Ch˜Ïng 6: Dịng th∏ - L¸c nâng & L¸c c£n Bài gi£ng cıa TS Nguyπn QuËc fi, nguyenquocy@hcmut.edu.vn Ngày 17 thỏng 11 nm 2015 Nẻi dung cản nm: Hm dũng, hm th v cỏc tớnh chòt Khỏi niêm lác nõng l¸c c£n Các dịng th∏ cÏ b£n ∞c tr˜ng cıa dịng bao quanh v™t - LĨp biên Bài tốn chÁng nh™p L¸c c£n: ma sát áp suòt Lác nõng: phõn bậ ỏp suòt b mt v dịng ch£y xốy / 40 Dịng l˛ t˜ng vs Dũng thác Potential Flow Potential Flow â S T Thoroddsen & © S T Thoroddsen Stanford University & Stanford University more b more blun increas increases leads leads to t downstrea downst a symmet a symm thethe differe dif ovals willw ovals the pressu the pre 6.6.3 Fl 6.6.3 Viscous Flow © Stanford University, Viscous Flow with permission © Stanford University, with As was no was theAs shape the doubletsha d it might doubleb used to re it migh used to / 40 c ! constant for r ! a, where a is the obtained from the where ps is the surface pressure Dòng Bernoulli l˛ t˜ng equation vs Dịng th¸c V6.6 Circular cylinder pressure can be expressed as c it follows that c ! for r ! a if V6.8 Circular cylinder with separation V6.7 Ellipse A comparison of this theoretica with a typical measured distrib the indicates upstreamthatpart the cylind which the of doublet streng flow a circular cylinder be thearound experimental results can Becau main flow separates from the the theoretical, frictionless fluc the cylinder 1see Chapter 92 and the corresponding velocity potenti The resultant force 1per u the pressure over the surface.f / 40 Dòng th∏ 2D: Hàm dòng ~ u, v V Xét dịng khơng quay, PT liên tˆc cho i∑u kiªn b£o tồn th∫ tích: u x Tìm hàm vơ h˜Ĩng u x, y : y v y r, : ur r ✓ u✓ r x thay vào PT liên tˆc: OK! V™y, có th∫ mơ t£ bi∏n u x, y , v x, y băng hm x, y hàm vơ h˜Ĩng x, y ˜Ịc gÂi hàm dòng v / 40 Dòng th∏ 2D: Hàm th∏ v™n tËc i∑u kiªn cho dịng khơng quay: !z Tìm hàm vơ h˜Ĩng u v x, y : x v x u y r, ✓ : ur u✓ y thay vào K không quay !z 0: satisfied! v™y có th∫ mơ t£ hai bi∏n u x, y , v x, y băng hm x, y ˜Òc gÂi th∏ v™n tËc r r ✓ x, y / 40 Dịng th∏ 2D: Dịng khơng nén ˜Ịc + khơng quay u v u x u x y v v y V™y y x2 y2 ∑u th‰a PT Laplace x v x !z u y 2 x2 y2 0 / 40 Dòng th∏ 2D: ChÁng nh™p nhi∑u dòng th∏ Nh˜ v™y, cho dịng th∏ 2D: có th∫ dùng ho∞c ∫ miêu t£ 1: th∏ v™n tËc cho dòng th∏ 2: th∏ v™n tËc cho dòng th∏ 2: th∏ v™n tËc cho dòng th∏ k∏t hỊp dịng th∏ t˜Ïng t¸ cho k∏t hỊp (chÁng nh™p) nhi∑u dịng th∏ có , 2, n: 1, 2, n n i i n V n Vi i i u v n i ui n i vi i / 40 Dòng th∏ 2D: PT Bernoulli cho dũng th mt phỉng năm ngang Nh˜ v™y, ã có u v bi∏t ho∞c , Làm ∫ tìm p? Dùng Pt Bernoulli cho dịng th∏ Ín ‡nh:: ~ V ~ r V ~ T¯ PT Euler cho dòng l˛ t˜ng: V rp, t ⇢ ~ ~ ~ ~ ~ r V ~ ~ ~ vÓi V rV V V r V r V V cho dòng th∏ 2 ~ V Dịng Ín ‡nh 0, v™y t ~ ~ rV V rp ⇢ Lßy d~s dx ~i dy ~j , rp d~s dp, ~ V ~ d~s r V d~s d V rV dp ⇢ dV 2 p1 ⇢ V12 p2 ⇢ V22 cho hai i∫m bßt kì / 40 Dịng th∏ 2D: Qua hª gi˙a ? ˜Ìng vĨi const ˜Ìng dịng (cho dịng Ín ‡nh) ˜Ìng vĨi const ˜Ìng ˜Ìng vĨi = const Øng Th∏ vĨi ˜Ìng =const Equipotential line ( φ = constant) d V d V d1 > d V1 < V V1 V2 d2 < d V2 > V Streamline (ψ = constant) / 40 sisofFluidFlow.qxd 2/17/12 4:40 PM Dịng th∏ 2D: Qua hª gi˙a dimensional flow The field can be or in terms of the stream fu expressed in terms c2 > c1 Thus of a stream function Page 287 deter l˜u l˜Òng? q 6.2 Conservation of Mass