloci y fluc ua io a d a e low i f eque cy. I eg al scales a e highly a iso opic a d a e defi ed i e ms of he o malized wo-poi flow veloci y co ela io s. The maximum le g h of hese scales is co s ai ed by he cha ac e is ic le g h of he appa a us. Fo example, he la ges i eg al le g h scale of pipe flow is equal o he pipe diame e . I he case of a mosphe ic u bule ce, his le g h ca each up o he o de of seve al hu d eds kilome e s.: The i eg al le g h scale ca be defi ed as L = ( 1 ⟨ u ′ u ′ ⟩ ) ∫ 0 ∞ ⟨ u ′ u ′ ( ) ⟩ d {\displays yle L=\lef ({\f ac {1}{\la gle u'u'\ a gle }}\ igh )\i _{0}^{\i f y }\la gle u'u'( )\ a gle \,d } whe e is he dis a ce be wee wo measu eme loca io s, a d u′ is he veloci y fluc ua io i ha same di ec io .[14] Kolmogo ov le g h scales Smalles scales i he spec um ha fo m he viscous sub-laye a ge. I his a ge, he e e gy i pu f om o li ea i e ac io s a d he e e gy d ai f om viscous dissipa io a e i exac bala ce. The small scales have high f eque cy, causi g u bule ce o be locally iso opic a d homoge eous. Taylo mic oscales The i e media e scales be wee he la ges a d he smalles scales which make he i e ial sub a ge. Taylo mic oscales a e o dissipa ive scales, bu pass dow he e e gy f om he la ges o he smalles wi hou dissipa io . Some li e a u es do o co side Taylo mic oscales as a cha ac e is ic le g h scale a d co side he e e gy cascade o co ai o ly he la ges a d smalles scales; while he la e accommoda e bo h he i e ial sub a ge a d he viscous sublaye . Neve heless, Taylo mic oscales a e of e used i desc ibi g he e m " u bule ce" mo e co ve ie ly as hese Taylo mic oscales play a domi a ole i e e gy a d mome um a sfe i he wave umbe space. Al hough i is possible o fi d some pa icula solu io s of he Navie –S okes equa io s gove i g fluid mo io , all such solu io s a e u s able o fi i e pe u ba io s a la ge Rey olds umbe s. Se si ive depe de ce o he i i ial a d bou da y co di io s makes fluid flow i egula bo h i ime a d i space so ha a s a is ical desc ip io is eeded. The Russia ma hema icia A d ey Kolmogo ov p oposed he fi s s a is ical heo y of u bule ce, based o he afo eme io ed o io of he e e gy cascade (a idea o igi ally i oduced by Richa dso ) a d he co cep of self-simila i y. As a esul , he Kolmogo ov mic oscales we e amed af e him. I is ow k ow ha he self-simila i y is b oke so he s a is ical desc ip io is p ese ly modified.[15] A comple e desc ip io of u bule ce is o e of he u solved p oblems i physics. Acco di g o a apoc yphal s o y, We e Heise be g was asked wha he would ask God, give he oppo u i y. His eply was: "Whe I mee God, I am goi g o ask him wo ques io s: Why ela ivi y? A d why u bule ce? I eally believe he will have a a swe fo he fi s ."[16] A simila wi icism has bee a ibu ed o Ho ace Lamb i a speech o he B i ish Associa io fo he Adva ceme of Scie ce: "I am a old ma ow, a d whe I die a d go o heave he e a e wo ma e s o which I hope fo e ligh e me . O e is qua um elec ody amics, a d he o he is he u bule mo io of fluids. A d abou he fo me I am a he op imis ic."