Three-dimensional analog of the Sokhotsky-Plemelj formulas and its application in the wing theory

A solution of a hydrodynamic problem of motion of an ideal incompressible fluid in a finite-thickness vortex layer is obtained. In the limiting case (infinitely thin layer), this layer transforms to a vortex surface. Formulas are derived for limiting values of the velocity vector of the fluid approaching this surface; these formulas extend the Sokhotsky-Plemelj formulas for a singular integral of the Cauchy type to a three-dimensional space. Three integral equations are derived on the basis of these formulas and the proposed method of modeling a finite-thickness wing by a closed vortex surface. It is shown that only one equation is left in the case of an infinitely thin wing, which corresponds to the condition of fluid non-penetration through the wing surface.     

Rotary aerodynamic derivatives in the asymptotic theory of a high-aspect-ratio wing

The linear problem of inviscid incompressible flow around a high-aspect-ratio wing at an angle of attack and in the presence of steady pitching and rolling rotation is considered. The main integral equation of the problem is reduced to a sequence of one-dimensional integral equations without use of the matched asymptotic expansions method. The first few terms of the series for the circulation distribution over the wing surface are calculated. For an elliptic high-aspect-ratio wing the corresponding aerodynamic forces are calculated. The derivatives of the aerodynamic coefficients of the wing with respect to the angle of attack and the angular velocities are determined. The asymptotic expressions obtained are compared with the results of numerical calculations of the corresponding derivatives using the discrete vortex method.     

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.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 124–131, October–December, 1981.     

Steady/unsteady aerodynamic analysis of wings at subsonic,sonic and supersonic Mach numbers using a 3D panel method

This paper treats the kernel function of an integral equation that relates a known or prescribed upwash distribution to an unknown lift distribution for a finite wing. The pressure kernel functions of the singular integral equation are summarized for all speed range in the Laplace transform domain. The sonic kernel function has been reduced to a form, which can be conveniently evaluated as a finite limit from both the subsonic and supersonic sides when the Mach number tends to one. Several examples are solved including rectangular wings, swept wings, a supersonic transport wing and a harmonically oscillating wing. Present results are given with other numerical data, showing continuous results through the unit Mach number. Computed results are in good agreement with other numerical results. Copyright © 2003 John Wiley & Sons, Ltd.     

Self‐propulsion of oscillating wings in incompressible flow

In this paper, we show that the oscillatory motion of an airfoil (wing) in an ideal fluid can determine the apparition of thrust. In the framework of the linearized perturbation theory, the pressure jump over the oscillating wing is the solution of a two‐dimensional hypersingular integral equation. Using appropriate quadrature formulas, we discretize the oscillatory lifting surface integral equation in order to obtain the jump of the pressure across the surface. Integrating numerically, we obtain the drag coefficient. For some oscillatory motions, if the frequency of the oscillations surpasses a certain value, the drag coefficient becomes negative, i.e. there appears a propulsive force. Copyright © 2007 John Wiley & Sons, Ltd.     

Determination of the aerodynamic characteristics of aircraft in the transonic velocity range

The problem of determining the integral aerodynamic characteristics of aircraft as a whole in the transonic velocity range is considered. An approximate method of their calculation is developed using the nonlinear transonic theory of small perturbations for three-dimensional flow over a body. The method of investigation consists in separating the flow region into two subregions (outer and inner), applying numerical methods of integrating the equations in those regions, and joining the solutions. The Murman-Cole method of calculating the pressure drag of an isolated wing is generalized to the case of a combination of wing and fuselage. Central Aerohydrodynamics Institute, Zhukovskii 140160. Translated from Prikladnaya Mekhanika i Tekhnicheskaya fizika, Vol. 39, No. 4, pp. 91–101, July–August, 1998.     

Supersonic flow over a slender wing with subsonic edges

The method of integral equations is generalized to calculate steady flow past wings with an arbitrary shape in plan with subsonic leading and trailing edges. The determination of the velocity potential in the leading part of the wing, where there is no influence of the vortex sheet, is reduced to the solution of a two-dimensional integral equation of the second kind. The trailing part, which is subject to the influence of the vortex sheet, is divided into a number of subregions, in which the calculation of the acceleration potential reduces to the solution of one-dimensional equations of the type of Fredholm equations of the second kind and to quadrature. The unique solvability of the obtained integral equations is investigated; it is shown that they can be solved by successive approximation. As an example, the solution to the problem of flow past a flat delta-shaped wing is found and compared with the exact solution to the problem found by the method of conic flows [4, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 119–127, September–October, 1981.I thank G. Yu. Stepanov for discussing the paper.     

Propagation of perturbations in the three-dimensional boundary layer on a delta wing in the viscid-inviscid interaction regime

The propagation of perturbations in the three-dimensional boundary layer on a planar delta wing in a hypersonic gas stream is investigated in the strong viscid-inviscid interaction regime. A characteristic associated with the induced pressure is found and an integral relation determining the velocity of its propagation is obtained. The directional diagrams of the propagation velocity of the characteristic surface in the boundary layer are determined for a series of constitutive parameters. This makes it possible to calculate the perturbation propagation velocities in the boundary layer when the velocity profiles in the longitudinal and transverse directions are known.     

