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Groundschool – Theory of Flight
Aerofoils and wings
Revision 58a — page content was last changed 18 August 2012
|The lift force is generated by a small pressure differential between the upper and lower surfaces of the wing, caused by the aerodynamic reaction to the wing motion through the atmosphere. The magnitude of the pressure differential, and the consequent momentum applied to the airflow, is generally dependent on the speed of the aircraft, the angle of attack and the physical characteristics of the wing. The wing centre of pressure moves fore and aft in response to changes in the aerodynamic reaction, thereby introducing pitching moments that affect the aircraft's trim. Drag induced by the generation of lift is modified by the plan form, the twist and the aspect ratio of the wing. Ailerons, flaps, and other lift and drag changing devices are fitted to the wing for control and performance purposes.|
Basic forces' module it was stated that when an aircraft is moving through the air, the consequent pressure changes or aerodynamic reactions to its motion will be acting at every location on its surface. We had a look at the formula for calculation of lift from the wings:
(Equation #1.1) Lift [ newtons] = CL × ½rV² × S
It is usual to substitute the symbol 'Q' to represent dynamic pressure [½rV²] so the expression above may be more simply presented as:
(Equation #4.1) Lift [newtons] = CL × Q × S
where Q × S is a force.
It is appropriate to state here that the formula is an approximation of the average lift from the wings. At any one time, the aerodynamic reactions will vary over the span of the wing and with the position at which the wing control surfaces are set.
The aerodynamic force has two sources: the frictional shear stress, or skin friction, that acts tangential to the surface at every point around the lifting body; and the pressure exerted perpendicular to the surface at every point. (At speeds over about 250 knots, flow compressibility introduces other factors.) The resultant net aerodynamic force is the sum of all those forces as distributed around the body. For wings, it is conventional to show the resultant force as acting from an aerodynamic centre and resolved into two components: that acting perpendicular to the flight path is the lift, and that acting parallel to the flight path is the drag. For propeller blades, the aerodynamic reaction is resolved into the thrust component and the propeller torque component. For rotor blades, a more complex resolution is necessary.
Note: normally the aerofoil is incorporated into a wing with upper and lower surfaces enclosing the load bearing structure. However, when designing a low speed minimum aircraft such as the Wheeler Scout there are advantages in using a 'single surface' cambered aerofoil wing, very similar to a hang glider wing. Such wings incorporate a rounded leading edge (formed by the aluminium tubing leading edge main spar) that directs the airflow into the upper and lower streams at all angles of attack. The slight camber is formed by battens sewn into sleeves in the 'sails'. Such wings are somewhere between a thin curved plate and a full aerofoil, and are similar in cross-section to a bird's wing. A parachute wing uses the ram air principle to form the aerofoil shape — see 'The ram-air parachute wing'.
Now we need to establish how that airflow actually produces the lifting force. John S Denker has published a web book 'See How it Flies' that has a particularly good section on lift generation with excellent illustrations. You should carefully read through section 3 'Airfoils and airflow' and particularly acquaint yourself with the Eulerian approach of 'streamlines' to visualise airflow. In the illustrative diagram at left, narrowing (A) of streamlines indicates accelerating local speed and decreasing local pressure — a favourable pressure gradient. Opening up (D) of streamlines indicates flow deceleration and increasing pressure — an adverse pressure gradient. The term 'free stream' is usually substituted for 'flight path' when discussing aerofoil characteristics because the aerofoil is presumed stationary, as in a wind-tunnel, and the airstream flows around it.
The following summarises the content of section 3 of 'See How it Flies':
• A flat plate, held at a small aoa, will generate an aerodynamic force — lift and drag — and indeed, some low momentum aircraft do use basically flat plates as their tailplane surfaces. As mentioned above, the shape of sail-type wings is somewhere between a plate and the more usual wing. However, for aircraft that cruise in the 50–150 knot range, a wing with a rounded leading edge, a sharp or square-cut trailing edge, a cambered upper surface and a flat or slightly cambered bottom surface — i.e. a full aerofoil section — will be far more efficient — aerodynamically and structurally — and more effective in performance. (The faster the aircraft, the more the aerofoil section tends to flatten out. So, for supersonic aircraft we are nearly back to the sharp-edged flat plate.)
• A cambered wing will still produce lift at zero, and slightly negative, geometric angles of attack, as shown in the lift coefficient diagram. The aoa where no lift — only drag — is produced is called the zero-lift aoa which, in the diagram, is nearly –2°. From that diagram you can infer that camber contributes a lift coefficient of about 0.2 and anything greater must be provided by aoa. Of course, this will vary with the amount of camber in a particular aerofoil. If the aoa was reduced below the zero-lift value, for example –4°, then the direction of lift would be reversed. The only time you would need such a negative aoa is when you are flying inverted, or performing aerobatics, neither of which are currently allowable in aircraft registered with the RA-Aus.
At the zero-lift aoa, all the aerodynamic force is acting parallel to the free stream and is mostly skin friction drag, with a less significant amount of pressure drag but the latter will increase as the aoa is increased. Pressure drag is explained in section 4.7 'Parasite drag'.
Cambered wings perform quite well in inverted flight, but are not as efficient as in normal flight because a higher aoa is needed to make up for the lower wing surface having the maximum camber when inverted. For this reason, aerobatic aircraft tend to use symmetrically shaped aerofoils — i.e. the 'camber' of the bottom surface balances the 'camber' of the top surface and aerodynamically the result is zero camber — thus such wings rely purely on the geometric aoa to produce lift.
• At positive angles of attack there is a stagnation point, or line, just under the leading edge of the aerofoil where some of the airflow has been brought to a standstill. The air molecules reaching that line, in the incoming stream, are equally likely to go under or over the wing. Stagnation pressure, the highest in the system, exists along the stagnation line. The location moves down and under the leading edge as aoa increases, up to the stalling aoa. Another more confined stagnation point exists at the trailing edge. If an imaginary line is drawn between the two stagnation points, the cross-sectional view of the division of the aerofoil into upper and lower flow areas becomes apparent.