[17][18] O se of u bule ce[edi ] The plume f om his ca dle flame goes f om lami a o u bule . The Rey olds umbe ca be used o p edic whe e his a si io will ake place The o se of u bule ce ca be, o some ex e , p edic ed by he Rey olds umbe , which is he a io of i e ial fo ces o viscous fo ces wi hi a fluid which is subjec o ela ive i e al moveme due o diffe e fluid veloci ies, i wha is k ow as a bou da y laye i he case of a bou di g su face such as he i e io of a pipe. A simila effec is c ea ed by he i oduc io of a s eam of highe veloci y fluid, such as he ho gases f om a flame i ai . This ela ive moveme ge e a es fluid f ic io , which is a fac o i developi g u bule flow. Cou e ac i g his effec is he viscosi y of he fluid, which as i i c eases, p og essively i hibi s u bule ce, as mo e ki e ic e e gy is abso bed by a mo e viscous fluid. The Rey olds umbe qua ifies he ela ive impo a ce of hese wo ypes of fo ces fo give flow co di io s, a d is a guide o whe u bule flow will occu i a pa icula si ua io .[19] This abili y o p edic he o se of u bule flow is a impo a desig ool fo equipme such as pipi g sys ems o ai c af wi gs, bu he Rey olds umbe is also used i scali g of fluid dy amics p oblems, a d is used o de e mi e dy amic simili ude be wee wo diffe e cases of fluid flow, such as be wee a model ai c af , a d i s full size ve sio . Such scali g is o always li ea a d he applica io of Rey olds umbe s o bo h si ua io s allows scali g fac o s o be developed. A flow si ua io i which he ki e ic e e gy is sig ifica ly abso bed due o he ac io of fluid molecula viscosi y gives ise o a lami a flow egime. Fo his he dime sio less qua i y he Rey olds umbe (Re) is used as a guide. Wi h espec o lami a a d u bule flow egimes: lami a flow occu s a low Rey olds umbe s, whe e viscous fo ces a e domi a , a d is cha ac e ized by smoo h, co s a fluid mo io ; u bule flow occu s a high Rey olds umbe s a d is domi a ed by i e ial fo ces, which e d o p oduce chao ic eddies, vo ices a d o he flow i s abili ies. The Rey olds umbe is defi ed as[20] R e = ρ v L μ , {\displays yle \ma h m {Re} ={\f ac {\ ho vL}{\mu }}\,,} whe e: ρ is he de si y of he fluid (SI u i s: kg/m3) v is a cha ac e is ic veloci y of he fluid wi h espec o he objec (m/s) L is a cha ac e is ic li ea dime sio (m) μ is he dy amic viscosi y of he fluid (Pa·s o N·s/m2 o kg/(m·s)). While he e is o heo em di ec ly ela i g he o -dime sio al Rey olds umbe o u bule ce, flows a Rey olds umbe s la ge ha 5000 a e ypically (bu o ecessa ily) u bule , while hose a low Rey olds umbe s usually emai lami a . I Poiseuille flow, fo example, u bule ce ca fi s be sus ai ed if he Rey olds umbe is la ge ha a c i ical value of abou 2040;[21] mo eove , he u bule ce is ge e ally i e spe sed wi h lami a flow u il a la ge Rey olds umbe of abou 4000. The a si io occu s if he size of he objec is g adually i c eased, o he viscosi y of he fluid is dec eased, o if he de si y of he fluid is i c eased. Hea a d mome um a sfe [edi ] Whe flow is u bule , pa icles exhibi addi io al a sve se mo io which e ha