KUNDU Fluid Mechanics 2E Episode 16 ppt

Figurn 15.24 Lift and induced drag on a wing element dy. Because the aspect ratio is assumed large, E is small. Each element dy of the finite wing may then be assumed to act as though it is an isolated two-dimensional section set in a stream of uniform velocity Ue, at an angle of attack a,. According to the Kutta-Zhukhovsky lift theorem, a circulation r supcrimposed on the actual resulrant velocity U, generates an elementary mrodynamic force dL, = pUJ dy, which acts normal to U,. This force may be resolved into two components, the conventional lift force dL normal to the direction of flight and a component dDi parallel to thc direction of flight (Figure 15.24). Therefore dL =dL,cos~=pU,rdyc0~~21pUrdy, dDi =dL,sin&=pUCrdysin&21pwrd4’. In general w, r, U,, E, and cyc are all Cunctions of y, so that for the entire wing (15.15) These expressions havc a simple interpretation: Whereas the interaction of U and r generates L, which acts normal to U, the interaction of w and r generales Di, which acts normal to 10. Induced Drag The drag force Di induced by the trailing vortices is called the induced drug, which is zero for an airfoil of infinite span. It arises because a wing of finite span continuously crcatcs trailing vortices and the rate dgeneration of the kinetic energy of thc vortices must equal the ratc of work done against the induced drag, namcly Di U. For this reason the induced drag is also known as the vortex drug. It is analogous to the wuve drug experienced by a ship, which continuously radiates gravity waves during its motion. As we shall see, the induccd drag is the largest part of the total drag experienced by an airfoil. A basic reason why there must be a downward velocity behind the wing is the following: The fluid exerts an upward lift force on the wing, and therefore be wing exerts a downward force on the fluid. The fluid must therefore constantly gain downward momentum as it goes past tbc wing. (See thc photograph of the spinning baqeball (Figure 10.25), which exerts an upward force on thc fluid.) For a given r(y), it is apparent that w(y) can be determined from Eq. (15.13) and Di can then be determined fom Eq. (15.15). However, r(y) itself depends on the distribution of w(y), essentially because the cffective angle of attack is changcd due to w(y). To see how r(y) may be cstirnated, Tist note that the lilt coefficient lor a two-dimensional Zhukhovsky airfoil is nearly CL = 217(a + b). For a finite wing we may assume (15.16) where (a - w/ V) is the effectivc angle of attack, -s(y) is the angle of attack for: zero lift (found from experimental dah such as Figure 15.18), and K is a constant whose value is nearly 6 for most airfoils. (K = 2;r for a Zhukhovsky airfoil.) An expression for the circulation can be obtained by noting that the lift coefficient is related to the circulation as CL L/($pV*c) = r/(;Vc), so that I’ = iVcCL. The assumption Eq. (151.6) is then equivalent to thc assumption that the circulation for a wing of finitc span is ( 15.1 7) For a given U, a, c(y), and #? (y), Eqs. (15.13) and (15.17) define an integral equation for dekrmjning r(y). (An integral equation is one in which the unknown function appears under an integral sign.) The problem can be solved numerically by iterativc techniques. Instead of pursuing this approach, in the next scction we shall assumc that r(y) is givcn. Lancheater versus Prandtl Thcre is some controversy in the literature about who should get more credit for developing modem wing theory. Since Prandtl in 1918 first published the thcory in a mathematical form, textbooks for a long time have called it the “Randtl Lifting Line Theory.” Lanchester was bitter about this, because he felt that his contributions werc not adequatcly recognizcd. The controversy has been discussed by von Karman (1 954, p. 50), who witncssed the dwclopment of the thcory. He givcs a lot ofcrcdit to Lanchester, but falls short of accusing his teacher Prandtl of bcing delibcrately unfair. Here we shall note a few facts that von Man brings up. Lanchester was thc first person to study a wing of finite span. He was also the h-st person to conceive that a wing can be rcplaced by a bound vortex, which