Finite Span Wing Research Articles

Unique features of the aerodynamics problems for a wing of finite span

Unique features of the aerodynamics problems for a wing of finite span

Lift-to-drag ratio at supersonic speeds

Wave lift-to-drag ratio is analyzed ignoring friction and using flows behind oblique shock waves and rarefaction waves. It is shown that the lift-to-drag ratio of an infinite oblique plate can surpass considerably that of triangular plates with subsonic, sonic, or supersonic edges. The simplest finite-span oblique wing is a wing with characteristic edges. However, when the normal-to-the-edge flow velocity component behind a shock reaches the speed of sound, the wing contracts into an edge, and other means must be used to exclude the end effect. Several possible variants are indicated. A straight wedge with side plates is the optimal shape for a lifting body with fixed volume, lift, length, and width. Under the same conditions, the cross-section of a pyramidal body formed by stream planes behind one or two plane shocks has practically no effect on the lift-to-drag ratio, while the region of high lift-to-drag ratio is much narrower than for a wedge. If a pyramid fails to provide the required lift-to-drag ratio, it is necessary to turn to forms that better fill the given area. Redistribution of lift between body and wing permits an improvement in the lift-to-drag ratio.

On Useful Shapes of Rigid Wings for Large-Amplitude Sculling Propulsion

If a wing of finite span moves through an incompressible and inviscid fluid, it will in general experience a resistance force, or equivalently, it leaves vorticity behind. We wonder if there are shapes of wings, which can move in a nontrivial way without leaving vorticity behind? We consider rigid, flat wings, hence without thickness and without curvature. Then we find, by numerical means, that wing shapes exist which can move tangentially to an arbitrary cylindrical surface without leaving vorticity behind. In fact, when we prescribe a chordlength distribution along the span, a unique wing shape with this property seems to exist. Such a resistanceless motion can be used as a base motion in the theory of sculling propulsion.

Problems of incompressible flow past thin blades and correction of their shape

Solutions to the problem of the flow of an ideal fluid past a blade or cascade of blades with low blocking factor are found in the framework of the first approximation of the theory of perturbations of the flow past infinitely thin arcs. Problems of correction of the shape of the blades are also considered. Problems associated with the application of perturbation theory in problems of flow past bodies are discussed in Van Dyke's monograph [1]. The present paper includes an example of realization of this theory for the thin curved blades that are widely used in compressor construction. Searches for effective methods for calculating the shape of blades to ensure necessary gas-dynamic properties, for example, a given distribution on the blades of the velocity of separationless flow, led to the appearance of algorithms based on the theory of small perturbations for a thin wing of finite span [2] and a single airfoil in a gas flow [3]. In such an approach, the problem of constructing the required profile can be formulated as a sequence of corrections of the boundary of the flow region with respect to small variations of the boundary values of the flow velocity. The paper contains a general formulation of the linear problem of the correction of the flow boundary, an algorithm for its solution in the case of thin blades in an incompressible flow, and analysis of the obtained solutions. Examples of calculations are presented.

Reply by the Authors to J.G. Leishman

Reply by the Authors to J.G. Leishman

Simple theories of dynamic stall that are helpful in interpreting computational results

This paper briefly reviews the phenomena of dynamic stall and comments on some of the theoretical work performed to help understand why the phenomena occurs for two-dimensional flow past an airfoil undergoing a pitching motion. A first look at how these understandings might be extended to three-dimensional flow fields past finite-span wings is also presented. In particular, a simple theory is developed that provides a rationale for why the experimental results from studies of finite-span wings undergoing dynamic stall closely resemble the dynamic-stall results for two-dimensional airfoils, while steady-flow results from finite-span wing studies are so dissimilar from those for two-dimensional airfoils.

On Optimum Large-Amplitude Sculling Propulsion by Wings of Finite Span

The efficiency of sculling propellers consisting of one or two wings is considered by means of a linearized theory. The thickness and finite span of the wings and their interaction are taken into account. The influence of the viscosity of water on the efficiency is investigated. For a prescribed mean value of thrust and amplitude of the motion of the wings the optimum frequency is determined such that the efficiency is maximum. Numerical results are given in the case that the wings move with "optimum" angles of incidence.

Problem of the supersonic flow around a thin finite span wing with completely subsonic leading edges

Problem of the supersonic flow around a thin finite span wing with completely subsonic leading edges

Numerical simulation of vortex sheet evolution

Recent research on the numerical simulation of vortex sheet evolution is discussed. Applications are presented to a periodic vortex sheet and to the vortex sheet shed behind a finite-span wing.