ces he a e of e e gy a d mome um excha ge be wee hem hus i c easi g he hea a sfe a d he f ic io coefficie . Assume fo a wo-dime sio al u bule flow ha o e was able o loca e a specific poi i he fluid a d measu e he ac ual flow veloci y v = (vx,vy) of eve y pa icle ha passed h ough ha poi a a y give ime. The o e would fi d he ac ual flow veloci y fluc ua i g abou a mea value: v x = v ¯ x ⏟ mea value + v x ′ ⏟ fluc ua io a d v y = v ¯ y + v y ′ ; {\displays yle v_{x}=\u de b ace {{\ove li e {v}}_{x}} _{\ ex {mea value}}+\u de b ace {v'_{x}} _{\ ex {fluc ua io }}\quad {\ ex {a d}}\quad v_{y}={\ove li e {v}}_{y}+v'_{y}\,;} a d simila ly fo empe a u e (T = T + T′) a d p essu e (P = P + P′), whe e he p imed qua i ies de o e fluc ua io s supe posed o he mea . This decomposi io of a flow va iable i o a mea value a d a u bule fluc ua io was o igi ally p oposed by Osbo e Rey olds i 1895, a d is co side ed o be he begi i g of he sys ema ic ma hema ical a alysis of u bule flow, as a sub-field of fluid dy amics. While he mea values a e ake as p edic able va iables de e mi ed by dy amics laws, he u bule fluc ua io s a e ega ded as s ochas ic va iables. The hea flux a d mome um a sfe ( ep ese ed by he shea s ess τ) i he di ec io o mal o he flow fo a give ime a e q = v y ′ ρ c P T ′ ⏟ expe ime al value = − k u b ∂ T ¯ ∂ y ; τ = − ρ v y ′ v x ′ ¯ ⏟ expe ime al value = μ u b ∂ v ¯ x ∂ y ; {\displays yle {\begi {alig ed}q&=\u de b ace {v'_{y}\ ho c_{P}T'} _{\ ex {expe ime al value}}=-k_{\ ex { u b}}{\f ac {\pa ial {\ove li e {T}}}{\pa ial y}}\,;\\\ au &=\u de b ace {-\ ho {\ove li e {v'_{y}v'_{x}}}} _{\ ex {expe ime al value}}=\mu _{\ ex { u b}}{\f ac {\pa ial {\ove li e {v}}_{x}}{\pa ial y}}\,;\e d{alig ed}}} whe e cP is he hea capaci y a co s a p essu e, ρ is he de si y of he fluid, μ u b is he coefficie of u bule viscosi y a d k u b is he u bule he mal co duc ivi y.[3] Kolmogo ov's heo y of 1941[edi ] Richa dso 's o io of u bule ce was ha a u bule flow is composed by "eddies" of diffe e sizes. The sizes defi e a cha ac e is ic le g h scale fo he eddies, which a e also cha ac e ized by flow veloci y scales a d ime scales ( u ove ime) depe de o he le g h scale. The la ge eddies a e u s able a d eve ually b eak up o igi a i g smalle eddies, a d he ki e ic e e gy of he i i ial la ge eddy is divided i o he smalle eddies ha s emmed f om i . These smalle eddies u de go he same p ocess, givi g ise o eve smalle eddies which i he i he e e gy of hei p edecesso eddy, a d so o . I his way, he e e gy is passed dow f om he la ge scales of he mo io o smalle scales u il eachi g a sufficie ly small le g h scale such ha he viscosi y of he fluid ca effec ively dissipa e he ki e ic e e gy i o i e al e e gy. I his o igi al heo y of 1941, Kolmogo ov pos ula ed ha fo ve y high Rey olds umbe s, he small-scale u bule mo io s a e s a is ically iso opic (i.e. o p efe e ial spa ial di ec io could be disce ed). I ge e al, he la ge scales of a flow a e o iso opic, si ce hey a e de e mi ed by he pa icula geome ical fea u es of he bou da ies ( he