Chapter 7: Boundary Layer Theory

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1 Proessor Fred Stern Fall 17 1 Chapter 7: Boundar Laer Theor 7.1. Introduction: Boundar laer lows: Eternal lows around streamlined bodies at high Re have viscous (shear and no-slip) eects conined close to the bod suraces and its wake, but are nearl inviscid ar rom the bod. Applications o BL theor: aerodnamics (airplanes, rockets, projectiles), hdrodnamics (ships, submarines, torpedoes), transportation (automobiles, trucks, ccles), wind engineering (buildings, bridges, water towers), and ocean engineering (buos, breakwaters, cables). 7. Flat-Plate Momentum Integral Analsis & Laminar approimate solution Consider low o a viscous luid at high Re past a lat plate, i.e., lat plate ied in a uniorm stream o velociti ˆ. Boundar-laer thickness arbitraril deined b 99% (where, 99% the value o at u.99). Streamlines outside 99% will delect an amount (the displacement thickness). Thus the streamlines move outward rom H at to Y H + at 1. is

2 Proessor Fred Stern Fall 17 Conservation o mass: ρ V nda CS H ρd + H+ ρud Assuming incompressible low (constant densit), this relation simpliies to Note: Y H ( + u ) d Y + ( u ) ud Y Y Y d H +, we get the deinition o displacement thickness: Y u 1 d ( a unction o onl) is an important measure o eect o BL on eternal low. To see this more clearl, consider an alternate derivation based on an equivalent discharge/low rate argument: Lam/3 Turb/8 d ud Flowrate between and o inviscid lowactual lowrate, i.e., inviscid low rate about displacement bod equivalent viscous low rate about actual bod d Inviscid low about bod d ud 1 u d w/o BL - displacement eectactual discharge For 3D low, in addition it must also be eplicitl required that is a stream surace o the inviscid low continued rom outside o the BL.

3 Proessor Fred Stern Fall 17 3 Conservation o -momentum: H + CS Y ( ) ( ) F D ρuv nda ρ d ρu ud Drag D ρ orce on CV (luid) H Y ρu d Fluid orce on plate - Plate Again assuming constant densit and using continuit: D ρ ρ D Y Y w u / d u d τ Y u u θ 1 d d Y u H d where, θ is the momentum thickness (a unction o onl), an important measure o the drag. C C D d θ d D θ ρ τ w 1 ρ C C 1 C d d τ w d ( C ) ρ D Per unit span dθ d dθ d Special case D momentum integral equation or p

4 Proessor Fred Stern Fall 17 4 Simple velocit proile approimations: u ( / / ) u() u() u () no slip matching with outer low se velocit proile to get C () and θ() and then integrate momentum integral equation to get (Re ) /3 θ /15 H /θ 5/ τ µ 15µ d d ρ / 5.5 / Re Re C / µ / dθ 1/ ρ d / ν / 1.83 / Re θ /.73 / Re C w D 3µ d ρ 1.46 / Re 1/ 1/ 1/ L 1/ C ( L) d d ( /15); 1% error, c. Blasius νν

5 Proessor Fred Stern Fall Boundar laer approimations, equations and comments, ρ,µ u v D NS, ρconstant, neglect g u + v u v t t + uu + uv + vu + vv 1 p + ν ( u ρ 1 p + ν ( v ρ + u + v ) ) Introduce non-dimensional variables that includes scales such that all variables are o O(1): / L Re L t t / L u u / υ v Re p p p ρ Re L / ν

6 Proessor Fred Stern Fall 17 6 The NS equations become (drop ) u + v 1 u + uu + vu p + u + u Re t ( v + uv + vv ) p + v + v t Re Re Re For large Re (BL assumptions) the underlined terms drop out and the BL equations are obtained. Thereore, -momentum equation reduces to p ie.. p p(, t) p ρ( + ) t D BL equations: u + v ; u t + uu Note: + vu ( t + ) +νu (1) (,t), p(,t) impressed on BL b the eternal low. () : i.e. longitudinal (or stream-wise) diusion is neglected. (3) Due to (), the equations are parabolic in. Phsicall, this means all downstream inluences are lost other than that contained in eternal low. A marching solution is possible. (4) Boundar conditions From Euler/Bernoulli equation or eternal low

