Prandtl's Lifting-line Theory Research Articles

An enhanced disk averaged CFD model for the simulation of horizontal axis tidal turbines

Simulating fully resolved Horizontal Axis Tidal Turbine (HATT) geometry for a time period great enough to resolve a fully developed wake, and accurately predict power and thrust characteristics, is computationally very expensive. The BEM-CFD method is an enhanced actuator disk and is able to reduce the computational cost by simulating a time averaged downstream velocity field. Current implementations fall short of accurately determining tip losses, which are a function of the hydrofoil geometry. This work proposes a method of addressing this shortfall by modifying the angle of attack to conform to the constraints outlined in Prandtl's lifting line theory, i.e. the zero lift angle of attack at the hydrofoil tip. The revised model is compared to existing BEM-CFD methods and validated against experimental data. The revised BEM-CFD method presented in this work shows a significant improvement over previous BEM-CFD methods when predicting power and thrust. The coefficient of power is reduced from 0.57 (approx. 30% above experiment) to 0.44 (approx. 3% above experiment). An increase in turbulence intensity in the rotor region, in particular at the wake boundary, improves the recovery of the wake without the addition of empirical turbulence source terms. Good correlation with experimental results for power, thrust and wake prediction, is observed. The model may also be applied to wind turbines.

Aerodynamic Loads over Arbitrary Bodies by Method of Integrated Circulation

A novel numerical method for the conversion of unstructured surface vorticity into directional integrated circulation has been presented. This method allows for the reduction of vorticity distributions on arbitrary three-dimensional bodies into their mathematically equivalent formulations in the Prandtl lifting-line theory. A discussion of its implementation into a practical numerical solver has been presented. This new solver was applied to simple wing shapes as well as complex aircraft geometries to demonstrate the applicability of such an approach to flow solvers. It was shown that the reduction to lifting-line theory was exact for simple shapes such as rectangular and elliptical wings. For complex aircraft geometries, such as leading-edge root extension and canard-equipped aircraft, it was shown that the method of integrated circulation allowed for accurate prediction of aircraft loads and the analysis of flow physics, in keeping with the Prandtl original concepts.

Aerodynamic modelling of flapping flight using lifting line theory

PurposeThe purpose of the work is to design a flapping wing that generates net positive propulsive force and vertical force over a flapping cycle operating at a given freestream velocity. In addition, an optimal wing is designed based on the comparison of the force estimated from the quasi‐steady theory, with the wind‐tunnel experiments. Based on the designed wing configuration, a flapping wing ornithopter is fabricated.Design/methodology/approachThis paper presents a theoretical aerodynamic model of the design of an ornithopter with specific twist distribution that results generation of substantial net positive vertical force and thrust over a cycle at non‐zero advance ratio. The wing has a specific but different twist distribution during the downstroke and the upstroke that maintains the designed angle of attack during the strokes. The wing is divided into spanwise strips and Prandtl's lifting line theory is applied to estimate aerodynamic forces with the assumptions of quasi‐steady flow and the wings are without any dihedral or anhedral. Spanwise circulation distribution is obtained and hence lift is calculated. The lift is resolved along the freestream velocity and perpendicular to the freestream velocity to obtain vertical force and propulsive thrust force. Experiments are performed in a wind tunnel to find the forces generated in a flapping cycle which compares well with the theoretical estimation at low flying speeds.FindingsThe estimated aerodynamic force indicates whether the wing geometry and operating conditions are sufficient to carry the weight of the vehicle for a sustainable flight. The variation of the aerodynamic forces with varying flapping frequencies and freestream velocities has been illustrated and compared with experimental data that shows a reasonable match with the theoretical estimations. Based on the calculations a prototype has been fabricated and successfully flown.Research limitations/implicationsThe theory does not take into account the unsteady effects and estimates the aerodynamic forces at wing level condition. It doesn’t predict stall and ignores structural deformations due to aerodynamic loads. The airfoil section is only specified by the chord, zero lift angle of attack, lift slope, profile drag coefficient and angle of attack as given inputs. To fabricate a light weight wing that maintains a very accurate geometric twist and camber distribution as per the theoretical requirement is challenging.Practical implicationsUseful for designing ornithopter wing (preferably bigger) involving an unswept rigid spar with flapping and twisting.Originality/valueThe novelty of the present wing design is the appropriate spanwise geometric twisting about the leading edge spar.

