An adjoint method for the calculation of remote sensitivities in supersonic flow

This paper presents an adjoint method for the calculation of remote sensitivities in supersonic flow. The goal is to develop a set of discrete adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the pressure distribution at an arbitrary location within the domain of interest. First, this paper presents the complete formulation and discretization of the discrete adjoint equations. The special treatment of the adjoint boundary condition to obtain remote sensitivities or sensitivities of pressure distributions at points remotely located from the wing surface are discussed. Secondly, we present results that demonstrate the application of the theory to a three-dimensional remote inverse design problem using a low sweep biconvex wing and a highly swept blunt leading edge wing. Lastly, we present results that establish the added benefit of using an objective function that contains the sum of the remote inverse and drag minimization cost functions.     

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.     

Homogenization and Enhancement for the G—Equation

We consider the so-called G-equation, a level set Hamilton–Jacobi equation used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably small spatial divergence, we prove that, as the size of the oscillations diminishes, the solutions homogenize (average out) and converge to the solution of an effective anisotropic first-order (spatio-temporal homogeneous) level set equation. Moreover, we obtain a rate of convergence and show that, under certain conditions, the averaging enhances the velocity of the underlying front. We also prove that, at scale one, the level sets of the solutions of the oscillatory problem converge, at long times, to the Wulff shape associated with the effective Hamiltonian. Finally, we also consider advection depending on position at the integral scale.     

Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous concentric spherical shell-in-cell model is studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin and Cunningham/Mehta-Morse models are considered. At the fluid-porous interfaces, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. The hydrodynamic drag force acting on the porous shell by the external fluid in each of the four boundary conditions on the cell surface is evaluated. It is found that the normalized mobility of the particles (the hydrodynamic interaction among the porous shell particles) depends not only on the permeability of the porous shells and volume fraction of the porous shell particles, but also on the stress jump coefficient. As a limiting case, the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres with jump.     

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.     

Wings of minimum drag for given lift

The variational problem of determining the optimal shape (camber and twist) of the midsurface of a wing having minimum wavedrag is examined in the linear formulation. It is shown that for wings with supersonic leading edge and straight trailing edge, whose shape is given in the form of a double polynomial, the over-all aerodynamic characteristics can be simply expressed in terms of the equation for the leading edge of the wing. This makes it possible not only to solve the variational problem by the Ritz method and obtain the minimum wave drag [1] but also to find the optimal shape of the wing. As examples we consider delta and double-delta wings.     

The minimum drag of thin wings at supersonic speed according to Kogan's theory

In the theory of thin lifting surfaces the minimum drag consistent with a given total lift occurs when the downwash, averaged between forward and reversed motion of the wing, has the same value at all points of the wing planform. In Kogan's theory the conditions for minimum drag are determined on the forward sloping characteristic surface touching the trailing edge of the wing and it is shown that such a surface plays the role of the Trefftz plane familiar in subsonic wing theory. This paper shows how Kogan's theory may be applied to determine the drag of elliptic wings at supersonic speed. It appears that such wings have lower drag than the conventional delta wing.     

水平地震时斜面水坝受到的非线性动水压力的全过程的解析解

本文给出在非线性Ｒａｙｌｅｉｇｈ阻尼作用下，由水平地震激发的水库内动水压力变化全过程的解析解：在激发初期其幅值较小，可用小参数法使控制方程线性化，从而求得各阶渐近控制方程的解析解；当动水压力的幅值增大到小参数展开不适用时，用ｖａｎ ｄｅｒｐｏｌ法求得非线性控制方程的渐近解，从而显示了动水压力的幅值从有序变化到发生突然跳跃的多值性的非线性本质。     

A receding contact plane problem between a functionally graded layer and a homogeneous substrate

In this paper, we consider the plane problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress–strain law and over a certain segment of its top surface is subjected to normal tractions while the rest of this surface is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using integral transforms, the plane elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact half-length. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact half-length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.     

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.     

Reduction of the Induced Drag of a Supersonic Wing

The oblique wing effect, i.e., a reduction in the wave drag for given lift, cannot be realized for a delta wing with supersonic leading edges owing to the lift reduction in the wing mid-section. To preserve the effect, the disturbances generated by the delta wing vertex must be eliminated by adding a body (wedge) to the wing by replacing the streamsurfaces behind the shock with rigid surfaces. Moreover, using wing tip deflection, and thereby reducing the wave drag to zero, makes it possible to obtain a lift- drag ratio close to that of the limiting, infinitely long flat plate.     

