Twist Distributions for Swept Wings, Part 1

Transcription

1 On the Wing... #161 Twist Distributions for Swept Wings, Part 1 Our curiosity got the better of us, and we asked Why are designers of swept wing tailless models placing proportionally more twist in the outboard portion of the wing? This series of articles will provide a comprehensive answer to that question. Our intense interest in tailless aircraft now spans twenty years. Over those two decades, we have built a number of plank type wings and several swept wings. As we explained in a recent column, there are advantages and disadvantages to both of these planforms. An introduction to twist distributions The impetus to begin designing our own swept wing tailless aircraft was the presentation given by Dr. Walter Panknin at the MARCS (Madison Area Radio Control Society) Symposium held in Dr. Panknin provided a relatively simple method for determining the geometric twist required for a stable planform when given the span, the root and tip chord lengths, the root and tip airfoil zero lift angles and pitching moments, the sweep angle of the quarter chord line, the design coefficient of lift, and the static margin. Dr. Panknin assumed that the wing twist would be imparted across the semi-span. That is, the root would be held at zero degrees and the tip twisted at some angle of washout, with the wing leading and trailing edges forming straight lines. Dr. Panknin s wing, the Flying Rainbow, along with Kurt Weller s Elfe II, utilized this type of twist distribution on tapered wings. In looking at other swept wings of that time period, we were also attracted to Hans-Jürgen Unverferth s CO2. The CO2 was different from the Flying Rainbow and the Elfe II in that the wing was not tapered but rather of constant chord. Additionally, CO2 utilizes a twist distribution in which the inner half of the semi-span has no twist at all. All of the geometric twist is in the outer half of the semi-span. While the actual twist angle is identical to that computed for the Panknin twist distribution, pitch stability is not adversely affected and in fact may be slightly better. More recently, Hans Jürgen and other swept wing designers have taken to imparting wing twist across three segments. From the root to one third of the semi-span there is no twist. About one third of the total twist is then put into the second third of the semi-span, and the remaining two thirds of the total twist is put into the wing between two thirds semi-span and the wing tip. Our curiosity got the better of us and we asked, Why are designers of swept wing tailless models placing proportionally more twist in the outboard portion of the wing? This series of articles will provide a comprehensive answer to that question. Page 1 of 10

2 Twist distributions for swept wings, Part 1 Lift distributions Nearly all aerodynamics text books devote pages to what is called the lift distribution. The lift distribution for any straight (quarter chord line at 90 degrees to the centerline) wing can be graphically represented by a curved line superimposed over a standard X-Y coordinate system. The lift distribution curve traces the local circulation the local coefficient of lift times the local geometric chord. How is the lift distribution determined? Let s start by taking a look at the construction of the elliptical lift distribution. Assign the aircraft wing tips to the points 1.0 and -1.0 on the Y-axis of the coordinate system. Draw a circular arc above the Y-axis using the aircraft wing tips to define the diameter. A semicircle is formed which has the radius b/2 (the semi-span) and the area π/2(b/2) 2 which in this specific case is simply π/2 = Now drop vertical lines from the semicircle circumference to the Y-axis. Mark the mean (halfway) point on each vertical. Connecting these identified points creates an ellipse. (See Figure 1). This elliptical lift distribution is predominantly promoted as being the ideal, as represented in the planform of the British Supermarine Spitfire fighter of the World War II era. Why would the designer want the lift distribution of his arbitrary wing to closely match that of the elliptical lift distribution? Because with the elliptical lift distribution, a discovery of Ludwig Prandtl in 1908 which he published in 1920, each small area of the wing is carrying an identical load and so is operating at the same local coefficient of lift, the downwash off the trailing edge of the wing is constant across the span, and the coefficient of induced drag (drag due to lift) is at its minimum point. To construct the lift distribution for an arbitrary wing without twist or sweep, lay out the wing outline over the elliptical lift distribution with chord lengths proportioned such that the area of the wing is equal to that of the ellipse (one half that of the semicircle, in this case π/4 = 0.785). Draw a curve along the mean of the ellipse and the wing planform outline. (See Figure 2.) With some graphical experimentation, we find that the lift distribution for a wing with a taper ratio of 0.45 almost exactly matches that of the elliptical lift distribution described by Prandtl. (See Figure 3.) The tapered planform has at least one advantage over the elliptical planform it s far easier to build. But the elliptical planform has a stall pattern in which the entire wing is subject to stalling at the same time. At high angles of attack, small gusts can serve to trigger a stall on any portion of the wing span. A tapered wing with a nearly identical lift distribution will tend to behave in the same way. Lift coefficient distributions As stated previously, the lift generated by any wing segment is directly proportional to the coefficient of lift and the local geometric chord. This means that there is also a coefficient of lift distribution. For Prandtl s elliptical wing lift distribution, as has been described here, the local coefficient of lift is identical across the span. On the other hand, if the taper ratio is zero (the wing tip comes to a point), the coefficient of lift at the wing tip will be zero only in a truly vertical dive, Page 2 of 10

