Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Flow due to an oscillating sphere and an expression for unsteady drag on the sphere at finite Reynolds number

Abstract: Unsteady flow due to an oscillating sphere with a velocity U0cosωt’, in which U0 and ω are the amplitude and frequency of the oscillation and t’ is time, is investigated at finite Reynolds number. The methods used are: (i) Fourier mode expansion in the frequency domain; (ii) a time-dependent finite difference technique in the time domain; and (iii) a matched asymptotic expansion for high-frequency oscillation. The flow fields of the steady streaming component, the second and third harmonic components are obtained with the fundamental component. The dependence of the unsteady drag on ω is examined at small and finite Reynolds numbers. For large Stokes number, e = (ωa2/2v)½ [Gt ] 1, in which a is the radius of the sphere and v is the kinematic viscosity, the numerical result for the unsteady drag agrees well with the high-frequency asymptotic solution; and the Stokes (1851) solution is valid for finite Re at e [Gt ] 1. For small Strouhal number, St = ωa/U0 [Lt ] 1, the imaginary component of the unsteady drag (Scaled by 6πU0pfva, in which Pf is the fluid density) behaves as Dml ∼ (h0Stlog St–h1St), m = 1,3,5… This is in direct contrast to an earlier result obtained for an unsteady flow over a stationary sphere with a small-amplitude oscillation in the free-stream velocity (hereinafter referred to as the SA case) in which D1∼ –h1 St (Mei, Lawrence & Adrian 1991). Computations for flow over a sphere with a free-stream velocity U0(1–α1+α1cosωt’) at Re = U02a/v = 0.2 and St [Lt ] 1 show that h0 for the first mode varies from 0 (at α1 = 0) to around 0.5 (at α1 = 1) and that the SA case is a degenerated case in which the logarithmic dependence of the drag in St is suppressed by the strong mean uniform flow.The numerical results for the unsteady drag are used to examine an approximate particle dynamic equation proposed for spherical particles with finite Reynolds number. The equation includes a quasi-steady drag, an added-mass force, and a modified history force. The approximate expression for the history force in the time domain compares very well with the numerical results of the SA case for all frequencies; it compares favourably for the PO case for moderate and high frequencies; it underestimates slightly the history force for the PO case at low frequency. For a solid sphere settling in a stagnant liquid with zero initial velocity, the velocity history is computed using the proposed particle dynamic equation. The results compare very well with experimental data of Moorman (1955) over a large range of Reynolds numbers. The present particle dynamic equation at finite Re performs consistently better than that proposed by Odar & Hamilton (1964) both qualitatively and quantitatively for three different types of spatially uniform unsteady flows.

Efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations

Abstract: A cell-vertex scheme for the three-dimensional Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing with locally varying coefficients, a multigrid method and carefully controlled dissipative terms, very good convergence rates are obtained for two- and three-dimensional flows. Details of the acceleration techniques, which are important for convergence on meshes with high aspect-ratio cells, are discussed. Emphasis is put on the analysis of the stability properties of the implicit smoothing of the explicit residuals with coefficients, which depend on cell aspect ratios.

Correspondence of variational finite-difference (relaxation) and imaginary-distance propagation methods for modal analysis.

Abstract: We demonstrate that relaxation algorithms for the determination of the lowest-order modes of a refractive-index profile are identical to imaginary-distance propagation procedures.

Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions

Abstract: An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the $L_\infty$ and $L_2$ norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.

PASVA3: An Adaptive Finite Difference Fortran Program for First Order Nonlinear, Ordinary Boundary Problems

Abstract: H e r e the f u n c t i o n s co, f, g t a k e v a l u e s i n ]Rd; f and g a r e a s s u m e d to be s u f f i c i e n t l y s m o o t h , and we a l s o a s s u m e t h a t p r o b l e m (1.1) h a s an i s o l a t e d s o l u t i o n ¢o*(t). In t h i s p a p e r w e w i l l p r e s e n t t he l a t e s t v e r s i o n of ou r v a r i a b l e o r d e r , v a r i a b l e s t ep , f i n i t e d i f f e r e n c e c o d e f o r p r o b l e m (1.1). Only t h o s e f e a t u r e s w h i c h h a v e n o t h i t h e r t o b e e n p u b l i s h e d w i l l be d i s c u s s e d in any d e t a i l .