Evaluation and Parameterization of Round Vortex Generator Jet Experiments for Flow Control

Many flow cases in fluid dynamics face undesirable flow separation due to ad-verse pressure gradients on wall boundaries. This occurs, for example, due togeometrical reasons as in a highly curved turbine-inlet duct or on flow-controlsurfaces such as wing trailing-edge flaps within a certain angle-of-attack range.Here, flow-control devices are often used in order to enhance the flow and delayor even totally eliminate flow separation. Flow control can e.g. be achieved byusing passive or active vortex generators (VGs) for momentum mixing in theboundary layer of such flows. This thesis focusses on such passive and activeVGs and their modelling for computational fluid dynamics investigations.First, a statistical VG model approach for passive vane vortex genera-tors (VVGs), developed at the Royal Institute of Technology Stockholm andthe Swedish Defence Research Agency, was evaluated and further improvedby means of experimental data and three-dimensional fully-resolved computa-tions. This statistical VVG model approach models those statistical vortexstresses that are generated at the VG by the detaching streamwise vortices.This is established by means of the Lamb-Oseen vortex model and the Prandtllifting-line theory for the determination of the vortex strength. Moreover, thisansatz adds the additional vortex stresses to the turbulence of a Reynolds-stresstransport model. Therefore, it removes the need to build fully-resolved three-dimensional geometries of VVGs in a computational fluid dynamics mesh. Usu-ally, the generation of these fully-resolved geometries is rather costly in termsof preprocessing and computations. By applying VVG models, the costs arereduced to that of computations without VVGs. The original and an improvedcalibrated passive VVG model show sensitivity for parameter variations suchas the modelled VVG geometry and the VVG model location on a flat plate inzero- and adverse-pressure-gradient flows, in a diffuser, and on an airfoil withits high-lift system extracted. It could be shown that the passive VG modelqualitatively and partly quantitatively describes correct trends and tendenciesfor these different applications.In a second step, active vortex-generator jets (VGJs) are considered. They were experimentally investigated in a zero-pressure-gradient flat-plate flow atTechnische Universitat Braunschweig, Germany, and have been re-evaluated for our purposes and a parameterization of the generated vortices was conducted. Dependencies of the generated vortices and their characteristics on the VGJsetup parameters could be identified and quantified. These dependencies wereused as a basis for the development of a new statistical VGJ model. This modeluses the ansatz of the passive VVG model in terms of the vortex model, theadditional vortex-stress tensor, and its summation to the Reynolds stress ten-sor. Yet, it does not use the Prandtl lifting-line theory for the determinationof the circulation but an ansatz for the balance of the momentum impact thatthe VGJ has on the mean flow. This model is currently under developmentand first results have been evaluated against experimental and fully-resolvedcomputational results of a flat plate without pressure gradient.

An analytical investigation of two-dimensional and three-dimensional biplanes operating in the vicinity of a free surface

In the present article, the classical two- and three-dimensional lifting theories are generalized to the biplane operating in proximity to a free surface. The singularity distribution method is employed to calculate the lifting force for a two-dimensional biplane subjected to wing-in-ground effect in the vicinity of a free surface, and the three-dimensional correction is carried out by the aid of the Prandtl lifting line theory. The essential techniques lie in finding the three-dimensional Green’s function for the system of horseshoe vortices operating above a free surface and ensuring numerical implementation. Extensive numerical examples are carried out to show the lift coefficient for the two- and three-dimensional biplanes in the vicinity of a free surface with the variation of the clearance-to-chord ratio and the height-to-chord ratio. Incidentally, the induced (inviscid) drag coefficients as well as the lift-to-drag ratio for a three-dimensional biplane are also computed. Good agreement can be found among results obtained from this study and the experiment.

Lifting line theory for wing-in-ground effect in proximity to a free surface

Although it has some limitations in applications, the classical Prandtl lifting line theory remains a standard methodology for evaluating lifting problems in free space. It is of theoretical interest in revealing lifting mechanisms. It is therefore, interesting to generalize the classical lifting line theory to cases more general than just the free space problem. In this article, we present the Prandtl lifting line theory for wing-in-ground effect (WIG) near a free surface. While, the fundamental methodology being similar to the classical lifting line theory, it turns out that the difficulty lies in finding the three-dimensional Green’s function for the system of horseshoe vortices operating over the deformable free surface. Linear free surface boundary conditions are applied to deal with the two-dimensional lifting problem solved by the singularity distribution method and the three-dimensional correction found by placing a system of horseshoe vortices on the wing. This approach was validated against published data. Excellent agreement is found among results obtained from this study, experiments and numerical simulations. Extensive numerical examples are carried out to show the features of lift coefficients in the vicinity of a free surface. As expected, the free surface can be represented by a rigid wall for the case of high velocity. Finally, the free surface effect on WIG is discussed.