3 Twist distributions for swept wings, Part 1 but otherwise it will be infinite because the wing tip chord is nil. Any time this wing is called upon to produce lift, the wing tip will be stalled. From this extreme example, we realize the tip chord cannot be too small, as it will then be forced to operate at a higher coefficient of lift, leading to a local stalling of the wing. (See Figure 4.) So called tip stalling can be inhibited by one or both of two methods. The first involves increasing the local chord near the wing tip, the second consists of imparting washout. As we intuitively know, enlarging the wing tip chord reduces the local coefficient of lift. An enlarged wing tip chord is not so efficient as the true elliptical planform, but the penalty for using a perfectly rectangular wing is just 7% and so it may be an acceptable trade-off for a machine designed for sport flying. Washout, on the other hand, while also reducing the coefficient of lift in the area of the wing tip, is good for only one speed. As the twist angle increases, the deleterious effects become stronger much more quickly as the coefficient of lift for the entire wing, C L, moves away from the design point. If washout is too great, the wing tips can actually be lifting downward at high speeds. This puts tremendous loads on the wing structure. Adverse yaw One other effect of utilizing the elliptical lift distribution comes about as we add control surfaces to the wing. Outboard ailerons, for example, create different coefficients of induced drag depending on whether the surface is moved up or down. The control surface moving downward creates more lift and hence more drag than the surface moving upward. When rolling into a turn, therefore, the aircraft is forced into a yaw away from the direction of the turn. (See Figure 5.) In conventional aircraft, this tendency can be reduced to some extent by what is called aileron differential. The upgoing control surface travels through a larger arc than the downgoing surface. While this tends to increase the drag on the downgoing wing, reducing adverse yaw to a great extent, many pilots find that some amount of rudder input is necessary to obtain a coordinated turn. Reduction of rudder input is an important consideration in the quest to reduce overall drag while maneuvering, but the associated induced drag from the fin and rudder, a low aspect ratio flying surface, cannot be entirely avoided. For a swept flying wing without vertical surface, elimination of adverse yaw is obviously imperative, but aileron differential cannot be used in this case because of its effect on pitch trim. Some other means of eliminating adverse yaw must be devised. Three major problems And so we are forced to solve three problems when designing a tailless aircraft: 1. achieve and hopefully surpass the low induced drag as exemplified by the elliptical lift distribution without creating untoward stall characteristics, 2. reduce the adverse yaw created by aileron deflection without adversely affecting the aircraft in pitch, and Page 3 of 10

4 Twist distributions for swept wings, Part 1 3. maintain an acceptable weight to strength ratio. A relevant historical tidbit The Wright brothers, along with their other accomplishments, were the first aircraft designers to determine that banking was necessary to turn, an idea which no doubt came from their experience with bicycles. While other early aviation pioneers had studied bird flight, the perspective of the Wrights while watching birds was very much different because of their cycling experiences. (Interestingly, their direct competitor, Glenn Curtiss, built and raced motorcycles.) The Wrights also had the ability to separate the major problem of controlled powered flight into manageable components. Propulsion was separated from the production of lift, and stability was separated from control, for example. In fact, their solution to the problem of flight incorporated only one integrated system, the wing, which provided lateral control, structure, and lift. It was Wilbur s twisting of the inner-tube box, through which the idea of wing warping was derived and the internal bracing of their wing structure was devised, which provided the insight needed to create a controllable flying machine capable of carrying a human pilot/passenger. But the flying machine they created, while tremendously successful, for all practical purposes ended the use of birds as models for aircraft design. As an indicator of this, the Wrights saw their early successes and records in powered flight quickly surpassed by the inventions of others. Curtiss, for example, solved the problem of banking turns with separate control surfaces rather than wing warping. His aileron system is still in use today. The Wright s separation of a huge problem into smaller more easily solved problems has continued to be the hallmark of aircraft design for 100 years, and aviation has made nearly unbelievable strides during that century. But there are a growing number of aircraft designers who wish to go back to the bird model. They wish to design an aircraft which is the minimum required for efficient controlled flight by integrating lift, stability and control into a single structural component. A bird is a biological system which has been very successful for a very long time. To be successful in the competitive environment of nature, a flying bird needs more than just lift, stability, and control. A bird must also be efficient at flying. That is, it must have a very low energy expenditure. Minimum drag while moving through the air is of course of major importance in this regard, as is a very light airframe because extra weight increases the energy drain on the system. We can see through direct observation that birds have no vertical surfaces, yet birds are able to make beautiful coordinated banked turns without any evidence of adverse yaw. Perhaps birds do not make use of Prandtl s elliptical lift distribution. What s next? As a prelude to future installments, let us ask a series of provocative questions: Page 4 of 10

5 Twist distributions for swept wings, Part 1 What if we found that the elliptical lift distribution does not lead to the minimum induced drag, as has been dogma in most aerodynamics texts since Prandtl introduced the concept in 1920? What if we found a way to produce induced thrust in addition to, and without increasing, the induced drag produced by the creation of lift? What if we could increase the wing span and aspect ratio without increasing the required strength of the spar at the wing root? What if the answers to all of the above questions are related? We ll cover all of this and more in future installments! Ideas for future columns are always welcome. RCSD readers can contact us by mail at P.O. Box 975, Olalla WA , or by at <bsquared@appleisp.net>. References: Anderson, John D. Jr. Introduction to flight. McGraw-Hill, New York, Anderson, John D. Jr. Fundamentals of aerodynamics. McGraw-Hill, New York, Bowers, Al. Correspondence within < list. Hoerner, Dr.-Ing. S.F. and H.V. Borst. Fluid-dynamic lift. Hoerner fluid dynamics, Vancouver WA USA, Horten, Dr. Reimar. Lift distribution on flying wing aircraft. Soaring June 1981, pp Hurt, H.H. Jr. Aerodynamics for naval aviators. Published as NAVWEPS 00-80T-80 by the U.S. Navy, Kermode, A.C. Mechanics of flight. Pitman, London, McCormick, Barnes W. Aerodynamics, aeronautics, and flight mechanics. John Wiley and Sons, New York, Raymer, Daniel P. Aircraft design: a conceptual approach. AIAA Education Series, Washington, DC, Shevell, Richard S. Fundamentals of flight. Prentice-Hall, Englewood Cliffs NJ USA, Simons, Martin. Model aircraft aerodynamics. Argus Books, Hemel Hempstead Great Britain, The White Sheet, Spring 1986, No. 36. Sean Walbank editor. White Sheet Radio Flying Club, Dorset/Somerset Great Britain. Page 5 of 10

