Cohen I.M., Kundu P.K. Fluid Mechanics

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702 Aerodynamics

The total lift force on a wing is then

L =



s/2

−s/2

ρU dy =

ρU

s. (15.19)

To determine the downwash, we ﬁrst ﬁnd the derivative of equation (15.18):

d

=−

4



− 4 y

From equation (15.13), the downwash at y

w(y

) =

4π



s/2

−s/2

d

− y



πs



s/2

−s/2

ydy

(y − y

)



− 4 y

Writing y = (y − y

) + y

in the numerator, we obtain

w(y

) =



πs





s/2

−s/2



− 4 y

+ y



s/2

−s/2

(y − y

)



− 4 y



The ﬁrst integral has the value π/2. The second integral can be reduced to a standard

form (listed in any mathematical handbook) by substituting x = y − y

. On setting

limits the second integral turns out to be zero, although the integrand is not an odd

function. The downwash at y

is therefore

w(y

) =



, (15.20)

which shows that, for an elliptic circulation distribution, the induced velocity at the

wing is constant along the span.

Using equations (15.18) and (15.20), the induced drag is found as



s/2

−s/2

ρw dy =

ρ

In terms of the lift equation (15.19), this becomes

ρU

πs

which can be written as

π

(15.21)

11. Results for Elliptic Circulation Distribution 703

where we have deﬁned the coefﬁcients (here A is the wing planform area)

 ≡

= aspect ratio

≡

(1/2)ρU

≡

(1/2)ρU

Equation (15.21) shows that C

→ 0 in the two-dimensional limit  →∞. More

important, it shows that the induced drag coefﬁcient increases as the square of the

lift coefﬁcient. We 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 determine the circumstances under which such a circulation can be established.

Consider an element dy of the wing (Figure 15.25). The lift on the element is

dL = ρU dy = C

ρU

cdy, (15.22)

where cdy is an elementary wing area. Now if the circulation distribution is elliptic,

then the downwash is independent of y. In addition, if the wing proﬁle is geomet-

rically similar at every point along the span and has the same geometrical angle of

attack α, then the effective angle of attack and hence the lift coefﬁcient C

will be

independent of y. Equation (15.22) shows that the chord length c is then simply pro-

portional to , and so c(y) is also elliptically distributed. Thus, an untwisted wing

with elliptic planform, or composed of two semiellipses (Figure 15.25), will generate

an elliptic circulation distribution. However, the same effect can also be achieved with

nonelliptic planforms if the angle of attack varies along the span, that is, if the wing

is given a “twist.”

Figure 15.25 Wing of elliptic planform.

704 Aerodynamics

12. Lift and Drag Characteristics of Airfoils

Before an aircraft is built its wings are tested in a wind tunnel, and the results are

generally given as plots of C

and C

vs the angle of attack. A typical plot is shown in

Figure 15.26. It is seen that, in a range of incidence angle from α =−4

◦

to α = 12

◦

the variation of C

with α is approximately linear, a typical value of dC

/dα being

≈0.1 per degree. The lift reaches a maximum value at an incidence of ≈15

◦

. If the

angle of attack is increased further, the steep adverse pressure gradient on the upper

surface of the airfoil causes the ﬂow to separate nearly at the leading edge, and a very

large wake is formed (Figure 15.27). The lift coefﬁcient drops suddenly, and the wing

is said to stall. Beyond the stalling incidence the lift coefﬁcient levels off again and

remains at ≈0.7–0.8 for fairly large angles of incidence.

The maximum lift coefﬁcient depends largely on the Reynolds number Re. At

lower values of Re ∼ 10

–10

, the ﬂow separates before the boundary layer undergoes

Figure 15.26 Lift and drag coefﬁcients vs angle of attack.

Figure 15.27 Stalling of an airfoil.

12. Lift and Drag Characteristics of Airfoils 705

transition, and a very large wake is formed. This gives maximum lift coefﬁcients <0.9.

At larger Reynolds numbers, say Re > 10

, the boundary layer undergoes transition

to turbulent ﬂow before it separates. This produces a somewhat smaller wake, and

maximum lift coefﬁcients of ≈1.4 are obtained.

