Statistical analysis of the transition to turbulent-laminar banded patterns in plane Couette flow

The transition between uniform turbulence in plane Couette flow and turbulent-laminar banded patterns is studied using numerical computations. Timeseries of the velocity along a line in the midplane along the pattern wavevector are Fourier transformed. When averaged in time, these spectra show diffuse maxima corresponding to streaks and longitudinal rolls with a wavelength near 4, and a sharper, higher, maximum corresponding to the turbulent-laminar pattern with wavelength 40. Probability distribution functions are computed for the Fourier component corresponding to wavelength 40. It is shown that this PDF is a Gaussian centered at 0 for a a uniform turbulent flow and that this maximum shifts to a finite value when a turbulent-laminar banded pattern appears.