Airplane wing

Abstract

The invention relates to a wing with two winglets (9-12) and a respective airplane. An upstream winglet (9, 11) broadens a region of inclined airflow and a downstream winglet (10, 12) produces a thrust contribution therein.

Claims

1. A wing for an airplane having a spanwise wing length from a base body of said airplane towards an outer wing end, and at least three winglets on said outer wing end connected to said wing, wherein an upstream first winglet of said at least three winglets precedes a downstream second winglet of said at least three winglets, said second winglet preceding an even more downstream third winglet of said at least three winglets in a flight direction of said airplane, said first winglet being arranged to produce a winglet tip vortex airflow additionally to a wing tip vortex airflow produced by said wing, such that said winglet tip vortex airflow and said wing tip vortex airflow are superposed to a combined vortex airflow in a plane between said first winglet and said second winglet and perpendicular to the flight direction of the airplane, wherein said first winglet is arranged to broaden a region of said combined vortex airflow by means of said superposition, and wherein said second winglet is adapted to produce an aerodynamic lift having a positive thrust component (Fxn), in said combined vortex airflow.

2. The wing of claim 1, wherein an air velocity angle relative to said flight direction of said combined vortex air flow, dependent on a distance from a region of maximum air velocity angle at said outer wing end, has an intermediate maximum air velocity angle at a select distance from said outer wing end and does not fall to values of said air velocity angle below 25% of a smaller one of said maximum air velocity angle and said intermediate air velocity angle, and also not below 25% of a larger one of said maximum air velocity angle, between said intermediate maximum air velocity angle and said outer wing end maximum.

3. The wing of claim 1, wherein said first winglet has a length of between 3% and 8% of said wing length.

4. The wing of claim 1, wherein said first winglet has an aspect ratio of between 3 and 7.

5. The wing of claim 1, wherein said second winglet has an aspect ratio of between 3 and 7.

6. The wing of claim 1, wherein said second winglet has an asymmetric wing profile for increasing said thrust component.

7. The wing of claim 1, wherein said first and said second winglets have an upward orientation relative to said wing in said flight direction.

8. The wing of claim 1, wherein said second winglet has a spanwise length of between 105% and 180% of a spanwise length of said first winglet.

9. The wing of claim 1, wherein an air velocity angle relative to said flight direction of said combined vortex airflow, dependent on a distance from a region of a maximum air velocity angle at said outer wing end, maintains a value of at least 25% of said maximum air velocity angle up to a value of said distance of at least 5% of said spanwise wing length of said wing.

10. The wing of claim 1, wherein said first winglet is upwardly inclined relative to said second winglet and said second winglet is upwardly inclined relative to said third winglet.

11. The wing of claim 1, wherein said third winglet is adapted to produce a lift having a positive thrust component.

12. The wing of claim 1, wherein said third winglet has a length of between 60% and 120% of said second winglet which is upstream of said third winglet.

13. An airplane having: two mutually opposed wings, each wing including a spanwise wing length from a base body of said airplane towards an outer wing end, and at least three winglets on said outer wing end and connected to said wing, wherein said at least three winglets comprise an upstream first winglet of said at least three winglets preceding a downstream second winglet of said at least three winglets, said second winglet preceding an even more downstream third winglet of said at least three winglets in a flight direction of said airplane, said first winglet being arranged to produce a winglet tip vortex airflow additionally to a wing tip vortex airflow produced by said wing, such that said winglet tip vortex airflow and said wing tip vortex airflow are superposed to a combined vortex airflow in a plane between said first winglet and said second winglet and perpendicular to a flight direction of the airplane, and wherein said first winglet is arranged to broaden a region of said combined vortex airflow by means of said superposition, and wherein said second winglet is adapted to produce an aerodynamic lift having a positive thrust component (F.sub.xn), in said combined vortex airflow.

14. A method comprising: mounting to a wing end of an airplane, a part comprising at least three winglets, the at least two winglets comprising an upstream first winglet of said at least three winglets positioned to precede a downstream second winglet of said at least three winglets, said first winglet being arranged to produce a winglet tip vortex airflow additionally to a wing tip vortex airflow produced by said wing, such that said winglet tip vortex and said wing tip vortex airflow are superposed to a combined vortex airflow in a plane between said first winglet and said second winglet and perpendicular to a flight direction of the airplane, and wherein said first winglet is arranged to broaden a region of said combined vortex airflow by means of said superposition, and wherein said second winglet is adapted to produce an aerodynamic lift having a positive thrust component (F.sub.xn), in said combined vortex airflow.

Description

(1) The invention will hereunder be explained in further details referring to exemplary embodiments below which are not intended to limit the scope of the claims but meant for illustrative purposes only.