• The behaviour of the airstream flowing around such a wing accords with Bernoulli's principle. As the air accelerates away from the stagnation line, the local airflow over the upper surface gains a greater speed than the lower. Consequently, to retain constancy, the static pressure on the upper surface will decrease, and on the lower surface it may decrease very slightly at low aoa but will increase as aoa increases.
There is another concept for explaining the pressure differential between upper and lower wing surfaces. Leonhard Euler was a mathematician who was a contemporary of, and collaborator with, Daniel Bernoulli. The Euler Equations (a special case of Newton's Third Law of Motion) express the relationship between flow velocity and the pressure fields in frictionless flow. Because the air particles follow the curved streamlines above the upper surface, there must be a centripetal force across the streamlines that accelerates the flow towards the centre of curvature. That force must be associated with a pressure gradient across the streamlines; i.e. ambient atmospheric pressure at some distance from the surface, grading to a lower pressure on the upper wing surface. For more information enter the terms 'Euler curvature airfoil OR aerofoil' into a search engine.
• The usual way of looking at the lift force is that the wing produces an upflow in the air in front of it and a downwash behind it. That downwash continuously imparts momentum — with a downward velocity component — to the air affected by the passage of the aircraft. As you will recall from the 'Basic forces' module the action of adding downward momentum will have an equal and opposite reaction, which in this case is an upward force applied to the wing. And, of course, the energy provided to impart momentum to the air comes from engine power; in a glider it would come from the gravitational potential energy of height. There is a distinction between the 'downflow' produced by the aerofoil and the additional 'downwash' produced by wing vortices (see below), the deflection of which increases with angle of attack. However, for our purposes we can treat all the momentum imparted to the airstream as 'downwash'.
You will also recall, from the 'Basic forces' module, that thrust is the reaction from the momentum imparted to a tube of air with the diameter of the propeller. The associated slipstream or 'prop wash' is the added momentum — quite apparent if you stand behind a stationary aircraft when 'running-up' the engine. Helicopter rotor blades are long, slender rotating wings — somewhere between variable pitch propeller blades and normal wings — and the momentum applied to the air — the 'rotor wash' — can be seen clearly by its effect on dust, vegetation and other objects (like parked ultralights) beneath a hovering helicopter. Similarly, a wing producing lift continuously accelerates a flattened tube of air with diameter approximating the wing span; the longitudinal downward inclination to the flight path of that flat tube increases as aoa increases. Some liken that concept to the wing acting as an airscoop.
• Another concept associated with the aerodynamic force — circulation theory — is a mathematical description of a 'bound vortex', which also fits in with the generation of the physical wing-tip vortices. Vorticity is rotary motion in a fluid, and you could regard 'circulation' as referring to the apparent flow rotation — upwash then downwash — around the upper/lower surfaces.
Note: there is a long-held and still-continuing argument, particularly in newsgroups and other internet venues, about the pros and cons of the various lift generation theories. None of the arguments put forward (often ill-informed) affect in any way how a light aircraft flies, how it should be safely and economically operated, or how it should be built; so it is best to ignore them unless you are particularly interested in the science of aerodynamics and skilled in mathematics.
Although it is still small in comparison with the ambient atmospheric pressure, it is this pressure differential resulting from the wing deflecting the air that initiates the lifting force; and this is true however lift theory may be expounded. Much work has been done in designing aerofoils that will maintain the required pressure difference in the targeted flight conditions.
We can calculate the net pressure difference for the Jabiru using the scenario in the 'Basic forces' module section 1.4; i.e. cruising at 6500 feet, airspeed 97 knots or 50 m/s, air density 1.0 kg/m³. The ISA atmospheric pressure at 6500 feet is about 800 hPa:
When the aircraft designer calculates the CL curve for an aircraft it must be related to a particular wing reference area. This may be the visible plan area of the wings but it could also include that area of the wings conceptually enclosed within the fuselage.
Note that the CL for an aerofoil will have a value perhaps 10–20% higher than the CL for any wing incorporating that aerofoil; this is discussed in the spanwise pressure gradient section. (The convention is to use a lower case 'L' [thus Cl ] when referring to the lift coefficient for an aerofoil to distinguish it from the lift coefficient for a wing, but I have retained CL for both.)
In level, non-manoeuvring flight, lift equals weight, so equation 4.1 can be restated as:
(Equation #4.2) CL = W / (Q × S)
The usable value of CL in a very light aircraft with low-aspect ratio wings without lift-enhancing devices might range between 0.1 and 1.6. (Unless it is a symmetrical aerofoil — same camber top and bottom — the lift coefficient range will be different for the same wing when in inverted flight.)
However, a very low CL value can be obtained momentarily if the wings are 'unloaded' in flight. This can be achieved by applying sufficient continuous forward pressure on the control column to attain a near-zero aoa such that the net pressure differential between the upper and lower wing surfaces is very low. This would imply low lift generation and reduced drag, so the thrust will accelerate the aircraft a little faster than normal.
Furthermore, a negative CL can be obtained by maintaining so much forward pressure on the control column that the aerodynamic force is reversed. If initially flying straight and level, the aircraft will 'bunt'; i.e. enter the first few degrees of an outside loop with the centripetal force for the turn being supplied by the reversed lift. (This reverses the direction of the wing loading and should never be attempted in weight-shift aircraft nor three-axis aircraft unless the three-axis manufacturer's flight manual allows such a manoeuvre.) And, of course a suitably equipped aircraft can be flown in inverted level flight — in which case the under-wing surface becomes the upper and a completely different CL range applies, because the cambered surface is now underneath and a higher aoa is necessary to maintain the lift required for level flight.
Incidentally many pilots utilise the low CL technique when landing a taildragger. The application of forward pressure on the control column after touchdown 'pegs' the aircraft down by reducing the aoa and thus generated lift, and thereby puts increased pressure on the tyres, and amplifies friction and any braking force applied. The same technique was used to bring military DC3 aircraft to a quick stop.
Lift = CL × Q × S
I suggest you try out what you have learned so far in an aerofoil flight test simulation program. You need a Java-enabled browser. Read the instructions carefully and reset the measurement units from pounds to newtons. In this case, airspeed will be shown in km/h but just mentally divide by two (and add 10%) to get knots — halve it again if you want m/s.
