This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ChE 37 4—Lecture 28—Boundary Layers NavierStokes Equations: — Complex: PDE, 3D, unsteady, nonlinear, 4 equations.
— Solve by simplifying: Inviscid, laminar, reduce dimensions, steady state. Boundary Layer Method. — Split flow into two regions that are matched at the interface: 1 An outer region that is inviscid. Solve the resulting Euler Equations.
 Many analytic solutions exist (especially in 2D) for complex geometries.
 But does not apply near walls.
2 An inner boundary layer region in reduced dimensions and simpliﬁed by dropping terms. Boundary layer region. — No gravity, 2D, Steady state, thin. — Scale the governing equations to determine properties of the ﬂow and the boundary layer equa
tions: Navier—Stokes equations (SS, no gravity), scale x,y with just L, 17 with U, and P with pUZ:
17 Vf)‘: —%VP + MVZU. Scale it to get: (17 V17)* = (VP)* + Rie(V217)*. — High Re gives no viscous term which makes no sense. Instead, we need two scales, L and 6, the
boundary layer thickness. — KEY RESULT: Need two scales, L and 6, boundary layers are thin. Continuity: 3—“ + Q = 0, scale to 3—“ * + M Q * 2 0.
81 (9y 8x 6U 3y — KEY RESULT: we}: 2 U6/L, and Uref << U.  Q E _ _lﬂ 32v 32”
Y—Momentum. uaz + ’Uay — p 8y + 12%; + 1/3112.
* ’k =0: *
_ av * av _ L2 8P 1 3% L2 621)
scaled~ (Max) + (Day) 6? (By) + Re (3:62) + 6 Re (all?) ' — KEY RESULT: g—l; = 0. (Pressure can vary along the length, but not through the boundary layer
thickness). This is because the boundary layer is thin and the streamlines are nearly parallel. X—Momentum: See exam 2, problem 2, where we dropped one of the viscous terms. 3n 871V__1_§.IZ 827;
u$+v3~y _ 96w +VW' — This and continuity are the laminar boundary layer equations. — KEY RESULT: ignore the % term (that is, we ignore diffusion of momentum in the downstream direction).
— Note: —%% = U%. (Just diﬁerentiate Solution proceedure: Solve U(X) for outer ﬂow using Inviscid equations; Solve Boundary layer equations
given U(x); Solve for wall stress, drag, etc. Bernoulli equation with respect to x). SEE POSTED SOLUTION OF THESE EQUATIONS FOR REFERENCE (not required).
Boundary Layers apply to balls, wings, jets, wakes, mixing layers. As for pipe ﬂow, we have laminar, transitional, and turbulent. Take Re : 5 x 105 as the cutoff between laminar and turbulent. Shear stress decreases with distance for laminar and turbulent, but wall stress (friction) is greater for
turbulent than for laminar. 51%; 29— Tsomam7 Lw/arr 10,4 ?rwim(~/ — NE}, 27%.
A P06} '5'? {frwt} noﬁlv‘acm gp‘ot § ” ’h‘ﬁxﬂéercj lo gala go, 22:5,»! no“); Ql‘ﬂx ctr ComP‘Qm’z‘m/D‘ 0 CPD ‘Hme/ '{o 486 “ Hal) AQQUMPWM
' {7 9014;.“ Jaw, @ Kama/M7 L47“ OFFIWIL '— F1914 aég‘umpp‘im '“ goldb‘w {334' )3 L, Arprowk U a 'g log.) (Ive/7 6L L2) \‘ A» ' Nita? HSNJV 7149‘ 3,1,, bwtlaﬂ (.1 1,4 ,7 3411:? t ‘
)1) $3 «3 1 "Vita/W CD Ody) F100 I '2. 2:31; Benawwu’ £7'x (75,5) M018)”, no /L w—p— 9‘
KW 4§rv$=~’'~‘7 *7 U) M
(7/32) «A f F J 17
my Aw» gamma 2 m, CD .__————' Jaimprﬁé/ Hg, In KL M” SD’“.
'9 Ri‘IUCC 'H: bin/7’ Dre? TEAM
ma r/(a‘IZ L\ T3 L ¥ (on35 ..._——v f
f_______________— EL )1 ~51;de . 7D 0671" Cﬂn+"nu)47
_ no —’ \/—— mam
? X, Marﬂ
' 40’
. 141;,“ W [7&11'77 £0 2 Origin». "U’;U€’“ L4 LA.»\i(1¢.:>m7}~/ [ 90!.u+;0v\ @ $0M 0v+MP10v3 mn JVW 1““ 75¢, Jaw +0 a)!» We» M74 ow ‘ ’4’ (yr/(’5’ Vfat/ [\lecé [Jig/L L ,7 11".“ g LaM'.~6‘/], L9“) wIZ—i 6} no //0LJ gePaﬂaJJWI MM
A9 1/”, Fan Pb”), W a. 7(4..wrvlw,
(£9 354
\u UM Tam img RC
>11 Raﬁ
Cf '3 YT“) l
v ""2 / —\>1A
“ELEM 70172 7 b J <1
é: O'ééLII/ ﬁt Lﬂr";"M
C g 0.017 7M5
43 .
mm Q: 09“ 6% 9+“? if, ‘I 6'7 94% “MW D/Sﬂma Dun
Flawl't7 (Law) [.Mua}w K“? may 6’»; laDL In,” P1539)
(WM) {0 ‘6 ...
View
Full Document
 Fall '12
 DavidLignell

Click to edit the document details