CHAPTER 3AIRFOIL AERODYNAMICS

3.1 FORCES AND MOMENTS

It is conventional to separate aerodynamic forces and moments into three forcecomponents (lift, drag, sideforce) and three moments (pitch, yaw, roll). Thesecomponents may be defined relative to the wind direction (wind axis system) or relative tothe vehicle centerline (body axis system) or a combination of the two. One must be carefulin the computation or use of force and moment data to use the proper axis system and to beconsistent in its use. The most commonly used system is the wind axis system where theforces are defined either along the free stream velocity vector or perpendicular to it asshown in Figure 3.1.

Figure 3.1. Wind Axis System

To many people this axis system appears inverted and somewhat unnatural. It was chosenprimarily because it is a standard right hand system. It is often more intuitive to invert partof the system to make the z axis point "ups and the x axis go with the wind; however, inthat arrangement the moments do not follow the right hand rule. Either system can be usedif one is careful in its use.

It is important to note that the axis system is aligned with the wind, rather than thehorizon or the vehicle axis. This is an easy source of confusion since it is common tovisualize the wind vector concurrent with the horizon or along the aircraft axis. Indeed, ina straight and level flight situation for an aircraft the free stream wind vector might coincidewith the vehicle axis and the horizon; however it is best not to think in terms of that specialcase. Some people prefer to think of the x axis shown in Figure 3.1 as lying along the pathof flight of the vehicle. Figure 3.2 illustrates the problem by showing a typical glidesituation for an aircraft.

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Figure 3.2. Aircraft in Glide

The three orthogonal forces are lift, L, drag, D, and side force, Y. Lift is defined as theforce along the negative z axis (normally "upward") and acting perpendicular to the freestream direction. Note that this is not necessarily upward with respect to the aircraft axis orthe horizon as indicated in Figure 3.2.

Drag is defined as the force in the direction of the relative wind or along the negative xaxis. Drag can always be thought of as the force which resists the motion of the vehicle.

Side force, given the symbol Y, is defined as mutually perpendicular to both lift anddrag and is positive outhe the right hand or starboard side of the vehicle.

The three moments, pitch (M), roll (LR) and yaw (N) are the moments which tend toresult in a rotation of the vehicle about the y, x or z axes respectively.

Pitching moment, M, is by far the most widely discussed of the three moments since itmust be considered in two dimensional (x, z plane) problems as well as in 3-D cases.Pitching moment is defined as positive when it tends to raise the nose of the vehicle. It isthis basic definition of the sense of pitch that requires the use of an "inverted" (zdownward) axis system in order to have a right had coordinate system. The pitchingmoment acts about the positive y axis.

The rolling moment, LR, is the moment which causes rotation about the x axis orcauses an aircraft to roll one wing up and the other down. Note that this definition of rollmay not coincide with that of an airplane pilot who thinks of roll as occurring about theplane's body axis rather than the wind axis.

Yawing moment, N. rotates the vehicle around the lift direction and is defined aspositive when it is clockwise or results in a nose right motion. Again the use of a bodyaxis rather than a wind axis may result in a different value for yawing moment.

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3.2 DIMENSIONAL ANALYSIS AND NON-DIMENSIONALCOEFFICIENTS

It is convenient in engineering work to deal with non-dimensional terms or unitlessnumbers rather than everyday dimensional terms. The resulting non-dimensionalparameters not only remain unchanged from one unit system to another but they are usuallymore meaningful in terms of the physics of a problem than conventional dimensionalnumbers.

It is possible to develop the concept of a non-dimensional force coefficient and toexamine the important physical parameter groupings on which aerodynamic forces dependby using a simple process known as dimensional analysis. Dimensional analysis, as usedhere, is simply a process of first identifying the parameters on which fluid forces dependand then grouping these parameters in such a way that the units or dimensions balance.Assume then that it is know that the forces one body in a fluid depend on the following:

1. The properties of the fluid itself, pressure, P, density, , viscosity, , and thespeed of sound (fluid's elastic properties), a. Note that one need not includetemperature since P and are considered.

2. The speed of the body relative to the fluid, V.

3. The acceleration of gravity, g.

4. The characteristic size or dimension of the body, l, or the distance of a bodyfrom a fluid boundary, also designated l.

Hence it can be said that the force on a body in a fluid is a function of all of the above,

F = f( ,V,l, ,g,P,a)

[3.1]

or to be completely general, the force is a function of each variable to some power,

F = f( A,VB ,lC , D ,g E ,P G,aH )

[3.2]

Since the left hand side of this equation, [3.1] has the dimensions of force, the righthand side must also have the same dimensions. Force has units of mass multiplied byacceleration (kilograms meters/seconds2 or slugs feet/seconds2). To be general then it canbe said that force has dimensions of (mass)x(length)/(time)2 and letting M, L, and Trepresent these physical dependencies one can write the dimensions for force as MLT-2.Likewise, one can write the dimensions for all the other terms in equation 3.2 in terms ofmass, length, and time:

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Parameter Dimensionvelocity (V) LT-1

length (L) Ldensity ( ) ML-3

viscosity ( ) ML-1T-1

pressure (P) ML-1T-2

gravitationalacceleration (g) LT-2

speed of sound (a) LT-1

force (F) MLT-2

Now, substituting these into equation [3.2] the result is a dimensional equation of theform:

MLT2 = ML−3( )ALT −1( )B

L( )CML− 1T− 1( )D

LT −2( )EML− 1T− 2( )G

LT −1( )H

[3.3]

The task is now to balance the above equation dimensionally; i.e.., the sum of the massexponents on the left side of the equation must equal those on the right, etc. Equatingexponents of mass, length and time respectively leads to three equations:

(Mass) 1 = A + D + G

(Length) 1 = -3A + B + C - D + E - G + H

(Time) 2 = B + D + 2E + 2G + H

[3.4]

Now, equations [Figure 3.4] give a set of three equations and seven unkowns. Theseequations may be solved for the values of any three of the unknowns in terms of theremaining four. Solving then for A, B, and C in terms of the remaining terms gives:

A = 1 - D - G

B = 2 - D - 2E - 2G - H

C = 2 - D + E

[3.5]

Substituting these solutions [Figure 3.2.5] into the original relationship [3.2.2] andgrouping all terms of like exponents gives:

F = V 2l2 Vl

− Dgl

V 2

EP

V 2

GVa

− H

[3.6]

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It is noted that each of the terms with an unknown exponent is a dimensionless term;

i.e., the term

Vl is unitless as are

gl

V2 , p

V 2 and V a . If the equation is divided

by V2 l2

, both side become unitless.

F V2l2 = f Vl

glV 2

P

V2

Va

[3.7]

Two conclusions can be drawn from equation [3.7]. The first is that the proper way tonondimensionalize a fluid force is to divide it by the fluid density, the square of the velocityand the square of the characteristic dimension of the body or the body's representative area.

Since the term 1

2V 2 represents the dynamic pressure of the fluid as found in Bernoulli's

equation, a factor of 1

2 is introduced into the relationship and a "force coefficient" is

defined as

CF = F1

2 V 2S(unitless)

[3.8]

where S is the representative area of the body. In two dimensional problems acharacteristic length (2-D area) is used in the denominator instead of S.