bends backward to foim the tip vortices. Last, Lanchestcr was the first to recognize that thc minimum power necessary to fly is that requircd to generate the kinctic energy field of the downwash field. It secms, then, that Lanchester had conceived all of the basic ideas of the wing theory, which he published in 1907 in the form of a book called ”Aerodynamics.” In Tact, a figurc from his book looks very similar to our Figure 15.21. Many ol these ideas werc cxplaincd by Lanchester in his talk at Gijttingen, long before Prandtl published his theory. Prandtl, his graduate student von Karman, and Carl Runge were all present. Runge, well-known €or his numerical integration scheme of ordinary differential equations, served as an interpreter, because ncithcr Lanchcstcr qor Prandtl could speak the other’s language. As von Karman said: “both Prandtl and Runge learned very much from these discussions.” However, Prdndtl did not want to recognize Lanchester for priority of ideas, saying that he conceivcd of thcm before he saw Lanchester’s book. Such controversies cannot bc scttlcd. And grcat mcn havc been involvcd in controversies before. For cxamplc, astTophyskist Stcphcn Hawking (1 988), who occupicd Newton’s chair at Cambridge (after Lighthill), described Newton to be a rather mean man who spent much of his later years in unfair attempts at discrediting Leibniz, in trying to force the Royal astronomer to release some unpublished data that he needed to verify his predictions, and in heated disputes with his lifelong nemesis Robert Hook. ln view of the fact that Lanchester’s book was already in print when Prandtl pub- lished his thcory, and the fact that Lanchcstcr had all the ideas but not a formal mathc- matical thcory, wc havc called it the “Li.liing Line Theory or Prandtl and Lanchester.” I 1. Resulk for Ellipdic C’imulalion Ilistribution Thc induced drag and other properties of a finite wing depend on thc distribution oT T(y). Tfie circulation distribution: however, dcpcnds in a complicated way on the wing planform, angle of attack, and so on. Tt can be shown that, for a given total lift and wing area, the induced drag is a minimum whcn thc circulation distribution is .:lliptic. (See, for e.g., Ashley and Landahl, 1965, Tor a proof.) Here we shall simply assume an elliptic distribution of the form (see Figure 15.22b) and deteminc thc rcsulting expressions for downwash and induced drag. The total lift Torce on a wing is then 7r s/2 L = 1 pWdy = puros. -VI2 4 - 4roy Jr To deteminc thc downwash, we first find thc dcrivative of Eq. (1.5.1 8): - dy S,/=@‘ (15.18) (15.19) Writing y = (y - yi) + y1 in the numerator, wc obtain The first integral has the valuc n/2. The second integral can be reduced to a standard form (listcd in any mathematical handbook) by substituting x = y - y1. On setting limits the second integral turns out to be zero, although the integrand is not an odd function. The downwash at y~ is thereforc (15.20) r0 W(Yl) = -9 2s which shows that, €or an elliptic circulation distribution, the induced velocity at the wing is constant along the span. Using Eqs. (15.18) and (15.20), the induced drag is found as Tn terms of the lift Eq. (15.19), this bccornes which can be written as I (15.21.) where we have defined the coefficients (hcrc: A is the wing planform area) 62 A A = - = aspect ratio Di L (1/2)pU2A' cL E (1/2)pU2A' CD; = Equation (15.21) shows that Cui + 0 in the two-dimensional limit A + m. More important, it shows that thc induced dmg coeflicient increases us rhe square of the liJr coe#cienl. Wc shall see in the following section that the induced drag generally makes the largest contribution to the total drag of an airfoil. Since an elliptic circulation distribution minimizes the induced drag, it is of inter- est to detcrmine the circumstances under which such a circulation can be established. Considcr an element dy of thc wing (Figurc 15.25). The lift on thc element is dL = pUI'dy = C~fpU~~dy, (1 5.22) where cdy is an elemcntary wing area. Now if the circulation distribution is clliptic, then the downwash is independent of y. In addition, if the wing profile is gcomel- rically similar at every point along the span and has thc same geometrical angle of D Figure 15.25 Wing of elliptic planibrm. attack a, ihen the egective angle or atlack and hence thc lift coerficient CL will be indcpendent of y. Equation (1 5.22) shows that the chord length c is then simply pro- portional to r, and so c(y) is also elliptically distributed. Thus, an untwisted wing with clliptic planform, or composed of two semiellipscs (Figure 15.25), will generate an elliptic circulation distribution. However, the same effcct can also be achieved with nonelliptic planrorms if the anglc of attack varies along the span, that is, if the wing is givcn a "twist." 