You can try this simple model out with a popular aerofoil, the NACA 2412, which is one of a series dimensioned by the U.S. National Advisory Committee for Aeronautics (the forerunner of NASA) in the 1920s and 1930s. The 2-4-12 (twenty-four twelve) has a camber of 2%  of chord with maximum camber occurring at 40%  of chord from the leading edge and a thickness/chord ratio of 12% .
Note that all dimensions are proportional to the chord so the same aerofoil section shape is retained throughout a wing even if it is tapered in plan form. The wing is thickest at the root and thinnest at the tip; i.e. it must also be tapered in thickness. Most aerofoils suitable for light aircraft have a camber of 2–4%, thickness ratio of 12–15% and the maximum thickness (not camber) occurring at around 30% of chord.
Now type the following data into the FoilSim boxes using the 'enter' key or use the sliders:
Size: chord 1 m, span 8 m (area 8 m²)
Shape: angle (of attack) 2°, camber 2%, thickness 12%
Flight test: speed 166 km/h (90 knots), altitude 1947 m (6400 feet)
Check the results displayed in the black boxes and in the plots. The static air pressure should be 80.0 kPa (800 hPa) and the lift is 4233 N. If you select 'surface pressure' from the output plots, you will see a plot of the pressure distribution across the chord for the upper (white line) and lower (yellow line) surfaces. Anything appearing above the green line (the atmospheric static pressure) can be regarded as a positive pressure pushing that surface at that point. Anything below the green line is a negative pressure pulling that surface at that point. The area between the two curves represents the magnitude of the differential pressure distribution. The horizontal axis indicates the percentage distance from the mid-chord position.
The pressure gradient plot for the upper surface shows a maximum decrease of around 1.5 kPa (15 hPa) close to the leading edge but changing to a slight positive increase in pressure at the trailing edge. The pressure gradient plot for the lower surface shows an increase in pressure under the leading edge, quickly changing to a decreased pressure of a few hPa then back to a positive pressure from mid-chord back. If you press the 'Save Geom' button, a data table will be displayed showing the pressure and local velocity readings at 19 X-Y coordinate positions on both the upper and lower surfaces.
If you now select 'surface velocity' for the output plot, you will see a plot of the local velocity distribution across the chord for the upper (white line) and lower (yellow line) surfaces. You can see that the local velocity increases to about 40% above the free stream velocity a very short distance downstream from the leading edge, then it gradually slows until local velocity is less than free stream velocity at the trailing edge.
Now change the airspeed to 110 km/h (60 knots) and the aoa to 12°, and look at the surface pressure and surface velocity plots again. Note the big increase in local velocity that is now some 2.5 times the free stream velocity a very short distance downstream from the leading edge. Also note the big increase in the pressure differential and that most (about 70%) is occurring within the first 25% of the chord.
You should do a little exploration starting with the aerofoil design, changing just one value at a time and noting the changes in the upper and lower pressure gradients. For instance change the camber from 2 to 4% (i.e. the NACA 4412 aerofoil) and see the lift generated increase to 6369 N with a CL now 0.74. You can do the same with the flight performance items under pilot control — aoa, altitude and airspeed. Of course, FoilSim doesn't provide any information concerning drag generation or pitching moment.
The innermost molecules of the moving air come into contact with the solid surface of the wing (and other parts of the aircraft) and are entrapped by the surface structure of the airframe materials. This is called the 'no-slip condition' and is common to all fluid flows. The interaction between those air molecules and the molecules of the solid surface transfers energy and momentum from the air molecules to the solid surface molecules — producing skin friction drag and shear stress that act tangentially to the surface. Those surface-interacting air molecules retreating from the surface consequently carry less momentum than they did on approach. In the very thin viscous sublayer adjacent to the solid surface, these molecules with reduced momentum move randomly into the fluid a small distance from the surface. The streamwise momentum per unit volume of the molecules that have interacted with the surface is less than the momentum a small distance from the surface. The random mixing of the two groups of molecules reduces the streamwise momentum of the molecules that have not directly interacted with the surface. This exchange of momentum between slower and faster molecules is the physical origin of air viscosity (the resistance to flow when a fluid is subject to shear stress) and of that viscous sublayer or boundary layer comprising the region between the wing surface and the unrestrained or inviscid outer stream. The diagram shows the velocity gradient within the boundary layer; the more turbulent the flow, the steeper the gradient and the greater the shear stress and friction.
The atmospheric boundary layer is similar but, of course, on a grander scale.
The extent of laminar flow and thus the location of the transition zone — where boundary flow is a mix of laminar and turbulent — depends on the designed aerofoil shape in profile, the angle of attack, contour variations (ripples, waviness) formed during construction and service, the flexibility of the wing's skin, surface roughness/cleanliness, porosity, and the pressure gradient along the wing chord. In the area where the pressure gradient is favourable (i.e. decreasing, thus the flow is accelerating), laminar flow will tend to continue, though becoming thicker, unless something trips it into the more irregular turbulent boundary layer flow — even paint stripes can trip laminar flow.
The laminae nearest the skin move slowly and cohesively, thus minimising skin friction drag. In the turbulent flow boundary layer, the air nearer the wing is moving faster and somewhat chaotically, thus greatly increasing skin friction drag. The transition zone tends to occur a particular distance downstream (for a combination of the preceding factors) rather than a percentage of chord even though the aerofoil might be designed for laminar flow for a particular percentage of chord.
The aerofoils used for light aircraft wings have very little laminar flow. But specialised high-speed aerofoils are designed to promote laminar flow over perhaps the first 30–40% of the wing chord by providing a favourable pressure gradient for at least that distance (i.e. maximum thickness at 40–50% of chord) and a properly contoured, very smooth, clean, non-flexing, seamless skin. The latter conditions are also important for minimising the thickness of the turbulent boundary layer flow with consequent reduction in skin friction drag and are achievable in composite construction. pitching moment. Downwash disappears and the rate of loss of lift will increase rapidly as the aircraft slows.
Aerodynamicists devote much effort to controlling and energising the boundary layer flow to delay separation and thus allow flight at lower speeds; for example, see vortex generators. More lift and much less pressure drag is generated in attached turbulent boundary layer flow than in partially separated flow.
