Flightlab Ground School 2. Two-Dimensional Aerodynamics · Bernoulli’s Theorem deals with...

Bill Crawford: WWW.FLIGHTLAB.NET 2.1

Flightlab Ground School2. Two-Dimensional Aerodynamics

Copyright Flight Emergency & Advanced Maneuvers Training, Inc. dba Flightlab, 2009. All rights reserved.For Training Purposes Only

Our Plan for Stall Demonstrations

We’ll do a stall series at the beginning of ourfirst flight, and tuft the trainer’s wing with yarnbefore we go. The tufts show complex airflowand are truly fun to watch. You’ll first see thetufts near the root trailing edge begin to wiggleand then actually reverse direction as the adversepressure gradient grows and the boundary layerseparates from the wing. The disturbance willwork its way up the chord. You’ll also see themovement of the tufts spread toward thewingtips. This spanwise movement can bemodified in a number of ways, but dependsprimarily on planform (wing shape as seen fromabove). Spanwise characteristics have importantimplications for lateral control at high angles ofattack, and thus for recovery from unusualattitudes entered from stalls. Our rectangular-planform trainers have excellent stallcharacteristics. Other planforms may need to becajoled into behaving as if they were rectangular,stalling first at the root while the ailerons keepflying.

We’ll first examine two-dimensional airfoilsections, then three-dimensional wing planforms.We’ll only spend a few minutes during ourflights watching the wing tufts, but those minutescan be full of information.

Remember Mass Flow?

Remember the illustration of the venturi fromyour student pilot days (like Figure 1)? Themajor idea is that the flow in the venturiincreases in velocity as it passes through thenarrows.

The Law of Conservation of Mass operates here:The mass you send into the venturi over a givenunit of time has to equal the mass that comes outover the same time (mass can’t be destroyed).This can only happen if the velocity increaseswhen the cross section decreases. The velocity isin fact inversely proportional to the cross sectionarea. So if you reduce the cross section area ofthe narrowest part of the venturi to half that ofthe opening, for example, the velocity mustdouble at that point.

For a fluid (like air), the density, ρ, times thecross section area, A, of the venturi times thevelocity, V, equals the mass airflow. Massairflow, in and out, remains constant, so:

ρAV = Constant

The above is known as the continuity equation.

Density (denoted by the Greek letter ρ,pronounced “roh”) is the mass of air (in slugs)per volume (one cubic foot). Air is consideredincompressible at the low, subsonic speeds wefly our trainers—so density doesn’t change forus, only velocity.

Figure 1 Velocity in a Venturi

Arrow length indicates velocity. Velocity actually starts rising before air enters the venturi.

(ρAV)A = (ρAV)B = (ρAV)C = constant

A B C

Two-Dimensional Aerodynamics

Bill Crawford: WWW.FLIGHTLAB.NET2.2

Remember Bernoulli?

Bernoulli’s Theorem deals with conservation ofenergy. It tells us that for an ideal fluid(incompressible and frictionless) the total energyof the flow in the venturi remains constant. If weconvert the total energy per unit volume of masstimes flow rate into pressures, the sum of thestatic pressure and the dynamic pressure willequal a constant total pressure. Static pressure isthe ambient pressure exerted by a column offluid at a given level. Dynamic pressure is thepressure exerted by a mass of fluid in motion. Inthe formula below, static pressure is PS. Dynamicpressure is 1/2 ρV2, or one-half the density, ρ, ofthe fluid times its velocity, in feet-per-second,squared. PT is the total pressure:

PS + 1/2 ρV2 = Constant PT

Or in English:

Static pressure + Dynamic pressure = Constant TotalPressure

So if velocity and thus dynamic pressureincreases, static pressure will decrease. In theventuri in Figure 2, the increase in dynamicpressure with velocity produces a decrease instatic pressure as the tube narrows. The staticpressure then rises again as the tube widensdownstream and the airflow slows down.

Of course, air isn’t an ideal fluid: It’scompressible and viscous. Compressibility

obviously becomes important approaching thespeed of sound, but for present purposes can beignored. But we won’t ignore viscosity for long,because the nature of the airflow within theboundary layer over a wing depends on friction.