1.2. I@ md !)rug Charackri.stics oJAi~foi1.s Before an aircrart is built its wings are tested in a wind tunnel, and the results are generally given as plots of C,. and CD vs the angle of attack. A typical plot is shown in Figure 15.26. It is seen that, in a range of incidcnce angle from a = -4' to a = 12", the variation of CL with a is approximatcly linear, a typical value of dCL/da being xO.1 per degree. Thc lift reaches a maximum value at an incidence of %IS". If the anglc of attack is increased further, thc steep adversc prcssure gradient on the upper surface of the airfoil causa the flow to separate nearly at thc lcading edge, and a very large wakc is rormed (Figurc 15.27). The lift coefficient drops suddenly, and thc wing is said to s/ull. Beyond thc stalling incidcnce the lift cocfficient levels off again and remains at aO.74.8 for rairly large anglcs of incidencc. The maximum liR coefficient dcpcnds largely on the Reynolds number Re. At lower values ofRe - 105-1 Oh, the flow separatcs before the boundary layerundergocs transition, and a very large wake is formcd. This givcs maximum lift cocfficients t0.9. At largcr Reynolds numbers, say Re > lo7, the boundary layer undergoes transition to turbulent flow before it separatcs. This produces a somewhat smaller wakc, and maximum lift coefficients of =z 1.4 are obtained. The angle of attack at zero lifi, denoted by -b here, is a function of the scclion camber. (For a Zhukhovsky airfoil, b = 2(camber)/chord.) The effect of increasing the airfoil camber is to raisc the entire graph of CL vs a, thus increasing thc maximum values of CL without stalling. A cambcrcd profile dclays stalling csscntially becausc 654 Aed-mumics -lo0 100 Figure 15.26 Ut and drag codkients vs angle of attack. F@re 15.27 Stalling of an airfoil. its leading edge points into the airstream while the rest of the airfoil is inclined to the stream. Rounding the airfoil nose is very helpful, for an airfoil of zero thickness would undergo separation at the leading cdge. Trailing edge flaps act to increase the camber when thcy are deployed. Then the maximum lift coefficient is increased, allowing for lower landing speeds. Various terms are in common usage to describe the different components of the drag. The total drag of a body can be divided into africrion drug due to the tangential stresses on the surface and pressure drag due to the normal stresses. The pressure drag can be furthcr subdivided into an induced drag and afiwm. drag. The induced drag is Lhc “drag due to lift” and results from the work done by the body to supply the kinetic energy of the downwash field as the trailing vortices incrcase in lcngth. The form drag is defined as thc parc of the total pressure drag that remains ah the induced drag is subtracted out. (Sometimes the skin friction and form drags are grouped together and called the projfe drug, which rcpresents the drag due to the “profile” alone and not due to the fmitcness of the wing.) The form drag depends strongly on hc shape and orientation of the airfoil and can be minimized by good design. In contrast, relatively little can be done about the induced drag if the aspect ratio is fixed. Normally thc induced drag constitutes the major part of the total drag of a wing. 4s Coi is ncarly proportional to Ci, and CL is nearly proportional to a, it rollows that Coi oc a2. This is why the drag cwfficient in Figure 15.26 seems to increase quadratically with incidence. For high-spced aircraft, the appearance of shock waves can adversely affect the behavior of thc lift and drag characteristics. In such caqes the maximumJlow speeds can be cbsc to or higher than the speed of