A Preliminary Study of the Burgers Equation with Symbolic Computation

A novel approach based on recursive symbolic computation is introduced for the approximate analytic solution of the Burgers equation. Once obtained, appropriate numerical values can be inserted into the symbolic solution to explore parametric variations. The solution is valid for both inviscid and viscous cases, covering the range of Reynolds number from 500 to infinity, whereas current direct numerical simulation (DNS) methods are limited to Reynolds numbers no greater than 4000. What further distinguishes the symbolic approach from numerical and traditional analytic techniques is the ability to reveal and examine direct nonlinear interactions between waves, including the interplay between inertia and viscosity. Thus, preliminary efforts suggest that symbolic computation may be quite effective in unveiling the “anatomy” of the myriad interactions that underlie turbulent behavior. However, due to the tendency of nonlinear symbolic operations to produce combinatorial explosion, future efforts will require the development of improved filtering processes to select and eliminate computations leading to negligible high order terms. Indeed, the initial symbolic computations present the character of turbulence as a problem in combinatorics. At present, results are limited in time evolution, but reveal the beginnings of the well-known “saw tooth” waveform that occurs in the inviscid case (i.e., Re=∞). Future efforts will explore more fully developed 1-D flows and investigate the potential to extend symbolic computations to 2-D and 3-D. Potential applications include the development of improved subgrid scale (SGS) parameterizations for large eddy simulation (LES) models, and studies that complement DNS in exploring fundamental aspects of turbulent flow behavior.