The close-approach problem associated with vortex-lattice methods was examined numerically with the objective of calculating velocities at arbitrary points, not just at midpoints, between the vortices. The objective was achieved using a subvortex technique in which a vortex splits into an increasing number of subvortices as it is approached. The technique, incorporated in a two-dimensional potential flow method using "submerged" vortices and sources, was evaluated for a cambered Joukowski airfoil. The method could be extended to three dimensions, and should improve non-linear methods, which calculate interference effects between multiple wings and vortex wakes, and which include procedures for force-free wakes.

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A subvortex technique for the close approach to a discretized vortex sheet

A method of predicting the minimum induced drag conditions in conventional or innovative lifting systems is presented. The method is based on lifting-line theories and the small perturbation acceleration potential. Assuming linearity and rigid wake aligned with the freestream, the optimal conditions are formulated using the Euler‐Lagrange integral equation subject to the conditions of fixed total lifting force and wing span. The Lagrange multiplier method is applied, and equations for the design optimum are obtained and solved directly. Particular attention is paid to the Hadamard finite-part integrals involved in the solution process. Munk’s drag theorems are also applied in order to verify the quality of the solutions. In this paper, where the theoretical/computational foundation is laid for the induced drag minimization of general lifting-line configurations, the case of the biplane under optimal conditions is extensively analyzed. It is demonstrated that, under optimal conditions, the two wings (which have the same wing span) have the same circulation distribution. Cases of finite, infinite, and infinitesimal distances between the wings are analyzed as well. It is shown that the optimal distribution is, in general, not elliptical. The proposed theoretical approach is general, and it offers a valuable tool for direct prediction of minimum induced drag in both planar and nonplanar lifting systems. Results obtained by the proposed approach shed light on some of the mathematical issues involved, and they can be used for verifying results obtained by numerical math-programming-based optimization.

/pdf/induced-drag-minimization-a-variational-approach-using-the-3ml7by9ol0.pdf

Induced Drag Minimization: A Variational Approach Using the Acceleration Potential

A problem of some practical importance concerns the interference between a wing surface in a free stream and a nearby streamwise vortex. This situation can arise for example, either in a helicopter rotor system, or when concentrated vortices, which are shed from a fuselage nose or from forward canard controls, interfere with an aft lifting surface. An approximate non-linear analytical study is described in Ref. 1 in which the three-dimensional loading on a two-dimensional non-swept wing underneath a streamwise line vortex in low speed flow (as shown in Fig. 1) is deduced. Since publication of Ref. 1 a number of linearised theoretical studies have come to light.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/20E2D76B5C41098647300BABC442755D/S0001924000088771a.pdf/div-class-title-some-experimental-results-of-the-effect-of-a-streamwise-vortex-on-a-two-dimensional-wing-div.pdf

Some experimental results of the effect of a streamwise vortex on a two-dimensional wing

The rolling up of the vortex sheet shed from a finite aspect ratio wing into discrete vortices has been extensively investigated in the past. Most of these investigations have, however, considered wings having an elliptic loading. The early design charts of Silvertstein et al gave downwash angles for a variety of flapped and unflapped aerofoils by considering the trailing vortex sheet to be composed of discrete line vortices but ignored the rolling up process that takes place.

The rolling up of a trailing vortex sheet

The three-dimensional vortex flow that develops around a close-coupled canard-wing configuration is characterized by a strong interaction between the vortex generated at the canard and the aircraft wing. In this paper, a theoretical potential flow model is devised to uncover the basic structure of the pressure and velocity distributions on the wing surface. The wing is modelled as a semi-infinite lifting-surface set at zero angle of attack. It is assumed that the vortex is a straight vortex filament, with constant strength, and lying in the freestream direction. The vortex filament is considered to be orthogonal to the leading-edge, passing a certain height over the surface. An incompressible and steady potential flow formulation is created based on the three-dimensional Laplace's equation for the velocity potential. The boundary-value problem is solved analytically using Fourier transforms and the Wiener-Hopf technique. A closed-form solution for the velocity potential is determined, from which the velocity and pressure distributions on the surface and a vortex path correction are obtained. The model predicts an anti-symmetric pressure distribution along the span in region near the leading-edge, and a symmetric pressure distribution downstream from it. The theory also predicts no vertical displacement of the vortex, but a significant lateral displacement. A set of experiments is carried out to study the main features of the flow and to test the theoretical model above. The experimental results include helium-soap bubble and oil-surface flow pattern visualization, as well as pressure measurements. The comparison shows good agreement only for a weak interaction case, whereas for the case where the interaction is strong, secondary boundary-layer separation and vortex breakdown are observed to occur, mainly owing to the strong vortex-boundary layer interaction. In such a case the model does not agree well with the experiments.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112096002704

