Unconditionally stable numerical scheme for natural convection problems

Abstract

Purpose

Both the importance of the natural convection in science and engineering and the shortage of publications in the field of numerical features of time-stepping schemes for the simulation of coupled heat and fluid flow problems motivate the present work. The paper aims to discuss these issues.

Design/methodology/approach

The paper presents the unconditionally stable time-stepping scheme for simulation of coupled problems of mass and heat transport. The paper is divided into two parts. The first part concerns the mathematical formulation of the scheme and discusses its implementation. The second part focuses on the numerical simulation and its results. A detailed investigation of the temporal order of the scheme with respect to the L2-norms of the errors of the pressure, velocity, temperature and divergence of velocity fields has also been given.

Findings

The work shows that it is possible to formulate a numerical scheme which is unconditionally stable with respect to the time step size. Moreover, application of the spectral element method for the spatial discretization results in a high order of approximation in space and very good overall accuracy. Furthermore, the investigation of the numerical features of the scheme showed that the formal temporal order of the scheme (formally second order) has been deferred very slightly and the order of 1.8-1.9 is achieved for all unknown fields.

Originality/value

The paper presents a new unconditionally stable scheme for simulation of unsteady flows with bidirectional coupling of heat transfer and the fluid flow. It also carefully investigates the numerical behaviour of the method.

Citation

Górecki, B. and Szumbarski, J. (2014), "Unconditionally stable numerical scheme for natural convection problems", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 24 No. 4, pp. 907-919. https://doi.org/10.1108/HFF-02-2013-0065