Unsteady viscous flow over a rotating stretchable disk with deceleration

In this work, the laminar unsteady flow over a stretchable rotating disk with deceleration is investigated. The three dimensional Navier–Stokes (NS) equations are reduced into a similarity ordinary differential equation group, which is solved numerically using a shooting method. Mathematically, two solution branches are found for the similarity equations. The lower solution branch may not be physically feasible due to a negative velocity in the circumferential direction. For the physically feasible solution branch, namely the upper solution branch, the fluid behavior is greatly influenced by the disk stretching parameter and the unsteadiness parameter. With disk stretching, the disk can be friction free in both the radial and the circumferential directions, depending on the values of the controlling parameters. The results provide an exact solution to the whole unsteady NS equations with new nonlinear phenomena and multiple solution branches.     

Lift maximization in the flow around a contour over a screen

The problem of lift maximization for a smooth contour of given length placed in a flow near a screen is analyzed. The distance between the contour and the screen is assumed to be given. Optimal contours are constructed, and the lift coefficient is derived as a function of the contour-screen separation. The results can be useful as accurate upper bounds for the lift coefficient of actual ekranoplan airfoils.     

A univalent spiral-vortex model for separated flow past a polygonal obstacle

Summary A free-streamline flow model for flow past a polygonal obstacle with a near-wake terminating in Tulin's double spiral vortices is constructed. The flows are univalent for a large class of geometries. In addition a criterion is given for determining the underpressure as function of the Reynolds number using the Stokes solution for diffusion of a vortex sheet, and an extension of Tulin and Hsu's matching theory to transitional flows.

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Sunto Si costruisce un modello di flusso con scia e vortici a doppia spirale alla Tulin per un ostacolo poligonale arbitrario. Il flusso risulta univalente per un' ampia classe di geometrie. Inoltre viene proposto un criterio per correlare il parametro del modello al numero di Reynolds del corrispondente flusso viscoso, combinando la soluzione di Stokes per la diffusione di uno strato di vortici con la teoria di Tulin e Hsu (1980).

     

基于外场实验的风力机叶片三维效应研究

针对一台33 kW水平轴风电机组开展了外场实验,得到其叶片7个断面翼型的压力分布曲线;基于求解时均N-S方程对风轮进行三维数值模拟,以及将叶片各断面作为二维翼型进行数值计算,分别得到各断面翼型的压力分布曲线及升阻力系数．通过将外场实验、三维和二维数值计算所得压力分布曲线及升阻力系数进行对比分析,研究了三维效应对风力机气动性能的影响．研究表明,从叶尖到叶根各断面翼型的压差先增大后逐渐减小,叶片表面压力分布曲线比较明显地反映了从叶尖到叶根流动分离的变化;叶片表面压力分布的三维数值计算结果较二维计算结果更加接近于外场实验值;风力机叶片表面的三维流动对叶片的气动性能影响较大,在叶尖和叶根部分尤为突出.     

On the theory of fugoid motions

In 1891 Zhukovslii in his paper “On soaring of birds” [1] solved the problem of the motion of a body of high lift — drag ratio in an atmosphere of constant density. In [2] this problem was considered in greater detail, but the basic assumption of a constant density was made here as well. There have recently appeared numerous papers concerning the analytical solution of the problem of entry into the atmosphere with orbital and escape velocities [3 to 5]. But these studies were concerned primarily with the problems of ballistic entry and entry with low lift — drag ratio. In considering oscillatory states, the authors limited their treatment to small angles between the trajectory and local horizon. In the present paper we consider the problem without imposing any limitations on the slope of the trajectory or initial velocity. The case examined will be that of a hypothetical glider spacecraft of sufficiently high lift — drag ratio. It is interesting to note that the solution of this problem reduces to the solution of Zhukovskii's problem, but for an atmosphere of variable density. The associated trajectories are termed “fugoid”. All of our assumptions about the parameters of such a glider are of a particular hypothetical character.     

An estimate of the curvature of level curves in a class of univalent functions

In this note a class of normalized functions which are univalent in the unit disk is considered. An exact solution of the problem of an upper estimate of curvature of level curves for functions of this class is obtained. In particular, an exact solution of this problem for univalent functions with real coefficients at the points of the interval (0, 1) is obtained.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 381–388, March, 1976.The author is thankful to the referee for taking interest in this note.     