6 Twist distributions for swept wings, Part 1 X semi-circle ellipse -Y -1 0 b b/2 +1 +Y Figure 1. Construction of elliptical lift distribution Page 6 of 10

7 Twist distributions for swept wings, Part 1 X ellipse -Y -1 0 b b/2 +1 +Y Figure 2. Construction of lift distribution for untwisted rectangular wing. Page 7 of 10

8 Twist distributions for swept wings, Part 1 X ellipse -Y -1 0 b b/2 +1 +Y Figure 3. Construction of lift distribution for untwisted wing with taper ratio of Page 8 of 10

9 Twist distributions for swept wings, Part 1 shape Lift CL downwash Figure 4. Three wing planforms and their associated lift distributions, coefficient of lift distributions, and downwash distributions. Page 9 of 10

10 Twist distributions for swept wings, Part 1 Figure 5a. Lift and drag profiles for untwisted rectangular wing in straight and level flight and no control surface deflection. roll to left yaw to right Figure 5b. Lift and drag profiles for untwisted rectangular wing with aileron deflection for left bank, no differential. Page 10 of 10

11 On the Wing... #162 Twist Distributions for Swept Wings, Part 2 Having defined and provided examples of lift distributions in Part 1, we now move on to describing the stalling patterns of untwisted and twisted wings, determining the angle of attack as from the location of the stagnation point, and how wing sweep affects the angle of attack across the semi-span. Stalling patterns for untwisted wings The lift generated by any wing segment is a product of the local coefficient of lift and the local chord length. Referring to Figure 1 (a reprint of Figure 4 from Part 1) we can see the results of this formula as applied to three wing planforms. The ideal lift distribution is the elliptical as shown in the left column. Note the local coefficient of lift (c l ) is identical across the entire span, as is the downwash. While the elliptical wing planform is efficient, it is difficult to build and, because the c l is the same across the span, all segments of the wing are equally susceptible to stalling. See Figure 2A. The rectangular wing, with its constant chord, Figure 2B, tends to stall at the root first. This is because the local coefficient of lift progressively decreases for those wing segments nearer the tip. This takes some of the load off them, inhibiting stalling. Note also from the middle column of Figure 1 that the rectangular wing tip vortex is quite large, indicating substantial outward flow across the lower surface, and substantial inward flow across the upper surface. The diamond planform (right column Figure 1 and Figure 2E), unless in a vertical dive (C L = 0) is stalled to some extent at all times. Note that although the local coefficient of lift at the wing tip tends to be infinite, the actual amount of lift generated is very low because of the diminishing chord, and the downwash in the tip region tends to zero. The stalling pattern for this wing planform grows inward from the trailing edge of the wing tip and toward the leading edge. From this information, it does not seem like a delta wing would be useful, but the airflow over a severely swept wing, which a delta is, is far different from the airflow over the straight wing described in this instance. Wings with large to moderate taper ratios, λ = ~0.4>, have stalling patterns approaching that shown in Figure 2C and tending toward that of the rectangular wing planform (Figure 2B). Wings with small taper ratios, λ = <~0.4, have stalling patterns approaching that of the highly tapered planform shown in Figure 2D and tending toward that of the diamond wing planform, Figure 2E. The most interesting stalling pattern, however, is that of the swept back wing, as depicted in Figure 2F. Although the wing tip has the same chord as the root, the stalling pattern is entirely different than that of the unswept rectangular wing. Page 1 of 13

