LIFT AND DRAG ON INFINITE SPAN AIRFOILS
LIFT AND DRAG ON INFINITE SPAN WINGS
I. Terminology.
The following picture illustrates terminology relevant to an airfoil, i.e, a wing cross-section. We assume the section is everywhere the same. (See Text Figure 3.2, p. 32.)
maximum camber – maximum distance between mean camber line and chord line
maximum thickness – maximum distance between upper and lower surface
An airfoil is symmetrical if the chord line and mean camber line are coincident.
Some of the most important airfoil parameters are
1. Shape of mean camber line
2. maximum thickness
3. location of maximum thickness
4. leading edge radius (curvature of leading edge)
II. Aerodynamic Force.
A lifting force on a wing develops when air passes over the wing, reducing static pressure on both sides (according to Bernoulli’s Principle). To create an upward pressure differential, the distance air travels along the upper surface must be greater than along the lower surface. (See Text Figure 3.5, p. 35.)
PU is pressure on upper surface; VU is air velocity across upper surface; distU is upper surface distance.
PL is pressure on lower surface; VL is air velocity across lower surface; distL is upper lower distance.
Figure 3.6 (text p. 36) provides additional insight into the creation of an upward lifting force on a wing.
[pic]
Text Figure 3.6, p. 36.
Aerodynamic Force (AF) – the vector result of forces acting on a wing in a relative wind at an angle of attack (AOA) ( (small alpha).
Note: here lift and drag are due to the wing only, and do not take into effect lift and drag created by other parts of the a/c. Also, since the span is infinite, there is no induced drag produced. (We will discuss induced drag later in the course.) See Text Figure 3.7, p. 36.
Definitions
• Flight Path – vector describing the magnitude and direction of the aircraft through the air mass; thus, a vector equal to the velocity vector of the aircraft.
• Relative Wind (RW)– a vector opposite and equal to the flight path vector; i.e., the speed and direction of the air passing over the aircraft (wing).
• AOA (() – angle between the chord line and the relative wind.
• Aerodynamic Force (AF) – as above; lift and drag are respectively the components perpendicular and parallel to the RW.
• Center of Pressure (CP) – point on an airfoil section where the AF acts.
• Laminar Flow – smooth, “parallel” flow of air over the wing; also called streamlined flow.
• Turbulent Flow – “opposite” of laminar flow, where streamlined flow breaks up.
Note: Figure 3.7 (p 36, text) is a little misleading, since it seems to imply that the AF is the source of all lift and drag on an airplane. This certainly isn’t true. We’re talking here just about forces created by infinite span wings, not by whole airplanes.
III. Pitching Moments Created by Lift
A moment M (ft-lbs) is a force F (lbs) applied to an object at a distance d (ft) from a fulcrum f. For a body to be at equilibrium, there can be no unbalanced moments acting on it. In Cartesian 3-space, moments can be resolved around the x, y, and z axes. Also, as we have seen previously, there can be no unbalanced forces along these axes acting on a body at equilibrium.
Thus we have the following properties of a body at equilibrium. Note that this is an extension of an earlier definition which neglects moments and applies to 2-space only:
(Fx = (Fy =(Fz = 0, and
(Mx = (My =(Mz = 0
Example. In the system illustrated below, the weight (force) x must be 66 2/3 # for the system to be in equilibrium. Note that if we assume the Cartesian z-axis comes out of the page at point f (fulcrum), then the balanced moments are around the z axis, i.e., (Mz = 0.
(Mz = 0
M+z = M-z
(100 #) (10 ft) = (x) (15 ft)
(1000/15) # = x
66 2/3 # = x
Since the CPs for upper and lower airfoil surfaces are often in different locations relative to the leading edge, pitching moments can develop when the AF is created. Note that these pitching moments develop for an asymmetrical airfoil, but not for a symmetrical airfoil, as illustrated in Figure 2.17 on the next page (text p. 33).
[pic]
(e)
Text Figures 3.9 & 3.10, and 3.11, p. 38.
