Showing papers on "Lift-induced drag published in 1991"

Drag reduction in nature

Abstract: Recent studies on the drag-reducing shapes, structures, and behaviors of swimming and flying animals are reviewed, with an emphasis on potential analogs in vehicle design. Consideration is given to form drag reduction (turbulent flow, vortex generation, mass transfer, and adaptations for body-intersection regions), skin-friction drag reduction (polymers, surfactants, and bubbles as surface 'additives'), reduction of the drag due to lift, drag-reduction studies on porpoises, and drag-reducing animal behavior (e.g., leaping out of the water by porpoises). The need for further research is stressed.

Base drag reduction by control of the three-dimensional unsteady vortical structures

Abstract: The present paper deals with the wake of a 2D body equipped with a drag reduction device. The device is a 3D trailing edge consisting of alternate segments of blunt base and spanwise cavity. The aerodynamic mechanisms acting on the near wake are studied in a water tunnel from schlieren observations by thermally marking large scale structures. The results show that the efficiency of the device is directly related to the presence of longitudinal vortices. An optimization of the shapes in subsonic compressible flow had led to a decrease of more than 40% of the total drag of the profile.

Spiroid-tipped wing

Abstract: The spiroid-tipped wing, in its basic form, comprises a wing-like lifting surface and a spiroidal tip device (i.e., spiroid) integrated so as to minimize the induced drag of the wing-spiroid combination and/or to alleviate noise effects associated with concentrated vorticity wakes that trail from lifting surfaces. The ends of the spiroid are attached to the wing tip at approproate sweep and included angles to form a continuous and closed extension of the wing surface. For a fixed wing aircraft the spiroid configuration on the right side is of opposite hand to that on the left side. The spiroid geometry incorporates airfoil cross sections with specified thickness, camber and twist. The airfoil thickness varies in relation to the local sweep angle being a minimum at an intermediate position where the sweep angle is zero. The camber and twist vary approximately linearly and change sign at some intermediate position between the spiroid ends so as to produce the optimum spiroid loading. Increasing the size of the spiroid in relation to the overall span of the lifting surface is used to further reduce drag and noise. The concept of the spiroid-tipped wing may include the use of more than one spiroid on each wing tip in any number of forms which may be selected to be adaptable to other design requirements and operational limitations. More generally the spiroid wing tip system is a generic geometric concept which can be adapted to achieve drag reduction and noise for most applications which incorporate wings or wing-like devices (i.e., lifting surfaces) such as helicopters, propellers, etc. including non-aeronautical applications.

On the Vortex Wake of an Animal Flying in a Confined Volume

Abstract: When an animal flies near a boundary, the airflows it generates interact with that boundary. These interactions may have a significant effect on flight performance, as measured by quantities such as the energy rate to sustain flight, or the circulation of the vortices bound on the wing or shed in the wake (or, equivalently, by the lift and induced drag coefficients). The problem of hovering and slow flight within a confined volume is considered by a theoretical model based on helicopter practice, and by flow visualization experiments. The wake takes the form of a strong recirculating flow within the volume, and because of this recirculation the boundaries appear to cause a large reduction in the induced power required for sustained flight, even when their distance from the animal is several times greater than the wingspan. The correction factor relative to ideal momentum jet theory is greater than for the hovering ground effect, forward flight ground effect, or wind tunnel wall interference problems at comparable distances. The flow pattern that develops in the presence of floor and wall interactions in hovering or slow flight includes a large-diameter vortex ring trapped underneath the animal; this vortex ring is conjectured to be analogous to that below a helicopter in slow descending flight in the `vortex ring state'. Performance measurements for animals in hovering flight within a confined volume may underestimate power for free hovering by a significant margin. Comparable boundary effects may also be important in confined forward flight. Because of speed-related changes in the wake, and the rise in induced power at lower speeds, the appropriate correction to total mechanical power is dependent on air speed, becoming progressively greater as speed reduces. Some wind tunnel measurements of total metabolic power have produced the apparently anomalous result that power is independent of flight speed within measurement error. These observations may be explained - at least in part - by boundary effects caused by interaction between the wake and the walls of the wind tunnel.