The second conclusion is that the nondimensional fluid force is dependent on thegroups of parameters on the right of equation [3.7]. These groups are known as "similarityparameters" and are important in relating nondimensional force coefficients found on onebody in a fluid to those on a geometrically similar body of different size. Technically,equation [3.7] says that for force coefficients on two geometrically similar bodies to beequal, each of the grouped terms or "similarity parameters" must be identical for the flowsaround the two bodies. It is obvious that it would be quite a task to make all of thesesimilarity parameters equal for tests on bodies of two different sizes or in different fluids.Fortunately, it is seldom necessary to match all four of these similarity parameters at thesame time as an examination of the meaning of each term will show.

The first term on the right of equation [3.7] is known as Reynolds number, Re.

Re = Vl

[3.9]

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Reynolds number is a parameter relating inertial effects in a fluid to viscous effects. This isan important parameter for flow similarity because it is found that laminar-turbulenttransition in a boundary layer is a function of Re and a body at low Re values may have asignificantly different behavior from one at high Reynolds number. The classic example ofReynolds number effects is found in the flow around a sphere or cylinder, where at low Reboundary layer separation occurs early, resulting in a high wake drag and at high Reseparation is delayed by turbulence and the wake drag is reduced.

Reynolds number is an important similarity parameter which must be considered inevery flow. However, there are some cases where it may be ignored. These are generallywhere flow separation occurs at a sharp corner on a body and the separation point will notbe influenced by the laminar or turbulent character of the flow. For this reason Re may notbe a factor when considering flows around some non-streamlined shapes. However, thesesituations are rare and Reynolds number is almost always the most important factor inconsiderations of flow scaling and similarity.

The second most important similarity parameter in most aerospace problems is Machnumber, M, where

M = Va

[3.10]

This parameter indicates the relevance of compressibility effects. Compressibility effectsoccur due to elastic compression and expansion of a fluid as it passes over a body. Theseare important only when compression or mach waves begin to form in a fluid as it flowsaround a body. Hence, Mach number similarity need not be considered at speeds givingMach numbers less than 0.5 or so. In water Mach number need not be considered sincewater is an incompressible fluid.

In some aerospace problems both Reynolds number and Mach number are importantbut it is impossible to satisfy both types of similarity at one time. Here, tests are usuallydone to examine separately the effects of each parameter and the resulting scaling of datamust be done using engineering judgement based on past experience and a thoroughunderstanding of the problem at hand.

The remaining two parameter groupings on the right side of equation [3.7] areencountered primarily in the fields of naval architecture or ocean engineering. The first ofthese is the inverse square root of a widely used similarity parameter known as Froudenumber (F), where

F = V gL

[3.11]

Froude number is the ratio of inertial forces to gravitational forces and it is essentially ameasure of the importance of the effects of a fluid boundary or interface on the forces onthe body. Here the term L refers to a distance which may be the distance of the vehicleabove or below the ground or air-water interface or the height of waves generated by a

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ship. The importance of Froude number is perhaps most easily understood whenconsidering the motion of a submarine below the surface. When a submarine is sufficientlyfar below the surface it can move without disturbing the surface; however, if it is close tothe surface, waves are generated. The energy present in these waves represents an energyloss by the submarine and consequently must be treated as part of the vehicle's drag. Inlike manner any vehicle moving over, under or through the air-sea interface which causessuch surface waves develops a wave drag and for proper simulation of this drag in testingFroude number for aerodynamic similarity, hydrodynamic problems often involve a needfor both Re and F similarity at the same time, a condition which may not be easilyachievable.

The last of the four similarity parameters developed in equation [3-7] is Euler number,

Euler number = PV 2

[3.12]

which is a measure of the ratio of inertial forces and pressure forces. This term assumesimportance when cavitation is a problem on ship hulls or propellers. Basically, cavitationoccurs when the pressures caused by motion of water around a body become low enoughto result in boiling of the water. Cavitation can result in loss of lift on hydrofoils, loss ofthrust on propellers, and high drag on hulls and is thus a very important phenomenon.Proper scaling of flows where cavitation may occur therefore nessitates the use of Eulernumber to insure similarity. In practice Ocean Engineers define a slightly different numbercalled the cavitation number, , where

= P − Pv

V 2

and Pv is the vapor pressure of the water. This is used in place of Euler number as thecavitation similarity parameter.

Example 3.1. An aircraft is designed to fly at 250 mph at an altitude of 25,000 feetwhere the pressure, temperature and density are standard. We wish to test a one-tenth scalemodel of this plane in a wind tunnel at sea level standard conditions. What problems mightwe have in achieving flow similarity?

The Reynolds number for the full scale aircraft would be

Re f = Vlf

and, at standard conditions for 25,000 feet and 250 mph we have

= 0.001066sl ft3

= 3.196x10-7 sl ft

V = 250mph = 367.5 fps

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This gives a Reynolds Number per foot of

Ref lf = 1.226x10 6 ft -1

Note: It is common practice in wind tunnel testing to speak of Reynolds Numberper foot.

To achieve this Reynolds Number on a one-tenth scale model in the wind tunnel atsea level conditions we need to make

Re f = Rem = SLVm1m SL = Re f 1 f = 1.223 ×1061b

Thus, the speed in the wind tunnel test section must be

Vm = 1.226x106 l f SL( ) SLlm

Using SL = 0.002376 sl ft3 , SL = 3.719 x 10−7 sl ftsec and l f lm = 10, weget:

Vm = 1919ft sec = 1305mph

But this supersonic speed quite obviously violates the Mach Number similarityrequirement! Does this mean that it is impossible to properly test for full scaleaerodynamics effects by using small scale models in a wind tunnel? Fortunately not.

One solution is to use a sealed, variable density wind tunnel. Most of the earlywing aerodynamics tests of the National Advisory Committee for Aeronautics ( NACA ),NASA's predecessor, were done in such a tunnel at Langley Field using 1/20th scalemodels tested in a tunnel pressurized to 20 atmospheres, giving a density twenty timesnormal and matching full scale Reynolds Numbers.

Another commonly used method, provided that the model scale is not too small, isto "fool" the flow into behaving like it would at a higher Reynolds Number. The primaryflow influence of Reynolds Number on a streamlined shape is to determine where on theshape the flow in the thin "boundary layer" next to the body changes from a smooth"laminar"" behavior to a turbulent behavior. This, in turn, partly determines where flowseparation from the surface might occur. We can "fool" the flow by forcing or tripping"boundary layer transition from laminar to turbulent flow by using a "trip strip" which canbe as simple as a line of fine sand grains glued to the surface.

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3.3 FORCE AND MOMENT COEFFICIENTS

In the preceeding section it was shown that thenondimensional force coefficient tookthe form

CF = Force12

V 2S

Hence, force coefficients can be defined for the aerodynamic forces of lift, drag andsideforce:

lift coefficient, CL = lift1

2V 2S

drag coefficient, CD = drag12

V 2S

side force coefficient, C Y = side force12

V 2S

As with all force coefficients these are unitless (nondimensional) and therefore are invariantfrom one unit system to another.