Suppose the following values hold Text Figures 3.9 & 3.10, d/e.
Calculate moments around leading edge of wing.
Nose down pitching moment: (6 ft) (10,000#) = 60,000 ft-#.
Nose up pitching moment: (2 ft) (5,000#) = 10,000 ft-#.
Net pitching moment is 50,000 ft-# nose down.
M+z = M-z
50,000 ft-# = (5,000 #) d
10 ft = d.
That is, the CP is 10 ft aft of the leading edge. The nose down pitching moment can be counteracted by applying a nose up moment at the horizontal tail.
M+z = M-z
50,000 ft-# = (28 ft) T
50,000/28 = T
1785.71 # = T
That is, 1786 pounds of downward lift must be applied at the horizontal tail to counter the nose down pitching moment created by the AF.
Exercise. Suppose the net lift developed by this airplane is 18,000 # acting at the CP 10 feet aft of the leading edge of the wing. Find the nose up force on the horizontal stabilizer 40 feet aft of the leading edge necessary to counteract the nose down pitching moment created by lift. _____________#.
IV. Aerodynamic Center.
The CP is not much used for calculations by aerospace engineers because
• CP experiences significant chordwise movement as AOA varies
• For cambered airfoils, a non-zero pitching moment still exists at zero lift, raising a paradox (contradiction), since then the moment arm d of the pitching moment L must have length d = (, since L = 0, and the pitching moment (d L) is non-zero.
As a consequence, aeronautical engineers devised the concept of the aerodynamic center (AC). The AC is the point on the airfoil (% of the chord) where lift acts and, for a given velocity, the pitching moment due to lift is constant. For subsonic airspeeds, position AC varies very little (23-27%) of chord aft of leading edge.
Key points about AC
• Lift acts at the AC
• Pitching moment is constant at AC for constant a/s
• 23-27% aft of leading edge for subsonic flight
• 50% aft of leading edge in supersonic flight (discussed later)
V. Lift Equation
Lift (and associated drag) depend mainly on
• Dynamic pressure
• Air density ratio (
• Planform wing area S in ft2
• Shape of airfoil section
• Air viscosity
• Compressibility effect
• AOA
Recall that airspeed and air density determine dynamic pressure q: Since ( = ( (0
q = ( V2 / 2 = (0 ( V2 / 2, with (0 a constant, and V in fps.
It is convenient to express lift (and drag) in terms of coefficients of lift (CL) and drag (CD). These dimensionless coefficients (we will see later) are a function of AOA. The lift equation is
L = CL q S = [pic],
where CL and q are as previously defined, and S is planform area in ft2. V is in ft/sec. If knots are to be used, then
L = [pic].
Solving for V, we have
[pic].
295.37 is a “magic number” which you will not need to memorize. The above formula for L will be given on quizzes and exams where it is needed to perform a calculation. However, you should be able to solve for V.
The coefficient of lift CL varies directly with AOA, with (CL)max (and maximum lift) occurring at the stall AOA (stall speed). That is, maximum lift is developed at the onset of stall. Figure 4.2 (text p. 44) shows a plot of CL vs. AOA for a wing having a symmetrical airfoil.
[pic]
Text Fig 4.2, p. 44.
Important Point. For a given a/c configuration (e.g., clean or dirty), stall always occurs at the same AOA. (Also, for a given configuration and weight, stall always occurs at the same EAS. More on this later.)
Assuming L = W in straight and level flight (approximately true in most flight conditions), the stall speed is
[pic].
From this, we see that stall speed in a given configuration varies directly with the square root of gross weight and inversely with the square root of density, since ( = ( / (0.
VI. Implications of Lift Equation for Steady State Flight
Wings Level Unaccelerated Flight (G = 1)
Let V1 and V2 be any two true airspeeds for a given aircraft. From the lift equation, again assuming that L = W, we see that when only lift (weight) varies
[pic].
In a similar manner, if only density varies
[pic].