The angle of attack at zero lift, denoted by −β here, is a function of the section

camber. (For a Zhukhovsky airfoil, β = 2(camber)/chord.) The effect of increasing

the airfoil camber is to raise the entire graph of C

vs α, thus increasing the maximum

values of C

without stalling. A cambered proﬁle delays stalling essentially because

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 edge. Trailing edge ﬂaps act to increase the camber

when they 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 a friction drag due to the tangential

stresses on the surface and pressure drag due to the normal stresses. The pressure

drag can be further subdivided into an induced drag and a form drag. The induced

drag is the “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 increase in length.

The form drag is deﬁned as the part of the total pressure drag that remains after

the induced drag is subtracted out. (Sometimes the skin friction and form drags are

grouped together and called the proﬁle drag, which represents the drag due to the

“proﬁle” alone and not due to the ﬁniteness of the wing.) The form drag depends

strongly on the 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 the induced drag constitutes the major part of the total drag of a wing.

As C

is nearly proportional to C

, and C

is nearly proportional to α, it follows

that C

∝ α

. This is why the drag coefﬁcient in Figure 15.26 seems to increase

quadratically with incidence.

For high-speed aircraft, the appearance of shock waves can adversely affect the

behavior of the lift and drag characteristics. In such cases the maximum ﬂow speeds

can be close to or higher than the speed of sound even when the aircraft is ﬂy-

ing at subsonic speeds. Shock waves can form when the local ﬂow speed exceeds

the local speed of sound. To reduce their effect, the wings are given a sweepback

angle, as shown in Figure 15.2. The maximum ﬂow speeds depend primarily on the

component of the oncoming stream perpendicular to the leading edge; this compo-

nent is reduced 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 ﬂies

at supersonic speeds, in which there is invariably a shock wave in front of the nose

of the fuselage, extending downstream in the form of a cone. Highly swept wings

are then used in order that the wing does not penetrate this shock wave. For ﬂight

speeds exceeding Mach numbers of order 2, the wings have such large sweepback

angles that they resemble the Greek letter ; these wings are sometimes called delta

wings.

706 Aerodynamics

13. Propulsive Mechanisms of Fish and Birds

The propulsive mechanisms of 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 case of a ﬁsh. It develops a forward thrust by horizontally oscillating

its tail from side to side. The tail has a cross section resembling that of a symmetric

airfoil (Figure 15.28a). One-half of the oscillation is represented in Figure 15.28b,

which shows the top view of the tail. The sequence 1 to 5 represents the positions of

the tail during the tail’s motion to the left. A quick change of orientation occurs at one

extreme position of the oscillation during 1 to 2; the tail then moves to the left during 2

to 4, and another quick change of orientation occurs at the other 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 these magnitudes so that the resultant ﬂuid velocity U

(relative to the tail) is inclined to the tail surface at a positive “angle of attack.” The

resulting lift L is perpendicular to U

and has a forward component L sin θ. (It is easy

to verify that there is a similar forward propulsive force when the tail moves from left

to right.) This thrust, working at the rate UL sin θ, propels the ﬁsh. To achieve this

propulsion, the tail of the ﬁsh pushes sideways on the water against a force of L cos θ ,

which requires work at the rate VLcos θ .AsV/U = tan θ, ideally the conversion

of energy is perfect—all of the oscillatory work done by the ﬁsh tail goes into the

translational mode. In practice, however, this is not the case because of the presence

of induced drag and other effects that generate a wake.

Most ﬁsh stay aﬂoat by controlling the buoyancy of a swim bladder inside their

stomach. In contrast, some large marine mammals such as whales and dolphins

Figure 15.28 Propulsion of ﬁsh. (a) Cross section of the tail along AA is a symmetric airfoil. Five

positions of the tail during its motion to the left are shown in (b). The lift force L is normal to the resultant

speed U

of water with respect to the tail.

13. Propulsive Mechanisms of Fish and Birds 707

develop both a forward thrust and a vertical lift by moving their tails vertically.

They are able to do this because their tail surface is horizontal, in contrast to the ver-

tical tail shown in Figure 15.28. A recent review by Fish and Lauder (2006) provided

evidence that leading edge tubercles as seen on humpback whale ﬂippers increase lift

and reduce drag at high angles of attack. This is because separation is delayed due

to the creation of streamwise vortices on the suction side. Cetacean ﬂukes or ﬂippers

and ﬁsh tail ﬁns as well as dorsal and pectoral ﬁns are ﬂexible and can vary their

camber during a stroke. As a result they are very efﬁcient propulsive devices.

