Induced Drag Coefficient Research Articles

Unified Approximation for Nonparabolic Drag Polars

where, by convention (excluding the k term), uppercase subscripts apply to wings and lowercase subscripts apply to airfoils. The first term on the right-hand side is the drag coefficient at zero lift and the second term represents the contribution of pressure drag (kpC 2 L for the airfoil) or pressure and induced (vortex) drag (kp iC l for the wing). For wings, both the pressure component of the profile drag coefficient kpC 2 L and the planform-dependent induced drag coefficient kiC 2 L vary with the square of the lift coefficient. This makes their separation difficult [thus the combination kp i parameter in Eq. (1b)]. Equations (1a) and (1b) are used as an approximation in performance estimates for preliminary design, as well as for interpretation and quantification of experimental and computational data. The form of Eq. (1) implies that the minimum drag of the configuration (airfoil or wing) occurs at zero lift. This is generally true for symmetrical airfoils or wings. For cambered airfoils or cambered section wings, a more representative approximation is of the form

Aerodynamic performances of complex shape wings

The task of calculation of optimum circulation distribution along wingspan of complex shape wings is considered. For solving this problem Glauert-Trefts’s equation and its modifications are used. Calculations are carried out for both sweptback and forward-swept wings. It is shown that optimum circulation distribution depends on the sweep angle χ and on the chord b(z) distribution along wingspan. Some aerodynamic coefficients such as induced drag coefficient C Di and pitching moment coefficient C mZ are calculated for wings of different shape. The comparison of wings performances is done. In order to obtain the minimum wing induced drag with the given lift force it is very important to determine how the circulation should change along the wingspan. Results obtained by E. K. Karafoli G.F. Burago and others are used. A set of theoretical generalizations and modifications of formulas for aerodynamic coefficients are obtained. These results permit to compare aerodynamic performances of sweptback and forward-swept wings. Modified Glauert-Trefts’s integral-differential equation is formulated for wings of complex shape.

Lifting-Line Theory of a Supercavitating Hydrofoil in Two-Dimensional Shear Flow

The existing lifting-line theory for a supercavitating hydrofoil in two-dimensional shear flow (where velocity varies in both spanwise and vertical directions) between two parallel planes is applied to partial cavitation under the same flow conditions. A lifting-line equation is derived from combining the assumption holding the lift coefficient two-dimensionally at any spanwise position with an up-wash velocity induced along a lifting-line. The hydrofoil for numerical examples is flat in section, rectangular in plan form, and has a 5 degree attack angle. Effects of two kinds of shear parameters, cavitation number, and taper ratio on local lift coefficient, total lift coefficient and induced drag coefficient are clarified through numerical calculations. Cavity length distributions are obtained on the basis of the assumption that the relationship between cavitation number and attack angle is two-dimensional at any spanwise position.

Lifting-Line Theory of a Supercavitating Hydrofoil in Both Vertical and Spanwise Shear Flow. 2nd Report. Application to Partial Cavitation.

The existing lifting-line theory for a supercavitating hydrofoil in both vertical and spanwise shear flow between two parallel planes is applied to partial cavitation in the same flow condition. A lifting-line equation is derived from combining the assumption holding lift coefficient two-dimensional at any spanwise position with the up-wash velocity distribution induced along a lifting-line. The hydrofoil for the numerical example is flat in section, rectangular in planform, and 5 degree in attack angle. Effects of two kinds of shear parameter, cavitation number, and taper ratio on local life coefficient, total lift coefficient and induced drag coefficient are clarified through the numerical calculations. The cavity length distribution is obtained on the basis of the assumption holding the pressure within cavity two-dimensional at any spanwise position.