size cha ac e izi g he la ge scales will be de o ed as L). Kolmogo ov's idea was ha i he Richa dso 's e e gy cascade his geome ical a d di ec io al i fo ma io is los , while he scale is educed, so ha he s a is ics of he small scales has a u ive sal cha ac e : hey a e he same fo all u bule flows whe he Rey olds umbe is sufficie ly high. Thus, Kolmogo ov i oduced a seco d hypo hesis: fo ve y high Rey olds umbe s he s a is ics of small scales a e u ive sally a d u iquely de e mi ed by he ki ema ic viscosi y ν a d he a e of e e gy dissipa io ε. Wi h o ly hese wo pa ame e s, he u ique le g h ha ca be fo med by dime sio al a alysis is η = ( ν 3 ε ) 1 / 4 . {\displays yle \e a =\lef ({\f ac {\ u ^{3}}{\va epsilo }}\ igh )^{1/4}\,.} This is oday k ow as he Kolmogo ov le g h scale (see Kolmogo ov mic oscales). A u bule flow is cha ac e ized by a hie a chy of scales h ough which he e e gy cascade akes place. Dissipa io of ki e ic e e gy akes place a scales of he o de of Kolmogo ov le g h η, while he i pu of e e gy i o he cascade comes f om he decay of he la ge scales, of o de L. These wo scales a he ex emes of he cascade ca diffe by seve al o de s of mag i ude a high Rey olds umbe s. I be wee he e is a a ge of scales (each o e wi h i s ow cha ac e is ic le g h ) ha has fo med a he expe se of he e e gy of he la ge o es. These scales a e ve y la ge compa ed wi h he Kolmogo ov le g h, bu s ill ve y small compa ed wi h he la ge scale of he flow (i.e. η ≪ ≪ L). Si ce eddies i his a ge a e much la ge ha he dissipa ive eddies ha exis a Kolmogo ov scales, ki e ic e e gy is esse ially o dissipa ed i his a ge, a d i is me ely a sfe ed o smalle scales u il viscous effec s become impo a as he o de of he Kolmogo ov scale is app oached. Wi hi his a ge i e ial effec s a e s ill much la ge ha viscous effec s, a d i is possible o assume ha viscosi y does o play a ole i hei i e al dy amics (fo his easo his a ge is called "i e ial a ge"). He ce, a hi d hypo hesis of Kolmogo ov was ha a ve y high Rey olds umbe he s a is ics of scales i he a ge η ≪ ≪ L a e u ive sally a d u iquely de e mi ed by he scale a d he a e of e e gy dissipa io ε. The way i which he ki e ic e e gy is dis ibu ed ove he mul iplici y of scales is a fu dame al cha ac e iza io of a u bule flow. Fo homoge eous u bule ce (i.e., s a is ically i va ia u de a sla io s of he efe e ce f ame) his is usually do e by mea s of he e e gy spec um fu c io E(k), whe e k is he modulus of he wavevec o co espo di g o some ha mo ics i a Fou ie ep ese a io of he flow veloci y field u(x): u ( x ) = ∭ R 3 u ^ ( k ) e i k ⋅ x d 3 k , {\displays yle \ma hbf {u} (\ma hbf {x} )=\iii _{\ma hbb {R} ^{3}}{\ha {\ma hbf {u} }}(\ma hbf {k} )e^{i\ma hbf {k\cdo x} }\,\ma h m {d} ^{3}\ma hbf {k} \,,} whe e û(k) is he Fou ie a sfo m of he flow veloci y field. Thus, E(k) dk ep ese s he co ibu io o he ki e ic e e gy f om all he Fou ie modes wi h k < |k| < k + dk, a d he efo e, 1 2 ⟨ u i u i ⟩ = ∫ 0 ∞ E ( k ) d k , {\displays yle {\ f ac {1}{2}}\lef \la gle u_{i}u_{i}\ igh \ a gle =\i _{0}^{\i