PhD Thesis

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Start date: 2009End date: 2009

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    A Study on Turbulence and MHD Turbulence
    (University of Rajshahi, 2009) Aziz, Md. Abdul; Sarker, M. Shamsul Alam; Azad, M. Abul Kalam
    Turbulent motions are very common in nature. 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 phenomena in high-speed aerodynamics. Turbulence occurs nearly everywhere; in the oceans, in the atmosphere, in rivers even in stars and galaxies. It occurs when an airplane hits an air pocket. Much like there are currents in the ocean, there are currents in the air. Winds disturbed by thunderstorms or mountains are just one of the many causes of turbulence. In turbulent flow, the 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 flows is that the turbulent fluctuations are random in nature. 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 flows does not satisfy the Navier-Stokes equations for a viscous fluid. Such flows, which occur at high Reynolds numbers, are often termed turbulent flows. Atmospheric scientists define "turbulence" as "a state of fluid flow in which the instantaneous velocities exhibit irregular and apparently random fluctuations." Those "irregular fluctuations" of the flow create the bumps. With sufficient disturbances the result is known as turbulence. The instability of laminar flow at a high Reynolds numbers, are causes disruption of the laminar pattern of fluid motion. In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. Turbulence is one of those few things that many don't understand. It's not a hard concept at all. 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 1s comparatively recent. Reynolds,0. [112] made the first systematic experimental investigation of turbulent flow. The turbulent motion of fluid was described by Reynolds [112], 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. The word "Turbulence" means: agitation, commotion, disturbance etc. 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. Taylor and Vonkarman [146) suggested that, "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. It is not sufficient just to say that turbulence is an irregular motion yet we do not have clear-cut definition of turbulence. In 1975, Hinze [51] gave the definition, "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". Turbulence is a form of movement which is characterized by an irregular or agitated motion. Both liquids and gases can exhibit turbulence, and a number of factors can contribute to the formation of turbulence. The addition "with time and space co-ordinates" is necessary; it is not sufficient to define turbulent motion as irregular in time alone. For instance, the case in which a given quantity of a fluid is moved bodily in an irregular way; the motion of each part of the fluid is then irregular with respect to time to a stationary observer, but not to an observer moving with the fluid. Nor is turbulent motion, a motion that is irregular in space alone, became a steady flow with an irregular flow pattern might then come under the definition of turbulence.
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    Laminar Flow of Incompressible Viscous Newtonian Fluid
    (University of Rajshahi, 2009) Haque, Md. Abdul; Rahman, M. Zillur; Alam, M. Shamsul
    The thesis entitled "Laminar Flow of Incompressible Viscous Newtonian Fluid" is being presented for the award of the degree of Doctor of Philosophy in Applied Mathematics. It is the outcome of my researches conducted in the Department of Applied Mathematics, University of Rajshahi, Bangladesh. The whole thesis consists of nine chapters. The first chapter is a general introductory chapter, giving the general information about Laminar Flow of Incompressible Viscous Newtonian Fluid. In chapter II, we have described about the basic concepts of incompressible viscous Newtonian fluid. Some fundamental equations are presented in this chapter. In Chapter III a laminar flow of incompressible viscous fluid has been considered. Here two numerical methods for solving boundary layer equation have been discussed; (i) Keller Box scheme, (ii) shooting method. Runge-Kutta method is used to solve the initial value problem. The shooting method is supported by a suitable example. The Chapter IV is divided into two parts. In Part: A an attempt has been made to investigate the velocity profile of unsteady laminar flow of incompressible viscous fluid. The method of separation of variable is used to determine the solutions of the governing differential equations. Time varying pressure gradient is considered for poiseuille flow. The velocity profiles for the various types of flow are shown by the figures. In Part: B a fully developed conducting flow of incompressible viscous Newtonian fluid between two parallel plates under the action of a parallel Lorentz force is considered. Analytic solutions for this type of flow are developed. The velocity profiles are presented in figures. In Chapter V, an attempt has been made to study the flow of a viscous incompressible fluid between two parallel porous plates. In case I, we have considered the flow of conducting fluid between two fixed porous plates in presence of a transverse magnetic field. Small suction and injection are imposed on the plates. The velocity of the fluid has been obtained under the three different cases, when pressure gradient is (i) varying linearly with time (ii) decreasing exponentially with time and (iii) varying periodically with time. In case I I of this chapter an attempt have been made to study the flow of a conducting viscous incompressible fluid between two porous plates in absence of pressure gradient force. One plate is at rest and the other plate is oscillating with a constant frequency. A small suction is imposed on the oscillating plate. A transverse magnetic field is also placed on the fluid. The velocity distribution has been investigated numerically with the help of finite difference method. In Chapter VI the laminar flow of Newtonian conducting fluid produced by a moving plate in presence of transverse magnetic field is investigated. The basic equation governing the motion of such flow is expressed in non-dimensional form. Analytic solution of the governing equation is obtained by Laplace transformation. Numerical solution of the dimensionless equation is also obtained with the help of Crank-Nicholson implicit scheme. Velocity profiles of the corresponding problem are shown in the graphs. The Chapter VII is also divided into two parts. In part: A, the temperature distributions of various types of parallel flow of incompressible viscous fluid have been considered. Temperature distribution near a heated plate is also discussed. Coefficients of heat transfer for various types flow has been investigated. In part: B of this chapter, we have considered unsteady MHD flows of an incompressible viscous fluid past an infinite vertical plate. The uniform flow is subject to a transverse applied magnetic field………………………………