12 Twist distributions for swept wings, Part 2 Lift distributions and stalling patterns of swept wings Figure 3 compares the elliptical lift distribution with representative lift distributions for swept forward and swept rearward wings. The swept back wing shows an increase of lift near the wing tips and a noticeable depression of lift near the wing root. The swept forward wing shows an increase in lift near the wing root, and depressed lift near the wing tip. Before speaking to why this is so, it should be mentioned that we can attempt to tailor the lift distribution of swept wings to closely approximate the lift distribution of the elliptical planform by modifying the taper ratio. Figure 4 shows in graphical terms the taper ratios required for this approximation as based on the sweep angle. While we can modify the lift distribution to more closely match the elliptical ideal by adjusting the taper ratio, the stalling pattern does not appreciably improve. The stalling pattern still tends to grow inboard from the wing tip. This is seen in Figure 5. The swept back wing, when stalled, tends to pitch up into a deeper stall as the center of lift moves forward when the rear of the wing is stalled. As the (elevon) control surfaces are normally placed outboard, they are in a stalled region of the wing. A swept forward wing will suffer from a somewhat similar malady. When the root of a swept forward wing stalls, the wing tips remain unstalled and the center of lift moves forward, pitching the nose up. Aileron control is maintained, but at the expense of a possible severe pitch up and deep stall. Despite having identical root and tip chords and sharing what some would consider dangerous stall behavior, we bring up these two cases as an example of how sweep can effect the air flow over the wing. The two swept wings in this example have different stall patterns caused by the imparted sweep. Sweep and angle of attack An airfoil which is creating lift demonstrates three important characteristics: The air going over the top of the section accelerates, the air going along the bottom decelerates. If the smoke stream is pulsed, these velocity differences are easily seen. Figure 6 was derived from a smoke tunnel photograph using this methodology. The acceleration differential is seen in the varying size of the pulses and the varying distances between them. (Some mixing of the smoke with clear air takes place because of turbulence caused by the boundary layer interfacing with air which is moving more rapidly.) The air rises toward the section as it approaches the leading edge. This is seen in Figure 6 as well. This portion of the air flow is called the upwash. The air is deflected downward aft of the airfoil section. The section acts as a vane, turning the air stream downward. Termed downwash, this flow is an important consideration in the design of conventional tailed aircraft as it influences the size and placement of the horizontal stabilizer. Page 2 of 13

13 Twist distributions for swept wings, Part 2 Going back to the second characteristic, there is a point near the leading edge where an air molecule actually comes to rest at the airfoil surface. This point is termed the stagnation point, and its location can be used to determine the section angle of attack. As the angle of attack increases from the zero lift angle, the stagnation point moves further aft along the bottom of the airfoil. See Figure 7. The air flow around a straight wing with an elliptical lift distribution is such that the location of the stagnation point remains consistent across the semi-span. On a swept back wing, we find any segment of the wing has an effect on the upwash of the section immediately downstream and hence outboard from it. The stagnation point thus moves rearward along the bottom of the lower surface, indicating an increasing angle of attack toward the wing tip. Figure 8 provides an exaggerated illustration of this behavior on an untwisted wing. Because of wing sweep, the effective angle of attack at the wing tip is greater than the effective angle of attack at the wing root. It s little wonder the wing tips are proportionally overloaded and subject to stalling. To maintain a constant angle of attack across the entire span, some amount of washout (leading edge down) must be imparted to the outer portion of the wing. This will reduce the tendency of the wing tips to stall first. A note about washout On a conventional tailed sailplane, it is common practice to place some amount of washout in the outer wing panel(s) to assist in reducing the tendency to tip stall. The problem with this methodology when used on a straight wing is that each spanwise wing segment is seeing the air approaching at the same angle, and the local angle of attack as defined by the location of the stagnation point is entirely dependent upon the segment angle of incidence. When the entire wing is called upon to generate very small coefficients of lift the root is flying at a relatively small angle of attack, and the wing tips may be flying at an angle of attack which is below the zero lift angle. The wing tip then generates lift in the downward direction. In the 1920 s and 1930 s, when sailplane designers were building wooden sailplanes with higher and higher aspect ratios, wings with insufficient torsional strength were destroyed by the aerodynamic forces generated by excessive wing twist. On a swept back wing, the angle of attack as seen by each wing segment increases toward the wing tip. For a specific coefficient of lift, washout can therefore be used to correlate the angle of attack of the wing tip with the angle of attack of the wing root. At some particular speed (C L ) the entire wing will be operating at the same local coefficient of lift (c l ) across the entire span. This is not quite as good as the lift distribution of an elliptical wing, which remains elliptical over a very large range of speeds, but it is a definite improvement over an untwisted swept wing. So long as the root is developing lift, the outboard segments will continue to see an increasing upwash. While required torsional strength is dictated by both sweep and twist, it is handled well with modern design and construction materials and methods. Are swept wings worth the effort? Page 3 of 13

14 Twist distributions for swept wings, Part 2 From what we ve said thus far, it would seem like getting a swept wing to perform in a fashion similar to the elliptical lift distribution, with its accompanying efficiency, would require a major effort. After all, the lift distribution is now dependent upon three variables sweep, taper and twist rather than simply taper and twist alone as with a straight planform. The addition of sweep to the design environment magnifies the number of complex computations required. At this point in our discussion, it would appear the only clear advantages to be derived from a tailless swept wing planform would come from either drag levels lower than those of a conventional tailed airplane or improved handling characteristics, both of which have the potential to significantly improve performance. Whether the gains to be achieved are worth the time and effort involved in obtaining them has always been open to question. A synthesis of concepts and technology may change that balance in the future. There are avenues of approach, first presented decades ago, which now look quite promising. The advent of low cost supercomputers which are able to quickly run the sophisticated software required to handle exceptionally complex iterative processes is bringing recent advancements in computational fluid dynamics to creative individuals outside the formal aircraft industry. What s next? As we mentioned in Part 1, there are three major hurdles to be overcome in order to design an efficient swept wing: (1) achieve and hopefully surpass the low induced drag as exemplified by the elliptical lift distribution without creating untoward stall characteristics; (2) reduce the adverse yaw created by aileron deflection without adversely affecting the aircraft in pitch; (3) maintain an acceptable weight to strength ratio. This column has focused on the first of these difficulties, and it would appear there may be acceptable solutions available. However, it would be quite valuable to not only achieve the high efficiency of the elliptical lift distribution, but to surpass it. Surprisingly, achieving that elusive goal may be one of the results of solving the second problem, the topic of the next installment. Ideas for future columns are always welcome. RCSD readers can contact us by mail at P.O. Box 975, Olalla WA , or by at <bsquared@appleisp.net>. References: Anderson, John D. Jr. Introduction to flight. McGraw-Hill, New York, Anderson, John D. Jr. Fundamentals of aerodynamics. McGraw-Hill, New York, Bowers, Al. Correspondence within < list, early Dommasch, Daniel O., Sydney S. Sherby and Thomas F. Connolly. Airplane aerodynamics. Putnam Publishing Corporation, New York, Hoerner, Dr.-Ing. S.F. and H.V. Borst. Fluid-dynamic lift. Hoerner fluid dynamics, Vancouver Washington USA, Page 4 of 13

