57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 1 Chapter 9 Flow over Immersed Bodies Flid lows are broadly categorized: 1. Internal lows sch as dcts/pipes, trbomachinery, open channel/river, which are bonded by walls or lid interaces: Chapter 8.. External lows sch as low arond vehicles and strctres, which are characterized by nbonded or partially bonded domains and low ield decomposition into viscos and inviscid regions: Chapter 9. a. Bondary layer low: high Reynolds nmber low arond streamlines bodies withot low separation. Re 1: low Re low (creeping or Stokes low) Re > 1,000: Laminar BL Re > 510 5 (Re crit ): Trblent BL b. Bl body low: low arond bl bodies with low separation.
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 3. Free Shear lows sch as jets, wakes, and mixing layers, which are also characterized by absence o walls and development and spreading in an nbonded or partially bonded ambient domain: advanced topic, which also ses bondary layer theory. Basic Considerations Drag is decomposed into orm and skin-riction contribtions: C D 1 1 V A S p p n îda S w t îda C Dp C
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 3 C t c L 1 p p n ĵda 1 V A S << 1 C > > C Dp streamlined body t c 1 C Dp > > C bl body Streamlining: One way to redce the drag redce the low separationredce the pressre drag increase the srace area increase the riction drag Trade-o relationship between pressre drag and riction drag Trade-o relationship between pressre drag and riction drag Beneit o streamlining: redcing vibration and noise
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 4 Qalitative Description o the Bondary Layer Flow-ield regions or high Re low abot slender bodies:
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 5 w = shear stress w rate o strain (velocity gradient) = y y 0 large near the srace where lid ndergoes large changes to satisy the no-slip condition Bondary layer theory and eqations are a simpliied orm o the complete NS eqations and provides w as well as a means o estimating C orm. Formally, bondary-layer theory represents the asymptotic orm o the Navier-Stokes eqations or high Re low abot slender bodies. The NS eqations are nd order nonlinear PDE and their soltions represent a ormidable challenge. Ths, simpliied orms have proven to be very sel. Near the trn o the last centry (1904), Prandtl pt orth bondary-layer theory, which resolved D Alembert s paradox: or inviscid low drag is zero. The theory is restricted to nseparated low. The bondary-layer eqations are singlar at separation, and ths, provide no inormation at or beyond separation. However, the reqirements o the theory are met in many practical sitations and the theory has many times over proven to be invalable to modern engineering.
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 6 The assmptions o the theory are as ollows: Variable order o magnitde U O(1) v <<L O() = /L x y 1/L O(1) 1/ O( -1 ) The theory assmes that viscos eects are conined to a thin layer close to the srace within which there is a dominant low direction (x) sch that U and v <<. However, gradients across are very large in order to satisy the no slip condition; ths, >>. y x Next, we apply the above order o magnitde estimates to the NS eqations.
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 7 p v x y x x y 1 1-1 1 - v v p v v v x y y x y 1 1-1 elliptic x v y 1 1 0 Retaining terms o O(1) only reslts in the celebrated bondary-layer eqations p v x y x y p 0 y v x y 0 parabolic
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 8 Some important aspects o the bondary-layer eqations: 1) the y-momentm eqation redces to i.e., p y 0 p = p e = constant across the bondary layer rom the Bernolli eqation: i.e., 1 p e U e p e U x e constant U x e Ths, the bondary-layer eqations are solved sbject to a speciied inviscid pressre distribtion ) continity eqation is naected 3) Althogh NS eqations are lly elliptic, the bondary-layer eqations are parabolic and can be solved sing marching techniqes 4) Bondary conditions = v = 0 y = 0 edge vale, i.e., inviscid low vale! = U e y = + appropriate initial conditions @ x i
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 9 There are qite a ew analytic soltions to the bondarylayer eqations. Also nmerical techniqes are available or arbitrary geometries, inclding both two- and threedimensional lows. Here, as an example, we consider the simple, bt extremely important case o the bondary layer development over a lat plate. Qantitative Relations or the Laminar Bondary Layer Laminar bondary-layer over a lat plate: Blasis soltion (1908) stdent o Prandtl x v y 0 Note: p x = 0 or a lat plate x v y y = v = 0 @ y = 0 = U @ y = We now introdce a dimensionless transverse coordinate and a stream nction, i.e., y U x y xu
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 10 y y U / U v x 1 U x Sbstittion into the bondary-layer eqations yields 0 Blasis Eqation 0 @ = 0 1 @ = 1 The Blasis eqation is a 3 rd order ODE which can be solved by standard methods (Rnge-Ktta). Also, series soltions are possible. Interestingly, althogh simple in appearance no analytic soltion has yet been ond. Finally, it shold be recognized that the Blasis soltion is a similarity soltion, i.e., the non-dimensional velocity proile vs. is independent o x. That is, by sitably scaling all the velocity proiles have neatly collapsed onto a single crve. Now, lets consider the characteristics o the Blasis soltion: U vs. y v U vs. y
