The Karhunen–Loéve decomposition of minimal channel flow

Minimal channel flow is analyzed by means of the Karhunen–Loeve (KL) decomposition. It is shown that the most energetic modes are streamwise rollers followed by outward tilted quasi-streamwise vortices. Both of these mode types have a strong similarity to structures seen in physical experiments. Temporal plots of roll energy, propagating energy, bulk velocity, and representational entropy have been obtained. Study of the evolution of these variables shows a consistent pattern of growth and decay in which entropy plays a key role in describing the events in the turbulent process. The roll and propagating modes are also shown to make independent contributions to the Reynolds stress with the roll modes dominating the profile near the walls and the propagating modes having larger values towards the channel center. A comparison of the KL dimension of this flow and a full channel flow shows that the dimension scales with box size, i.e., it confirms the assertion that dimension is an extensive variable.