A numerical model of turbulent shear flow behind a prolate spheroid

A finite element based computer model is developed to simulate turbulent shear flow. The results obtained from the model are compared with numerical finite difference based calculations and experimentally determined values associated with flow behind a prolate spheroid. The results obtained utilizing a velocity-pressure formulation are, in the region immediately downstream of the spheroid, markedly better than those obtained under the standard assumption of zero pressure gradient boundary layer flow.     

A numerical model for predicting bubble formation in a 3D fluidized bed

Fluidized bed systems have the potential to be widely used in the power generation, mineral processing and chemical industries. One factor limiting their increased use is the lack of adequate design techniques for scaling such systems. A model has been developed for simulating gas–solid fluidized bed plant. The model uses a multiphase Eulerian–Eulerian technique to predict the transient behaviour of fluidized bed systems. The commercial CFD code CFX is used as the computational framework for solving the discretized equations. To overcome the problem of accurate geometrical representation experienced in previous models a body fitted grid system is employed. The model is used to predict isothermal flow in a three-dimensional bubbling fluidized bed. Predictions of the three-dimensional model show bubble formation with gas bubbles or voids preferentially moving along the centre of the bed. Predicted behaviour is qualitatively consistent with experimental observations.     

Free boundary fluid flow in a trough: A numerical approach

This paper examines the slow flow of a viscous liquid in an open rectangular container, one side (the base) of which moves steadily along its own plane, thereby providing the driving force the liquid needs. Unlike the two vertical sides that are rigid and stationary, the top side is left open so that the upper part of the liquid is in contact with air and is being controlled by surface tension and gravity. A numerical procedure for obtaining solutions for the cases when the capillary numbers are small is provided and the curves of the free boundaries obtained here are presented for some flow parameters. The deviation of the shape of the free boundary is observed to be strongly dependent on the aspect ratio of the boundary (i.e., the ratio of the vertical to horizontal spread of the liquid) with its curvature changing sign in the interval [1, 1.5].     

Instantaneous rate of change: a numerical approach

The calculus reform movement has encouraged numerical and graphical approaches to functions in addition to the more traditional analytical approach. While valiant efforts have been made to use these other approaches in newer calculus curricula, more numerical approaches should be introduced. Research on student learning in calculus indicates that particular numerical approaches hold promise for students' learning of instantaneous rate of change. Numerical approaches involving the average rate of change over successively smaller intervals can be used to obtain the instantaneous rate of change for a given function at a given value of x. These approaches can help students appreciate the fundamental relationship between average and instantaneous rates of change. They can also be used to obtain general expressions for the derivative of most elementary functions. Standard computer spreadsheet programs facilitate this process and make numerical approaches a more viable option for calculus instruction. These are underutilized resources for instruction in calculus, even in reform or other new calculus curricula.     

A new approach to numerical differentiation

In this paper we consider the numerical differentiation of functions specified by noisy data. A new approach, which is based on an integral equation of the first kind with a suitable compact operator, is presented and discussed. Since the singular system of the compact operator can be obtained easily, TSVD is chosen as the needed regularization technique and we show that the method calls for a discrete sine transform, so the method can be implemented easily and fast. Numerical examples are also given to show the efficiency of the method.     

A numerical approach to proving projectivity

Summary We present a nonconstructive method which uses intersection numbers and linear space theory for proving the existence of projective embeddings of suitable algebraic schemes, and we apply it to establish Chevalley's conjecture that a complete nonsingular variety such that any finite number of points is contained in an open affine subset is projective. In memory of Guido Castelnuovo in the recurrence of the first centenary of his birth.     

Finite element calculations of flow past a spherical bubble rising on the axis of a cylindrical tube

The velocity and pressure fields of a Newtonian fluid with homogeneous and constant physical properties flowing around a sphere on the axis of a cylindrical tube with no slip, free slip and partial slip at the sphere surface and no slip at the cylinder wall have been calculated by solving the Navier-Stokes equations and the continuity equation using the finite element technique with the penalty function method. Terminal rise velocities of spherical air bubbles in water have been calculated as function of the bubble radius and some conclusions have been drawn about the nature of the interface. Finally, the influence of the presence of a cylindrical wall on the drag force has been determined and a new empirical equation is derived for the wall correction factor for a sphere rising with free slip at its surface at low Reynolds number.     

On the performance of West’s bubble test: A simulation approach

In this research we examine the ability of West’s bubble test [1] in detecting speculative bubbles using Brock’s (1982) [2] intertemporal general equilibrium model of asset pricing as the basis for a simulation study. In this setting, (1) the economy, by construction is efficient and produces the maximally possible amount of welfare for society, and (2) asset prices reflect the utility-maximizing behavior of consumers and the profit-maximizing behavior of firms. We find that the West’s bubble test flag as “bubbles” in the simulated data yet the data is produced from an economy in which markets are efficient in welfare production.     

