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.
A direct simulation of turbulent flow in a channel is analyzed by the method of empirical eigenfunctions (Karhunen–Loève procedure, proper orthogonal decomposition). This analysis reveals the presence of propagating plane waves in the turbulent flow. The velocity of propagation is determined by the flow velocity at the location of maximal Reynolds stress. The analysis further suggests that the interaction of these waves appears to be essential to the local production of turbulence via bursting or sweeping events in the turbulent boundary layer, with the additional suggestion that the fast acting plane waves act as triggers.
The bifurcation structure and asymptotic dynamics of even, spatially periodic solutions to the time‐dependent Ginzburg‐Landau equation are investigated analytically and numerically. All solutions spring from unstable periodic modulations of a uniform wavetrain. Asymptotic states include limit cycles, two‐tori, and chaotic attractors. Lyapunov exponents for some chaotic motions are obtained. These show the solution strange attractors to have a fractal dimension slightly greater than 3.
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