very short, and the balls never reach the speeds in the turbulent range. There-fore, the surfaces of table tennis balls are made smooth.

Once the drag coefficient is available, the drag force acting on a bodyin cross-flow can be determined from Eq. 11–5 where A is the frontal area(A ! L D for a cylinder of length L and A ! pD 2/4 for a sphere). I t shouldbe kept in mind that free-stream turbulence and disturbances by other bod-ies in the flow (such as flow over tube bundles) may affect the drag coeffi-cients significantly.

EXAMPLE 11–4 Drag Force Acting on a Pipe in a River

A 2.2-cm-outer-diameter pipe is to span across a river at a 30-m-wide sec-tion while being completely immersed in water (Fig. 11–38). The averageflow velocity of water is 4 m/s and the water temperature is 15°C. Determinethe drag force exerted on the pipe by the river.

SOLUTION A pipe is submerged in a river. The drag force that acts on thepipe is to be determined.Assumptions 1 The outer surface of the pipe is smooth so that Fig. 11–34can be used to determine the drag coefficient. 2 Water flow in the river issteady. 3 The direction of water flow is normal to the pipe. 4 Turbulence inriver flow is not considered.Properties The density and dynamic viscosity of water at 15°C are r! 999.1 kg/m3 and m ! 1.138 " 10#3 kg/m · s.Analysis Noting that D ! 0.022 m, the Reynolds number is

The drag coefficient corresponding to this value is, from Fig. 11–34, CD !1.0. Also, the frontal area for flow past a cylinder is A ! LD. Then the dragforce acting on the pipe becomes

Discussion Note that this force is equivalent to the weight of a mass over500 kg. Therefore, the drag force the river exerts on the pipe is equivalent tohanging a total of over 500 kg in mass on the pipe supported at its ends30 m apart. The necessary precautions should be taken if the pipe cannotsupport this force. If the river were to flow at a faster speed or if turbulentfluctuations in the river were more significant, the drag force would be evenlarger. Unsteady forces on the pipe might then be significant.

11–7 ■ LIFTL ift was defined earlier as the component of the net force (due to viscousand pressure forces) that is perpendicular to the flow direction, and the liftcoefficient was expressed as

(11–6)C L !F L

12rV

2A

! 5275 N " 5300 N

F D ! C D A rV 2

2! 1.0(30 " 0.022 m2)

(999.1 kg/m3)(4 m/s)2

2a 1 N1 kg $ m/s2b

Re !VDn

!rVDm

!(999.1 kg/m3)(4 m/s)(0.022 m)

1.138 " 10#3 kg/m $ s! 7.73 " 104

587CHAPTER 11

CD

Smooth Rough Surface,Re Surface e/D ! 0.0015

2 " 105 0.5 0.1106 0.1 0.4

FIGURE 11–37Surface roughness may increase or

decrease the drag coefficient of aspherical object, depending on the

value of the Reynolds number.

R iver

30 m

Pipe

FIGURE 11–38Schematic for Example 11–4.

where A in this case is normally the planform area, which is the area thatwould be seen by a person looking at the body from above in a directionnormal to the body, and V is the upstream velocity of the fluid (or, equiva-lently, the velocity of a flying body in a quiescent fluid). For an airfoil ofwidth (or span) b and chord length c (the length between the leading andtrailing edges), the planform area is A # bc. The distance between the twoends of a wing or airfoil is called the wingspan or just the span. For an air-craft, the wingspan is taken to be the total distance between the tips of thetwo wings, which includes the width of the fuselage between the wings(F ig. 11–39). The average lift per unit planform area F L /A is called the wingloading, which is simply the ratio of the weight of the aircraft to the plan-form area of the wings (since lift equals the weight during flying at constantaltitude).

A irplane flight is based on lift, and thus developing a better understandingof lift as well as improving the lift characteristics of bodies have been thefocus of numerous studies. Our emphasis in this section is on devices suchas airfoils that are specifically designed to generate lift while keeping thedrag at a minimum. But it should be kept in mind that some devices such asthe spoilers and inverted airfoils on racing cars are designed for the oppo-site purpose of avoiding lift or even generating negative lift to improve trac-tion and control (some early cars actually “took off” at high speeds as aresult of the lift produced, which alerted the engineers to come up withways to reduce lift in their design).

For devices that are intended to generate lift such as airfoils, the contribu-tion of viscous effects to lift is usually negligible since the bodies arestreamlined, and wall shear is parallel to the surfaces of such devices andthus nearly normal to the direction of lift (F ig. 11–40). Therefore, lift inpractice can be taken to be due entirely to the pressure distribution on thesurfaces of the body, and thus the shape of the body has the primary influ-ence on lift. Then the primary consideration in the design of airfoils is min-imizing the average pressure at the upper surface while maximizing it at thelower surface. The Bernoulli equation can be used as a guide in identifyingthe high- and low-pressure regions: P ressure is low at locations where theflow velocity is high, and pressure is high at locations where the flow veloc-ity is low. A lso, lift is practically independent of the surface roughness sinceroughness affects the wall shear, not the pressure. The contribution of shear

588FLUID MECHANICS

Planformarea, bc

A ngle ofattack

Chord, c

Span, b

FL

FD

a

FIGURE 11–39Definition of various terms associatedwith an airfoil.

