Lecture-22_(12-7-10)

Lecture-22_(12-7-10) - APPH 4200 Physics of Fluids Boundary...

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Unformatted text preview: APPH 4200 Physics of Fluids Boundary Layers (Ch. 10) December 7, 2010 Answer: A pioneer in the mathematical development of aerodynamics who conceived the idea of a fluid “boundary layer”, considered by many as “the greatest single discovery in fluid dynamics.” Question: Who was Ludwig Prandtl? 1 Ludwig Prandtl Physics Today, Dec 2005 Feb 4, 1875 – Aug 15, 1953 German Scientist 2 3 4 5 6 (1 o ,l ~ .. s:N "" ~ ~ ~1 1\° ~ ..r "V i. * "l\- ~ il CJ \\ s.) , II ~ - ~~ ~ rJ . ii r .. ø- lI ~ 0' % c "ft Øi .(J ~~ .( -l -! '\1 '01- i ,l ~1 (11 . cl -- ~ \~ ~~ ~ ~ () i1 " .a ~~ "\ "'i (. 1\ C V\ ~ ~ r ;: fe ""'0t rC t ~ ni l ..) ~ ri ~ ~ tI - I( .. l-.%~ ~ t -I ~ I; ~'\~ci '\ r ) 01- r \- )l H \J ~ t ,. '1 ¡ ~ c "" 1 ~ ~ -( \.. ~ \ Sl '" ~ .( '\ '" "l ~r i. ~~l i\ \ ~ "' ,\;t nt g '\ ~ (f :: tl \:; i d '\ \t-~ '" c .. ." 1\ '\ ~ l '\ e .. r '" C t .. C) c '\ ( II '\ po ~ ( ç: - "' (j ~ ~ ~ l' ì, i "' 6- ~ ~ ~ "" ~ (\ Ñ o r ~L .. 'n "n "' \1 I '\ '\ r ~ c. ~ ~~ .. "\ "' ,.c- )~ 0, ..-- ~1 p (l ~ ,~ C) ~ -l' c Ç) '\ ~t r( '\ i l~ "'t l~ ~~ e "" \. t" () ~L 1 V) .. ~ . l" t pC Uniform Flow Across a Stationary Flat Plate (Blasius, 1908) 7 -l r :i o - 0\ ( t"0 ~lf :' - (\ C ,,'" c f ~ lf .. ~l t. J j~~ ... t. ~ "" ? (/ J ~~ 0(, c, ~ f\ ~ L \\ \'N .. I Do " ~ C ~ 0 "" N r)( ft in c V' 0 in ~fl ~ l t ~ .. ~ r ß~ t t: "'t ~ ? -/ ~ (V ~ .. l l ~ ~ "'.. ~ ~ "" t t, 1. " C. .. ~ V\ i~ ~ ~ ~ "Ì c -0 /" II ~ r'\1 ¡. ,.r ri~ ~ ~ r r' ii &r \\ ~ . .c \ .. .. ~S OA\ ~ ~ ). f- 1 J. -v V .. " l.c -" ,. 1 '1 C, (¡I . £:1 Cl .l i J. -0 ~ '1 1- £: ~ -( \). '" ,- Ñ\ \A £ ~. " "' c; "1 I.c ~ "~ IV .l )~ ~1~ )C .c ~). .. r \ f '" . ') ",I; rc, \~. . i -I () ~ S rC \ ~ t "\ ! \. c= ~ ~\ ~ ~ ~ :: - How big is the boundary layer? 8 "" 1- I '1 l ~ :,.r: ~\L '" ~ + 't f- l) \ ~ ;: .. x~ y f\ y. "" .c Iv! ~ ~ + "i ~ ~)~ ~ \. 'V \~ ~I- i ,) l \) .. ro ~ , \ , 1 I \ , , \ , L , ';' ;: ~ ", ~ 1 rn '1 \) "" 1l. I l) 'I ~~ ~G tv " -- t... ~I ~ ~ 0. (' .. , t " ,. ~ t. .. ,. . l' r (. è" 0 ~ ~ l ~~) f t frJ iJ)kJ t"( ,.,t ~ e rl ~ ~ -- s: 1- +~t ~ ~ rr "t ~~ ~ 9,~ -c r- . ~ I\'l,x 01 o \ L.. iv .- , v - . + ~ t= IJ I ~ ,.s: .. ~) ~ ~ IfL", IV I L", ~.,c. ,,~ V\ ~ .. "' ~ , lJ) l. "' l- I 11 rat f Iv , 'I T J: ~ ~ \ ~¡: 1- 1U ~ s: ~. -(') ~ rv L. ~ ~ ;ì ~ j ~ l ~ -. C' - c :r 0 What are the magnitudes of the terms in Navier-Stokes? 9 -( u. L .( 1\ ~ -Ç -t ') ~ i ,i )' -f )J 1 ~ "" ~ ~)~ -( -o I) .c ~ ). ~-(-t L \ ~ \J \?. .. .) n ~ i N -ç¡ "" 9-, ~N ~ (l ..