Three-Dimensional Bodies of Minimum Total Drag in Hypersonic Flow

The problem of constructing three-dimensional bodies of minimum total drag is studied within the framework of a local interaction model. Under certain assumptions, this model can be adopted to describe the distributions of both pressure and skin friction on the body during its high-speed motion through gases and dense media. Without any constraints on the possible drag law within the scope of the accepted model, the optimum shapes providing the minimum drag are found without any simplifying assumptions regarding their geometry. It is shown that, for a given base area and specified limitations on the body size, one can construct an infinite number of optimum forebody shapes. It is proved that the desired shapes are formed by combinations of surface parts whose normal makes a certain constant angle with the direction of motion. The optimum angle is determined by the velocity and medium characteristics in terms of the constants of the drag law. A method of optimum shape design is proposed; in particular, it allows one to construct optimum bodies like missiles with aft feather and optimum bodies with a circular base. All the bodies constructed have the same minimal total drag for the given base area. Even for asymmetrical bodies, the acting force has no component in a plane perpendicular to the direction of motion. Special attention is paid to the particular case of the minimum drag body design in hypersonic flow, when the pressure on the body is specified by the Newton formula. A comparative study of the results obtained for Newtonian flow shows that the proposed shapes are more effective in providing a drag reduction than bodies found to be optimum in earlier studies under special simplifying assumptions.     

The construction of a nose shape of minimum drag for specified external dimensions and volume using Euler equations

The problem of constructing the axisymmetric nose shape which gives minimum wave drag for a specified volume and external dimensions is solved by a direct method using Euler equations. As in the Newton's formula approximation, the optimum contours together with the front faces – a segment of the boundary extremum along the longitudinal coordinate and the gently sloping segment of the bilateral extremum – may contain a cylindrical end part with a horizontal segment of the boundary extremum with respect to the maximum admissible radial coordinate. In the direct method, the required parameters (“controls”), which define the shape of the optimum contour, are the radii corresponding to the points of the segment of a bilateral extremum, including the radius of the face for fixed abscissas. For each aspect ratio (the ratio of the length to the radius of the base), when a certain value of the volume coefficient (the ratio of the volume to the volume of a cylinder of maximum external dimensions) is exceeded, the optimum nose shape is completed by a rear cylindrical part. The optimum nose shape, which begins from a certain initial contour, that satisfies the limitations of the problem, is constructed after a finite number of cycles. In each cycle, all the controls are corrected, and together with the directions of the change, their increments are found, while the information necessary for this for any number of controls is obtained after three direct calculations. One other advantage of the method is the rapid, close to quadratic, convergence. The nose shapes constructed are compared with the nose shapes that are optimum in the Newton's formula approximation.     

The optimal conical twist of a delta wing

The problem of reducing the drag of a wing at a specified lift in a supersonic flow is investigated. A solution for a delta wing is obtained in a simplified formulation of the optimization problem and a theoretical analysis. It is shown that the optimal conical wing is formed by elements of elliptical cones and planes. Numerical modelling of the flow of a non-viscous non-heat-conducting gas past the wing is performed, and the results of the theoretical analysis and direct optimization are compared. ©2012     

The effective mass of a point mass moving along a string on a Winkler foundation

A simple in form and physically clear asymptotic solution of the problem of the motion (without friction) of a point mass acted upon by a specified external force on a string on a Winkler foundation is obtained, taking into account the wave drag on the motion. It is shown that the point mass moves along the string in the same way as a point with a variable velocity-dependent mass would move when acted upon solely by an external force (ignoring drag).     

Optimum flat-top airfoils in viscous hypersonic flow

The problem of determining the shape of the lower surface of a flat-top airfoil which maximizes the lift-to-drag ratio at hypersonic speeds is considered. The body is assumed to be slender with the flat upper surface parallel to the free-stream direction. The total drag is made up of pressure drag, base drag, and skin-friction drag. The local pressure coefficient used is that of the tangent-wedge approximation; the base pressure coefficient is assumed constant; and the local skin-friction coefficient is that for laminar flow. The extremal problem is solved with the use of the methods of the calculus of variations, and results are obtained by using numerical techniques on the resulting equations. It is seen that the optimum shape is a flat plate followed by a slightly convex shape which is nearly a wedge. The results for these airfoils are compared with those of the optimum wedge and the optimum flat plate-wedge. For all practical purposes, either of the above simple shapes can be used in place of the variational optimum, the latter being the more nearly identical.This research was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, US Air Force, Grant No. AF-AFOSR-69-1744.     