Aspect ratio = wing span² / wing area
It is conventional to use the symbol 'b' to represent span, so the equation above is written as:
(Equation #4.3) A = b² / S
The Jabiru's aspect ratio (span 7.9 m, area 8.0 m²) = 7.9 × 7.9 / 8 = 7.8, whereas an aircraft like the Thruster would have an aspect ratio around 6. Consequently you would expect such an aircraft to induce much more drag at high angles of attack, and thus slow much more rapidly than the Jabiru.
And incidently, the mean chord (not the mean aerodynamic chord) of a wing is span/aspect ratio. A high-performance sailplane wing designed for minimum induced drag over the CL range might have a wingspan of 22 m and an aspect ratio of 30, thus a mean chord of 0.7 m. There are a few ultralight aeroplanes, designed to have reasonable soaring capability, that have aspect ratios around 16–18, but most ultralights would have an aspect ratio between 5.5 and 8, and averaging 6.5. General aviation aircraft have an aspect ratio between 7 and 9, probably averaging around 7.5. Note that the higher the aspect ratio in powered aircraft, the more likely is wingtip damage on landing.
Note that 'wing area' includes the nominal extension of the wing shape into and through the fuselage. This would appear quite apt for a parasol wing or a high-wing aircraft, but will no doubt seem odd for a mid or low wing. It is just a means for consistent application/comparison between aircraft designs.
The span loading is the aircraft weight divided by the wingspan = W/b. The term sometimes refers to the loads applying at specified stations along the span.
Where these two surface airflows with different spanwise velocities recombine past the trailing edge, they initiate a sheet of trailing vortices. These are weakest near the fuselage and strongest at the wingtips, and roll up into two large vortices, centred just inboard and aft of each wingtip. The vortices increase in magnitude as aoa and lift increase, and so increase the vertical component of, and the momentum imparted to, the downwash. As the centre of each vortex is a little inboard of the wingtip, the vortices also have the effect of reducing the effective wing span, the effective wing area and probably the effective aspect ratio.
The vortices also affect the air ahead of the aircraft by reducing the magnitude of the upflow in front of the wing and thus modifying (decreasing) the effective wing aoa, with the greatest effect near the wing tip and little effect near the wing root. When a wing is at a low CL aoa the airstream affected by the wing has a slight downward flow. When it is at maximum CL aoa, that airstream has a more substantial downward flow contributed by the vortices.
Because of the reduction in the effective aoa, the wing must fly at a greater aoa to achieve the same lift coefficient that a two-dimensional aerofoil will achieve in the laboratory. Also, the wing tip vortices have a decreasing effect with increasing aspect ratio. This is demonstrated in the diagram where there are three (exaggerated) CL and aoa curves plotted. On the left is the laboratory curve for an aerofoil, in the middle the curve for a high aspect ratio wing utilising the same aerofoil and the curve on the right is for a low aspect ratio version. The red horizontal line connects with a particular CL value, say 1.2. The vertical red lines indicate a different aoa for each curve at the same CL, thus the high aspect ratio wing must fly at a higher aoa and the low aspect ratio wing must fly at a still higher aoa for either to achieve CL 1.2. Or to put it another way, at any aoa the wings produce less lift than the laboratory aerofoil.
Also apparent from the diagram is that a higher aspect ratio has the effect of a higher rate of lift increase, as aoa increases, than lower aspect ratio wings. A high aspect ratio wing will have a higher CLmax but a lower stalling aoa than a low aspect ratio wing utilising the same aerofoil. Induced drag has a direct relationship to aspect ratio; see section 4.6.
Wing-tip vortices make up most of the wake turbulence created by an aircraft in flight and are certainly the most hazardous to following aircraft. They are usually referred to as wake vortices in the context of air traffic and are the same as other atmospheric vortices in that there is a central low pressure core that is often visible as condensation trails when an aircraft pulls higher g in a humid atmosphere. Read the New Zealand Civil Aviation Authorities booklet 'Wake Turbulence'.
Induced drag is least at minimum aoa and greatest at maximum aoa. It is often said that the induced drag is the energy dissipated to induce lift; i.e. if CL is increased, induced drag increases, so thrust must be increased to provide additional energy — if the aircraft's flight path is to continue as before. For example, if the pilot wants to increase aoa and maintain the same airspeed (as in a constant rate level turn), then thrust must be increased to counter the increase in induced drag.
There is a point in an aircraft's flight envelope where, because of the increasing induced drag, the slower you want to fly the greater the power you must apply — known as 'flying the back of the power curve' — which is opposite to the norm of applying power to fly faster.
Induced drag is minimised if the spanwise distribution of the lift forces can be made to present an elliptically shaped pattern, as shown in the diagram, and that aerodynamic load is equally distributed over the wing so that all areas of the wing contribute to load sharing. (This idealised lift force distribution diagram presents a head-on view of the whole wing without any representation of — or distortion by — the fuselage.) .
Elliptical spanwise lift distribution will provide a desirable uniform downwash along the span, and can be achieved by choice of wing plan form and/or by twisting the wing to provide something near an elliptical distribution in a speed band selected by the designer.
High aspect ratio elliptically shaped (in plan form) wings generally achieve spanwise elliptical lift distribution; however, because of the compound skin curvatures they are the most difficult and time-consuming to construct. Low aspect ratio constant chord (i.e. rectangular) wings without twist are the easiest to construct but generate the most induced drag; however, the introduction of twist makes such a wing much more efficient. Medium aspect ratio wings with a medium taper ratio plus twist are probably the most used shape.
Taper ratio is the ratio of the tip chord to the wing root chord. 'Medium taper' would indicate that the tip chord is greater than 50% of the root chord.
Sailplane designers have demonstrated that the most effective high aspect ratio wing is one that has a straight (i.e. non-tapered) trailing edge with a leading edge that is increasingly tapered in sections from root to tip. incidence, 3–4° or so, and thus lower aoa than the inboard sections in all flight conditions. The main reason for wing twist is to reduce induced drag (see section 'Elliptical lift force distribution') and particularly so at a cruising angle of attack or perhaps the climb speed angle of attack. Another reason is to improve the stall characteristics of the wing so that flow separation begins near the wing roots and moves out towards the wingtips.
