DEVELOPMENT OF EFFICIENT ALGORITHMS FOR GDQ RESULTANT EQUATIONS AND APPLICATIONS TO SIMULATE NATURAL CONVECTION IN ECCENTRIC ANNULI

Abstract

This research covers two topics: the development of numerical techniques for the solution of GDQ (generalized differential quadrature) resultant equations and simulation of natural convection in arbitrary eccentric annuli.

A high-order temporal scheme, a GDQ temporal block-marching scheme, is developed. The scheme is applied to the unsteady Navier-Stokes equations in the ?- ? formulations at Re=l, which has exact solutions. The present results are compared with both the exact solutions and those obtained from the standard four-stage Runge­Kutta scheme. It was found that the scheme is flexible and easy to apply. We can adjust the accuracy, computational effort and storage by changing the block size ?t and mesh size within each block. Compared to the four-stage Runge-Kutta scheme, the present scheme can both save the computational effort and obtain more accurate solutions for unsteady problems.

The iterative expressions of some conventional iterative methods, namely the CG, J­ OE (or Jacobi), Gauss-Seidel and SOR iterative methods, are derived for the GDQ algebraic equations. Simple rules are derived for working out these iterative expressions. The performance of J-OE (or Jacobi), Gauss-Seidel and SOR iterative methods on GDQ and second-order FD (finite difference) algebraic equation systems is compared. It was found that, among the above-mentioned three iterative methods, SOR method gives the fastest convergence rate for both GDQ and FD algebraic equation systems. However, for GDQ algebraic equations, the effect of relaxation factor is reduced. The Gauss-Seidel method keeps the same efficiency for both GDQ and FD algebraic equations. Natural convection in the arbitrary eccentric annuli is numerically investigated by using GDQ discretization technique and SOR iterative method. The GDQ method includes PDQ (polynomial based differential quadrature) and FDQ (Fourier expansion­ based differential quadrature) schemes. The former can be used to get more accurate results for problems with wall boundary conditions and the latter has advantages for problems with periodical boundary conditions. The present results agree very well with other numerical and experimental results. Side way eccentricity of the annuli produces a global circulation of flow through the annuli. The amount of circulation is governed by the uniqueness of the pressure field (single-value pressure distribution) in the annuli. The relationship between the global circulation in the flow-field and the inclination of plume the thermal field is analysed. The results suggest that global circulation, flow separation and the space between the two cylinders on the top of the inner cylinder have significant effects on plume inclination. The natural convection between eccentric cylinders for Ra= I04 and radius ratio of rr=2.6 is systemically investigated and analysed; including the effects of inner cylinder position on the overall Nusselt number, local Nusselt number distribution along the inner cylinder surface, flow field and temperature field.