FL

Va

FD

Direction ofwall shear

Directionof lift

FIGURE 11–40For airfoils, the contribution ofviscous effects to lift is usuallynegligible since wall shear is parallelto the surfaces and thus nearly normalto the direction of lift.

to lift is usually only significant for very small (lightweight) bodies that canfly at low velocities (and thus very low Reynolds numbers).

Noting that the contribution of viscous effects to lift is negligible, weshould be able to determine the lift acting on an airfoil by simply integrat-ing the pressure distribution around the airfoil. The pressure changes in theflow direction along the surface, but it remains essentially constant throughthe boundary layer in a direction normal to the surface (Chap. 10). There-fore, it seems reasonable to ignore the very thin boundary layer on the air-foil and calculate the pressure distribution around the airfoil from the rela-tively simple potential flow theory (zero vorticity, irrotational flow) forwhich net viscous forces are zero for flow past an airfoil.

The flow fields obtained from such calculations are sketched in F ig.11–41 for both symmetrical and nonsymmetrical airfoils by ignoring thethin boundary layer. A t zero angle of attack, the lift produced by the sym-metrical airfoil is zero, as expected because of symmetry, and the stagnationpoints are at the leading and trailing edges. For the nonsymmetrical airfoil,which is at a small angle of attack, the front stagnation point has moveddown below the leading edge, and the rear stagnation point has moved up tothe upper surface close to the trailing edge. To our surprise, the lift pro-duced is calculated again to be zero— a clear contradiction of experimentalobservations and measurements. Obviously, the theory needs to be modifiedto bring it in line with the observed phenomenon.

The source of inconsistency is the rear stagnation point being at the uppersurface instead of the trailing edge. This requires the lower side fluid tomake a nearly U -turn and flow around the trailing edge toward the stagna-tion point while remaining attached to the surface, which is a physicalimpossibility since the observed phenomenon is the separation of flow atsharp turns (imagine a car attempting to make this turn at high speed).Therefore, the lower side fluid separates smoothly off the trailing edge, andthe upper side fluid responds by pushing the rear stagnation point down-stream. In fact, the stagnation point at the upper surface moves all the wayto the trailing edge. This way the two flow streams from the top and thebottom sides of the airfoil meet at the trailing edge, yielding a smooth flowdownstream parallel to the trailing edge. L ift is generated because the flowvelocity at the top surface is higher, and thus the pressure on that surface islower due to the Bernoulli effect.

589CHAPTER 11

(a) I rrotational flow past a symmetricalairfoil (zero lift)

(b) I rrotational flow past anonsymmetrical airfoil (zero lift)

(c) A ctual flow past anonsymmetrical airfoil (positive lift)

Stagnationpoints

Stagnationpoints

Stagnationpoints

FIGURE 11–41I rrotational and actual flow past symmetrical and nonsymmetrical two-dimensional airfoils.

The potential flow theory and the observed phenomenon can be reconciledas follows: Flow starts out as predicted by theory, with no lift, but the lowerfluid stream separates at the trailing edge when the velocity reaches a certainvalue. This forces the separated upper fluid stream to close in at the trailingedge, initiating clockwise circulation around the airfoil. This clockwise cir-culation increases the velocity of the upper stream while decreasing that ofthe lower stream, causing lift. A starting vortex of opposite sign (counter-clockwise circulation) is then shed downstream (Fig. 11–42), and smoothstreamlined flow is established over the airfoil. W hen the potential flow the-ory is modified by the addition of an appropriate amount of circulation tomove the stagnation point down to the trailing edge, excellent agreement isobtained between theory and experiment for both the flow field and the lift.

I t is desirable for airfoils to generate the most lift while producing theleast drag. Therefore, a measure of performance for airfoils is the lift-to-drag ratio, which is equivalent to the ratio of the lift-to-drag coefficientsC L /C D . This information is provided by either plotting C L versus C D for dif-ferent values of the angle of attack (a lift–drag polar) or by plotting the ratioC L /C D versus the angle of attack. The latter is done for a particular airfoildesign in F ig. 11–43. Note that the C L /C D ratio increases with the angle ofattack until the airfoil stalls, and the value of the lift-to-drag ratio can be ofthe order of 100.