Numerical study of self-induced wing rock of a low-aspect-ratio delta wing in the presence of external disturbances

Self-induced wing rock of a delta wing, in particular, in the presence of external disturbances are studied by means of numerical simulations of a separated flow of an ideal incompressible fluid around a delta wing. The results obtained are compared with experimental data. The vortex nature and the mechanism of self-induced oscillations are studied. Regions of synchronization of the aerodynamic self-oscillatory system in the presence of external disturbances are identified. Methods of suppression of self-induced wing rock are proposed.     

Surface integral analogy approaches for predicting noise from 3D high-lift low-noise wings

Three surface integral approaches of the acoustic analogies are studied to predict the noise from three concep- tual configurations of three-dimensional high-lift low-noise wings. The approaches refer to the Kirchhoff method, the Ffowcs Williams and Hawkings （FW-H） method of the permeable integral surface and the Curle method that is known as a special case of the FW-H method. The first two approaches are used to compute the noise generated by the core flow region where the energetic structures exist. The last approach is adopted to predict the noise specially from the pressure perturbation on the wall. A new way to con- struct the integral surface that encloses the core region is proposed for the first two methods. Considering the local properties of the flow around the complex object-the actual wing with high-lift devices-the integral surface based on the vorticity is constructed to follow the flow structures. The surface location is discussed for the Kirchhoff method and the FW-H method because a common surface is used for them. The noise from the core flow region is studied on the basis of the dependent integral quantities, which are indicated by the Kirchhoff formulation and by the FW-H formulation. The role of each wall component on noise contribution is analyzed using the Curle formulation. Effects of the volume integral terms of Lighthill＇s stress tensors on the noise pre-diction are then evaluated by comparing the results of the Curle method with the other two methods.     

Aerodynamics and mechanisms of elementary morphing models for flapping wing in forward flight of bat

Large active wing deformation is a significant way to generate high aerodynamic forces required in bat's flapping flight. Besides the twisting, elementary morphing models of a bat wing are proposed, including wing-bending in the spanwise direction, wing-cambering in the chordwise direction, and wing area-changing. A plate of aspect ratio 3 is used to model a bat wing, and a three-dimensional unsteady panel method is used to predict the aerodynamic forces. It is found that the cambering model has great positive influence on the lift, followed by the area-changing model and then the bending model. Further study indicates that the vortex control is a main mechanism to produce high aerodynamic forces. The mechanisms of aerodynamic force enhancement are asymmetry of the cambered wing and amplification effects of wing area-changing and wing bending. Lift and thrust are generated mainly during downstroke, and they are almost negligible during upstroke by the integrated morphing model-wing.     

Methodologies for weight estimation of fixed and flapping wing micro air vehicles

One of the important steps in the sizing process of fixed and flapping wing micro air vehicles (MAVs) is weight estimation of the electrical and structural components. In order to enhance the flight performance and endurance of MAVs, it is required to carefully estimate their weight with a minimum error. In this study, methodologies to estimate the weight of fixed and flapping wing MAVs are proposed. After dividing the total weight of the MAV into weights of structural and electrical components, these two weights are separately identified. The weight of the MAV electrical components is estimated by using engineering design techniques and the weight of the structure is identified by using statistical and computational methods. The proposed methodology for structural weight estimation is based on calculating the percentage of the used material in the construction of different parts of MAVs and then presenting the weight of each part in terms of the wing surface. The proposed computational method gives the exact estimation for the weight of each structure component, such as wing, tail, fuselage, and etc. Based on the offered method for weight estimation of MAVs, the weight estimation of a fixed wing MAV with inverse Zimmerman planform and a flapping wing MAV named “Thunder I” are experimentally shown. This developed methodology gives guidelines for weight estimation and determination of the structural weight percentages in order to design and fabricate efficient fixed and flapping wing MAVs.     

基于卷积神经网络和松鼠优化算法的机翼结构混合优化设计#br#

针对大展弦比机翼的结构轻量化优化设计,提出了一种高效的布局和尺寸混合优化方法．在CFD/CSD气动弹性计算的基础上,对不同的结构变量进行统一编码,使用一维卷积神经网络建立代理模型,并使用松鼠优化算法建立了混合优化模型进行搜索寻优．以某型太阳能无人机的机翼结构优化为例,优化结果表明翼肋的布局变量和翼梁的尺寸变量之间存在着耦合关系,使用松鼠优化算法相比于遗传算法节省了35 %~45 %的计算成本,且混合优化后的结构比原始结构减重4.1 %,验证了该方法的有效性．     

Numerical solution of the problem of the flow of a supersonic gas stream over the upper surface of a delta wing in the expansion region

A numerical method is described for the calculation of supersonic flow over the arbitrary upper surface of a delta wing in the expansion region. The shock wave must be attached everywhere to the leading edge of this wing from the side of the lower surface. The stream flowing over the wing is assumed to be nonviscous. A problem with initial conditions at some plane and with boundary conditions at the wing surface and the characteristic surface is set up for the nonlinear system of equations of gas dynamics. The difference system of equations, which approximates the original system of differential equations on a grid, has a second order of accuracy and is solved by the iteration system proposed in [1]. The initial conditions are determined by the method of establishment of self-similar flow. A number of examples are considered. Comparison is made with the solutions of other authors and with experiment.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 76–81, November–December, 1973.The author thanks A. S. II'ina who conducted the calculations and V. S. Tatarenchik for advice.     

