Finite element solutions of nonsteady incompressible viscous flow between two rotating concentric spheres

The problem of unsteady, incompressible viscous flow between two rotating concentric spheres has been investigated here. The full Navier-Stokes equations in terms of the velocity components u, v w and the pressure p, using spherical coordinates for axially symmetric flow, were solved by means of the finite element method in the spatial dimension and the alternating-direction method in the time dimension using Glowinski's algorithm. The element used is an annular-sector-type element with a bilinear approximation for the velocity components and with constant pressure within the element. Reynolds numbers in the range from 1 to 1000, gap size 0.5 and different combinations of the angular velocity of the inner and outer spheres were studied. In some of these cases a steady-state solution was possible, while in others only a transient solution was possible. This method proved to be successful and powerful in predicting the behavior of the flow for these nonlinear-type problems.     

Improved solution of steady,viscous, incompressible flow between two rotating spheres

A refined solution is presented for the analysis of viscous, incompressible, steady flow between two rotating spheres. A new method, used previously for simpler problems only, is adapted to this problem. The method allows the use of small grid spacing and thus yields improved accuracy.     

Numerical studies of steady,viscous, incompressible flow between two rotating spheres

A new numerical method, developed for the study of secondary flow in a curved tube, is adapted and extended to the study of viscous, incompressible, steady flow between two rotating spheres. The Navier-Stokes equations are approximated by a triple sequence of linear problems, each of which has a diagonally dominant coefficient matrix. Computer examples are described and discussed.     

On the flow between two rotating shrouded discs

Numerical solutions to the full Navier Stokes equations for the flow enclosed between two rotating discs and a sidewall are presented. The sidewall remains fixed throughout, while we consider examples in which: (i) the bottom disc is rotating and the top disc is fixed; (ii) the two discs rotate in the same sense; (iii) the two discs rotate in opposite senses. We present solutions up to a Reynolds number of 1200, using second order central finite differencing.     

A collocation finite element-finite difference solution for developing isothermal flow between two parallel plates

This paper presents a finite element-finite difference method for the solution of the boundary layer equations for developing flow between two parallel plates. Due to the parabolic nature of the equations it was possible to discretize the transverse flow direction with one-dimensional Hermite cubic finite elements and the axial flow direction with a backward finite difference approximation. The collocation finite element-finite difference approximation was found to be appropriate for the modeling of the non-linear convection terms in the axial momentum equation. The resulting system of mixed linear and non-linear algebraic equations was solved using the Newton-Raphson method. Several numerical experiments were conducted to study the behavior of the solution with respect to the element size and number, order of finite difference approximation, and the marching step size.     

Numerical solution of laminar recirculating flow between shrouded rotating disks

In this paper we develop a method to obtain the numerical solution of the problem of recirculating flow between shrouded rotating disks. The major difficulty of this type of problem is in obtaining convergence at high Reynolds numbers. With the technique developed in this paper we have obtained convergence for Reynolds numbers up to 10,000. The procedure can be extended to higher Reynolds numbers if desired. The contours of the stream function, vorticity function and angular velocity are presented for Reynolds numbers of 500, 2000, 5000 and 10000. The method is applicable to any problem which has similar equations of motion.     

The solution of the poisson-boltzmann equation between two spheres: Modified iterative method

We present a modification on the successive overrelaxation (SOR) method and the iteration of the Green's function integral representation for the solution of the (nonlinear) Poisson-Boltzmann equation between two spheres. In comparison with other attempts, which approximate the geometry or the nonlinearity, the computations here are done for the full problem and compared with those done by the finite element method as a typical method for such problems. For the parameters of general interest, while the SOR method does not work, and the iteration of the integral representation is limited in its convergence, our modification to these iterative schemes converge. The modified SOR surpasses both methods in simplicity and speed; it is about 100 times faster than the modified iteration of the integral representation, with the latter being still simpler and faster than the finite element method. These two examples further illustrate the advantage of our recent modification to iterative methods, which is based on an analytical fixed point argument.     

