scholarly journals Motion of an infinite elliptic cylinder in fluids with constant vorticity

1937 ◽  
Vol 158 (895) ◽  
pp. 522-535 ◽  
Keyword(s):  
Equations Of Motion ◽  
Elliptic Cylinder ◽  
The Body ◽  
Uniform Rotation ◽  
Constant Vorticity ◽  
Uniform Shear ◽  
Potential Motion ◽  
Irrotational Motion ◽  
Shear Motion ◽  
Solid Bodies

An extension of Kirchoff’s theory of the motion of solid bodies in irrotationally moving liquids to the case of motion in liquids in which a vorticity is present does not exist. Only a few isolated cases of such motion are known. Bearing on the consideration of this paper, there is an important work by Taylor which expresses the additional pressure effect on a system of cylinders moving in a perfect liquid without rotation when the whole system is rotated uniformly about an axis. Taylor’s theory reduces the problem of such motion to one of irrotational motion. In the present paper the motion of a perfect liquid having constant vorticity, and in which a cylinder of any cross-section is moving in any manner, has been considered. The pressure integral can be obtained in a simple form, referred to axes fixed in the body, which is very suitable for calculation. It is shown, whenever the pure potential motion of the liquid for the rotation of the cylinder and the solution of a definite potential problem or the corresponding Green’s function can be found, the formula can be applied to calculate the motion of the cylinder in liquids with constant vorticity. Two important cases of constant vorticity are uniform shear motion along a direction and uniform rotation about an axis. In the present paper the former case is considered in detail for an elliptic cylinder. The case of uniform rotation being covered by Taylor’s result it is only verified that the present method gives the same result as Taylor’s formulae. There are some simple free motions of an elliptic cylinder in a liquid with uniform shear motion which have been discussed in the paper. 2—Equations of Motion Referred to Axes Fixed in the Body and the Pressure Integral It is first necessary to write down the equations of motion referred to a system of axes fixed in the body having both translation and rotation. These equations are obtained below following a method of Taylor.

scholarly journals A mechanical method for solving problems of flow in compressible fluids

1928 ◽  
Vol 121 (787) ◽  
pp. 194-217 ◽  
Keyword(s):  
Classical Theory ◽  
Speed Of Sound ◽  
High Speed ◽  
The Body ◽  
Compressible Fluids ◽  
Mechanical Method ◽  
Vortex Sheets ◽  
Dissipation Of Energy ◽  
Irrotational Motion ◽  
Solid Bodies

At the present time the chief study of aerodynamical laboratories is concerned with the steady flow of air past solid bodies at speeds which are so low that the effect of compressibility is inappreciable. In recent years, however, the rapid increase in the speed of aircraft has very much increased the importance of the study of the effect of compressibility on air flow. The highest speeds of aircraft at the present time are of the order of 280 miles per hour or 400 feet per second, i. e ., 0.4 of the speed of sound. The tips of the propellers of these high speed machines may move at speeds as high as 1.3 times the speed of sound. At low speeds when air behaves like an incompressible fluid, the classical theory of hydrodynamics which is concerned with irrotational motion predicts that a body moving steadily will experience no resistance or lift. In the simpler form of the theory of Prandtl the motion is still irrotational, but the existence of a circulation round the body and of vortex sheets trailing down-wind permit the possibility of resistance and lift. At speeds higher than that of sound it is known that a nearly discontinuous wave is formed in front of the body. This wave has been photographed and has been studied theoretically by many writers. It involves a dissipation of energy so that even on the classical theory which involves no viscosity, bodies moving at speeds higher than that of sound should have a resistance.

scholarly journals Experiments on the motion of solid bodies in rotating fluids

1923 ◽  
Vol 104 (725) ◽  
pp. 213-218 ◽  
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Steady Motion ◽  
The Body ◽  
Two Dimensional ◽  
Experimental Conditions ◽  
Axis Of Rotation ◽  
Rotating Liquid ◽  
Steady Motions ◽  
Solid Bodies