Präzisionversus Exaktheit: Konfligierende Ideale der angewandten mathematischen Forschung: Das Beispiel der Tragflügeltheorie

It is often supposed that mathematical argument provides a model of precision for the sciences. In contrast to this view, the present article proposes to distinguish between mathematical exactness as a (historically variable) ideal regulating the inner standards of mathematical argumentation and precision as a (historically variable) norm governing the relation between products of mathematical reasoning in scientific contexts and empirical or practical data. By discussing a major achievement in the mathematization of flight, Ludwig Prandtl's lifting line theory of wings, it is shown that exact reasoning does not necessarily lead to scientific precision, and that the achievement of precision may even require to loosen existing standards of exactness. It is argued that the main contribution of mathematical argument to generating precision in science lies in its capacity to provide sophisticated tools for the production of data, rather than in its adherence to an ideal of exactness.

On an extension of Prandti's lifting line theory to curved wings

In this paper we propose an extension of Prandtl's lifting line theory to curved wings. The aim is to evidence the effect of the curved geometry on the aerodynamic efficiency. By means of an analytical-numerical integration methodology, leading asymptotically to an exponential convergence, we show that the influence of such a geometry on the overall aerodynamic performance must be considered as a first order issue also in the limits of a linear approach.

An unsteady lifting-line theory

A lifting-line theory is developed for wings of large aspect ratio undergoing time-harmonic oscillations, uniformly from high to low frequencies. The method of matched asymptotic expansions is used to enforce the compatibility of two approximate solutions valid far from and near the wing surface. The far-field velocity potential is expressed as a distribution of normal dipoles on the wake, and its expansion near the wing span leads to an expression for the oscillatory downwash. The near-field flow is two-dimensional. A particular solution is obtained from strip theory and a homogeneous component is added to account for the spanwise hydrodynamic interactions. The compatibility of the inner and outer solutions leads to an integral equation for the distribution of circulation along the wing span. In the zero-frequency limit it reduces to that in Prandtl's lifting-line theory, and for high frequencies it tends to the two-dimensional strip theory. Lift computations are presented for an elliptic and a rectangular wing of aspect ratio A = 4.

Numerical lifting line theory applied to drooped leading-edge wings below and above stall

A numerical iterative solution to the classical Prandtl lifting-line theory, suitably modified for poststall behavior, is used to study the aerodynamic characteristics of straight rectangular finite wings with and without leading-edge droop. This study is prompted by the use of such leading-edge modifications to inhibit stall/spins in light general aviation aircraft. The results indicate that lifting-line solutions at high angle of attack can be obtained that agree with experimental data to within 20%, and much closer for many cases. Therefore, such solutions give reasonable preliminary engineering results for both drooped and undrooped wings in the poststall region. However, as predicted by von Karman, the lifting-line solutions are not unique when sectional negative lift slopes are encountered. In addition, the present numerical results always yield symmetrical lift distributions along the span, in contrast to the asymmetrical solutions observed by Schairer in the late 1930's. Finally, a series of parametric tests at low angle of attack indicate that the effect of drooped leading edges on aircraft cruise performance is minimal.

Simplified Unsteady Aerodynamic Concepts with Application to Parameter Estimation

A simplified aerodynamic force model based on the physical principle of Prandtl's lifting line theory and trailing vortex concept has been developed to account for unsteadiness in the aircraft dynamics. The wake is assumed to be compressed to a single shed vortex element of appropriate strength moving downstream at a speed sufficient to approximate the Wagner function. Results are presented illustrating the ability of the simplified theory to duplicate exact solutions in unsteady aerodynamics. Further, consideration is given to the utility of the model in a parameter identification application.

Lifting-Line Theory of Oblique Wings

The study extends Prandtl's lifting-line theory to planar wings involving swept and curved centerlines. Attention is focused on distinct features of such a theory, with special emphasis on an oblique wing in a steady incompressible potential flow. The analysis presented is compared with exact solutions derived from an inverse method and with results from a panel method.