High lift/low drag wing and missile airframe

Abstract: An air launched, air-to-surface missile which has extended range and reduced radar cross section for low observability is disclosed. The design uses a triangular cross section fuselage, for low profile drag and reduced weight, a very high aspect ratio, such as 22.5, folding wing for low induced drag and three folding tail fins for less profile drag. The wing is a composite structure of graphite/epoxy sparcaps and 2024 aluminum alloy core with glass/epoxy skins and full depth Nomex.

Experimental investigation on the effect of crescent planform on lift and drag

Abstract: Lift and drag forces were compared for elliptic and crescent wing models at cruise and climb conditions in the NASA Langley 14 X 22 ft subsonic tunnel. The force measurements were obtained for an angle-of-atta ck range from - 3 to 10 deg at a Reynolds number (based on the freestream conditions and the average wing chord) of about 1.7 x IO 6. The experiment and the accuracy of the measurements are discussed in detail. In addition, lift and drag measurements are summarized for high angle-of-attack conditions. The results indicate that, for attached flow conditions, the crescent wing with its highly swept tips generated less lift-dependent drag than the elliptic wing for given lift force, wing span, and freestream conditions. The drag reduction is thought to be the result of a favorable influence of trailing wake deformation on the pressure distribution of the crescent wing.

Aerodynamic characteristics of a rotorcraft airfoil designed for the tip region of a main rotor blade

Abstract: A wind tunnel investigation was conducted to determine the 2-D aerodynamic characteristics of a new rotorcraft airfoil designed for application to the tip region (stations outboard of 85 pct. radius) of a helicopter main rotor blade. The new airfoil, the RC(6)-08, and a baseline airfoil, the RC(3)-08, were investigated in the Langley 6- by 28-inch transonic tunnel at Mach numbers from 0.37 to 0.90. The Reynolds number varied from 5.2 x 10(exp 6) at the lowest Mach number to 9.6 x 10(exp 6) at the highest Mach number. Some comparisons were made of the experimental data for the new airfoil and the predictions of a transonic, viscous analysis code. The results of the investigation indicate that the RC(6)-08 airfoil met the design goals of attaining higher maximum lift coefficients than the baseline airfoil while maintaining drag divergence characteristics at low lift and pitching moment characteristics nearly the same as those of the baseline airfoil. The maximum lift coefficients of the RC(6)-08 varied from 1.07 at M=0.37 to 0.94 at M=0.52 while those of the RC(3)-08 varied from 0.91 to 0.85 over the same Mach number range. At lift coefficients of -0.1 and 0, the drag divergence Mach number of both the RC(6)-08 and the RC(3)-08 was 0.86. The pitching moment coefficients of the RC(6)-08 were less negative than those of the RC(3)-08 for Mach numbers and lift coefficients typical of those that would occur on a main rotor blade tip at high forward speeds on the advancing side of the rotor disk.

Drag calculations of wings using Euler methods

Abstract: Several techniques for the calculation of drag using Euler-equation formulations are discussed and compared. Surface-pressure integration (a nearfield technique) as well as two different farfield calculation techniques are described and applied to three-dimensional flow-field solutions for an aspect-ratio-7 wing with attached flow. The present calculations are limited to steady, low-Mach-number flows around three-dimensional configurations in the absence of active systems such as surface blowing/suction and propulsion. Although the main focus of the paper is the calculation of aerodynamic drag, the calculation of aerodynamic lift is also briefly discussed. Three Euler methods are used to obtain the flowfield solutions. The farfield technique based on the evaluation of a wake-integral appears to provide the most consistent and accurate drag predictions.

Mechanics on the Tip Clearance Loss of Impeller Blades

Abstract: For predicting the tip clearance loss of turbomachines, different equations are published in the literatures based on differnt principles. In 1986 the present author posturated a new theory where the pressure loss consisted of two parts, one was the pressure loss induced by the drag force of the leaked flow and the other was the pressure loss to support the axial pressure difference without blades in the tip clearance zone. There were comments such as the two losses were the same loss looked from two different view points, or at least a part of the former was included in the latter or vice versa.In this paper the pressure loss due to the tip clearance is examined based on a macroscopic balance of forces and the two kinds of loss are derived. Furthermore, it is made clear that the former comes from the induced drag which is parallel to the blade while the latter comes from the missing blade force normal to the blade in the clearance zone. Because these two forces are mutually perpendicular, the two losses are entirely different in nature and they do not even partially overlap to each other. It is also made clear quantitatively, how the loss of the kinetic energy of leaked flow is related to the induced drag of the clearance flow.Copyright © 1990 by ASME