A NUMERICAL STUDY OF CYLINDERS IN WAVES AND CURRENTS

The research on combination flow of planar oscillatory flow plus an in-line steady stream is of importance to the situation of structures in waves and current. The combination flow has not been studied extensively. There is still little information about the effect of current and about combined effect of current and waves on hydrodynamic loading of the structures. The present study investigates the combination flow around a circular cylinder using a vortex-based method incorporating vortex moving particles (discrete vortices) with a finite-difference scheme for the vorticity diffusion. The main attention is paid to the effects of a small current on in-line fluid forces and vortex patterns in the wake. Morison's equation and an equation with two drag terms are examined. The results show that the presence of a small current in an oscillatory flow can reduce the drag coefficient significantly. Morison's equation gives reasonably good predictions for the in-line forces for an oscillatory flow plus a small current. The current tends to bring the whole vortex wake downstream and tries to form the stable Karman asymmetrical form in the downstream wake. The present results show certain agreement with some previous experimental results.     

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.     

Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axi-Symmetric Punch

In this paper we consider the problem of adhesive frictionless contact of an elastic half-space by an axi-symmetric punch. We obtain integral equations that define the tractions and displacements normal to the surface of the half-space, as well as the size of the contact regions, for the cases of circular and annular contact regions. The novelty of our approach resides in the use of Betti’s reciprocity theorem to impose equilibrium, and of Abel transforms to either solve or substantially simplify the resulting integral equations. Additionally, the radii that define the annular or circular contact region are defined as local minimizers of the function obtained by evaluating the potential energy at the equilibrium solutions for each pair of radii. With this approach, we rather easily recover Sneddon’s formulas (Sneddon, Int. J. Eng. Sci., 3(1):47–57, 1965) for circular contact regions. For the annular contact region, we obtain a new integral equation that defines the inverse Abel transform of the surface normal displacement. We solve this equation numerically for two particular punches: a flat annular punch, and a concave punch.     

Modification of static-wing tip vortex via a slender half-delta wing

The modification of the tip vortex generated by a rectangular NACA 0012 wing via a tip-mounted slender half-delta wing (HDW) was attempted experimentally at Re=2.81×10⁵. In addition to the increase in lift with increasing HDW deflection, compared to the baseline wing, the roll-up process of the tip vortex was also found to be significantly modified, as a result of the breakdown of the HDW vortex. The addition of the HDW also caused an increased total drag. Fortunately, the lift-induced drag was found to be reduced compared to its baseline counterpart for 0° and 5° HDW deflections. The change in the lift-induced drag also translates into a virtually unchanged profile drag, regardless of HDW deflection. In short, the largest lift-induced drag reduction achieved by the zero-deflection HDW resulted in an improved lift-to-drag ratio, at high angle-of-attack range, compared to the baseline wing.     

Wings with optimal characteristics in hypersonic flow

We present some exact results and results obtained by the local variation method [1–3] from numerical solutions of variational problems on the symmetric wing of minimum drag and the wing of maximum L/D in hypersonic flow; a modification of the method of local variations is proposed for the numerical solution of variational problems with isoperimetric constraints. The Newton method [4] was used to calculate the pressure distribution over the wing surface.The author wishes to thank M. N. Kogan and O. G. Fridlender for many helpful discussions.     

Effect of momentum transfer condition at the interface of a model of creeping flow past a spherical permeable aggregate

Creeping flow past an isolated, spherical and permeable aggregate has been studied adopting the Stokes equation to model the fluid external to the aggregate and the Brinkman equation for the internal flow. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used along with the continuity of velocity components and continuity of the normal stress. Using Faxen’s laws, drag and torque are calculated for different flow conditions and it is observed that drag and torque not only change with the permeability of the porous region, but as stress jump coefficient increases, the rate of change in behavior of drag and torque increases.     

On the Torsion of a Class of Nonlinear Fiber Composite Cylinders

This paper presents an analysis of the torsion of a solid or annular circular cylinder consisting of nonlinear material in the form of an elastic matrix with embedded unidirectional elastic fibers parallel to the cylinder axis. The specific class of composite considered is one for which nonlinear fiber-matrix interface slip is captured by uniform cohesive zones of vanishing thickness. Previous work on the effective antiplane shear response of this material leads to a stress–strain relation depending on the interface slip together with an integral equation governing its evolution. Here, we obtain an approximate single mode solution to the integral equation and utilize it to solve the torsion problem. Equations governing the radial distributions of shear stress and interface slip are obtained and formulae for torque–twist rate are presented. The existence of singular surfaces, i.e., surfaces across which the slip and the shear stress experience jump discontinuities are analyzed in detail. Specific results are presented for an interface force law that allows for interface failure in shear.