With twist, the sections near the wing root reach the stalling aoa first, thus allowing effective aileron control even as the stall progresses from inboard to outboard. This is usually achieved by building geometric twist into the structure by rotating the trailing edge, so providing a gradual decrease in aoa from root to tip. Washout reduces the total lift capability a little but this disadvantage is more than offset by the wing twist improving elliptical lift distribution and thus decreasing induced drag.
Another form of washout — aerodynamic twist — might be attained by using an aerofoil with a higher stalling aoa in the outboard wing sections.
Aircraft incorporating washout tend to not drop a wing during an unaccelerated stall. Instead, there is a tendency to just 'mush' down sedately then drop the nose and regain flying speed. The turbulent wake from airflow separation starting at the wing root buffets the tailplane, thus providing some warning of the oncoming stall before it is fully developed. Also, washout is usually applied, for aerodynamic balance, to the swept wings utilised in weight-shift ultralights. However, geometric washout can cause problems at excessive speed.
Induced drag = (k × CL² / A) × Q × S where A is the wing aspect ratio [b²/S] and k is related to a span effectiveness ratio.
So, induced drag is directly proportional to CL² and inversely proportional to dynamic pressure [Q], and might comprise 50% of total drag at maximum angle of climb speeds. The lower the span loading [W/b](i.e. the greater the physical span or the 'effective' span), the lesser the induced drag at all angles of attack. This results in a decrease in the thrust needed, particularly for climb — or an increase in the potential energy of height for a sailplane. Various wingtip designs, such as Hoerner wingtips, have the effect of moving the vortices slightly further outboard, thereby increasing the effective span and thus reducing the span loading and induced drag.
The information in the following box may only be of interest to aircraft homebuilders, so skip it if you wish and go to the next part .
Induced drag = k × CL² / A × ½rV² × S
For the Jabiru, k = 1/(3.14 × 0.8)= 0.4, aspect ratio [A] is 7.8 and the CL at that speed is 0.4.
= 0.4 × (0.4 × 0.4 / 7.8) × (0.5 × 1.0 × 50 × 50) × 8.0
= 0.4 × 0.02 × 1250 × 8 = 80 newtons
If you repeat the CL calculation in section 1.4 using the Jabiru's stall speed at 6500 feet, say a TAS of 25 m/s, you will find that CLmax is 1.6. Now if you repeat the induced drag calculations, you will find it has increased fourfold:
Induced drag = 0.4 × (1.6 × 1.6 / 7.8) × (0.5 × 1.0 × 25 × 25) × 8.0
= 0.4 × 0.33 × 312.5 × 8 = 330 newtons
boundary layer air flow section above. The parasite drag constitutes much of the total aircraft drag at minimum aoa (i.e. high speed) but comparatively little at maximum aoa (minimum speed). Refer to the diagram in section 1.6. When associated with airflow around an aerofoil, the parasite drag is termed profile drag.
Pressure drag or form drag is the net pressure differential of those points on the wing; for example, where a component of the pressure acts in the fore and aft direction, and that pressure differential tends to retard the aircraft. Pressure drag, like skin friction, applies to all parts of the aircraft 'wetted' by the airflow. It is greatest for any part of the airframe that presents a flat surface perpendicular to the flow and least for a streamlined shape that has a fineness ratio (i.e. length to breadth) between 3:1 and 4:1.
There are two specially named classes of parasite drag: interference and cooling drag. Interference drag occurs at the junctions of airframe structures; for example, the junction of the wings and fuselage or the junction of the undercarriage legs and fuselage. The boundary and outer streamflows interfere with each other at the intersections and cause considerable turbulent drag. Interference drag for a well-designed composite aircraft might be 5–10% of total parasite drag but can be very much higher. The cross-flow associated with unbalanced flight (slip/skid) exacerbates interference drag.
If interference drag potential is ignored by the designer, vortex development can occur at the wing/fuselage junctions, effectively splitting the spanwise lift distribution into two separate elliptical patterns; this is particularly so with low-wing configurations but not so much with high wings. The problem is minimised, and total parasite drag considerably decreased, by careful design to reduce the number of junctions, and to use fillets and fairing to direct a smooth airflow around the remainder. Usually the most visible evidence of an interference drag reduction program is the large wing root fillet used in low wing aircraft as seen in the AR-5 photograph.
Engine cooling drag is normally associated with the cooling airflow for engines enclosed in a drag reducing cowling. The cooling airflow is designed to be efficiently directed from an air intake through a system of baffles for optimum engine cooling, and perhaps to utilise the energy of the added heat to provide a little thrust at the cowling exit point. Where the engine is not cowled, there is a great deal of parasite drag that certainly cools the engine but would not be specially classed as cooling drag.
section 1.6 you will see that the drag varies with the airspeed which means, of course, that it varies with angle of attack. The diagram on the left is a plot of the fixed lift value divided by the total drag value; i.e. the L/D ratio, at varying aoa for a reasonably efficient aircraft. It can be seen that L/D [L over D] improves rapidly between zero or negative aoa up to 4–5° then drops off until the stall angle, where the deterioration rate accelerates. Note that a non-aerobatic light aircraft in normal flight would not experience these low L/D values at aoa between 0° and 2°.
The maximum L/D for light aeroplanes — a measure of the aerodynamic efficiency of the aircraft — is possibly between 8 and 12. Some of the ultralights designed with wide span, high aspect ratio wings to provide some soaring capability have a maximum L/D around 30. High-performance sailplanes that are built with very wide span, slender, high aspect ratio wings have the greatest L/D, at 40 –50, and thus the greatest efficiency. Powered parachutes have a L/D ratio around 3.
There is a limit to the thrust that the engine/propeller can provide (i.e. the drag that it can match) thus there is also a minimum L/D at which maximum engine power is required to maintain constant altitude. Consequently, there will be a minimum aoa (maximum airspeed) and a maximum aoa (minimum airspeed) at which an aircraft can maintain level flight. As there may not be much range between minimum and maximum L/D, the minimum L/D can be quite significant for ultralight aircraft, where a range of engines, some with rather low power, may be utilised in the same model. An under-powered aircraft will perform very badly at the back of the power curve.
