An investigation of strange attractor theory and small-scale turbulence

Chaotic solutions to Burgers' equation for the small-scale turbulence structure have been obtained. The equation for the small-scale turbulent component of the velocity is derived from an unfiltered additive decomposition of Burgers' equation by utilizing a modally truncated local Galerkin approximation. The solutions become temporally chaotic with the structure of a strange attractor after a finite sequence of bifurcations. The topological structure of the attracting sets associated with the bifurcations is displayed via Poincaremaps, and the computed small-scale velocities and power spectra are compared with experimental data. The transitions leading to turbulence defined by the sequence of bifurcations, and the qualitative features of the solutions, show remarkable similarity to certain laboratory observations of Rayleigh-Benard convection. In addition to the periodic, quasi-periodic, phase-locked, and chaotic stages observed experimentally, an intermittent structure and indications of a subharmonic component are observed in the calculations. The rich bifurcation structure computed is a function of the bifurcation subparameter representing pressure gradient with the main bifurcation parameter Re fixed.