Tendency to occupy a statistically dominant spatial state of the flow as a driving force for turbulent transition

TL;DR

This paper proposes a model where the tendency of flow to occupy a dominant spatial state drives the transition from laminar to turbulent flow, linking statistical mechanics with fluid dynamics.

Contribution

It introduces a simple analytical model that explains turbulent transition as a result of localized structures forming at high Reynolds numbers, based on maximum entropy principles.

Findings

Reynolds number correlates with formation of localized structures

Model explains turbulent transition and flow properties

Flow instability initiates structural rearrangement to a dominant state

Abstract

The transition from laminar to turbulent fluid motion occurring at large Reynolds numbers is generally associated with the instability of the laminar flow. On the other hand, since the turbulent flow characteristically appears in the form of spatially localized structures (e.g., eddies) filling the flow field, a tendency to occupy such a structured state of the flow cannot be ruled out as a driving force for turbulent transition. To examine this possibility, we propose a simple analytical model that treats the flow as a collection of localized spatial structures, each of which consists of elementary cells in which the behavior of the particles (atoms or molecules) is uncorrelated. This allows us to introduce the Reynolds number, associating it with the ratio between the total phase volume for the system and that for the elementary cell. Using the principle of maximum entropy to calculate the most probable size distribution of the localized structures, we show that as the Reynolds number increases, the elementary cells group into the localized structures, which successfully explains turbulent transition and some other general properties of turbulent flows. We show that the basic assumptions underlying the model, i.e. that the particles are indistinguishable and elementary volumes of phase space exist in which the state of the particles is uncertain, are involved in the derivation of the Navier-Stokes equation. Taking into account that the model captures essential features of turbulent flows, this suggests that the driving force for the turbulent transition is basically the same as in the present model, i.e. the tendency of the system to occupy a statistically dominant state plays a key role. The instability of the flow at high Reynolds numbers can then be a mechanism to initiate structural rearrangement of the flow to find this state.