We can use the '1-in-60' rule to calculate the angle of the glide path relative to the ground; for example:
L/D = 12, then 60/12 = 5° glide path angle.
If the aircraft is maintained in a glide at a degraded L/D, then the glide path will be steeper: L/D = 8, then 60/8 = 7.5° glide path angle. This is one effect of using flaps (see section 4.11).
Be aware that quoted L/D ratios rarely take into account the considerable drag generated by a windmilling propeller.
The aoa associated with maximum L/D decides the best engine-off glide speed [Vbg] for distance and the best speed for range [Vbr] according to the operating weight of the aircraft. But because of the flat shape of the curve around maximum L/D, these speeds are more akin to a small range of speeds rather than one particular speed.
FoilSim aerofoil flight test simulation program, the static pressures around the aerofoil are given in the output plot that shows the pressure distribution pattern changing with the aoa. It is convenient to sum that distribution and represent it as one lift force vector acting from the centre of pressure [cp] of the aerofoil or wing for each aoa; much the same way as we sum the distribution of aircraft mass and represent it as one force acting through the centre of gravity. The plot on the left is a representation of the changing wing centre of pressure position with aoa. The cp position is measured as the distance from the leading edge expressed as a percentage of the chord. (Please note the diagram is not a representation of the pitching moment.)
At small aoa (high cruise speed) the cp is located around 50% chord. As aoa increases (speed decreases) cp moves forward reaching its furthest forward position around 30% chord at 10–12° aoa, which is usually around the aoa for Vx, the best angle of climb speed. With further aoa increases, the cp now moves rearward; the rate of movement accelerates as the stalling aoa, about 16°, is passed. Most normal flight operations are conducted at angles between 3° and 12°, thus the cp is normally positioned between 30% and 40% of chord.
The movement of the cp of the lift force changes the pitching moment of the wing, a rotational force applied about some reference point — the leading or trailing edges for example — which, in isolation, would result in a rotation about the aircraft's lateral axis. The consequence of the rotation is a further change in aoa and cp movement that, depending on the cp starting position may increase or decrease the rotation. Thus a wing by itself is inherently unstable and will change the aircraft's attitude in pitch — i.e. the aircraft's nose will rotate up or down about its lateral axis, which may be reinforced or countered by the action of the lift/weight couple — so there must be a reacting moment/balancing force built into the system provided by the horizontal stabiliser and its adjustable control surfaces. This will be discussed further in the Stability and Control modules.
The pitching moment is consistently nose-down, changing in magnitude as airspeed changes. When plotted on an aerofoil wind tunnel data graph, the moment coefficient Cmc/4 is a roughly horizontal line for most of the angle of attack range, but the straight line may have a slight slope if the actual aerodynamic centre varies a little from the 25% chord location.
Pitching moment equation:
(Equation #4.6) Pitching moment [ Mc/4 ] = Cmc/4 × ½rV² × S × c
The pitching moment equation is much the same as the lift and drag equations with the addition of the mean aerodynamic chord [c] for the moment arm; using SI units the result is in N·m. As the coefficient is always negative and nearly constant (up to the stall), then V² is the significant contributor to the nose-down pitching torque, which must be offset by tailplane forces to keep the aircraft in balanced flight. However, high torsion loads may still exist within the wing structure; see aerodynamic effects of flight at excessive speed.
The concept of the aerodynamic centre is useful to designer/builders, because it means the centre of application of lift can be assumed fixed at 25% chord and only the lift force changes. For non-rectangular wings, a mean aerodynamic chord [MAC] for the wing has to be calculated; see ascertaining mean aerodynamic chord graphically — in that diagram the aerodynamic centre position [ac] is shown on the root chord line. module 6), and so do parts of a well-designed fuselage. Consequently the aerodynamic centre for the aircraft as a whole, known as the neutral point, will not be in the same location as the wing aerodynamic centre but — for a tailplane aircraft — behind it and on the fuselage centreline. This is the fixed point from which net lift, drag and aircraft pitching moment are assumed to act.
section 1.4 that the pilot cannot change the shape of the wing aerofoil. But this, like many statements made regarding aeronautics, needs qualification. In fact, the pilot manoeuvres the aircraft in the lateral plane by altering the effective camber of the outboard sections of the wings. And remember in the last paragraphs of section 4.1 above, using FoilSim, we found that altering camber from 2% to 4% produced a substantial increase in CL and lift.
If you examine the Seafire photograph, in section 4.6, you will see that each wing has a separated section at the outboard trailing edge. These are ailerons, hinged to the main wing so that they can move down or up and linked, via control rods or cables, to left/right movement of the pilot's control column. The control column is a simple lever which amplifies forces applied by the pilot. Thus the pilot can, in effect, increase or decrease the camber of the outer portion of each wing; as shown by the effective chord lines in figures A and B at left. The ailerons are interconnected so that downward movement — a camber increase — in one is combined with an upward movement — a camber 'reflex' — in the other. The aileron movement then increases the lift generated by the outer section of one wing whilst decreasing that from the other, thus the changed lift forces (at a distance from the aircraft's longitudinal axis) impart a rolling moment in the lateral plane about that axis. This rolling moment is primarily used to initiate a turn but other manoeuvres depend on the amount and timing of aileron movement; more about this in the 'Control' module; see 'Control in a turn'.
Ailerons span perhaps the outer 35% of each wing and occupy perhaps the aft 20% of the wing chord at that location. High-speed aircraft may have two sets: a normal outer wing set used only for low-speed flight (because of the moment of force they are capable of applying at high speed) and a second, high-speed set of spoiler-type ailerons located at the inboard end of the wing. moment arm.
Aileron drag can have an opposite yaw effect. When an aircraft is turning at low speed and the pilot applies aileron to roll upright, the downwards movement of the aileron on the lower wing might take the aoa, on that part of the wing, past the critical aoa. Thus that section of wing — rather than increasing lift and making the wing rise — will stall and lose lift. The aircraft, instead of straightening up, will roll into a steeper bank. Although the wing section may be stalled, CL and thus induced drag will still be fairly high, so there will be a substantial yaw toward the lower wing which pulls the nose down and increases the rate of descent. There is potential for other aileron-induced problems when turning at low speeds; see 'Control in a turn'.
