The dimension of attractors underlying periodic turbulent Poiseuille flow

Abstract

Using a coarse grained (16 × 33 × 8) numerical simulation, a lower bound on the Lyapunov dimension, Dλ, of the attractor underlying turbulent, periodic Poiseuille flow at a pressure-gradient Reynolds number of 3200 has been calculated to be approximately 352. These results were obtained on a spatial domain with streamwise and spanwise periods of 1.6π, and correspond to a wall-unit Reynolds number of 80. Comparison of Lyapunov exponent spectra from this and a higher-resolution (16 × 33 × 16) simulation on the same domain shows these spectra to have a universal shape when properly scaled. Using these scaling properties, and a partial exponent spectrum from a still higher-resolution (32 × 33 × 32) simulation, we argue that the actual dimension of the attractor underlying motion on the given computational domain is approximately 780. The medium resolution calculation establishes this dimension as a strong lower bound on this computational domain, while the partial exponent spectrum calculated at highest resolution provides some evidence that the attractor dimension in fully resolved turbulence is unlikely to be substantially larger. These calculations suggest that this periodic turbulent shear flow is deterministic chaos, and that a strange attractor does underly solutions to the Navier–Stokes equations in such flows. However, the magnitude of the dimension measured invalidates any notion that the global dynamics of such turbulence can be attributed to the interaction of a few degrees of freedom. Dynamical systems theory has provided the first measurement of the complexity of fully developed turbulence; the answer has been found to be dauntingly high.