After looking at the behavior of velocity andpressure in a venturi, it’s tempting to declare thatthe reason airflow accelerates over the top of awing, and static pressure consequently decreases,is that a wing is just one half of a venturi, withthe mass of the atmosphere playing a rollequivalent to the other half. That’s a valid wayof thinking about it, and easy to visualize. Butaerodynamics is a subject in which alternativevisualizations exist side-by-side. A different butnot necessarily contradictory understanding ofthe acceleration of flow has to do with the ideaof circulation. Circulation in turn gives us a wayof visualizing the generation of wingtip vortices,and understanding how the strength of thevortical flow relates to the particular conditionsunder which the wing was producing lift. We’llreturn to this farther on.

Figure 2 Pressure Decreases as Velocity Increases

Longer arrows indicate greater pressure drop relative to the freestream.

Pressure rise



Streamlines

An airfoil section is equivalent to a hypotheticalwing of infinite span (therefore no tips) or to awing model in a wind tunnel, when the modelextends right to the tunnel walls. Because of theabsence of the spanwise flow induced by thepresence of wingtips, no tip vortex and novariations in downwash behind the wing occur.A two-dimensional representation is all you needto depict what’s happening.

The streamlines generated by injecting smokeinto a wind tunnel, or calculated in a computersimulation (Figure 3), allow us to visualize notjust the direction of flow around a wing, but alsoits velocity and pressure. The flow direction atany instant or point along a streamline is alwaystangential to the line. An important feature ofstreamlines is that air particles never cross them;adjacent pairs of streamlines thus behave like thewalls of a flexible tube. When the flowaccelerates, the resulting decrease in staticpressure within them causes the streamtubes tocontract, and the streamlines move closertogether. The distance between streamlines isthus an indication of relative velocity and static

pressure. Notice in the figure how thestreamtubes passing over the leading edgecontract, indicating an accelerating flow and apressure decrease. As they move down the wingthey expand, indicating a decelerating airflowand a pressure rise. This is shown in closer detailin Figure 5.

Note the upwash in the airflow ahead of thewing, and the downwash behind. In the two-dimensional case illustrated, the upwash anddownwash angles are equal. In the three-dimensional case of a finite wing, the tip vortexcan add significantly to the downwash behindthe wing, as we’ll see later.

Upwash ahead of wing

Downwash behind wing

Streamlines move closer together as flow accelerates and static pressure decreases.

Figure 3 Streamlines Upwash/Downwash

Streamlines move apart as flow decelerates and static pressure increases.



Pressure Distribution

Figure 4 shows the surface pressure distributionalong an airfoil section at three different anglesof attack. The longer arrows representincreasingly higher or lower static pressures, asindicated, relative to the static pressure of thefreestream, undisturbed air ahead of the section.The velocity over the forward, upper part of thewing increases as the angle of attack increases,resulting in a greater decrease in local staticpressure, as well as a shifting of the pressurepattern. The illustrations show how the point oflowest static pressure (longest arrow) movesforward as angle of attack, α, increases.

Notice that at low α (as in the top illustration)the static pressure can drop below freestreamunder the wing as well as above. Lift results aslong as the overall reduction in pressure abovethe wing is greater.

NACA 2412 4 Deg. Angle ofAttack

NACA 241212 Deg. Angleof Attack

Decreasedpressure

Decreasedpressure

NACA 24120.3 Deg. Angleof Attack

Figure 4Pressure Distributionversus Angle of Attack

Increased pressure



Figure 5 illustrates streamlines and surfacepressures together. Pressure is highest at thestagnation point (where it equals static plusdynamic pressure). Here airflow comes to a stop,and the streamlines split on either side to followeither the upper or lower surface. (Stagnationpressure is what the pitot tube senses. The staticside of the pitot system eliminates the staticcomponent, and the remaining dynamic pressureappears as an airspeed indication.)

