Conservation Of Kinetic Energy Research Articles

The choice of conservation equations for plume models

Theories on the behavior of turbulent buoyant plumes generally form three classes: (1) strictly self-similar models valid only in a uniform environment; (2) quasi-similar models based on flux conservation of mass, momentum, and heat; and (3) quasi-similar models based on conservation of momentum, heat, and kinetic energy of mean motion. Few direct comparisons of the various numerical solutions for the different models have been reported; and little attempt has been made to compare the formulation of models (2) and (3) in spite of the fact that such a comparison must form the basis for an understanding of their differences. The following contribution has been stimulated by yet another solution published recently by Fox in this journal, but differs from earlier papers in that it concentrates wholly on a discussion of the equations used in the two quasi-similarity approaches. The relationship between the field and gross-flux forms of the equations of motion is considered. In particular, it is shown that the use of the flux equations for vertical momentum and kinetic energy is not equivalent to the use of flux equations for mass and vertical momentum, and that the latter pair is to be preferred.

Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I

The integral constraints on quadratic quantities of physical importance, such as conservation of mean kinetic energy and mean square vorticity, will not be maintained in finite difference analogues of the equation of motion for two-dimensional incompressible flow, unless the finite difference Jacobian expression for the advection term is restricted to a form which properly represents the interaction between grid points, as derived in this paper. It is shown that the derived form of the finite difference Jacobian prevents nonlinear computational instability and thereby permits long-term numerical integrations.

A Heuristic Theory of Large-Scale Turbulence and Long-Period Velocity Variations in Barotropic Flow

By combining an equation for the conservation of zonal momentum with the eddy vorticity equation for nondivergent barotropic flow through a channel, it has been found possible to express the time variations of the longitude-averaged zonal wind in terms of the mean zonal wind itself and certain statistics of the eddy velocity field. In large-scale turbulence-where the characteristic dimension of the eddies is comparable with the entire “width” of the flow— these eddy statistics take on the properties of autocorrelation functions; thus, they depend primarily on the overall scale of the eddies and intensity of turbulence, rather than on details of the flow pattern. Changes in the scale and intensity of large-scale turbulence are related to changes in the mean zonal flow through the principles of conservation of total kinetic energy and variance of vorticity. Under a general hypothesis regarding the “stability” of the eddy statistics, the equations for the zonally-averaged motion comprise a complete system of nonlinear equations, which can be integrated by finite-sum and-difference methods. Accordingly, these equations constitute a mathematical theory of large-scale “free” turbulence, and provide the basis for a numerical method of predicting the zonally-averaged motion.
 With the assumption that the eddy scale is invariant, the equation governing the mean zonal wind takes a form related to that of the classical wave equation. Numerical integrations of this equation verify that extremes of mean zonal wind (e.g., pronounced jets) maintain their identity over long periods of time, and migrate either northward or southward (or in both directions at once) in a very regular way—a phenomenon that has been observed by Riehl bt al. (1950). The results also show that a strong jet in the mean westerlies tends to split into two weaker jets, which then move in opposite directions to produce the separated double jet structure typical of widespread blocking. The latter has been described as a characteristic denouement from strong zonal flow by Cressman (1950). The theoretical speed of migration of extremes in the mean zonal wind is found to agree closely with the observed speed in a case where published data are available.
 Since these features of long-period velocity variations—the most striking and regular discovered to date—have a characteristic time scale of a week or two, numerical prediction methods based on the present theory show some promise of aiding in the preparation of extendedrange forecasts.