Finally, if only CL varies (e.g., because of a configuration change),
[pic]
Turning Accelerated Flight (G > 1)
Since G = L / W = 1 / cos (, L = W / cos (. Thus, if only bank angle and G change,
[pic].
Since the cosine of an acute angle decreases as the angle increases, we can conclude that increasing angle of bank increases stall speed, a fact well known to pilots and one which has accounted for many stall-spin incidents in the past, and will account for many more in the future.
VII. Summary of Equations and Their Application
[pic]
These equations will not be given on quizzes or tests. You must be able to reconstruct them from the lift equation given above. An easy way to do this for the first three is as follows. First solve the lift equation for V, giving a radical on the right hand side containing symbols L (= W), CL, and (. Now realize that if the quantity that varies is in the numerator of the radical (W = L) in the lift equation, then W2 goes in the numerator of the radical in the V2 / V1 equation, and otherwise ((, CL), it ((2 or (CL)2) goes in the denominator. (This is true just if the left hand side of the equation is V2 / V1.)
Example. Suppose an airplane stalls at 130 kts TAS at 22,000#. What is its stall speed at 18,000# in the same configuration?
[pic]kts TAS
Note that TAS stall at the same density altitude decreases (increases) with a decrease (increase) in gross weight. Since EAS = TAS at sea level in a standard atmosphere, this helps you remember that EAS stall speed also varies with gross weight. On the other hand, the stall AOA corresponding to (CL)max remains constant for the same aircraft configuration.
Example. Suppose an airplane stalls at sea level in a standard atmosphere at 120 kts TAS, for a given gross weight. What is the EAS stall speed at sea level and the EAS and TAS stall speed at 20,000’, in a standard atmosphere at the same gross weight and in the same configuration?
Since EAS = TAS at sea level in a standard atmosphere, the stall speed EAS at sea level is 120 kts. At 20,000’, the density ratio (from the standard atmosphere table given in Part I of the notes) is 0.5328. Thus
[pic]kts TAS.
At 20,000’, [pic]kts EAS. However, we did not actually need to perform this calculation, since we know that an aircraft in steady state wings level flight at a constant gross weight always stalls at the same EAS.
Note that if there is no pitot-static system error or compressibility correction, the IAS = EAS, and an aircraft in steady state 1G flight always stalls at the same IAS. However, although the airspeed you see in the cockpit doesn’t vary, the TAS (speed through the air mass) increases quite dramatically as altitude increases as density
Example. The density ratio at 35,000’ in a standard atmosphere is 0.3099. Thus an airliner cruising at 525 kts TAS is actually indicating (assuming IAS = EAS) only 525 ((0.3099) = 292 kts. And it flies like it’s going at this slow speed rather than the high speed. Suppose the stall speed indicated of this airplane is 140 kts at sea level in a standard atmosphere. (This of course is the EAS stall speed at any altitude.) But at 35,000’ (still assuming IAS = EAS), the TAS stall speed is 140 / ((0.3099) = 251 kts. Be sure you understand all the implications of this example.
Example. Suppose an airplane has (CL)max = 1.2 in the clean configuration, and 1.55 in the dirty configuration. If the TAS clean configuration stall speed is 150 kts at sea level in a standard atmosphere, what is the EAS stall speed in the dirty configuration under the same conditions? What is the TAS and EAS stall speed in the dirty configuration at 10,000’ in a standard atmosphere and at the same gross weight?
At sea level in a standard atmosphere, EAS = TAS, so
[pic]kts TAS (EAS)
At 10,000’, the EAS dirty stall speed dirty is 132.0 kts, the same as at sea level. Since the density ratio is 0.7385 at 10,000’, the TAS dirty stall speed at 10,000’ is
[pic]kts TAS.
Example. Suppose gross weight is 20,000#, wing planform area S = 400 ft2, and (CL)max = 1.15 in the clean configuration, and 1.35 in the dirty configuration. Find the sea level stall speed in both configurations.
[pic]=113.3233158.
There are at least two ways to compute Vdirty (they are mathematically equivalent):
1) [pic]= 104.5926844.