f y }E(k)\,\ma h m {d} k\,,} whe e .mw-pa se -ou pu .sf ac{whi e-space: ow ap}.mw-pa se -ou pu .sf ac. io ,.mw-pa se -ou pu .sf ac . io {display:i li e-block;ve ical-alig :-0.5em;fo -size:85%; ex -alig :ce e }.mw-pa se -ou pu .sf ac . um,.mw-pa se -ou pu .sf ac .de {display:block;li e-heigh :1em;ma gi :0 0.1em}.mw-pa se -ou pu .sf ac .de {bo de - op:1px solid}.mw-pa se -ou pu .s -o ly{bo de :0;clip: ec (0,0,0,0);heigh :1px;ma gi :-1px;ove flow:hidde ;paddi g:0;posi io :absolu e;wid h:1px}1/2⟨uiui⟩ is he mea u bule ki e ic e e gy of he flow. The wave umbe k co espo di g o le g h scale is k = 2π/ . The efo e, by dime sio al a alysis, he o ly possible fo m fo he e e gy spec um fu c io acco di g wi h he hi d Kolmogo ov's hypo hesis is E ( k ) = K 0 ε 2 3 k − 5 3 , {\displays yle E(k)=K_{0}\va epsilo ^{\f ac {2}{3}}k^{-{\f ac {5}{3}}}\,,} whe e K 0 ≈ 1.44 {\displays yle K_{0}\app ox 1.44} would be a u ive sal co s a . This is o e of he mos famous esul s of Kolmogo ov 1941 heo y, a d co side able expe ime al evide ce has accumula ed ha suppo s i .[22] Ou of he i e ial a ea, o e ca fi ds he fo mula [23] below : E ( k ) = K 0 ε 2 3 k − 5 3 exp ⁡ [ − 3 K 0 2 ( ν 3 k 4 ε ) 1 3 ] , {\displays yle E(k)=K_{0}\va epsilo ^{\f ac {2}{3}}k^{-{\f ac {5}{3}}}\exp \lef [-{\f ac {3K_{0}}{2}}\lef ({\f ac {\ u ^{3}k^{4}}{\va epsilo }}\ igh )^{\f ac {1}{3}}\ igh ]\,,} I spi e of his success, Kolmogo ov heo y is a p ese u de evisio . This heo y implici ly assumes ha he u bule ce is s a is ically self-simila a diffe e scales. This esse ially mea s ha he s a is ics a e scale-i va ia i he i e ial a ge. A usual way of s udyi g u bule flow veloci y fields is by mea s of flow veloci y i c eme s: δ u ( ) = u ( x + ) − u ( x ) ; {\displays yle \del a \ma hbf {u} ( )=\ma hbf {u} (\ma hbf {x} +\ma hbf { } )-\ma hbf {u} (\ma hbf {x} )\,;} ha is, he diffe e ce i flow veloci y be wee poi s sepa a ed by a vec o (si ce he u bule ce is assumed iso opic, he flow veloci y i c eme depe ds o ly o he modulus of ). Flow veloci y i c eme s a e useful because hey emphasize he effec s of scales of he o de of he sepa a io whe s a is ics a e compu ed. The s a is ical scale-i va ia ce implies ha he scali g of flow veloci y i c eme s should occu wi h a u ique scali g expo e β, so ha whe is scaled by a fac o λ, δ u ( λ ) {\displays yle \del a \ma hbf {u} (\lambda )} should have he same s a is ical dis ibu io as λ β δ u ( ) , {\displays yle \lambda ^{\be a }\del a \ma hbf {u} ( )\,,} wi h β i depe de of he scale . F om his fac , a d o he esul s of Kolmogo ov 1941 heo y, i follows ha he s a is ical mome s of he flow veloci y i c eme s (k ow as s uc u e fu c io s i u bule ce) should scale as ⟨ ( δ u ( ) ) ⟩ = C ( ε ) 3 , {\displays yle {\Big \la gle }{\big (}\del a \ma hbf {u} ( ){\big )}^{ }{\Big \ a gle }=C_{ }(\va epsilo )^{\f ac { }{3}}\,,} whe e he b acke s de o e he s a is ical ave age, a d he C would be u ive sal co s a s. The e is co side able evide ce ha u bule flows devia e f om his behavio . The scali g expo e s devia e f om he /3 value p edic