(2) FIG. 1 shows a plan view of an airplane according to the invention including six winglets schematically drawn;

(3) FIG. 2 is a schematic diagram for explaining the creation of a thrust by a winglet;

(4) FIG. 3a, b are schematic illustrations of the air velocity distribution in a tip vortex;

(5) FIG. 4 is a schematic perspective view of a wing according to the invention;

(6) FIG. 5 is a schematic front view of a wing tip according to the invention including two winglets;

(7) FIG. 6 is a diagram showing two graphs of an inclination angle dependency on distance relating to FIG. 5;

(8) FIG. 7 is a schematic side view to explain the gamma angles of two winglets of an embodiment;

(9) FIG. 8 is a front view of the same winglets to explain the delta angles;

(10) FIG. 9 is a plan view of an Airbus A320 main wing;

(11) FIG. 10 is a front view of said wing;

(12) FIG. 11 is a side view of said wing;

(13) FIG. 12 is a side view to explain reference lines used for simulations in the embodiment;

(14) FIG. 13 is a top view to illustrate the same reference lines;

(15) FIGS. 14 to 17 are diagrams illustrating beta angles at varying distances from the main wing tip for various simulations in the embodiment;

(16) FIG. 18 is a front view of three winglets according to an embodiment of the invention showing their dihedral angles;

(17) FIG. 19 is another front view of two winglets for explaining a relative dihedral angle;

(18) FIG. 20 is a schematic drawing for explaining a bending of a first winglet;

(19) FIG. 21 is a side view of sections of a main wing and three winglets for explaining angles of inclination;

(20) FIG. 22 combines a front view and a top view for explaining a sweepback angle of a winglet;

(21) FIG. 23 is a top view onto three winglets in a plane for explaining the shape;

(22) FIG. 24 is a perspective drawing of a complete airplane according to the invention;

(23) FIG. 25 is a top view onto three winglets at a main wing tip of said airplane;

(24) FIG. 26 is a side view of the three winglets of FIG. 25; and

(25) FIG. 27 is a front view thereof.

(26) FIG. 1 is a plan view of an airplane 1 having two main wings 2 and 3 and two horizontal stabilizers 4 and 5 as well as a vertical tail 6 and a fuselage or base body 7. FIG. 1 shall represent an Airbus model A 320 having four propulsion engines, not shown here. However, in FIG. 1, the main wings 2 and 3 each have three winglets 8, 9, 10, respectively. Two respective winglets sharing a reference numeral are mirror symmetrical to each other in an analogous manner as both main wings 2 and 3 and the base body 7 are mirror symmetric with regard to a vertical plane (perpendicular to the plane of drawing) through the longitudinal axis of the base body.

(27) Further, an x-axis opposite to the flight direction and thus identical with the main airflow direction and a horizontal y-axis perpendicular thereto are shown. The z-axis is perpendicular and directed upwardly.

(28) FIG. 2 is a schematic side view of an airfoil or profile (in FIG. 2 a symmetric standard wing airfoil, in case of the A 320 an asymmetric airfoil) of a main wing 2 and an airfoil (for example NACA 2412, a standard asymmetric wing airfoil or RAE 5214, an asymmetric wing airfoil for transonic flight conditions) of an exemplary winglet W which is just for explanation purposes.

(29) A solid horizontal line is the x-axis already mentioned. A chain-dotted line 13 corresponds to the chord line of the main wing 2 (connecting the front-most point and the end point of the profile), the angle alpha there between being the angle of attack of the main wing.

(30) Further, a bottom line 14 of the profile of winglet W (which represents schematically one of winglets 8, 9, 10) is shown and the angle between this bottom line 14 and the bottom line of the main wing profile is gamma, the so-called angle of incidence. As regards the location of the definition of the chord lines along the respective span of the wing and the winglets reference is made to what has been explained before.

(31) FIGS. 3a and b illustrate a tip vortex as present at any wing tip during flight. The fields of arrows at the right sides symbolize the component of the airflow velocity in the plane of drawing as regards direction and magnitude (arrow length). FIG. 3a shows a point of x=2.5 m (x=0 corresponding to the front end of the wing tip) and FIG. 3b relating to a downstream location of x=3.4 m. It can be seen that the tip vortex “develops with increasing x” and that the vortex is quite concentrated around the wing tip and quickly vanishes with increasing distance therefrom. This statement relates to almost any direction when starting from the wing tip with no qualitative but also small quantitative differences.

(32) Further, FIGS. 3a and b illustrate that the wing tip vortex principally adds some upward component to the airflow velocity together with some outward component in the lower region and some inward component in the upper region. With this in mind, it can be understood that FIG. 2 shows a local flow direction having an angle beta to the flight direction x. This local flow direction (components perpendicular to the plane of drawing of FIG. 2 being ignored) attacks the symbolic winglet W and causes a lift L.sub.n thereof as shown by an arrow. This lift is perpendicular to the flow direction by definition. It can be seen as a superposition of a vertically upward component and a positive thrust component F.sub.xn,L.

(33) Principally the same applies for the drag D.sub.n of the winglet W. There is a negative thrust component of the drag, namely F.sub.xn,D. The thrust contribution of the winglet W as referred to earlier in this description is thus the difference thereof, namely F.sub.xn=F.sub.xn,L−F.sub.xn,D and is positive here. This is intended by the invention, namely a positive effective thrust contribution of a winglet.

(34) FIG. 4 shows the main wing 2 and exemplary two winglets of FIG. 2, namely 8 and 9. Wing 2 is somewhat inclined relative to the y-axis by a so called sweepback angle and has a chord line length decreasing with the distance from the base body 7 from a root chord line length cr to a tip chord line length ct. At a wing outer end 15, winglets 8 and 9 are mounted, compare also FIG. 5.