Experimental and nonlinear vortex lattice method results for variouswing-canard configurations

Experimental measurements of the aerodynamic characteristics of five close-coupled wing-canard configurations up to moderately high angles of attack are used to evaluate the ability of the nonlinear vortex lattice method (NLVLM) to calculate the aerodynamic characteristics of such configurations. The longitudinal aerodynamic coefficients, the rolled-up vortex trajectories, and the pressure distributions are compared for the five wing-canard configurations, as well as for the three wing planforms for which experimental data is available. The investigation includes the effects of varying the canard deflections and the canard positions relative to the wing of the various configurations. The lift- and induced-drag coefficients which are evaluated by the NLVLM for the wing-canard configurations are found to be in good agreement with the experimental data up to the angle of attack of 20-25 deg. The NLVLM works even better for wings alone in that the pitching moment can also be well-predicted. Therefore, this tool can potentially be used in the by analysis process employed during the preliminary design for wing-canard configurations. Nomenclature AR = aspect ratio b - wing span, mm CD = induced-drag coefficient based on wing planform area CL = lift coefficient based on wing planform area Cm = pitching moment coefficient based on wing area and wing root chord Cp = pressure coefficient when Ap is the lifting

Lifting-Line Theory for Supercavitating Hydrofoil in Shear Flow. (Continuing Report, Application to Partial Cavitation).

An existing lifting-line theory for a supercavitating hydrofoil in spanwise shear flow between parallel plane walls is applied to partial cavitation in the same condition. A lifting-line equation is derived from an assumption holding two-dimensional at any spanwise position, and the method of solution of the lifting-line equation is applied, in which an eigenvalue problem of the Sturm-Liouville type should be solved. The method to obtain the cavity spanwise distribution, in which the solution of type lifting-line equation is used, is also applied. The wing for the numerical example is flat in section, rectangular in planform, and 5 degrees in attack angle. Effects of the main stream velocity profile, aspect ratio and cavitation number on local lift coefficient, total lift coefficient and induced drag coefficient for force characteristics, and on spanwise distribution in cavity length and its three-dimensional effect for cavity characteristics are clarified.

Interference Drag of a Turbulent Junction Vortex

The interference drag identified with the junction of a streamlined cylindrical body and a flat plate was investigated. The junction drag was calculated from a set of detailed, self consistent, high quality data using a control volume approach. The drag for the isolated flat plate and streamlined cylinder making up the junction was calculated using boundary-layer solvers together with surface pressure measurements. For the particular and relatively thick body under consideration, the results show a significant increase in drag due to the junction. These and other available results indicate that the interference drag has a systematic dependence on the thickness to chord ratio. The junction vortex wake increases the downstream flat plate drag significantly. Because of this effect, a unique value for the drag force, drag coefficient, or induced drag coefficient for a junction vortex flow would require that the geometry be specified in detail. The induced drag and the total pressure losses identified with the junction are also reported.

Blockage corrections at high angles of attack in a wind tunnel

Recent developments have increased the need to investigate flight conditions at high angles of attack. Because these conditions have to be studied in wind tunnels, it is obvious that the blockage corrections must be assessed as accurately as possible. With this in mind, we thought of comparing the results obtained from the same model in two different wind tunnels: the Aeritalia wind tunnel and the Emmen F + W 32.4 m2. The size of the Emmen makes the wall and blockage corrections negligible. Therefore, in practice, with any differences in the results coming from tests conducted in the two wind tunnels, it is only necessary to make corrections to those coming from the Aeritalia wind tunnel. Since the Emmen wind tunnel provided us with the exact results, our work consisted of establishing a method of correction whose validity could be verified. Initially we tried to use Maskell's formula introducing constant coefficients according to test conditions but the results were not satisfactory. We then developed Maskell's formula and decided to use the flat-plate pressure coefficient as a base pressure coefficient—which is needed in Maskell's formula—when no such coefficient can be deduced from the model. In all these cases, the induced drag coefficient was expressed as a function of the lift coefficient. Final corrections for lift and moment coefficients were very satisfactory, though the drag coefficient was slightly overcorrected.