2) [pic]= 104.5926844.
Example. Suppose an aircraft stalls wings level at 120 kts at sea level in a standard atmosphere. Find the sea level stall speeds and G forces in the same configuration at 30, 45, and 60 degrees angle of bank.
Since [pic],
[pic] and
[pic].
Note that since cos 0 = 1, when V1 is the wings level stall speed, then V2 = V1 / ((cos (). Observe too that the EAS stall speed is 120 kts at all altitudes under consideration.
Exercise (This is the form in which exam questions typically appear!).
An airplane has a gross weight of 18,000 #. Its sea level dirty configuration TAS stall speed is 105 knots, and the best power off glide speed is 125 knots. (CL)max = 1.5 in the clean configuration and 2.1 in the dirty configuration. The density ratio at 15,000 feet MSL is 0.6292. Assume a standard atmosphere. Each problem should be considered to be independent of any other problem; i.e., start each problem using the basic data given above.
A. Find the TAS dirty configuration stall speed at 15,000 feet. _____________ kts.
B. Find the EAS dirty configuration stall speed at 15,000 feet. _____________ kts.
C. Find the sea level TAS stall speed in the clean configuration. _____________ kts.
D. Find the sea level TAS dirty configuration stall speed at 22,000 # gross. _____________ kts.
E. Find the sea level TAS dirty configuration stall speed at 75o angle of bank. _____________ kts.
F. Find the sea level TAS dirty configuration stall speed at 15,000 # gross. _____________kts.
VIII. Effect of Weight on Clean Configuration Power-Off Best Glide Speed
The best glide speed is the speed that gives the maximum gliding distance. Recall from Unit I Notes (p. 6) that D = W sin a and L = W cos a, where a is the glide angle. Thus we have
tan a = (W sin a) / W cos a = D / L
from which it follows that the glide angle a is minimum when D/L is minimum, or L/D is maximum
( (L/D)max ). Then, since the maximum glide distance results when the glide angle is minimum, the best glide angle occurs at (L/D)max. (Compare Text Figure 13.1, p. 204.)
Also, from the above diagram, since tan a = (absolute altitude) / (glide distance) = D/L, the glide distance at the glide angle a is
glide distance = absolute altitude / (D/L) = (absolute altitude) (L/D).
This means the best glide distance (achieved at (L/D)max) is
Best glide distance = (absolute altitude) ( L/D)max ).
(We will discuss (L/D)max and the Lift-Drag Curve later in the course.)
Facts about Best Glide Angle
• Minimum glide angle is independent of W (this is counterintuitive!)
• An a/c glides the same distance from the same altitude regardless of weight
• The best glide speed varies with weight, but AOA for best glide speed is constant
Example. Suppose (L/D)max for an airplane is 12.0 in the clean configuration and 5.0 in the dirty configuration. Then the maximum glide distance from 10,000’ AGL in the clean configuration is
GD = (absolute altitude) (L/D)max = 10,000’ (12.0) = 120,000’ = 120,000 / 6076 = 19.75 nm.
In the dirty configuration, the maximum glide distance from 10,000’ is (10,000) (5.0) / 6076 = 8.23 nm.
Exercise. Suppose (L/D)max for an is 10.22 in the clean configuration and 6.75. in the dirty configuration. Find the glide distance clean and dirty from 7,500’ AGL in statute miles and nautical miles.
Exercise. Suppose a plane glides 30 miles from 15,000’ AGL clean, and 20 miles dirty. Find (L/D)max in the clean and in the dirty configuration.
IX. Types of Drag and Drag Equation
Drag developed by a wing is usually categorized as being either parasite drag or induced drag. A third type of drag is shock wave drag, which develops in transonic and supersonic flight. Induced drag is drag created when lift is created by a finite span wing. (We discuss induced drag in detail in Unit Three.)
Parasite drag of course is developed on parts of the aircraft other than the wing, as follows:
Parasite Drag. Results when an aircraft produces resistance to airflow. Includes some of the drag associated with producing lift, but not induced drag. There are four subcategories of parasite drag.