(35) FIG. 5 shows the wing 2 and the winglets 8 and 9 in a projection on a y-z-plane and the length b of main wing 2 (b being measured from the centre of base body 7 at y=0 along the span of main wing 2 as explained before) and respective lengths b1 and b2 of winglets 8 and 9, respectively. For simplicity, wing 2 and winglets 8 and 9 are shown straight and horizontal, only. However, an inclination relative to wing 2 around an axis parallel to the x-axis would not lead to qualitative changes.

(36) FIG. 6 shows a diagram including two graphs. The vertical axis relates to beta (compare FIG. 2), namely the angle of inclination of the local airflow direction in a projection on a x-z-plane.

(37) The horizontal line shows “eta”, namely the distance from outer wing end 15 divided by b, the length of main wing 2.

(38) A first graph with crosses relates to the condition without winglets 8 and 9 and thus corresponds to FIGS. 3a and b, qualitatively. The second graph showing circles relates to an airflow distribution downstream of first winglet 8 and thus upstream of second winglet 9 (the first graph relating to the same x-position). The graphs result from a computer simulation of the airflow distribution (such as FIGS. 3a and b).

(39) It can easily be seen that the first graph shows a maximum 16 closely to outer wing end 15 whereas the second graph has a maximum 17 there, an intermediate minimum at around eta=1.025 and a further maximum 18 at around eta=1.055, and decreases outwardly therefrom. Further, the second graph drops to a value of more than 50% of its smaller (left) maximum and more than 40% of its larger (right) maximum whereas it drops to a value of still more than 25% of its larger maximum at about eta=1.1, e.g. at a distance of about 10% of b from outer wing end 15. This angle distribution is a good basis for the already described function of winglet 9, compare FIG. 2.

(40) Simulations on the basis of the airplane type Airbus A320 have been made. They will be explained hereunder. So far, the inventors achieve around 3% reduction of the overall drag of the airplane with three winglets as shown in FIG. 1 by means of the thrust contribution of the winglets and a small increase of the overall lifting force (in the region of maybe 1% lift increase). The lift increase enables the airplane to fly with a somewhat lower inclination (compare alpha in FIG. 2) which leads to a further reduction of the overall drag. These simulations have been made by the computer programme CFD (computational fluid dynamics) of ANSYS.

(41) As a general basic study, computer simulations for optimization of the thrust contribution of a two winglet set (first and second winglet) with a standard NACA 0012 main wing airfoil and a NACA 2412 winglet airfoil and without any inclination of the winglet relative to the main wing (thus with a setup along FIGS. 4 and 5) have shown that an aspect ratio 5 is a good choice. Although higher aspect ratios are more efficient in an aerodynamic sense, they have a smaller area and thus, produce smaller forces (and thus a small thrust). In other words, within the limitation of a length b2 (span) of 1.5 m (for the A320), a substantial winglet area is preferred. On the other hand, a too low aspect ratio increases the drag and decreases the efficiency in an amount that finally reduces the effective thrust by means of an increased drag. All in all, the CFD simulations repeatedly showed optimum values around 5.

(42) On this basis, the length b1 of the upstream first winglet 8 for the A320 has been chosen to be ⅔, namely 1 m in order to enable the downstream second winglet 9 to take advantage of the main part of the broadened vortex region, compare again the setup of FIGS. 4 and 5 and the results in FIG. 6.

(43) The mean chord length results from the length of the fingers and from the fixed aspect ratio. As usual for airplane wings, there is a diminution of the chord line length in an outward direction. For the first upstream winglet 8, the chord line length at the root is 400 mm and at the top is 300 mm, whereas for the downstream second winglet 9 the root chord length is 600 mm and the tip chord length 400 mm. These values have been chosen intuitively and arbitrarily.

(44) For the winglets, instead of the above mentioned (readily available) NACA 2412 of the preliminary simulations, a transonic airfoil RAE 5214 has been chosen which is a standard transonic airfoil and is well adapted to the aerodynamic conditions of the A320 at its typical travel velocity and altitude, compare below. The Airbus A320 is a well-documented and economically important model airplane for the present invention.

(45) The most influential parameters are the angles of incidence gamma and the dihedral angle delta (namely the inclination with respect to a rotation around an axis parallel to the travel direction). In a first coarse mapping study, the mapping steps were 3° to 5° for gamma and 10° for delta. In this coarse mapping, a first and a second but no third have been included in the simulations in order to have a basis for a study of the third winglet.

(46) FIG. 7 illustrates the angle gamma, namely gamma 1 of winglet 8, the first winglet, and gamma 2 of winglet 9, the second winglet, both shown as airfoils (compare FIG. 2) and with their chord lines in relation to the main wing airfoil and its chord line. FIG. 8 illustrates the angle delta in a perspective as in FIG. 5, but less schematic. Again, delta 1 is related to the first winglet 8 and delta 2 to the second winglet 9. The structures in the left part of FIG. 8 are transient structures as used for the CFD simulations. These structures do not correspond to the actual A320 main wing to which the winglets, the slim structures in the middle and the right, have to be mounted but they define a pragmatic model to enable the simulation.