The low pressure generated by the accelerationof the flow over the leading edge creates asuction that draws the airflow forward from thestagnation point—against the general, leading-to-trailing-edge flow. Below stall, any increase in αfurther accelerates the upper-surface flow andfurther decreases the pressure around the leadingedge. Because the increasing suction pullsadditional lower-surface air forward, thestagnation point actually moves aft along thelower surface as α increases. It also moves aftwhen you extend the flaps or deflect an ailerondown.

Stagnation point

Leading edge streamlines and surface pressures of NACA 2412 at 12 Deg. Angle of Attack. Arrows pointing away from surface show relative pressure drop. (The scale is reduced compared to the earlier figures to make the arrows fit the frame. Arrows pointing toward surface show pressure increase.

Figure 5 Streamlines and Surface Pressures



Pressure Gradient

Air flowing over the top of the wing initiallymoves through an area of decreasing staticpressure (Figure 6). Since higher pressures flowtoward lower, this favorable pressure gradientencourages the airflow. But once past the pointof highest velocity and lowest static pressure onthe airfoil, local static pressures begin to rise(although still remain negative) and the pressuredistribution tends to retard the flow. This adversepressure gradient becomes steeper and occupiesmore of the airfoil as the angle of attack, α,increases and the low-pressure over the wingintensifies and shifts forward (Figure 7).

Figure 8 defines the concept of coefficient ofpressure, CP, and shows how its distributionchanges along the chord as α rises. Negativevalues mean local static pressures lower thanfreestream static pressure; positive values meanlocal static pressures higher than freestreamstatic. Note how the adverse gradient increasesover the top of the wing when α rises, asindicated by the increasing negative slope.

As α rises, the adverse gradient increasinglyretards the airflow within the boundary layer,until the boundary layer ultimately separatesfrom the surface, like a sheet blown away frombeneath (Figure 9). As adverse pressure rises theseparation point moves forward; lift drops andthe airfoil stalls.

CP = Local Static Surface Pressure – Freestream Static Pressure/Freestream Dynamic Pressure

5 Deg. AOA Top Pressure 1 Deg. AOA Top Pressure

1 Deg. AOA Bottom Pressure 5 Deg. AOA Bottom Pressure

CP –2.0

CP 0.0

CP +1.0

Leading Edge Trailing Edge

NACA 2412 Coefficient of Pressure, CP

Suction peak. Adverse gradient begins here.

Figure 8 Coefficient of Pressure, CP

Figure 6 Pressure Gradient

Favorable pressure Adverse pressure

Figure 7 Increase in Adverse Pressure with Angle of Attack

Favorable pressure gradient

Adverse pressure gradient resists airflow.

Location of minimum static surface pressure

Figure 9 Separation

Boundary layer separation at stall

Adverse pressure strips the boundary layer from the wing.



Figure 10 Boundary Layer

NACA 2412

Transition from laminar to turbulent flow (separation bubble) is approximately here at a 2 deg. AOA for NACA 2412. Transition point will move forward as AOA increases.

Boundary layer thickness exaggerated. Thickness is about 1% of distance traveled. The point of boundary layer separation moves forward

as AOA increases.

Separation Point: Flow velocity reaches zero, then reverses direction and boundary layer separates from wing.

Turbulent boundary layer

Velocity profile

Laminar Boundary Layer: Laminar shear layers remain distinct, no intermixing.

Boundary Layer

In exaggerated scale, Figure 10 shows theregions of the boundary layer atop the wing, andthe changing velocity profile within theboundary layer, as indicated by the length of thearrows. In both the laminar and turbulentregions, the viscosity of air causes the particlesright next to the wing to come to a halt, due tofriction. Their velocity relative to the wing iszero. The particles flowing immediately aboveare slowed down almost to zero by friction, andthey in turn slow the particles above them.Initially, a profile of shear layers (lamina)develops, with the velocity of the layersincreasing with their distance from the surface asthe effect of friction diminishes. By definition,the boundary layer is the area from the surfaceout to the point where the flow reaches 99percent of the freestream velocity.

The viscous, frictional forces generated withinthe boundary layer are responsible for thecomponent of drag known as skin friction drag.Outside of the boundary layer, viscosity has noimportant effect when it comes to predictingairfoil characteristics.