One problem which appears when using force coefficients is the area, S, to be used inthe denominator. This is merely a representative area which is characteristic of the body orcase being considered. For example, when using lift coefficient the area S used is almostalways the planform or projected area of the wing. This is logical since the wing is the liftproducing element of the vehicle. The best area to use for drag coefficient is not as obviousand any one of several areas will be found in common usage; wing planform area whenconsidering drag wings, fuselage or hull cross sectional when considering such shapes, or"wetted" (total surface) area when skin friction drag is being treated. Similar variations willbe found when defining characteristic areas for side force coefficients. Hence, it isimportant that one be extremely careful in interpreting data in coefficient form sincethemagnitude of the resulting coefficient may vary greatly depending on the choice of arepresentative area. For the same reason extreme care must be exercised in combining

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coefficients. The drag coefficients of various parts of a vehicle cannot merely be added toget a total drag coefficient unless all the coefficients are based on a common area

Force coefficients can also be similarly defined in two dimensions where only a twodimensional section of a body such as an airfoil section is under consideration. In thesecases only lift and drag are considered and the forces are two dimensional forces; that is,they are expressed in units of force-per-unit-length such as force-per-foot or meter of wingspan. Therefore, to obtain a nondimensional coefficient only a characteristic length isneeded in the denominator.

CL2− D= L

1

2V 21

CD2 − D= D

12

V 21

3.13

The length commonly used for two dimensional lift and drag coefficients is the wing chord(distance from leading-to-trailing edge) when dealing with airfoils.

Moment coefficients are similarly defined except that an additional length factor isneeded in the coefficient denominator. The length most commonly used is the wing's meanchord c or the length of the body.

CM = M12

V 2S c

3.14

In like manner coefficients for rolling and yawing moment may be defined with appropriatecharacteristic lengths.

In two dimensions the pitching moment coefficient becomes

CM2 −D= M

12

V 2c2

3.15

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3.4 AIRFOIL GEOMETRY

In order to discuss airfoil aerodynamics it is necessary to have a grasp of theterminology commonly used to define the geometry of a wing. This terminology will bediscussed in reference to Figure 3.3 which shows the "planform" view of a wing andFigure 3.4 which shows a typical airfoil section.

Figure 3.3. Wing Planform Geometry

Figure 3.4. Wing Section Geometry

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The wing span, b, is the tip-to-tip dimension perpendicular to the fuselage centerlineindlucing the width of the fuselage.

Several chords may be defined for a 3-D wing. The root chord, Co , is the distancefrom the airfoil leading edge to trailing edge taken parallel to the fuselage centerline at hewing-fuselage junction. The tip chord CT is the same dimension taken at the wing tip.Two

different definitions of a mean or average chord are in common usage. The mean chord cis defined as

c = ∫ ob 2 cdy∫ o

b 2 dy

[3.16]

where y is measured from fuselage centerline along the span to the wing tip. Also definedis the "aerodynamic mean chord", cA ,

c A = ∫ob 2 c2dy∫o

b 2 cdy

[3.17]

Wing area, S, is usually thought of as the planform area rather than the actual surfacearea. The planform area includes the imaginary area between the wings in the fuselage.When this fuselage area is to be ignored, ie., the planform area of the exposed wing only isto be used, this is referred to as the net wing area SN .

A term which is of some importance in airfoil fluid dynamics is the "aspect ratio"AR . Aspect ratio is a measure of the "narrowness" of the wing planform and is the ratio ofthe wing span to the mean chord

AR = b c

[3.18a]

which is often written as

AR = b2

bc= b2

S

[3.18b]

a form which is easier to use. Aspect ratio will be found to be a measure of the 3-Defficiency of a wing.

74

While some airfoils have a rectangular planform, most are swept and/or tapered to somedegree. Wing sweep can be measured at the leading or trailing edge or at some other pointsuch as a line along the quarter chord (C/4 behind the leading edge) and the sweep angle isgiven the symbol Λ . Taper is defined in terms of a taper ratio where = C r Co . Thewing angle of attack ( ) shown in Fig. 3.4 is defined as the angle between the chord lineand the fluid velocity vector.

Wing camber is usually expressed as percent camber of an airfoil section where percentcamber is the percentage of the airfoil chord represented by the maximum perpendiculardistance between the chord and camber lines. Wing thickness is also defined in terms ofpercent chord.

Figure 3.5 illustrates wing dihedral which is defined as a dihdral angle due to theinclination of the wings from a common plane. Dihedral is built into wings for rollstabaility purposes and the dihedral angle is defined as 2 , the sum of the inclination anglesof both wings. A negative dihedral is called "anhedral".

Figure 3.5. Wing Dihedral

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3.5 NACA AIRFOIL DESIGNATIONS

In the early days of wing development wing shapes were often given names andnumbers in a very nonsystematic manner. Sections were names after researchers such asClark or Eiffel or after laboratories or research groups (RAF, Gottingen) and each groupdeveloped its own series of airfoil shapes. In an attempt to systematize airfoil research anddesignation, the National Advisary Committee for Aeronautics or NACA (NASA'sforerunner), devised a systematic scheme of testing and classifying airfoil sections.Hundreds of airfoil sections were thoroughly tested and catalogues. These airfoils weredesignated by four, five, and six digit numbers according to basic shape and performance.The data from these tests is reported in numerous NACA publications and some of it willbe referenced in later sections. To properly use this data one should understand themeaning of the NACA airfoil designations.

Most of the NACA airfoils fall into the four, five or six digit airfoil series as explainedin examples below.

NACA 24122 - The maximum camber of the mean line is 0.02c4 - the position maximum camber is 0.4c12 - the maximum thickness is 0.12c

NACA 230212 - the maximum camber of the mean line is approximately 0.02c (also the

design lift coefficient is 0.15 times the first digit for this series)30 - the position of the maximum camber is at 0.30/2 = 0.15c21- the maximum thickness is 0.21c

NACA 632 -215 (laminar flow series)6 - series designation3 - the maximum pressure is at 0.3c2 - the drag coefficient is near its minimum value over of lift coefficients of

0.2 above and below the design CL2 - the design lift coefficient is 0.215 - the maximum thickness is 0.15c

There are other series of airfoil sections besides the ones given above, however, theseare most common. There is a new class of airfoil which will be discussed later that is beingdeveloped out of the "super critical" design of wing. NASA has now began tosystematically number these shapes in categories of low, medium and high speed airfoilswith shapes given designations such as LS(1)-0417 and MS (1)-0313 etc. Hence, theprocess of systematically defining and designating airfoil shapes still continues.

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3 .6 . PITCHING MOMENT AND ITS TRANSFER

Before a discussion of airfoil characteristics can be meaningful a better understandingof pitching moment is helpful. Since most of the later discussion of airfoils will centeraround the dimensional case the only moment of concern will be the pitching moment. It isobvious that pitching moment can be defined as acting about any chosen point on theairfoil, however, its value will vary depending on the point of definition. It is thereforeconvenient to choose some sort of standard reference points for definition of pitchingmoment which will be meaningful physically. One then needs to be able to transfer apitching moment which has been measured or calculated at a given point on the airfoil toone of the chosen reference points or any other location.

In order to transfer the pitching moment all forces and moments acting on the airfoilmust be known. In two dimensions this means that one must know the lift, drag andpitching moment at some point. Suppose, for example, that wing has been mounted in awind tunnel and the lift, drag and pitching moment measured at the mounting point (a), adistance, a, behind the airfoil's leading edge, as shown in Figure 3.6 (a) and that one needsto know the pitching moment about a different point x where a structural member is to belocated.

Figure 3.6. Transfer of Pitching Moment

The forces are invariant in the transfer, however, their effects on the moment must beconsidered.

La = Lx

Da = Dx

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To examine the effect of the moment transfer take the moments in each case relative to acommon reference point, the leading edge. In case a, the lift and drag produce momentsaround the leading edge of

a( ) Lcos( )and a( ) Dsin( )

Since both moments are counterclockwise they are considered negative.

Thus, the entire moment about the leading edge is

M LE = Ma - Lacos - Dasin

Likewise for wing b

MLE = M x -Lxcos -Dxsin

Since these two moments must be identical, equating [3.19] and [3.20] gives

M x = Ma - Lcos = Dsin( ) a - x( )

converting to coefficient form by dividing by 1

2V 2c2 gives

CMx = CMa − CL cos + CD sin( ) ac

− xc

As mentioned previously, there are some cases of special interest regarding placementof pitching moment. One such case is the point where the pitching moment becomes zeroand another is the point where the moment becomes a constant over a wide range of angleof attack or lift. Both of these may be important both aerodynamically and structurally inthe design of a wing. These two points are called the center of pressure and theaerodynamic center, respectively.

The aerodynamic center is probably the most commonly used reference point forforces and moment on an airfoil. If one were to measure the forces and pitching moment onan airfoil over a wide range of angles of attack or lift coefficient and then for each value ofCL look at the values of CM at various chordwise positions on the wing, one special pointwould be found where CM was virtually constant for all values of CL This point is theaerodynamic center.

78

The aerodynamic center is defined as the point along the chord where the pitchingmoment coefficient is constant and independent of the lift coefficient. Note that CM isconstant at the aerodynamic center and not necessarily zero. There are some limits to thisdefinition since at high angles of attack where CL does not change linearly with theaerodynamic center may shift; however, the concept of an aerodynamic center is veryuseful and valid over the normal operating range of CL 's for an airfoil.

It is therefore useful to find an equation which will locate the aerodynamic center oncethe forces and pitching moment about some point on the airfoil have been found.Returning to equation [3.22] and assuming that the unknown position x is the aerodynamiccenter

CMxac= CMa − CL cos + CD sin( ) a

c− xac

c

[3.23]

Several assumptions can be used to simplify this equation based on the alreadymentioned fact that the definition for the aerodynamic center is only meaningful formoderate angles of attack. At these angles cos is approximately ten times the magnitudeof sin ,

cos ≈ 10sin

Also at moderate one normally finds that CL is about twenty times the magnitude ofCD ,

CL ≈ 20CD

therefore,

CLcos ≈ 200C Dsin

and the latter term can be neglected. Using this and assuming Cos is approximatelyunity gives

CMxac= CMa

− CLac

− xac

c

[3.24]

Now the definition for aerodynamic center can be utilized. This definition states that atthe aerodynamic center CM does not vary as CL is changed, or

dCMac

dCL

= 0

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To use this the derivative of equation [3.23] is taken with respect to CL ,

dCMac

dCL

= dCMa

dCL

− dCL

dCL

ac

− xac

c

By definition then the term on the left becomes zero and the derivative of CL with respectto itself is unity; thus, the equation can be rearranged as

xac

c= a

c− dCMa

dCL

[3.24]

Using the equation 3.24 the position of the aerodynamic center can be found as adistance xac from the leading edge by finding the value of the derivative of the knownmoment coefficient with respect to the lift coefficient. This can be easily obtained byplotting a graph of CM versus CL and measuring the slope of the curve. Such a curve willbe essential linear over a normal range of angle of attack for most common airfoils.

EXAMPLE 3.2 For a particular airfoil section the pitching moment coefficient about apoint 1/3 chord behind the leading edge varies with the lift coefficient in the followingmanner:

CLK 0.2 0.4 0.6 0.8

CM K − 0.02 0.00 + 0.02 + 0.04

Find the aerodynamic center and the value of CMo .

It is seen that CM varies linearly with CL , value of dCM dCL being

0.04 − −0.02( )0.80 − 0.20

= + 0.060.60

= +0.10

Therefore, from equation 3.12, with a/c = 1/3

xac

c= 1

3− 0.10 = 0.233

The aerodynamic centre is therefore at 23.3% chord behind the leading edge.

Plotting CM against CL gives the value of CMo , the value of CM when CL = 0, as-0.04.

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Theoretically, on a flat plate or circular arc airfoil in subsonic flow the aerodynamiccenter is exactly one forth of the chord from the leading edge or at a point called the"quarter chord". In practice it is usually from 23 to 25% of the chord from the leading edgeat incompressible speeds. For this reason it is popular to define aerodynamic forces andpitching moments about the quarter chord as is the practice in may reports of tabulatedairfoil aerodynamic data.

When speeds are high enough for compressibility to become a factor (M ≥ 0.5) theaerodynamic center will start to move rearward along the airfoil. The theoretical position ofthe aerodynamic center in supersonic flow is at 50% chord. This shift will be examined in alater section where compressibility effects on airfoils are discussed.

The center of pressure, while not necessarily a fixed point over a range of orCL , is important since this is the point where the moment disappears. It is the point, notnecessarily on the chord, where the pitching moment is zero for a particular value of lift.Returning to equation [3.20] the following relation can be written relating the moment at theleading edge, MLE to the moment at the center of pressure MMP .

M LE = MMCP − Lcos +Dsin( )kCPC

Now, recognizing that by definition MCP = O and writing XCP as kCPC where kCP is thefraction of the chord to the center of pressure, the result is

MLE = MCP − L cos a + Dsin a( )kCPC .

Writing the same equation with the moment about the aerodynamic center in the equationgives

MLE = Mac − Lcos + Dsin( )Xac

Now equating the two equations [3.25] and [3.56] and dividing by 1

2V 2c2 to get the

relation in coefficient form gives

CMac − CL cos + CD sin( ) Xacc

= − CL cos + CD sin( )kcp

Solving for kcp, the position of the center of pressure as a fraction of the chord,

kcp = Xacc

− CMac

CL cos + CD sin

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Here again, since is relatively small and CL > > CC , the term CD sin can beneglected and cos is assumed to be unity, giving

kcp = Xacc

− CMac

CL

Hence a relation has been developed which will determine the position of the center ofpressure for any value of CL once the pitching moment at the aerodynamic center and thelocation of the aerodynamic center known.

Since CMac is almost always negative it can be seen from equation [3.28] that

kcp −Xac

c will be positive, indicating that the center of pressure is almost always behind

the aerodynamic center.

Example 3.3 For the airfoil section of Example 3.2, plot a curve showing theapproximate variation of center of pressure position with lift coefficient, for lift coefficientsbetween zero and unity.

For this case, kcp ≅ 0.233 − −0.04CL

≅ 0.233 + 0.04CL

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The corresponding curve is shown above. It shows that kcp tends asymptotically to Xac asCL increases, and tends to infinity behind the airfoil as CL tends to zero. For values ofCL less that 0.05 the center of pressure is actually behind the airfoil.

For a symmetrical section (zero camber) and for some special camber lines, the pitchingmoment coefficient about the aerodynamic center is zero. It then follows, that kcp = xac ;i.e., the center of pressure and the aerodynamic center coincide, and that for moderateincidences the center of pressure is therefore stationary at about the quarterchord point.

3.7 AIRFOIL AERODYNAMIC PERFORMANCE

Now that the meaning of forces and moments on airfoils have been discussed and theterminology of airfoil geometry is understood, the effects of changes in airfoil geometryand fluid flow behavior can be explored. The discussion that follows will be very generaland many exceptions can undoubtedly be found to the examples given; however, the airfoilbehavior discussed will be that typical of most airfoils.

Any discussion of airfoils should begin with the simple symmetrical airfoil whichtypically exhibits an aerodynamic behavior similar to that shown in Figure 3.7.Theoretically the lift coefficient for a two dimensional airfoil increases linearly with a curveslope of 2 per radian when plotted against . When the CL curve begins to becomenon-linear, flow separation is beginning along the upper surface of the airfoil. Asseparation or stall progresses the curve slope decreases until a peak is reached and beyondthis angle of attack CL will decrease. At this peak the value of lift coefficient is termedCLmax The nature of the stall region will vary with leading edge radius, Reynolds numberand other factors which will be discussed later.

Fig. 3.7. Symmetrical airfoil characteristics.

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The pitching moment coefficient shown is for the quarter chord position and since thisis roughly the position for the aerodynamic center for an airfoil in incompressible flow,CMc 4 is constant over the range of where dCL d is constant.

The drag coefficient for the symmetrical airfoil is seen to be at a minimum at a CL ofzero which corresponds to a zero angle of attack. Drag coefficient rises rapidly in the stallregion as wake drag increases.

The effects of camber on an airfoil are illustrated in Figure 3.8.

Fig. 3.8. Effects of Camber

The primary effect of camber on an airfoil is to shift the CL vs curve to the left whichmeans that the cambered airfoil produces a finite lift at a zero nominal angle of attack. Thecambered airfoil experiences a zero lift at some negative angle of attack designated LO .

Stall usually occurs at a lower nominal angle of attack for the cambered airfoil, howeverCLmax will usually be somewhat increased over that for a symmetrical airfoil of comparablethickness and leading edge radius.

Increasing camber is seen to produce a larger negative CMc 4 and to cause shift in the

drag polar to correspond to the CL shift.

The effects of changes in airfoil thickness depend largely on where the maximumthickness of the airfoil occurs. It is somewhat intuitive that a thicker section will produce ahigher drag coefficient, however, the primary effect of thickness is on the wings stallbehavior and the lift curve slope. The effect of thickness on the lift curve slope varies withthe distribution of the thickness along the chord as shown in Figure 3.9

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Fig. 3.9 Effect of Thickness on Lift Curve Slope

For the NACA 4 and 5 digit series airfoils the slope of the lift curve is seen to decreaseslightly as thickness increases (the theoretical slope should be 2 radian or0.10966/degree)/ whereas, for the 63 series airfoil it shows an increase.

The thickness can also either increase or decrease CLmax as illustrated in Figure 3.10.An increase in thickness up to a point can lead to a higher CLmax because of its effect onimproving the leading edge radius, however further increases increase the likelihood ofearlier separation over the rear of the airfoil.

Figure 3.10. Thickness effects on CLmax

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The most significant effect of thickness is to be seen near the airfoil's leading edge interms of leading edge radius. Too small a leading edge radius can result in an abrupt stallwith a sudden decrease in lift coefficient. A larger leading edge radius can smooth the stalland give an increase in CLMax as shown in Figure 3.11.

Figure 3.11

The effects of Reynolds number on airfoil performance are due to the influence ofReynolds number on the airfoil boundary layer and on flow separation. These effects havebeen discussed to some extent earlier. At low values of Re a laminar boundary layer existsover a large distance from the leading edge of the airfoil. A laminar boundary layer is verypoor at resisting separation over the region of the airfoil where the flow is slowing downand the pressure is increasing, and early flow separation may result, giving larger drag andearly stall. At high Re values the boundary layer goes turbulent at an early point on theairfoil and because of the ability of a turbulent boundary layer to resist flow separation,wake drag may be reduced and stall delayed. These effects are illustrated in Figure 3.12for a typical airfoil.

Figure 3.12. Reynolds Number Effects on Lift and Drag

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One of the more important effects of Reynolds number is its influence on the near stallregion at high angles of attack. Increasing Reynolds number almost always increasesCLMax and it can also have a significant effect on the nature of stall as shown in Figure3.13. For a particular airfoil it is seen that at low Re a fairly smooth stall exists with arelatively low CLMax . As Re is increased a higher CLMax is obtained but the stall is asharp one. Here the high Re flow is able to resist separation but when it finally occurs itsuddenly covers a large portion of the airfoil. At even higher Re the turbulent boundarylayer is able to resist a sudden leading edge separation and an ever higher CLMax isachieved with a somewhat improved stall behavior.

Figure 3.13. Effect of Re on Stall

The effect shown in Figure 3.13 is also a function of the airfoil thickness andparticularly the leading edge radius even small increases in Re may increase CLMax whileon a thin airfoil with small leading edge radius it may be necessary to go to very highvalues of Re before significant CLMax increases occur, as shown in Figure 3.14. A smallleading edge radius produces very large flow accelerations at high angles of attack resultingin large pressure deficits. The subsequent pressure rise as the flow slows down again overthe airfoil's upper surface is also very rapid and this large "adverse" pressure gradientcauses flow separation. Very high Reynolds numbers are required to resist the separationinducing effects of this large pressure gradient. on a thicker airfoil with a larger leadingedge radius the pressure gradient is not as large and it is not necessary to reach as large avalue of Re to control separation.

Fig. 3.14. Effects of thickness and Re on CLMax

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Compressibility effects on an airfoil must be considered at Mach numbers above 0.5 orso. Even at relatively modest subsonic speeds the local velocities of air over a wing mayapproach the speed of sound. When the free stream velocity is great enough that the localflow at some point on the upper surface of the airfoil reaches the speed of sound the airfoilis said to have reached its "critical Mach number, Mcrit ." For example, the NACA 4412airfoil flying in air at sea level standard conditions will begin to experience locallysupersonic flow at about 20 - 30% of the chord behind the leading edge at a speed of 592fps or 180m/s. Since the speed of sound is 1117 fps at sea level the wing is flying at aMach number of 0.53. Hence its critical Mach number Mcrit = 0.53.

Once a region of supersonic flow beings to form on the airfoil a shock wave will formas the flow "shocks" down to subsonic flow. In a shock wave the flow deceleration occursvery suddenly resulting in a sharp, essentially instantaneous, pressure increase. It is thispressure increase or compression wave which results in the sonic boom when the wave isstrong enough to reach the ground. The sudden pressure increase in the shock wave actsas an almost infinite adverse pressure gradient on the wing's boundary layer, resulting inalmost certain flow separation. This process is illustrated in Figure 3.15.

Figure 3.15. Transonic Flow Patterns Around an Airfoil

As this supersonic flow region and accompanying shock wave grow the separatedwake also grows, increasing drag and changing the manner in which lift is produced on thewing. This causes the aerodynamic center to shift and large changes in the pitchingmoment about the quarter chord. These changes with Mach number are shown in Figure3.16 as changes in the slope of the lift curve, drag coefficient and moment coefficient aboutthe quarter chord.

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Figure 3.16. Compressibility Effects on Lift Curve Slope, Drag and Pitching Moment

It is these large changes which created the myth of the "sound barrier" in the early daysof transonic flight. For an aircraft designed for subsonic flight these changes could easilylead to disaster when the wing reached and passed its critical Mach number in a high speeddive. When these speeds were reached there was a sudden sharp rise in drag whichrequired thrust beyond that available in the engine technology of the day, and worse, thesudden aerodynamic center shift and flow separation caused severe stress in the structureand loss of control. As shown in Figure 3.16-c the value of CMc 4 may go from a negativeto a positive value, resulting in a complete reversal of the stability behavior of the aircraftand control behavior. The result was often a disaster where the aircraft went out of controland broke up in mid-air. Once powerplants were large enough to handle the drag rise andaircraft were designed to handle the moment shifts it became possible to fly in and throughthe transonic regime.

3.8 FLAPS AND HIGH LIFT DEVICES

In order to fly, an aircraft must produce enough lift to counteract its weight. From theequation for lift coefficient [3.12] it can be seen that the lift generated by a wing is afunction of its area, (S), the square of its velocity (V∞ ), the air density ( ), and the liftcoefficient (CL ).

Lift = 12

V∞2SCL

Hence, lift can be increased by increasing any one of these four parameters. This presentslittle problem in high speed flight since for a given wing and angle of attack the liftincreases as the square of the speed. There are however, problems at lower speeds.

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The lower limit for the airspeed of a wing is its stall speed VSTALL . This is the speedat which a wing of given area will stall at a given altitude (or density). Stall occurs whenthe lift coefficient has reached its maximum (CLMax). Thus for a given wing area, densityand weight of aircraft, the minimum flying speed is given by:

V∞min =L

1

2SCLmax

=2W

SCL max

This is obviously then the speed which will result in the shortest takeoff or landing roll.Naturally, it would not be safe to land or takeoff an aircraft at CLMax conditions since it ison the verge of stall; thus, a slightly higher speed is used to give a safety factor. Thisspeed is usually figured to be 1.3VSTALL . However, the fact still remains that theminimum safe flying speed and, hence, the minimum landing or takeoff distance isdetermined by CLMax for a wing at a given altitude, weight and wing area.

In the early days of aviation, when more lift was needed to keep landing and takeoffdistances within limits, a larger wing area was used. However, this also increased the dragof the aircraft and limited its cruise speed. A more highly cambered wing or one whichwas relatively thick to give a larger leading edge radius could also be used to increaseCLMax but these also increased drag and limited cruise speed. Therefore, the cruise speedof an aircraft was effectively limited by the length of runway available for the takeoff andlanding. What was needed was a way to use a minimal drag wing for high speed cruiseand then increase wing area and/or camber for takeoff and landing to reduce the requiredtakeoff and landing speeds.

The answer to this requirement was the flap. The basic idea behind the use of the flapis that of developing a variable camber airfoil which will normally have a low camber, lowdrag shape but can have its camber increased when a higher maximum lift coefficient isneeded for low speed flight. Adjustable camber airfoils were built and flown as early as1910 and the hinged flap can be traced to as early as 1914 in England.

Theory can show that a change in camber near the trailing edge of an airfoil has a muchgreater effect on CL than changes at any other position. Hence flaps are, in their simplestform, merely hinged portions of the airfoil's trailing edge as shown in Figure 3.17. Thetypical effects of flap deflection are also shown in the figure to be essentially the same asadding camber.

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Figure 3.17. Aerodynarnic Effects of the Flap .

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There are many different designs for flaps, some more effective than others. Theeffectiveness of a variety of flaps is illustrated in Figure 3.18.

Designation Diagram CLMaxat CLMax(degrees)

L Dat CLMax

ReferenceNACA

Basic airfoilClark Y

1.29 15 7.5 TN 459

.30cPlain flap

deflected 45o

1.95 12 4.0 TR 427

.30cSlotted flap

deflected 45o

1.98 12 4.0 TR 427

.30cSlit flap

deflected 45o

2.16 14 4.9 TN 422

.30c hinged at .80cSplit flap (zap)deflected 45o

2.26 13 4.43 TN 422

.30c hinged at .90cSplit flap (zap)deflected 45o

2.32 12.5 4.45 TN 422

.30cFowler flap

deflected 40o 2.82 13 4.55 TR 534

.40cFowler flap

deflected 40o

3.09 14 4.1 TR 534

Fixed slot 1.77 24 5.35 TR 427

Handley Pageautomatic slot

1.04 28 4.1 TN 459

Fixed slot and.30c plain flapdeflected 45o

2.18 19 3.7 TR 427

Fixed slot and.30c slotted flapdeflected 45o

2.26 18 3.77 TR 427

Handley Page slot and.40c Fowler flapdeflected 40o

3.36 16 3.7 TN 459

Data taken form NACA 7 x 10 ft tunnel, wing AR=6, Re=609,000

Fig. 3.18. Effectiveness of Flaps and Slots on a Clark Y airfoil.

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Some of the flaps have slots between the flap and the main wing and some both deflect andextend. The Fowler flap, which deflects and extends to open a slot is seen to be the mosteffective of the single flap systems, increasing CLMax by a factor of 2.4.

The leading edge slot and flap is also illustrated in Figure 3.18. It is seen that theleading edge flap and slot can be very effective in increasing CLMax by itself and its usewith trailing edge flaps improves the performance of the wing even further.

The leading edge flap, slot of "slat" does not achieve its effectiveness through a changein camber, because a camber change at the leading edge has very little effect; but works byreducing the likelihood of flow separation and stall at high angles, much like an increasedleading edge radius or increased Reynolds number would. Figure 3.19 illustrates the basicinfluence of the slot on an airfoils performance.

Figure 3.19. Effect of Leading Edge Slot

Figure 3.20 shows several types of leading edge devices which have been used onwings. The first four do not incorporate slots and achieve their effect essentially byproviding an easier path for the flow to follow over the leading edge. This reduces the lowpressure peak experienced by the upper surface of the airfoil at high angles of attack sincethe flow does not have to accelerate as rapidly to get around the leading edge.Subsequently the following adverse pressure gradient is not as strong and separation is lesslikely.

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Figure 3.20. Leading Edge Devices

The other two devices shown in Figure 3.20 incorporate a slot of some type. Theleading edge flap with a slot is much more effective than one without it. The action of theleading edge slot has often been incorrectly explained as that of a nozzle which directs highspeed air into the wing's boundary layer, "energizing" it so that it can continue on aroundthe wing without separating. However, an examination of the flow through the slot willshow that it is, in fact, a very low speed flow and that, instead of adding high speed air tothe flow above the wing, it slows the flow.

The slot effect is best understood if the main wing and the leading edge flap or slat areviewed as two separate wings. If these two wings (A and B in Figure 3.21) were testedalone at the orientation to the free stream shown, a pressure distribution similar to thatshown by the dashed lines in the plot would result. When the two are placed in proximityto each other however, the blockage of flow through the slot slows the flow over thebottom of wing A and the leading edge of wing B. This reduction in the velocity of theflow has two effects. The most important effect is that by slowing the flow over theleading edge of wing B, the low pressure peak is reduced in magnitude, thus reducing theseverity of the adverse pressure gradient which follows and delaying separation. Thismeans the wing can go to higher angles of attack before stall. As is always the case,however, this benefit is not free. The solid line on the CP graph shows that the newpressure distribution reduces the lift generated by the wing.

The loss of lift on wing B is more than counteracted by the increase i lift on wing A dueto the reduced velocity and hence, higher pressure below wing A. In actuality, the flowaround wing B results in an effective large positive angle of attack on wing A. Thepressure distribution on wing A thus changes as shown in Figure 3.21 and the lift on wingA changes from negative to a large positive value.

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The net effect of the flow over the two wings is a slotted airfoil which can go to highangles of attack before separation and stall. In actuality, the total flow is a bit morecomplicated than that just described and includes boundary layer interaction anddownstream influences. However, the essentials of the flow fit the above description.

Figure 3.21. Action of a Leading Edge Slot.

Slotted trailing edge flaps work on the same principle as that just described for theleading edge slot. In this case the slots allow the trailing edge flaps to be deflected to verylarge angles without flap stall. Modern transport aircraft which need to be able to cruise athigh speeds and land in reasonable distances often use multiple flap and slot systems suchas the one shown in Figure 3.22.

Fig. 3.22. Geometry of Leading Edge Slat and Triple Slotted Flap

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A number of other devices exist which are intended to increase the maximum liftcoefficient of an airfoil. Most of these devices are in some way dependent on power fromthe aircraft and are essentially boundary layer control devices. With non-power augmenteddevices such as mechanical flaps and slats it is possible to achieve lift coefficients in therange of 3.5 to 4.0. According to inviscid theory the upper limit on CL is 4 or about12.5. Of course, this theory does not account for the effects of viscosity in retarding theflow in the boundary layer and subsequent flow separation. By attempting to control theboundary layer to prevent separation it is possible to achieve higher maximum liftcoefficients.

The two primary types of power augmented boundary layer control high lift devicesused have been suction and blowing. Boundary layer suction can be effective indelaying separation on an airfoil. Suction is applied through a slot in the wing surface inthe region where the adverse pressure gradient would be likely to lead to separation. Thesuction pulls away the "stale" boundary layer and, essentially a new boundary layer begins.It is possible to roughly double CLmax for an airfoil by using suction properly. Theimprovement in CLmax increases as the suction increases.

A certain amount of power is required to create the suction used to control the boundarylayer and this power usage must be accounted for. Since power used in a vehicle is usuallythat used for propulsion to overcome drag and the use of suction for boundary layer controlmust be considered as part of the total vehicle power requirement, this power is usuallytreated as a drag penalty. Hence the question must always be asked whether or not the gainin lift is worth the price paid.

Boundary layer blowing may be accomplished in several different ways and is morecommon than suction as a boundary layer control device. Boundary layer control byblowing is usually accomplished by introducing a jet through a slot such that the jet istangential to the boundary layer (Figure 3.23). This jet "energizes" the boundary layer byintroducing a high speed stream and by entraining fluid from outside the boundary layer.Like suction similarly placed, boundary layer blowing on the wing's upper surface canessentially double the CLmax of the airfoil. Again, the power required for blowing must beconsidered as a penalty and is usually treated as drag.

Blowing is often used to control the flow over the flaps where fluid is blown over theflap either from a slot in the aft portion of the main airfoil or from a slot in the leading edgeof the flap itself. This can be extremely effective in increasing CLmax for a wing bypreventing flow separation over the flaps at high wing angles of attack and high flapdeflection angles (Figure 3.24).

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Fig. 3.23. Boundary Layer

Figure 3.24. Internally Blown Flaps

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A type of blowıng with sinıilar effect is found in the externally flow flap (Figure 3.25)where the jet engine or fan exhaust flows over the airfoil enhancing the normal flap, slot,slat action to increase CLmax . Engine placement is critical in this system and the exhaustmust be designed to cover as much of the wing as possible.

Many other types of boundary layer control, high lift devices exist or have beeninvestigated. These are too numerous to mention within the scope of this text. Theseinclude the augmentor wing upper surface blowing, the Coanda effect, jet flaps and otherdevices.

Figure 3.25. Externally Blown Flaps.

3.9 LAMINAR FLOW AIRFOILS

An examination of boundary layer behavior would show that a laminar boundary layercauses less skin friction drag than a turbulent boundary layer; however, the turbulentboundary layer is much better at resisting flow separation. Knowing this, NACAresearchers in the late 1930's and 1940's designed a low drag series of airfoils calledlaminar flow airfoils. These airfoils, know as the NACA 6-series airfoils, were designedto take advantage of the low skin friction of a laminar boundary layer by encouraging alaminar flow over the first 30 to 50 percent of the airfoil surface.

In order to maintain laminar flow over a large portion of an airfoil something must bedone to prevent laminar-turbulent transition and to prevent flow separation, which occursrather easily in a laminar boundary layer. In the laminar flow airfoil this can be achieved atlow to moderate angles of attack by shaping the airfoil to produce a favorable pressuregradient over the area where laminar flow is required. A favorable pressure gradient, thatis one where pressure is decreasing and flow is accelerating, has the effect of suppressingthe development of turbulence in the boundary layer. Laminar-turbulent transition in aboundary layer is caused by the growth and merger of small scale turbulence disturbancesin the flow. The likelihood of the growth of these disturbances is spoken of in terms ofboundary layer stability and a favorable pressure gradient is a stabilizing one while anadverse gradient is de-stabilizing.

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The most straightforward way of creating a favorable pressure gradient on an airfoil isto shape the wing in such a way that the flow accelerates relatively slowly over a largeportion of the wing. This is done primarily by reducing the leading edge radius to cutdown on large accelerations there, and by moving the maximum thickness of the airfoilrearward. This will result in easier separation at high angles of attack and a lower CLmaxbut will give a lower drag at more moderate angles of attack hence, saving fuel in a longdistance cruise condition. Figure 3 26 shows the difference in wing shape and pressuregradient between the conventional NACA 0015 airfoil and its laminar flow counterpart, theNACA 653 -015.

Figure 3.26. Conventional and Laminar Flow Airfoils

Figure 3.27 shows the result of this change in comparing an NACA 2415 airfoil with aNACA 642 -415 wing. Both airfoils are 15% thickness and have the same camber,however the latter has a region of reduced drag coefficient over a range of CL from 0 to0.5. This area of reduced CD is known as the "drag bucket" and it is this "bucket" that isthe primary characteristic of laminar flow airfoils. It should be noted that at CLs above thebucket, the drag of the laminar flow airfoil is actually higher than its conventionalcounterpart because of the effect of the sharper leading edge at high , however thereduction of drag coefficient by a factor of two in the drag bucket region may be moreimportant in a given wing design than the effects at higher angles of attack.

