Mean large-scale linear response of a turbulent flow: scalar transport

Previous analyses of the mean large-scale linear response of a simulated turbulent channel flow in the plane defined by its two homogeneous directions (turbulent Hele-Shaw flow) are here extended with the inclusion of scalar transport. The same fundamental questions can be asked of the passive scalar (temperature or contaminant concentration, say) as of the main flow's velocity, namely whether the low-frequency and low-wavenumber response function of each quantity exhibits emerging behaviour that can be described by an equivalent large-scale differential equation. This is basically what every turbulence model aims to do, with the distinction that here the relevant information is extracted from a DNS. The extraction technique, already described, e.g., in Luchini, Quadrio & Zuccher, Phys. Fluids 18, 121702 (2006), consists of forcing the system with an externally generated wideband noise-like signal and taking suitable correlations. 

Adding a passive scalar is particularly interesting in this respect because the basic equation of its transport (in the absence of gravitational or other feedback on the fluid's velocity) is linear by itself. For the extraction technique used here this implies that no upper bound is imposed on the amplitude of the external forcing. The problem is therefore in a sense easier than for the momentum equations. Nonetheless, the simplification is not such that a large-scale turbulent diffusion coefficient could be predicted in advance, just as it cannot for momentum, and the overall statistics of the passive scalar are known to be qualitatively similar to those of velocity. This is so much more astounding if one realizes that linearity combined with boundedness of the scalar implies that the no unstable eigenvalues can be present, whereas common wisdom places the origin of chaos in turbulence seen as a dynamical system in the combination of unstable eigenvalues, which the linearized velocity equation is known to possess, with boundedness.