{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture_28_notes

# Lecture_28_notes - ChE 37 4—Lecture 28—Boundary Layers...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ChE 37 4—Lecture 28—Boundary Layers Navier-Stokes 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'? {fr-wt} noﬁlv‘acm gp‘ot § ” ’h‘ﬁxﬂéercj lo gala go, 22:5,»! no“); Ql‘ﬂx ctr ComP‘Q-m’z‘m/D‘ 0 CPD ‘Hme/ '{o 486 “ Hal) AQQUMPWM ' {7 9014;.“ Jaw, @ Kama/M7 L47“ OFFIWIL '— F1914 aég‘ump-p‘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 ¥ (on-35 ..._——--v f f_______________—- EL )1 ~51;de . 7D 0671" Cﬂn+"nu)47 _ no —’ \/—— mam ? X, Mar-ﬂ ' 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 gePaﬂa-JJWI 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 / —-\>1-A “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

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern