SOLUTION OF BOUNDARY LAYER
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1 SOUION OF BOUNDARY AYER EQUAIONS Prabal alkar Aociate Proeor Department o Mechanical Engineering II Delhi [email protected] P.alkar/Mech-IID
2 Bonar laer Approimation X momentm: ρ μ P Appling Newton n law in the -irection, we get -momentm eqation ρ μ P h,p P P P() Hence, For a late plate, ince U contant an otie the bonar laer, X- momentm eqation gie P P P.alkar/Mech-IID hereore, or low oer a lat plate, the prere remain contant oer the entire plate (both inie an otie the bonar laer).
3 Bonar laer oer a lat plate P.alkar/Mech-IID
4 Pal Richar Heinrich Blai (88 97) wa a German Fli Dnamic Engineer. He wa one o the irt tent o Prantl. he continit an momentm eqation were irt ole in 98 b the German engineer H. Blai, a tent to. Prantl. Born Die Fiel Alma mater Doctoral aior 9 Agt 88 Berlin, German 4 April 97 (age 86) Hambrg, Wet German Fli mechanic an mechanical engineering Unierit o Göttingen wig Prantl hi wa one b tranorming the two partial ierential eqation into a ingle orinar ierential eqation b introcing a new inepenent ariable, calle the imilaritariable. he ining o ch a ariable, aming it eit, i more o an art than cience, an it reqire to hae a goo inight o the problem. P.alkar/Mech-IID
5 he hape o the elocit proile remain the ame along the plate. Blai reaone that the nonimenional elocit proile / hol remain nchange when plotte againt the nonimenional itance /δ, where δ i the thickne o the local elocit bonar laer at a gien. hat i, althogh both δ an at a gien ar with, the elocit at a ie / δ remain contant Blai wa alo aware rom the work o Stoke that δ i proportional to P.alkar/Mech-IID
6 Scale Anali Diiing b to epre the relt in imenionle orm gie P.alkar/Mech-IID
7 Similarit Variable he igniicant ariable i /δ, an we ame that the elocit ma be epree a a nction cto o othi aabe.wete ariable. then hae We hi eine make, / / δ Here, i calle the imilarit ariable, an g() i the nction we eek a a oltion P.alkar/Mech-IID
8 Variable ranormation A tream nction wa eine ch that: to get ri o continit eqation Ψ Ψ to get ri o continit eqation Ψ ( ) Ψ ) g( ) ( where ( ) ( ) Ψ Ψ Ψ Ψ Ψ P.alkar/Mech-IID
9 Dierentiating the preio eqation with repect to an, Sbtitting thee relation into the momentm eqation an, Sbtitting thee relation into the momentm eqation an impliing, we obtain hi i a thir-orer nonlinear ierential eqation. hi wa the tem o two PDE i conerte to one ODE hi wa the tem o two PDE i conerte to one ODE. P.alkar/Mech-IID
10 Blai Soltion he ale o correponing to /.99 i 5. 5 δ P.alkar/Mech-IID δ Re Signiicance o,,
11 P.alkar/Mech-IID
12 he hear tre at the wall can be etermine rom: τ w μ μ τ w ρμ.. ρ Re ocal kin riction coeicient become C, τ ρv w ρ τ w.664re P.alkar/Mech-IID Note that nlike the bonar laer thickne, wall hear tre an the kin riction coeicient ecreae along the plate a -/.
13 Energ Eqation Introce a non-imenional temperatre θ(, ) (, ) Sbtittion gie an energ eqation o the orm: θ θ θ α emperatre proile or low oer an iothermal lat plate are imilar like the elocit proile. h, we epect a imilarit oltion or temperatre to eit. Frther, the thickne o the thermal bonar laer i proportional to / Uing the chain rle an btitting the an epreion into the energ eqation gie θ θ θ α P.alkar/Mech-IID
14 Uing the chain rle an btitting the an epreion into the energ eqation gie θ θ θ α / i replace b θ θ θ Pr Compare For Pr θ ( ) an θ ( ) an h we concle that the elocit an thermal bonar laer coincie, an the nonimenional elocit an temperatre proile (/ an θ) areientical or tea, incompreible, laminar low o a li with contant propertie an Pr oer an iothermal lat plate he ale o the temperatre graient at the race (Pr )?? θ P.alkar/Mech-IID.
15 θ θ Pr hi eq. i ole or nmero ale o Prantl nmber. For Pr >.6, the nonimenional temperatre graient at the race i on to be proportional to Pr / Pr >.6 θ. Pr θ(, ) (, ) δ he temperatre graient at the race i θ ( ) ( ) θ ( ). Pr P.alkar/Mech-IID
16 P.alkar/Mech-IID hi oltion i gien b Pohlhaen
17 he local conection coeicient can be epree a: k q h. Pr k ( ) ( ). Pr ( ) An the local Nelt nmber become h N k. Pr Re Pr >.6 Soling the thermal bonar laer eqation nmericall or the temperatre proile or ierent Prantl nmber, an ing the einition o the thermal bonar laer, it i etermine that δ δ t Pr P.alkar/Mech-IID δ t δ Pr Pr 5. Re
18 Non-imenionalization Non imenionalization p μ ρ μ ρ k ρ p k c an V p,p V, V,, ρ P.alkar/Mech-IID
19 p Continit: p Re Momentm: Pr Re Energ: With the bonar conition: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),,,,,,,,,,,, P.alkar/Mech-IID
20 P.alkar/Mech-IID Geometricall Similar
21 Fnctional orm o Friction an Conection Coeicient i Momentm: p Re For a gien geometr, the oltion or can be epree a (,,Re ) hen the hear tre at the race become τ μ μ V μ V (, Re ) (, Re ) (, Re ) (, Re ) τ μv C, ρvv ρ V Re P.alkar/Mech-IID
22 Energ: he oltion or can be epree a Re Pr g (,,Re,Pr) Uing the einition o, the conection heat traner coeicient become h k k( ) ( ) k Nelt nmber: N h k g (, Re,Pr) Note that the Nelt nmber i eqialent to the imenionle temperatre graient at the race, an th it i properl reerre to a the imenionle heat traner coeicient P.alkar/Mech-IID
23 P.alkar/Mech-IID Aerage riction coeicient
24 Renol Analog Renol Analog When Pr (approimatel the cae or gae) an P/ (e g For lat Re or gae) an P / (e.g. For lat plate) Re h N k N N V V μ μ μ τ, N Re V N V V C ρ μ ρ τ, N Re C (Pr) Re V ρv ρ, St C (Pr) P.alkar/Mech-IID Pr Re N V c h St p ρ
25 Clinton-Colbrn Analog Alo calle moiie Renol analog C.664Re, N. Pr Re aking the ratio between C, an N Vali or.6<pr<6 P.alkar/Mech-IID C, Re C, N Pr Pr jh ρcpv h Colbrn j-actor Althogh thi relation i eelope ing relation or laminar low oer a lat plate (or which P/ ), eperimental tie how that it i alo applicable approimatel or trblent t low oer a race, een in the preence o prere graient. For laminar low, howeer, the analog i not applicable nle P/. hereore, it oe not appl to laminar low in a pipe
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