Non-normal energy amplifications in stratified turbulent channels

Abstract

The influence of stable and unstable stratification on the amplification of coherent structures in turbulent channel flows is investigated by computing the linear response to stochastic forcing. The responses to momentum and thermal forcing are considered separately. Consistently with results of previous direct numerical simulations, the influence of the mean flow stratification on stochastic forcing amplifications is found to be non-negligible only for streamwise-elongated large-scale structures. Unstable stratification is found to enhance the peak variance of the response, except for the velocity response to thermal forcing, and to increase the spanwise wavelength of the most amplified structures. Stable stratification induces opposite effects. The spanwise wavelengths maximizing the different types of variance amplifications, all converge to approximately 6h when approaching the linear instability threshold where large-scale coherent rolls become linearly unstable. In the presence of even moderately unstable stratification, the profiles of turbulent buoyancy and momentum fluxes and of rms vertical velocity of all types of most amplified stochastic responses are nearly indistinguishable from those of the critical mode becoming unstable at the critical Richardson number. The two most energetic POD modes always contribute to more than 90% of the variance of the response, except for the thermal response to thermal forcing. We conclude that the same mechanism underlies the onset of the instability of coherent large-scale rolls at the critical Richardson number and the amplification of coherent large-scale structures at subcritical Richardson numbers.