Scale-space energy density for inhomogeneous turbulence based on filtered velocities

The energy spectrum is commonly used to describe the scale dependence of turbulent fluctuations in homogeneous isotropic turbulence. In contrast, one-point statistical quantities, such as the turbulent kinetic energy, are mainly employed for inhomogeneous turbulence models. Attempts have been made to describe the scale dependence of inhomogeneous turbulence using the second-order structure function and two-point velocity correlation. However, unlike the energy spectrum, expressions for the energy density in the scale space fail to satisfy the requirement of being non-negative. In this study, a new expression for the scale-space energy density based on filtered velocities is proposed to clarify the reasons behind the negative values of the energy density and to obtain a better understanding of inhomogeneous turbulence. The new expression consists of homogeneous and inhomogeneous parts; the former is always non-negative, while the latter can be negative because of the turbulence inhomogeneity. Direct numerical simulation data of homogeneous isotropic turbulence and a turbulent channel flow are used to evaluate the two parts of the energy density and turbulent energy. It was found that the inhomogeneous part of the turbulent energy shows non-zero values near the wall and at the centre of a channel flow. In particular, the inhomogeneous part of the energy density changes its sign depending on the scale. A concave profile of the filtered-velocity variance at the wall accounts for the negative value of the energy density in the region very close to the wall.

Inhomogeneous Turbulent Dispersion across the Nearshore Induced by Surfzone Eddies

In various oceanic regions, drifter-derived diffusivities reach a temporal maximum and subsequently decrease. Often, these are regions of inhomogeneous eddies, however, the effect of inhomogeneous turbulence on dispersion is poorly understood. The nearshore region (spanning from the surfzone to the inner shelf) also has strong cross-shore inhomogeneous turbulence. Nearshore Lagrangian statistics are estimated from drifter trajectories simulated with a wave-resolving two-dimensional Boussinesq model with random, normally incident, and directionally spread waves. The simulation is idealized and does not include other (wind, tidal, Coriolis) processes. The eddy field cross-shore inhomogeneity affects both the mean cross-shore drift and cross- and alongshore diffusivities. Short-time diffusivities are locally ballistic, and the mean drift is toward the eddy velocity variance maximum. The diffusivities reach a maximum and subsequently decrease, that is, are subdiffusive. The diffusivity maximum and time to maximum are parameterized taking into account the eddy field inhomogeneity. At long times, the cross- and alongshore diffusivities scale as t−1/2 and t−1/4, respectively, which is related to the offshore decay of the eddy intensity. A diffusion equation, with a space-dependent Fickian diffusivity related to the eddy velocity variance, reproduced the short-, intermediate-, and long-time behavior of the mean drift and cross-shore diffusivity. The small Middleton parameter, indicating fixed float dispersion, suggests the Eulerian time scale can parameterize the Lagrangian time scale in this region. Although this idealized simulation had no mean currents, and thus no shear dispersion or mixing suppression, inhomogeneous turbulence effects may be relevant in other regions such as the Antarctic Circumpolar Current (ACC) and western boundary current extensions.