The initial, laminar region of the boundary layeris very thin. The flow remains layered—there’sno interchange of fluid particles across thelamina. Moving downstream from the leading

edge, the laminar flow gains kinetic energy as itaccelerates through the favorable pressuregradient. The favorable gradient also helps dampout irregularities in the flow. But the flow beginsto slow as it enters the destabilizing resistance ofthe adverse gradient. The laminar boundary layerseparates from the wing, becomes turbulent, andthen reattaches, forming a separation bubble as itmakes the transition. Roughness on the wingsurface can cause the boundary layer to tripprematurely from laminar to turbulent flow.

Whether the laminar flow reattaches as turbulentflow depends on Reynolds number—on the ratioof the inertial forces to the viscous forces withinthe flow. Low Reynolds numbers are associatedwith laminar flow (viscous forces prevail), highReynolds numbers with turbulent flow (viscousforces unimportant, inertial forces dominant). Ifthe Reynolds number is too low, a detachedlaminar flow won’t reattach as turbulent flow.Up to a point, an airfoil produces highermaximum lift at higher Reynolds numbers,because the reattached turbulent flow can betterresist the adverse gradient and remain attached tothe wing at higher angles of attack. Wind tunneldata for airfoil sections often includes curves fordifferent Reynolds numbers and surface textures.(Reynolds number is more complicated, but theabove gives you a start if the concept is new.)



In Figure 11, you can see that the velocitygradient curve in the initial turbulent region(dashed line) starts out shallower than in thelaminar region, but as the curve rises becomessteeper and the boundary layer grows thicker.The turbulent boundary layer produces a lotmore drag than the laminar layer (laminar-flowwings achieve their low drag by moving thelowest pressure point farther back along thechord, thus extending the laminar region). On theother hand, a turbulent layer stays attached to thewing better than a laminar layer as angle ofattack rises, because the turbulence transferskinetic energy down to the surface, which helpsovercome the adverse pressure gradient. (Thisturbulent energy boost is the principle behindvortex generators. The vortices add energy to theflow, delaying separation caused by adversepressure.) Figure 10 shows how the velocityprofile of the turbulent boundary layer changesalong the wing chord as the retarding effects ofthe adverse gradient accumulate.

The tufts on the trainers can give you an idea ofhow the turbulent boundary layer increases inthickness. The tufts themselves would trip anylaminar to turbulent flow. At low angle of attackthe trailing tips of the first row of tufts should liefairly quietly against the surface of the wing, butby the second row the flow will have becomemore turbulent and the tips will shakeperceptibly. The tips will show increasingmovement, bouncing between the wing and thetop of the boundary layer, as you look fartherback along the chord. You can easily see how theboundary layer thickens as it goes downstream.

More important, the tufts show how flowreversal and turbulent boundary layer separationmove up the chord from the trailing edge. You’llbe able to see the effect of the adverse pressuregradient intensify as angle of attack rises. Theturbulent boundary layer’s separation point willmove forward along the chord and the tufts willreverse direction, the free ends actually pointingtoward the leading edge. The reversal of the tuftsin order back up the chord means that the wing isgenerating a larger and larger turbulent wake.Pressure drag is rising rapidly.

Laminar Boundary Layer Initial reattached

turbulent boundary layer, with higher velocity near the surface.

Intermixing in the turbulent boundary area brings high-kinetic-energy (high inertia) particles down toward the surface, increasing the boundary layer’s ability to overcome the adverse pressure gradient. Intermixing also sends low-energy (low inertia) particles up to the top, delaying the return to freestream velocity and causing the boundary layer to become thicker. The higher velocities closer to the surface increase friction in the turbulent boundary layer, and thus friction drag.

Turbulent mixing

Velocity

Thic

knes

s

Figure 11 Velocity Profiles

Adverse Pressure



The Lift Curve

Figures 12 and 13 show typical lift curves, whichplot angle of attack against coefficient of lift(CL). You may have grown accustomed to usingthe term coefficient of lift (or just as happy notusing it) without quite remembering how it’sderived. It’s just an indication of how efficientlyan airfoil shape turns dynamic pressure (definedearlier as one-half the density of the air timesvelocity squared: 1/2ρV2) into lift at any givenangle of attack. S stands for total wing area insquare feet. L stands for lift in pounds.