Tip vortex/airfoil interaction for a low reynolds number canard/wing configuration

Abstract: The effects of the vortical wake shed by a finite span canard on a low Reynolds number airfoil were examined. Aerodynamic performance was evaluated through direct measurements of lift, drag, and 1/4-chord pitching moment. Spanwise static pressure and surface film visualization data were also acquired. A reduction in the down-stream airfoil drag coefficient and an increase in its lift/drag were noted in the presence of the canard for a wide range of configurations

Flight test of a half-scale unmanned air vehicle

Abstract: A half-scale unmanned air vehicle was flight tested to provide lift and drag data for correlation with the fullscale vehicle. Additional work was carried out to determine if wing drag could be reduced with an improved surface finish and a trailing-edge modification. Ground tests for power and thrust using a torque stand and a lowspeed wind tunnel supported the flight tests for the determination of engine and propeller parameters. A panel method was used to predict the induced drag behavior of the tested air vehicle. Parasite drag was predicted by build-up methods. The predicted parasite drag for the half-scale fell short of flight-tested values by about 25%, but that predicted for the full-scale vehicle correlated well with the flight data. Within the scatter of data, the induced drag correlated satisfactorily. A comparison was made to lift and drag data extracted from full-scale, idlepower glide tests, and correlation was poor. Implications are that residual thrust prevents these glide-test data from accurately representing the full-scale drag polar.

Aerodynamic fairing for reducing drag of structural openings

Abstract: The named fairing reduces the aerodynamic drag of structural openings, increases the flow rate of a vent in a moving body, reduces the accumulation of toxic fumes in a cavity on a moving body and reduces the noise generated by a cavity on a moving body. Reducing the drag of structural openings such as cowling exits, landing gear wells, control surface gaps and vents by placing fairings downstream and external to basic contour has not yet been done or marketed commercially. There have been many devices marketed that are located at the forward side of the openings. These devices protrude outside contour to deflect airflow over the opening. In doing so these devices produce low pressure which acts on the aft side of the device; the force developed because of this pressure points in the downstream direction, causing a drag penalty. None of the known devices reduce pressure without causing large amounts of drag. The above mentioned fairing produces low pressure but its resultant force points perpendicular to stream flow and hence produces very little adverse pressure drag. In this approach, the above mentioned fairing is unique.

Aerodynamic characteristics of the HL-20 and HL-20A lifting-body configurations

Abstract: The data show that the HL-20 is longitudinally and laterally stable over the test range from Mach 10 to 0.2. At hypersonic speeds it has a trimmed lift/drag ratio of 1.4. This values gives the vehicle a cross range capability similar to that of the Space Shuttle. At subsonic speeds, the HL-20 has a trimmed lift/drag ratio of about 3.6. Replacing the flat plate outboard fins with fins having an airfoil shape, increased the maximum trimmed L/D to 4.3. Preliminary evaluation of configuration modifications (the HL-20A series), indicates that trim at higher values of lift at hypersonic speeds could be achieved with an L/D of about 1.0. In the supersonic range, the lift and directional stability characteristics were improved. The untrimmed subsonic L/D was increased to 5.8 with airfoil fins.

Determination of balloon drag

Abstract: The evolution of an empirical drag relationship that has stimulated rethinking regarding the physics of balloon drag phenomena is discussed. Combined parasitic drag from all sources in the balloon system are estimated to constitute less than 10 percent of the total system drag. It is shown that the difference between flight-determined drag coefficients and those based on the spherical assumption should be related to the square of the Froude number.

