Symbolic Computation of Turbulence and Energy Dissipation in the Taylor Vortex Model

Using a classic example proposed by G. I. Taylor, we reconsider through the use of computer algebra, the mathematical analysis of a fundamental process in turbulent flow, namely: How do large scale eddies evolve into smaller scale ones to the point where they are effectively absorbed by viscosity? The explicit symbolic series solution of this problem, even for cleverly chosen special cases, requires daunting algebra, and so numerical methods have become quite popular. Yet an algebraic approach can provide substantial insight, especially if it can be pursued with modest human effort.

The specific example we use dates to 1937 when Taylor and Green⁸ first published a method for explicitly computing successive approximations to formulas describing the three-dimensional evolution over time of what is now called a Taylor–Green vortex.

With the aid of a computer algebra system, we have duplicated Taylor and Green's efforts and obtained more detailed time-series results. We have extended their approximation of the energy dissipation from order 5 in time to order 14, including the variation with viscosity.

Rather than attempting additional interpretation of results for fluid flow (for which, see papers by Brachet et al.,^2,3 we examine the promise of computer algebra in pursuing such problems in fluid mechanics.