A numerical approach to degenerate parabolic equations

Summary. In this work we propose a numerical approach to solve some kind of degenerate parabolic equations. The underlying idea is based on the maximum principle. More precisely, we locally perturb the (initial and boundary) data instead of the nonlinear diffusion coefficients, so that the resulting problem is not degenerate. The efficiency of this method is shown analytically as well as numerically. The numerical experiments show that this new approach is comparable with the existing ones. Received January 20, 1999 / Revised version received February 28, 2000 / Published online July 25, 2001     

Integral points in rational polygons: a numerical semigroup approach

In this paper we use an elementary approach by using numerical semigroups (specifically, those with two generators) to give a formula for the number of integral points inside a right-angled triangle with rational vertices. This is the basic case for computing the number of integral points inside a rational (not necessarily convex) polygon.     

A numerical approach to cyclic-service queueing models

An iterative numerical technique for the evaluation of queue length distributions is applied to multi-queue systems with one server and cyclic service discipline with Bernoulli schedules. The technique is based on power-series expansions of the state probabilities as functions of the load of the system. The convergence of the series is accelerated by applying a modified form of the epsilon algorithm. Attention is paid to economic use of memory space.     

Update to the MUSIG model in ANSYS CFX for reliable modelling of bubble coalescence and breakup

The MUSIG (Multiple Size Group) model in the commercial CFD code ANSYS CFX is a population balance approach for describing binary bubble coalescence and breakup events. It is widely used in the simulation of poly-dispersed bubbly flows. The purpose of this work is to identify the internal inconsistencies in the discrete method that is applied for the solution of the population balance equation in MUSIG, and to propose an internally consistent one for discretising the source and sink terms that result from bubble coalescence and breakup. The new formulation is superior to the existing ones in preserving both mass and number density of bubbles, allowing arbitrary discretisation schemes and is free of costly numerical integrations. The numerical results on the evolution of bubble size distributions in bubbly flows reveal that the inconsistency in the original MUSIG regarding bubble breakup is non-negligible for both academic and practical cases. The discussion on the effect of internal inconsistency as well as updates to the model presented in this work are necessary and important for calibration of bubble coalescence and breakup models using the MUSIG approach.     

On a numerical approach to Stefan-like problems

Summary This paper is devoted to the numerical analysis of a bidimensional two-phase Stefan problem. We approximate the enthalpy formulation byC ⁰ piecewise linear finite elements in space combined with a semi-implicit scheme in time. Under some restrictions related to the finite element mesh and to the timestep, we prove positivity, stability and convergence results. Various numerial tests are presented and discussed in order to show the accuracy of our scheme.This work is supported by the Fonds National Suisse pour la Recherche Scientifique.     

On the availability of a warm standby system: a numerical approach

We consider a basic duplex system characterized by warm standby and attended by two general heterogeneous repairmen. In order to derive computational results for the point availability of the engineering system, we first employ a stochastic process endowed with time-dependent transition measures satisfying coupled partial differential equations. However, an explicit evaluation of the (exact) solution is in general excluded. Therefore, we also propose a numerical solution of the equations. Our methodology is based on new modification of the first-order upwind scheme applied to a semiinfinite region. As an application, we consider the important case of Weibull–Gnedenko repair and provide an in-depth analysis of some key features of the duplex system.     

A numerical approach for a semilinear parabolic equation with a nonlocal boundary condition

A semilinear reaction-diffusion problem with a nonlocal boundary condition is studied. This paper presents a new and very easy implementable numerical algorithm for computations. This is based on a suitable linearization in time and on the principle of linear superposition. Any method for the space discretization (FEM was taken in this analysis) can be chosen. The derived algorithm is implicit and it does not need any iteration scheme to get a solution with the nonlocal boundary condition. Stability analysis has been performed and the optimal error estimates have been derived. Numerical results have been compared with other known techniques.     

When to rush a ‘behind’ in Australian rules football: a dynamic programming approach

In Australian rules football, points are scored when the ball passes over the goal line. Six points are awarded for a goal when the ball passes between the two centre posts, and one point for a ‘behind’, when the ball passes between a centre post and an adjacent outer post. After a behind, the defending team has a free kick from the goal line. It may be worthwhile, particularly in the closing stages of a game, for a defending team voluntarily to concede a behind, by themselves passing the ball between the two outer posts, either to avert the possibility of an imminent goal or to increase the probability of scoring a goal themselves. A dynamic programming model is used to analyse this situation.     

A numerical approach to a minimum problem with applications in image segmentation

A numerical approach to the problem: minF ^λ(E), whereF ^λ(E)=P(E,R ⁿ )+λ|Ω/E|, is considered. The functionalF ^λ is approximated, using techniques of Γ-convergence, with a sequence of functionals that are successively discretized by finite differences. A relation between the index of the approximating sequence and the meshsize of the domain is found.

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Riassunto Viene presentato un approccio numerico del problema: minF ^λ(E), doveF ^λ(E)=P(E,R ⁿ )+λ|Ω/E|. Il funzionaleF ^λ viene approssimato, usando tecniche di Γ-convergenza, con una successione di funzionali, successivamente discretizzati con differenze finite. Viene trovata una relazione tra l'indice della successione approssimante e il passo del reticolo del dominio.