Nonlinear Diffusion and Decay of an Expanding Turbulent Blob

Abstract

Turbulence, left unforced, decays and invades the surrounding quiescent fluid. Though ubiquitous, this simple phenomenon has proven hard to capture within a simple and general framework. Experiments in conventional turbulent flow chambers are inevitably complicated by proximity to boundaries and mean flow, obscuring the fundamental aspects of the relaxation to the quiescent fluid state. Here, we circumvent these issues by creating a spatially-localized blob of turbulent fluid using eight converging vortex generators focused towards the center of a tank of water, and observe its decay and spread over decades in time, using particle image velocimetry with a logarithmic sampling rate. 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 these dynamics within the framework of a nonlinear diffusion equation, which predicts that the ideal quiescent-turbulent fluid boundary is sharp and propagates non-diffusively, driven by turbulent eddies while decaying with characteristic scaling laws. We find direct evidence for this model within the expansion phase of our turbulent blob and use it to account for the detailed behavior we observe, in contrast to earlier studies. 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.