1. Skin friction drag—caused by viscous air in any boundary layer (wing, fuselage, tailplane, engine nacelle, &c.)
2. Form drag (also called pressure drag)—caused by air piling up in front of a non-streamlined aircraft surface
3. Interference drag—caused by interference of boundary layers from different part of the aircraft.
4. Leakage drag—caused by air flowing from high to low pressure, as for example from inside a pressurized fuselage to outside, or from lower to upper wing surface, e.g., where flaps attach to wings.
The drag equation for a wing is similar to the lift equation. We discuss drag again later in the course.
D = CDqS = CD(V2S / 295.37.
ANGLE OF ATTACK INDICATOR
• A vane or slot aligns with the relative wind
• Measures angle ( between RW and wing chord line
• Cockpit instrument graduated in units (vice degrees) for universal use
• Sometimes used in association with “donut” indicator (called “stall indexer” in text)
Uses of AOA Indicator
• Stall, max range, max endurance AOAs fixed for given a/c and configuration regardless of weight or altitude
• Handy way to stay on correct speed: just fly AOA
• Can be used with auto throttle. In that case, attitude controls airspeed and throttle controls vertical speed (rate of climb or descent). (This is not intuitive and is the same as in a “manual” approach.) Note: you may hear a contrary analysis in AS310, involving the ostensible difference between how Air Force and Navy pilots control an aircraft during approach to landing.
STALLS
I. Causes and Types of Stalls
When air passes over a wing, or other smooth surface, a boundary layer (BL) develops, in which friction from the surface slows airflow in the immediate vicinity of the surface to a relative velocity of almost zero. As distance from the surface increases, relative air velocity also increases. Points where relative velocity is equal to the free airflow lie on the upper surface of the boundary layer.
[pic]
Figure 4.7 (text p. 49)
BL on a typical wing varies from 1/250 of an inch at the leading edge to ¾ of an inch near the trailing edge. A wing is stalled when the boundary layer begins to separate significantly from its upper surface. This usually occurs toward the trailing edge of the wing first, as shown in the illustration below. For a given configuration, stall always occurs at CL(max) and at the same AOA.
Kinds of Stalls. Three types of stalls have been identified.
1. Conventional (1-G flight) as described above.
2. Accelerated—sudden increase of AOA at high speed causes high G forces. BL separation also occurs at the leading edge of the wing.
3. Shock wave induced stall (discussed later in course)—at transonic airspeeds (Mach 0.9-1.3), BL separates aft of a shock wave which forms due to air compression. Typically causes a pitching moment on an aircraft.
[pic]
Figure 17.4 (text, p. 282)
Often (but not always), stalls cause a nose down pitching moment due to the AC being aft of the aircraft CG (center of gravity). To understand stalls better, must consider boundary layer theory.
II. Boundary Layer Theory.
Toward the end of the 19th century, Osborne Reynolds studied fluid flow along a tubular surface. He found out that it is possible to predict, with some precision, where laminar flow becomes turbulent flow.
• Laminar flow—smooth fluid flow composed of parallel layers.
• Turbulent flow—parallel fluid layers begin to intermix, causing turbulence.
Figures 4.5 and 4.6 (text p. 90) illustrate the difference between these two flow types. Note also in Figure 4.5 that laminar flow becomes turbulent flow as flow distance increases.
[pic]
Figure 4.5 (Text P. 49)Boundary layer composition
[pic]
Figures 24.6 (Text p. 49) Smoke pattern
Reynolds Number Re can be used to predict when laminar flow becomes turbulent.
Re = Vx /( where V is flow velocity in ft/sec, x is flow distance downstream in feet, and ( (small nu) is kinematic viscosity.
Viscosity of a fluid is its resistance to flow. Kinematic viscosity is viscosity taking into consideration the density of the fluid resisting flow. The kinematic viscosity of air increases with altitude, lowering Re.