Ideal incompressible flow over a thin wing of finite span with large-amplitude oscillations

Ideal incompressible flow over a thin wing of finite span with large-amplitude oscillations

Energetics and optimum motion of oscillating lifting surfaces of finite span

The energetics of an unswept wing of finite span oscillating harmonically in combined pitch and heave in in viscid incompressible flow are determined in closed form. The calculations are based on a recently developed low-frequency unsteady lifting-line theory. The energetic calculations for the wing consist of sectional and total values of thrust, leading-edge suction force, power required to maintain the wing oscillations, and energy-loss rate due to vortex shedding in the wake, where the latter quantity is only defined for the entire wing. These results are used to analyse the optimum motion of a wing oscillating harmonically: optimum motion minimizes the power input for fixed average total thrust. The optimum solution is found to be unique (at least for low reduced frequencies), in contrast to the two-dimensional optimum, which is non-unique. Numerical results are presented for the energetics and optimum motion of an elliptic wing.To understand better the structure of the known solution for the optimum motion of an oscillating two-dimensional airfoil, the solution is recast in terms of the normal modes of the energy-loss-rate matrix. It is found that one of the modes, termed here the ‘invisible mode’, plays a central role in the optimum solution and is responsible for its non-uniqueness. The three-dimensional optimum, which is unique, does not have an invisible mode.

Bounded flow sound a supercavitating wing of finite span

Bounded flow sound a supercavitating wing of finite span

Neutral Stability of Laminar Boundary Layers along the Concave Wall

Neutral stability of Gortler vortices was experimentally and theoretically examined. By towing a concave wall of 500mm radius of curvature at a constant speed in a water tank, the neutral stability of the vortices in a low wavenumber region was experimentally determined by observing the behaviour of a given disturbance, which was introduced with a row of wings of finite span, by the hydrogen bubble technique. The neutral curves were calculated for two cases of parallel flow and non-parallel flow models. The present measurements are explained by the Smith model better than by the Gortler model.

Flow past wing-body junctions

The three-dimensional laminar flow past a junction formed by a thin wing protruding normally from a locally flat body surface is considered for wings of finite span but short or long chord. The Reynolds number is taken to be large. The leading-edge interaction for a long wing has the triple-deck form, with the pressure due to the wing thickness forcing a three-dimensional flow response on the body surface alone. The same interaction describes the flow past an entire short wing. Linearized solutions are presented and discussed for long and short two-dimensional wings and for certain three-dimensional wings of interest. The trailing-edge interaction for a long wing is different, however, in that the three-dimensional motions on the wing and on the body are coupled together and in general the coupling is nonlinear. Linearized properties are retrieved only for reduced chord lengths. The overall flow structure for a long wing is also discussed, including the traditional three-dimensional corner layer, which is shown to have an unusual singular starting form near the leading edge. Qualitative comparisons with experiments are made.

Hypersonic flow past a delta wing at large angles of attack

The thin shock layer method /1, 2/ is used to investigate the previously unknown mode of flow past a delta wing of finite span, at angles of attack close to π 2 . The flow problem is formulated and analytic expressions are obtained for the gas-dynamic functions together with the equations expressing the relationships between the form of the wing surface and the shock wave. A method is given for solving inverse problems of flow past actual wings with an attached shock wave. If the angle of attack remains finite, when the ratio ε of the densities on the shock wave tends to zero, the shock wave will remain attached to the sharp leading edge of the wing at any finite sweep-back angle. The basic results of the study of such a flow were given in /3/. On the other hand, when the angles of attack are close to π/2, a flow with a detached shock wave results. Below it is shown that when ε ⪡ 1, a flow past a delta wing exists for the range of angles of attack close to ε/2, α = π/2 − ε 1 2 A , with the shock wave attached to the wing tip, but attached to or detached from the leading edge, depending on the sweep-back angle. This mode falls between the two modes mentioned above, and lends itself to analytic study.

Calculation of the flow of an ideal fluid past a wing of finite thickness

The problem of irrotational flow past a wing of finite thickness and finite span can be reduced by Green's formula to the solution of a system of Fredholm equations of the second kind on the surface of the wing [1]. The wake vortex sheet is represented by a free vortex surface. Besides panel methods (see, for example, [2]) there are also methods of approximate solution of this problem based on a preliminary discretization of the solution along the span of the wing in which the two-dimensional integral equations are reduced to a system of one-dimensional integral equations [1], for which numerical methods of solution have already been developed [3–6]. At the same time, a discretization is also realized for the wake vortex sheet along the span of the wing. In the present paper, this idea of numerical solution of the problem of irrotational flow past a wing of finite span is realized on the basis of an approximation of the unknown functions which is piecewise linear along the span. The wake vortex sheet is represented by vortex filaments [7] in the nonlinear problem. In the linear problem, the sheet is represented both by vortex filaments and by a vortex surface. Examples are given of an aerodynamic calculation for sweptback wings of finite thickness with a constriction, and the results of the calculation are also compared with experimental results.

