Finite size analysis of a double crossover in transitional wall turbulence

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 $E$.