Hypersonic airfoils of maximum lift-to-drag ratio

The problem of determining the slender, hypersonic airfoil shape which produces the maximum lift-to-drag ratio for a given profile area, chord, and free-stream conditions is considered. For the estimation of the lift and the drag, the pressure distribution on a surface which sees the flow is approximated by the tangent-wedge relation. On the other hand, for surfaces which do not see the flow, the Prandtl-Meyer relation is used. Finally, base drag is neglected, while the skin-friction coefficient is assumed to be a constant, average value. The method used to determine the optimum upper and lower surfaces is the calculus of variations. Depending on the value of the governing parameter, the optimum airfoil shapes are found to be of three types. For low values of the governing parameter, the optimum shape is a flat plate at an angle of attack followed by slightly concave upper and lower surfaces. The next type of solution has a finite thickness over the entire chord with the upper surface inclined so that the flow is an expansion. Finally, for the last type of solution, the upper surface begins with a portion which sees the flow and is followed by an inclined portion similar to that above. For all of these solutions, the lower surface sees the flow. Results are presented for the optimum dimensionless airfoil shape, its dimensions, and the maximum lift-to-drag ratio. To calculate an actual airfoil shape requires an iteration procedure due to the assumption on the skin-friction coefficient. However, simple results can be obtained by assuming an approximate value for the skin-friction coefficient.This research was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under AFOSR Grant No. 69-1744.     

MILU-CG方法和绕旋转圆柱流动的数值研究

本文采用以修正的不完全LU分解作预处理器的共轭梯度法(MILU-CG),结合高阶隐式差分格式,改进了作者(1992)提出的基于区域分解、有限差分法与涡法杂交的数值方法(HDV).系统地研究了雷诺数Re=1000,200,旋转速度比α∈(0.5,3.25)范围内,绕旋转圆柱从突然起动到充分发展,长时间内尾流旋涡结构和阻力、升力系数的变化规律.计算所得流线与实验流场显示相比,完全吻合.首次揭示了临界状态时的旋涡结构特性,并指出最佳升阻比就在该状态附近得到.     

二元拉伐尔喷管不可压缩流动精确解

在物理平面上,仔细分析沿拉伐尔喷管中心线和喷管型线的流动,可以发现拉伐尔喷管流动的上下两半部分在速度平面中是两个相同的具有尾缘点前后错开的双尾的裂缝厚翼型。该两个翼型处在不同的黎曼面内。翼型的内部在复位势平面中可转绘成无限长的条带。利用这些结果得到了二元拉伐尔喷管内不可压缩位势流动的精确解。精确解对任意给定的收缩比n₁、扩张比n₂和喉部壁面曲率半径R*都适用。作为应用的举例,给出了一些典型的喷管型线,喷管内的流速分布以及不同瞬间流体质点的所在位置。     

Numerical bifurcation analysis of static stall of airfoil and dynamic stall under unsteady perturbation

By the finite element method combined with Arbitrary-Lagrangian-Eulerian (ALE) frame and explicit Characteristic Based Split Scheme (CBS), the complex flows around stationary and sinusoidal pitching airfoil are studied numerically. In particular, the static and dynamic stalls are analyzed in detail, and the natures of the static stall of NACA0012 airfoil are given from viewpoint of bifurcations. Following the bifurcation in Map, the static stall is proved to be the result from saddle-node bifurcation which involves both the hysteresis and jumping phenomena, by introducing a Map and its Floquet multiplier, which is constructed in the numerical simulation of flow field and related to the lift of the airfoil. Further, because the saddle-node bifurcation is sensitive to imperfection or perturbation, the airfoil is then subjected to a perturbation which is a kind of sinusoidal pitching oscillation, and the flow structure and aerodynamic performance are studied numerically. The results show that the large-scale flow separation at the static stall on the airfoil surface can be removed or delayed feasibly, and the ensuing lift could be enhanced significantly and also the stalling incidence could be delayed effectively. As a conclusion, it can be drawn that the proper external excitation can be considered as a powerful control strategy for the stall. As an unsteady aerodynamic behavior of high angle of attack, the dynamic stall can be investigated from viewpoint of nonlinear dynamics, and there exists a rich variety of nonlinear phenomena, which are related to the lift enhancement and drag reduction.     

The legendre condition in optimum problems of supersonic gasdynamics : PMM vol. 39, n 6, 1975, pp. 1032–1042

The necessary Legendre condition for problems of optimum (in the sense of minimum wave drag) supersonic flow past bodies is obtained. Plane and axisymmetric flows are considered on the assumption of imposition of isoperimetric constraints of a general form. Shock-free flows and flows with attached shock waves are investigated. The method here proposed is used for deriving the second order condition in the particular case when it is possible to pass to the reference contour, and which has been earlier obtained by Shmyglevskii [1] and then by Guderley and others [2].     