7 Proessor Fred Stern Fall 17 7 inlet matching Solution b marching X No slip No slip: u (,, t) v(,, t) Initial condition: (,,) Inlet condition: u (,, t) u known given at Matching with outer low: u(,, t) ( t, ) (5) When appling the boundar laer equations one must keep in mind the restrictions imposed on them due to the basic BL assumptions not applicable or thick BL or separated lows (although the can be used to estimate occurrence o separation). (6) Curvilinear coordinates

Proessor Fred Stern Fall 17 8 Although BL equations have been written in Cartesian Coordinates, the appl to curved suraces provided << R and, are curvilinear coordinates measured along and

8 Proessor Fred Stern Fall 17 8 Although BL equations have been written in Cartesian Coordinates, the appl to curved suraces provided << R and, are curvilinear coordinates measured along and normal to the surace, respectivel. In such a sstem we would ind under the BL assumptions p ρu R Assume u is a linear unction o : u Or dp ρ d R p( ) p() ρ 3R p ρ ; thereore, we require << R 3R

9 Proessor Fred Stern Fall 17 9 (7) Practical use o the BL theor For a given bod geometr: (a) Inviscid theor gives p() integration gives L and D (b) BL theor gives (), τ w (), θ(),etc. and predicts separation i an (c) I separation present then no urther inormation must use inviscid models, BL equation in inverse mode, or NS equation. (d) I separation is absent, integration o τ w () rictional resistance and bod +, inviscid theor gives p() or drag +, can go back to () or more accurate BL calculation including viscous inviscid interaction (8) Separation and shear stress At the wall, u v 1 st derivative u gives τ w u 1 p µ τ w µ u w τ w separation nd derivative u depends on p

10 Proessor Fred Stern Fall 17 1 Inlection point 7.4. Laminar Boundar Laer - Similarit solutions (D, stead, incompressible): method o reducing PDE to ODE b appropriate similarit transormation (RRee tttttttttt ) u v uu + + vu + νu BCs: u (, ) v(,) u (, ) ( ) + inlet condition

11 Proessor Fred Stern Fall For Similarit ( ) F ( ) g( ) u, Or in terms o stream unction ψ : u epect g( ) related to ( ) ψ v ψ For similarit ψ ( ) g( ) ( η) η g( ) u BC: u v u ψ ' v ψ ' g + g gη ( ) (, ) ( ) () () (,) ( ) g( ) () + ( ) g ( ( ) g ( ) () ( ( ) g( ) + ( ) g ( ) ) () ) () () (, ) ( ) ( ) ( ) ( ) ( ) 1 Write boundar laer equations in terms o ψ ψ ψ ψ ψ + νψ Substitute Assemble them together: ψ ψ '' ''' g g ' '' ψ ηg / ' ' '' g ' '' ( ) η ( g + g g η )( g) + ν g ''' ( g ) g

12 Proessor Fred Stern Fall 17 1 ' ' '' g g g '' g + ν g '' ''' ( g) + ν ''' ''' g g + ν ν '' ' ( g) + ( 1 ) C1 C Where or similarit C 1 and C are constant or unction η onl i.e. or a chosen pair o C 1 and C ( ), ( ) (Potential low is NOT known a priori) ''' '' ' Then solution o ( 1 ) ( ) u,, g can be ound + C + C gives ( η) 1 u µ τ w µ g w '' ( ),,,θ, H, C, C D The Blasius Solution or Flat-Plate Flow constant C Then d d C Let gg ν C ν ( g ) 1 ( ) [ ] 1 C, then g( ) ν g Cν η 1 ν

13 Proessor Fred Stern Fall ''' + 1/ '' ' ' ( ) ( ), ( ) 1 Blasius equations or Flat Plate Boundar Laer Solutions b series technique or numerical u.99 when η Re ' ( 1 ) u ϑ 1 d dη ' ' ϑ ( 1 ) dη Re ϑ 1.78 Re u u θ.664 θ 1 d Re So, H. 59 θ '' u µ ( ) τ w.664 θ τ w µ C w ν 1 ρ Re C D D ρ L 1 L C d L 1.38 Re L ; Re L L ν ;