ed by he heo y, becomi g a o -li ea fu c io of he o de of he s uc u e fu c io . The u ive sali y of he co s a s have also bee ques io ed. Fo low o de s he disc epa cy wi h he Kolmogo ov /3 value is ve y small, which explai he success of Kolmogo ov heo y i ega ds o low o de s a is ical mome s. I pa icula , i ca be show ha whe he e e gy spec um follows a powe law E ( k ) ∝ k − p , {\displays yle E(k)\p op o k^{-p}\,,} wi h 1 < p < 3, he seco d o de s uc u e fu c io has also a powe law, wi h he fo m ⟨ ( δ u ( ) ) 2 ⟩ ∝ p − 1 , {\displays yle {\Big \la gle }{\big (}\del a \ma hbf {u} ( ){\big )}^{2}{\Big \ a gle }\p op o ^{p-1}\,,} Si ce he expe ime al values ob ai ed fo he seco d o de s uc u e fu c io o ly devia e sligh ly f om he 2/3 value p edic ed by Kolmogo ov heo y, he value fo p is ve y ea o 5/3 (diffe e ces a e abou 2%[24]). Thus he "Kolmogo ov −5/3 spec um" is ge e ally obse ved i u bule ce. Howeve , fo high o de s uc u e fu c io s, he diffe e ce wi h he Kolmogo ov scali g is sig ifica , a d he b eakdow of he s a is ical self-simila i y is clea . This behavio , a d he lack of u ive sali y of he C co s a s, a e ela ed wi h he phe ome o of i e mi e cy i u bule ce. This is a impo a a ea of esea ch i his field, a d a majo goal of he mode heo y of u bule ce is o u de s a d wha is eally u ive sal i he i e ial a ge. See also[edi ] .mw-pa se -ou pu .div-col{ma gi - op:0.3em;colum -wid h:30em}.mw-pa se -ou pu .div-col-small{fo -size:90%}.mw-pa se -ou pu .div-col- ules{colum - ule:1px solid #aaa}.mw-pa se -ou pu .div-col dl,.mw-pa se -ou pu .div-col ol,.mw-pa se -ou pu .div-col ul{ma gi - op:0}.mw-pa se -ou pu .div-col li,.mw-pa se -ou pu .div-col dd{page-b eak-i side:avoid;b eak-i side:avoid-colum } As o omical seei g A mosphe ic dispe sio modeli g Chaos heo y Clea -ai u bule ce Diffe e ypes of bou da y co di io s i fluid dy amics Eddy cova ia ce Fluid dy amics Da cy–Weisbach equa io Eddy Navie –S okes equa io s La ge eddy simula io Hage –Poiseuille equa io Kelvi –Helmhol z i s abili y Lag a gia cohe e s uc u e Tu bule ce ki e ic e e gy Mesocyclo es Navie –S okes exis e ce a d smoo h ess Rey olds umbe Swi g bowli g Taylo mic oscale Tu bule ce modeli g Velocime y Ve ical d af Vo ex Vo ex ge e a o Wake u bule ce Wave u bule ce Wi g ip vo ices Wi d u el Refe e ces a d o es[edi ] .mw-pa se -ou pu . eflis {fo -size:90%;ma gi -bo om:0.5em;lis -s yle- ype:decimal}.mw-pa se -ou pu . eflis . efe e ces{fo -size:100%;ma gi -bo om:0;lis -s yle- ype:i he i }.mw-pa se -ou pu . eflis -colum s-2{colum -wid h:30em}.mw-pa se -ou pu . eflis -colum s-3{colum -wid h:25em}.mw-pa se -ou pu . eflis -colum s{ma gi - op:0.3em}.mw-pa se -ou pu . eflis -colum s ol{ma gi - op:0}.mw-pa se -ou pu . eflis -colum s li{page-b eak-i side:avoid;b eak-i side:avoid-colum }.mw-pa se -ou pu . eflis -uppe -alpha{lis -s yle- ype:uppe -alpha}.mw-pa se -ou pu . eflis -uppe - oma {lis -s yle- ype:uppe - oma }.mw-pa se -ou pu . eflis -lowe -alpha{lis -s yle- ype:lowe -alpha}.mw-pa se -ou pu . eflis -lowe -g eek{lis -s yle- ype:lowe -g eek}.mw-pa