Three-surface aircraft - Optimum vs typical

Comparisons of the induced drag for a three-surface general aviation aircraft were made using a vortex lattice method and Prandtl-Munk theory. Substantial differences between the two prediction methods have been shown in the presence of practical nonelliptic spanwise load distributions. At the same time, a parametric study using the vortex lattice method has been carried out to determine the sensitivity of lift to induced-drag ratio to different design variables. Using the resulting trends, a three-surface general aviation aircraft has been modeled and compared with its equivalent canard and conventional configurations. It has been shown that although the three-surface geometry is more efficient than a canard configuration, it remains inferior to a conventional design. Nomenclature M. = aspect ratio B — canard load/tail load b = span CL = lift coefficient CD = drag coefficient CD. = induced-drag coefficient DJ = induced drag e = efficiency factor G =gap, vertical distance to the wing/wing span, positive above the wing L =lift q = dynamic pressure S .= stagger, horizontal distance to the wing/wing span S =planform area V = airspeed a = geometric angle of attack X = taper ratio A = sweep angle cr =PrandtPs mutual induced-drag factor

Analytical Prediction of Vortex Lift

General integral expressions are derived for the nonlinear lift and pitching moment of arbitrary wing planforms in subsonic flow. The analysis uses the suction analogy and an assumed pressure distribution based on classical linear theory results. The potential flow lift constant and certain wing geometric parameters are the only unknowns in the integral expressions. Results of the analysis are compared with experimental data and other numerical methods for several representative wings, including ogee and double-delta planforms. The present method is shown to be as accurate as other numerical schemes for predicting total lift, induced drag, and pitching moment. b c c CL CD Cm CT Cs cc, ccd E2 Nomenclature = aspect ratio = wing span =chord = reference length = lift coefficient = drag coefficient = pitching moment coefficient = thrust coefficient = suction coefficient = section lift coefficient = section induced drag coefficient = section suction coefficient = pressure loading coefficient = drag = proportionality constant, Eq. (32) = proportionality constant, Eq. (53) = chordwise function, Eq. (44) ff(rj) = span wise f unction, Eq. (28) K = potential constant L =lift loading constant, Eq. (5) S = suction force SR = reference area s = suction force per unit length T = leading edge thrust, Eq. (7) T' = leading edge thrust per unit length V = freestream speed Wj = downwash velocity component, Eq. (11) a. = angle of attack F = vorticity p = freestream density £ = nondimensional chordwise coordinate 77 = nondimensional spanwise coordinate A = leading edge sweep angle Subscripts P = potential flow E =edge / = induced VLE = leading edge vortex VSE = side edge vortex

A Theoretical Note on the Lift Distribution of a Non-planar Ground Effect Wing

SummaryThe problem of a non-planar wing of finite span very close to the ground is considered by the method of matched asymptotic expansions. This method is based on the work of Widnall and Barrows, in which a planar wing very close to the ground was examined in detail. A simple analytic solution, to first-order approximation, is obtained for a non-planar wing which is uncambered. Expressions for the lift coefficient, induced angle and induced drag coefficient, which are valid for small ground clearance and moderately small aspect ratio, are derived for the case when the configuration of the wing projected onto a transverse plane normal to the free stream is elliptic. The problem of the optimum lift distribution around the wing and the rolling-moment coefficient for the inclined flat wing are discussed.The distribution of camber and of aerofoil angle of attack is given analytically for an optimally loaded non-planar wing, to first-order approximation. Moreover, it is seen that, to first-order approximation, the aerodynamic forces given for the small aspect ratio are independent of the wing planform and depend only on the wing configuration at the trailing edge projected onto a transverse plane.

翼端吹き出しを有する高循環三次元翼の非線型理論

This paper deals with the nonlinear theory on the highly loaded wing of the finite span with the tip blowing. The authors have published the paper on the nonlinear theory of the ordinary three-dimensional wing without the tip blowing which will be available even in such a high circulation region as PRANDTL'S linear theory cannot be applied, and the present investigation is carried out on the basis of the previous results.It is concluded that the ratio of the lift coefficient to the physical aspect ratio will increase and that of the induced drag coefficient to the physical aspect ratio will decrease by the pertinent wing tip blowing in comparison with the case without it.