Investigation on vortex shedding of a swept-back wing

Effects of Reynolds number and angle of attack on the vortex shedding of a finite swept-back wing are experimentally studied. The cross-sectional profile of the wing is NACA 0012, and the sweep-back angle is 15° The time series signals detected by hot-wire in the wake region shows four distinct behaviors: laminar, subcritical, transitional, and supercritical. The derived Strouhal number curves are significantly different in these four behaviors. In addition, the statistical properties of turbulence, that is, the power spectrum density function, probability density function, correlation coefficient, Lagrangian integral time scales, and length scales are also presented in this paper.     

Suppression of the wing-rock phenomenon using nonlinear controllers

This paper deals with the design of nonlinear controllers for the wing-rock phenomenon of a delta wing aircraft. A fifth order dynamic model is used to describe this phenomenon. A state transformation is introduced such that the transformed dynamic model is in a form which is suitable for a variety of control designs. A feedback linearization control scheme and a sliding-mode control (SMC) scheme are then proposed to suppress wing rock oscillations. It is shown that the two controllers successfully suppress the undesired oscillations and guarantee the asymptotic convergence of all system trajectories to their desired values. The effectiveness of the proposed controllers is verified through simulation studies.

     

Effects of mass and chordwise flexibility on 2D self-propelled flapping wings

A self-propelled flexible flapping wing 2D numerical model undergoing a combined pitching and heaving motion is presented. Since such freely moving foil experiences zero net thrust, a definition of efficiency for this kind of problem is proposed and discussed against other formulations found in the literature. It is also shown that the deviation motion of wings such as that found in natural flyers is likely a consequence of the fluid–structure dynamics of the wings. The passive deviation motion observed in numerical simulations is either a consequence of a feathering mechanism referred to as rigid feathering or of the inertial displacement caused by the wing deformation. The effects of flexibility on the performance of the wing are also presented. It is found that flexibility may significantly enhance the efficiency in pressure-driven deformation cases. The rigid feathering mechanism is found to have an effect similar to that of the feathering caused by wing flexibility on the performances of pressure-driven deformation cases.     

Velocity field generated by wing vibrations propagating along the surface

The velocity field generated by wing vibrations propagating along an elastic wing surface with finite velocity is studied.The gasdynamic problem is reduced to a mixed boundary-value problem with a moving boundary for the three-dimensional wave equation. The solution is obtained in closed form when the wing travels at supersonic velocity following an arbitrary law, the vibration propagation front is an arbitrary curve displacing along the wing surface, and the wing edges are supersonic.     

Nondimensional frequency scaling of aerodynamically-tensioned membranes

Membrane wings have applications that involve low Reynolds number flyers such as micro air vehicles. The time-averaged and time-dependent deformations of the membrane affect the aerodynamic characteristics of the wing, primarily in the region beyond the maximum aerodynamic efficiency of the wing. This paper investigates an appropriate nondimensional vibration frequency scaling of a spanwise tensioned membrane with free (unattached) leading and trailing edges at low Reynolds numbers relative to nondimensional aeroelastic parameters. Silicone rubber membranes with varying spanwise pre-tension, aerodynamic tension (due to wing angle-of-attack and flow dynamic pressure), modulus of elasticity, span, and thickness are studied. Experimental results are compared to a proposed scaling that simplifies the aerodynamic loading as a uniform pressure distribution acting on the membrane. Data is further compared and discussed relative to previous published results of membrane wings with finite wing spans (three-dimensional flow) and fixed (rigid) leading edges.     

Calculating the effect of gusts on an arbitrary thin wing

A numerical method of calculating the unsteady flow about a thin wing moving in an ideal incompressible medium is developed on the basis of the lifiting surface scheme. The variation of the boundary conditions on the wing surface with time and coordinates may be arbitrary. Therefore, the method makes it possible to examine the aperiodic motion of a wing as a rigid body, consider any wing deformations, analyze the wing entry into a gust, study the effect of a weak shock wave on the wing, etc. In addition, practically no limitation is imposed on the shape of the thin lifting surface: the method is applicable to monoplane wings of any planform, to annular wings, to systems of similar wings, etc.Studies in which the effect of a gust on a wing is analyzed have been reviewed in [1, 2]. Without dwelling on this review, we note that at subsonic speeds an effective solution of the problem has been obtained only for a profile.The author wishes to thank E. P. Kapustina for working the examples.