~ ) \I ~\ ~ \) ~ ~ ~ -r ~\~ \ ~ -Ç ') tV 9v \ ~ -( k~\~ ~ ,. 7I .. 1 ,. '" '" 0.1 t ~ tI ~ v -- ,i Co, . ~ f' .~ ~ . l" t Co ~ m C? ~ . -- .c + ~~ y.\ .,ç ~ II "" rJ y. ~ )1U,. -( c: (. n "( ., ~ ~ .. l;I~ lJ,i; -t "" f l. i '- l: ~ Ñ1 t r c, V) 0 -l "" 1 -1 0 t ~ ~ f1 , i' Jl ~ .. 11 ~ )( L "' to t (' r~ ¥ ~ ~ C) ') ~ i' 0. UT r t r1 ~ 'Ti J ~ i~ ~ ~ t. f) c t~ '" 11 ~ r; ~~ t .. ~ v) . ~ ~ t" '" -l ." ,. .i '" c V\ - 1) r c; Blasius Flat Plate Solution 10 ~ ~ ., Q -0 J I ~ ¡. ~I i t( 'Å ~~ ~\)o t" l' ~ I ti ,. Cõ " ~~ ~I~ (i ì- ;~ ~\ 1. ~r r- Ý Boundary Conditions ~ yj f- " \. "- Q l' Q l- l\ ~ ~ (I Cl ~ T\ x (( 0 ~ 's& f U 'J'I \\ Q (:J ., Q '- n ! i. 'i ~ ~ t' T\ i- t- ( ,Cl l' "( \( & J f f"J , Vi 'Ì ).. ~ )'t æ. , ~ . .? (, l- f. v 0' l 1I ,, ~ ~ ~ t t \J I.. l" () tt~ ~~ ") ~ \l t (\ ~ '\ ~ .., ¿ J ~ ~ A, ~ ~ ~ VI )-I~ ~ \11 :r ~ )C ~ ~-l .. " ). .. "- ~ ~ ¡oJ .. ~ ( .. ~ ~ ~ ,r Ul ~ ~ ~~ ~ 11 ~ \ "i ~I t ~ ~ tn ras: ~ i.r 1\ ~ ~ ~i~ L -J ~)~ 01 + r- )C ~ ~ "' )t ~ ~,~ ~)\L 'i '" ~l ~ ( I II " I) ') -t ?Jl~ tv \ ~ -- ~ ~ -" '-\~ -L -f .. " r ri -f f" N l" 19- ~~ ~\~ 1 t\~N ..( -Ç ~ \~ q. \l V\ - ~ ;i t "' :i "" f ~ ~ l' '- :i ~ -ç L Ili i ': ~ ~i~ i II -- l ~ :: l' t "' \J f ~ II )& (Y II ~ ? .. "6~ ri ~ ~ f v "j o~ t, '- .I ': '- 'i ~ -. ~ I-\ 11. ': ~ 'X' "t ~ -( 'I ..t .. o~ .. ~ ~ 1 -( -f 1\ l.19. ~ s: t 6'1 r (\ ~ r ~ ! r '" ) "- V) \ () (' Ì' V\ ;; i. t .. .. \) ~ \) - Finding the self-similar Stream function… 12 ~ 0 Q. l ~ :f ~ 0 '), " .. (j \\ ~" + .. ~.. ~ rJ~ " ~ ~ ~ '" \- II ~\~ )l ~ ~\ i!: ~ -) t ç: .. .. " ~ ii .. ~.. ~ '- ~\~ '" ~ ..\f ~ ~ 'N ~ t' .. " ~" C1 )-( ii: II y.. \J ~ , r:) .. ~I~\~ ~~ ~.. ?t s: ~ i .. ~".. ~ ': ~\~ l. ~ cA\- ß.r rJ \ ~~ tJ \~ -( ii f' .. '1 IV f- \ \l q. \~ "l "' ~ ItJ "r y. .. v I\ ~~ --\ iJ (continued…) ~ g J ': l"r ~ c \I ~ :: ~ ~)~ -- l\ ~ ~ , ~\~ ~ ~ '" C. .. i ~ ~ t ~ -: t. f) . '" ~ ~ '" ,. V\ ~, f ;i What is f(η)? 13 I Ñ\~ 1) -t \J () ¿ ~ " \I d ~ f:, \, Jl to , 0( ~ \. " 1 ~ "' rl ~ -- C\ ~ P -( i~ l. II 1: N ! \J ~ ~f' ~ o,l~ ?J r: ~ ~ II ~ -f r r ~ r if V\ ~ .. il cI ~ Jl - ,. V) t. f l-", 0( \) (\ 8t: ¡ 'bÃl ~ -(.f ,."' '" ~ -( '" ~ t: L)~ f' )- ri \. ( ~ .. ~ fJ '1 () ~ ~ "' (\ "i .. "' t1 0 'J II ~\- II ,~ r;~ ~ L\~ I) )l s: l- ;0 ~ ~ ~ "" 1 Ll ~ -h ~)~ , 8 s: I) ~\~ ~)~ )c ~ ~ IJ t i" \' ( ~ . ~ ~ ~ b \)) ~ ì r ~ ~ L L e. ~ ~ 1 i .. ( ., e r ~ . (J ,.c i" "Ì -: .. ~ \~ ') e V' -( - '- .."" ~ -Ç ~\~ -( .. 