Re increases as V increases and/or as the fluid flow distance x increases. Thus, on a wing, points near the trailing edge have a higher Reynolds number than points near the leading edge.
Effect of High Re
High Re is associated with turbulent flow flow, and low Re with laminar flow, as shown in Figure 4.9 (text p. 51).
[pic]
Figure 4.9 (Text p. 51)Reynolds number effects on airflow on a smooth flat plate
Figure 4.9 shows that skin friction drag is higher in turbulent flow than in laminar flow. However, total drag can actually be less, as shown in Figure 2.33 (Text 1st edition, p. 46). Also, high Re and associated turbulent flow energize the BL and delay BL separation, hence increase CL and lift by allowing higher AOA before stall, as shown in Figure 2.27 (Text 1st edition p. 42).
Figure 2.33 Reynolds number effects on CD Figure 2.27 Reynolds number effect on CL-( curves
Turbulent vs. Laminar BL Flow on a Sphere
The relationship between turbulent flow and delayed BL separation is vividly illustrated by instances of smooth and dimpled spheres in airflow. Laminar flow on a smooth sphere causes early BL separation, and a wide wake, which increases total drag. A dimpled sphere, e.g. a golf ball, creates turbulent flow and delays BL separation, resulting in less total drag, even though skin friction drag is higher. See Figure 5.3 (text p. 53)
[pic]
Figure 5.3 (Text p. 53) Sphere wake drag: (a) smooth sphere; (b) rough sphere
Turbulent BL Flow on a Wing
Airflow separation on the aft portion of a wing is promoted by an adverse presser gradient. This causes the BL flow to slow and separate from the wing, as illustrated in Figures 4.10 and 4.11 (text pp. 51-52).
[pic]
Figure 4.10 (Text p. 51) Adverse pressure gradient
[pic]
Figure 4.11 (Text p. 52) Airflow separation velocity profiles
This situation can be remedied by increasing turbulent flow, which delays BL separation and the onset of stall. Vortex generators on airliner wings (tabs extending from upper wing surface parallel to airflow) are an example of one method used to achieve this desirable effect. Turbulent flow “energizes” the “dead air” on aft portion of upper wing surface, as reflected in Figure 4.8 (text p. 50).
[pic]
Figure 4.8 (Text p. 50) Velocity profiles for laminar and turbulent flow
HIGH LIFT DEVICES
Vortex generators fall into the category of high lift devices and are just one of several approaches to delaying BL separation.
1. Vortex generators—as described above. See Figure 4.15 (text p. 55).
2. BL Control—blow air compressed by engine along wing upper surface. See Figure 4.18 (text, p. 57).
3. Leading edge slots—high-pressure air from below wing flows upward through slot and along upper surface of wing (free BL control). See Figures 4.16 – 4.17 (text p. 56).
[pic]
Figure 4.15 (Text p. 55)-Vortex Generators
Figure 4.18 (text p. 57). Effect of blowing BLC on CL-( curve.
Figure 4.16 (text p. 56). Fixed slot operation.
Figure 4.17 (text p. 56). Effect of “energy adder” on CL-( curve.
-----------------------
upper surface
leading edge
trailing edge
lower surface
mean camber line chord line
(drawn halfway between (straight line from leading to
upper and lower surfaces) trailing edge)
leading edge radius
PU VU distU distU > distL
VU > VL
PU < PL
Vin
Vout
PL VL distL
L AF
D
relative wind
(
100 # f x #
10 ft 15 ft
10,000#
5,000#
(net lift)
6’ d’
CP
2’
5,000#
28’
5000 #
10’
T #
D
glide path (x-axis)
L
W cos a
W sin a
W
a (glide angle)
a
a
absolute
altitude
glide distance
(
1. On speed (plus or minus 1.5 knots)
2. Slightly slow (1.5- 3 knots slow)
3. Slightly fast (1.5- 3 knots fast)
4. Very fast (more than 3 knots fast)
5. Very slow (more than 3 knots slow)
1 2 3 4 5
[pic]
[pic]
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