The three-dimensional interaction of a streamwise vortex with a large-chord lifting surface: theory and experiment

A wing of finite span in flight generates trailing vorticity which “rolls up” into two “discrete” vortices. During the rolling up process the trailing vorticity is displaced vertically downward (assuming that the aircraft is in horizontal flight) under the influence of the wing circulation and the self-induced downwash velocities. In this note a method is presented for the prediction of the spatial distribution of the trailing vorticity during the rolling up process, thus giving the overall downwash field behind the wing.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/1B0118BEBCEE21778AD41F9E6B0AC802/S0001924000046133a.pdf/a-numerical-method-for-calculating-the-trailing-vortex-system-behind-a-swept-wing-at-low-speed.pdf

A Numerical Method for Calculating the Trailing Vortex System behind a Swept Wing at Low Speed

The aerodynamics of a circular jet emerging normally from a wall into a cross flow is critically assessed. Some general trends are obtained when viscous effects are neglected. But viscous effects dominate the real flow behaviour. Attempts to predict these real effects are reviewed and their anomalies highlighted. The importance of understanding and specifying the jet exit conditions is stressed, which is relevant to complete numerical solutions.

A review of the aerodynamics of a jet in a cross flow

Practical problems often arise in which the aerodynamic loadings on wings and fins, due to nearby free vortices, are required. For example, aerodynamic loads on helicopter blades are affected by the tip vortices which are continuously being shed and which lie beneath the rotor plane. And vortices shed from the nose of an aircraft fuselage can induce significant side loads on a fin. The estimation of the aerodynamic load distribution on a moving wing in the neighbourhood of a free vortex (whose axis is primarily in the free stream direction) is an interesting problem which, as far as the author is aware, has not been discussed previously in the literature. Standard techniques for conventional linearised wing theory cannot be applied directly to this problem because the trailing vorticity shed from the wing trailing edge is affected by the flow field of the free vortex and so the mathematical application of the Kutta trailing edge condition is more complex.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/645489FA684B2DA3D18425D1077DD703/S000192400004567Xa.pdf/div-class-title-aerodynamic-loading-induced-on-a-two-dimensional-wing-by-a-free-vortex-in-incompressible-flow-div.pdf

Aerodynamic Loading Induced on a Two-Dimensional Wing by a Free Vortex in Incompressible Flow

One of the best known results in finite wing theory, which is quoted in aerodynamic text books, states that the distance between two rolled up vortices behind a wing with an elliptic spanwise load distribution is sπ/2 , where s is the wing semi-span. Within the framework of linear wing theory this result is correct. But it is of interest to enquire how this result is affected when the standard linear model becomes less valid, for example, at higher wing lift coefficients. In this note, it is assumed that a continuous trailing vortex sheet rolls up into two discrete vortices. These two discrete vortices are assumed to have cores of finite radius; inside the cores the flow is taken to be solid body rotation, while the flow outside the cores is the standard irrotational vortex field. It is assumed that this rolling up process takes place far downstream of the finite wing which generates the trailing vorticity.

On the Rolling Up of a Trailing Vortex Sheet

Summary The aim of axiomatic aerodynamic modelling is introduced, namely to understand the validity of aerodynamic modelling in the context of aircraft dynamics with particular emphasis on understanding the relationship between full-scale flight behaviour as compared with that predicted from data based on wind tunnel tests.

G. J. Hancock

Papers

A Numerical Method for Calculating the Trailing Vortex System behind a Swept Wing at Low Speed

A review of the aerodynamics of a jet in a cross flow

Aerodynamic Loading Induced on a Two-Dimensional Wing by a Free Vortex in Incompressible Flow

On the Rolling Up of a Trailing Vortex Sheet

Part 1 — Introduction to the concept of axiomatic aerodynamic modelling