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 11 5 x Re x U x Re x vale o y where /U =.99
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 1 i.e., c w U (0) x / U U w 0.664 Re x x see below : Local riction coe. C Note: = plate width = plate length b bl L 0 c dx = c 1.38 Re ( L ) L : Friction drag coe. U L Wall shear stress: w 3 0.33U or 0.33 U x Re w x x Other: * x 1 dy 1.708 displacement thickness 0 U Re x measre o displacement o inviscid low de to bondary layer x 1 dy 0.664 momentm thickness 0 U U Re x measre o loss o momentm de to bondary layer H = shape parameter = * =.5916
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 13
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 14 Qantitative Relations or the Trblent Bondary Layer -D Bondary-layer Form o RANS eqations x v y 0 x v y x p e y y v reqires modeling Momentm Integral Analysis Historically similarity and AFD methods sed or idealized lows and momentm integral methods or practical applications, inclding pressre gradients. Modern approach: CFD. To obtain general momentm integral relation which is valid or both laminar and trblent low y 0 For lat plate or or general case momentm eqation ( v) continity dy w U 1 c d dx H U du dx dp dx U du dx du lat plate eqation 0 dx
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 15 0 U 1 U dy momentm thickness * H shape parameter * 1 0 U dy displacement thickness Can also be derived by CV analysis as shown next or lat plate bondary layer. Momentm Eqation Applied to the Bondary Layer Consider low o a viscos lid at high Re past a lat plate, i.e., lat plate ixed in a niorm stream o velocityui ˆ. Bondary-layer thickness arbitrarily deined by y = 99% 99% is the vale o y at = 0.99U). Streamlines otside 99% (where, will * delect an amont (the displacement thickness). Ths the streamlines move otward rom y H at x 0 to * y Y H at x x1.
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 16 Conservation o mass: V nda =0= 0 0 CS H Udy H Assme incompressible low (constant density): UH Sbstitting * dy U U dy UY U dy Y Y Y dy 0 0 0 Y H * * deines displacement thickness: Y 0 1 U dy is an important measre o eect o BL on external low. Consider alternate derivation based on eqivalent low rate: δ δ* Lam=/3 δ* Trb=/8 Udy * 0 dy * Flowrate between and o inviscid low=actal lowrate, i.e., inviscid low rate abot displacement body = viscos low rate abot actal body Inviscid low abot δ* body * * Udy Udy dy 1 dy 0 0 0 0 U w/o BL - displacement eect=actal discharge For 3D low, in addition it mst also be explicitly reqired that is a stream srace o the inviscid low contined rom otside o the BL. *
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 17 Conservation o x-momentm: H Y x 0 0 CS F D V nda U U dy dy Drag D U H Y 0 = Flid orce on plate = - Plate orce on CV (lid) Again assming constant density and sing continity: H Y dy 0 U Y Y 0 0 0 dy D U / U dy dy dx D Y 1 0 U dy U U where, is the momentm thickness (a nction o x only), an important measre o the drag. C D D 1 x c dx U x x x 0 d w c c xc D 1 dx U d dx c w U x d dx w Per nit span d dx Special case D momentm integral eqation or dp/dx = 0
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 18 Simple velocity proile approximations: U ( y / y / ) (0) = 0 (δ) = U y (δ)=0 no slip matching with oter low Use velocity proile to get C () and () and then integrate momentm integral eqation to get (Re x ) δ* = δ/3 θ = δ/15 H= δ*/θ= 5/ U / w U / d d c ( / 15) 1 / U dx dx 15 dx d U 30 dx U / x 5.5 / R e R e Ux / ; x 1/ x * 1/ / x 1.83 / R e x / x 0.73 / R e 1/ x 1/ D L C 1.46 / R e C ( L ) 10% error, c. Blasis
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 19 Approximate soltion Trblent Bondary-Layer Re t = 5 10 5 3 10 6 or a lat plate bondary layer Re crit 100,000 c d dx as was done or the approximate laminar lat plate bondary-layer analysis, solve by expressing c = c () and = () and integrate, i.e. assme log-law valid across entire trblent bondary-layer * 1 ln y * B neglect laminar sb layer and velocity deect region at y =, = U U * 1 ln c * 1 / B Re c c 1 / 1 / or.44 ln Re 5 c () c 1 / 6.0 Re power-law it
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 0 Next, evalate d dx d dx 0 U 1 U dy can se log-law or more simply a power law it 7 U 7 y w or 1 / 7 c x 1 U Re 0.16 Re x d 7 U U dx 7 1 / 6 d 9.7 dx 1/ 7 6 / 7 x almost linear c 0.07 Re 1/ 7 x 0.031 7 C c L Re 6 1/ 7 L Note: cannot be sed to obtain c () since w These ormlas are valid or a lly trblent low over a smooth lat plate rom the leading edge. Assming the transition rom laminar to trblent occrs at Re larger than 10 5, those ormlas in general give better reslts or siciently large Reynolds nmber Re L > 10 7. d dx i.e., mch aster growth rate than laminar bondary layer
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 1 Alternate orms by sing the same velocity proile /U = (y/) 1/7 assmption bt sing an experimentally determined shear stress ormla w = 0.05U (/U) 1/4 are: x shear stress: 1/5 0.37 Re x 1/ 5 0.09 U w Re 0.058 0.074 c C 1/ 5 Re Re 1/5 x x L These ormlas are valid only in the range o the experimental data, which covers Re L = 5 10 5 10 7 or smooth lat plates. Other empirical ormlas or smooth lat plates are as ollows: Total shear-stress coeicient Local shear-stress coeicient L c c ( ).98 log Re log Re x. 65 L.73.3 For the experimental/empirical ormlas, the bondary layer is sally tripped by some roghness or leading edge distrbance, to make the bondary layer trblent rom the leading edge.