On an analytic approach to three-dimensional contact problems of elasticity theory

A special kernel approximation is applied to the fundamental two-dimensional integral equation of the contact problem about the frictionless indentation of a rigid stamp into an elastic half-space whereupon it is successfully reduced to a form containing just one-dimensional singular Cauchy-type integrals for a broad class of contact domains. The idea of the method is borrowed from the theory of finite span wings. In the case of a rectangular contact domain the equation obtained decomposes into two one-dimensional integro-differential equations. Considered as examples are the cases of a square stamp and a rectangular stamp with the ratio 1 2 between the sides. Numerical results are compared with those from papers in which numerical methods of solving the problem under consideration were used.

A semi-analytic approach to the self-induced motion of vortex sheets

The rolling-up of the trailing vortex sheet produced by a wing of finite span is calculated as a series expansion in time. For a vorticity distribution corresponding to a wing with cusped tips, the shape of the sheet is found by summing the series using Padé approximants. The sheet remains analytic for some time but ultimately develops an exponential spiral at the tips. The centroid of vorticity is conserved to high accuracy.

Three-Dimensional Oscillatory Piecewise Continuous-Kernel Function Method-Part I: Basic Problems

The two-dimensional subsonic, piecewise continuous-kernel function method used for studying either oscillatory or steady flows is extended in the present work to three-dimensi onal problems involving finite-span wings. The work treats questions associated with the choice of spanwise pressure polynomials, spanwise collocation points, and numerical integration techniques that need be faced by this method. A subsequent paper contains results which confirm the accuracy of the method, its rapid convergence, and its very high efficiency in terms of computational time. The method is tested for a limited class of geometrical discontinuities (i.e., at the wing root only). A third paper contains additional results which relate to a wider class of problems associated with geometrical discontinuities. EOMETRICAL discontinuities have become common in modern airplane wings, including not only control surface deflections but also wing chord discontinuities such as those existing at the root of a delta wing (discontinuity in the first derivative of the chord along the span), at leading-edge extensions, or at wing surface break points. Since these geometrical discontinuities lead to pressure singularities at the same geometrical locations, it is necessary to know the exact form of these pressure singularities if the use of the kernel function method (KFM) is contemplated. The lattice methods (i.e., the vortex or doublet) can successfully cope with unknown pressure singularities if their location is known, but they require a relatively large number of unknowns (boxes) for convergence which at times leads to a relatively large residual error at the converged values.1'2 In Refs. 1 and 2, a different method is proposed which represents the pressure distribution by a set of piecewise continuous polynomials spanning the regions between adjoining singularities (also referred to as boxes) and employs the KFM for solution of the pressure coefficients. It is shown in Refs. 1 and 2 where this two-dimensional problem was treated that such an ap- proach, referred to as the piecewise continuous-kernel func- tion method (PCKFM), has the ability to treat pressure discontinuities in a manner similar to the doublet-lattice method, with the added accuracy and rapid convergence characteristics of the kernel function method. In addition, it is not essential to determine the nature or form of the pressure singularities that might exist along some of the boundaries forming each box. However, to accelerate convergence, pressure singularities are assumed to be known only along the boundaries of the wing; or more specifically, the form of the leading-edge (LE), trailing-edge (TE), and wing-tip pressure singularities are assumed to be known and are treated in the analysis. All other pressure singularities are ignored during the analysis and their consideration is limited to the deter- mination of the boundaries for the different boxes. The basic problems associated with the two-dimensional PCKFM were treated in Refs. 1 and 2. These problems in- cluded the determination of orthogonal pressure polynomials for boxes with different known pressure singularities along their boundaries, the determination of the collocation points associated with the assumed pressure polynomials, the determination of the desired number of boxes, and the number of orthogonal polynomials required in each box. These problems will be addressed again while attempting to extend the PCKFM to wings with finite spans. Additional problems arising in three-dimensi onal flow configurations involve the formulation of numerical techniques which are required for the successful application of the method. These techniques, which are useful beyond the methods described in this work, will be developed herein. Wing-Tip Pressure Singularity and Associated

Influence of transverse slits on the hydrodynamic coefficients of a wing of finite span in stationary and nonstationary motion near a wall

The method of matched asymptotic expansions is used to construct an approximate solution to the problem of the influence of narrow transverse slits on the hydrodynamic coefficients of a thin rectangular wing moving near a wall. The flow in the neighborhood of a slit is described by a local asymptotic solution satisfying the condition of continuity of the pressure on the leading edge of the slit and matched to the main solution. Results of the calculations illustrate the influence of the slits on the hydrodynamic characteristics of the wing at different Strouhal numbers and aspect ratios.