Flight of Birds and Insects

Now consider the ﬂight of birds, who ﬂap their wings to generate both the lift to

support their body weight and the forward thrust to overcome the drag. Figure 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. The important point is that the upstroke is inclined at a greater angle

to the airstream than the downstroke. As the ﬁgure shows, the downstroke develops a

lift force L perpendicular to the resultant velocity of the air relative to the wing. Both

Figure 15.29 Propulsion of a bird. A cross section of the wing is shown during upstroke and downstroke.

During the downstroke, a lift force L acts normal to the resultant speed U

of air with respect to the wing.

During the upstroke, U

is nearly parallel to the wing and very little aerodynamic force is generated.

708 Aerodynamics

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 velocity

is then nearly parallel to the wing. Birds therefore do most of the work during the

downstroke, and the upstroke is “easy.”

A recent study by Liu et al. (2006) provides the most complete description to date

of wing planform, camber, airfoil section, and spanwise twist distribution of seagulls,

mergansers, teals, and owls. Moreover, ﬂapping as viewed by video images from free

ﬂight was digitized and modeled by a two-jointed wing at the quarter chord point.

The data from this paper can be used to model the aerodynamics of bird ﬂight.

Using previously measured kinematics and experiments on an approximately

100X upscaled model, Ramamurti and Sandberg (2001) calculated the ﬂow about a

Drosophila (fruit ﬂy) in ﬂight. They matched Reynolds number (based on wing-tip

speed and average chord) and found that viscosity had negligible effect on thrust and

drag at a ﬂight Reynolds number of 120. The wings were near elliptical plates with

axis ratio 3:1.2 and thickness about 1/80 of the span. Averaged over a cycle, the mean

thrust coefﬁcient (thrust/[dynamic pressure × wing surface]) was 1.3 and the mean

drag coefﬁcient close to 1.5.

14. Sailing against the Wind

People have sailed without the aid of an engine for thousands of years and have

known how to arrive at a destination against the wind. Actually, it is not possible

to sail exactly against the wind, but it is possible to sail at ≈40–45

◦

to the wind.

Figure 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

. If the sail is properly

oriented, this gives rise to a lift force perpendicular to U

and a drag force parallel to

. The resultant force F can be resolved 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 lateral component does not

cause much sideways drift because of the shape of the hull. It is clear that the thrust

decreases as the angle θ decreases and normally vanishes when θ is ≈40–45

◦

. The

energy for sailing comes from the wind ﬁeld, which loses kinetic energy after passing

through the sail.

In the foregoing discussion we have not considered the hydrodynamic forces

exerted by the water on the hull. At constant sailing speed the net hydrodynamic force

must be equal and opposite to the net aerodynamic force on the sail. The hydrodynamic

force can be decomposed into a drag (parallel to the direction of motion) and a

lift. The lift is provided by the “keel,” which is a thin vertical surface extending

downward from the bottom of the hull. For the keel to act as a lifting surface, the

longitudinal axis of the boat points at a small angle to the direction of motion of the

boat, as indicated near the bottom right part of Figure 15.30. This “angle of attack”

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

lateral aerodynamic force on the sail would topple the boat around its longitudinal axis.

Exercises 709

Figure 15.30 Principle of a sailboat.

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

coming from the east, we can ﬁrst 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).

Exercises

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

=−



dx +



dx,

where p

and p

are the pressures on the upper and the lower surfaces and c is the

chord length. Show that this relation can be rearranged in the form

≡

(1/2)ρU







where C

≡ (p − p

∞

)/(

ρU

), and the integral represents the area enclosed in a

vs x/c diagram, such as Figure 15.8. Neglect shear stresses. [Note that C

is not

exactly the lift coefﬁcient, since the airstream is inclined at a small angle α with the

x-axis.]

710 Aerodynamics

2. The measured pressure distribution over a section of a two-dimensional airfoil

at 4

◦

incidence has the following form:

Upper Surface: C

is constant at −0.8 from the leading edge to a distance

equal to 60% of chord and then increases linearly to 0.1 at the trailing edge.

Lower Surface: C

is constant at −0.4 from the leading edge to a distance

equal to 60% of chord and then increases linearly to 0.1 at the trailing edge.

Using the results of Exercise 1, show that the lift coefﬁcient is nearly 0.32.