On generalized matrix approximation problem in the spectral norm

In this paper theoretical results regarding a generalized minimum rank matrix approximation problem in the spectral norm are presented. An alternative solution expression for the generalized matrix approximation problem is obtained. This alternative expression provides a simple characterization of the achievable minimum rank, which is shown to be the same as the optimal objective value of the classical problem considered by Eckart–Young–Schmidt–Mirsky, as long as the generalized problem is feasible. In addition, this paper provides a result on a constrained version of the matrix approximation problem, establishing that the later problem is solvable via singular value decomposition.     

The conical deformation of a delta wing with sonic leading edges

The optimal conical deformation of a delta wing with sonic leading edges is determined in a linear formulation of the problem. The drag of the wing caused by the creation of the lifting force is taken as the objective function. It is established that a superelliptical distribution of the local angle of attack over the wing span corresponds to the minimum drag. A representation in the form of a hypergeometric function is found for the directrix of the wing in a cross section. The results obtained are compared with the results of a numerical investigation within the Euler model.     

The construction of symmetric nose shapes of minimum wave drag

The problem of minimizing the wave drag of axisymmetric noses in the supersonic flow of an inviscid, non-heat-conducting perfect gas is considered. A procedure is proposed for constructing the extremal generatrix by step-by-step solution of single parameter problems using assumptions concerning the local relation between the geometric parameters and the gas-dynamic functions. A unique generatrix, from which parts are separated which form noses of arbitrary length, corresponds to the specified free-stream conditions. Comparison of the noses obtained and the noses which are optimal in the exact formulation of the problem shows that the aerodynamic characteristics are close when there is an appreciable difference between the geometric parameters.     

A self-consistent field method applied to the dynamics of viscous suspensions

A method for the approximate solution of the problem of many bodies of spherical form in a viscous fluid is developed in the Stokes approximation. Using a purely hydrodynamic approach, based on the use of the concept of a self-consistent field, the classical boundary value problem is reduced to a formal procedure for solving a linear system of algebraic equations in the tensor coefficients, which occur in the solution obtained for the velocity field and pressure of the liquid. A procedure for the approximate solution of this system of equations is constructed for the case of dilute suspensions, when the ratio of the size of the dispersed particles to the characteristic distance between them is a small parameter. Finally, the initial boundary value problem is reduced to solving a recurrent system of equations, in which each subsequent approximation for all the required quantities depends solely on the previous approximations. The system of recurrent equations obtained can be solved analytically in any specified approximation with respect to a small parameter. It is shown that this system of equations contains in itself all possible physical formulations of the problems, and, within the frameworks of the mathematical procedure constructed, they are distinguished solely by a set of specified and required functions. The practical possibilities of the method are in no way limited by the number of dispersed particles in the fluid.     

Scheduling groups of tasks with precedence constraints on three dedicated processors

We study the problem of scheduling groups of tasks with precedence constraints on three dedicated processors. Each task requires a specified set of processors. Up to three precedence constraints are considered among groups of tasks requiring the same set of processors. The objective of the problem is to find a nonpreemptive schedule which minimizes the maximum completion time (makespan). This scheduling problem is equivalent to the problem of finding an extension of the constraint graph (i.e. the graph which represents the conflicts between tasks and the precedence constraints) to a comparability graph with minimum (over all the extensions) maximum clique weight. The problem is NP-hard in the strong sense. A normal schedule is such that all the tasks requiring the same set of processors are scheduled consecutively. With a normal schedule the problem reduces to the quotient graph of the constraint graph. In this paper we obtain tight approximation results for the minimum makespan of a normal schedule through tight results on the minimum increase of the maximum clique weight when the (partially oriented) quotient graph is extended to a comparability graph.     

Necessary extremum conditions for differential inclusion with aftereffect

Necessary extremum conditions for optimization problem on the minimum of a functional specified at the right-hand end of the trajectory of differential inclusion with aftereffect are obtained. The constraints, specified at the left-hand end of the trajectory, imply that the initial function belongs to some rnultivalued mapping. The problem is considered at the fixed interval of time     

On the effective viscosity of a dilute emulsion of gas bubbles

The problem of the motion of a rigid spherical body in a homogeneous emulsion of gas bubbles is considered in the Stokes approximation, using the self-consistent field method. An expression is obtained for the correction factor in the Stokes formula for the drag of the body in the first approximation with respect to the volume concentration of the dispersed phase. An analytical relation between the correction factor and the ratio of the sizes of the bubbles and the body is found. It is shown that, in the limit when this ratio tends to zero, the correction factor obtained is identical to Taylor's result for the effective viscosity of an emulsion of gas bubbles. In the case of non-point bubbles, the coefficient on the volume concentration in the expression for the effective viscosity of the emulsion can be considerably different from Taylor's result. A similar conclusion was also obtained in the case of the problem of the motion of a spherical bubble of arbitrary size in an emulsion of gas bubbles.     

