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Adaptation of the Theodorsen theory to the representation of an airfoil as a combination of a lifting line and a thickness distribution
The theory provides a direct method for resolving an airfoil into a lifting line and a thickness distribution as well as a means of synthesizing thickness and lift components into a resultant airfoil and computing its aerodynamic characteristics. Specific applications of the technique are discussed.
Transonic lifting line theory - Numerical procedure for shock-free flows
An algorithm for computing transonic lifting-line theory without shocks is presented; the numerical procedure relies on two-dimensional analyses developed to solve inviscid flow equations for slender airfoils. From a series of solutions to two-dimensional problems in which span effects appear parametrically, the three-dimensional potential field characterization is obtained. Numerical results are given for a lifting wing with an elliptic (spanwise) distribution of chord and a NACA-0012 cross section (a freestream Mach number of 0.63 and an angle of attack of 2 deg are assumed).
Limitations of Lifting-Line Theory for Estimation of Aileron Hinge-Moment Characteristics
Hinge-moment parameters for several typical ailerons were calculated from section data with the aspect-ratio correction as usually determined from lifting-line theory. The calculations showed that the agreement between experimental and calculated results was unsatisfactory. An additional aspect-ratio correction, calculated by the method of lifting-surface theory, was applied to the slope of the curve of hinge-moment coefficient against angle of attack at small angles of attack. This so-called streamline-curvature correction brought the calculated and experimental results into satisfactory agreement.
Nonlinear Large Deflection Theory with Modified Aeroelastic Lifting Line Aerodynamics for a High Aspect Ratio Flexible Wing
This paper investigates the effect of nonlinear large deflection bending on the aerodynamic performance of a high aspect ratio flexible wing. A set of nonlinear static aeroelastic equations are derived for the large bending deflection of a high aspect ratio wing structure. An analysis is conducted to compare the nonlinear bending theory with the linear bending theory. The results show that the nonlinear bending theory is length-preserving whereas the linear bending theory causes a non-physical effect of lengthening the wing structure under the no axial load condition. A modified lifting line theory is developed to compute the lift and drag coefficients of a wing structure undergoing a large bending deflection. The lift and drag coefficients are more accurately estimated by the nonlinear bending theory due to its length-preserving property. The nonlinear bending theory yields lower lift and span efficiency than the linear bending theory. A coupled aerodynamic-nonlinear finite element model is developed to implement the nonlinear bending theory for a Common Research Model (CRM) flexible wing wind tunnel model to be tested in the University of Washington Aeronautical Laboratory (UWAL). The structural stiffness of the model is designed to give about 10% wing tip deflection which is large enough that could cause the nonlinear deflection effect to become significant. The computational results show that the nonlinear bending theory yields slightly less lift than the linear bending theory for this wind tunnel model. As a result, the linear bending theory is deemed adequate for the CRM wind tunnel model.
A Note about Self-Induced Velocity Generated by a Lifting-Line Wing or Rotor Blade
This report presents an elementary analysis of the induced velocity created by a field of vortices that reside in the wake of a rotor blade. Progress achieved by other researchers in the last 70 years is briefly reviewed. The present work is presented in four stages of complexity that carry a lifting-line representation of a fixed wing into a single-blade rotor. The analysis leads to the conclusion that the lifting rotor's spiraling vortex wake structure has very high induced power when compared to the ideal wing. For an advanced ratio of one-half, induced power is on the order of 10 times that of the wing when the comparison is made at wingspan equal to rotor diameter and wing and rotor having equal lift.
Lifting-line theory of oblique wings in transonic flows
Three-dimensional corrections to the nonlinear mixed flow admitted by a high-aspect-ratio swept wing of practical interest are analyzed by solving a perturbation problem and matching its solution to that of an outer flow. The latter is identified with a linear solution involving a lifting line, but the centerline of the planform is not required to be straight and unyawed. The existence of a similarity in the three-dimensional flow structure for a certain oblique-wing geometry is demonstrated, along with a solution to the reduced problem in a high-subcritical case.
A multipurpose lifting-line flow solver for arbitrary wind energy concepts
Abstract. In this work, we extend the AeroDyn module of OpenFAST to support arbitrary collections of wings, rotors, and towers. The new standalone AeroDyn driver supports arbitrary motions of the lifting surfaces and complex turbulent inflows. Aerodynamics and inflow are assembled into one module that can be readily coupled with an elastic solver. We describe the features and updates necessary for the implementation of the new AeroDyn driver. We present different case studies of the driver to illustrate its application to concepts such as multirotors, kites, or vertical-axis wind turbines. We perform verification and validation of some of the new features using the following test cases: elliptical wings, horizontal-axis wind turbines, and 2D and 3D vertical-axis wind turbines. The wind turbine simulations are compared to existing tools and field measurements. We use this opportunity to describe some limitations of current models and to highlight areas that we think should be the focus of future research in wind turbine aerodynamics.