Some years ago it was pointed out by Prof. Proudman that all slow steady motions of a rotating liquid must be two-dimensional. If the motion is produced by moving a cylindrical object slowly through the liquid in such a way that its axis remains parallel to the axis of rotation, or if a two-dimensional motion is conceived as already existing, it seems clear that it will remain two-dimensional. If a slow three-dimensional motion is produced, then it cannot be a steady one. On the other hand, if an attempt is made to produce a slow steady motion by moving a three-dimensional body with a small uniform velocity (relative to axes which rotate with the fluid) three possibilities present themselves:— ( a ) The motion in the liquid may never become steady, however long the body goes on moving. ( b ) The motion may be steady but it may not be small in the neighbourhood of the body. ( c ) The motion may be steady and two-dimensional. In considering these three possibilities it seems very unlikely that ( a ) will be the true one. In an infinite rotating fluid the disturbance produced by starting the motion of the body might go on spreading out for ever and steady motion might never be attained, but if the body were moved steadily in a direction at right angles to the axis of rotation, and if the fluid were contained between parallel planes also perpendicular to the axis of rotation, it seems very improbable that no steady motion satisfying the equations of motion could be attained. There is more chance that ( b ) may be true. A class of mathematical expressions representing the steady motion of a sphere along the axis of a rotating liquid has been obtained. This solution of the problem breaks down when the velocity of the sphere becomes indefinitely small, in the sense that it represents a motion which does not decrease as the velocity of the sphere decreases. It seems unlikely that such a motion would be produced under experimental conditions.

Subcritical Flow without Circulation past a Swept Semi-Infinite Elliptic Cylinder

1977 ◽  
Vol 28 (2) ◽  
pp. 142-148 ◽  
Author(s):  
G J Clapworthy
Keyword(s):  
Body Shape ◽  
Velocity Potential ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Elliptic Cylinder ◽  
The Body ◽  
Infinite Domain ◽  
Working Space ◽  
Mach Number Distribution ◽  
Grid Points

SummaryThe numerical calculation of the subcritical, steady, three-dimensional, potential flow past a semi-infinite swept elliptic cylinder attached to a plane wall is described. The full equations of motion are written in terms of the velocity potential and the exact body-surface conditions are applied. The equations are expressed in coordinates defined by the body shape and further transformations are necessary to reduce the infinite domain to a finite working space and to concentrate the grid points in regions of greatest variation of potential. The resulting equations are solved using a finite-difference scheme. The Mach number distribution for a typical case is presented.

The unsteady motion of solid bodies in creeping flows

1995 ◽  
Vol 303 ◽  
pp. 83-102 ◽  
Author(s):  
J. Feng ◽  
D. D. Joseph
Keyword(s):  
Equations Of Motion ◽  
Elliptic Cylinder ◽  
Unsteady Motion ◽  
Circular Cylinders ◽  
Simple Shear Flow ◽  
Total Force ◽  
Creeping Flows ◽  
Steady Solutions ◽  
Unsteady Effects ◽  
Solid Bodies

In treating unsteady particle motions in creeping flows, a quasi-steady approximation is often used, which assumes that the particle's motion is so slow that it is composed of a series of steady states. In each of these states, the fluid is in a steady Stokes flow and the total force and torque on the particle are zero. This paper examines the validity of the quasi-steady method. For simple cases of sedimenting spheres, previous work has shown that neglecting the unsteady forces causes a cumulative error in the trajectory of the spheres. Here we will study the unsteady motion of solid bodies in several more complex flows: the rotation of an ellipsoid in a simple shear flow, the sedimentation of two elliptic cylinders and four circular cylinders in a quiescent fluid and the motion of an elliptic cylinder in a Poiseuille flow in a two-dimensional channel. The motion of the fluid is obtained by direct numerical simulation and the motion of the particles is determined by solving their equations of motion with solid inertia taken into account. Solutions with the unsteady inertia of the fluid included or neglected are compared with the quasi-steady solutions. For some flows, the effects of the solid inertia and the unsteady inertia of the fluid are importanty quantitatively but not qualitatively. In other cases, the character of the particles' motion is changed. In particular, the unsteady effects tend to suppress the periodic oscillations generated by the quasi-steady approximation. Thus, the results of quasi-steady calculatioins are never uniformly valid and can be completely misleading. The conditions under which the unsteady effects at small Reynolds numbers are important are explored and the implications for modelling of suspension flows are addressed.

