Co-spectral Budget Models Link Energy Distribution in Eddies to Bulk Flow Properties

Around the turn from the 19th to the 20th century onwards, a large number of semi-empirical formulae have been independently introduced to describe turbulent flow properties in micrometeorology, hydrology, hydraulics, and many branches in engineering. These formulae remain the corner stone of textbooks and working professional tool-kits alike. Their success at packing a large corpus of experiments dealing with flow conveyance above gravel beds, pipes and channels, momentum and scalar mixing in stratified boundary layers (including evaporation and sensible heat) explains their wide usage in hydraulics, atmospheric and climate models today. They continue to serve as 'work-horse' equations for flow and transport in natural systems operating at Reynolds numbers that are simply too large for direct numerical simulations. Despite all these successes, what is evidently missing is a link between these equations and the most prominent features of the flows they aim to describe: turbulent fluctuations and eddies. Establishing this link has been drawing significant research attention over the past 20 years and frames the scope of this presentation. To date, two approaches have been proposed that bridge turbulent energetics (or fluctuations) to bulk flow (or macroscopic) properties: the spectral link and the co-spectral budget model. These links connect facets of the universal properties of the turbulent energy distribution in eddies to bulk flow properties (e.g. log-law for the mean velocity profile, stability correction functions in stratified flows, relations between the turbulent Prandtl number and the gradient Richardson number, etc...). The talk reviews the theoretical aspects associated with current spectral links and co-spectral budget models and highlights their prospects to offer derivations of commonly used expressions originating from experiments and observations. These approaches are beginning to delineate contours of a unified fluctuation-dissipation like relation that may describe conveyance and mixing laws by turbulence independently developed in disjointed disciplines. A practical outcome is that the many semi-empirical formulae now in use in micrometeorology, hydrology, hydraulics, and geomorphology/sediment transport may finally profit from developments and advances in turbulence theory. As illustration, the talk will focus on connections between spectral descriptions of turbulence and the mean velocity profile in wall-bounded flows. The co-spectrum is first derived using a standard model for the wall normal velocity variance and a linear Rotta-like return-to-isotropy closure scheme modified to include the isotropozition of the production for the pressure-strain effects scale-by-scale. For analytical tractability, the wall-normal velocity spectrum considered here is idealized and includes only two regimes: large scales where energy is uniformly distributed (resembling splashing effects due to the presence of the wall) and an inertial subrange regime characterized by a -5/3 power-law scaling predicted from Kolmogorov's theory. The transition between the two regimes is assumed to occur at a wavenumber that scales with distance from the wall resulting in a spectrum that is continuous but not smooth. The derived co-spectrum is shown to exhibit a -7/3 scaling in the inertial subrange and gradually flattens out at large scales. The approach also provides a relation between well-established constants such as the von Karḿan and Kolmogorov constants, and the Rotta constant known to vary with the flow configuration. Depending on the choices made about small-scale intermittency corrections and the energy injection mechanism into the inertial cascade, the logarithmic mean velocity profile or a power-law profile with an exponent that depends on the intermittency correction can be derived for near infinite Reynolds number. This finding offers a new perspective on a long standing debate about the shape of the mean velocity profile in the equilibrium region. The effects of finite Reynolds number are then introduced by accounting for the ratio of the integral scale to the Kolmogorov micro-scale. For these cases, the co-spectral budget model suggests that the shape of the mean velocity profile is logarithmic but is amended by power-law corrections that vary with the Reynolds number. The same co-spectral budget is then generalized to include the effects of stable stratification so as to predict the mixing efficiency associated with eddy diffusivity for heat, or equivalently the turbulent Prandtl number. This example is selected because stably stratified flows are fraught with complex dynamics originating from the scalewise interplay between shear generation of turbulence and its dissipation by density gradients. The co-spectral budget approach recovers the near-universal relation between the turbulent Prandtl number and the gradient Richardson number, which encodes the relative importance of buoyancy dissipation to mechanical production of turbulent kinetic energy. This relation is shown to be derivable solely from the co-spectral budgets for momentum and heat fluxes with no tunable constants. The only constants that arise in the aforementioned derivation are the Kolmogorov constant for the one-dimensional vertical velocity energy spectrum, the Kolmogorov-Obukhov-Corrsin constant for the temperature spectrum, the Rotta constant, and a constant linked to the isotropization of the production that was predicted from Rapid Distortion Theory. Refinements to the linear Rotta closure scheme that include contributions from second and third invariants scale-by-scale are possible but are outside the scope of this presentation.