c + dc c2 q c u dy A y q B c1 – v dx c1 x (a) (b) ■ Figure 6.8 The flow between two streamlines c2 < c1 of flow An infinite number of streamlines make up a particular flow field, since for each constant q dq value assigned can be drawn dx udy vdxto c a streamlinedy d The actual numericalyvalue associatedxwith a particular streamline is not of particular significance, but the change in the value of c is related to the volume rate of flow cConsider two hange in the of the stream on is related volume rate w The right-hand side of then Eq sub c1 The r Thus,Vthe rate ∂c of fl c c1 C dq 287 y u= ∂c ∂y q c1 x Thu The relative value of c2pro with in the margin and funt In cylindrical coordin equa dimensional flow reduces t V closely spaced streamlines, shown in2 Fig 6.8a The lower streamline is designated c and the upper v one c # dc Let dq represent the volume rate of flow 1per unit width perpendicular to the x–y y ∂c vr = _1r_ u flow never crosses streamlines, since by Streamlines plane2 passing between the two streamlines Note that θ and ∂the velocity x definition the velocity is tangent to the streamline From conservation of mass we know that the equations inflow, dq, crossing the arbitrary surface AC of Fig 6.8a must equal the net outflow through surfaces V AB and BC Thus, ∂c vθ = – ∂r dq ! u dy " v dx q d q const v™n tËc t´ng, or in terms of the stream function kho£ng cách gi£m dq ! 0c dy # 0c dx 0y The right-hand side of Eq 6.38 is equal to dc so that 0x ∂c vr = _1r_ ∂θ (6.38) dime alo the figu as sh componen alon conti axisy r dime θ flows The as shown by the figure 10 / 40 isin gt coefficient is an integral of the local friction coefficient Dịng th¸c: odies.qxd 2/20/12 9:31 PM CDf ! Page 503 LĨp biên tßm phØng: Phân bË v™n tËc Table 9.2 Flat Plate Momentum Integral Results for Vario Laminar Flow Velocity Profiles Biên t¶ng: DRe1x" 2"x Profile Character 9.2 Boundary Layer Characteristics 0.06 x = 5.25 ft x = 6.76 ft c09FlowoverImmersedBodies.qxd x = 8.00 ft 0.05 U = 89 ft/s; airflow y, ft 0.04 Transitional 0.03 503 a Blasius solution b Linear u"U ! y"d Parabolic 2/20/12 c.9:31 PM Page 501 u"U ! 2y"d # y "d2 d Cubic u "U ! 31 y"d2 "2 # y"d2 "2 e Sine wave u"U ! sin 3p1 y"d2 "24 9.2 5.00 3.46 5.48 4.64 4.79 Layer Ch Boundary 1.0 Linear Turbulent Cubic 0.02 Sine wave 0.01 y δ 0.5 Blasius Laminar 0 0.2 0.4 0.6 u U 0.8 Parabolic ■ Figure 9.14 Typical boundary layer profiles on a flat plate for laminar, transitional, and turbulent flow (Ref 1) 0 0.5 u U Transition from laminar to turbulent flow also involves a noticeable change in the shape of the boundary layer velocity profile Typical profiles obtained in the neighborhood of the transition 1.0 ■ Figure 9.12 Typi boundary layer profiles momentum integral equ 26 / 40 u 0x !v 0y "# r 0y !na 0x2 ! 0y2 b Dịng whichth¸c: express Newton’s second law In addition, the conservation of mass equation, Eq incompressible flow is Ma sát b∑ m∞t tßm phØng LĨp biên tßm phØng: 0u 0v ! "0 0x 0y Biên t¶ng: δ ⌧w Re x cr x τw Laminar 0.332U Turbulent x Lác cÊn ma sỏt tßm phØng: b 0` ⌧w dx 2 ⇢U characteristics b` ⇢Uof boundary b` ■ Figure 9.9 Typical laye thickness and wall2shear stress for laminar and turbulent CDf x Df boundary layers 27 / 40 ρ U b! " ! D ■ Figure 9.19 coefficient From dimensional considerations 1see Drag