Spectral link and macroscopic non-universality in turbulent plane Couette flow

Abstract

In fully developed turbulent plane Couette flow the classical experimental data of frictional factor f vs. Reynolds number Re display well-known disparities which have long remained unexplained. These disparities are accompanied by previously unnoticed disparities in the wakes, and only in the wakes, of the attendant meanvelocity profiles (MVPs). To help explain these experimental data, we apply the model of the "spectral link" for MVPs to turbulent plane Couette flows. The model links the dissipative range, the inertial range, and the energetic range of the standard phenomenological model of the spectrum of turbulent kinetic energy to, respectively, the buffer layer, the log layer, and the wake of the MVPs. By assessing the empirical data using the spectral model, we argue for the existence, in plane Couette flow, of multiple states of turbulence which differ from one another only at the largest lengthscales in the flow, corresponding to the energetic range, where the spectrum is subject to finite-domain effects. Thus, the multiplicity of turbulent states is entirely consistent with small-scale universality, and the experimental data on plane Couette flow pose no challenge to the phenomenological theory of turbulence. Our findings might apply more broadly to a general class of flows engendered by moving boundaries.

Spectral link and wave breaking enhanced dissipation of turbulent kinetic energy

“…Then, the turbulent shear stress produced by an eddy of size s scales as −ρsv s du dy . By identifying the dominant eddies that transfer momentum across w y and produce the turbulent shear stress acting on w y are the eddies of size s = y (Gioia et al 2010), the turbulent shear stress τ t at a depth y below the ocean surface can be written as (Gioia et al 2010, Zhang 2019) where κ τ is a dimensionless proportionality constant determined by κ τ = κ 4/3 κ −1/3 ε (Gioia et al 2010) and v y is the dominant eddy of the size s = y. When writing the turbulent shear stress in the form of turbulent viscosity, it reads τ t = −ρν t du dy .…”

Shape and scaling of the mean-velocity profile in thermally-stratified plane-Couette flows

“…To solve the equations mentioned above, we need a closure model for the turbulent shear stress, τ t . To that end, we invoke the spectral link (Gioia et al 2010, Zhang 2019.…”

“…Setting β d = 5.2 and κ = 0.4 (which yields ỹv = 3.41), and β e = 2, (Note the empirical values of β d and β e are of the order 1 to 10 (Pope 2000). To understand how the spectral coefficients β d and β e affect the MVPs, please refer to Gioia et al (2010), Zhang (2019).) in figure 2, we plot the theoretical predictions for the MVPs in wall variables for a fixed Re τ and for various values of h/L 0 spanning unstable and stable stratification.…”