One obvious way to change the lift and drag characteristics of an airfoil isto change the angle of attack. On an airplane, for example, the entire planeis pitched up to increase lift, since the wings are fixed relative to the fuse-lage. A nother approach is to change the shape of the airfoil by the use ofmovable leading edge and trail ing edge flaps, as is commonly done in mod-ern large aircraft (F ig. 11–44). The flaps are used to alter the shape of thewings during takeoff and landing to maximize lift and to enable the aircraftto land or take off at low speeds. The increase in drag during this takeoffand landing is not much of a concern because of the relatively short timeperiods involved. Once at cruising altitude, the flaps are retracted, and thewing is returned to its “normal” shape with minimal drag coefficient andadequate lift coefficient to minimize fuel consumption while cruising at aconstant altitude. Note that even a small lift coefficient can generate a largelift force during normal operation because of the large cruising velocities ofaircraft and the proportionality of lift to the square of flow velocity.

The effects of flaps on the lift and drag coefficients are shown in F ig.11–45 for an airfoil. Note that the maximum lift coefficient increases fromabout 1.5 for the airfoil with no flaps to 3.5 for the double-slotted flap case.But also note that the maximum drag coefficient increases from about 0.06

590FLUID MECHANICS

Clockwisecirculation

Counterclockwisecirculation

Startingvortex

FIGURE 11–42Shortly after a sudden increase inangle of attack, a counterclockwisestarting vortex is shed from the airfoil,while clockwise circulation appearsaround the airfoil, causing lift to be generated.

120

100

80

60

40

20

0

–20

– 40 –4–8 0a degrees

4 8

NA CA 64(1) – 412 airfoilRe = 7 × 105 Stall

–––C L

C D

FIGURE 11–43The variation of the lift-to-drag ratiowith angle of attack for a two-dimensional airfoil.F rom Abbott, von D oenhoff, and Stivers (1945).

FIGURE 11–44The lift and drag characteristics of anairfoil during takeoff and landing canbe changed by changing the shape ofthe airfoil by the use of movable flaps.P hoto by Yunus Ç engel. (a) F laps extended (takeoff) (b) F laps retracted (cruising)

for the airfoil with no flaps to about 0.3 for the double-slotted flap case.This is a fivefold increase in the drag coefficient, and the engines must workmuch harder to provide the necessary thrust to overcome this drag. Theangle of attack of the flaps can be increased to maximize the lift coefficient.A lso, the leading and trailing edges extend the chord length, and thusenlarge the wing area A. The Boeing 727 uses a triple-slotted flap at thetrailing edge and a slot at the leading edge.

The minimum flight velocity can be determined from the requirement thatthe total weight W of the aircraft be equal to lift and C L # C L , max. That is,

(11–24)

For a given weight, the landing or takeoff speed can be minimized by maxi-mizing the product of the lift coefficient and the wing area, C L , max A . Oneway of doing that is to use flaps, as already discussed. A nother way is tocontrol the boundary layer, which can be accomplished simply by leavingflow sections (slots) between the flaps, as shown in F ig. 11–46. Slots areused to prevent the separation of the boundary layer from the upper surfaceof the wings and the flaps. This is done by allowing air to move from thehigh-pressure region under the wing into the low-pressure region at the topsurface. Note that the lift coefficient reaches its maximum value C L # C L , max,and thus the flight velocity reaches its minimum, at stall conditions, whichis a region of unstable operation and must be avoided. The Federal AviationA dministration (FA A ) does not allow operation below 1.2 times the stallspeed for safety.

A nother thing we notice from this equation is that the minimum velocityfor takeoff or landing is inversely proportional to the square root of density.Noting that air density decreases with altitude (by about 15 percent at 1500m), longer runways are required at airports at higher altitudes such as Den-ver to accommodate higher minimum takeoff and landing velocities. Thesituation becomes even more critical on hot summer days since the densityof air is inversely proportional to temperature.

End Effects of Wing TipsFor airplane wings and other airfoils of finite size, the end effects at the tipsbecome important because of the fluid leakage between the lower and upper

W # F L # 12 C L , max rV

2min A → Vmin #B 2W

rC L , max A

591CHAPTER 11

0 5–5 2010

Double-slottedflap

Slotted flap

C lean (no flap)

A ngle of attack, (deg.)

3.48

2.67

1.52

C L max

C L

C Da

15 0 0.150.05 0.300.250.10 0.20

3.5

3.0

2.5

2.0

1.5

1.0

0.5

C L

3.5

3.0

2.5

2.0

1.5

1.0

0.5C lean (no flap)

Slotted flap

Double-slottedflap

FIGURE 11–45Effect of flaps on the lift and drag

coefficients of an airfoil.F rom Abbott and von D oenhoff, for NAC A 23012

(1959).