14 Proessor Fred Stern Fall v ' η Re << 1 or Re >> 1 Oseen Blasius CD 3-6 (3 rd edition,vicous lows) ReL <1 1<Re<Retr~3 1 6 LE Higher order correction C D 1.38/ Re + L.3/ Re L Similar breakdown occurs at Trailing edge. From triple deck theor the correction is +.661/ 7 / 8 Re L Re small thereore local breakdown o BL approimation

15 Proessor Fred Stern Fall 17 15

16 Proessor Fred Stern Fall Falkner-Skan Wedge Flows ' ( 1 ) ''' '' + C1 + C ' ' ( ) ( ), ( ) 1 C g ν ( g) C g ν 1 g Consider ( ) (Blasius Solution: C, C 1 1) gg + g gg + g ( g) g g 1 ν C νc Hence ( g ) ( C C ) 1 g ν, C Choose C 1 1 and C arbitrarc, Integrate g ν ( C) Combine C g ν C 1 C C ln ln + k C C ( C ) Then ( ) k g ( ) 1 C ( C) C ν k g ν ( η) g ( ) η u '( η) Similarit orm o BL eq.

17 Proessor Fred Stern Fall Change constants η m ( ) k g ' ( 1 ) m +1 ϑ ''' '' + + β, + 1 Solutions or β 1. m β β m m, β ' ' ( ) ( ) ( ) 1 Separation ( τ w ) Solutions show man commonl observed characteristics o BL low: The parameter β is a measure o the pressure gradient, dp d. For β >, dp d < and the pressure gradient is avorable. For β <, the dp d > and the pressure gradient is adverse. Negative β solutions drop awa rom Blasius proiles as separation approached Positive β solutions squeeze closer to wall due to low acceleration Accelerated low: Decelerated low: τ ma ma near wall τ moves toward

18 ME:516 Proessor Fred Stern Fall 17 Chapter 7 18

19 Proessor Fred Stern Fall Momentum Integral Equation Historicall similarit and other AFD methods used or idealized lows and momentum integral methods or practical applications, including pressure gradients. Momentum integral equation, which is valid or both laminar and turbulent low: τ ρ θ ( BL orm o momentum equation + ( u ) continuit)d w 1 dθ C + d u 1 H ; θ 1 u Momentum: u d; d uu + vu ( + H ) θ d d p + ρ 1 τ ρ The pressure gradient is evaluated orm the outer potential low using Bernoulli equation 1 p+ ρ constant 1 p + ρ p ρ d For lat plate equation d

20 Proessor Fred Stern Fall 17 ( u )( u + v ) uu + uv u v uu, 1 τ ρ 1 + vu τ + uu + uv u + v ρ Continuit uu vu + ( u u ) + ( u) + ( v vu) 1 τ d ( τ τ ) / ρ uv u + u + v w ρ ( u) d + ( u) d + ( v vu) τ w u u 1 d ρ + θ + θ + ( u) d ( θ + ) C dθ 1 + d d d C dθ + d ( + H ) θ d d, H θ τ w ρ 1 C θ + θ ( + H )

21 Proessor Fred Stern Fall 17 1 Historicall two approaches or solving the momentum integral equation or speciied potential low (): 1. Guessed Proiles. Empirical Correlations Best approach is to use empirical correlations to get integral parameters (,,θ, H, C, C D ) ater which use these to get velocit proile u/ Thwaites Method θ Multipl momentum integral equation b ν τ wθ θ dθ θ d + µ ν d ν d ( + H ) LHS and H are dimensionless and can be correlated with pressure gradient parameter λ θ d ν d as shear and shape-actor correlations τθ w S ( λ) ( λ+.9) µ.6 H H( ) 5 ai i / θ λ (.5 λ) a i (, 4.14, -83.5, 854, -3337, 4576) Note i θ dθ ν d 1 d d θ ϑ Substitute above into momentum integral equation

22 Proessor Fred Stern Fall 17 1 d θ S ( λ ) + ( + H ) d λ ν d( λ / ) S λ( + H ) λ F d [ ] ( λ) ( ).45 6 F λ λ based on AFD and EFD Deine z θ ν so that d λ z d dz.45 6λ.45 6z d dz d + 6 z.45 d d 1 d 6 z. 5 d i.e. ( ) 45 z 6.45 ( ) 5 d + C d d. 45ν θ θ + 6 θ and () known rom potential low solution Complete solution: 5 d λ λ ( θ ) θ ν d d