(47) FIG. 9 shows a plan view onto a main wing of the A320, the wing tip is oriented downward and the base body is not shown but would be on top. FIG. 9 shows a main wing 20 of the A320 which actually has a so called fence structure, namely a vertical plate, at the end of the wing which has been omitted here, because it is to be substituted by the winglets according to the invention.

(48) FIG. 10 shows the main wing 20 of FIG. 9 in a front view, in FIG. 11 shows the main wing 20 in a side view (perspective perpendicular to the travel direction—X). The somewhat inclined V geometry of the main wings of the A320 can be seen in FIGS. 10 and 11.

(49) A typical travel velocity of 0.78 mach and a typical travel altitude of 35,000 feet has been chosen which means an air density of 0.380 kg/m.sup.3 (comparison: 1.125 kg/m.sup.3 on ground), a static pressure of 23.842 Pa, a static temperature of 218.8 K and a true air speed (TAS) of 450 kts which is 231.5 m/s. The velocity chosen here is reason to a compressible simulation model in contrast to the more simple incompressible simulation models appropriate for lower velocities and thus in particular for smaller passenger airplanes. This means that pressure and temperature are variables in the airflow and that local areas with air velocities above 1 Mach appear which is called a transsonic flow. The total weight of the aircraft is about 70 tons. A typical angle of attack alpha is 1.7° for the main wing end in in-flight shape. This value is illustrated in FIG. 2 and relates to the angle between the chord line of the main wing at its tip end to the actual flight direction. It has been determined by variation of this angle and calculation of the resultant overall lifting force of the two main wings. When they equal the required 70 to, the mentioned value is approximately correct.

(50) In this mapping, a certain parameter set, subsequently named V0040, has been chosen as an optimum and has been the basis for the following more detailed comparisons.

(51) The gamma and delta values of winglets 8 and 9 (“finger 1 and finger 2”) are listed in table I which shows that first winglet 8 has a gamma of −10° and a delta of −20° (the negative priority meaning an anti-clockwise rotation with regard to FIGS. 7 and 8) whereas second winglet 9 has a gamma of −5° and a delta of −10°. Starting therefrom, in the third and fourth line of table I, gamma of the first winglet 8 has been decreased and increased by 2°, respectively, and in the fifth and sixth lines, delta of first winglet 8 has been decreased and increased by 10°, respectively. The following four lines repeat the same schedule for second winglet 9. For comparison, the first line relates to a main wing without winglet (and without fence). In the column left from the already mentioned values of gamma and delta, the numbers of the simulations are listed. V0040 is the second one.

(52) From the sixth column on, that is right from the gamma and delta values, the simulation results are shown, namely the X-directed force on an outward section of the main wing (drag) in N (Newton as all other forces). In the seventh column, the Z-directed force (lift) on this outward section is shown. The outward section is defined starting from a borderline approximately 4.3 m inward of the main wing tip. It is used in these simulations because this outward section shows clear influence of the winglets whereas the inward section and the base body do not.

(53) The following four columns show the drag and the lift for both winglets (“finger 1 and 2” being the first and second winglet). Please note that the data for “finger 1” in the first line relates to a so-called wing tip (in German: Randbogen) which is a structure between an outward interface of the main wing and the already mentioned fence structure. This wing tip is more or less a somewhat rounded outer wing end and has been treated as a “first winglet” here to make a fair comparison. It is substituted by the winglets according to the invention which are mounted to the same interface.

(54) The following column shows the complete lift/drag ratio of the wing including the outward and the inward section as well as the winglets (with the exception of the first line).

(55) The next column is the reduction achieved by the two winglets in the various configurations with regard to the drag (“delta X-force”) and the respective relative value is in the next-to-last column.

(56) Finally, the relative lift/drag ratio improvement is shown. Please note that table I comprises rounded values whereas the calculations have been done by the exact values which explains some small inconsistencies when checking the numbers in table I.

(57) It can easily be seen that V0040 must be near a local optimum since the drag reduction and the lift drag ratio improvement of 2.72% and 6.31%, respectively, are with the best results in the complete table. The small decrease of gamma of the first winglet 8 (from −10 to −8) leads to the results in the fourth line (V0090) which are even a little bit better. The same applies to a decrease of delta of the second winglet 9 from −10° to 0°, compare V0093 in the next-to-last line. Further, a reduction of delta of the first winglet 8 from −20° to −30° leaves the results almost unchanged, compare V0091. However, all other results are more or less remarkably worse.

(58) FIG. 12 shows a side view in the perspective of FIG. 11 but with the two winglets added to the main wing in FIG. 11 and, additionally, with two hatched lines for later reference (reference lines for air velocity angle) and FIG. 13 shows a plan view onto the main wing tip and the two winglets with the same reference lines as in FIG. 12. Both reference lines are upstream of the respective leading edge of the winglet by 10 cm and are parallel to said leading edge.