3. The Zhukhovsky transformation z = ζ + b

/ζ transforms a circle of radius

b, centered at the origin of the ζ -plane, into a ﬂat plate of length 4b in the z-plane.

The circulation around the cylinder is such that the Kutta condition is satisﬁed at the

trailing edge of the ﬂat plate. If the plate is inclined at an angle α to a uniform stream

U, show that

(i) The complex velocity in the ζ -plane is

w = U



ζe

−iα

iα



i

2π

ln (ζ e

−iα

where  = 4πUb sin α. Note that this represents ﬂow over a circular cylinder

with circulation, in which the oncoming velocity is oriented at an angle α.

(ii) The velocity components at point P (−2b, 0) in the ζ -plane are [

U cos α,

U sin α].

(iii) The coordinates of the transformed point P



in the xy-plane are [−5b/2, 0].

(iv) The velocity components at [−5b/2, 0]in the xy-plane are [U cos α,3U sin α].

4. In Figure 15.13, the angle at A



has been marked 2β. Prove this. [Hint : Locate

the center of the circular arc in the z-plane.]

5. Consider a cambered Zhukhovsky airfoil determined by the following

parameters:

a = 1.1,

b = 1.0,

β = 0.1.

Using a computer, plot its contour by evaluating the Zhukhovsky transformation. Also

plot a few streamlines, assuming an angle of attack of 5

◦

6. A thin Zhukhovsky airfoil has a lift coefﬁcient of 0.3 at zero incidence. What

is the lift coefﬁcient at 5

◦

incidence?

7. An untwisted elliptic wing of 20-m span supports a weight of 80,000 N in a

level ﬂight at 300 km/hr. Assuming sea level conditions, ﬁnd (i) the induced drag and

(ii) the circulation around sections halfway along each wing.

Literature Cited 711

8. The circulation across the span of a wing follows the parabolic law

 = 



1 −



Calculate the induced velocity w at midspan, and compare the value with that obtained

when the distribution is elliptic.

Literature Cited

Anderson, John D., Jr. (1998). A History of Aerodynamics, London: Cambridge University Press.

Anderson, John D., Jr. (2007). Fundamentals of Aerodynamics, New York: McGraw-Hill.

Ashley, H. and M. Landahl (1965). Aerodynamics of Wings and Bodies, Reading, MA: Addison-Wesley.

Fish, F. E. and G. V. Lauder (2006). “Passive and Active Control by Swimming Fishes and Mammals.”

Annual Rev. Fluid Mech. 38: 193–224.

Hawking, S. W. (1988). A Brief History of Time, New York: Bantam Books.

Herreshoff, H. C. and J. N. Newman (1986). “The study of sailing yachts.” Scientiﬁc American 215 (August

issue): 61–68.

von Karman, T. (1954). Aerodynamics, New York: McGraw-Hill. (A delightful little book, written for the

nonspecialist, full of historical anecdotes and at the same time explaining aerodynamics in the easiest

way.)

Kuethe, A. M. and C. Y. Chow (1998). Foundations of Aerodynamics: Basis of Aerodynamic Design,

New York: Wiley.

Lighthill, M. J. (1986). An Informal Introduction to Theoretical Fluid Mechanics, Oxford, England:

Clarendon Press.

Liu, T., K. Kuykendoll, R. Rhew, and S. Jones (2006). “Avian Wing Geometry and Kinematics.” AIAA J.

44: 954–963.

Ramamurti, R. and W. C. Sandberg (2001). “Computational Study of 3-D Flapping Foil Flows.” AIAA

Paper 2001–0605.

Supplemental Reading

Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.

Karamcheti, K. (1980). Principles of Ideal-Fluid Aerodynamics, Melbourne, FL: Krieger Publishing Co.

Prandtl, L. (1952). Essentials of Fluid Dynamics, London: Blackie & Sons Ltd. (This is the English edition

of the original German edition. It is very easy to understand, and much of it is still relevant today.)

Printed in New York by Hafner Publishing Co. If this is unavailable, see the following reprints in

paperback that contain much if not all of this material:

Prandtl, L. and O. G. Tietjens (1934) [original publication date]. Fundamentals of Hydro and Aero-

mechanics, New York: Dover Publ. Co.; and

Prandtl, L. and O. G. Tietjens (1934) [original publication date]. Applied Hydro and Aeromechanics,

New York: Dover Publ. Co. This contains many original ﬂow photographs from Prandtl’s laboratory.