23 Proessor Fred Stern Fall 17 3 τ wθ S µ θh ( ) λ ( ) λ Accurac: mild p ± 5% and strong adverse p (τ w near ) ± 15% i. Pohlhausen Velocit Proile: u 3 4 ( η) aη + bη + cη + dη with η a, b, c, d determined rom boundar conditions 1) u, u ) ν u, u, u No slip is automaticall satisied. F ( η) η η η G 6 d Λ ν d 3 ( η) ( 1 η) 3 p + η µ 4 u pressure gradient parameter related to λ λ Λ 945 ( Λ) + Λ F( η) + ΛG( η) Λ 97, 1 Λ 1 (eperiment: separation Λ separation -5) Proiles are airl realistic, ecept near separation. In guessed proile methods u/ directl used to solve momentum integral equation numericall, but accurac not as good as empirical correlation methods; thereore, use Thwaites method to get λ, etc., and then use λ to get Λ and plot u/.

24 Proessor Fred Stern Fall 17 4 ii. Howarth linearl decelerating low (eample o eact solution o stead state D boundar laer) Howarth proposed a linearl decelerating eternal velocit distribution ) 1 L ( as a theoretical model or laminar boundar laer stud. se Thwaites s method to compute: a) X sep b) C. 1 L Note - /L Solution θ 6.45ν 1 6 L νl 1 d L L can be evaluated or given L, Re L (Note: θ, θ L ) λ θ d ν d L 1

25 Proessor Fred Stern Fall 17 5 λ sep X sep.9 L.13 3% higher than eact solution.1199 C L. 1 i.e. just beore separation λ.661 S C 1.99 C Re (.99) Re ( λ ) θ θ Compute Reθ in terms i Re L θ θ L θ L νl Re 1 θ Reθ Re L L C.57 νl.661 L 6 [( 1.1) 1].57 Re (.99) 1 1/ Re L.661 Re L L 1.77 Re ϑl.661 L To complete solution must speci Re L

Proessor Fred Stern Fall 17 6 Consider the comple potential a a iθ F( z) z r e a ϕ Re[ F ( z) ] r cosθ a ψ Im[ F ( z) ] r sin θ Orthogonal rectangular hperbolas ϕ : asmptotes ± ψ : asmptotes, V v v V

26 Proessor Fred Stern Fall 17 6 Consider the comple potential a a iθ F( z) z r e a ϕ Re[ F ( z) ] r cosθ a ψ Im[ F ( z) ] r sin θ Orthogonal rectangular hperbolas ϕ : asmptotes ± ψ : asmptotes, V v v V r θ ϕ ϕ eˆ r ar cosθ r ar sin θ v 1 + ϕθ eˆ r θ π θ (low direction as shown) ( cosθiˆ + sinθˆj ) + v ( sinθiˆ + cosθˆj ) ( v cosθ v sinθ ) iˆ + ( v sinθ + v cosθ ) ˆj r r θ Potential low slips along surace: (consider 1) determine a such that v r v r θ r θ at rl, al cos( 9) al, i.e. v at L-r: cos( 9) ( ) ) let ( ) r a( L ) v r Or : ( ) a( L ) L ( L ) θ 9 ) θ 9 (1 ) L a L

27 Proessor Fred Stern Fall 17 7

28 Proessor Fred Stern Fall 17 8

29 Proessor Fred Stern Fall Turbulent Boundar Laer 1. Introduction: Transition to Turbulence Chapter 6 described the transition process as a succession o Tollmien- Schlichting waves, development o Λ - structures, vorte deca and ormation o turbulent spots as preliminar stages to ull turbulent boundar-laer low. The phenomena observed during the transition process are similar or the lat plate boundar laer and or the plane channel low, as shown in the ollowing igure based on measurements b M. Nishioka et al. (1975). Periodic initial perturbations were generated in the BL using an oscillating cord. 5 For tpical commercial suraces transition occurs at Re, tr However, one can dela the transition to Re, tr 3 1 with care in polishing the wall.. Renolds Average o D boundar laer equations u u + u ; v v + v ; w w + w ; p p + p ; Substituting u, v and w into continuit equation and taking the time average we obtain,