Three-dimensional theory on supercavitating hydrofoils near a free surface

Supercavitating hydrofoils of large aspect ratio operating near a free surface are investigated, assuming an inviscid and irrotational flow with the effects of gravity and surface tension neglected. The flow near the foil, treated as two-dimensional, is solved by a nonlinear free-streamline theory, then a three-dimensional ‘downwash’ correction is made using Prandtl's lifting-line theory. The strength of the lifting-line vortex is determined by information from the two-dimensional solution through a matching procedure, in which the inverse of aspect ratio is used as a small parameter for asymptotic expansions. The analysis incorporates a free-surface reference level to determine the submergence depth of the foil. The present method can be applied to any type of foil having an arbitrary planform or profile shape, including a rounded leading edge, a twist and even a small dihedral angle, within the assumption of large aspect ratio. Numerical computations made on rectangular flat-plate hydrofoils show excellent agreement of results with existing experimental data, even for large angles of attack and relatively low aspect ratios. The pressure distributions, shapes of the cavity and free surface are also calculated as a function of spanwise position.

A Three-Dimensional Theory for Calculating Circulation in Axial Turbomachines (Direct Problems)

In this work the steady-state spanwise circulation distribution of thin, slightly cambered radial blades of finite length is calculated using the method of singularities. The analysis extends the method of Scholz [1] for two-dimensional cascades to the three-dimensional case of radial blades of finite length. The effect of the casing enclosing the cascade is introduced by the method of images. The present analysis uses the generalized cylindrical coordinates without the restriction of the Prandtl lifting line theory. Comparisons show that for large hub-tip ratios, the use of the lifting line approximation will result in large errors. For small tip clearance or large length-chord ratio the present results reduce to the two-dimensional cascade solution.

Interference of Wing and Multipropellers

A systematic procedure is presented for the analysis of the interference of a wing with multipropellers. The assumptions are made that the radius of the propeller and the chord are of the same order and both are much smaller than the span. Using the chord to span ratio as the small expansion parameter, the local two-dimensional solution for the sectional lift to angle-of-attack relationship is uncoupled from the outer three-dimensional solution in the spirit of Prandtl's lifting line theory. Details of the nonuniform stream behind the propellers enter into the local sectional analysis. For the outer solution, the planform of the wing reduces to the lifting line and the stream behind the propellers reduces to a thin sheet of a jet carrying the sectional momentum gained across the propellers and supporting a pressure difference across the sheet. By representing the thin sheet of a jet as a vortex sheet, an integral equation is obtained. When the momentum gain across the propellers is not too large, the integral equation can be further simplified. Several numerical examples are presented to demonstrate the versatility of the analysis.

Theoretical and experimental aerodynamics of the sailwing

The sailwing is a unique type of semiflexible foldable wing. A brief description of its construction, basic properties, and some past research, is used to introduce an analytical and experimental study of its aerodynamic characteristics. Emphasis is placed on an approximate structural analysis which treats the nonlinear behavior of the sail deflection. Twodimensional flexible airfoil theory and Prandtl lifting-line theory are used to establish the aerodynamic loading. The results allow prediction of the induced tensions, the nonlinear life curve, induced drag, and aeroelastic divergence of the sail chordwise deflection mode. Selected experimental results are presented for comparison with the theory, along with a brief discussion of the implications of the study regarding flight vehicle applications of the sailwing.

Linear aerodynamic theory of rotor blades.

A theory is developed for predicting the spanwise lift distribution of rotor blades in forward flight. The guiding thought of Reissner's theory for oscillating wings of finite span is followed, that is, the theory starts with the lifting-surf ace theory and then equations of the lifting-line theory are derived by mathematical approximation. The equation of the liftingsurface theory is exact within the scope of the linear theory. An approximation equivalent to that of Weissinger's L method is adopted rather than that of Prandtl's lifting-line theory. Numerical computations are carried out for the case of an NACA experiment. Fairly good agreements are obtained with the experimental values. However, the theory seems to fail to predict the abrupt lift changes, which are probably due to the nonlinear effect.

Lifting-line theory as a singular-perturbation problem

Abstract : The method of matched asymptotic expansions, recently developed for treating singularperturbation problems, is applied to the flat unswept lifting wing of high aspect ratio. This yields a simplified equivalent of Prandtl's lifting-line theory, with the solution of an integral equation replaced by quadratures. The next approximation is calculated in general terms. Specific application is made to cusped, lenticular, elliptic, and rectangular planforms, and comparison drawn where possible with previous work. Additional non-uniformities at tips and other discontinuities are described, and procedures outlined for their correction. (Author)