Department of Mechanical Engineering

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IIT Jodhpur

This paper deals with the radially flowing film produced by a vertical liquid jet impinging on a horizontal surface. Typically, circular hydraulic jump is formed at some radius. A new integral method to calculate the film flow is proposed. Flow visualization pictures and film thickness measurements for a range of flow rates are presented. A newly discovered phenomenon of surface wave-induced transition is discussed.

World University root node

Journal of the Indian Institute of Science

On radial film flow on a horizontal surface and the circular hydraulic jump

In this article we address the problem of the existence of superconvergence points for finite element solutions of systems of linear elliptic equations. Our approach is quite different from all other studies of superconvergence. We prove that the existence of superconvergence points can be guaranteed by a numerical algorithm, which employs a finite number of operations (provided that there is no roundoff-error). By employing this approach, we can reproduce all known results on superconvergence of finite element solutions for linear elliptic problems and we can obtain many new results. Here, in particular, we address the problem of the superconvergence points for the gradient of finite element solutions of Laplace's and Poisson's equations and we show that the sets of superconvergence points are very different for these two cases. We also study the superconvergence of the components of the gradient of the displacement, the strain and stress for finite element solutions of the equations of elasticity. For Laplace's and Poisson's equations (resp. the equations of elasticity), we consider meshes of triangular as well as square elements of degree p, 1 ≤ p ≤ 7 (resp. 1 ≤ p ≤ 4). For the meshes of triangular elements we investigate the effect of the geometry of the mesh by considering four mesh patterns that typically occur in practical meshes, while in the case of square elements, we study the effect of the element type (tensor-product, serendipity, or other). © 1996 John Wiley & Sons, Inc.

Numerical Methods for Partial Differential Equations

Computer-Based Proof of the Existence of Superconvergence Points in the Finite Element Method; Superconvergence of the Derivatives in Finite Element Solutions of Laplace's, Poisson's, and the Elasticity Equations

A new model is proposed to explain the Nielsen's equilibrium condition. This model takes into account the effects of both turbulent entrainment and diffusion on the growth/erosion of the gradient zone. The existing turbulent entrainment model is modified to make it applicable near the equilibrium condition. The new model predictions indicate the dependence of the equilibrium condition on the mixed layer depth, apart from the salinity and temperature gradients in the gradient zone. © 1995.

Solar Energy

Modeling the dynamics of the mixed layer in solar ponds

In this paper we introduce a new definition of superconvergence - tne η%-superconvergence, which generalizes the classical idea of superconvergence to general meshes. We show that this new definition can be employed to determine the regions of least-error in any element in the interior of any grid by using a computer-based approach. We present numerical results for the standard displacement finite element method for the scalar equation of orthotropic heat-conduction, for meshes of conforming triangles of degree p, 1 ≤ p ≤ 5, and elements in the interior of the mesh. The results demonstrate that, unlike classical superconvergence, η%-superconvergence is applicable to the complex grids which are employed in practical engineering computations. © 1995.

Computer Methods in Applied Mechanics and Engineering

η%-Superconvergence of finite element approximations in the interior of general meshes of triangles

In Babuška et al. (Comput. Meth. appl. Mech. Engng, 114, 307-378 (1994); Int. J. numer. Meth. Engng 37, 1073-1123 (1994)) we presented a model study of a posteriori error estimators in the interior of finite element meshes using a computer-based methodology. In this paper we investigate further the quality of element-residual error estimators. We analyzed several versions of the element-residual estimator and based on this study we propose recipes for robust estimators. © 1995.

Computers and Structures

A model study of element residual estimators for linear elliptic problems: The quality of the estimators in the interior of meshes of triangles and quadrilaterals

In References 1 and 2 we showed that the error in the finite‐element solution has two parts, the local error and the pollution error, and we studied the effect of the pollution error on the quality of the local error‐indicators and the quality of the derivatives recovered by local post‐processing. Here we show that it is possible to construct a posteriori estimates of the pollution error in any patch of elements by employing the local error‐indicators over the mesh outside the patch. We also give an algorithm for the adaptive control of the pollution error in any patch of elements of interest. Copyright © 1995 John Wiley & Sons, Ltd

International Journal for Numerical Methods in Engineering

A posteriori estimation and adaptive control of the pollution error in the h‐version of the finite element method

Experimental results are presented that show that the translational velocities of piston generated vortex rings often undergo oscillations, similar to those recently discovered for drop generated rings. An attempt has been made to minimize uncertainties by utilizing both dye and hydrogen bubbles for visualization and carefully repeating measurements on the same ring and on different realizations under the same nominal piston conditions. The results unambiguously show that under most conditions, both for laminar and turbulent rings and for rings generated from pipes and orifices, the oscillations are present. The present results, together with the earlier results on drop generated rings, give support to the view that translational velocity oscillations are probably an inherent feature of translating vortex ring fields. © 1995 American Institute of Physics.

Physics of Fluids

Translational velocity oscillations of piston generated vortex rings

We present, results from a two dimensional numerical simulation of a stably stratified layer heated from below. The aim of the simulation is to study the mechanism of erosion of a stably stratified medium by the convective velocity. We solve transport equations for momentum, heat and species using Patankar's SIMPLER method. Tracer particles are used to get flow pattern inside the mixed layer and in the gradient zone. We have observed randomly moving thermals issuing from the bottom heated surface and from the interface separating the mixed layer and the gradient zone. Stable fluid from the gradient zone gets entrained into the mixed layer with the falling thermal from the interface. This is the main mechanism of the turbulent entrainment. Results from the numerical simulation indicate the need to include the effects of the turbulent entrainment in modelling the equilibrium condition.

Lecture Notes in Physics

Study of the erosion of stably stratified medium heated from below

A numerical methodology which determines the quality (or robustness) of a posteriori error estimators is described. The methodology accounts precisely for the factors which affect the quality of error estimators for finite element solutions of linear elliptic problems, namely, the local geometry of the grid and the structure of the solution. The methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations. Copyright © 1994 John Wiley & Sons, Ltd

Validation of a posteriori error estimators by numerical approach

In this paper, we present a methodology for checking the local quality of recipes for the recovery of stresses or derivatives from finite element solutions of linear elliptic problems. The methodology accounts precisely for the factors which affect the local quality of the recovered quantities, namely, the geometry of the grid, the polynomial degree and the type of the elements, the coefficients of the differential equation and the class of solutions of interest. We give examples of how the methodology can be used to obtain precise conclusions about the quality of a class of recipes, based on least-squares patch-recovery, in the interior of complex grids, like the ones employed in engineering computations. By using this approach, we were able to discover recipes which are much more robust than the ones which are currently in use in the various finite element codes. © 1994.