Multispectral Reduction of Two-Dimensional Turbulence

Abstract

Turbulence is a chaotic motion of fluid that can be described by the Navier--Stokes equations or even highly simplified shell models. Under the continuum limit, standard shell models of turbulence are shown to reduce to a common evolution equation that reproduces many predictions of the classical Kolmogorov theory. In the spectral domain, the quadratic advective nonlinearity of the Navier--Stokes equations appears as a convolution, which is often calculated using pseudospectral collocation. An implicit dealiasing method, which removes spurious contributions from wave beating in these convolutions more efficiently than conventional dealiasing techniques, is investigated. Even with efficient dealiasing, the simulation of highly turbulent flow is still a formidable task. Decimation schemes such as spectral reduction replace the many degrees of freedom in a turbulent flow by a limited set of representative quantities. A new method called multispectral reduction is proposed to overcome a significant drawback of spectral reduction: the requirement that all scales be decimated uniformly. Multispectral reduction, which exploits a hierarchy of synchronized spectrally reduced grids, is applied to both shell models and two-dimensional incompressible turbulence.