APPH 4200 Physics of Fluids Boundary Layers (Ch. 10) Answer: A pioneer in he mahemaical developmen of aerodynamics who conceived he idea of a fluid boundary layer, considered by many as he greaes single discovery in fluid dynamics. Quesion: Who was Ludwig Prandl? 1
Ludwig Prandl Physics Today, Dec 2005 Feb 4, 1875 Aug 15, 1953 German Scienis 2
3
4
5
6
Uniform Flow Across a Saionary Fla Plae (Blasius, 1908) '\ Ç) c -l' C), (l 1 p i l "l\- * r "V i. 1\ 1 "" s:n, l o. 1 V) " () L e "" \. l "' i l r( '\ -- (1, s.) \\ CJ. - rj ) 0,,.clI "n ø- r r ( ii c " Øi f -- ".a i 1 () cl 0' (11 1, l i '01- '\1 -Ð -l.( %.(J \ '\ci r '" C "\ "'i (. 1\ C V\ -I l-.% I - I( l ) ri rc ni r ;: "" f e '0 "' "\ '\ '\ '\ ( po '\ "' \1 l '\ e." 1\ '\ d '" '\ c \ - g '\ (f \:; l i :: "' i\,\; \ n l '\ '" "l ri. '\ I ;.( '" ç: ṭ. 1 "" c '1,. r) \J -( \ \ Sl )l H 01- r \-. 7
How big is he boundary layer? - 0 0 c "Ì V' in c (V "" " -/ \ 0 rc,. C.? Do I i..... )... )C.c. 1 /" ) '1 :1 ( I r \ f '". ') ",I; C, V\ \\ -l r :i 1 o - 0\ (.l l l, 1..c \ "" OA\ I L C. " S :' - (\ c ii IV " '1 1- "' & r "1.c -0,,'" C " lf 0.. ""? (/ r r' 0(, c, J f\ f c; l,.r j lf r i "' ".. J \A 'N \ \ \ in Ñ\ r)( f : "' ß N,- : r l fl 1 r- \). '\. -( '" -- V v " - f- 1 J. ). l-.c,. J..l i Cl 8
Wha are he magniudes of he erms in Navier-Sokes? -(') ro x ) \....... rn ", ;: 1 l) \ I 1.. - r- ;:, s:,.s: v -c IJ I ' f- 1- \ : 1U =, ';' +.- iv 9, + T s: \ J: rl '" -- \) \L :,.r: Iv, 'I ij)kj frj, ra f "(,.,,) l '1 l 11 y 'V \ I, I "" 1- "' l- "" 1-, lj) l. l. I l) "i ) \. "' '1 \) +, 'I I\'l,x 01 0 G (. r l' L Iv! I f L",.,c. IV I L",,, V\,. f\ y. + "".c, 1- \ o \ L Ie l è",..., I. rr, " -- v " (' 0. ) f i " 9
Blasius Fla Plae Soluion.. - C? m,.. f1 v) f'. Co,, i v -- Co \ 1 -( k- l" 9v \ ') V -Ç \) \ \ -r 0.1 I '" '",.,. 7I l: \ \I l. i '- \ ) - (l lj,i; "" f y. ç l;i "(., "" 9-, N n -ç - N 1\.( L u. -Ç -(.) ') L \ \J \?. ). -(-.c "",i I) )J 1 ) )' -f -( -o - -1 0 1 "" -l 11 r; Jl 0 f) c V) c,., + Ñ1. - '" i' r J 'Ti.c i -- r1 "" n (. -( c: )1U,. rj y. U T r i' 0. ') C) 11 (' r o "' L )( i i 10
Boundary Condiions ; \ 1. ì- (i I Cõ I " i,. ( 'Å " I i l' \)o Ý r Q Q l' l\ l- 0 's& yj & f- " \. "- ( ( T\ x f Cl (I T\ i- - (,- Cl l' ' 'i U i. "( \(! J f n 'J'I '- Q f"j., Vi, (:J Q 'Ì \\ )- ) ' æ.., l- v f. 0' l 1I,, \11 :r )C -l.. ). ".? (, oj "- ( \J l" () I )-I VI A, J '\, \l (\ ") - r Ul, 11
Finding he self-similar Sream funcion q. \ \l \N ( -Ç 1 \ l" 19- n " I \ "i '-\ ': 11. I-\ -. s: ri 1 \ "6? ;i f" N i \J ri -f -L -f "' l.19. -( -f Ili - L l' 1 V\ -ç :i r -" v \ --?Jl ') - f I) " i.r )\L :: '- l l' -- ' I i f (Y o i " ras: "" :i ), 'i '" l ( I "' r- )C + 1\ ) 01 i L -J ': o -(, '-.I ': )& '- 'i 'X' "' f "j v 12