Dissociation effects in hypersonic flow past a circular cone at an angle of attack

The analytical solution to the steady, compressible, non-viscous, inviscid hypersonic flow past a circular cone at an angle of incidence, with an attached Shockwave, in the presence of dissociation of air in the shock layer, has been obtained here under the assumption of thermal equilibrium. Expression for the velocity, pressure, temperature, density, velocity of air, Mach number, pressure, drag and lift coefficients have been obtained both in the shocklayer outside the vortical layer and on the surface of the cone inside the vortical layer.     

Efficient Aerodynamic Shape Optimization in MDO Context

The aerospace industry is increasingly relying on advanced numerical flow simulation tools in the early aircraft design phase. Today's flow solvers, which are based on the solution of the compressible Euler and Navier-Stokes equations, are able to predict aerodynamic behaviour of aircraft components under different flow conditions quite well [1]. Within the next few years numerical shape optimization will play a strategic role for future aircraft design. It offers the possibility of designing or improving aircraft components with respect to a pre-specified figure of merit, subject to geometrical and physical constraints. Here, aero-structural analysis is necessary to reach physically meaningful optimum wing designs. The use of single disciplinary optimizations applied in sequence is not only inefficient but in some cases is known to lead to wrong, non-optimal designs [2]. Although multidisciplinary optimizations (MDO) are possible with classical approaches for sensitivity evaluations by means of finite differences, these methods are extremely expensive in terms of calculation time, requiring the reiterated solution of the coupled problem for every design variable. However, adjoint approaches allow the evaluation of these sensitivities in an efficient way and lead to high accuracy. Firstly, we present the development and application of a continuous adjoint approach for single disciplinary aerodynamic shape design. This approach was previously developed at the German Aerospace Center (DLR) [3] and was the starting point for the extension to aero-structural wing designs. Secondly, we describe the adjoint approach and its implementation for the evaluation of the sensitivities for coupled aero-structure optimization problems [4] and its application to the drag reduction of the AMP wing by constant lift while taking into account the static deformation of this wing caused by the aerodynamic forces (see figures). Finally, we show the application of the coupled aero-structural adjoint approach for the Breguet formula of aircraft range, where in addition to the lift to drag ratio the weight of the AMP wing is taken into account (see figures). (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the flow resistance of wide surface structures

The possibility of skin-friction drag reduction in channel flows due to surface structures is investigated numerically. In this context, surface structures with a high width to height ratio compared to the typical dimensions of riblets are studied in the laminar as well as in the turbulent flow regime. In general, it is found that a reduction of the flow resistance is possible in both flow regimes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Viscous flow over a shrinking sheet with a second order slip flow model

In this paper, viscous flow over a shrinking sheet is solved analytically using a newly proposed second order slip flow model. The closed solution is an exact solution of the full governing Navier–Stokes equations. The solution has two branches in a certain range of the parameters. The effects of the two slip parameters and the mass suction parameter on the velocity distribution are presented graphically and discussed. For certain combinations of the slip parameters, the wall drag force can decrease with the increase of mass suction. These results clearly show that the second order slip flow model is necessary to predict the flow characteristics accurately.     

The design of symmetric, optimal supersonic and hypersonic flow profiles for arbitrary isoperimetric conditions

A simple and accurate approach to the design of symmetric profiles which are optimal in the supersonic and hypersonic flow with an attached shock is developed. Besides dimensional constraints, the bodies being optimized can satisfy arbitrary isoperimetric conditions. The approach which has been developed uses a modification of the “shock-expansion” method (SEM). The modified shock-expansion method (MSEM), unlike SEM, does not lead to a physically absurd result, that is, to a finite change in the flow parameters when the slope of the contour is solely changed at the leading point of the body. This makes MSEM suitable for solving two-dimensional variational problems in gas dynamics, by reducing any of them to a certain extension of the Lagrange problem for systems which are described by ordinary differential equations. The possibilities of the approach which has been developed are illustrated using examples of profiles which achieve a minimum wave drag coefficient, C_x. Profiles designed using the MSEM are compared with those obtained using the Newtonian model and linear theory and with wedges while the C_x values found for them using the above-mentioned approximate models and MSEM are compared with the results of the numerical integration of Euler's equations.     