Wing

Slot

F lap

FIGURE 11–46A flapped airfoil with a slot to prevent

the separation of the boundary layerfrom the upper surface and to increase

the lift coefficient.

surfaces. The pressure difference between the lower surface (high-pressureregion) and the upper surface (low-pressure region) drives the fluid at thetips upward while the fluid is swept toward the back because of the relativemotion between the fluid and the wing. This results in a swirling motionthat spirals along the flow, called the tip vortex, at the tips of both wings.Vortices are also formed along the airfoil between the tips of the wings.T hese distributed vortices collect toward the edges after being shed fromthe trailing edges of the wings and combine with the tip vortices to form two streaks of powerful trailing vortices along the tips of the wings(F igs. 11–47 and 11–48). Trailing vortices generated by large aircraft con-tinue to exist for a long time for long distances (over 10 km) before theygradually disappear due to viscous dissipation. Such vortices and theaccompanying downdraft are strong enough to cause a small aircraft to losecontrol and flip over if it flies through the wake of a larger aircraft. There-fore, following a large aircraft closely (within 10 km) poses a real dangerfor smaller aircraft. This issue is the controlling factor that governs thespacing of aircraft at takeoff, which limits the flight capacity at airports. Innature, this effect is used to advantage by birds that migrate in V-formationby utilizing the updraft generated by the bird in front. I t has been determinedthat the birds in a typical flock can fly to their destination in V-formationwith one-third less energy. M ilitary jets also occasionally fly in V-formationfor the same reason.

Tip vortices that interact with the free stream impose forces on the wingtips in all directions, including the flow direction. The component of theforce in the flow direction adds to drag and is called induced drag. Thetotal drag of a wing is then the sum of the induced drag (3-D effects) andthe drag of the airfoil section.

The ratio of the square of the average span of an airfoil to the planformarea is called the aspect ratio. For an airfoil with a rectangular planform ofchord c and span b, it is expressed as

(11–25)

Therefore, the aspect ratio is a measure of how narrow an airfoil is in theflow direction. The lift coefficient of wings, in general, increases while the

AR #b2

A#

b2

bc#

bc

592FLUID MECHANICS

FIGURE 11–48A crop duster flies through smoky airto illustrate the tip vortices producedat the tips of the wing.NASA L angley Research C enter.

FIGURE 11–47Trailing vortices from a rectangularwing with vortex cores leaving thetrailing edge at the tips.C ourtesy of The Parabolic P ress, Stanford,C alifornia. U sed with permission.

drag coefficient decreases with increasing aspect ratio. This is because along narrow wing (large aspect ratio) has a shorter tip length and thussmaller tip losses and smaller induced drag than a short and wide wing ofthe same planform area. Therefore, bodies with large aspect ratios fly moreefficiently, but they are less maneuverable because of their larger moment ofinertia (owing to the greater distance from the center). Bodies with smalleraspect ratios maneuver better since the wings are closer to the central part.So it is no surprise that fighter planes (and fighter birds like falcons) haveshort and wide wings while large commercial planes (and soaring birds likealbatrosses) have long and narrow wings.

The end effects can be minimized by attaching endplates or winglets atthe tips of the wings perpendicular to the top surface. The endplates func-tion by blocking some of the leakage around the wing tips, which results ina considerable reduction in the strength of the tip vortices and the induceddrag. Wing tip feathers on birds fan out for the same purpose (F ig. 11–49).

The development of efficient (low-drag) airfoils was the subject of intenseexperimental investigations in the 1930s. These airfoils were standardizedby the National A dvisory Committee for A eronautics (NA CA , which is nowNA SA ), and extensive lists of data on lift coefficients were reported. Thevariation of the lift coefficient C L with the angle of attack for two airfoils(NA CA 0012 and NA CA 2412) is given in F ig. 11–50. We make the follow-ing observations from this figure:

• The lift coefficient increases almost linearly with the angle of attack a,reaches a maximum at about a # 16°, and then starts to decrease sharply.This decrease of lift with further increase in the angle of attack is calledstall, and it is caused by flow separation and the formation of a wide wakeregion over the top surface of the airfoil. Stall is highly undesirable sinceit also increases drag.

• A t zero angle of attack (a # 0°), the lift coefficient is zero forsymmetrical airfoils but nonzero for nonsymmetrical ones with greater

593CHAPTER 11

FIGURE 11–49Induced drag is reduced by

(a) wing tip feathers on bird wings and (b) endplates or other

disruptions on airplane wings.(a) © Vol. 44/P hotoD isc. (b) C ourtesy Schempp-H irth. U sed by permission.

(b) Winglets are used on this sailplane to reduceinduced drag.

(a) A bearded vulture with its wing feathersfanned out during flight.

–5 0 5

A ngle of attack, , degrees

10 15 20

2.00

1.50

1.00

0.50

0

–0.50

NA CA 2412 sectionC L

Va

Va

a

NA CA 0012 section

FIGURE 11–50The variation of the lift coefficient

with the angle of attack for asymmetrical and a nonsymmetrical

airfoil.F rom Abbott (1932).

curvature at the top surface. Therefore, planes with symmetrical wingsections must fly with their wings at higher angles of attack in order toproduce the same lift.

• The lift coefficient can be increased by severalfold by adjusting the angleof attack (from 0.25 at a# 0° for the nonsymmetrical airfoil to 1.25 at a # 10°).

• The drag coefficient also increases with the angle of attack, oftenexponentially (F ig. 11–51). Therefore, large angles of attack should beused sparingly for short periods of time for fuel efficiency.

