scholarly journals Relaxation methods applied to engineering problems XII. Fluid motions characterized by 'free' stream-lines

1946 ◽  
Vol 240 (815) ◽  
pp. 117-161 ◽  
Keyword(s):  
Conformal Transformation ◽  
Free Stream ◽  
Two Dimensional ◽  
Relaxation Methods ◽  
Inviscid Fluids ◽  
Engineering Problems ◽  
Steady Motions ◽  
Stream Lines ◽  
In Virtue Of ◽  
Hydrodynamical Theory

This paper deals with problems which are like the percolation problems of Part VII (Shaw & Southwell 1941) in that a double condition, imposed at a boundary initially unknown, replaces the more usual single condition at a specified boundary. They relate to ‘free’ stream-lines in the hydrodynamical theory of inviscid fluids. For plane two-dimensional (steady) motions, the device of conformal transformation has led in the hands of Helmholtz, Kirchhoff and Rayleigh to a variety of solutions; but up to the present it has not taken account of gravity, and it would not seem capable of extension to motions characterized by axial symmetry. Relaxation Methods, in virtue of their tentative approach, here deal successfully with some problems hitherto unsolved.

scholarly journals III. The theory of free stream lines

1890 ◽  
Vol 47 (286-291) ◽  
pp. 129-133
Keyword(s):  
Free Stream ◽  
Complex Variables ◽  
Two Dimensional ◽  
Limited Application ◽  
Stream Lines ◽  
General Method

The chief object of the paper is to give a general method for the problem of free stream lines in two-dimensional motion of liquids when the boundaries are plane. The method rests on the transformation from one diagram to another by means of functions of complex variables, and, so far, is similar to that given by Kirchhoff in his ‘Vorlesungen,’ which is, however, of very limited application. The first part is devoted to general theorems of transformation.

Two-dimensional incompressible fluid jet penetration

1967 ◽  
Vol 29 (2) ◽  
pp. 273-287 ◽  
Author(s):  
D. F. Hopkins ◽  
J. M. Robertson
Keyword(s):  
Free Stream ◽  
Density Ratio ◽  
Two Dimensional ◽  
Principal Characteristics ◽  
Yield Information ◽  
Jet Penetration ◽  
Stream Lines ◽  
Kinetics Of ◽  
Kinematic Solution ◽  
Flow Case

The principal characteristics of jet penetration are the appearance of free stream-lines at the sides of the jet and of a dividing streamline, which separates the jet and penetrated fluid. Kinematic analysis of such flow via free-streamline theory and the notched hodograph is developed with one unspecified parameter, the ratio of jet to counterstream velocity in the steady flow case. The kinetics of the problem, appearing when the jet and penetrated (or counterstream) fluid differ in density, is simply related to the kinematic solution via the square root of the density ratio. Experiments, both steady state and transient with several liquids, are presented which generally verify the theory. The experiments also yield information on the magnitude of the parameter and indicate its variation with the density ratio.

scholarly journals Relaxation methods applied to engineering problems V. Conformal transformation of a region in plane space

1940 ◽  
Vol 238 (796) ◽  
pp. 453-475 ◽  
Keyword(s):  
Conformal Transformation ◽  
Functional Relation ◽  
Mathematical Expression ◽  
Sufficient Accuracy ◽  
Relaxation Methods ◽  
Isotropic Cylinder ◽  
Standard Problem ◽  
Plane Space ◽  
Engineering Problems ◽  
Construct Maps

1. Part III of this series (Christopherson and Southwell 1938) brought within the scope of relaxation methods problems governed by Poisson’s or Laplace’s equation in two variables. Its first example (torsion of an isotropic cylinder) was a standard problem in potential theory, calling for the evaluation within a given region of a planeharmonic function defined as having specified values at the boundary. This was solved without difficulty, and with more than sufficient accuracy for practical purposes. Conformal transformation, the problem considered here, although essentially similar presents some questions of detail which were not encountered in Part III. Regarded as a weapon of the analyst, it is a means whereby problems relating to specified regions in plane space may be reduced to problems which concern regions of simpler shape (e.g. circles or rectangles) and can be solved in terms of known functions of ordinary (e.g. polar or Cartesian) co-ordinates. Contours of these co-ordinates by intersection divide the simpler region into rectangles: conformal transformation entails a similar division, or “ mapping”, of the specified region by intersecting contours of two conjugate planeharmonic functions, , which in turn serve as co-ordinates to define the position of any point. Orthodox mathematics presents the transformation in an equation of the type which, when the form of is known, expresses a one-to-one relation between points in the first region and in the second; but this functional relation is of no concern to the practical computer provided that he can construct “ maps” of which it is the mathematical expression, and for this it is only necessary to have and evaluated at nodal points of some regular “ net” . The procedure whereby contours are constructed is so obvious as not to require description.

scholarly journals Relaxation methods applied to engineering problems - Free transverse vibrations of membranes, with an application (by analogy) to two-dimensional oscillations in an electro-magnetic system

1945 ◽  
Vol 239 (810) ◽  
pp. 488-500 ◽  
Keyword(s):  
High Frequency ◽  
Magnetic System ◽  
Mathematical Form ◽  
Two Dimensional ◽  
Relaxation Methods ◽  
Electromagnetic Oscillation ◽  
Transverse Vibrations ◽  
Engineering Problems ◽  
Electro Magnetic

Membranes whether uniform or non-uniform in density are easily treated by a technique similar to that of Part VII B , and with an accuracy more than sufficient for practical purposes. An equation of the same mathematical form governs certain practically-important types of high-frequency electromagnetic oscillation, and here the illustrative example treated has direct importance for design.

