A Legendre spectral element model for sloshing and acoustic analysis in nearly incompressible fluids

A new spectral finite element formulation is presented for modeling the sloshing and the acoustic waves in nearly incompressible fluids. The formulation makes use of the Legendre polynomials in deriving the finite element interpolation shape functions in the Lagrangian frame of reference. The formulated element uses Gauss-Lobatto-Legendre quadrature scheme for integrating the volumetric stiffness and the mass matrices while the conventional Gauss-Legendre quadrature scheme is used on the rotational stiffness matrix to completely eliminate the zero energy modes, which are normally associated with the Lagrangian FE formulation. The numerical performance of the spectral element formulated here is examined by doing the inf-sup test oil a standard rectangular rigid tank partially filled with liquid The eigenvalues obtained from the formulated spectral element are compared with the conventional equally spaced node locations of the h-type Lagrangian finite element and the predicted results show that these spectral elements are more accurate and give superior convergence The efficiency and robustness of the formulated elements are demonstrated by solving few standard problems involving free vibration and dynamic response analysis with undistorted and distorted spectral elements. and the obtained results are compared with available results in the published literature (C) 2009 Elsevier Inc All rights reserved

Rheology of dense sheared granular flows

Rapid granular flows are defined as flows in which the time scales for the particle interactions are small compared to the inverse of the strain rate, so that the particle interactions can be treated as instantaneous collisions. We first show, using Discrete Element simulations, that even very dense flows of sand or glass beads with volume fraction between 0.5 and 0.6 are rapid granular flows. Since collisions are instantaneous, a kinetic theory approach for the constitutive relations is most appropriate, and we present kinetic theory results for different microscopic models for particle interaction. The significant difference between granular flows and normal fluids is that energy is not conserved in a granular flow. The differences in the hydrodynamic modes caused by the non-conserved nature of energy are discussed. Going beyond the Boltzmann equation, the effect of correlations is studied using the ring kinetic approximation, and it is shown that the divergences in the viscometric coefficients, which are present for elastic fluids, are not present for granular flows because energy is not conserved. The hydrodynamic model is applied to the flow down an inclined plane. Since energy is not a conserved variable, the hydrodynamic fields in the bulk of a granular flow are obtained from the mass and momentum conservation equations alone. Energy becomes a relevant variable only in thin 'boundary layers' at the boundaries of the flow where there is a balance between the rates of conduction and dissipation. We show that such a hydrodynamic model can predict the salient features of a chute flow, including the flow initiation when the angle of inclination is increased above the 'friction angle', the striking lack of observable variation of the volume fraction with height, the observation of a steady flow only for certain restitution coefficients, and the density variations in the boundary layers.

Secondary flows induced by the rotation of a sphere or coaxial cones in micropolar fluids

The micropolar fluids like Newtonian and Non-Newtonian fluids cannot sustain a simple shearing motion, wherein only one component of velocity is present. They exhibit both primary and secondary motions when the boundaries are subject to slow rotations. The primary motion, as in Non-Newtonian fluids, characterized by the equation due to Rivlin-Ericksen, Oldroyd, Walters etc., resembles that of Newtonian fluid for slow steady rotation. We further notice that the micro-rotation becomes identically equal to the vorticity present in the fluid and the condition b) of "Wall vorticity" can alone be satisfied at the boundaries. As regards, the secondary motion, we notice that it can be determined by the above procedure for a special class of fluids, namely that for which j0(n2-n3)=4 n3/l2. Moreover for this class of fluids, the micro-rotation is identical with the vorticity of the fluid everywhere. Also the stream function for the secondary flow is identical with that for the Newtonian fluid with a suitable definition of the Reynolds number. In contrast with the Non-Newtonian fluids, characterized by the equation due to Rivlin-Ericksen, Oldroyd, Walters etc., this class of micropolar fluids does not show separation. This is in conformity with the statement of Condiff and Dahler (3) that in any steady flow, internal spin matches the vorticity everywhere provided that (i) spin boundary conditions are satisfied, (ii) body torques and non-conservative body forces are absent, and (iii) inertial and spin-inertial terms are either negligible or vanish identically.

A general treatment of two dimensional plane flows, for a micropolar fluid

In this paper we have studied the flow of a micropolar fluid, whose constitutive equations were given by Eringen, in two dimensional plane flow. In two notes, we have discussed the validity of the boundary condition v=a ω and its effect on the entire flow field. We have restricted our study to the case when Stokes' approximation is valid, i. e. slow motion for it is difficult to uncouple the equations in the most general case.