The position of the drag bucket can be changed as shown in Figure 3.28 with the firstdigit in the second series of three numbers in the NACA designation referring to the valueof CL at the center of the drag bucket. This gives the designer an important tool, allowingher or him to select a laminar flow airfoil which is best for his or her design. If an aircraftis to be designed for long distance cruise and a CL of 0.2 is needed in cruise, the designercan select the laminar flow airfoil which will position the drag bucket around the CL = 0.2range, giving a low drag cruise and a high fuel economy in cruise. Likewise if a highperformance aircraft requires low drag during maneuvers and climbs where CL is high, thedrag bucket can be placed there.

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Fig. 3.27: Drag Bucket for a Laminar Airfoil

Figure 3.28. Placement of Drag Bucket

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Laminar flow airfoils have been used on almost all transport aircraft and on manygeneral aviation aircraft designed since about 1940. They can greatly reduce drag in cruiseand with proper use of flaps and other high lift devices it is still possible to achieve highCLmax values when needed for landing and takeoff. It should be noted however, that it ispossible to loose the drag bucket if sufficient dirt or roughness accumulates on the forwardpart of the airfoil and, hence, such wings must be kept clean to work at their best.

3.10 SUPERCRITICAL AIRFOILS

The problem of drag rise, moment shift and lift loss during transonic flow wasdiscussed earlier. When the critical Mach number is reached, a shock wave forms on theairfoil's upper surface and the flow is very likely to separate due to the large adversepressure gradient imposed by the shock. Anything which can be done to increase thecritical Mach number; i.e., delay the onset of the shock wave to a higher speed, will allowhigher speed subsonic flight with low drag. Hence a higher Mcrit will allow flight athigher speed on a given amount of engine power or flight at the same speed as wings withlow Mcrit at reduced power and fuel levels.

The most obvious way to delay the onset of an upper wing surface shock is to reducethe curvature of the upper surface in such a way that the flow does not accelerate to as higha speed as it would with a conventional or even a laminar flow airfoil. With a "flatter"upper surface the region of supersonic flow may be spread over a larger portion of theairfoil and the flow does not accelerate to as high a supersonic speed. This accomplishestwo things, it delays the onset of the shock until a more aft position where any resultingseparation will affect less of the airfoil and it gives a weaker shock which is less likely tocause separation. Because of the lower supersonic speeds over the airfoil the pressuresdeveloped are not as low and lift due to the flow over the upper surface may be reducedfrom that on a conventional airfoil. Hence, the new airfoil is designed with a large cambercreated by a cusp in the lower surface at the trailing edge which creates enough lift to makeup for that lost on the upper surface. The resulting design is shown in Figure 3.29.

Figure 3.29. Supercritical Airfoils

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The supercritical airfoil has proved to be very effective at lower speeds as well as in thetransonic regime. As of the mid 1970's supercritical airfoils were appealing on transportdesigns for production in the 1980's and beyond. Low speed versions of the airfoil havebeen tested on general aviation aircraft and are appearing on new models of such airplanes.the airfoil design has an advantage of giving low drag with a relatively thick wing section(13 to 21% thickness) and thus provides room for improved structure or fuel capacity. Thelarge thickness and large leading edge radius generally result in a higher CLmax in lowspeed applications.

The primary drawback for the supercritical airfoil design has been the large negativepitching moment caused by the large magnitude of lift generated near its trailing edge.Another problem has been in the manufacture of the sharp trailing edges which result fromthe trailing edge cusp. It is, however, possible to design around these problems and thesupercritical airfoil is expected to be the wing used on most aircraft for the foreseeablefuture.

3.11 THREE DIMENSIONAL EFFECTS

Most of the discussion in the previous sections of this chapter has dealt with twodimensional airfoil behavior. A two dimensional airfoil is, of course, only a section orslice of a 3-D airfoil and two dimensional airfoil aerodynamics must be considered the idealcase. Three dimensional effects can often be thought of as simply factors which limit thenormal 2D performance of an airfoil. The two primary three dimensional factors whichaffect wing performance are aspect ratio, AR, and wing sweep.

As mentioned in section 3.4 aspect ratio is a measure of the ratio of the span to themean chord. Aspect ratio has a significant effect on the performance of the total wingbecause of wing tip losses. Ideally, the lift generated by an untapered wing would beconstant at every point along the span. However, due to flow around the wing tip there arelift losses near the tip as shown in Figure 4.30. Since there is a lower pressure on top ofthe wing than below the fluid will flow around the wingtip to the area of lower pressure.This, therefore reduces the lift generated near the wingtip and must be considered as a 3-Dloss. This flow around the wingtip is also the source of the trailing vortex, a swirling flowcoming off of each wingtip.

The loss of lift at the tip of a 3-D wing goes inboard over some percentage of thespan. Hence, for a short stubby wing with low AR a greater percentage of the total wingarea experiences some tip loss than in a high aspect ratio wing. Therefore a high aspectwing will produce more lift than a low aspect ratio wing of the same area and airfoilsection. The 3-D wing has a higher drag coefficient than the 2D airfoil and this additionaldrag is inversely proportional to aspect ratio. The net result is that a high aspect ratio airfoilhas higher lift and lower drag than a low aspect ratio wing of the same area and airfoilshape.

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Figure 3.30. 3-D Airflow Effects

Wing sweep primarily affects the performance of a 3-D airfoil in the transonic flowregime. The primary effect of sweeping the wing is to raise the critical Mach number of theairfoil. The shock wave, which develops on an airfoil in transonic flow, is developed inresponse to the component of the free stream to the line down the quarter chord of thewing. Hence, for a given free stream velocity, the greater the wing sweep the lower will bethe component of velocity normal to the quarter chord line. It is not until this normalcomponent of the flow reaches the critical Mach number that the drag rise will occur andeven when it does occur the drag rise is not as great as in the unswept case. Figure 3.31shows the effect of varymg degrees of sweep on one senes of wings.

Figure 3.31. Sweepback Effect on Drag Rise vs. Mach No.

As is the case with almost any change that is of benefit in some way a price must bepaid for the favorable effects of sweep. This price is sweep-induced cross flow tip stall.The span-wise flow which results from the sweep can cause severe adverse pressuregradients at the wing tips and tip stall. Severe structural as well as aerodynamic problemscan result.

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Summary

A review of this chapter will show that almost all phenomena which occur on airfoilscan be understood in terms of simple pressure-velocity behavior and their effects on thebehavior and their effects on the behavior of the boundary layer. All that is needed for aphysical understanding of airfoil aerodynamics is an appreciation of the meaning of thepressure-velocity relationship called Bernoullis equation. This, combined with a physicalfeel for the meaning of laminar, turbulent and separated boundary layers can explain theaerodynamic behavior of flow about any shape. Other texts will show how to take thissimple physical understanding and build it into a useful mathematical description of fluidflows.

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