Calculation of flow between two rotating coaxial disks by differentiation with respect to an actual parameter

A method which allows to find the dependence of the solution $\text{x(ξ, s)}$ on the parameter $\text{s}$ for a nonlinear boundary value problem will be presented. The calculation of the dependence of $\text{x(ξ, s)}$ on $\text{s}$ is performed in a non-iterative way. The technique is employed to solve a difficult two-point boundary value problem: steady state flow of an incompressible viscous fluid between two rotating coaxial disks.     

Finite difference solution of differential equations with stochastic inputs

This paper discusses application of two numerical methods (central difference and predictor corrector) for the solution of differential equations with deterministic as well as stochastic inputs. The methods are applied to a second order linear differential equation representing a series RLC netowrk with step function, sinusoidal and stochastic inputs. It is shown that both methods give correct answers for the step function and sinusoidal inputs. However, the central-difference method of solution is recommended for stochastic inputs. This statement is justified by comparing the auto-correlation and cross-correlation functions of the central-difference solution (with stochastic inputs) with the corresponding theoretical values of a continuous system. It is further shown that the more common predictor-corrector methods, although suitable for solution of differential equations with regular inputs, diverge for stochastic inputs. The reason is that these methods, by the application of several point integral formulas, use a high degree of smoothing on the variable and its derivatives. Inherent in the derivation of these integral formulas is the assumption of the continuity of the variable and its derivatives, a condition which is not satisfied in problems with stochastic inputs.Note that the second order differential equation chosen here for numerical experiments can be solved by classical methods for all of the given inputs, including the probabilistic inputs. The classical methods, however, unlike the numerical solutions, can not be extended to nonlinear differential equations which frequently arise in the digital simulation of engineering problems.     

Influence of viscous dissipation in a fluid between concentric rotating spheres

The steady motion of a viscous fluid contained between two concentric spheres which rotate about a common axis with different angular velocities is considered. A second-order method which was introduced previously by the authors to obtain numerical solutions of a class of Navier-Stokes problems coupled with a superimposed thermal field is successfully extended here to investigate the influence of the internal generation of heat by viscous dissipation on the thermal field. The resulting thermal field solutions are presented for various values of the Eckert number and the rotation ratio. It is shown that the inclusion of heat generation by viscous dissipation significantly alters the thermal field behavior, while the relative rates of rotation of the two spheres do not change the general character of the thermal field.     

Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions

Approximate solutions are considered for the extended Fisher-Kolmogorov (EFK) equation in two space dimension with Dirichlet boundary conditions by a Crank-Nicolson type finite difference scheme. A priori bounds are proved using Lyapunov functional. Further, existence, uniqueness and convergence of difference solutions with order O(h²+k²) in the L^∞-norm are proved. Numerical results are also given in order to check the properties of analytical solutions.     

Asymptotic equivalence between two difference systems

In this paper, we study the asymptotic equivalence between the linear system Δx(n) = A(n)x(n) and its perturbation Δy(n) = A(n)y(n)+g(n, y(n)) by using the comparison principle and supplementary projections. Furthermore, we establish some asymptotic properties for the nonlinear system Δx(n) = f(n, x(n)).     

Confidence intervals for the difference between two means

This paper compares three confidence intervals for the difference between two means when the distributions are non-normal and their variances are unknown. The confidence intervals considered are Welch-Satterthwaite confidence interval, the adaptive interval that incorporates a preliminary test (pre-test) of symmetry for the underlying distributions, and the adaptive interval that incorporates the Shapiro-Wilk test for normality as a pre-test. The adaptive confidence intervals use the Welch-Satterthwaite interval if the pre-test fails to reject symmetry (or normality) for both distributions; otherwise, apply the Welch-Satterthwaite confidence interval to the log-transformed data, then transform the interval back. Our study shows that the adaptive interval with pre-test of symmetry has best coverage among the three intervals considered. Simulation studies show that the adaptive interval with pre-test of symmetry performs as well as the Welch-Satterthwaite interval for symmetric distributions. However, for skewed distributions, the adaptive interval with pre-test of symmetry performs better than the Welch-Satterthwaite interval.     