(59) FIG. 14 is a diagram comparable to FIG. 6, namely showing the angle beta on the vertical axis and the distance from the main wing tip along the reference lines just explained. The basic parameter set and simulation V0040 is represented by circles, V0046 is represented by triangles, and V0090 is represented by diamonds. The solid lines relate to the reference line upstream of the first winglet 8 and the dotted lines to the other one, upstream of the second winglet 9 and downstream of the first winglet 8. Table I clarifies that V0046 has a reduced gamma of the first winglet 8 and V0090 an increased gamma of the first winglet 8 with a step size 2°.

(60) First of all, the graphs show that the first winglet 8 produces a significantly “broadened” vortex region, even upstream of the first winglet 8 as shown by the solid lines. In contrast to FIG. 6, there is no pronounced second maximum (18 in FIG. 6) but a more or less constant beta angle between 0.5 m and about 1.2 m. The respective length of the main wing is 16.35 m which means for example an eta of 1.031 for 1.5 m and of 1.07 for 1.2 m, approximately (compare FIG. 6).

(61) This beta value is in the region of 9° which is in the region of 70% of the maximum at 0° (both for the reference line between both winglets, i. e. the dotted graph). Further, with the reduced gamma value, V0046 (triangles) shows an increased beta upstream of the first winglet 8 and a decreased beta downstream thereof.

(62) Contrary to that, with increased gamma, V0090 shows an increased beta downstream of the first winglet 8 and a decreased beta upstream thereof. Thus, the inclination gamma (angle of incidence) can enhance the upwards tendency of the airflow in between the winglets, in particular for places closer to the main wing tip than 1 m, compare FIG. 14. In this case, the beta values above a distance of 1 m are not deteriorated thereby. The results in table I show, that the overall performance of this parameter set is even a little bit better than V0040. This is obviously due to a reduced overall drag (although the angle of incidence has been increased), i. e. by a stronger contribution to the overall thrust.

(63) On the other hand, a reduction of the gamma value from 10° to 8° and thus from V0040 to V0046 clearly leads to substantially deteriorated results, compare table I. Consequently, in a further step of optimization, gamma values higher, but not smaller than 10° and possibly even a little bit smaller than 12° could be analyzed.

(64) Further, FIG. 15 shows an analogous diagram, but for V0040 in comparison to V0092 and V0091. Here, the angle delta of the first winglet 8 has been varied from −20° to −10° and to −30°, compare table I and FIG. 8. Obviously, this has little impact on the air velocity angle (beta) distribution upstream of the first winglet 8 (solid lines) but it has an impact on the airstream angles downstream thereof (dotted lines). Again, the beta values increase a little bit for distances below 1 m by increasing the delta value, namely for V0091. The respective performance results in table I are almost identical with those of V0040 and obviously the beta values in FIG. 15 as well.

(65) On the other hand, decreasing the delta value to −10 and thus bringing both winglets in line (as seen in the flight direction) qualitatively changes the dotted graph in FIG. 15. The beta values are reduced up to about 1 m, namely the length of the first winglet 8, and are clearly increased above that distance value. Seemingly, the second winglet 9 is somewhat in the lee of the first winglet 8 up to 1 m and “sees” the winglet tip vortex thereof at distances above 1 m. In summary, this does not improve the results but leads to some deterioration, as table I shows. The inventors assume that the beta increase at distances above 1 m does not compensate for the beta decrease at smaller distances.

(66) FIG. 16 shows another analogous diagram, now relating to a variation of the gamma angle of the second winglet 9. Again, this obviously has not much impact on the beta values upstream of the first winglet 8 (solid lines), but has a substantial impact on the beta values in between both winglets (dotted lines). Here, the beta values increase with a small decrease of gamma from 5° to 3° and, in the opposite, they decrease with an increase of gamma from 5° to 7°. In a similar manner as the solid lines in FIG. 14, a turning into the airstream of the winglet obviously decreases the inclination of the airstream upstream of the winglet. The results in table I clearly show that both variations, V0038 and V0042 decrease the performance results. In particular, the reduction of beta between both winglets by an increase of gamma of the second winglet 9 substantially deteriorates the lift/drag improvement. Further, a too strong inclination of the winglet does produce more lift but also produces over-proportionally more drag and thus leads to a deterioration.

(67) Obviously, with a next step of optimization, the gamma value of the downstream winglets should be left at 5°.

(68) Finally, FIG. 17 relates to a variation of the delta angle of the second winglet 9 and leads to similar results as FIG. 15: for V0094, the delta values of both winglets are −20° and again the second 9 winglet seems to be in the lee of the upstream winglet and shows a strong impact by the winglet tip vortex thereof which leads to comparatively bad results, in particular with regard to the lift drag ratio. Increasing the delta difference between both winglets by V0093 does not change much in the beta values and leads to similar (somewhat improved) results in table I. Again, with a next step of optimization, the range of delta for the second winglet 9 between 0° and −10° is interesting.

(69) On the basis of the above results, further investigations with three winglets and again based on what has been explained above in relation to the A320 have been conducted. Since the number of simulations feasible in total is limited, the inventors concentrated on what has been found for two winglets. Consequently, based on the comparable results with regard to the drag reduction of more than 2.7 and the lift/drag ratio for the complete wing (compare the fourth-last and second-last column in table I), the parameters underlying V0040, V0090, V0091, and V0093 were considered in particular. Consequently, simulations with varying values for the angle of incidence gamma and the dihedral angle delta of the third winglet were performed on the basis of these four parameter sets and were evaluated in a similar manner as explained above for the first and second winglet.