30 Proessor Fred Stern Fall 17 3 u v + w + z u ' ' v + ' w + z Similarl or the momentum equations and using continuit (neglecting g), DV ρ Dt Where τ ij p + τ u µ Assume i j u + Laminar i j ij ρu a. ( ) << which means u ' i u ' j Turbulent v << <<, b. mean low structure is two-dimensional: w, z ' Note the mean lateral turbulence is actuall not zero, w, but its z derivative is assumed to vanish. Then, we get the ollowing BL equations or incompressible stead low: u u + v u u v + e d d e + 1 τ ρ Continuit -momentum p ' v ρ -momentum

31 Proessor Fred Stern Fall Where e is the ree-stream velocit and: u τ µ ρu Note: The equations are solved or the time averages u and v The shear stress now consists o two parts: 1. irst part is due to the molecular echange and is computed rom the time-averaged ield as in the laminar case;. The second part appears additionall and is due to turbulent motions. The additional term is new unknown or which a relation with the average ield o the velocit must be constructed via a turbulence model. Integrate - momentum equation across the boundar laer ' v ' ' ( ) v p pe ρ So, unlike laminar BL, there is a slight variation o pressure across the turbulent BL due to velocit luctuations normal to the wall, which is no more than 4% o the stream velocit and thus can be neglected. The Bernoulli relation is assumed to hold in the inviscid ree-stream: dpe / d ρ ede / d Assume the ree stream conditions, e ( ) is known, the boundar conditions: No slip: u (, ) v(,) Free stream matching: u(, ) ( ) e 3. Momentum Integral Equations valid or BL solutions The momentum integral equation has the identical orm as the laminar-low relation: dθ θ d e τ C w + ( + H ) d d e ρ e

32 Proessor Fred Stern Fall 17 3 For laminar low: (, H, θ ) are correlated in terms o simple parameter C θ λ υ d e d For Turbulent low: ( θ ) cannot be correlated in terms o a single parameter. C, H, Additional parameters and relationships are required that model the inluence o the turbulent luctuations. There are man possibilities all o which require a certain amount o empirical data. As an eample we will review the π β method. 4. Flat plate boundar laer (zero pressure gradient) a. Log law analsis o Smooth lat plate Assume log-law can be used to approimate turbulent velocit proile and use to get C C () relationship 1 u u / u ln + B κ ν where κ.41 and B 5 At (edge o boundar laer) 1 u e / u ln + B κ ν However: e / u 1/ 1/ τ 1/ ρ / / e C w e e ρ ρ C u e ϑ ϑ u e Re C C 1/ 1/ 1 ln Re κ C 1/ 1/ + B

33 Proessor Fred Stern Fall Following a suggestion o Prandtl, we can orget the comple log law and simpl use a power-law approimation: C.Re 1/ 6 b. se u/ proile to get θ, C,,, and H or smooth plate dθ τ w ρ d dθ or : C d 1 ρ C LHS: From Log law or RHS: se u / e C dθ to get d.re 1/ 6 Eample: u / e 7 u u ( / ) 1/ θ (1 ) d e e 7 7 dθ 1/ 6 7 d C. Re ρ e d 7 d 1/ 6 d(re ) d(re ) 1 Re 9.7 d(re ) 1/ 6 d(re ) Re 9.7 Assuming that: at or Re at Re : 1/ 7 6 / 7 /.16 / Re or : Turbulent BL has almost linear growth rate which is much aster than laminar BL which is proportional to 1/.