Momentum and energy diffusion in the turbulent wake of a cylinder

A detailed experimental investigation of the turbulent motion in the wake of a circular cylinder, 0.953 cm. diameter placed in a n air-stream of velocity 1280 cm.sec. -1 , has been carried out with particular reference to those quantities determining the transport of turbulent energy and mean stream momentum. At distances of 80, 120 and 160 diameters down-stream from the cylinder, direct measurements have been made of mean flow velocity, turbulent intensity , viscous dissipation, energy diffusion, scale, and form factors of the velocity components and their spatial derivatives.These observations show that, except close to the wake centre, the flow at a point fixed with respect to the cylinder is only intermittently turbulent, due to the passage of the point of observation through jets or billows of turbulent fluid emitted from the inner wholly turbulent core of the wake. Further consideration of the results indicates that the turbulent motion within the jets is solely responsible for the turbulent transfer of momentum , while diffusion of turbulent energy and of heat is carried out by the bulk movement of the jets. Most probably, the jets are initiated b y local fluctuations of pressure inside the turbulent core, and in the late r stages of their development that are slowed down by adverse pressure gradients. The existence of pressure-velocity correlations of sufficient magnitude is demonstrated by using the equation for the conservation of kinetic energy in the wake, all terms of which are know n excepting the one involving the pressure-velocity correlation, which is the n obtained by difference. While the conception of jets of turbulent fluid is more convenient for following the physical processes in the wake, the alternative but equivalent description that the turbulent motion consists of a motion of scale small compared with the mean flow superimposed on a slower turbulent motion whose scale is large compared with the mean flow may be used. A formal explanation of this two-stage turbulent structure in terms of the Fourier representation of the velocity field is suggested, which relates the structure to the presence of a quasi-constant source of energy of nearly fixed wave-number, and to the free boundary which allows an unlimited range of wave-numbers. It is expected that this type of motion will occur in all systems of turbulent shear flow with a free boundary, such as wakes, jets and boundary layers.

Collisions of α-particles with fluorine nuclei

Present knowledge concerning individual collisions of swift particles and nuclei is derived mainly from the researches of Blackett and his co-workers; it has reference to the collisions of α-particles with hydrogen, helium, nitrogen, oxygen and argon nuclei—and of H-particles with oxygen nuclei. Its content is briefly as follows: conservation of kinetic energy and momentum has been found to hold, to the accuracy of measurement, in all save obviously disintegration collisions (disintegration has been found, amongst the above, only for nitrogen), and an empirical relation between velocity and range has been derived for each of the five types of recoil atom in question. The present paper describes an investigation of the collisions of α-particles with fluorine nuclei by the same general method; it establishes the appropriate rangevelocity relation, though it does not afford any positive information concerning the inelastic (disintegration) collisions which from other evidence are known to occur. Less complete data establishing the range-velocity relation for carbon recoil atoms have been obtained as a bye-product in the course of the work. Fluorine has been employed in the present work very conveniently in the form of the saturated carbon compound, carbon tetrafluoride. In general the application of expansion chamber methods to fresh elements has been greatly retarded on account of the chemical activity of the simpler chemical compounds available. Of the lighter elements carbon has received less attention than might otherwise have been profitable presumably because it appeared unlikely that any disintegration effects would be observed, but, for the rest, boron, silicon, phosphorus, sulphur and chlorine, in order to be examined, must have been employed in the form of highly active gases. In such conditions technical difficulties arise in the construction of a suitable chamber and the finding of a convenient substitute to employ in place of the usual water vapour. With elementary fluorine these difficulties are doubtless effectively insuperable; with carbon tetrafluoride, on the other hand, they entirely disappear. This compound has been reported at various timesf in the past, but it remained for Lebeau and Damiens in 1926-30 and Ruff and Keim in the latter year, definitely to establish its properties, both chemical and physical. The most important, in view of its employment in the expansion chamber may be summarized as follows: boiling point, —130° C .; ratio of specific heats, C p /C v , 1·17; molecular stopping power, 2·75 ( v. inf .); action on water, glass, copper, silver, lead, nil; very slow absorption by some oils. Except in a fully equipped laboratory the preparation of the gas cannot be easily undertaken, since elementary fluorine is required. Professor Lebeau, however, kindly undertook to provide a sufficiently large sample for the work to be described. For this most generous gift of 2 litres of the gas I wish here to offer my sincere thanks.