Fluid flow and heat transfer past two spheres in a cylindrical tube

The fluid flow and the heat transfer for a row of spheres in a cylindrical tube is modelled by considering the flow past two spheres in a long tube. The problem is solved numerically by a finite element method, using a velocity-pressure formulation for the Navier-Stokes equations.Results are obtained for Reynolds number up to 200, with Prandtl numbers of 0.72 and 7.0, for a range of sphere sizes and sphere separations. It was found that as the distance between the spheres was decreased a circulatory region of flow appeared between the spheres for a given Reynolds number. This eddy led to poor heat transfer in this region. Increasing the Reynolds number was found not to improve the situation as the eddy grew in size and caused poorer heat transfer. This was found to be true with even the widest gap sizes considered.     

The finite element solution of laminar combined convection from two spheres in tandem arrangement

The purpose of this investigation is to study the flow and heat transfer characteristics of laminar combined convection from two isothermal spheres of the same diameter in tandem arrangement, the distance between the sphere centers being twice the value of the diameter. The full Navier-Stokes and energy equations are solved by a finite element method. The variations of surface shear stress, pressure, and Nusselt number are obtained for gases having a Prandtl number of 0.7 over the entire sphere surface, including the zone beyond the separation point. The predicted values of the average Nusselt number, the location of the separation point, and the friction drag coefficient are also presented. It is found that the behavior of the upstream sphere is close to that of a single sphere.     

Numerical solution of potential flow equations with a predictor-corrector finite difference method

We develop a numerical solution algorithm of the nonlinear potential flow equations with the nonlinear free surface boundary condition.A finite difference method with a predictor-corrector method is applied to solve the nonlinear potential flow equations in a two-dimensional (2D) tank.The irregular tank is mapped onto a fixed square domain with rectangular cells through a proper mapping function.A staggered mesh system is adopted in a 2D tank to capture the wave elevation of the transient fluid.The finite difference method with a predictor-corrector scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition.We present the numerical results of wave elevations from small to large amplitude waves with free oscillation motion,and the numerical solutions of wave elevation with horizontal excited motion.The beating period and the nonlinear phenomenon are very clear.The numerical solutions agree well with the analytical solutions and previously published results.     

Finite difference analysis of plate response to random loads

The application of the finite difference method to analyse the response of plates subjected to stationary random loading is presented. The discrete forced dynamical system generated by the finite difference approximations is analysed by a generalised harmonic analysis. The power spectral density (PSD) for displacement at each node is calculated directly for particular frequencies and subsequently these PSDs are integrated numerically over the frequency range to yield the mean square response. Both viscous and structural damping have been considered. For illustrative purposes the random excitation is taken to be clipped white noise with uniform spatial correlation. To validate the numerical procedure, further verification was obtained by comparing the numerical results with an analytic and with another numerical solution.     

Finite difference procedure for solution of poisson equation over complex domains with Neumann boundary conditions

A generalized finite difference scheme for solving Poisson equation over multiply connected domain bounded by irregular boundaries at which Neumann boundary conditions are specified, is presented in this paper. The method used to treat the Neumann condition is a six-point gradient approximation method given by Greenspan[6]. The method is generalized to treat all types of grid intersections with the boundary. An efficient computational procedure is devised by eliminating the calculations at the boundary during the interations.The scheme is applied to the problem of forced convection heat transfer in a fully developed laminar flow through seven and nineteen rod-cluster assemblies. Fluid properties are assumed to be uniform. In arriving at the fast converging and efficient method from computational point of view, different iterative techniques, overrelaxation methods and boundary treatments were tried. The results of computations and the computer times are reported in the present paper.     

Computing the velocity of a rotating flow

To compute the time-dependent flow of a rotating incompressible fluid we consider the velocity–vorticity formulation of the Navier–Stokes equations in cylindrical coordinates. In the numerical method employed the velocity field at each time-step is found as the least squares solution of an overdetermined system of linear equations, Ax=b. We consider how to compute x using the preconditioned conjugate gradient algorithm for least squares (PCGLS) on a distributed parallel computer. The various aspects of using a parallel computer are discussed, and results for a wide range of parallel computers are presented. The parallel speed-up depends on the architecture but is typically about 80% of the number of processors used.