sound even when the aircraft is flying at subsonic speeds. Shock waves can form when thc local flow speed exceeds the local specd dsound. To reduce their effect, the wings are given asweepbackangle, as shown in Figure 15.2. The maximum flow speeds depcnd primarily on the component of the oncoming stream perpendicular to the leading edge; this component is rcduced as a result of the sweepback. As a result, increased flight speeds are achievable with highly swept wings. This is particularly true when the aircraft fits at supersonic speeds, in which there is invariably a shock wave in rmnt of the nose of the fuselage, extending downstream in the €om of a cone. Highly swept wings are hen used in ordcr that the wing does not pcnetrate this shock wave. For flight spceds exceeding Mach numbers of order 2, thc wings have such large sweepback angles hat they resernblc the Greek letter A; thcse wings are somctimes called delta wings. 13. Pmpulxive Mechnniumw of’l+ish and Bids The propulsive mechanisms or many animals utilize the aerodynamic principle of lift generation on winglike surfaces. We shall now describe some of the basic ideas of this interesting subject, which is discussed in more detail by Lighthill (1986). Locomotion of Fish First consider the caqe of a fish. It develops a forward thrust by horizontally oscillating its tail fmm side tu side. The tail has a cross section resembling that of a symmctric airfoil (Figure 15.28a). One-half of the oscillation is represented in Figure 15.28bb, which shows the top view of tbe tail. The sequence 1 to 5 represents the positions of the tail during the tail’s motion to the left. A quick change of orientutiun occurs at one extreme position of thc oscillation during 1 to 2; the tail then moves to the Icft during 2 to 4, and another quick change of orientation occurs at the othcr extreme during 4 to 5. Suppose the tail is moving to the left at speed V, and the fish is moving forward at speed U. The fish controls thesc magnitudes so that the resultant fluid velocity U, (relative to the tail) is inclined to the tail surface at a positive “angle of attack.” Thc resulting lift L is perpendicular to U, and has a [orward component L sin 8. (It is casy to verify that there is a similar forward propulsive force when he tail moves from IcIt to right.) This thrust, working at the rate U L sin 8, propels the fish. To achieve this propulsion, the tail of thc Esh pushcs sideways on the water against a force of L cos 8, which rcquires work at the ratc VLcosO. As V/U = tan0, idcally the conversion or energy is perfect-all of thc oscillatory work done by the fish tail goes into the (b) Top view of tail motion Figure 15.28 Propulsion of fish. (a) Cross section of the Pail along AA is a symmetric airfoil. Fivc positions of Ihc tail during its motion 10 the left lirc shown in (b). The lin force I, is normal to the resulkml speed U, of water with respect 10 the tail. translational mode. In practice, however, this is not the case because of the presence of induced drag and other effccts that generate a wake. Most fish stay afloat by controlling the buoyancy of a swim hladdcr inside their stomach. In contrast, some large marinc mammals such as whales and dolphins develop buth a forward thrust and a vertical lift by moving their tails vem'cally. They arc able to do this bccause thcir tail surface is horizonrul, in contrast to thc vertical tail shown in Figure 15.28. night of Birds Now consider the flight of birds, who flap their wings to gencrate horh the lift to support their body weight and the forward thrust to overcome hc drag. Figurc 15.29 shows a vertical section of the wing positions during the upstroke and downstroke of the wing. (Birds have cambered wings, but this is not shown in the figure.) The angle of inclination of the wing with the airstream changes suddenly at the end of each stroke, as shown. Thc important point is that the upstroke is inclincd at a greater angle to the airstream than the downstroke. As the figure shows, thc downstroke dcvelops a lift force L perpendicular to the ~sultant velocity of thc air relative LO the wing. Both a forward thrust and an upward force result from the downstroke. In contrast, very little aerodynamic