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 Finally, composite ormlas that take into accont both the initial laminar bondary layer and sbseqent trblent bondary layer, i.e. in the transition region (5 10 5 < Re L < 8 10 7 ) where the laminar drag at the leading edge is an appreciable raction o the total drag: or ( ) with transitions at Re t = 5 10 5 or all cases.
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 3 Bl Body Drag Drag o -D Bodies First consider a lat plate both parallel and normal to the low C Dp 1 S 1 V A p p n î 0 C 1 1 V A S w t îda 1.33 = 1 / Re L laminar low.074 = 1 / 5 Re L trblent low low pattern where C p based on experimental data vortex wake typical o bl body low
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 4 C Dp 1 1 S V A 1 = C p A da S p p n îda = sing nmerical integration o experimental data C = 0 For bl body low experimental data sed or C D.
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 5 In general, Drag = (V, L,,, c, t,, T, etc.) rom dimensional analysis c/l C D Drag t Re, Ar,,, T, etc. 1 L L V A scale actor
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 6 Potential Flow Soltion: sin r a r U U 1 p V 1 p r p U 1 U 1 p p C p sin 4 1 a r C srace pressre Flow Separation Flow separation: The lid stream detaches itsel rom the srace o the body at siciently high velocities. Only appeared in viscos low!! Flow separation orms the region called separated region r r 1 r
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 7 Inside the separation region: low-pressre, existence o recirclating/backlows viscos and rotational eects are the most signiicant! Important physics related to low separation: Stall or airplane (Recall the movie yo saw at CFD-PreLab!) Vortex shedding (Recall yor work at CFD-Lab, AOA=16! What did yo see in yor velocity-vector plot at the trailing edge o the air oil?)
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57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 9
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57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 37 Terminal Velocity Terminal velocity is the maximm velocity attained by a alling body when the drag reaches a magnitde sch that the sm o all external orces on the body is zero. Consider a sphere sing Newton Second law: F F F F m a d b g when terminal velocity is attained F a 0 : or F F F d b g 1 V C A 0 D p Sphere lid V Sphere Z For the sphere Ap d 4 and V Sphere d 6 3 The terminal velocity is: 43 sphere lid V 0 C D lid d 1 Magns eect: Lit generation by spinning Breaking the symmetry cases the lit!
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 38 Eect o the rate o rotation on the lit and drag coeicients o a smooth sphere: Lit acting on the airoil Lit orce: the component o the net orce (viscos+pressre) that is perpendiclar to the low direction
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 39 Variation o the lit-to-drag ratio with angle o attack: The minimm light velocity: Total weight W o the aircrat be eqal to the lit W F L 1 C L,max V min A V min C W L,max A
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 40 Eect o Compressibility on Drag: CD = CD(Re, Ma) Ma U a speed o sond = rate at which ininitesimal distrbances are propagated rom their sorce into ndistrbed medim Ma < 1 Ma 1 Ma > 1 Ma >> 1 sbsonic transonic (=1 sonic low) spersonic hypersonic < 0.3 low is incompressible, i.e., constant C D increases or Ma 1 de to shock waves and wave drag Ma critical (sphere).6 Ma critical (slender bodies) 1 For U > a: pstream low is not warned o approaching distrbance which reslts in the ormation o shock waves across which low properties and streamlines change discontinosly
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 41