There are a number of configurations which, used singly or jointly, reduce aileron drag. For example, differential ailerons, where the down-going aileron moves through a smaller angle than the up-going aileron or Frise ailerons, where the leading edge of the up-going aileron protrudes below the wing undersurface, increasing parasite drag on the down-going wing.
Seafire photo, are the flaps. Plain flaps are also a hinged section of the wing — as in figures C and D in the aileron diagram above — but move only (and jointly) downward usually to fixed predetermined positions, each position providing varying degrees of increased lift coefficient and increased drag coefficient that the designer thought appropriate. For instance, for one particular aircraft, at 5° deflection there is a good increase in CL with only slight increase in drag. At 15° the drag increase starts to equate with the increase in the CL, whereas at 25° or 30° the increase in drag is much greater than the increase in CL; at 45° the flap is starting to act as an airbrake.
The change in camber (over perhaps 50–60% of the wing span and 20–25% of the wing chord) caused by lowering flaps in flight, without changing other control positions, has effects which will vary according to the amount of deflection employed:
As we saw in FoilSim, the effect of increasing camber is an increase in CL (the ratio of lift to dynamic pressure or airspeed) at all aoa. This is shown in the plot at the left. At an aoa of 6° CL is about 1.0 with flaps lowered — about 50% greater than the CL of 0.65 with flaps raised. What this means is that the minimum controllable flight speed is lower with flaps deployed.
So, returning to the equation:
lift = CL × ½rV² × S
thus for lift to remain constant if CL increases then V² must decrease. Consequently, the stall speed is also lower with flaps deployed.
(Incidently, this diagram shows that the zero lift aoa for this wing occurs at –2°.)
Note that the flapped section will stall at a lower aoa than the unflapped section. Generally the flapped wing area, being the inboard section of the wing, represents a very large proportion of the total wing area — check the Seafire photo. Also, even if the flapped section has passed its stalling angle, it is still producing lots of lift. Providing there is sufficient thrust available to overcome the big increase in drag, the aircraft can still maintain height and stability because the wing outboard section and ailerons are not stalled.
Bear in mind that to maintain the same airspeed and altitude after lowering flaps, that thrust, if available, must be increased to counter the additional drag from the lowered flaps. Similarly, when flaps are raised, the aircraft will initially sink due to the loss of lift unless the pilot takes compensating control action; this is particularly important when a landing approach is discontinued and a go-around initiated.
Now what aoa are we measuring? If you look at figure C (in the drawing in section 4.10) which represents the unflapped part of the wing, you can see that it has an aoa of about 5° or so whereas, at the same time, the flap extended section of wing (figure D) has a considerably greater aoa. As the flapped section will still have a stalling aoa around 16° we can surmise that this flapped wing section is going to stall when the unflapped section is only at 13° or so. (The horizontal axis of the plot shows only the aoa of the unflapped wing.) However, we also have to take into account the increased downwash and thus the change in effective aoa associated with it, so the effect of flaps is not as straight-forward as implied in the preceding.
The most common (because of its simplicity) is the plain flap, which might provide a 0.5 increase in CLmax with a large increase in drag when fully deflected. The split flap provides slightly more increase in lift but a larger increase in drag, and is more difficult to construct and thus probably not worth the effort.
The slot incorporated into the junction between the main wing and the plain flap in the slotted flap arrangement allows airflow from under the wing to energise (i.e. accelerate and smooth) the turbulent boundary layer flow over the upper surface of the lowered flap. This provides better downstream boundary layer adherence, and thus allows a larger angle of attack to be achieved before stall, with higher CL and lower drag than the plain flap. Ailerons may also be 'slotted' for improved performance.
The rearward extension of the Fowler flap as it is deflected increases wing area as well as camber, so it provides the best increase in lift of all the simpler systems — although perhaps even a single-element Fowler flap like that shown is not that simple to construct.
Some slat/slot systems also have the effect of increasing wing area thus reducing W/S and stall speed.
Leading edge slots combined with long-span slotted flaps, as used in STOL aircraft, allow a critical aoa much greater than the usual 16°. They can perhaps double the maximum CL of the basic wing, which allows much lower landing speeds but requires flight at the back of the power curve. Fixed leading edge slots work particularly well with a tailwheel configuration in a 'utility' aircraft such as the Slepcev Storch, but in a touring aircraft they have no value unless the pilot intends operating into very small, rough airstrips. There are simple automatic slat/slot systems where the slat is stowed when flying at lower angles of attack but pops out to form the slot when a particular angle of attack is reached. There are also retractable slat/slot systems that provide STOL capability when required without sacrificing cruise performance, except for the weight increase due to the more complex operating system.
I suggest now you have a look at the diagrams in Anatomy of a STOL aircraft.
Airbrakes or speedbrakes have a similar but more effective function. They are often vertically mounted plates, pairs of which are incorporated into the wing structure and which protrude from the upper and lower wing surfaces when activated. They create a lot of drag but little or no change in pitch, so the pilot must lower the nose to maintain approach speed. Airbrake or spoiler configurations are sometimes associated with flap systems that are primarily directed to lift generation, rather than lift generation plus drag creation. Such flap systems would have maximum downward deflection of perhaps 20°.
Military aircraft utilise very complex flaperon/spoileron systems.
The next module in this Flight Theory Guide discusses engine and propeller performance.
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Things that are handy to know
Notes for homebuilders
• The parasite drag coefficient. The equation for calculation of the total parasite drag for an aircraft is:
Parasite drag [newtons] = CDp × ½rV² × S
Unlike the lift coefficient, the parasite drag coefficient CDp is more or less a constant — the ratio of drag to dynamic pressure — and thus provides a means for comparing the relative aerodynamic 'cleanness' of two aircraft. The coefficient is usually in the range 0.03 to 0.08 for fixed-undercarriage aircraft.
• There is another value, the 'equivalent flat plate area' [FPA] used by aircraft, motor vehicle and structural engineers who are concerned with the calculation of air resistance. FPA is often quoted in aviation magazines when comparing the parasite drag efficiency of an aircraft with other similar aircraft, and it is usually stated in terms of square feet.