34 Proessor Fred Stern Fall Other properties: 1/7 C.7 / Re 1/7 6/7 13/7.135µ ρ τ w, turb 1/7 1/7 7 CD.31/ Re L C ( L) H / θ 1.3 τ w,turb decreases slowl with, increases with ρ and and insensitive to µ c. Inluence o roughness The inluence o roughness can be analzed in an eactl analogous manner as done or pipe low i.e. + 1 u ln ( + u + B+ B ε ) κ υ B( ε ) ln(1 +.3 ε ) κ i.e. rough wall velocit proile shits downward b a constant amount B ( ε + + ) which, increases with ε εu / ϑ A complete rough-wall analsis can be done using the composite loglaw in a similar manner as done or a smooth wall i.e. determine C () and θ() rom and equate using momentum integral equation d C ( ) θ ( ) d Then eliminate to get C (, ε / ) However, analsis is complicated: solution is Fig For ull roughlow a curve it to the C and C D equations is given b,

Proessor Fred Stern Fall 17 35 Fig. 7.6 Drag coeicient o laminar and turbulent boundar laers on smooth and rough lat plates. C C D (.87 + 1.

35 Proessor Fred Stern Fall Fig. 7.6 Drag coeicient o laminar and turbulent boundar laers on smooth and rough lat plates. C C D ( log ) ε L ( log ) ε.5.5 Full rough low Again, shown on Fig along with transition region curves developed b Schlichting which depend on Re t

36 Proessor Fred Stern Fall Boundar laer with pressure gradient u uu + v + vu u τ µ ( p / ρ) ρu v + 1 τ ρ The pressure gradient term has a large inluence on the solution. In particular, adverse pressure gradient (i.e. increasing pressure) can cause low separation. Recall that the momentum equation subject to the boundar laer assumptions reduced to p i.e. p p e constant across BL. That is, pressure (which drives BL equations) is given b eternal inviscid low solution which in man cases is also irrotational. Consider a tpical inviscid low solution (chapter 8)

37 Proessor Fred Stern Fall Even without solving the BL equations we can deduce inormation about the shape o the velocit proiles just b evaluating the BL equations at the wall ( ) u pe µ where pe d -ρ e d e which, shows that the curvature o the velocit proile at the wall is related to the pressure gradient. Eect o Pressure Gradient on Velocit Proiles Point o inlection: a point where a graph changes between concave upward and concave downward. The point o inlection is basicall the location where second derivative u o u is zero, i.e. (a) avorable gradient: p <, >, u < No point o inlection i.e. curvature is negative all across the BL and BL is ver resistant to separation. Note u ()< in order or u to merge smoothl with.

38 Proessor Fred Stern Fall (b) zero gradient: p u (c) weak adverse gradient: p >, <, u > PI in low, still no separation

39 Proessor Fred Stern Fall (d) critical adverse gradient: p >, <, u >, u PI in low, incipient separation (e) ecessive adverse gradient: p >, <, u >, u < τ w < PI in low, backlow near wall i.e. separated low region i.e. main low breaks awa or separates rom the wall: large increase in drag and loss o perormance: H separation 3.5 laminar.4 turbulent

40 Proessor Fred Stern Fall π-β Method C dθ + d ( H ) θ d d With () known there are three unknowns θ, H and C. At least two additional relations are needed. se Cole s law o the wake Π u ln + B+ ( / ) κ κ η / π 3 ( η) sin ( η) 3η η π κa/ se correlation or C (Reθ, H) and Reθ(Π) 1.33H.3e C C (Re, ) H θ H + (log Re ) 1 θ 1+Π Reθ ep( κλ κ B Π) κπ Net relate Π to Clauser equilibrium pressure gradient parameter dp a ( Π) H β H θ τ d ( H 1) w Π+.4Π Π+ 1.5Π a( Π ) κ(1 +Π) Momentum integral is solved with initial condition or θ( ) and integrated to + iterativel using above relations. Estimated θ gives Reθ and Π, β gives H, and lastl C evaluated using Reθ and H. Iterations required until all relations satisied and then proceed to net.

Proessor Fred Stern Fall 17 41 7. 3-D Integral methods Momentum integral methods perorm well (i.e. compare well with eperimental data) or a large class o both laminar and turbulent D lows.

41 Proessor Fred Stern Fall D Integral methods Momentum integral methods perorm well (i.e. compare well with eperimental data) or a large class o both laminar and turbulent D lows. However, or 3D lows the do not, primaril due to the inabilit o correlating the cross low velocit components. The cross low is driven b outer potential low (,z). p, which is imposed on BL rom the z 3-D boundar laer equations uu uw u + + vu + vw + v + w + wu + ww z ; closure equations z z ( p / ρ) + ϑu ( p / ρ) + ϑw z ( u v ); ( v w ); Dierential methods have been developed or this reason as well as or etensions to more comple and non-thin boundar laer lows.