(70) Simultaneously, data with regard to the in-flight shape of the main wing of the A320 were available with the main impact that the chord line at the wing end of the main wing is rotated from the so-called jig shape underlying the calculations explained above by about 1.5°. This can be seen by the slightly amended gamma values explained below. Still further, data relating to the drag of the complete airplane for different inclinations thereof were available, then, so that the impact of an improvement of the overall lift (by a lift contribution of the winglets as well as by an increase of the lift of the main wing due to a limitation of the vortex-induced losses) on the overall drag due to a variation of the inclination of the airplane could be assessed.

(71) The results (not shown here in detail) showed that the V0091 basis proved favourable. The respective embodiment will be explained hereunder.

(72) FIG. 18 shows a front view of the winglets 8, 9, 10 of this embodiment as seen in the x-direction and illustrates the dihedral angles delta 1, 2, 3 of the three winglets. The upper most winglet is the first one, the middle winglet is the second one, and the lowest winglet is the third downstream one. FIG. 18 shows qualitatively, that a substantial, but limited relative dihedral angle between the succeeding winglets has proven to be advantageous also for the three winglet embodiment.

(73) Taking this opportunity, FIG. 19 explains the definition of a relative dihedral angle. In the same perspective as FIG. 18, the first and the second winglet are shown together with two radii r1 and r2 of different size. The meeting point of a vertical and the horizontal line is the root R (at the splitting point horizontally and the meeting of the leading edges vertically) and one vertex of an isosceles triangle shown, the other two vertices of which are on the leading edges of the two winglets and referred as V1 and V2. The angle between the line R-V1 and the line R-V2 is the relative dihedral angle if taken as an average over all radii ri possible within the shorter one of the two winglets, namely the first one.

(74) The visible difference between the line R-V1 from the leading edge of the first winglet is connected to the bending of the first winglet to be explained hereunder which is also the background of the deviation between the line for delta 1 and the first winglet in FIG. 18.

(75) In this connection, the inventors have found that average relative dihedral angles in this sense from 5° to 35° with more preferred lower limits of 7°, 9°, 11°, 13° and 15° and more preferred upper limits of 33°, 31°, 29°, 27°, and 25°, are preferred both with regard to the first and second winglets and to the second and third winglets (if any) in a general sense and also independently of the embodiments. A certain synergy between the winglets can be upheld whereas a too much “in the lee” position of a downstream winglet can be avoided.

(76) FIG. 20 illustrates the above mentioned bending of the first winglet which is so to say a distribution of a part of the dihedral angle along a certain portion of the spanwise length. Actually, in FIG. 20, a leading edge L is schematically shown to start from a root R and to be bent along a circular arch shape B extending over one third (330 mm) of its length with a radius of 750 mm and an arch angle of −15°. Already at the start of R the leading edge of the first winglet has a dihedral angle of −20°. This means that outwards of the bending, the dihedral angle for the second and third third of the length of the first winglet is actually −35°. In an average along the complete spanwise length of the first winglet from R to its outward end, an average dihedral angle of about −30° results, −15° of which have been “distributed” along the arch as described.

(77) The reason is that in this particular embodiment, a straight leading edge of the first winglet with a dihedral angle of −30° has made it somewhat difficult to provide for a smooth transition of a leading edge to that one of the main wing end (in the so-called fairing region) whereas with −20° dihedral angle, the smooth transition has not caused any problems. Therefore, in order to enable an average value of −30°, the solution of FIG. 20 has been chosen.

(78) In general, it is within the teaching of this invention to use winglet shapes that are not straight along the spanwise direction such as shown in FIG. 20. They could even be arch shaped along the complete length as pointed out before. What is most relevant in the view of the inventors, is the relative dihedral angle in an average sense. If for example, a first and a second winglet would both be arch shaped in a similar manner so that the isosceles triangle construction explained earlier with a fixed vertex at the root would be inclined more and more with increasing length of the equal sides thereof due to the curvature of the winglet leading edges, the relative dihedral angle according to this construction might even remain almost constant along the leading edges. Still, at a certain portion along the spanwise length of for example the second winglet, the proximate portion along the spanwise length of the first winglet would be positioned relative to the second winglet in a manner that is well described by the relative dihedral angle (remember the somewhat rotationally symmetrical shape of the vortex at the wing end) and is well described by the triangle construction.

(79) The absolute dihedral angles of the second and the third winglet in this embodiment are delta 2=−10° and delta 3=+10° wherein these two winglets of this embodiment do not have an arch shape as explained along FIG. 20. Consequently, the relative dihedral angle between the first and the second winglet is 20°, is the same as the relative dihedral angle between the second and the third winglet, and the first winglet is more upwardly inclined than the second winglet, the second winglet being more upwardly inclined than the third winglet, compare FIG. 18. The angle delta 1 shown in FIG. 18 is the starting dihedral angle at the root of the first winglet, namely −20° instead of the average value of −30°.