force is developed during the upstroke, as the resultant vclocity is then nearly parallel to the wing. Birds thcreforc do most of the work during the downstroke, and the upstroke is "easy." 14. LYuiling against he Mnd People have sailed without the aid of an engine €or thousands of years and havc known how to arrive at a destination against the wind. Actually, it is not possiblc L b \ \ V U downstroke Figure 15.29 Propulsion of bird. A cmss 2 2 ection of thc wing is shown during upstroke and downslrokc. During thc downs&ke. a lirt hrcc I. acts nod to thc resultant spccd 0, of air with respcct to ihc wing. During tbc upstroke. Ur is ncarly pwdllel to lhc wing and wry lilllc adynamic romc is generated. to sail cxactly against the wind, but: it is possiblc lo sail at ~4045” to the wind. Figurc 15.30 shows how this is made possible by the aerodynamic lift on the sail, which is a piece of large stretched cloth. The wind speed is U, and the sailing speed is V, SO that the apparent wind speed relative to the boat is U,. II the sail is properly orientcd, this givcs rise to a lift force perpendicular to U, and a drag force parallel to UT. The rcsultant forcc F can be rcsolved into a driving component (thrust) along the motion of the boat and a lateral component. The driving component performs work in moving the boat; most of this work goes into overcoming the frictional drag and in generating the gravity waves that radiate outward. The latcral componcnt does not cause much sideways drift because of the shape of the hull. It is clcar that the thrust decrcases as thc angle 0 dccrea9es and normally vanishes whcn 0 is ~40-45’. The energy for sailing comes from the wind field, which loses kinetic energy aftcr passing througb thc sail. In the foregoing discussion we havc not considered the hydrodynamic forces cxerted by the water on the bull. At constant sailing spccd the net hydrodynamic ibrce must bc equal and opposite to thenei aerodynamic force onthe sail. The hydrodynamic force can be dccornposed inlo a drag (parallel to the dirccrion of motion) and a lift. Thc lift is provided by the “keel,” which is a thin vcrlical surface extending downward from the bottom 01 the hull. For thc keel to act as a lifting surfacc, the longitudinal axis or the boat points at a small angle to thc direction o€ motion or the boat, as indicatcd near thc bottom right part of Figure 15.30. This “angle of attack” 658 Aennly7uamicmr Fiprc 15.30 Principlc ora sailboat. is generally <3" and is not noticeable. The hydrodynamic lift developed by the keel opposes the aerodynamic lateral force on the sail. It is clear that without the keel the latcral aerodynamic force on the sail would topple the boat around its longitudinal axis. To arrive at a destination directly against the wind, one has to sail in a zig-zag path, always maintaining an angle of %45" to the wind. For example, if the wind is corning from the east, we can fist proceed northeastward as shown, then change the orientation of the sail to proceed southeastward, and so on. In practice, a combination of a number of sails is used for effective maneuvering. The mechanics of sailing yachts is discussed in Herreshoff and Newman (1966). l!kCfViSt?# 1. Consider an airfoil section in the xy-plane, the x-axis being aligned with the chordline. Examine the pressure forces on an element ds = (dx, dy) on the surface, and show that the net force (per unit span) in the y-direction is Fy = - lc pu dx + lf'pl dx? where pu and pl are the pressures on the upper and the lower surfaces and c is the chord lenglh. Show that this relation can be rearranged in the form where C, = (p - pm)/($pV2), and the integral represents the ma enclosed in a C, vs x/c diagram, such as Figure 15.8. Neglect shear stresses. [Note that Cy is not [...]... “compression wavc” (for which the fluid pressure rises after the passage of the wave) must movc the fluid in the dircction of propagation, as shown in Figure 16. la In contrast, an “expansion wave” moves the fluid ”backwards.” To make the analysis steady, we superimpose a velocity c, dirccted to h c right, on the entire system (Figure 16. lb) The wave is now stationary, and the fluid enters the wave with velocity... volume flow rate in m3/s.) Equation (16. 