Analysis and design of planar and non-planar wings for induced drag minimization

Abstract: The goal of the work was to develop and validate computational tools to be used for the design of planar and non-planar wing geometries for minimum induced drag. Because of the iterative nature of the design problem, it is important that, in addition to being sufficiently accurate for the problem at hand, they are reasonably fast and computationally efficient. Toward this end, a method of predicting induced drag in the presence of a non-rigid wake is coupled with a panel method. The induced drag prediction technique is based on the Kutta-Joukowski law applied at the trailing edge. Until recently, the use of this method has not been fully explored and pressure integration and Trefftz-plane calculations favored. As is shown in this report, however, the Kutta-Joukowski method is able to give better results for a given amount of effort than the more common techniques, particularly when relaxed wakes and non-planar wing geometries are considered. Using these tools, a workable design method is in place which takes into account relaxed wakes and non-planar wing geometries. It is recommended that this method be used to design a wind-tunnel experiment to verify the predicted aerodynamic benefits of non-planar wing geometries.

An assessment of current methods for drag-due-to-lift minimization at supersonic speeds

Abstract: The estimation and minimization of drag-due-to-lift at supersonic speeds has been examined in this study. Correlations of theory with experimental data are used to assess the applicability and limitations of the linearized theory. The role of leading-edge thrust and the use of twist and camber to develop distributed thrust are also discussed. A semiempirical design and estimation method which takes into account the shortcomings of the linear theory is presented. The use of this method will allow the design of more nearly optimum lifting surfaces and provide an accurate prediction of their level of performance. A preliminary examination is made of the use of an Euler code for estimation of the aerodynamic characteristics of a twisted and cambered wing.

Winglets for sailplanes

Abstract: There is a good deal of reliable experimental evidence that winglets can increase sailplane performance with measured improvements of 2 to 5 points in glide ratio. Adding winglets to an exi:sting sailplane will increase skin friction drag which is most important at high speed. However, with properly designed winglets, the speed at which skin friction drag of the winglets cancels out the induced drag reduction is well above normal cruising speed, and even at higher speeds the drag increment is almost too small to measure. The concept behind the 15 meter span limitation is not violated by winglets because winglets only provide a finite performance increase. Winglets are inherently small in size-which limits both their cost and the cost of supporting structures. ln contrast, performance increase due to incieased span is limited only by weight, structural problems and cost. Winglets provide a substantial improvement in aerodynamic efficiency without an increase in wing span, both for the Open Class case where span is limited by structural considerations and for 15 meter and Standard Classes where there is an arbitary span limit. Winglets are particularly attractive for retrofit on the many aircraft which already have short span extensions, such as the ASW-20, Nimbus, Ventus, DG-600, LS-3, for example. The winglets allow the low speed advantage of the span extension to be retained while at the same time keeping the high speed advantage of the shorter span.

Effect of Induced Drag Damping on the Near Resonant Responses of a Deepwater Tripod Tower Platform

Abstract: Near-resonant responses of a deepwater platform to wave excitation are controlled by inducing additional hydrodynamic damping in the system. The study used a device designed to induce flow-separation around a vertical column at low Kc numbers. The results of two experimental investigations are presented in this paper. In the first experiment, in-line forces on a 0.3m diameter vertical cylinder were measured without and with the attachment of the flow-separation device. At low Kc numbers the drag coefficients increased by five times due to the attached device to the cylinder, whereas the inertial coefficients were found to be insensitive to the device. The second experiment investigated the wave induced responses of a 8.6m high hydroelastic model of a deepwater tripod tower platform in a 200x12x7m wave tank. A significant reduction in responses at and close to resonance, due to the induced drag damping, observed in the experimental measurements shows that the device is very effective for damping applications. The experimental findings are also supported by analytical results.

Super/hypersonic aerodynamic characteristics for a transatmospheric vehicle concept having a minimum drag forebody

Abstract: Experimental longitudinal and lateral-directional aerodynamic characteristics were obtained for a generic transatmospheric vehicle concept having a replaceable minimum drag forebody shape. The alternate forebody tested was a 1/4-power series body. Tests were made over a range of Mach numbers from 2 to 10 at a nominal Reynolds number, based on a length of 2.3 x 10 to the 8th and angles of attack from -4 to 20 deg. The minimum drag forebody provided significant improvements in minimum drag and L/D for the configuration as well as a longitudinally stabilizing increment. Although the baseline configuration is longitudinally unstable, the L/D improvements at low to moderate angles of attack would enhance cruise performance. Varying wing incidence angles was demonstrated as an effective horizontal trim device without significant trim drag penalties.