(80) As regards the angles of incidence, reference is made to FIG. 21 showing a side view and sections through the three winglets 8, 9, 10, and the main wing 2. The sectional planes are different, naturally, namely 10% outward of the spanwise length of the winglets from the respective splitting positions, and 10% inward in case of the main wing 2, to provide for undisturbed chord lines. The chord lines and the respective angles gamma 1, 2, 3 are shown in FIG. 21. The angles are gamma 1=−9° for the first winglet, gamma 2=−4° for the second winglet and gamma 3=−1° for the third winglet, all being defined relative to the main wing chord line at the described outward position and in the in-flight shape of the winglets and of the main wing (all parameters explained for this embodiment relating to the in-flight shape).

(81) FIG. 21 also shows the respective rotating points on the chord line of main wing 2 as well as on the chord line of the respective winglet 8, 9, 10. In terms of the respective chord line length of the winglets, the rotating points are approximately at a third thereof. In terms of the chord line length of main wing 2, the rotating point of the first winglet is at 16.7% (0% being the front most point on the chord line), the rotating point of the second winglet is at 54.8%, and the rotating point of the third winglet is at 88.1%.

(82) FIG. 22 illustrates the sweepback angle epsilon of a representative winglet 9, namely the angle between the leading edge thereof and a direction (y in FIG. 22) being horizontal and perpendicular to the flight direction. Herein, winglet 9 is thought to be horizontal (delta and gamma being zero in a fictious manner). alternatively, the spanwise length of winglet 9 could be used instead of its actual extension in the y-direction when being projected onto a horizontal plane. Please note that also the arch shape of winglet 8 as explained along FIG. 22 would be regarded to be unrolled. In other words, the spanwise length includes the length of the arch.

(83) In the present embodiment, the sweepback angle of the main wing 2 is 27.5°. Variations starting from this value showed that an increased sweepback angle of 32° is preferable for the winglets, in other words 4.5° sweepback angle relative to the main wing's sweepback angle. This applies for the second and for the third winglets 9, 10 in this embodiment whereas for the first winglet 8, the sweepback angle has been increased slightly to 34° in order to preserve a certain distance in the x-direction to the leading edge of the second winglet 9, compare the top view in FIG. 25 explained below.

(84) FIG. 23 is a fictious top view onto the three winglets 8, 9, 10, to explain their shape. It is fictious because the dihedral angles and the angles of incidence are zero in FIG. 23 and the arch shape of the first winglet 8 is unrolled. FIG. 23, thus, shows the respective spanwise length b1, 2, 3. It further shows the chord line lengths cr1, 2, 3, at 10% of the spanwise length outward of the splitting points (these being at the bottom of FIG. 23) as well as the tip chord line lengths ct1, 2, 3, at 10% inward of the winglets' tips.

(85) The actual values are (in the order first, second, third winglet): a root chord length cr of 0.4 m, 0.6 m, 0.4 m; a tip chord length ct of 0.3 m, 0.4 m, 0.25 m; a spanwise length b of 1 m, 1.5 m, 1.2 m. This corresponds to a root chord length cr of approximately 25% of the main wing chord length at its end (as defined), approximately 37% and approximately 25%; a tip chord length relative to the root chord length of 75%, 67% and 63%; and a spanwise length relative to the spanwise main wing length (16.4 m) of 6.1%, 9.2%, 7.3%, respectively.

(86) Please note that the angle of sweepback as shown in FIG. 23 is no rotating operation result. This can be seen in that the chord line lengths cr and ct remain unchanged and remain in the x-z-plane, in other words horizontal in FIG. 23. This is necessary in order not to disturb the airfoil by the introduction of the sweepback angle.

(87) Still further, FIG. 23 shows a rounding of the respective outer forward corner of the winglets' shape. This rounding relates to the region between 90% and 100% of the spanwise length wherein the chord line length is continuously reduced from 90% to 100% spanwise length by 50% of the chord line length such that in the top view of FIG. 23 an arch shape is generated. It is common practice to use roundings at the outer forward corners of wings to avoid turbulences at sharp corner shapes. By the just explained reduction of the chord line length in the outer 10% of the spanwise length, the qualitative nature of the airfoil can be preserved.

(88) The airfoil used here is adapted to the transonic conditions at the main wing of the A320 at its typical travel velocity and travel altitude and is named RAE 5214. As just explained this airfoil is still valid in the outer 10% of the spanwise length of the winglets.

(89) Still further, this trailing edge (opposite to the leading edge) of the winglets is blunt for manufacturing and stability reasons by cutting it at 98% of the respective chord line length for all winglets.

(90) The transformation of the shapes shown in FIG. 23 to the actual 3D geometry is as follows: first, the sweepback angles are introduced which are already shown in FIG. 23. Second, the bending of the first winglet along the inner third of its spanwise length with the radius of 750 mm and the angle of 15° is introduced. Then, the winglets are inclined by a rotation by the angle of incidence gamma. Then, the dihedral angles are adjusted, namely by inclining the first winglet by 20° upwardly (further 15° being in the bending), the second winglet by 10° upwardly and the third winglet by 10° downwardly.