Section 7.7.12 for an ellipse with the characteristic area either the frontal d " Cm/U area, A ! bD, or the planform area, A ! b/ (Ref 5) (9.38) where the value of the constant C depends on the shape of the body If we put Eq 9.38 into dimensionless form using the standard definition of the drag coefficient, we obtain U After all, it is the planform area on which the shear stress acts, rather than the much smaller 1for thin d " d !2Cm/U 1rU 2b/!22,2Cis bodies2 frontal area drag coefficient based on the planform area, C f(U, ",ellipse !) ! = The CD " D2 " " also shown in Fig 9.19 Clearly the drag obtained by using either of these drag coefficients be Re rU 2/would rU / the same They merely represent two different ways to package the same information rU/!m The where use on of the pressure,the rUdrag in the definition of the drag coeffi!2, on The amount of streamlining can haveRea " considerable effect the dynamic drag Incredibly, cienttois scale somewhat the case creeping the two two-dimensional objects drawn in Fig.misleading 9.20 is theinsame The of width of theflows wake 1Re for 12 because it introduces the ! Dũng thác: Lác cÊn: Hê sậ l¸c c£n more blunt s increases wi the streamlined strut is very thin, on the order of that for the much smaller diameter circular cylinder leads to sep D Reynolds Number Dependence Another parameter on which the drag coefficient can downstream be very dependent is the Reynolds number The main categories of Reynolds number dependence 1are 112 very2 low Reynolds number flow, 122 moderate Reynolds number flow 1laminar boundary a symmetric layer2, and 132 very large Reynolds number flow 1turbulent boundary layer2 Examples of these three ⇢U A are discussed below the differenc 2situations Low Reynolds number flows 1Re 12 are governed by a balance between viscous and presPotential Flow ■ Figure 9.20 Two objects of considerably different size that have the same drag force: (a) circular sure forces Inertia effects are negligibly small In such instances the drag on a three-dimensional ovals will gi cylinder C ! 1.2; (b) streamlined strut C ! 0.12 CDp CDfto be a function of the upstream velocity, U, the body ©size,S./,T.andThoroddsen body is expected the viscosity, & the pressure m Thus, for a small grain of sand settling in a lake 1see figure in the margin2 Stanford University " a = "b CD U, ρ U, ρ 10 D Diameter = D (a) CD Re < CDf Df ⇢U A " ! U CD (b) D Dp CDp D d " f 1U, /, m2 From dimensional considerations 1see Section 7.7.12 (9.38) ⇢U A d " Cm/U where the value of the constant C depends on the shape of the body If we put Eq 9.38 into di2 mensionless form using the standard definition of the drag coefficient, we obtain d 2Cm/U 2C " " Hình d§ng v™t th∫, Re, C Ma, , /✏ "`Re rU / FrrU ! = f(U, ", !) D 2 2 where Re " rU/!m The use of the dynamic pressure, rU2 !2, in the definition of the drag coefficient is somewhat misleading in the case of creeping flows 1Re 12 because itViscous introducesFlow the © Stanford University, with " a = "b permission U, ρ U, ρ 6.6.3 Flow As was note the shape of doublet desc it might be e used to repr 10 D A doublet combined with a uniform flow ■ Figure 9.20 Two objects of considerably different size that have the same drag force: circular can(a)be used to repcylinder C ! 1.2; (b) streamlined strut C ! 