The Resolution of d'Alembert's Paradox and Thermodynamic Admissibility

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Gerald G. Kleinstein
Keyword(s):  
Experimental Data ◽  
Potential Theory ◽  
Continuum Mechanics ◽  
Equations Of Motion ◽  
Second Law Of Thermodynamics ◽  
Experimental Conditions ◽  
Flow Past ◽  
Potential Motion ◽  
Irrotational Motion ◽  
Flow Past A Sphere

The d'Alembert paradox, annunciated in 1752, was established after it was shown that the result of a net zero drag, obtained by applying potential theory to the incompressible flow past a sphere, was in contradiction with experimental results which showed a positive drag. Interpreting the result as a flaw in the theory, resulted in the declaration of the paradox. Following d'Alembert, we assume a potential motion, and proceed to analyze the consequences of this assumption using the global principles of continuum mechanics. We show that if the fluid is inviscid, the potential motion is thermodynamically admissible, the drag is zero, and the motion can persist indefinitely. Although no conventional fluid is available to either falsify (or validate), this result experimentally, in principle, the theory could be tested by using a superfluid, such as liquid Helium. If the fluid is viscous, we show that the potential irrotational motion is thermodynamically inadmissible, it is in violation of the second law of thermodynamics, and hence such a motion cannot persist. Indeed, observations show that when a rigid body is impulsively set into motion, an irrotational motion may exist initially but does not persist. Any flow property which is derived from a thermodynamically inadmissible motion cannot be expected to be in agreement with experimental data. As an illustration we show that the drag, calculated from the viscous potential solution of the flow past a sphere, is zero. We submit that since the theory of continuum mechanics predicts that no agreement between results obtained from viscous potential theory and experimental data can be expected, there is no room for a paradox once a contradiction is in fact observed. In nature, or under experimental conditions, the nonpersistence of the thermodynamically inadmissible motion proceeds in a breakup of the irrotational motion which transforms into a rotational and obviously admissible motion. We show that by selecting boundary conditions, required in the solution of the differential equations of motion, such that they satisfy the Clausius–Duhem jump conditions inequality, the thermodynamic admissibility of the solution is a priori assured. We also show the vorticity distribution at the wall associated with the particular choice of boundary condition.

Rotating elliptic cylinders in a viscous fluid at rest or in a parallel stream

1977 ◽  
Vol 79 (1) ◽  
pp. 127-156 ◽  
Author(s):  
Hans J. Lugt ◽  
Samuel Ohring
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Numerical Solutions ◽  
Elliptic Cylinder ◽  
The Body ◽  
Oscillatory Behaviour ◽  
Incompressible Fluid Flow ◽  
Transient Period ◽  
Parallel Stream ◽  
Rotating Bodies

Numerical solutions are presented for laminar incompressible fluid flow past a rotating thin elliptic cylinder either in a medium at rest at infinity or in a parallel stream. The transient period from the abrupt start of the body to some later time (at which the flow may be steady or periodic) is studied by means of streamlines and equi-vorticity lines and by means of drag, lift and moment coefficients. For purely rotating cylinders oscillatory behaviour from a certain Reynolds number on is observed and explained. Rotating bodies in a parallel stream are studied for two cases: (i) when the vortex developing at the retreating edge of the thin ellipse is in front of the edge and (ii) when it is behind the edge.