Looking at the formula, it’s clear that the morelift generated for a given combination ofdynamic pressure and wing area, the greater theCL. The formula for dynamic pressure istypically shortened to q, so that the abovebecomes:

You can determine q in the cockpit, in poundsper square foot, simply by multiplying the squareof the indicated airspeed in miles per hour by0.0025577. That’s q. Then multiply the result bythe wing area, found in the aircraft manual, anddivide the product into the aircraft weight (sincelift equals weight in equilibrium flight). Theresult is your current CL for the wing as a whole(the coefficient at a given section along the span,called Cl to make the distinction when necessary,is often different than that of the wing as awhole). At a constant IAS, your CL must slowlydecrease as you shed fuel weight and need lesslift. Any change in airspeed (thus in q) requires achange in CL for level flight.

It’s good aerodynamics, and good pilotingtechnique, never to think about lift without alsoconsidering its inevitable pal, drag. Substitutingdrag for lift, we derive the coefficient of drag,CD, just as above. D stands for drag in pounds:

In a lift curve, as in Figure 11, the coefficient oflift initially shows a linear increase with angle ofattack. A slope of about 0.1 in lift coefficientincrease for each degree increase in angle is

typical for all two-dimensional airfoil sectioncurves. Variations in slope come from planform,as we’ll see.

The curve begins to shallow and reverse asairflow separation occurs on the upper surface ofthe airfoil. The point where maximum CL versusα is reached (CLmax) marks the stalling angle ofattack.

At CLmax, depending on airfoil shape, airflowseparation may have already reached some 20 to50 percent of the chord. Notice that an airfoilwill still produce lift, and a lot more drag, evenpast the stalling angle of attack.

The airfoil in Figure 12 is symmetrical, andcharacteristically produces no net lift (and itsminimum drag) at zero angle of attack. Acambered airfoil, as in Figure 13, can producelift at small negative angles of attack. Its liftcurve shifts up and to the left (producing a higherCLmax), compared to a symmetrical airfoil’s. Thedrag curve shifts to the right.

- 0 Angle of Attack, α +

Figure 12 Symmetrical Airfoil

Coe

ffic

ient

s of L

ift a

nd D

rag

Symmetrical Airfoil

Cord line

CLmax

CD

CL

SV2/1LC

2Lρ

=

qSLCL =

qSDCD =

CL

CD


Figure 13 Cambered Airfoil

Coef

ficie

nts o

f Lift

and

Dra

g

Camber line

Lift due to AOA

Lift due to camber



Note how the rates of change in the lift and dragcurves vary with angle of attack. The lift curveshows a constant rate until near its peak, whenthe slope diminishes so that a given change inangle of attack produces a smaller change in lift.The drag curve is different. At low angles ofattack, drag doesn’t change much, but as theangle of attack approaches stall, drag increases atan accelerating rate. You learn this, at leastimplicitly, when you learn to land an airplane.The technique of balancing the rates of change oflift and drag at high angles of attack is one ofaviation’s foundation skills. The technique candepend on whether the wing is long or short,swept or straight, as we’ll note.

Airfoil thickness, the amount of camber andposition of maximum camber along the chord,the radius of the leading edge, plus Reynolds andMach numbers, are all factors that affect thevelocities and pressures and therefore theaerodynamic forces generated by an airfoil. Theyalso affect the characteristics of the boundarylayer, its profile and turbulent transition, and itsseparation from the wing at high angles of attack.

Increasing the camber of an airfoil increases itsCLmax. It also tends to increase the adversepressure gradient as angle of attack rises, whichin turn encourages earlier boundary layerseparation, making the section then stall at alower angle of attack, relative to one with lesscamber. A wing in which camber has beenincreased by lowering the flaps will stall at alower angle of attack (See Figure 17).