FPA is calculated as CDp times the wing area divided by the CDp for a flat plate. However, it is assumed that the CDp for a flat plate held at 90° to the airstream = 1 (in fact it is about 20% greater, but that is of no real consequence) so the flat plate CDp is omitted from the calculation, thus:
FPA = CDp × S ft²
For example, the FPA for the run-of-the-mill two or four-seater fixed-undercarriage general aviation aircraft would be around 6 ft² with CDp of 0.03 to 0.05, and the retractables around 4–5 ft² with CDp of 0.02 to 0.03. FPA of a very clean, high-performance general aviation aircraft like a Mooney model, is around 3 ft² with CDp about 0.015. Some very clean, high-performance GA kit-built aircraft have FPA less than 2. Note that FPA does not represent the frontal cross-section area of the aircraft.
One of the smallest known FPA is not associated with a general aviation aircraft but with an owner-designed and built ultralight! Californian Mike Arnold's 65 hp two-stroke Rotax 582 powered AR-5 held the world speed record, in the under 300 kg FAI efficiency Class C1-A/0 of 213 mph in August 1992. This handsome little glass-epoxy aircraft has an FPA of 0.88 ft² with CDp about 0.016. It demonstrates the efficiency that can be achieved — an unmatchable 3.3 mph per hp — in an ultralight design when the home designer/builder pays the utmost attention to detail. Note the drag reduction achieved by the beautifully shaped engine cowling, the wing root fillet and the minimisation of the junctions of undercarriage leg fairing and wheel cover. The choice of a fibreglass/foam lay-up composite structure also facilitates the drag reduction program.
Don't let anyone tell you ultralights have to be slow and draggy!
• 'Separation bubbles' or 'laminar flow separation bubbles'. In laminar flow, sometimes the laminar flow boundary layer separates from the wing surface then reattaches itself a short distance downstream. This forms a 'bubble' of stagnant air with a significant spanwise dimension that changes the aerodynamic thickness of the wing which, in effect, increases pressure drag. Bubbles may also cause increased turbulent flow to be generated downstream of the reattachment point. Aircraft designers avoid laminar separation at cruising speeds by inducing a turbulent — but attached — boundary layer where necessary. Separation bubbles that increase drag may also occur on the fuselage and tailplane. See vortex generators below.
• Reynolds number. Occasionally, in reference to boundary layer laminar to turbulent flow and flow separation characteristics you may see mention of the critical Reynolds number [Re]. Re is a measure of the relative influence of viscous and inertia effects on boundary layer behaviour. Like aspect ratio Re is a dimensionless quantity, i.e. it has no units of measurement.
For rough estimates Re in airstreams = air density/air viscosity × airflow velocity × flow distance.
ISA sea-level density is 1.225 kg/m³ and standard viscosity is 0.0000179 kg/m/s, so standard air density/air viscosity = 68 459 (say 70 000).
If the Re is estimated for an average flow across the entire wing at a particular airspeed, then the equation can be simplified to:
Re = velocity (m/s) × the mean aerodynamic chord (m) × 70 000.
Thus, the wing chord Reynolds number for an aircraft with a MAC of 1.2 m flying at 97 knots (50 m/s) is roughly 50 × 1.2 × 70 000 = 4 200 000. When that same aircraft is cruising at 78 knots (40 m/s), Re would be about 3 360 000.
For a particular wing and wing surface condition, there is a critical boundary layer Re, above which the laminar flow will transition to turbulent flow. In slow flight speed, the critical boundary layer Re will be attained a particular distance downstream from the leading edge stagnation point; as airspeed increases (and in accordance with the equation above), that distance must shorten.
• The vortex generators [VGs] used in a few light aircraft designs, particularly short take-off and landing [STOL] aircraft — or as post-delivery 'add-ons' — are small boundary layer control devices with a swept-back leading edge and a vertical trailing edge, the chord at the base of the device is two to four times its height. VGs are machined from an aluminium 'T' extrusion or formed in polycarbonate, a row of 10 or 15 of which are usually spaced along the upper surface of each wing, probably close to the transition zone. Each VG is carefully sited, with a specific angle of attack (perhaps 15° or more) to the local airflow and with sufficient height to just intrude into the free-stream flow. So situated, they induce fast-rotating, highly organised, downstream vortices (much the same principle as wingtip vortices) which mix the high-speed free-stream airflow into the slow-moving, surface boundary layer flow; entraining and re-energising that flow so that the chordwise pressure gradient profile on the upper surface is decreased (see boundary layer flow). Consequently, surface pressure is decreased, so the pressure differential — and thus the lift coefficient — is increased at all wing aoas. VGs are often paired, to produce counter-rotating vortices. The VGs also delay boundary layer flow separation at high aoa; i.e. VGs lower the stall speed while improving the aircraft's low-speed behaviour. But there is likely to be minimum warning of onset of the stall, and stall behaviour may be more violent.
Appropriately sized and sited wing vortex generators can be effective at providing good manoeuvring control of the aircraft when operating low and slow, and provide a greater CLmax, and improve aileron performance and aircraft climb performance. They are sometimes also used on the horizontal and vertical stabilisers mounted just forward of the rudder/elevator hinge lines where they have the effect of allowing greater control surface deflection before separation occurs. VGs are also useful in locations where interference drag is a problem. The use of VGs in light aircraft may slightly degrade performance at the upper end of the speed range, probably depending on the amount of additional turbulence they generate outside the normal turbulent boundary flow.
• The term burble is sometimes used to describe a turbulent stream. For example a disturbance emanating from something on the fuselage can induce a turbulent streamflow that affects the tailplane. There may also be a separation of flow at the junctions of structural components, which causes interference drag.
Groundschool – Flight Theory Guide modules
| Flight theory contents | 1. Basic forces | 1a. Manoeuvring forces | 2. Airspeed & air properties |
| 3. Altitude & altimeters | [4. Aerofoils & wings] | 5. Engine & propeller performance | 6. Tail assembly surfaces |
| 7. Stability | 8. Control | 9. Weight & balance | 10. Weight-shift control | 11. Take-off considerations |
| 12. Circuit & landing | 13. Flight at excessive speed | 14. Safety: control loss in turns |
| Operations at non-controlled airfields | Safety during take-off & landing |
Copyright © 2002 — 2012 John Brandon [contact information]