Wha is f( )? i ç: \ C. -) ß.r rj \ l. \ i!: -- \ ij ': \ y. I\, '" \ v " l \ -- 0 rj )l "r :: '" \- i \f IJ ca\- ) -? s: 'N "l "' + I- q. \ \I c \ '" f- \ \l ': \ '1 IV f' J " l"r g (j :f '- ii \\ " r:) -(, ii J \ 0 y '- l Q. \ J ), " ii: C1 )-( ' " " 13
Describing he boundary layer \J y - ( - U V' -Ç -: \I 'J 0 J \ - )-- "" d 1 "' f:, "i r;,.c \, (\ '- ", ') e "Ì i" "' ( J \' V) i" () '1 - rl \,. r. Jl \- ( fj () "' 1 ci - ;0 il l- I Ñ\ \. Jl " V\ ( 1) - o, 0( f' )- ri \. - if -( '" : L) )l s: r I) 8:,."' 'bãl '" -(.f r L\ \ \J f' (\ -( P. f l-", -- C\ -f r 'i -Ç \) I) IJ 0( 8 s: )., e i 1 e. L L \ )c, "" 1 N b Ð \)) 1: l.?j r: ) -h -( i o,l L l. ( r ì 14
Numerical Soluion In[8]:= Plo@gBlasius@hD, 8h, 0, hbig ê 2<, PloLabel Ø "g@hd", PloRange Ø All, AxesLabel Ø 8"h", "uêu "<D uêu 1.0 g@hd 0.8 Ou[8]= 0.6 0.4 0.2 2 4 6 8 10 12 h 15
Long Fla Plaes: Transiion o Turbulence ') y Ô(x) xc. -- x Rex - 1 Rex;;;;! similariy: insabiliy: ransiion: urbulence leading edge region: i. B L. iniial condiion disurbances flow becomes Xo aminar.. equaions valid; iniial condiion forgoen grow and increasingly full N-S equaions a Xo required inerac irregular downsream Figure 10.11 Schemaic depicion of flow over a semiinfinie fla plae. 16
Long Fla Plaes: Transiion o Turbulence 10-2..,.,. " " " liolly 5 x 10-3 - D M)pU2L -- " " 4irÒlJJ. - _ el1-2 x 10-3 I o.a 6.,,.,,, ", 10-3 105 1() 10' 109 109 UL/v Figure 10.12 Measured drag coeffcien for a bounda layer over a fla plae. The coninuous line shows he drag coeffcien for a plae on which he flow is pary lamar and parly urbulen, wih he ransiion ang place a a posiion where he local Reynolds number is 5 x 105. The dashed lines show he behavior if he boundar layer was eiher compleely lainar or compleely urbulen over he enire lengh of he plae. 17
On-Line Video Fluid Mechanics (Boundary Layers par 1) (From Harvard s Abernahy: hp://www.seas.harvard.edu/direcory/fha) hp://www.youube.com/wach?v=7skwxeuxiom&feaure=relaed Fluid Mechanics (Boundary Layers par 2) hp://www.youube.com/wach?v=49usvafkm40&feaure=relaed Fluid Mechanics (Boundary Layers par 3) hp://www.youube.com/wach?v=wex72jextgm&feaure=relaed 18
Summary Prandl s hin boundary layer resolved he apparen conradicion beween he usefulness of Euler s inviscid flow and he realiy of he no-slip boundary condiion For Reynolds numbers up o around 10 5, he boundary layer is laminar For faser flows, or longer objecs, he flow becomes urbulen. 19