Nonlinear diffusion and decay of a blob of turbulence spreading into a quiescent fluid

Abstract

Turbulence, left unforced, evolves under its own dynamics, invading surrounding quiescent fluid as it decays. A ubiquitous and familiar phenomenon, this fundamental aspect of turbulence has resisted the marriage of principled theory and experiment with no universal law yet capturing its evolution. Conventional flow chamber experiments have been hampered by boundary effects or strong mean flows that obscure the intrinsic dynamics of relaxation to quiescence. To circumvent these limitations, we create a spatially localized blob of turbulence using eight converging vortex generators focused at the center of a water tank, and observe its decay and expansion over decades in time using particle image velocimetry with logarithmic time sampling. The blob initially expands and decays until it reaches the walls of the tank and eventually transitions to a second regime of approximately spatially uniform decay. We interpret the turbulent dynamics as an interplay of nonlinear diffusion with associated steep fronts separating the turbulent and quiescent regions, and nonlinear decay, as described by the Kolmogorov–Barenblatt equation. We find direct evidence for this model within the expansion phase and decay phases of our turbulent blob and use it to account for the detailed behavior we observe. Our work provides a detailed spatially resolved narrative for the behavior of turbulence once the forcing is removed, and demonstrates unexpectedly that the turbulent cascade leaves an indelible footprint far into the decay process.