Numerical modelling of swirling momentumless turbulent wake dynamics

2018 ◽  
pp. 37-48
Author(s):  
Андрей Геннадьевич Деменков ◽  
Геннадий Георгиевич Черных
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Mathematical Models ◽  
Equations Of Motion ◽  
Turbulent Wake ◽  
The Body ◽  
Jet Flows ◽  
Reynolds Stresses ◽  
Bodies Of Revolution ◽  
Averaged Equations

С применением математической модели, включающей осредненные уравнения движения и дифференциальные уравнения переноса нормальных рейнольдсовых напряжений и скорости диссипации, выполнено численное моделирование эволюции безымпульсного закрученного турбулентного следа с ненулевым моментом количества движения за телом вращения. Получено, что начиная с расстояний порядка 1000 диаметров от тела течение становится автомодельным. На основе анализа результатов численных экспериментов построены упрощенные математические модели дальнего следа. Swirling turbulent jet flows are of interest in connection with the design and development of various energy and chemical-technological devices as well as both study of flow around bodies and solving problems of environmental hydrodynamics, etc. An interesting example of such a flow is a swirling turbulent wake behind bodies of revolution. Analysis of the known works on the numerical simulation of swirling turbulent wakes behind bodies of revolution indicates lack of knowledge on the dynamics of the momentumless swirling turbulent wake. A special case of the motion of a body with a propulsor whose thrust compensates the swirl is studied, but there is a nonzero integral swirl in the flow. In previous works with the participation of the authors, a numerical simulation of the initial stage of the evolution of a swirling momentumless turbulent wake based on a hierarchy of second-order mathematical models was performed. It is shown that a satisfactory agreement of the results of calculations with the available experimental data is possible only with the use of a mathematical model that includes the averaged equations of motion and differential equations for the transfer of normal Reynolds stresses along the rate of dissipation. In the present work, based on the above mentioned mathematical model, a numerical simulation of the evolution of a far momentumless swirling turbulent wake with a nonzero angular momentum behind the body of revolution is performed. It is shown that starting from distances of the order of 1000 diameters from the body the flow becomes self-similar. Based on the analysis of the results of numerical experiments, simplified mathematical models of the far wake are constructed. The authors dedicate this work to the blessed memory of Vladimir Alekseevich Kostomakha.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

Experimental and numerical investigation of railroad vehicle braking dynamics

2009 ◽  
Vol 223 (3) ◽  
pp. 255-267
Author(s):  
C Mellace ◽  
A P Lai ◽  
A Gugliotta ◽  
N Bosso ◽  
T Sinokrot ◽  
...  
Keyword(s):  
Equations Of Motion ◽  
The Body ◽  
Computer Algorithms ◽  
Coordinate Systems ◽  
State Equations ◽  
Relative Coordinates ◽  
Linear State ◽  
Cartesian Coordinate ◽  
Railroad Vehicle ◽  
Braking Dynamics

One of the important issues associated with the use of trajectory coordinates in railroad vehicle dynamic algorithms is the ability of such coordinates to deal with braking and traction scenarios. In these algorithms, track coordinate systems that travel with constant speeds are introduced. As a result of using a prescribed motion for these track coordinate systems, the simulation of braking and/or traction scenarios becomes difficult or even impossible. The assumption of the prescribed motion of the track coordinate systems can be relaxed, thereby allowing the trajectory coordinates to be effectively used in modelling braking and traction dynamics. One of the objectives of this investigation is to demonstrate that by using track coordinate systems that can have an arbitrary motion, the trajectory coordinates can be used as the basis for developing computer algorithms for modelling braking and traction conditions. To this end, a set of six generalized trajectory coordinates is used to define the configuration of each rigid body in the railroad vehicle system. This set of coordinates consists of an arc length that represents the distance travelled by the body, and five relative coordinates that define the configuration of the body with respect to its track coordinate system. The independent non-linear state equations of motion associated with the trajectory coordinates are identified and integrated forward in time in order to determine the trajectory coordinates and velocities. The results obtained in this study show that when the track coordinate systems are allowed to have an arbitrary motion, the resulting set of trajectory coordinates can be used effectively in the study of braking and traction conditions. The results obtained using the trajectory coordinates are compared with the results obtained using the absolute Cartesian-coordinate-based formulations, which allow modelling braking and traction dynamics. In addition to this numerical validation of the trajectory coordinate formulation in braking scenarios, an experimental validation is also conducted using a roller test rig. The comparison presented in this study shows a good agreement between the obtained experimental and numerical results.

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