A unified viscous theory of lift and drag of 2-D thin airfoils and 3-D thin wings

Abstract: A unified viscous theory of 2-D thin airfoils and 3-D thin wings is developed with numerical examples. The viscous theory of the load distribution is unique and tends to the classical inviscid result with Kutta condition in the high Reynolds number limit. A new theory of 2-D section induced drag is introduced with specific applications to three cases of interest: (1) constant angle of attack; (2) parabolic camber; and (3) a flapped airfoil. The first case is also extended to a profiled leading edge foil. The well-known drag due to absence of leading edge suction is derived from the viscous theory. It is independent of Reynolds number for zero thickness and varies inversely with the square root of the Reynolds number based on the leading edge radius for profiled sections. The role of turbulence in the section induced drag problem is discussed. A theory of minimum section induced drag is derived and applied. For low Reynolds number the minimum drag load tends to the constant angle of attack solution and for high Reynolds number to an approximation of the parabolic camber solution. The parabolic camber section induced drag is about 4 percent greater than the ideal minimum at high Reynolds number. Two new concepts, the viscous induced drag angle and the viscous induced separation potential are introduced. The separation potential is calculated for three 2-D cases and for a 3-D rectangular wing. The potential is calculated with input from a standard doublet lattice wing code without recourse to any boundary layer calculations. Separation is indicated in regions where it is observed experimentally. The classical induced drag is recovered in the 3-D high Reynolds number limit with an additional contribution that is Reynold number dependent. The 3-D viscous theory of minimum induced drag yields an equation for the optimal spanwise and chordwise load distribution. The design of optimal wing tip planforms and camber distributions is possible with the viscous 3-D wing theory.

Effect of induced drag damping on the near resonant response of a deepwater platform

Abstract: Occurrence of resonance in the design of deepwater structures is unavoidable under all operating conditions. When the wave forces on the structure are dominated by the inertial component of loading, the hydrodynamic damping in the system is low. It is also known that the structural response can be reduced if energy can be dissipated through the fluid-media via the mechanism of flow-separation since it increases the fluid component of the total damping. Thus the resonant response could be suitably reduced by the generation of increased flow-separation. In view of certain new types of structures which possess one or more large diameter circular cylinders crossing the free surface, this dissertation describes an innovative technique to control the response at frequencies near resonance to moderate waves. -- When the Keulegan-Carpenter (KC) number for the flow is less than 4. the time available for the development of vortices behind a circular cylinder is less than adequate to form a stable wake. Consequently the drag forces on the structure are small. The first objective of this study was to increase the drag forces on a 0.3 m diameter circular vertical cylinder at low KC numbers (KC ≤ 2) for regular and irregular waves. The dissertation proposes a physical device to induce flow-separation that will increase the drag forces experienced by the circular cylinder. -- An experimental set-up was made to measure the wave forces on a simple vertical cylinder. The measured wave elevations and the wave forces were used to fit the Morison wave force formula, in the least squares sense, and the values of the inertial (Cm) and drag (Cd) force coefficients were determined. To study the effect of the physical device on the circular cylinder, the experiment was repeated by attaching the device to the cylinder. The experimental results revealed that, at low KC numbers, the value of Cd for the circular cylinder with the device increased by a factor of 4, irrespective of the wave height and wave period. The device did not significantly increase the value of the inertial coefficient of the main cylinder. -- The significant increase in the drag forces due to the attachment of the device could be successfully used to control the resonant response of certain deepwater structures. To investigate this, the dynamic response of an offshore platform model was tested in a wave tank, both with and without the device. The three-dimensional platform model was a 1/50th scale structure possessing the key features of a typical deep water tripod tower platform. The scaling considerations required for a hydroelastic model study in a wave tank were utilized to fabricate the 8.6 m tall hydroelastic model. The vibration properties of the model structure were determined both in air and water. To reliably estimate the modal parameters, modal testing and analysis were used. Parallel studies were conducted to analytically determine the natural frequencies and mode shapes, using a finite element method. The natural frequencies obtained by the two methods were in good agreement. -- The response of the physical model to resonant wave excitation was investigated for both regular and random waves and the experimental results were complemented by analytical results. Excellent correlation was obtained over the entire range of test conditions. In the analytical study concerning the effect of the device on the dynamic response of the structure to waves, a relative velocity formulation was used in the Morison forcing term and the nonlinear equations of motion were solved in the time domain. The experimental values of the Cm and Cd coefficients, with and without the device, were utilized in the analytical study. -- Both experimental and analytical results support the potential application of the device in controlling the resonant response of large diameter structures. Measured experimental results demonstrate that with the device the longitudinal acceleration response was reduced by a maximum of 32% for resonant wave conditions. Findings show that, as the wave height increases under resonant excitation, the percentage reduction due the induced drag damping also increases. -- Key words: Dynamics of offshore structures; Deepwater structures; Induced drag damping; Hydrodynamic damping; Hydroelastic model; Hydroelastic response

