A Study on Turbulent and Magneto-hydrodynamic turbulent Flow in Incompressible Fluid

Abstract

The first chapter is a general introductory chapter and gives the general idea of turbulence, distribution functions and their principal concepts. Some results and theories which are needed in the subsequent chapters have been included in this chapter. Types and examples of turbulence, different stages of Reynolds number, Reynolds equation, averaging rules, Coriolis effect etc have been briefly discussed. Distribution functions, Joint distribution functions, equation of motion of dust particles, spectral representation of turbulence and Fourier Transformation of the Navier-Stockes equation have also been discussed. Lastly, a brief review of the past researchers related to this thesis have also been studied in this chapter. Throughout the work we have considered the flow of fluids to be isotropic and homogeneous. The notions generally adopted are those used by Taylor, Vonkarman, Hinze, Reynolds, Deissler, Sarker, Kisore, Batchelor, Coriolis and Lundgren. The Second chapter consist of two parts. In part A, we have studied the decay of temperature fluctuations in dusty fluid homogeneous turbulence prior to the final period considering correlations between fluctuating quantities at two- and three- point. In this part we have tried to solve the correlation equations by converting it to spectral form by taking their Fourier transform. Lastly, by integrating the energy spectrum over all wave numbers, the energy decay law of temperature fluctuations in homogeneous turbulence before the final period in presence of dust particle is obtained. In part B, we have studied the decay of temperature fluctuations in dusty fluid homogeneous turbulence before the final period in presence of Coriolis force and have considered correlations between fluctuating quantities at two- and three- points by neglecting the fourth order correlation in comparison to the second and third order correlations. The correlation equations for two- and three- point in a rotating system in presence of dust particles are obtained and these equations are converted to spectral form by taking their Fourier transforms. Finally by integrating the energy spectrum over all wave numbers, the energy decay law of temperature fluctuations in homogeneous dusty fluid turbulence before the final period in presence of Coriolis force is obtained. The Third chapter consists of two parts. In part A, we have studied the joint distribution functions for simultaneous velocity, temperature, concentration fields in turbulent flow undergoing a first order reaction in presence of Coriolis force. The various properties of the constructed joint distribution functions have been discussed. In this chapter we have tried to derive the transport equations for one and two point joint distribution functions of velocity, temperature, concentration in convective turbulent flow due to first order reaction in presence of coriolis force. In part B, we have an attempt to derive the transport equation for the joint distribution function of certain variables in convective turbulent flow undergoing a first order reaction in a rotating system in presence of dust particles. Equations for the evolution of one- point and two- point joint distribution function for velocity, temperature and concentration in convective turbulent flow field undergoing first- order reaction in a rotating system in presence of dust particles have been derived. Finally we have made a result with comparison of the equation for one- point distribution function in the case of zero coriolis force in the absence of the dust particles and negligible diffusivity. In Chapter four, we have studied the statistical theory of certain variables for three- point distribution functions in MHD turbulent flow in a rotating system in presence of dust particles. In this chapter we have made an attempt to derive the transport equations for evolution of distribution functions for simultaneous velocity, magnetic, temperature and concentration fields in MHD turbulent flow due to Coriolis force in presence of dust particles and various properties of the distribution function have been discussed. In Chapter five, we have made an attempt to discuss the summary about the whole thesis.