It’s usually best if the separation of the turbulentboundary layer from the wing surface movesslowly forward along the chord as angle of attackrises near the stall. This allows a relativelygradual, parabolic change in the slope of the liftcurve as the section approaches CLmax, andresults in less abrupt stall characteristics, betteropportunity for aerodynamic stall warning(which also depends on wing planform and taildesign), and less roll-off tendency if one wingstarts to stall before the other. (See Figures 14and 15.)

But abrupt turbulent boundary layer separation issometimes useful. Modern, high-performanceaerobatic wings often have large leading-edgeradii and then become flat from the point ofmaximum thickness back to the trailing edge.The profile of these “ice cream cone” wingsplaces the location of maximum thickness

forward on the chord, which limits the possibleregion of favorable pressure gradient and causesearly laminar-to-turbulent boundary layertransition. Compared to wings designed forbetter laminar flow, they tend toward higherdrag. Drag helps speed control in vertical downlines, but the major aerobatic benefit is thetendency for the stall separation point to remainnear the trailing edge as angle of attackincreases, and then suddenly to move forward upthe chord. This allows the wing to hold airflowattached during high-g maneuvers, but stallabruptly for quick snap roll, spin, and tumblingentries. Wing tufts show how rapidly theseparation point advances and how quickly thestall breaks, as you’ll see in our ground-schoolvideo.

- 0 Angle of Attack α +

Coe

ffic

ient

of L

ift

Stall Buffet/Warning Onset

Stall

Good stall warning margin, less potential roll-off

More gradual CL change with α

Figure 14 Gradual Stall



Leading-Edge Stall

A thin airfoil with a sharp leading edge radiuscan suffer sudden leading-edge stall (orsimultaneous leading-edge and trailing-edgestall) due to the sudden bursting of the leading-edge separation bubble. The bubble occurs wherethe laminar flow separates from the wing andreattaches as turbulent flow. As angle of attackrises, the bubble follows the suction peak (Figure8) forward. As the curvature of the wingincreases, the detached laminar flow can nolonger “make the turn” necessary to reattach asturbulent flow. The bubble bursts, causing rapid,complete boundary layer separation along theentire chord. This sharpens the peak of the liftcurve, with the result that, near stall, a smallchange in angle of attack produces a sudden dropin CL. This can lead to an abrupt stall withlimited or no aerodynamic warning, and to asudden roll-off when inevitable variations inwing surface or contour cause the separationbubble to burst on one wing ahead of the other.

Depending on how the airfoil section variesalong the span, it’s possible for one part of awing to have a trailing-edge stall, while anotherpart has a leading-edge stall, or demonstrates acombined leading-edge/trailing-edge stall.

Learjet wings built before the introduction of theCentury III wing section had roll-off problemsdue to asymmetrically bursting separationbubbles. A stick pusher was necessary to keepthe wing out of stall territory. In addition toadding inboard stall strips and stall fences,improving the pusher-off stall characteristicsinvolved breaking the bubble into stable,spanwise segments by mechanically tripping thelaminar flow with small triangular shapesattached to the outboard leading edge, ahead ofthe ailerons. Placed on the chord ahead of thenormal laminar separation point, the trianglescaused the flow to become turbulent andreattach, preventing a spanwise, continuousseparation bubble from forming and thus frombursting. This dramatically reduced roll-off atstall.

Separation bubble (greatly exaggerated) with circulating flow.

Bursting the separation bubble can cause leading edge stall and abrupt boundary layer separation.

- 0 Angle of Attack α +

Coe

ffic

ient

of L

ift

Rapid CL change with α. Poor stall warning and potential roll-off.

Figure 15 Abrupt Stall

Laminar flow separates.

Reattaches as turbulent flow.



Flaps

When you start to lower the flaps on our trainers,you’ll see the tufts begin to show airflowseparation along the trailing edge. The flapsincrease local wing camber, which adds a secondlow-pressure peak (as shown in Figure 16) and asteep adverse pressure gradient at the trailingedge. If you increase the flap deflection further,or increase angle of attack, the adverse gradientover the flap becomes more severe. Flowreversal and separation occur and the tufts startdancing.