Evaluation of wingsail interactions for windship propulsion

Abstract: An evaluation is presented of the interaction of several windship propulsion systems. Two calculation methods are used, both based on Prandtl's lifting line theory, and calculated and experimental results are compared. The inverse, or «Prandtl optimum» method (in which the minimum induced drag is found for a given lift), using normal axis doublets in the «Trefftz plane», yields quite a good approximation to the direct method, which provides a more complete calculation and solves the problem with horse-shoe vortices

Applications of low lift to drag ratio aerobrakes using angle of attack variation for control

Abstract: Several applications of low lift to drag ratio aerobrakes are investigated which use angle of attack variation for control. The applications are: return from geosynchronous or lunar orbit to low Earth orbit; and planetary aerocapture at Earth and Mars. A number of aerobrake design considerations are reviewed. It was found that the flow impingement behind the aerobrake and the aerodynamic heating loads are the primary factors that control the sizing of an aerobrake. The heating loads and other loads, such as maximum acceleration, are determined by the vehicle ballistic coefficient, the atmosphere entry conditions, and the trajectory design. Several formulations for defining an optimum trajectory are reviewed, and the various performance indices that can be used are evaluated. The 'nearly grazing' optimal trajectory was found to provide the best compromise between the often conflicting goals of minimizing the vehicle propulsive requirements and minimizing vehicle loads. The relationship between vehicle and trajectory design is investigated further using the results of numerical simulations of trajectories for each aerobrake application. The data show the sensitivity of the trajectories to several vehicle parameters and atmospheric density variations. The results of the trajectory analysis show that low lift to drag ratio aerobrakes, which use angle of attack variation for control, can potentially be used for a wide range of aerobrake applications.

Winglet effectiveness on low aspect ratio wings at supersonic mach numbers

Abstract: A computational study has been conducted on two wings of aspect ratios 1.244 and 1.865, each having 65-deg leading edge sweep angles, to determine the effects of nonplanar winglets at supersonic Mach numbers. A design Mach number of 1.62 was selected. The winglets studied were parametrically varied in alignment, length, sweep, camber, and thickness to determine the effects of winglet geometry on predicted performance. For the computational analysis, an existing Euler code that employed a marching technique was used. The results indicated that the possibility existed for wing-winglet geometries to equal the performance of wing-alone bodies in supersonic flows with both bodies having the same semispan length. The performance parameters of main interest were the lift-to-pressure drag ratio and the pressure drag coefficient as functions of lift coefficient. The lift coefficient range for this study was from -0.20 to 0.70 with emphasis on the range of 0.10 to 0.22.

Aerodynamic parameter studies and sensitivity analysis for rotor blades in axial flight

Abstract: The analytical capability is offered for aerodynamic parametric studies and sensitivity analyses of rotary wings in axial flight by using a 3-D undistorted wake model in curved lifting line theory. The governing equations are solved by both the Multhopp Interpolation technique and the Vortex Lattice method. The singularity from the bound vortices is eliminated through the Hadamard's finite part concept. Good numerical agreement between both analytical methods and finite differences methods are found. Parametric studies were made to assess the effects of several shape variables on aerodynamic loads. It is found, e.g., that a rotor blade with out-of-plane and inplane curvature can theoretically increase lift in the inboard and outboard regions respectively without introducing an additional induced drag.