15 Twist distributions for swept wings, Part 2 Horten, Dr. Reimar. Lift distribution on flying wing aircraft. Soaring June 1981, pp Hurt, H.H. Jr. Aerodynamics for naval aviators. Published as NAVWEPS 00-80T-80 by the U.S. Navy, Jones, Bradley. Elements of Practical Aerodynamics, third edition. John Wiley & Sons, New York, Kermode, A.C. Mechanics of flight. Pitman, London, Lennon, A.G. Andy. R/C model airplane design. Motorbooks International, Osceola Wisconsin USA, Masters, Norm. Correspondence within < list, early Raymer, Daniel P. Aircraft design: a conceptual approach. AIAA Education Series, Washington, DC, Shevell, Richard S. Fundamentals of flight. Prentice-Hall, Englewood Cliffs NJ USA, Smith, H.C. Skip. The illustrated guide to aerodynamics, second edition. TAB Books, Blue Ridge Summit Pennsylvania USA, The White Sheet, Spring 1986, No. 36. Sean Walbank editor. White Sheet Radio Flying Club, Dorset/Somerset Great Britain. Page 5 of 13

16 Twist distributions for swept wings, Part 2 shape Lift C L downwash Figure 1. Three wing planforms and their associated lift distributions, coefficient of lift distributions, and downwash distributions. Page 6 of 13

17 Twist distributions for swept wings, Part 2 Elliptical Rectangular, λ = 1.0 A B Moderate taper, λ = 0.5 High taper, λ = 0.25 C D Pointed tip, λ = 0.0 Swept back E F Figure 2. Stalling patterns of various untwisted wing planforms. Page 7 of 13

18 Twist distributions for swept wings, Part Swept forward Elliptical 1.0 Swept back b / Figure 3. Representative lift distributions for swept forward and swept back wings compared to ideal elliptical lift distribution. Page 8 of 13

19 Twist distributions for swept wings, Part Taper ratio required, λ = c tip /c root 1.0 Rectangular 0.5 Angle of sweep, Λ Figure 4. Taper ratios theoretically required for near-elliptical lift distribution for swept wings. Page 9 of 13

20 Twist distributions for swept wings, Part 2 Swept forward Swept back Direction of flight Figure 5. Stalling patterns of swept back and swept forward untwisted wing planforms. Page 10 of 13

21 Twist distributions for swept wings, Part 2 Figure 6. Air flow over a section as visualized through pulsed smoke streams. Page 11 of 13

22 Twist distributions for swept wings, Part 2 α = ~0 α = ~5 α = ~10 α = ~15 Figure 7. Rearward movement of the stagnation point and increased upwash ahead of wing with changes in angle of attack. Page 12 of 13

23 Twist distributions for swept wings, Part 2 Tip Direction of flight Right semi-span ß, sweep angle Root Figure 8. Movement of the stagnation point and changes to effective angle of attack along the semi-span of a swept back wing, exaggerated. Page 13 of 13

24 On the Wing... #163 Twist Distributions for Swept Wings, Part 3 In Part 1 we defined and provided examples of lift distributions. Part 2 examined stalling patterns of various planforms and introduced the notion that sweep angle and coefficient of lift can affect the angle of attack of outboard wing segments. Three consistent themes have been underlying the discussion thus far: (1) achieve and hopefully surpass the low induced drag exemplified by the elliptical lift distribution without creating untoward stall characteristics, (2) reduce adverse yaw created by aileron deflection without adversely affecting the aircraft in pitch, and (3) maintain an acceptable weight to strength ratio. In Part 3 we will describe a method of achieving the second goal. Sweep and twist Figure 1 (reprint of Figure 8, Part 2) shows the increasing upwash which affects outboard segments of a swept untwisted wing as it produces lift. Although exaggerated in the diagram, the overall tendency is clear and does appear in practice. While there are several ways of reducing the tendency for the wing tip to stall, like careful consideration of airfoils or addition of wing fences, there are advantages to imparting some twist to the wing in the form of washout (leading edge down). Figure 2 illustrates the case where the wing is twisted such that each wing segment has the same angle of attack as related to the oncoming air flow. Since the increasing upwash ahead of the wing is directly proportional to the amount of lift produced by inboard wing segments, this illustration is obviously accurate for only one aircraft velocity and attitude. The general concept is, however, very important. Vectors Mass, length, pressure and time can be defined by single real numbers. The length of a spar for a two meter sailplane, as an example, may be 39 inches. As there is a unit of measurement, inches in this case, the spar length is a scalar quantity. The number which provides the magnitude, 39, is considered a scalar. Force, on the other hand, has both a magnitude and a direction, and is therefore classified as a vector quantity. A five pound brick resting on a table in a gravitational field may be represented as shown in Figure 3A and 3B. If another five pound brick is placed on the first brick, the situation can be depicted as in Figure 3C. Note that the arrowhead always indicates the direction of the force, while the length of the line indicates the magnitude of the force. There are two basic forces of interest to aerodynamicists - lift and drag. In a wind tunnel, the investigator may measure the lift and drag of the airfoil by setting up two scales. One scale will measure the lift generated by the section through a balance system which has its axis vertical to the tunnel test section and hence the air flow. Another scale is set up with its axis parallel to the air flow to measure drag. Page 1 of 17