Optimum lift-drag ratio for a ram wing tube vehicle

Munk's theorem specifying the downwash condition for minimum drag is generalized to include lifting surfaces operating in proximity to solid boundaries. A simple method for finding the optimum lift distribution on a wing above an infinite flat plane is developed. The optimum configuration for a ram wing in a tube is found, and by means of a simple transformation this is mapped into the previously obtained solution for a wing in ground effect. An expression for the induced power required is calculated, and it is shown that there is a favorable effect on this requirement for the case of tube vehicles which have significant blockage ratios. Experimental results are presented which demonstrate that at very small clearances the theory must be modified to include viscous effects, and because of these effects ram wings in tubes usually have lower induced power requirements than the inviscid theory would indicate. Nomenclature AR = aspect ratio A t = cross-sectional area of tube b = wing span = vehicle width Cof — induced drag coefficient CL = lift coefficient Di = induced drag F = volume flux

A Note on the Induced Drag of Jet-Flapped Wings

The main purpose of this note is to explain the apparent discrepancy which arises when the change in effective aspect ratio due to blowing, as predicted by the theoretically derived expression for the induced drag coefficient, is compared with what one intuitively expects from the general flow picture. If the jet sheet is considered as a chordwise extension of the wing (mechanical flap analogy), the general flow picture suggests that the effective aspect ratio of a wing without blowing is reduced by blowing. Maskell and Spence's relation for the induced drag coefficient, however, indicates that the effect of jet blowing is to increase the effective aspect ratio.

Lagrange Multipliers and Quasi-Steady Flight Mechanics

A general method is presented for investigating optimum conditions of the quasi-steady mechanics of flight. For motion in a vertical plane the problem of extremizing an arbitrarily specified function of altitude, Mach Number, lift, path inclination, engine control parameter, and thrust inclination is considered for an aircraft which has to satisfy the equations of motion and 3 additional arbitrary constraints. A generalized solution is obtained in a determinantal form. The characteristic of this solution is that it unifies into a single equation the results of a large segment of the previous contributions to the quasi-steady mechanics of flight. Particular problems such as maximum speed, maximum range, maximum endurance, ceiling, steepest ascent, best rate of climb, flattest descent, etc., are thus all covered by the same determinantal equation. The optimum ratio (R) of induced drag to zero-lift drag is evaluated for arbitrary relationships between zero-lift drag coefficient, induced drag coefficient, thrust, specific fuel consumption, and Mach Number. Design problems are also investigated, such as those associated with the selection of the best wing surface or the best aspect ratio for a given performance to be optimized. Finally, the paper is completed with the study of the optimum flight conditions for curvilinear motion in a horizontal plane. Another general determinantal equation is derived, from which, as a particular case, the turning flight with maximum angle of bank, maximum angular velocity, or minimum radius of curvature is investigated.

Aerofoil Theory for Swallow Tail Wings of Small Aspect Ratio

SummaryA method is developed for the calculation of the aerodynamic forces acting on a “swallow tail” wing (Figs. l(a) and l(b)) of small aspect ratio. For a given incidence, lift and induced drag coefficient are, within the limits of the theory, proportional to aspect ratio and independent of Mach number. The exact solution is derived for the limiting case of small sweepback of the trailing edge. A numerical method is indicated for the more general case.

Induced Drag of a Twisted Wing

in which CL is the wing lift coefficient, A the aspect ratio and a the induced-drag factor. Although valid for an untwisted wing (section zero-lift chords all parallel), Eq. (1) gives incorrect values for a twisted wing. For example, CDI = 0 in Eq. (1) when CL = 0, while the induced-drag coefficient of a twisted wing is never zero. The correct formula for the induced-drag coefficient of a twisted wing can be derived from the theory of spanwise air-load distribution given in references 1 and 2. In this theory the section lift coefficient Cx a t any airfoil section along the span is expressed as a Fourier series, namely (p. 7-29)

Minimum Induced Drag

IT is fairly well known that for any given wing arrangement—monoplane, biplane, monoplane with end‐plates corresponding with an increasingly popular type of tail unit, etc.—there is a definite lift distribution associated with the minimum value of induced drag of which a particular span is capable. The theory giving the appropriate lift distribution and, what is perhaps of more general practical interest, the resulting induced drag coefficient, is perhaps not so well known and while numerical results for arrangements in common use have been published, novel arrangements require a knowledge of the underlying theory for their evaluation, and this is in any case useful as giving a clear grasp and understanding of the problem.