A Cartesian grid technique based on one‐dimensional integrated radial basis function networks for natural convection in concentric annuli

Abstract

This paper reports a radial-basis-function (RBF)-based Cartesian grid technique for the simulation of two-dimensional buoyancy-driven flow in concentric annuli. The continuity and momentum equations are represented in the equivalent stream function formulation that reduces the number of equations from three to one, but involves higher-order derivatives. The present technique uses a Cartesian grid to discretize the problem domain. Along a grid line, one-dimensional integrated RBF networks (1D-IRBFNs) are employed to represent the field variables. The capability of 1D-IRBFNs to handle unstructured points with accuracy is exploited to describe non-rectangular boundaries in a Cartesian grid, while the method's ability to avoid the reduction of convergence rate caused by differentiation is instrumental in improving the quality of the approximation of high-order derivatives. The method is applied to simulate thermally-driven flows in annuli between two circular cylinders and between an outer square cylinder and an inner circular cylinder. High Rayleighnumber solutions are achieved and they are in good agreement with previously published numerical data.

An Effective Integrated-RBFN Cartesian-Grid Discretization for the Stream Function–Vorticity–Temperature Formulation in Nonrectangular Domains

“…By substituting (19) into (13), a boundary condition for the vorticity at a boundary point on a horizontal grid line will be computed by…”

“…Since integration is a smoothing operator, the latter has higher approximation power than the former in the handling of derivative functions (e.g. [16,18,19,20]). …”

“…Instead of using conventional schemes, 1D-IRBFN approximation schemes [18] are utilised in the present work. Unlike our previous work [19] The remainder of the paper is organised as follows. Section 2 gives a brief review of the governing equations.…”

Numerical study of stream-function formulation governing flows in multiply-connected domains by integrated RBFs and Cartesian grids

“…Over the last twenty years, radial-basis-function networks (RBFNs) have been developed to solve different types of differential problems encountered in applied mathematics, science and engineering (e.g. [1,2,3,4,5,6,7,8,9]). …”

“…In [4,5,6,7,8], the RBF approximations are constructed using integration (integrated RBFNs (IRBFNs)) rather than the usual differentiation. This approach has the ability to overcome the problem of reduced convergence rates caused by differentiation and to provide effective ways to implement derivative boundary values.…”