A class of one-dimensional steady-state flows in radiation-gas-dynamics

The study of steady-state flows in radiation-gas-dynamics, when radiation pressure is negligible in comparison with gas pressure, can be reduced to the study of a single first-order ordinary differential equation in particle velocity and radiation pressure. The class of steady flows, determined by the fact that the velocities in two uniform states are real, i.e. the Rankine-Hugoniot points are real, has been discussed in detail in a previous paper by one of us, when the Mach number M of the flow in one of the uniform states (at x=+∞) is greater than one and the flow direction is in the negative direction of the x-axis. In this paper we have discussed the case when M is less than or equal to one and the flow direction is still in the negative direction of the x-axis. We have drawn the various phase planes and the integral curves in each phase plane give various steady flows. We have also discussed the appearance of discontinuities in these flows.

Transient MHD stagnation flow of a non-Newtonian fluid due to impulsive motion from rest

The transient boundary layer flow and heat transfer of a viscous incompressible electrically conducting non-Newtonian power-law fluid in a stagnation region of a two-dimensional body in the presence of an applied magnetic field have been studied when the motion is induced impulsively from rest. The nonlinear partial differential equations governing the flow and heat transfer have been solved by the homotopy analysis method and by an implicit finite-difference scheme. For some cases, analytical or approximate solutions have also been obtained. The special interest are the effects of the power-law index, magnetic parameter and the generalized Prandtl number on the surface shear stress and heat transfer rate. In all cases, there is a smooth transition from the transient state to steady state. The shear stress and heat transfer rate at the surface are found to be significantly influenced by the power-law index N except for large time and they show opposite behaviour for steady and unsteady flows. The magnetic field strongly affects the surface shear stress, but its effect on the surface heat transfer rate is comparatively weak except for large time. On the other hand, the generalized Prandtl number exerts strong influence on the surface heat transfer. The skin friction coefficient and the Nusselt number decrease rapidly in a small interval 0 < t* < 1 and reach the steady-state values for t* >= 4. (C) 2010 Published by Elsevier Ltd.

Parametric Analysis Of Turbulent Wall Jets

The concept of a fully-developed flow based on the hypothesis of selective memory is here applied to general wall-jet type flows. In the presence of a (constant) external stream, the free-stream velocity and the jet momentum flux are taken to be the chief quantities governing the development of the wall jet: two additional nondimensional parameters, representing a momentum flux Reynolds number and the relative momentum defect in the initial boundary layer, are shown to have only a secondary effect on the fully-developed flow. The standard correlations so determined are also found to predict quite well the flow development in Gartshore and Newman's experiments on wall jets in adverse pressure gradients; possible reasons for this somewhat surprising result are discussed. Finally it is shown, by application to the still-air case, that the parameters discovered in incompressible flow are, with appropriate but straightforward modification, successful in describing compressible wall jets also.

A new method of producing local enhancement of buoyancy in liquid flows

We describe here a novel method of generating large volumetric heating in a liquid. The method uses the principle of ohmic heating of the liquid, rendered electrically conducting by suitable additives if necessary. Electrolysis is prevented by the use of high frequency alternating voltage and chemically treated electrodes. The technique is demonstrated by producing substantial heating in an initially neutral jet of water. Simple flow visualisation studies, made by adding dye to the jet, show marked changes in the growth and development of the jet with heat addition.

Generators contribution towards loads and line flows - A case study

n many parts of the world, the goal of electricity supply industries is always the introduction of competition and a lowering of the average consumer price. Because of this it has become much more important to be able to determine which generators are supplying a particular load, how much use each generator is making of a transmission line and what is generator's contribution to the system losses. In this paper a case study on generator contributions towards loads and transmission flows are illustrated with an equivalent 11-bus system, a part of Indian Southern Grid, based on the concepts of circuit flow directions, for normal and network contingency conditions.

Correlations and the Ring-Kinetic Equation in Dense Sheared Granular Flows

A formal way of deriving fluctuation-correlation relations in dense sheared granular media, starting with the Enskog approximation for the collision integral in the Chapman-Enskog theory, is discussed. The correlation correction to the viscosity is obtained using the ring-kinetic equation, in terms of the correlations in the hydrodynamic modes of the linearised Enskog equation. It is shown that the Green-Kubo formula for the shear viscosity emerges from the two-body correlation function obtained from the ring-kinetic equation.