A second‐order approximation to natural convection for large Rayleigh numbers and small Prandtl numbers
Authors: Shay William A. , Schultz David H. 
Published: August 2005

The problem under investigation is that of fluid flow within an enclosed rectangular cavity. It is assumed that one wall is maintained at a constant temperature T1 (hot wall) and the other wall is maintained at a constant temperature T0 (cold wall). At the remaining walls, two separate cases are studied. In the first, an adiabatic boundary condition is assumed. That is, the normal derivative of the temperature function is assumed to be 0. In the second, it is assumed the temperature varies linearly from T0 to T1.

The purpose of this paper is the application of a second order numerical technique to the problem of fluid flow within a heated closed cavity. The method is a modification of a method developed by Shay1 and applied to the driven cavity problem. In order to test the viability of this technique, it was decided to extend the technique to the problem of natural convection in a square. Jones2 proposed that this problem is suitable for testing techniques that may be applied to a wide range of practical problems such as reactor insulation, cooling of radioactive waste containers, solar energy collection and others.3

The technique makes use of second‐order finite difference approximations to all derivatives in the governing equations. Furthermore, second‐order approximations are also used to determine boundary vorticities and, when the adiabatic boundary condition is used, for the boundary temperatures as well. In some works, where second‐order approximations are used at interior points, second‐order boundary approximations have been sacrificed in favour of a more stable, but first‐order boundary approximation.

The current approximations are generated by writing the unknown value of a function at a given interior node as a linear combination of unknown function values at all of the neighbouring nodes. Then the function values at these neighbouring nodes are expanded in a Taylor series about the given node. Through appropriate regrouping of terms and the use of the equations to the solved, constraints are imposed on the coefficients of the linear combination to yield a second‐order approximation. As it turns out, there are more unknowns than constraints and, as a result, we are left with some freedom in choosing coefficients. In this work this freedom was used to choose coefficients in such a way as to maximize stability of the resulting system of equations. In other words, the approximations to the governing partial differential equation are individually determined at each point dependent on the direction of flow in order to generate the best possible stability. This idea is analogous to that used in the derivation of the upwind method. However, the current method is second‐order accurate where the upwind method is only first‐order accurate. Thus, what is generated is an easily implemented second‐order method that yields a system of equations that has proved easy to solve.

The system of equations is solved via the method of successive overrelaxation. The stability of the method is shown in the convergence for a wide range of Rayleigh numbers, Prandtl numbers and mesh sizes. Level curves of the stream, vorticity and temperature functions are provided for Rayleigh numbers (Ra) as large as 100,000, Prandtl numbers (Pr) as small as 0.0001, and mesh sizes as small as 0.0125. Values of the Nusselt number have also been calculated through the use of Simpson's rule, and a second order approximation to the normal derivative of the temperature along the cold wall. Comparisons are made with other current works to aid in the verification of this methods' accuracy and also with the first‐order upwind method to demonstrate superiority over the first‐order method.