Enhanced inverse-cascade of energy in the averaged Euler equations

Abstract

For a particular choice of the smoothing kernel, it is shown that the system of partial differential equations governing the vortex-blob method corresponds to the averaged Euler equations. These latter equations have recently been derived by averaging the Euler equations over Lagrangian fluctuations of length scale $\a$, and the same system is also encountered in the description of inviscid and incompressible flow of second-grade polymeric (non-Newtonian) fluids. While previous studies of this system have noted the suppression of nonlinear interaction between modes smaller than $\a$, we show that the modification of the nonlinear advection term also acts to enhance the inverse-cascade of energy in two-dimensional turbulence and thereby affects scales of motion larger than $\a$ as well. This latter effect is reminiscent of the drag-reduction that occurs in a turbulent flow when a dilute polymer is added.

Figures (3)

• Figure 1: The evolution of kinetic energy E with time for $k_\alpha=\infty$ (solid line), $k_\alpha=42$ (dotted line), $k_\alpha=21$ (dashed line), and $k_\alpha=14$ (dot-dashed line). An increase in $\alpha$, for identical forcing and dissipation, results in an overall reduced viscous behavior. In the inset is shown the evolution of enstrophy Z for the same time interval and for the same four values of $\alpha$ and with the same line types as for kinetic energy. While there is a significant difference between zero and non-zero $\alpha$ cases, the dependence on the actual value of $\alpha$ itself is rather weak.
• Figure 2: Stationary wavenumber-energy spectra (log-log scale) for the forced-dissipative simulations of the averaged Euler equations with zero and nonzero $\alpha$. $k_\alpha=\infty$ (solid line), $k_\alpha=42$ (dotted line), $k_\alpha=21$ (dashed line), and $k_\alpha=14$ (dot-dashed line). The insert shows the same plot with a linear-linear scale for the first ten wavenumbers. The enhanced inverse cascade of energy and the suppressed energy level at smaller scales with increasing $\alpha$ is evident.
• Figure 3: The spectra for the four cases in Fig. \ref{['spec1']} are replotted along with the spectra for the same four cases with resolution reduced by 25% and 50% in each direction. The sets of spectra for each $\alpha$ are offset by a decade each to improve clarity. The degree of non-resolution of the flow with the reduced resolution is indicated by the difference between that spectrum and the spectrum for the fully resolved case. With a 25% reduction in resolution, the flows are almost resolved, while with a 50% reduction, the flows are not fully resolved anymore. The degree of non-resolution is independent of $\alpha$ to the lowest order.