The word turbulence is employed to label many different physical phenomena, which exhibit the common characteristics of disorder and complexity. It is the ubiquitous presence of spontaneous (intrinsic) fluctuations, distributed over a wide range of length and time scales, that makes turbulence a worthwhile research topic. The very nature of the turbulent fluctuations is extremely peculiar. Turbulence has to do with non-linearity; there is no hint of the non-linear solutions in the linearized approximations, and strong departure from thermodynamic equilibrium.
Most natural and industrial flows are turbulent; turbulence lies at the core of so much of what we observe. Turbulence generally eclipses the laminar (steady and regular) behavior of the flow, and contributes to a largely enhanced energy dissipation, mixing, heat and mass transfer, etc. In astrophysics, interstellar turbulence is said to be one of the key ingredients of modern theories of star formation.

Modelling prospecting aims at elaborating numerically tractable mathematical models of turbulence.

It is devoted to the statistical description of turbulent motions, also called turbulent eddies. By analogy with a molecule, an eddy may be seen as a glob of fluid of a given size, or (spatial) scale, that has a certain structure and life history of its own. The turbulent activity of the bulk is the net result of the interactions between the eddies. It is characteristic of turbulence that these interactions are unpredictable in details; however, statistically distinct properties can be identified and profitably examined.

Helium-II at (finite) temperature below the transition T=2.17K can be described as the superposition of two interacting fluids. A normal fluid which has non-zero viscosity and an inviscid superfluid, in which vorticity is confined to quantized vortices. The large-scale dynamics of these two co-penetrating fluids obey respectively the Navier-Stokes and Euler equations (at low mach number) coupled together by a mutual friction force. This is the general framework of the so-called two-fluid model initiated by Landau and Tisza in 1941. Like in classical fluids, turbulent dynamics can be generated in superfluid helium II. This is refered to as superfluid turbulence in the literature. Since the pioneering experiments by Viven about half a century ago, quantum turbulence has been investigated mostly experimentally with the motivation to examine to what extent quantum turbulence resembles or differs from classical turbulence in ordinary fluids. Related theoretical and numerical works rely mainly on simplified one-way coupling (no back reaction) between the normal and the superfluid; wall boundary conditions are also often ignored. The studies have yielded significant progress, however, we are still a long way from a fulll understanding. In this regard, it is admitted that improved numerical simulations can be helpful.

The prohibitive cost of direct numerical simulations (DNS) of turbulent engineering flows motivate the development of simplified models, requiring less computation, but still relevant (to some degree) for reproducing the large-scale dynamics. Under these circumstances, the modelling of turbulent flows near a solid boundary is of special interest. A boundary affects the kinetics of the flow through different mechanisms. The most prominent is that related to the mean shear, which is extreme at the boundary and responsible for the production of streamwise vortices and streaky structures, which eventually detach and sustain turbulence in the bulk. Thus, it is thought that understanding how the mean shear impacts on fluid motions is a key to improving the capabilities of numerical models.

It is desirable to account explicitly for the mean-flow gradients in the modeling of the subgrid-scale viscosity, also the canonical assumption of homogeneous and isotropic turbulence should be abandoned. This is particularly relevant for high-Reynolds flows, when the resolution is almost comparable to the large-scales of the flow. These features naturally call for a generic procedure to estimate numerically the mean flow and evaluate the subgrid-scale viscosity as the simulation progresses. Two distinc time-domain smoothing algorithms are proposed to estimate the mean flow, namely, an exponentially weighted moving average (exponential smoothing) and an adaptive low-pass Kalman filter. These algorithms highlight the longer-term evolution or cycles of the flow but erase short-term fluctuations. Indeed, it is our assumption that the mean flow (in the statistical sense) may be approximated to the low-frequency component of the velocity field, and that the turbulent part of the flow adds itself to this unsteady mean.