An approximation theory of matrix rank minimization and its application to quadratic equations

Matrix rank minimization problems are gaining plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization problem can be approximated to any level of accuracy via continuous optimization (especially, linear and nonlinear semidefinite programming) problems. One of the main results in this paper shows that if the feasible set of the problem has a minimum rank element with the least Frobenius norm, then any accumulation point of solutions to the approximation problem, as the approximation parameter tends to zero, is a minimum rank solution of the original problem. The tractability under certain conditions and convex relaxation of the approximation problem are also discussed. An immediate application of this theory to the system of quadratic equations is presented in this paper. It turns out that the condition for such a system without a nonzero solution can be characterized by a rank minimization problem, and thus the proposed approximation theory can be used to establish some sufficient conditions for the system to possess only zero solution.     

Axisymmetric bodies of minimum drag in hypersonic flow

The problem of determining the longitudinal contour of a slender, axisymmetric body in hypersonic flow which has minimum drag is considered. The pressure distribution is assumed to be Newtonian, while the skin-friction distribution is for laminar flow and depends on body geometry. This investigation is conducted with the method of steepest descent, whose feasiblity is demonstrated by solving minimum drag problems having known analytical solutions.     

Approximating largest convex hulls for imprecise points

Assume that a set of imprecise points in the plane is given, where each point is specified by a region in which the point will lie. Such a region can be modelled as a circle, square, line segment, etc. We study the problem of maximising the area of the convex hull of such a set. We prove NP-hardness when the imprecise points are modelled as line segments, and give linear time approximation schemes for a variety of models, based on the core-set paradigm.     

解任意形状扁轴对称体Stokes流动的连续奇点分布法

本文以Sampson球形无穷级数作为基本奇点,采用分段等强度和分段二次抛物分布两种体内连续分布法解任意形状扁轴对称体的Stokes流动.通过扁球的无界绕流问题,对这两种方法的收敛性,精度和适用范围做了检验和比较.结果表明,在一定的范围内,无论是阻力系数或压力分布,它们的计算结果都和精确解符合得很好,而且,随着分布函数逼近程度的提高,其收敛性得到改善,适用范围也随之扩大.作为一般算例,分别用这两种方法解决了卡西尼扁卵形体的绕流问题,得到了一致的结果.最后,用分段二次连续分布法计算了具有一定生理意义的红细胞体的Stokes流动,首次得到了它的阻力系数和表面压力分布.     

An asymptotic method of solving transient dynamic contact problems

Using a special approximation in the complex plane of the symbol of the kernel of the contact-problem integral equation, an asymptotic form of its solution is constructed which is the fundamental solution of the transient dynamic plane contact problem of the impact of a rigid punch with an elastic half-plane for short interaction times. The proposed approximation of the kernel symbol enables it to be approximated in the complex plane with any previously specified accuracy. Unlike existing approaches [1, 2, etc.], the approximation of the kernel symbol of the integral equation employed here enables the solution of this problem to be obtained in the form of simple formulae not containing singular quadratures.     

The flow pattern in a liquid layer and the spectrum of the boundary-value problem when the surface tension depends non-linearly on the temperature

The flow of a liquid in a plane channel on the bottom of which a specified temperature distribution is maintained while the free surface is thermally isolated is considered. The surface tension of the liquid depends quadratically on the temperature. The system of Navier-Stokes and heat conduction equations possess a self-similar solution which leads to the non-linear eigenvalue problem of finding the flow temperature fields in the channel. The spectrum of this problem is investigated analytically for low Marangoni numbers (the second approximation) and numerically in the limiting case of an ideally heat conducting liquid for any Marangoni number. The pattern of the thermocapillary flow in the layer is analysed as a function of the parameter values. The non-uniqueness of the solution, which is typical for problems of this kind, is established. The results are compared with those obtained previously in the first approximation with respect to the Marangoni number.     

An optimal control problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions with objective derivative

We study a problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions with smoothed Jaumann objective derivative. We prove the existence of an optimal solution yielding the minimum of a specified bounded lower semicontinuous quality functional. To establish the existence of an optimal solution, we use the topological approximation method for studying problems of hydrodynamics.     

A fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array

In this paper, we propose a fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array. Our problem is mathematically a boundary value problem of the Stokes equation with periodic boundary conditions, to which it is difficult to give a good approximation by the ordinary fundamental solution method. Our method gives an approximate solution by a linear combination of the periodic fundamental solutions. In addition, we can compute the drag forces on the obstacles by using the data obtained in our method. Numerical examples for the problems of flows past spheres show the effectiveness of our method.