Finite size analysis of a double crossover in transitional wall turbulence

TL;DR

This study analyzes the finite size effects of two crossovers in transitional wall turbulence, identifying a critical-like transition and a smeared first-order transition through numerical simulations and phase transition phenomenology.

Contribution

It provides a detailed finite size analysis of two consecutive crossovers in wall turbulence, linking them to phase transition phenomena and offering new insights into their nature.

Findings

The first crossover resembles a critical phase transition.

The second crossover behaves like a smeared first-order transition.

Finite size effects influence the sharpness of the transitions.

Abstract

This article presents the finite size analysis of two consecutive crossovers leading laminar-turbulent bands to uniform wall turbulence in transitional plane Couette flow. Direct numerical simulations and low order modeling simulations of the flow are performed. The kinetic energy $E$ of the turbulent flow and the order parameter $M$, a measure of the spatially organised modulation of turbulence, are sampled and processed in view analytical results from the phenomenology of phase transitions. The first crossover concerns the loss of spatial organisation of turbulence in the flow. In the band phase, the order parameter $M$ decreases continuously with the Reynolds number $R$ toward a small value, while its response function $\chi_M$ displays a maximum at the crossover. In the uniform phase, the order parameter $M$ and its variance $\sigma$ decrease toward zero following mean field field scalings $M,\sigma \propto 1/\sqrt{L_xL_z(R-R_c)}$ as $R$ is increased. The kinetic energy $E$ is an affine function of $R$ except in a small range where a sharp increase is detected, which corresponds to the second crossover. In this range, spatial and temporal coexistence of the uniform turbulence phase and laminar-turbulent bands phase is observed. This sharp increase is concomitant with a maximum of the response function of the kinetic energy. The finite size analysis reveals that the jump does not steepen and that the maximum of response function of $E$ saturates as size is increased. The first crossover is formally identical to a critical phenomenon in condensed matter. The second crossover is in agreement with a first order phase transition smeared by finite noise. The analytical analysis of this phenomenon assuming a non interacting gas of fronts between domain of the two phases provides a scaling of the response function consistent with that of $E$.