0.12 resent flow around Diameter = D (a) D (b) and for the D 28 / 40 Dũng thác: Lác cÊn: Hê sậ lác cÊn: ẫnh hng ca hỡnh dĐng 2.5 ã Flat plate normal to flow 2.0 U UD Re = v = 105 b = length D 1.5 CD ! 1.0 CD = " ρ U bD Flat plate parallel to flow 0.5 CD = 0 " ρ U b! 2 ! D ■ Figure 9.19 D with the characteris area, A ! bD, or th (Ref 5) 29 / 40 dies.qxd 3/12/12 8:29 PM Page 520 Dịng th¸c: L¸c cÊn: Hê sậ lác cÊn: ẫnh hng ca sậ Re apter ■ Flow over Immersed Bodies 400 200 100 60 40 A 20 CD 10 24 CD = _ Re B Smooth cylinder C D 0.6 0.4 Smooth sphere 0.2 0.1 0.06 –1 10 E 100 101 102 103 104 105 106 107 UD Re = ρ µ ( a) 30 / 40 crease with increasing flight speed of an airplane The changes in CD due to a to changes in both Re and Ma Dịng th¸c:The precise dependence of the drag coefficient on Re and Ma is generally efficient 132 However, the following simplifications are often justified For low Mach n depenLác cÊn: Hê sậ lác cÊn: ẫnh hng ca sậ Ma efficient is essentially independent of Ma as is indicated in Fig 9.23 For this si h num- h to ly 0.5 3.0 U 2.5 D ! CD = ρU2 bD 2.0 1.5 1.0 b = length 4D U D 0.5 ■ Figure 0 0.5 Ma 1.0 cient as a ber for tw subsonic fl 31 / 40 Dòng thác: Lác cÊn: Hê sậ lác cÊn: ẫnh hng ca sË Froude apter ■ Flow over Immersed Bodies 0.0015 ! U Hull with no bow bulb 0.0010 U "w CD = ρ U2 !2 CDw Hull with bow bulb 0.0005 Design speed, Fr = 0.267 0.1 0.2 0.3 0.4 U Fr = √!g ■ Figure 9.26 Typical drag coefficient data as a function of Froude number and hull characteristics 32 / 40 Dịng th¸c: L¸c cÊn: Hê sậ lác cÊn: ẫnh hng ca er ■ Flow over Immersed Bodies Ỵ nhám 0.6 ε = relative roughness D 0.5 Golf ball ! CD = _ π D2 ρU2 0.4 0.3 0.2 ε = 1.25 × 10–2 ε = (smooth) D 0.1 D ε = × 10–3 D ε = 1.5 × 10–3 D 4 × 10 105 × 105 106 × 106 _ Re = UD v ■ Figure 9.25 The effect of surface roughness on the drag coefficient of a sphere in the Reynolds 33 / 40 V9.19 Drag on a Dịng th¸c: truck L¸c c£n: Hê sậ lác The aerodynamic drag on automobiles provides an example of the use of adding componen drag forces The power required to move a car along a level street is used to overcome the rollin resistance and the aerodynamic drag For speeds above approximately 30 mph, the aerodynam drag becomes a significant contribution to the net propulsive force needed The contribution of th c£n: V™t th∫ ph˘c t§pof car 1i.e., front end, windshield, roof, rear end, windshield peak, rea drag due to various portions roof!trunk, and cowl2 have been determined by numerous model and full-sized tests as well as b 0.8 0.6 CD 0.4 V9.20 Automobile streamlining 0.2 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 Year ■ Figure 9.27 The historical trend of streamlining automobiles to reduce their aerodynamic drag and increase their miles per gallon (adapted from Ref 5) 34 / 40 (c) ■ Figure E9.14 Dịng th¸c: Note that the pressure contribution to the lift coefficient Lác nõng: Hê sậ lác nõng is 0.88, whereas that due to the wall shear stress is only 1.96! 1Re1! 