Lift Generated by SpinningYou have probably experienced giving a spin to a tennis ball or making adrop shot on a tennis or ping-pong ball by giving a fore spin in order toalter the lift characteristics and cause the ball to produce a more desirabletrajectory and bounce of the shot. Golf, soccer, and baseball players alsoutilize spin in their games. The phenomenon of producing lift by the rotationof a solid body is called the M agnus effect after the German scientist Hein-rich M agnus (1802–1870), who was the first to study the lift of rotatingbodies, which is illustrated in F ig. 11–52 for the simplified case of irrota-tional (potential) flow. W hen the ball is not spinning, the lift is zero becauseof top–bottom symmetry. But when the cylinder is rotated about its axis, thecylinder drags some fluid around because of the no-slip condition and theflow field reflects the superposition of the spinning and nonspinning flows.The stagnation points shift down, and the flow is no longer symmetric aboutthe horizontal plane that passes through the center of the cylinder. The aver-age pressure on the upper half is less than the average pressure at the lowerhalf because of the Bernoulli effect, and thus there is a net upward force(lift) acting on the cylinder. A similar argument can be given for the lift gen-erated on a spinning ball.

The effect of the rate of rotation on the lift and drag coefficients of asmooth sphere is shown in Fig. 11–53. Note that the lift coefficient stronglydepends on the rate of rotation, especially at low angular velocities. The effectof the rate of rotation on the drag coefficient is small. Roughness also affects

594FLUID MECHANICS

0.020

0.016

0.012

0.008

0.004

0

C D

NA CA 23015section

0 4

Va

8 12A ngle of attack, (degrees)

16 20a

FIGURE 11–51The variation of the drag coefficient ofan airfoil with the angle of attack.F rom Abbott and von D oenhoff (1959).

Stagnationpoints

Stagnationpoints

H igh velocity,low pressure

L ift

L ow velocity,high pressure

(b) Potential flow over a rotating cylinder(a) Potential flow over a stationary cylinder

FIGURE 11–52Generation of lift on a rotating circularcylinder for the case of “idealized”potential flow (the actual flowinvolves flow separation in the wake region).

the drag and lift coefficients. In a certain range of the Reynolds number,roughness produces the desirable effect of increasing the lift coefficient whiledecreasing the drag coefficient. Therefore, golf balls with the right amount ofroughness travel higher and farther than smooth balls for the same hit.

EXAMPLE 11–5 Lift and Drag of a Commercial Airplane

A commercial airplane has a total mass of 70,000 kg and a wing planformarea of 150 m2 (Fig. 11–54). The plane has a cruising speed of 558 km/hand a cruising altitude of 12,000 m, where the air density is 0.312 kg/m3.The plane has double-slotted flaps for use during takeoff and landing, but itcruises with all flaps retracted. Assuming the lift and the drag characteristicsof the wings can be approximated by NACA 23012 (Fig. 11–45), determine(a) the minimum safe speed for takeoff and landing with and without extend-ing the flaps, (b) the angle of attack to cruise steadily at the cruising alti-tude, and (c) the power that needs to be supplied to provide enough thrustto overcome wing drag.

SOLUTION The cruising conditions of a passenger plane and its wing char-acteristics are given. The minimum safe landing and takeoff speeds, theangle of attack during cruising, and the power required are to be determined.Assumptions 1 The drag and lift produced by parts of the plane other thanthe wings, such as the fuselage drag, are not considered. 2 The wings areassumed to be two-dimensional airfoil sections, and the tip effects of thewings are not considered. 3 The lift and the drag characteristics of the wingscan be approximated by NACA 23012 so that Fig. 11–45 is applicable. 4The average density of air on the ground is 1.20 kg/m3.Properties The densities of air are 1.20 kg/m3 on the ground and0.312 kg/m3 at cruising altitude. The maximum lift coefficients CL, max of thewings are 3.48 and 1.52 with and without flaps, respectively (Fig. 11–45).Analysis (a) The weight and cruising speed of the airplane are

The minimum velocities corresponding to the stall conditions without andwith flaps, respectively, are obtained from Eq. 11–24,

Then the “safe” minimum velocities to avoid the stall region are obtained bymultiplying the values above by 1.2:

Without flaps:

With flaps: Vmin 2, safe # 1.2Vmin 2 # 1.2(46.8 m/s) # 56.2 m/s # 202 km/h

Vmin 1, safe # 1.2Vmin 1 # 1.2(70.9 m/s) # 85.1 m/s # 306 km/h

V min 2 #B 2WrC L , max 2A

#B 2(686,700 N )

(1.2 kg/m3)(3.48)(150 m2) a1 kg ' m/s2

1 Nb # 46.8 m/s

Vmin 1 #B 2WrC L , max 1A

#B 2(686,700 N )

(1.2 kg/m3)(1.52)(150 m2) a1 kg ' m/s2

1 Nb # 70.9 m/s

V # (558 km/h)a 1 m/s3.6 km/h

b # 155 m/s

W # mg # (70,000 kg)(9.81 m/s2)a 1 N1 kg ' m/s2b # 686,700 N

595CHAPTER 11

0.8

0.6

0.4

0.2

00 1 2 3 4 5

D

Smooth sphere

v

VDRe = = 6 × 104

CD

, CL

V

F D

D 2C D = 12

p4

rV 2

F L

D 2C L = 12

p4

rV 2

V12vD /

n

FIGURE 11–53The variation of lift and drag

coefficients of a smooth sphere withthe nondimensional rate of rotation

for Re # VD /n# 6 & 104.F rom G oldstein (1938).

