Some Theoretical Studies on Turbulence and Magneto-Hydrodynamic Turbulence

Abstract

The theory of turbulent motion has received considerable attention in recent developments of high-speed jet aircraft, plasma physics and chemical engineering. The formation of a turbulent boundary layer is one of the most frequently encountered pheno111ena in high-speed aerodynamics. It is common experience that the flow observed in nature such as rivers and winds usually differ from stream flow or laminar flow of a viscous fluid. The mean motion of such flow does not satisfy the Navier-Stockes equations for a viscous fluid. Such flows, which occur at high Reynolds numbers, are often termed turbulent flows. In turbulent flow, the steady motion of the fluid is steady so far as the temporal mean values of velocities and the pressures are concerned where as actually both velocities and the pressures are irregularly fluctuating. The velocity and pressure distributions in turbulent flows as well as the energy losses are determined mainly by turbulent fluctuations. The essential characteristic of turbulent !lows is that the turbulent fluctuations are random in nature. The instability of laminar flow at a high Reynolds numbers, are causes disruption of the laminar pattern of fluid motion. With sufficient disturbances the result is known as turbulence. The irregular, chaotic motion of fluid particle characterizes turbulence. It is far too complicated to be known in complete detail. This fact has been recognized by all theories extant today start with a stochastic formulation of the phenomenon. At least, the technical people understand the meaning of turbulence. The use of the word "Turbulence" to characterize a certain type of flow, namely, the counterpart of streamline motion is comparatively recent. Reynolds, 0 [I 02] made the first systematic experimental investigation of turbulent flow. The turbulent motion of fluid was described by Reynolds [I 02], one of the pioneers in the study of turbulent flows as "sinuous motion" because fluid particles in turbulent flow appeared to follow sinusoidal or irregular paths. Turbulence is rather a familiar notion; yet it is not easy to define in such a way as to cover the detailed characteristic comprehended in it and to make the definition agree with the modern view of it held by professionals in this field of applied science. The word "Turbulence" means: agitation, commotion, disturbance etc. This definition is, however, too general, and does not suffice to characterize turbulent fluid motion in the modern sense. In 1937, Taylor and Vonkarman [ 128) gave the following definition: " Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surface or even when neighboring streams of the same fluid flow past or over one another". According to this definition, the flow has to satisfy the condition of irregularity. But this irregularity is a very important feature. Because of irregularity, it is impossible to describe the motion in all details as a function of time and space co-ordinates. But fortunately turbulent motion is irregular in the sense that it is possible to describe it by laws of probability. It appears possible to indicate distinct average values of various quantities, such as velocity, pressure, temperature, etc and this is very important. If turbulent motion were entirely irregular, it would be inaccessible to any mathematical treatment. Therefore, it is not sufficient just to say that turbulence is an irregular motion yet we do not have clear-cut definition or turbulence. This is rather difficult. Hinze [41) suggested that, "Turbulent fluid motion is an irregular condition of flow in which various quantities show a random variation with time and space co-ordinates, so that statistically distinct average values can be discerned"……………………….