Reply by the Authors to J.G. Leishman

Abstract: This model can be extended to finite span wings, i.e., three dimensions. For a helicopter, the relationship between u and A/7 cannot be derived directly from Eq. (9) in Ref. 1. One then must question if the leading-edge suction model is valid for helicopters as an extension of two-dimensional theory to three dimensions. Eqs. (10-20) in Ref. 1 show that this extension is possible. The second purpose was to show the difference between the induced drag per unit length predicted by the two models, i.e., the induced angle model and the leading-edge suction model. As mentioned in our paper, this difference must exist for helicopter rotor blades. It is emphasized that either lifting-line theory or lifting-surface theory could be used in both models. Our purpose was to compare the two theories and not to calculate the induced power exactly. The latter needs further correlation. In Ref. 1 we stated that "lifting-line theory is not adequate in describing helicopters in forward flight." Leishman considers this to be rather "misleading and largely untrue." His reason is that the lifting-line theory, even the classical momentum theory, will do a reasonable job in predicting the induced power. But the sense of the words "describing helicopters" can include several meanings, such as lift distribution, drag distribution, blade vibration, noise, ambient flowfield, etc. The rotor performance is only one item; it is not the unique item of these areas. There are many codes that use the lifting-line theory to predict the lifting distribution, including the very complicated free wake analysis. But we cannot be satisfied with their results, especially at higher harmonic loading (see Ref. 2, for example). In fact, it is Professor Leishman himself who pointed out many of the deficiencies of lifting-line theory in his comments. We consider that if Professor Leishman agreed on the aforemention sense of the phrase "not adequate in describing helicopters," his comment about this problem would not be presented. It appears that Professor Leishman does not believe Table 1 in Ref. 1, because of the lower values of/. His reason is that lifting-line theory can predict induced drag reasonably. However, this is a misunderstanding. Both factors, yW/U\ and (ya-S), come from the same lifting-line theory. The only modification is the two-dimensional Theodorsen function. Since in Ref. 3, which was used for Table 1 in Ref. 1, it was assumed that the number of blades is infinite, then the theory used is a steady theory. There are three factors for the case when/ 5* 1.0. The first is the inherent difference between the two models, which we wanted to show by using Table 1 in our paper. For example, if the unsteady modifications make the slope of the lift curve decrease by 3%, or the angle of attack increase by 3%, then/will decrease by 6% (with S/ya == 0.5). The second is the lower than usual slope of the lift curve, 5.73, used in Ref. 3 as usual. The third perhaps is the problem in the theory used to calculate the lift distribution. Our purpose for presenting Table 1 in Ref. 1 was to show the first factor, but indeed/showed the combined effect of the three factors already mentioned. When we use the leadingedge suction model to predict induced power, practically, we must choose the theoretical value 2ir in using lifting-line theory or lifting-surface theory for the lift distribution and angle of attack, or an alternative method. In this sense, we thank Professor Leishman for his comments. It is emphasized that since the values of/are far from 1.0, then perhaps this means there may be some problems in the lifting-line theory used. It certainly does not mean that the induced power is too low. However, we do not consider that the two-dimensional Theodorsen function and Table 1 in Ref. 1 were used and presented arbitrarily. We now present Table la which shows the difference between the two models. It is the rotor of a CH-34, at /* = 0.0873, \l/ = 90 deg (y, W, and a are calculated). The calculation is a free wake analysis, using lifting-surface theory, completely unsteady, and similar to that used in Ref. 4. Although the lift distribution does not show good agreement with experiment, the calculated result still can be used. Since the values for / are not far from 1.0, they may reveal the difference between the two models because the slope of the lift curve is lower than 2-jr. In fact, the comparison between the two models (i.e., the values of/) is so simple that those who doubt the results can do the calculation themselves with whatever data is at hand. We acknowledge Leishman's point that in describing helicopters the leading-edge sweep angle should be included in the leading-edge suction. When the blade is at azimuths not at 90 or 270 deg, the sweep angle exists due to the pcosil/ term. Usually, in existing lifting-line or lifting-surface theory applications for predicting the lift distribution, or in the liftingline/blade element analysis, the term ^cosi/' is omitted. We do not have an available method to include /ucosi/'. It is unreasonable to use the lift in which the j-icosi/' is not included, or the leading-edge suction in which the JKCOS^ is included for Table 1 in Ref. 1. Our purposes were to demonstrate that the leading-edge suction model could be used for helicopters and to show the difference between the two models; a complicated planform (e.g., tip sweep) was beyond the scope of Ref. 1. Coincidentally, in Ref. 5, for incompressible unsteady flow, we obtained theoretically