(91) Please note that the above transformation procedure does not relate to the jig shape and to the geometry as manufactured which is slightly different and depends on the elastic properties of the main wing and the winglets. These elastic properties are subject of the mechanical structure of the wing and the winglets which is not part of the present invention and can be very different from case to case. It is, however, common practice for the mechanical engineer to predict mechanical deformations under aerodynamic loads by for example finite elements calculations. One example for a practical computer program is NASTRAN.

(92) Thus, depending on the actual implementation, the jig shape can vary although the in-flight shape might not change. It is, naturally, the in-flight shape that is responsible for the aerodynamic performance and the economic advantages of the invention.

(93) Table II shows some quantitative results of the three winglet embodiment just explained (P0001). It is compared to the A320 without the invention, but, in contrast to table I, including the so-called fence. This fence is a winglet-like structure and omitting the fence, as in table I, relates to the improvements by the addition of a (two) winglet construction according to the invention to a winglet-free airplane whereas table II shows the improvements of the invention, namely its three winglet embodiment, in relation to the actual A320 as used in practice including the fence. This is named B0001.

(94) The lift to drag ratios for both cases are shown (L/D) in the second and third column and the relative improvement of the invention is shown as a percentage value in the forth column. This is the case for six different overall masses of the airplane between 55 t and 80 t whereas table I relates to 70 t, only. The differences between the masses are mainly due to the tank contents and thus the travel distance.

(95) Table II clearly shows that the lift to drag improvement by the invention relative to the actual A320 is between almost 2% in a light case and almost 5% in a heavy case. This shows that the invention is the more effective the more pronounced the vortex produced by the main wing is (in the heavy case, the required lift is much larger, naturally). In comparison to table I, the lift to drag ratio improvements are smaller (around 6.3% for the best cases in table I). This is due to the positive effect of the conventional fence included in table II and to the in-flight deformation of the main wing, namely a certain twist of the main wing which reduces the vortex to a certain extend. For a typical case of 70 t, the drag reduction of an A320 including the three winglet embodiment of the invention compared to the conventional A320 including fence is about 4% (wing only) and 3% (complete airplane), presently. This improvement is mainly due to a thrust contribution of mainly the second winglet and also due to a limited lift contribution of the winglets and an improved lift of the main wing by means of a reduction of the vortex. As explained earlier, the lift contributions allow a smaller inclination of the complete airplane in travel flight condition and can thus be “transformed” into a drag reduction. The result is about 3% as just stated.

(96) For illustration, FIGS. 24 to 27 show the 3D shape of the A320 and three winglets, namely a perspective view in FIG. 24 of the complete airplane, a top view onto the main wing end and the winglets in FIG. 25 (against the z-direction), a side view (in y-direction) in FIG. 26, and finally a front view (in x-direction) in FIG. 27.

(97) The figures show smooth transitions in the fairing region between the main wing end and the winglets and also some thickening at the inward portion of the trailing edges of the first and second winglets. These structures are intuitive and meant to avoid turbulences.

(98) TABLE-US-00001 TABLE I Outboard Outboard Fin- Fin- Fin- Fin- Complete Ratio section of section of ger 1 ger 1 ger 2 ger 2 wing Lift/ wing X- wing Z- X- Z- X- Z- Ratio delta drag Drag Force Force Force Force Force Force Lift/ X- reduc- improve- Finger 1 Finger 2 (Sim) (Sim) (Sim) (Sim) (Sim) (Sim) Drag Force tion ment Run CFDC γ δ γ δ [N] [N] [N] [N] [N] [N] [—] [N] [%] [%] V204b_L02 839 68862 −38 6331 0 0 22.9 V0040_A245_L02 −10 −20 −05 −10 730 67992 −160 1805 −244 4653 24.4 −476 −2.72 6.31 V0046_A245_L02 −08 −20 −05 −10 731 68172 −151 2339 −200 4202 24.3 −422 −2.41 5.91 V0090_A245_L02 −12 −20 −05 −10 733 67839 −137 1230 −281 5135 24.4 −486 −2.78 6.32 V0092_A245_L02 −10 −10 −05 −10 719 67718 −162 1748 −223 4632 24.3 −469 −2.68 6.16 V0091_A245_L02 −10 −30 −05 −10 743 68214 −150 1716 −266 4741 24.4 −475 −2.71 6.32 V0038_A245_L02 −10 −20 −03 −10 753 68711 −173 1916 −146 5931 24.3 −368 −2.10 6.09 V0042_A245_L02 −10 −20 −07 −10 711 67221 −150 1683 −227 3272 24.2 −468 −2.67 5.44 V0093_A245_L02 −10 −20 −05 +00 709 67910 −146 1821 −240 4594 24.4 −479 −2.73 6.34 V0094_A245_L02 −10 −20 −05 −20 754 68031 −165 1683 −249 4576 24.3 −461 −2.64 5.96

(99) TABLE-US-00002 TABLE II P0001 vs B0001 - wing only Ratio Lift/Drag improvement m [t] P0001 L/D B0001 L/D [%] 55.0 27.7 27.1 1.9 60.0 27.1 26.3 2.8 65.0 25.8 24.9 3.5 70.0 24.1 23.1 4.1 75.0 22.3 21.3 4.5 80.0 20.5 19.6 4.7