A study of an arbitrary Stokes flow past a fluid coated sphere in a fluid of a different viscosity

A general method to discuss the problem of an arbitrary Stokes flow (both axisymmetric and non-axisymmetric flows) of a viscous, incompressible fluid past a sphere with a thin coating of a fluid of a different viscosity is considered. We derive the expressions for the drag and torque experienced by the fluid coated sphere and also discuss the conditions for the reduction of the drag on the fluid coated sphere. In fact, we show that the drag reduces compared to the drag on a rigid sphere of the same radius when the unperturbed velocity is either harmonic or purely biharmonic, i.e., of the form ${r^2\vec{\textbf{v}}}$ , where ${\vec{\textbf{v}}}$ is a harmonic function. Previously Johnson (J Fluid Mech 110:217–238, 1981), who considered a uniform flow showed that the drag on the fluid coated sphere reduces compared to the drag on the uncoated sphere when the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4. We show that this result is true when the undisturbed velocity is harmonic or purely biharmonic, uniform flow being a special case of the former. However, we illustrate by an example that the drag may increase in a general Stokes flow even if this ratio is greater than 4. Moreover, when the unperturbed velocity is harmonic or purely biharmonic, and the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4 for a fixed value of the viscosity of the ambient fluid, we determine the thickness of the coating for which the drag is minimum.     

Steady Compressible Euler Equations of Concentration Layers for Hypersonic-limit Flows Passing Three-dimensional Bodies and Generalized Newton-Busemann Pressure Law*

For stationary hypersonic-limit Euler flows passing a solid body in three-dimensional space, the shock-front coincides with the upwind surface of the body, hence there is an infinite-thin layer of concentrated mass, in which all particles hitting the body move along its upwind surface. By proposing a concept of Radon measure solutions of boundary value problems of the multi-dimensional compressible Euler equations, which incorporates the large-scale of three-dimensional distributions of upcoming hypersonic flows and the small-scale of particles moving on two-dimensional surfaces, the authors derive the compressible Euler equations for flows in concentration layers, which is a stationary pressureless compressible Euler system with source terms and independent variables on curved surface. As a by-product, they obtain a formula for pressure distribution on surfaces of general obstacles in hypersonic flows, which is a generalization of the classical Newton-Busemann law for drag/lift in hypersonic aerodynamics.     

Axisymmetric nose shapes of specified aspect ratio,optimum or close to optimum with respect to wave drag

The problem of constructing optimum or close to optimum nose-shapes of bodies of revolution of fixed aspect ratio in a supersonic flow is solved within the framework of a perfect (inviscid and non-heat-conducting) gas. Their contour includes the front face, that is, the boundary extremum section with respect to the length and, adjacent to it, the smooth, slightly sloping section that makes a corner. In case of low aspect ratios, the slightly sloping section is the result of the exact solution of a variational problem. In the case of aspect ratios which exceed a certain value, depending on the free-stream Mach number M_∞, the exact solution requires the introduction of small internal breaks with corner points where even the dominant one of these only has a weak effect on the drag value. Contours which are referred to as “close to optimum” do not satisfy the optimality condition, which defines the dominant corner. In the examples (1.2 ≤ M_∞ ≤ 10) for which calculations were carried out, conical nose shapes were found to be far worse that the optimum ones. For contours which are optimum in the approximation of Newton's formula and also, optimum blunt and pointed, power-law nose shapes, the situation occurs for low-aspect ratios and low supersonic Mach numbers (pointed, power-law contours can only be successfully constructed for fairly high aspect ratios). The fact that the front face is a section of a boundary extremum is shown by comparing the drags of bodies obtained with different permissible variations of the front face. An alternative proof, which is not limited by the actual form of the variation in the front face, can be obtained from the solution of the conjugate problem, formulated within the framework of the general method of Lagrange multipliers. This problem is also of interest in its own right, in particular, on account of the singularities, revealed during its formulation, in the reflection of the discontinuities of the Lagrange multipliers from the sonic line with parts of them becoming infinite at the point of reflect.     

Die Begriffe Auftrieb,Widerstand und Zirkulation bei Schaufeln von Strömungsgittern

Summary The force on a wing of a cascade can be divided in the two components lift and drag like the force on a single wing. The magnitude and direction of the lift component are calculated. The change of the flow direction is caused not only by the lift but also by the drag. Approximately the flow can be calculated as the field of vortices with a circulation corresponding to cascade wings without drag.