Tutorial

Scale-Energy Budget in Inhomogeneous Flows

C.M. Casciola
Dipartimento di Meccanica e Aeronautica
Università di Roma "La Sapienza"

Actual turbulent flows can hardly be classified as spatially homogeneous or isotropic. Nonetheless these idealizations allow to identify certain universal features associated with the small scale motions almost invariably observed in a variety of different conditions. The single most significant aspect is a flux of energy through the spectrum of inertial scales related to the phenomenology commonly referred to as the Richardson cascade. Inhomogeneity, inherently present in most flows of technical significance, generates additional energy fluxes of a different nature, corresponding to the spatial redistribution of turbulent kinetic energy. Traditionally the spatial flux is associated to a single point observable, namely the turbulent kinetic energy density. The flux through the scales instead is classically related to two-points statistics, given in terms of energy spectrum or, equivalently, in terms of the second order moment of the velocity increments.

Purpose of the present tutorial is to introduce an audience of young researches to the use of proper tools able to reconcile the two views, in the space of scale and in physical space, respectively. The starting point is a suitably generalized form of the classical Kolmogorov equation, whereby a scale-by-scale balance for the turbulent fluctuations in inhomogeneous environments is evaluated. The procedure examines in detail how the energy associated to a specific scale of motion - hereafter called the scale-energy - is transferred through the spectrum of scales and, simultaneously, how the same scale of motion exchanges energy with a properly defined spatial flux. Several examples of applications will be discussed using data sets taken either from direct numerical simulations (DNS) and, wherever possible, from experiments of moderate Reynolds number inhomogeneous flows, such as channel flows, pipe flows and jets.

Special consideration will be deserved to discuss the same concepts in the larger context of complex fluids, to enable the treatment of active substructures, such as those occurring in flows of long chain polymers near a solid wall or in chemically reacting systems. The detailed scale-by-scale balance will be applied to the different regions of each flow in the various ranges of scales, to understand how - i.e. through which mechanisms, at which scales and in which regions of the flow domain - turbulent fluctuations in the carrier fluid and in the substructure are generated and sustained, providing rigorous support to specific conceptual models.