"A High-Order Method for Heat Transfer in a Fluid Between Rotating Concentric
Spheres"
by
David H. Schultz,
Steve Schwengels, R. N. Pralhad and Gunol Kocamustafaogullari
The purpose of this paper is to present an improved approximation technique
to obtain the numerical solution of the class of Navier-Stokes equation.
The particular problem under study is that of the flow of a viscous
incompressible fluid, with heat transfer, between concentric rotating
spheres. This involves the solution of four PDEs.
The application of a new technique to the rotating spheres problem is
appropriate for several reasons. First reaseachers in numerical analysis
and fluid dynamics are always interested in numerical techniques that are
accurate and stable for a wide range of parameters. Secondly, fluid motion
inside rotating containers has a wide variety of applications in engineering,
such as the study of gyroscopes and centrifuges. Applications also arise in
geophysics, where atmospheric and oceanic circulations are studied.
The spherical geometry approach was chosen because of its applications
and because the Navier-Stokes equations, written in Spherical co-ordinates,
are a particularly difficult set of PDEs to solve, and represent a good
benchmark for numerical techniques.
Many papers have been written on the numerical solution of steady,
viscious, imcompressible flow between two rotating spheres. However, these
papers have either been based only on first-order methods or have produced
results only for small Reynolds number.
In this paper we will present a second-order method which converges for
both large and small Reynolds numbers and Prandtl numbers. This technique
allows us to generate more accurate solutions with larger Reynolds numbers
than has been possible with other methods.
This problem, with heat transfer, has been studied in only one other
paper, and only for very small Reynolds numbers. We present results for
Reynolds numbers up to 25000, and Prandlt numbers from .01 to 100.0.