By studying Figure 16 you’ll gain some insightinto how flaps (and ailerons and rudders) work.Fundamentally, they modify lift by changing thevelocity of the airflow. Notice how the increasein camber over the rear of the airfoil affects theoverall pressure pattern. Camber increases localflow velocity, which decreases local surfacestatic pressure. The result is a more favorablepressure gradient immediately forward of thedeflected surface, which in turn causes air toaccelerate over the wing ahead, resultingultimately in lower surface pressures at theleading edge. At the same time, airflow below

the wing slow down, resulting in higherpressures there. (Notice the rearward shift instagnation point that follows flap deployment.The flap-induced increase in leading-edgesuction pulls more air forward from beneath thewing, sending the stagnation point further aft.)

Figure 17 shows how the lift curve changes withflaps of different types. The Fowler flap is themost efficient because it produces the greatestincrement in lift with the least increment in drag.Flaps shift the lift curve up and to the left as youincrease the deflection angle. The shift is theresult of the increase in camber. With theexception of the Fowler flap’s curve, whichbecomes steeper, the slope of the lift curveremains unchanged with flap deployment.Maximum CL increases with flaps, but occurs ata lower angle of attack than when the flaps areup. The zero-lift angle of attack becomes morenegative (nose down).

Figure 16 Flap Deployment

Stagnation point

Stagnation point

Favorable pressure gradient

Adverse gradient



Because it reaches a higher maximum coefficientof lift, CLmax, with flaps extended, the wing willstall at a slower speed for a given aircraft weight.The benefit is disproportional, however, becauselarge increases in CLmax are necessary to gain anysubstantial decrease in stall speed. For example,a 50 percent increase in CLmax produces only an18 percent decrease in stall speed. A 100 percentincrease in CLmax reduces stall speed by 30percent, but at the expense of a large increase indrag.

Any type of flap is less effective on a thin wingthan on a wing of greater thickness. Flaps arealso less effective on swept wings compared tostraight wings when their hinge line follows thesweep angle. You’ll often see that the flapsthemselves are not swept on an otherwise sweptwing.

Arrow shows direction of shift as flap deployment increases.


Figure 17 Lift Curve shift withFlaps

Coe

ffic

ient

s of L

ift

Fowler

Slotted

Plain

Basic section, flaps up

Split

Plain Flap

Split Flap

Fowler Flap

Slotted Flap



Circulation

Airflow passing over the top of a wing speedsup, while airflow passing beneath the wing slowsdown. These changes in velocity producepressure differences between top and bottom,and thus lift.

Figure 18 shows what happens when yousubtract the freestream flow from the acceleratedflow over the top of the wing and from thedecelerated flow beneath. The result reveals anembedded circulatory flow. (The circulation hasa positive value above the wing; a negative valuebeneath.)

Circulation of this type doesn’t mean thatindividual air particles actually travel completelyaround the wing, only that a circulatory tendencyabout the wing exists at any moment. Thecirculation is outside the boundary layer andextends well above and below the wing.

Although the idea of circulation pre-dates theWright brother’s first powered flight (thebrothers weren’t aware of it) and has greatmathematical use, it’s never been popular as ameans of explaining lift to pilots. Imaginingcirculation around a wing isn’t easy. It’s easier tothink of airfoils in wind-tunnel terms, the airmoving in streamlines past a fixed airfoil thataccelerates the air by means of an easilyvisualized venturi effect. In terms of the forcesand moments produced, it doesn’t matterwhether the air or the wing moves. But in actualflying it’s of course the wing that does thetraveling. At subsonic speeds, it telegraphs itsapproach by the pressure wave it sends ahead,and the flow field starts accelerating ordecelerating even before the wing arrives. Try toshift your frame of reference and to think interms of the effects that the wing carries alongwith it, as it whizzes by, and of the effects itleaves behind in originally stationary air.

Circulation depends on airfoil design (leadingedge radius, maximum thickness and its location,maximum camber and its location). Circulationincreases with angle of attack up to the stall, andalso with wing camber as modified by flaps. Anaileron deflected down increases circulation overthe affected part of the wing. One deflected upreduces circulation.

Other factors remaining equal, the greater thecirculation the greater the velocity difference

above and below the wing, and thus the greaterthe pressure difference and the resulting lift.