25 Twist distributions for swept wings, Part 3 The investigator can rotate the airfoil section through negative and positive angles of attack relative to the air flow. As the angle of attack increases or decreases, both lift and drag will vary. Regardless of the angle of attack, generated lift is always measured perpendicular to the air flow and drag parallel to the air flow. Figure 4A demonstrates how two vectors having the same source may be resolved into a single vector by constructing a simple parallelogram. Since lift and drag are always perpendicular to each other, they can always be resolved into a single vector by means of a rectangle (a parallelogram which has intersections of 90 degrees). We can also perform this operation in reverse. That is, given a single vector and the angle(s) of the parallelogram, the separate component vectors may be derived As an example, we know that the lift vector is always perpendicular to the air flow and the drag vector is always parallel to it. By constructing the requisite rectangle on the resultant, we can define the lift and drag vectors. This process is shown in Figure 4B. We can perform a similar procedure on the weight vector, thereby establishing two separate component vectors one parallel to the direction of flight and one perpendicular to it. The upper illustrations in Figure 5 provide a depiction of the vectors involved in sustained, constant velocity flight. The upper illustration, Figure 5A, shows a powered aircraft in straight and level flight. The weight of the aircraft, W, is counteracted by the generated lift, L. The drag, D, is counteracted by the generated thrust, T. There is a single vector, R 1, which can represent the combined lift and drag forces, and a single vector R 2 which can represent the combined thrust and weight vectors. These two resultant vectors are calculated by constructing a parallelogram using the two known vectors. R 1 and R 2 are of equal magnitude and opposite direction in this case, and the aircraft is therefore flying at a constant velocity. If thrust is increased, as shown in Figure 5B, the T vector length increases, indicating increased thrust, thus changing the shape of the parallelogram. The aircraft accelerates horizontally. To maintain straight and level flight after application of additional thrust, aircraft trim must be adjusted so the wing continuously generates only enough lift to exactly match the aircraft weight. R 2 becomes longer and rotates forward. The drag force D then increases as the aircraft velocity increases. Drag will increase until it exactly matches thrust R 1 becomes the same length as, and in opposite direction to, R 2. Once drag and thrust are again equal, the aircraft is once more stabilized in straight and level flight. The aircraft velocity will be greater and constant, the amount of lift will be unchanged, the coefficient of lift will be lower, and the wing will be operating at a lower angle of attack. The lower illustrations in Figure 5 depict the case of a powerless aircraft of the same design. It is in gliding flight. In Figure 5C the aircraft is moving forward at a constant velocity and slight downward angle. We know the direction of the air flow, so R 1 can be resolved into the lift and drag vectors which are perpendicular to each other, as described previously. The resultant vector, R 1, is of exactly the same magnitude as R 2 and in the opposite direction, so the aircraft is flying at constant velocity. There is no engine to generate thrust so the weight W alone forms R 2. R 2, however, can be dissociated into two component vectors. One component vector, parallel to D, can be denoted T (thrust), the other can remain unnamed. Page 2 of 17

26 Twist distributions for swept wings, Part 3 Consider the flight path and note that the lift vector remains at ninety degrees to the air flow and the drag vector remains parallel to the air flow. This is the same as seen in the previously described powered example. As the glide angle steepens, the portion of the weight which is considered thrust increases. At the same time, the lift decreases and the drag increases. See Figures 6A and 6B. To help explain this, take a look at the extreme. Figure 6C shows the glider in a sustained true vertical dive. The wing is operating at the zero lift angle of attack and so lift has been reduced to nothing. Drag makes up all of R 1 and weight makes up all of R 2. If in a vertical dive we adjust the angle of attack so that it matches what was required for straight and level flight, the lift will be the same as during straight and level flight and it will be oriented exactly in the horizontal. See Figure 6D. The drag vector will also be the same length as before the change in attitude and will remain parallel to the air flow. The resultant R 1 is rotated nearly ninety degrees from the vertical. The lift force immediately begins accelerating the wing horizontally while the weight accelerates the aircraft vertically downward. As the horizontal speed increases, the air flow changes direction so there is a reduction in the angle of attack. If we consistently maintain the initial angle of attack, the aircraft will pull out of the dive. In Figure F4D, the aircraft has just been put into a steep dive from straight and level flight. The aircraft is assumed to be flying at the same speed as before the change in attitude. The weight vector can be broken down into its two component parts, as was done previously, and the thrust component is accelerating the aircraft in the direction of flight. The lift and drag vectors remain oriented to the direction of flight. R 1, the resolution of the lift and drag vectors, is rotated forward of the vertical, indicating that a portion of R 1 is directed in the horizontal direction. This small force is denoted in the illustration as T i, induced thrust. If the angle of attack is held constant, the aircraft will pull out of the dive, just as in the previous example. Induced thrust We ve used the term induced thrust in the previous paragraph, and there are some readers who may not believe that such a thing exists, despite having a knowledge of induced drag. Perhaps one of the best examples of induced thrust is the action of a winglet. A very large number of aerodynamics texts describe winglets in detail, so we will not do so here. What we want to bring into focus is the production of induced thrust by the winglet. The upper illustration of Figure 7 shows a wing from the rear, with the winglet structure defined by phantom lines. The air flow is shown traveling outboard along the bottom surface of the wing and inboard across the upper surface. The velocity of this movement is generally greater near the wing tip as shown by the lengths of the lines. The air flow outboard of the wing tip is very close to circular, but remember, the free stream velocity is added to this circular motion, so the resultant air flow meets the winglet at an angle. The lift and drag vectors are shown in the lower illustration. Note the now familiar rotation of the resultant in reference to the winglet MAC/4 axis. (MAC/4 is the 25% chord point of the mean aerodynamic chord and is the origin for the winglet lift and drag vectors, just as for any wing segment. The MAC/4 axis and the yaw axis are in parallel planes in the presented examples.) The vector T i is the induced thrust generated by the winglet. Page 3 of 17