Limitations of lifting-line theory for estimation of aileron hinge-moment characteristics
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A least-squares procedure for the solution of the lifting-line integral equation
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A modified lifting line theory for wing-propeller interference
An inviscid incompressible model for the interaction of a wing with a single propeller slipstream is presented. The model allows the perturbation quantities to be potential even though the undisturbed flow is rotational. The governing equations for the spanwise lift distribution are derived and a simple method of solving these is indicated. Spanwise lift and induced drag distribution for two cases are computed.
Correction of the lifting-line theory for the effect of the chord
It is shown that a simple correction for the chord of a finite wing can be deduced from the three-dimensional potential flow around an elliptic plate. When this flow is compared with the flow around a section of an endless plate, it is found that the edge velocity is reduced by the factor 1/E, where E is the ratio of the semiperimeter to the span. Applying this correction to the circulation brings the theoretical lift into closer agreement with experiments.
Results of the integration of a transonic full-potential analysis program with a free-wake lifting-line program for hovering rotors
The hovering performance predictions of the TFAR1 and OPLIN codes and the experimental data are discussed. The TFAR1 program solves the full-potential equation in a rotor-fixed coordinate system by use of the line relaxation method. The OPLIN program calculates the positions of wake vortices and rotor performance using the influence-coefficient-and-lifing-line method. The two programs are combined by adding the induced velocities from the OPLIN code to the near flow-field of the rotor from the TFAR1 code. Results show that the TFAR1 program converges better with the downwash-coupling method than the twist correction method to include the wake downwash.
Linearized lifting-surface and lifting-line evaluations of sidewash behind rolling triangular wings at supersonic speeds
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Summary and recent results from the NASA advanced High Speed Propeller Research Program
Advanced high-speed propellers offer large performance improvements for aircraft that cruise in the Mach 0.7 to 0.8 speed regime. The current status of the NASA research program on high-speed propeller aerodynamics, acoustics, and aeroelastics is described. Recent wind tunnel results for five 8- to 10-blade advanced models are compared with analytical predictions. Test results show that blade sweep was important in achieving net efficiencies near 80 percent at Mach 0.8 and reducing near-field cruise noise by dB. Lifting line and lifting surface aerodynamic analysis codes are under development and some initial lifting line results are compared with propeller force and probe data. Some initial laser velocimeter measurements of the flow field velocities of an 8-bladed 45 deg swept propeller are shown. Experimental aeroelastic results indicate that cascade effects and blade sweep strongly affect propeller aeroelastic characteristics. Comparisons of propeller near-field noise data with linear acoustic theory indicate that the theory adequate predicts near-field noise for subsonic tip speeds but overpredicts the noise for supersonic tip speeds. Potential large gains in propeller efficiency of 7 to 11 percent at Mach 0.8 may be possible with advanced counter-rotation propellers.
Lift and Drag of Wings with Small Span
The lift coefficient of!a wing of small span at first shows a linear increase for the increasing angle of attack, but to a lesser degree then was to be expected according to the theory of the lifting line; thereafter the lift coefficient increases more rapidly than linearity, as contrasted with the the theory of the lifting line. The induced drag coefficient for a given lift coefficient, on the other hand, is obviously much smaller than it would be according to the theory. A mall change in the theory of the lifting line will cover these deviations.
Line-vortex theory for calculation of supersonic downwash
The perturbation field induced by a line vortex in a supersonic stream and the downwash behind a supersonic lifting surface are examined to establish approximate methods for determining the downwash behind supersonic wings. Lifting-lines methods are presented for calculating supersonic downwash. A bent lifting-line method is proposed for computing the downwash field behind swept wings. When applied to triangular wings with subsonic leading edges, this method gives results that, in general, are in good agreement with the exact linearized solution. An unbent lifting-line method (horseshoe-vortex system) is proposed for unswept wings. This method is applied to determine downwash behind rectangular wings with aspect ratios of 2 and 4. Excellent agreement with exact linearized theory is obtained for both aspect ratios by placing the lifting line at the 1/2-chord point. The use of lifting-lines therefore appears promising for obtaining estimates of the downwash behind supersonic wings.
Wake model for helicopter rotors in high speed flight
Two alternative approaches are developed to calculate blade-vortex interaction airloads on helicopter rotors: second order lifting-line theory, and a lifting surface theory correction. The common approach of using a larger vortex core radius to account for lifting-surface effects is quantified. The second order lifting-line theory also improves the modeling of yawed flow and swept tips. Calculated results are compared with wind tunnel measurements of lateral flapping, and with flight test measurements of blade section lift on SA349/2 and H-34 helicopter rotors. The tip vortex core radius required for good correlation with the flight test data is about 20 percent chord, which is within the range of measured viscous core sizes for helicopter rotors.