558 km/h

70,000 kg

12,000 m

150 m2, double-flapped

FIGURE 11–54Schematic for Example 11–5.

since 1 m/s # 3.6 km/h. Note that the use of flaps allows the plane to takeoff and land at considerably lower velocities, and thus on a shorter runway.

(b) When an aircraft is cruising steadily at a constant altitude, the lift mustbe equal to the weight of the aircraft, FL # W. Then the lift coefficient isdetermined to be

For the case with no flaps, the angle of attack corresponding to this value ofCL is determined from Fig. 11–45 to be a " 10°.

(c) When the aircraft is cruising steadily at a constant altitude, the net forceacting on the aircraft is zero, and thus thrust provided by the engines mustbe equal to the drag force. The drag coefficient corresponding to the cruisinglift coefficient of 1.22 is determined from Fig. 11–45 to be CD " 0.03 forthe case with no flaps. Then the drag force acting on the wings becomes

Noting that power is force times velocity (distance per unit time), the powerrequired to overcome this drag is equal to the thrust times the cruisingvelocity:

Therefore, the engines must supply 2620 kW of power to overcome the dragon the wings during cruising. For a propulsion efficiency of 30 percent (i.e.,30 percent of the energy of the fuel is utilized to propel the aircraft), theplane requires energy input at a rate of 8733 kJ/s.Discussion The power determined is the power to overcome the drag that actson the wings only and does not include the drag that acts on the remainingparts of the aircraft (the fuselage, the tail, etc.). Therefore, the total powerrequired during cruising will be much greater. Also, it does not considerinduced drag, which can be dominant during takeoff when the angle of attackis high (Fig. 11–45 is for a 2-D airfoil, and does not include 3-D effects).

EXAMPLE 11–6 Effect of Spin on a Tennis Ball

A tennis ball with a mass of 0.125 lbm and a diameter of 2.52 in is hit at45 mi/h with a backspin of 4800 rpm (Fig. 11–55). Determine if the ballwill fall or rise under the combined effect of gravity and lift due to spinningshortly after being hit in air at 1 atm and 80°F.

SOLUTION A tennis ball is hit with a backspin. It is to be determinedwhether the ball will fall or rise after being hit.Assumptions 1 The surfaces of the ball are smooth enough for Fig. 11–53to be applicable. 2 The ball is hit horizontally so that it starts its motion hor-izontally.

# 2620 kW

Power # Thrust & V elocity # F D V # (16.9 kN )(155 m/s)a 1 kW1 kN ' m/s

b # 16.9 kN

F D # C D A rV 2

2# (0.03)(150 m2)

(0.312 kg/m3)(155 m/s)2

2a 1 kN1000 kg ' m/s2b

C L #F L

12 r

V 2A

#686,700 N

12 (0.312 kg/m3)(155 m/s)2(150 m2)

a1 kg ' m/s2

1 Nb # 1.22

596FLUID MECHANICS

45 mi/h

4800 rpm

Ballm = 0.125 lbm

FIGURE 11–55Schematic for Example 11–6.

Properties The density and kinematic viscosity of air at 1 atm and 80°F arer # 0.07350 lbm/ft3 and n # 1.697 & 10$4 ft2/s.Analysis The ball is hit horizontally, and thus it would normally fall underthe effect of gravity without the spin. The backspin generates a lift, and theball will rise if the lift is greater than the weight of the ball. The lift can bedetermined from

where A is the frontal area of the ball, which is A # pD 2/4. The translationaland angular velocities of the ball are

Then,

From Fig. 11–53, the lift coefficient corresponding to this value is CL# 0.21. Then the lift force acting on the ball is

The weight of the ball is

which is more than the lift. Therefore, the ball will drop under the combinedeffect of gravity and lift due to spinning with a net force of 0.125 $ 0.036# 0.089 lbf.Discussion This example shows that the ball can be hit much farther by giv-ing it a backspin. Note that a topspin has the opposite effect (negative lift)and speeds up the drop of the ball to the ground. Also, the Reynolds numberfor this problem is 8 & 104, which is sufficiently close to the 6 & 104 forwhich Fig. 11–53 is prepared.

Also keep in mind that although some spin may increase the distance trav-eled by a ball, there is an optimal spin that is a function of launch angle, asmost golfers are now more aware. Too much spin decreases distance byintroducing more induced drag.