27 Twist distributions for swept wings, Part 3 We can extend the notion of induced thrust from a winglet to the outer segment of a lifting swept wing. Consider Figure 8A. In this case, an airfoil is generating some lift while the air flow is precisely horizontal. This is a situation identical to that when an airfoil with a zero lift angle of some negative value is set in a wind tunnel at zero degrees angle of incidence to the air flow. Note that the lift vector is vertical (ninety degrees to the air flow) and the drag vector is parallel to the air flow. The resultant is rotated at an angle behind the vertical quarter chord axis. In the wind tunnel, as the airfoil angle of attack is increased, the lift vector remains perpendicular to the air flow, the drag vector remains parallel to air flow, and the axis remains vertical, perpendicular to the air flow. In Figure 8B, the air flow is coming from below at an angle of five degrees. The lift and drag vectors have rotated to match the air flow, and the resultant coincides with the vertical MAC/4 axis. Figure 8C shows the case where the air flow is coming up at an angle of ten degrees. The lift and drag vectors (and the resultant, of course) have rotated forward of the axis. Figure 8D shows two situations which take place at an air flow angle of 15 degrees. We ve shown a single lift vector and two drag vectors. If the drag is low, the resultant (R 1 ) remains well ahead of the axis. If the drag is excessive, however, the resultant (R 2 ) rotates behind the axis. This is an important concept to keep in mind. The case of the outer segment of a twisted swept wing is shown in Figure 8E. The air flow is coming up at an angle of ten degrees and the airfoil is set at an angle of incidence of minus five degrees. As the wing section sees an angle of attack of five degrees, the lift is of the same magnitude as in Case 8B, but the resultant is rotated to a direction nearly identical to that of Case 8C. It may be helpful to consider the outer portion of a swept back wing to be a flattened winglet, as the effects of the two are essentially identical. Winglets, and swept wings with washout, can take advantage of the rotated R 1 because the angle of attack of the airfoil section can be held constant. The induced thrust which is produced may not seem like much of a force, but consider that if a wing section has an L/D of 20:1, R 1 must rotate forward of the vertical just 2.86 degrees in order for that part to get a free ride. If R 1 can be rotated forward beyond 2.86 degrees, that portion of the wing is actually producing thrust. And as the L/D increases, the required angle of rotation gets smaller. See Figure 9 and Table 1. Induced thrust and aileron deflection And now the part you ve been waiting for... Take a look at Figure 10.. This illustration is of the outer segment of a twisted swept back wing with aileron installed. When the aileron is in neutral position, the resultant vector is directly over the projected yaw axis. When the aileron is deflected downward, the lift is increased substantially. The resultant is rotated forward of the axis. This induced thrust actually pushes the wing forward. When the aileron is deflected upward, the lift vector decreases in magnitude, reducing the induced thrust. (If the aileron deflection is large enough, the lift vector changes direction.) The resultant of the lift and drag vectors rotates behind the axis, pulling the wing backward. Page 4 of 17