2 " 0.001 The Reynolds number dependency of CL is quite minor The lift is pressure dominated Recall from Example d§ng v™tdrag th∫, Ma, Fr , ✏ ` L this is Hình 9.9Cthat also true for the on aRe, similar shape From Eq with A " 20 ft # 50 ft " 1000 ft , we obtain the lift for the assumed conditions as L CL 2AC2L " 12 10.00238 slugs!ft3 2130 ft!s2 11000 ft2 210.8812 l " 12rU ⇢U A ! ~ U2 U There is a ground Clearl The lift force the fluid forc apart Fr : có m∞t thống Ma : Ma ! or 0.8 A typical device designed to produce lift doe ✏: £nh h˜ng lên D nhi∑u hÏn is different onlên theL top and bottom surfaces For larg tributions are usually directly proportional to the d Re: không nhi∑u Hence, as indicated being of secondary importance the lift is proportional to thed§ng square the airspeed CL Hình v™tofth∫ in Fig 9.32 Clearly the symmetrical one cannot prod Because of the asymmetry of the nonsymmetric airfo lower surfaces are different, and a lift is produced 35 / 40 Dịng th¸c: L¸c nâng: Do phân bË áp sußt verImmersedBodies.qxd 532 2/20/12 9:33 PM Page 532 Chapter ■ Flow over Immersed Bodies Denotes p > p0 Denotes p < p0 U, p0 36 / 40 airfoils that have too large an angle of attack 1see Fig 9.182 As is indicated in Fig 9.33,αup tain point, the lift coefficient increases rather steadily with the angle of attack If a is too l boundary layer on the upper surface separates, the flow over the wing develops a wide, t wake region, the lift decreases, and the drag increases This condition, as indicated by the f the margin, termed stall Such conditions are extremely dangerous if they occur while the Lác nõng: Hiên tềngis "stall" is flying at a low altitude where there is not sufficient time and altitude to recover1903 from the W e angles of he boundary parates and g stalls Dịng th¸c: 1.4 1.2 !=7 1.0 Not stalled !=3 0.03 0.8 0.6 !=1 CL 0.4 0.02 0.2 Stalled CD 0.01 –0.2 –0.4 –10 10 α , degrees 20 –10 150 lb "f Ty are indic wing len the wing In in aspec !=1 ! nor than ten term !=7 the wing highly e however or acrob Al 10 α , degrees lift, they (b) the visco (a) Courtesy of NASA ■ Figure 9.33 Typical lift and drag coefficient data as a function of angle of attack and the37aspect / 40 9FlowoverImmersedBodies.qxd Dịng th¸c: 2/22/12 5:40 PM Page 538 L¸c nâng: Mơ tÊ băng l thuyt th lu - Xoỏy tá Énh ỴImmersed nhĨtBodies khơng k∫? 538 h˜ng Chapter ca Flow over Mụ tÊ băng l thuyt th l˜u? α=0 !=0 (a) α>0 !=0 (b) α>0 !>0 (c) + = “(b) + circulation = (c)” (d) ■ Figure 9.36 Inviscid flow past an airfoil: (a) symmetrical flow past the symmetrical airfoil at a zero angle of attack; (b) same airfoil at a nonzero angle of attack—no lift, flow near trailing edge not realistic; (c) same conditions as for (b) except circulation has been added to the flow—nonzero lift, realistic flow; (d) superposition of flows to produce the final flow past the airfoil the flow pattern of Fig 9.36b is changed to that of Fig 9.36c2, and 122 the average velocity on the 38 / 40 s.qxd 2/20/12 9:34 PM Page 539 Dịng th¸c: L¸c nâng: Xốy mÙi cánh - Xốy móng ng¸a 9.4 Lift 539 V9.25 Wing tip vortices U B Bound vortex A Trailing vortex ( a) Bound vortex Low pressure A B High pressure Trailing vortex (b) ■ Figure 9.37 Flow past a finite-length wing: (a) the horseshoe vortex system produced by the bound vortex and the trailing vortices; (b) the leakage of air around the wing tips produces the trailing vortices (Photograph courtesy of NASA.) swirl airfoil and R seem a flow p erage wing t to the stream reason trailin of 25 rately along lift T shoe v vortex erate v bility aircraf gin cl flew t 39 / 40 Dịng th¸c: L¸c nâng: Xốy móng ng¸a: Winglets 40 / 40