No discussion on lift and drag would be complete without mentioning thecontributions of Wilbur (1867–1912) and Orville (1871–1948) W right. TheW right B rothers are truly the most impressive engineering team of all time.Self-taught, they were well informed of the contemporary theory and prac-tice in aeronautics. They both corresponded with other leaders in the fieldand published in technical journals. W hile they cannot be credited withdeveloping the concepts of lift and drag, they used them to achieve the first

W # mg # (0.125 lbm)(32.2 ft/s2)a 1 lbf32.2 lbm ' ft/s2b # 0.125 lbf

# 0.036 lbf

F L # (0.21)p(2.52/12 ft)2

4 (0.0735 lbm/ft3)(66 ft/s)2

2a 1 lbf32.2 lbm ' ft/s2b

vD2V

#(502 rad/s)(2.52/12 ft)

2(66 ft/s)# 0.80 rad

v # (4800 rev/min)a2p rad1 rev

b a1 min60 sb # 502 rad/s

V # (45 mi/h)a5280 ft1 mi

b a 1 h3600 s

b # 66 ft/s

F L # C L A

rV 2

2

597CHAPTER 11

powered, manned, heavier-than-air, controlled flight (F ig. 11–56). They suc-ceeded, while so many before them failed, because they evaluated anddesigned parts separately. Before the W rights, experimenters were buildingand testing whole airplanes. W hile intuitively appealing, the approach didnot allow the determination of how to make the craft better. W hen a flightlasts only a moment, you can only guess at the weakness in the design.Thus, a new craft did not necessarily perform any better than its predeces-sor. Testing was simply one belly flop followed by another. The W rightschanged all that. They studied each part using scale and full-size models inwind tunnels and the field. Well before the first powered flyer was assem-bled, they knew the area required for their best wing shape to support aplane carrying a man and the engine horsepower required to provide an adequate thrust with their improved impeller. The W right B rothers not only showed the world how to fly, they showed engineers how to use theequations presented here to design even better aircraft.

598FLUID MECHANICS

FIGURE 11–56The W right B rothers take flight atK itty Hawk.National A ir and Space M useum/SmithsonianInstitution.

SUMMARY

In this chapter, we study flow of fluids over immersed bodieswith emphasis on the resulting lift and drag forces. A fluidmay exert forces and moments on a body in and about vari-ous directions. The force a flowing fluid exerts on a body inthe flow direction is called drag while that in the directionnormal to the flow is called l i ft. The part of drag that is duedirectly to wall shear stress tw is called the skin fr iction dragsince it is caused by frictional effects, and the part that is duedirectly to pressure P is called the pressure drag or form dragbecause of its strong dependence on the form or shape of thebody.

The drag coefficient C D and the l i ft coefficient C L aredimensionless numbers that represent the drag and the liftcharacteristics of a body and are defined as

where A is usually the frontal area (the area projected on aplane normal to the direction of flow) of the body. For platesand airfoils, A is taken to be the planform area, which is thearea that would be seen by a person looking at the body fromdirectly above. The drag coefficient, in general, depends onthe Reynolds number, especially for Reynolds numbers below104. A t higher Reynolds numbers, the drag coefficients formost geometries remain essentially constant.

A body is said to be streamlined if a conscious effort ismade to align its shape with the anticipated streamlines in theflow in order to reduce drag. Otherwise, a body (such as abuilding) tends to block the flow and is said to be blunt orbluff. A t sufficiently high velocities, the fluid stream detachesitself from the surface of the body. This is called flow separa-tion. W hen a fluid stream separates from the body, it forms aseparated region between the body and the fluid stream. Sep-

C D #F D

12rV

2A and C L #

F L12rV

2A

aration may also occur on a streamlined body such as an air-plane wing at a sufficiently large angle of attack, which is theangle the incoming fluid stream makes with the chord (theline that connects the nose and the end) of the body. F lowseparation on the top surface of a wing reduces lift drasticallyand may cause the airplane to stall.

The region of flow above a surface in which the effects ofthe viscous shearing forces caused by fluid viscosity are felt iscalled the velocity boundary layer or just the boundary layer.The thickness of the boundary layer, d, is defined as the dis-tance from the surface at which the velocity is 0.99V . Thehypothetical line of velocity 0.99V divides the flow over aplate into two regions: the boundary layer region, in which theviscous effects and the velocity changes are significant, and theirrotational outer flow region, in which the frictional effectsare negligible and the velocity remains essentially constant.

For external flow, the Reynolds number is expressed as

where V is the upstream velocity and L is the characteristiclength of the geometry, which is the length of the plate in theflow direction for a flat plate and the diameter D for a cylin-der or sphere. The average friction coefficients over an entireflat plate are

L aminar flow:

Turbulent flow:

I f the flow is approximated as laminar up to the engineeringcritical number of Recr # 5 & 105, and then turbulent

C f #0.074Re1/5

L 5 & 105 ) ReL ) 107

C f #1.33Re1/2

L ReL ( 5 & 105

ReL #rVLm

#VLn

CHAPTER 11599

beyond, the average friction coefficient over the entire flatplate becomes

A curve fit of experimental data for the average friction coef-ficient in the fully rough turbulent regime is

Rough surface:

where e is the surface roughness and L is the length of theplate in the flow direction. In the absence of a better one, thisrelation can be used for turbulent flow on rough surfaces forRe * 106, especially when e/L * 10$4.

