P.: Numerical challenges for turbulence computation: Statistical equipartition and the method of spectral reduction

Abstract. Numerical issues in the implementation of spectral reduction, a new method for the computation of statistical moments of homogeneous turbulence, are examined. The method implements a coarse graining in Fourier space and exploits the fact that statistical moments are much smoother functions of wave number than the underlying uctuating velocities. A notable feature of this turbulence model is the existence of a control parameter (bin size) that can be varied to increase the accuracy of the approximation. The inviscid version of spectral reduction satises a Liouville theorem and yields statistical equipartition solutions. However, if the wavenumber bins are of nonuniform size (as is desirable for eciency), an additional bin-dependent rescaling of time by the relative bin area must be introduced to obtain the correct equipartition. This rescaling of the time derivative term drastically increases the stiness of the spectrally reduced equations. The prospect of developing an implicit nonlinear integrator for this highly stiened convection problem is examined. Key words. homogeneous turbulence, statistical closures, spectral reduction, sti dierential equations AMS subject classications. 76F05, 34A6