The circulation needed to produce lift of a givenvalue increases as airspeed or air densitydecreases. The pertinent formula, in plainEnglish, is:

Lift = Density x Freestream Velocity x Circulation x Span

Freestream component

Circulatory components: The flow around the wing minus the freestream

Figure 18 Circulation

Actual airflow velocities



Bound Vortex/Tip Vortex

The circulation around an airfoil is called thebound vortex (Figure 19). On an actual three-dimensional wing, as opposed to a two-dimensional section, the bound vortex in effectturns the corner and becomes a trailing, tipvortex. The tip vortex is no longer an embeddedflow bound to and carried along by thecoordinates of the wing, but now a “free” or“true” vortex that remains attached to the samefluid particles and continues circulating longafter the wing is gone. Because the tip vortex isgenerated by the pressure difference between thetop of the wing and the bottom, and by thetendency of the airflow to try to even out thatdifference by “leaking” around the wingtip, tipvortex strength is a function of circulation.

At low speeds and high angles of attack, whencirculation necessarily rises, tip vortices increasein intensity. The heavier the aircraft andtherefore the greater the required lift (lift equalsweight in steady flight) the stronger thecirculation has to be at a given airspeed.Consequently, heavy airplanes flying at slowapproach speeds produce the most dangerous tipvortices. One reason to describe circulation is togive pilots a better sense of how potentiallydangerous tip vortices are generated, and howtheir strength depends on the components of thelift formula:

Lift = Density x Freestream Velocity x Circulation x Span

As the formula indicates, the circulation neededto generate lift equal to aircraft weight is also afunction of wingspan. The longer the span theless circulation required. Consequently, at thesame aircraft weight, long wings produce lessintense tip vortices than short wings. This in turnlowers induced drag, as we’ll see.

In principle, a vortex can’t suddenly terminate inmid air. It has to form a continuous enclosedloop. Figure 19 shows the starting vortex thatcompletes the loop. Figure 20 shows the role thestarting vortex plays in getting circulation going.

Bound Vortex Trailing Tip Vortex

Figure 19 Vortex System

Starting Vortex



The Kutta Condition

The top of Figure 20 shows the theoreticalstreamline flow around an airfoil that wouldoccur in the absence of friction and the resultingviscosity. The streamlines coming out frombeneath the wing reverse direction. The rearstagnation point lies on the top of the wing,forward of the trailing edge.

That can’t happen in nature, however, becausethe viscosity of the air prevents it from makingthe necessary sharp turn. Instead, as a wingbegins to move forward from a standstill, astarting vortex forms. Because nature also saysthat the total circulation within any arbitrarilydefined area must remain constant, an oppositevortex begins to form, the bound vortex. Thewing begins to develop the circulation describedearlier, and the airflow leaves the trailing edgesmoothly, as shown in the bottom illustration.The starting vortex is left behind. This smoothdeparture is known as the Kutta condition. Theupshot is that as angle of attack, camber, orairspeed change, the wing develops whatevercirculation is necessary to maintain the Kuttacondition. An increase in angle of attack, forexample, causes a new starting vortex to formand be left behind. If angle of attack isdecreased, a stopping vortex of opposite sign isformed and shed. A stopping vortex forms if theaircraft accelerates (since circulation required forlift goes down as airspeed increases). Ultimately,the vortices left behind break down due tofriction and turbulence.

Bound vortex

Starting vortex

Figure 20 Kutta Condition

Inviscid Flow

Real viscid airflow requires meeting the Kutta condition. The circulation generated causes the front and rear stagnation points to shift as shown.

Stagnation point

In the absence of viscosity, the low pressure ahead draws the rear stagnation point forward.

Stagnation point

Starting vortex circulation

But in Viscid Flow

Starting vortex induces circulation of the bound vortex.

+ = Constant

Defined area

Flightlab Ground School 2. Two-Dimensional Aerodynamics · Bernoulli’s Theorem deals with...

Documents

Transcript of Flightlab Ground School 2. Two-Dimensional Aerodynamics · Bernoulli’s Theorem deals with...