28 Twist distributions for swept wings, Part 3 In an aileron induced turn, adverse yaw in a swept wing planform can be reduced or eliminated entirely by means of manipulating the lift and drag vectors of the outer portion of the wing through appropriate wing twist. When the wing tips are lifting downward, aileron deflection acts to reduce adverse yaw. This case can be envisioned by inverting the vector diagram for a (normal) upward lifting wing. We ve done the inverting and placed the results in Figure 11. Reducing adverse yaw Figure 12 examines the case of the unswept wing with an elliptical lift distribution with aileron deflection for a left turn. (This diagram is a reprint of Figure 5 from Part 1.) The aileron deflection increases the drag of the wing semi-span having the downward deflected aileron and decreases the drag of the wing semi-span having the aileron deflected upward. This causes a roll to the left and a yaw to the right. This adverse yaw requires a compensating rudder deflection. Figure 12 also examines the case of the swept wing which utilizes a lift distribution which is not elliptical but which does allow for coordinated turns by eliminating adverse yaw through induced thrust. The wing semi-span with the upward deflected aileron generates more drag than the wing semi-span with the downward deflected aileron. The wing rolls and yaws to the left. In this case no compensating rudder deflection is required. Swept wings without a vertical surface, like many of the Horten designs, can use wing twist in conjunction with sweep to produce coordinated turns, particularly at low speed (high C L ), as when thermalling. There may be some disadvantages to this methodology when flying at high speed (low CL), but the detrimental effects can be controlled by careful design of the ailerons, including their location, size, and deflection angles. Coming in Part 4 The next installment will devote some space to the relationships between aileron configurations, wing lift distributions, and adverse and proverse yaw. And now that we have a method of reducing or eliminating adverse yaw, we can back up a bit and take a look at what wing sweep, increased upwash and wing twist can do for the first of those three points we keep mentioning, our quest to reduce induced drag. Ideas for future columns are always welcome. RCSD readers can contact us by mail at P.O. Box 975, Olalla WA , or by at <bsquared@appleisp.net>. References: Bowers, Al. Correspondence within < list, early Galè, Ferdinando. Tailless tale. B 2 Streamlines, Olalla Washington USA, Gullberg, Jan. Mathematics: from the birth of numbers. W.W. Norton & Co., New York, Hoerner, Dr.-Ing. S.F. and H.V. Borst. Fluid-dynamic lift. Hoerner fluid dynamics, Vancouver Washington USA, Page 5 of 17

29 Twist distributions for swept wings, Part 3 Horten, Dr. Reimar. Lift distribution on flying wing aircraft. Soaring June 1981, pp Jones, Bradley. Elements of Practical Aerodynamics, third edition. John Wiley & Sons, New York, Jones, Robert T. Wing theory. Princeton University Press, Princeton New Jersey USA, Kermode, A.C. Mechanics of flight. Pitman, London, Masters, Norm. Correspondence within < list, early McCormick, Barnes W. Aerodynamics, aeronautics, and flight mechanics. John Wiley and Sons, New York, Nickel, Karl, and Michael Wohlfahrt. Tailless aircraft in theory and practice. American Institute of Aerodynamics and Astronautics, Washington D.C., Shevell, Richard S. Fundamentals of flight. Prentice-Hall, Englewood Cliffs New Jersey USA, Simons, Martin. Model aircraft aerodynamics. Argus Books, Hemel Hempstead Great Britain, Smith, H.C. Skip. The illustrated guide to aerodynamics, second edition. TAB Books, Blue Ridge Summit Pennsylvania USA, The White Sheet, Spring 1986, No. 36. Sean Walbank editor. White Sheet Radio Flying Club, Dorset/Somerset Great Britain. Page 6 of 17

30 Twist distributions for swept wings, Part 3 Tip Direction of flight Right semi-span ß, sweep angle Root Figure 1. Movement of the stagnation point along the span of a swept back wing, exaggerated. Page 7 of 17

31 Twist distributions for swept wings, Part 3 Tip Direction of flight Right semi-span ß, sweep angle Root Figure 2. Alignment of wing sections to local flow by means of geometric twist, exaggerated. Page 8 of 17

32 Twist distributions for swept wings, Part 3 brick, five pounds A B C Figure 3. Examples of vector quantities. L L R 1 L R 1 R 1 A D D D D R 1 L R 1 L R 1 L B D D D D Figure 4. Calculation of resultant vectors Page 9 of 17

33 Twist distributions for swept wings, Part 3 L R 1 T D A L R 1 R 2 W T D B L R 1 R 2 W T D C L R 1 T i R 2 T D D R 2 Figure 5. Force vectors on powered and unpowered aircraft which are otherwise identical. Page 10 of 17

34 L R 1 T D A R 1 R 2 L D B T R 2 D C R 1 L D D W W Figure 6. Force vectors on powered and unpowered aircraft which are otherwise identical. Page 11 of 17

35 Twist distributions for swept wings, Part 3 L R Air flow α winglet MAC/4 T i D Figure 7. Induced thrust, T i, generated by winglet. Page 12 of 17

36 Twist distributions for swept wings, Part 3 L R D A. L R L R D B. D C. L R 1 R 2 Direction of flight D 1 D 2 L R D. D E. Figure 8. Rotation of lift vector caused by upwash, and a single case of the effect of twist on that rotation. Page 13 of 17

37 Twist distributions for swept wings, Part 3 vertical vertical L R 1 L R 1 R 1 D i L D i vertical T i γ D D D Figure 9. Force vectors demonstrating reduction of induced drag, D i, and development of induced thrust, T i, by rotation of lift and drag resultant, R 1. Table 1: L/D and required rotation of R 1 for D i = 0 L/D γ, R 1 vertical 10: degrees 20: degrees 30: degrees 40: degrees Page 14 of 17

38 Twist distributions for swept wings, Part 3 L R D Neutral aileron L R Direction of flight D Down aileron L R D Up aileron Figure 10. Generalized representation of the direction and strength of forces when the outboard aileron of a twisted swept wing is deflected. Page 15 of 17

39 Twist distributions for swept wings, Part 3 D Neutral aileron L R D Up Aileron L R Direction of flight L R D Down aileron Figure 11. Generalized representation of the direction and strength of forces when the outboard aileron of a downward lifting wing segment is Page 16 of 17