'i -Ç II -~ tJ \ - ~ )-- ~.. U ~~ y '" ~ §) i -j cl ~ .. ~ " ö ~ ~ t' ~ t' ~ ~ t L c, ç: Describing the boundary layer… 14 2 Numerical Solution Blasius_Boundary.nb In[8]:= Plot@gBlasius@hD, 8h, 0, hBig ê 2<, PlotLabel Ø "g@hD", PlotRange Ø All, AxesLabel Ø 8"h", "uêU¶ "<D g@hD uêU¶ 1.0 0.8 Out[8]= 0.6 0.4 0.2 2 4 6 8 10 12 h 15 (flat plate). Recr "- 1 06 Long Flat Plates: Figure 1 0.1 i schematically depicts the flow regimes on a semi-infite flat plat. finite Rex = U x/v "- 1, the full Navier-Stokes equations are reuied todesbe the I Transition to Turbulence leading edge region properly. As Rex gets large at the downstream limit of edge region, we can locate Xo as the maximal upstream extent of the boun i . ') Ô(x) y xc. Rex - 1 leading edge region: Rex;;;;! similarity: -- x instability: transition: turbulence i . B L . initial Xo aminar . . equatIons condition disturbances flow becomes valid; initial condition forgotten grow and increasingly full N-S equations at Xo required interact irregular downstream Figure 10.11 Schematic depiction of flow over a semiinfinite flat plate. 16 in Figure 10.1 1. The exact point at which the observed drag deviates from the wholly laminar behavior depends on experimental conditions and the transition shown in Figure 10.1 2 is at Long Flat Plates: Recr = 5 x 105. Transition to Turbulence 10-2 " . .,.,. 5 x 10-3 - " " .. ~liolly " " 4irÒlJJ. -- - - _ el1t D M)pU2L It ~o.a 2 x 10-3 ,., , ~~, ~6. , " 10-3 105 1() , 10' 109 109 UL/v Figure 10.12 Measured drag coeffcient for a bounda layer over a flat plate. The continuous line shows the drag coeffcient for a plate on which the flow is pary lamar and parly turbulent, with the transition tang place at a position where the local Reynolds number is 5 x 105. The dashed lines show the behav ior if the boundar layer was either completely lainar or completely turbu lent over the entire length of the plate. 17 On-Line Video Fluid Mechanics (Boundary Layers part 1) (From Harvard’s Abernathy: http://www.seas.harvard.edu/directory/fha) http://www.youtube.com/watch?v=7SkWxEUXIoM&feature=related Fluid Mechanics (Boundary Layers part 2) http://www.youtube.com/watch?v=49UsvAFKm40&feature=related 18 Summary • Prandtl’s thin boundary layer resolved the apparent contradiction between the usefulness of Euler’s inviscid flow and the reality of the no-slip boundary condition • For Reynolds numbers up to around 105, the boundary layer is laminar • For faster flows, or longer objects, the flow becomes turbulent. 19 ...
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This document was uploaded on 10/18/2011.

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