The transition from steady to weakly turbulent flow in a close-packed ordered array of spheres

The sequence of transitions in going from steady to unsteady chaotic flow in a close-packed face-centred cubic array of spheres is examined using lattice-Boltzmann simulations. The transition to unsteady flow occurs via a supercritical Hopf bifurcation in which only the streamwise component of the spatially averaged velocity fluctuates and certain reflectional symmetries are broken. At larger Reynolds numbers, the cross-stream components of the spatially averaged velocity fluctuate with frequencies that are incommensurate with those of the streamwise component. This transition is accompanied by the breaking of rotational symmetries that persisted through the Hopf bifurcation. The resulting trajectories in the spatially averaged velocity phase space are quasi-periodic. At larger Reynolds numbers, the fluctuations are chaotic, having continuous frequency spectra with no easily identified fundamental frequencies. Visualizations of the unsteady flows in various dynamic states show that vortices are produced in which the velocity and vorticity are closely aligned. With increasing Reynolds number, the geometrical structure of the flow changes from one that is dominated by extension and shear to one in which the streamlines are helical. A mechanism for the dynamics is proposed in which energy is transferred to smaller scales by the dynamic interaction of vortices sustained by the underlying time-averaged flow.