se -ou pu . eflis -lowe - oma {lis -s yle- ype:lowe - oma } ^ .mw-pa se -ou pu ci e.ci a io {fo -s yle:i he i }.mw-pa se -ou pu .ci a io q{quo es:"\"""\"""'""'"}.mw-pa se -ou pu .id-lock-f ee a,.mw-pa se -ou pu .ci a io .cs1-lock-f ee a{backg ou d:li ea -g adie ( a spa e , a spa e ),u l("//upload.wikimedia.o g/wikipedia/commo s/6/65/Lock-g ee .svg") igh 0.1em ce e /9px o- epea }.mw-pa se -ou pu .id-lock-limi ed a,.mw-pa se -ou pu .id-lock- egis a io a,.mw-pa se -ou pu .ci a io .cs1-lock-limi ed a,.mw-pa se -ou pu .ci a io .cs1-lock- egis a io a{backg ou d:li ea -g adie ( a spa e , a spa e ),u l("//upload.wikimedia.o g/wikipedia/commo s/d/d6/Lock-g ay-al -2.svg") igh 0.1em ce e /9px o- epea }.mw-pa se -ou pu .id-lock-subsc ip io a,.mw-pa se -ou pu .ci a io .cs1-lock-subsc ip io a{backg ou d:li ea -g adie ( a spa e , a spa e ),u l("//upload.wikimedia.o g/wikipedia/commo s/a/aa/Lock- ed-al -2.svg") igh 0.1em ce e /9px o- epea }.mw-pa se -ou pu .cs1-subsc ip io ,.mw-pa se -ou pu .cs1- egis a io {colo :#555}.mw-pa se -ou pu .cs1-subsc ip io spa ,.mw-pa se -ou pu .cs1- egis a io spa {bo de -bo om:1px do ed;cu so :help}.mw-pa se -ou pu .cs1-ws-ico a{backg ou d:li ea -g adie ( a spa e , a spa e ),u l("//upload.wikimedia.o g/wikipedia/commo s/4/4c/Wikisou ce-logo.svg") igh 0.1em ce e /12px o- epea }.mw-pa se -ou pu code.cs1-code{colo :i he i ;backg ou d:i he i ;bo de : o e;paddi g:i he i }.mw-pa se -ou pu .cs1-hidde -e o {display: o e;fo -size:100%}.mw-pa se -ou pu .cs1-visible-e o {fo -size:100%}.mw-pa se -ou pu .cs1-mai {display: o e;colo :#33aa33;ma gi -lef :0.3em}.mw-pa se -ou pu .cs1-fo ma {fo -size:95%}.mw-pa se -ou pu .cs1-ke -lef ,.mw-pa se -ou pu .cs1-ke -wl-lef {paddi g-lef :0.2em}.mw-pa se -ou pu .cs1-ke - igh ,.mw-pa se -ou pu .cs1-ke -wl- igh {paddi g- igh :0.2em}.mw-pa se -ou pu .ci a io .mw-selfli k{fo -weigh :i he i }Ba chelo , G. 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Ex e al li ks[edi ] Ce e fo Tu bule ce Resea ch, Scie ific pape s a d books o u bule ce Ce e fo Tu bule ce Resea ch, S a fo d U ive si y Scie ific Ame ica a icle Ai Tu bule ce Fo ecas i e a io al CFD da abase iCFDda abase Tu bule flow i a pipe o YouTube Fluid Mecha ics websi e wi h movies, Q&A, e c Joh s Hopki s public da abase wi h di ec ume ical simula io da a Tu Base public da abase wi h expe ime al da a f om Eu opea High Pe fo ma ce I f as uc u es i Tu bule ce (EuHIT) Au ho i y co ol Ge e al I eg a ed Au ho i y File (Ge ma y) Na io al lib a ies F a ce (da a) U i ed S a es Japa O he Mic osof Academic Re ieved f om "h ps://e .wikipedia.o g/w/i dex.php? i le=Tu bule ce&oldid=1056443657" Ca ego ies: Tu bule ceCo cep s i physicsAe ody amicsChaos heo yT a spo phe ome aFluid dy amicsFlow egimesHidde ca ego ies: Pages wi h missi g ISBNsA icles wi h sho desc ip io Sho desc ip io ma ches Wikida aCS1 Russia -la guage sou ces ( u)A icles wi h GND ide ifie sA icles wi h BNF ide ifie sA icles wi h LCCN ide ifie sA icles wi h NDL ide ifie sA icles wi h MA ide ifie s

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