Relaxation methods applied to engineering problems XIV. Plastic straining in two-dimensional stress-systems

1950 ◽  
Vol 242 (850) ◽  
pp. 379-414 ◽  
Keyword(s):  
Plane Stress ◽  
Worked Examples ◽  
Shear Stresses ◽  
Two Dimensional ◽  
Plastic Shear ◽  
Principal Stresses ◽  
Relaxation Methods ◽  
Plastic Distortion ◽  
Engineering Problems ◽  
Shear Strains

Only experiment can decide the criterion of elastic failure, and the relation of stress to strain during plastic distortion, in real materials such as steel; and (since stress is not directly measurable) it can do this only by verifying relations deduced from theory in regard to total displacements and resultant actions. Consequently practical value attaches to computational methods whereby, on the basis of some assumed criterion, relations of that kind can be formulated. This paper draws conclusions regarding two-dimensional systems (plane stress and plane strain) from the ‘Mises-Hencky hypothesis’, according to which failure occurs when (р 2 - р 3 ) 2 + (р 3 - р 1 ) 2 + (р 1 - р 2 ) 2 = const. (р 1 , р 2 , р 3 denoting the principal stresses), and from the relation ∆ ϒ1 : ∆ ϒ2 : ∆ ϒ3 = q 1 : q 2 : q 3 assumed to hold during the subsequent plastic distortion (∆ ϒ1 , ∆ ϒ2 , ∆ ϒ3 denoting the incremental plastic shear-strains and q 1 , q 2 , q 3 the principal shear stresses). Its methods could be applied to other hypotheses. In its worked examples some regions remain elastic while in other regions (here termed enclaves ) the strain is partly plastic. Such cases present special difficulty in an orthodox treatment.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

Two- and Three-Dimensional Simulations of the Flow Around a Cylinder Fitted With Strakes

2012 ◽  
Author(s):  
Bruno S. Carmo ◽  
Rafael S. Gioria ◽  
Ivan Korkischko ◽  
Cesar M. Freire ◽  
Julio R. Meneghini
Keyword(s):  
Reynolds Number ◽  
Vortex Shedding ◽  
Free Stream ◽  
Three Dimensional ◽  
The Body ◽  
Vortex Wake ◽  
Two Dimensional ◽  
Flow Around A Cylinder ◽  
Three Dimensional Simulations ◽  
Angle Of Rotation

Two- and three-dimensional simulations of the flow around straked cylinders are presented. For the two-dimensional simulations we used the Spectral/hp Element Method, and carried out simulations for five different angles of rotation of the cylinder with respect to the free stream. Fixed and elastically-mounted cylinders were tested, and the Reynolds number was kept constant and equal to 150. The results were compared to those obtained from the simulation of the flow around a bare cylinder under the same conditions. We observed that the two-dimensional strakes are not effective in suppressing the vibration of the cylinders, but also noticed that the responses were completely different even with a slight change in the angle of rotation of the body. The three-dimensional results showed that there are two mechanisms of suppression: the main one is the decrease in the vortex shedding correlation along the span, whilst a secondary one is the vortex wake formation farther downstream.

Nonlinear Aspects of Subcritical Shallow-Water Flow Past Two-Dimensional Obstructions

1978 ◽  
Vol 22 (04) ◽  
pp. 203-211
Author(s):  
Nils Salvesen ◽  
C. von Kerczek
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Flow Problem ◽  
Free Stream ◽  
The Body ◽  
Two Dimensional ◽  
Third Order ◽  
Submerged Body ◽  
The Mean ◽  
The Waves

Some nonlinear aspects of the two-dimensional problem of a submerged body moving with constant speed in otherwise undisturbed water of uniform depth are considered. It is shown that a theory of Benjamin which predicts a uniform rise of the free surface ahead of the body and the lowering of the mean level of the waves behind it agrees well with experimental data. The local steady-flow problem is solved by a numerical method which satisfies the exact free-surface conditions. Third-order perturbation formulas for the downstream free waves are also presented. It is found that in sufficiently shallow water, the wavelength increases with increasing disturbance strength for fixed values of the free-stream-Froude number. This is opposite to the deepwater case where the wavelength decreases with increasing disturbance strength.

scholarly journals V. On the theory of free stream lines

1890 ◽  
Vol 181 ◽  
pp. 389-431 ◽  
Keyword(s):  
Free Stream ◽  
Fluid Motion ◽  
Steady Motion ◽  
Two Dimensions ◽  
Mathematical Solution ◽  
The Subject ◽  
Negative Pressures ◽  
Stream Lines ◽  
Salient Edges

The attention of mathematicians was first called to the subject of the present paper by a memoir of Helmholtz’s in 1868, on “Discontinuous Fluid Motion.” In discussing the steady motion of liquids past salient edges of fixed obstacles, it is found that the assumptio of continuity of the motion leads to negative pressures in the liquid. Helmholtz showed, in the paper above-mentioned, that some cases of this kind could be solved by assuming a surface of discontinuity, on one side of which the liquid is at rest, and he gave a mathematical solution of one case where the motion is in two dimensions.

Optimal design for the free-stream water wheel: A two-dimensional study

Energy ◽  
2021 ◽  
Vol 214 ◽  
pp. 118880
Author(s):  
Olivier Cleynen ◽  
Sebastian Engel ◽  
Stefan Hoerner ◽  
Dominique Thévenin
Keyword(s):  
Optimal Design ◽  
Free Stream ◽  
Stream Water ◽  
Two Dimensional ◽  
Water Wheel
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