13) then becomes e2 + TU? - e l 1 2 - 1 2 =PIVI - ~ 2 + Q ~ 2 (16. 14) It is apparent that plul is the work donc (per unit mass) by the surroundings in pushing fluid into the control volumc Similarly, p21.9 is the work done by the fluid inside thc control volume on the surroundings in pushing fluid out of the control volume Equation (16. 14) therefore has a simple meaning... rcsultant force exerted on the fluid by thc walls The momentum principle (16. 18) is applicable even when there are frictional and dissipative processes (such as shock waves) within the control volume: If frictional processes are absent, then Eq (16. 18) reduces to the Eu1e.r equation (16. 16) To see this, consider an infinitesimal area change between sections l and 2 (Figure 16. 3) Thcn the averagc pressure... ,p o / p , po/p, and A/A* at a point can bc determined f o rm Eqs (16. 20)- (16. 23) if the local Mach number is known For y = 1.4, these ratios arc tabulated in Table 16. 1 The reader should examine this table at this point Examples 16. 1 and 16. 2 given later will illustrate the use of this table 00 0.02 0.04 0.06 0.08 01 0.12 0.14 0 .16 0.18 0.2 0.22 0.24 0.26 0.28 03 0.32 0.34 0.36 0.38 0.4 0.42 0.44... 0.01 28 0.1747 0 I735 0.0 I25 0.01 23 0.1724 0.0121 0.1712 0.01 19 0.1700 0.01 17 0 .168 9 0.01 15 0 .167 8 0.01 13 0 I667 AlA’ 11.1077 I I 3068 1 1.509I 11.7147 1 1.9234 12.1354 12.3508 12.5695 12.79I6 13.0172 13.2463 13.4789 13.715 I 13.9549 14.1984 14.4456 14.6965 14.9513 15.2099 15.4724 15.7388 16. 0092 16. 2837 16. 5622 16. WY 17.1317 17.4228 17.7181 18.0178 18.3218 18.6303 18.9433 19.2608 19.5828 19.9095... equation (16. 16) Tntegrating along a streamline, we obtain the Bernoulli equation for a compressible flow: I-uz 2 +J = const., (16. 17) which agrees with Eq (4.78) For adiabaticfrictionless flows the Bemuulli equation is identical to the energy equation To see this, note that this is an isentropic flow, so that the T dS equation T d S = dh - v d p , gives dh = d p / p Then the Euler equation (16. 16) becomes... incomprcssibility assumption requires that ap u- ax au < p-ax < sp su P U or that - < - < (16. 1) h s s u r e changes can be estimated from the definition of c, giving sp 2 sp (16. 2) SP - 2 c 1 (1 6.3) The Euler equalion requircs usu- P By combining Eqs (16. 2) and (16. 3), we obtain _ -sp p u2su c2u’ From comparison with Eq (16. 1) we see that the density changcs are negligiblc if 112 -=M2 < I c2 < The constant... of x-momentum, and the f second term represents the rate a i d o w of x-momentum Simplifyingthe momentum equation, we obtain d p = pcdu (16. 7) Eliminating du between Eqs (16. 6) and (16. 7), we obtain (16. 8) If thc amplitude of the wave is iniinitesimal, then each fluid particle undergocs a nearly isentropic process as the wave passes by The basic reason for this is that the irreversible entropy production... cxpression for the speed o i propagation of sound Figure 16. l a shows an infinitcsimal pressurc pulse propagating to the l d t with speed c into a still fluid The fluid properties ahead ofthe wave are p, T, and p , while the flow moving wavc P T P / I L I P+@ T+dT P+dP I (a) 4-h u=o I I Figure 16. 1 Propagation ora sound wavc: (a) wavc propagating into still fluid; and (h) stationary wavc + + + speed is u... kt)lc 16. 1: 1xiim)pii:I:lo~of Brft:ct Cas ( y = I 4) 673 5 .4nici-lhloci[y lkl(hw.9 in (~ri(?-~~irri~n~ion(il b'lou 676 Iwrihnpir Exrirriplc 16. 1 679 6 rVorn7d Shock Mire 680 tl Normal Shock I'rnpriptiug iii R S d \ic.diiirri 683 Slm:k Stmctiirc 684 I h k h w m 9 685 c l \ e g nYomlc 685 ol-let ( : o r i \ c ~ ~ n t - l ~ i v : ~ ~ i 685 Soxxlc ~ Exumplc 16. 2 687 lirblc 16. 2: . (16. 1) hssure changes can be estimated from the definition of c, giving (16. 2) 2 sp 21 c sp. The Euler equalion requircs SP usu- (1 6.3) P By combining Eqs. (16. 2) and (16. 3),. the fluid pressure rises after the passage of the wave) must movc the fluid in the dircction of propagation, as shown in Figure 16. la. In contrast, an “expansion wave” moves the fluid. pcdu. (1 6.7) Eliminating du between Eqs. (16. 6) and (16. 7), we obtain (16. 8) If thc amplitude of the wave is iniinitesimal, then each fluid particle undergocs a nearly isentropic