Surface roughness, in general, increases the drag coeffi-cient in turbulent flow. For blunt bodies such as a circularcylinder or sphere, however, an increase in the surface rough-ness may decrease the drag coefficient. This is done by trip-ping the flow into turbulence at a lower Reynolds number,and thus causing the fluid to close in behind the body, nar-rowing the wake and reducing pressure drag considerably.

I t is desirable for airfoils to generate the most lift whileproducing the least drag. Therefore, a measure of perfor-mance for airfoils is the l i ft-to-drag ratio, C L /C D .

C f # a1.89 $ 1.62 log e

Lb$2.5

C f #0.074Re1/5

L$

1742ReL

5 & 105 ) ReL ) 107

The minimum safe flight velocity of an aircraft can bedetermined from

For a given weight, the landing or takeoff speed can be mini-mized by maximizing the product of the lift coefficient andthe wing area, C L , max A .

For airplane wings and other airfoils of finite size, thepressure difference between the lower and the upper surfacesdrives the fluid at the tips upward. This results in a swirlingeddy, called the tip vortex. Tip vortices that interact with thefree stream impose forces on the wing tips in all directions,including the flow direction. The component of the force inthe flow direction adds to drag and is called induced drag.The total drag of a wing is then the sum of the induced drag(3-D effects) and the drag of the airfoil section.

I t is observed that lift develops when a cylinder or spherein flow is rotated at a sufficiently high rate. The phenomenonof producing lift by the rotation of a solid body is called theM agnus effect.

Some external flows, complete with flow details includingplots of velocity fields, are solved using computational fluiddynamics, and presented in Chap. 15.

V min #B 2WrC L , max A

599CHAPTER 11

REFERENCES AND SUGGESTED READING

1. I . H . A bbott. “The Drag of Two Streamline Bodies asA ffected by Protuberances and A ppendages,” NAC AReport 451, 1932.

2. I . H . A bbott and A . E . von Doenhoff. Theory of WingSections, I ncluding a Summary of A ir foil D ata. New York:Dover, 1959.

3. I . H . A bbott, A . E . von Doenhoff, and L . S. Stivers.“Summary of A irfoil Data,” NAC A Report 824, L angleyField, VA , 1945.

4. J . D . A nderson. F undamentals of Aerodynamics, 2nd ed.New York: M cGraw-H ill, 1991.

5. R. D. B levins. Applied F luid D ynamics H andbook. NewYork: Van Nostrand Reinhold, 1984.

6. S. W. Churchill and M . Bernstein. “A CorrelatingEquation for Forced Convection from Gases and L iquidsto a C ircular Cylinder in Cross F low,” J ournal of H eatTransfer 99, pp. 300–306, 1977.

7. C. T. Crowe, J . A . Roberson, and D. F. E lger. E ngineeringF luid M echanics, 7th ed. New York: Wiley, 2001.

8. S. Goldstein. M odern D evelopments in F luid D ynamics.L ondon: Oxford Press, 1938.

9. J . Happel. L ow Reynolds Number H ydrocarbons.Englewood C liffs, N J : Prentice Hall, 1965.

10. S. F. Hoerner. F luid-D ynamic D rag. [Published by theauthor.] L ibrary of Congress No. 64, 1966.

11. G. M . Homsy, H . A ref, K . S. B reuer, S. Hochgreb, J . R .K oseff, B . R . M unson, K . G. Powell, C . R . Robertson,S. T. Thoroddsen. M ulti-M edia F luid M echanics (CD).Cambridge U niversity Press, 2000.

12. W. H . Hucho. Aerodynamics of Road Vehicles. L ondon:Butterworth-Heinemann, 1987.

13. B. R . M unson, D. F. Young, and T. Okiishi. F undamentalsof F luid M echanics, 4th ed. New York: Wiley, 2002.

14. M . C . Potter and D. C . Wiggert. M echanics of F luids, 2nded. U pper Saddle R iver, N J : Prentice Hall, 1997.

15. C. T. Crowe, J . A . Roberson, and D. F. E lger. E ngineeringF luid M echanics, 7th ed. New York: Wiley, 2001.

16. H. Schlichting. Boundary L ayer Theory, 7th ed. NewYork: M cGraw-H ill, 1979.

17. M . Van Dyke. An A lbum of F luid M otion. Stanford, CA :The Parabolic Press, 1982.

18. J . Vogel. L ife in M oving F luids, 2nd ed. Boston: WillardGrand Press, 1994.

19. F. M . W hite. F luid M echanics, 5th ed. New York:M cGraw-H ill, 2003.