42 Proessor Fred Stern Fall Separation What causes separation? The increasing downstream pressure slows down the wall low and can make it go backward-low separation. dp d > adverse pressure gradient, low separation ma occur. dp d < avorable gradient, low separation can never occur Previous analsis o BL was valid beore separation. Separation Condition u τ w µ Note: 1. Due to backlow close to the wall, a strong thickening o the BL takes place and BL mass is transported awa into the outer low. At the point o separation, the streamlines leave the wall at a

43 Proessor Fred Stern Fall certain angle. Separation o Boundar Laer Notes: 1. D to E, pressure drop, pressure is transormed into kinetic energ.. From E to F, kinetic energ is transormed into pressure. 3. A luid particle directl at the wall in the boundar laer is also acted upon b the same pressure distribution as in the outer low (inviscid). 4. Due to the strong riction orces in the BL, a BL particle loses so much o its kinetic energ that is cannot manage to get over the pressure gradient rom E to F. 5. The ollowing igure shows the time sequence o this process: a. reversed motion begun at the trailing edge b. boundar laer has been thickened, and start o the reversed motion has moved orward considerabl. c. and d. a large vorte ormed rom the backlow and then soon separates rom the bod.

Proessor Fred Stern Fall 17 44 Eamples o BL Separations (two-dimensional) Features: The entire boundar laer low breaks awa at the

44 Proessor Fred Stern Fall Eamples o BL Separations (two-dimensional) Features: The entire boundar laer low breaks awa at the point o zero wall shear stress and, having no wa to diverge let or right, has to go up and over the resulting separation bubble or wake. 1. Plane wall(s) Thin wall (a). Plane stagnation-point low: no separation on the streamlines o smmetr (no wall riction present), and no separation at the wall (avorable pressure gradient) (b).flat wall with right angle to the wall: low separate, wh?. Diuser low:

negative pressure gradient ma re-laminarize a low (b) a strong positive pressure gradient causes a

45 Proessor Fred Stern Fall Turbulent Boundar Laer (a) (b) Inluence o a strong pressure gradient on a turbulent low (a) a strong negative pressure gradient ma re-laminarize a low (b) a strong positive pressure gradient causes a strong boundar laer top thicken. (Photograph b R.E. Falco) Eamples o BL Separations (three-dimensional) Features: unlike D separations, 3D separations allow man more options.

Proessor Fred Stern Fall 17 46 There are our dierent special points in separation: (1). A nodal Point, where an ininite number o surace streamlines merged tangentiall to the separation line ().

46 Proessor Fred Stern Fall There are our dierent special points in separation: (1). A nodal Point, where an ininite number o surace streamlines merged tangentiall to the separation line (). A saddle point, where onl two surace streamlines intersect and all others divert to either side (3). A ocus, or spiral node, which orms near a saddle point and around which an ininite number o surace streamlines swirl (4). A three-dimensional singular point, not on the wall, generall serving as the center or a horseshoe vorte. 1. Boundar laer separations induced b ree surace (animation) CFDSHIP-IOWA

Proessor Fred Stern Fall 17 47. Separation regions in corner low 3.

47 Proessor Fred Stern Fall Separation regions in corner low 3. 3D separations on a round-nosed bod at angle o attack Video Librar (animations rom Multi-media Fluid Mechanics, Homs, G. M., etc.)

48 Proessor Fred Stern Fall Conditions Producing Separation Separations on airoil (dierent attack angles) Leading edge separation Separations in diuser Eect o bod shape on separation Laminar and Turbulent separation

49 Proessor Fred Stern Fall Flow over clinders: eect o Re Flow over spheres: eect o Re Flow over edges and blunt bodies Flow over a truck

50 Proessor Fred Stern Fall 17 5 Eect o separation on sports balls

Chapter 7: Boundary Layer Theory

Chapter 7: Boundary Layer Theory Proessor Fred Stern Fall 14 1 Chapter 7: Boundar Laer Theor 7.1. Introduction: Boundar laer lows: Eternal lows around streamlined bodies at high Re have viscous (shear and no-slip) eects conined close