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

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 fluctuating 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 satisfies a Liouville theorem and yields statistical equipartition solutions. However, if the wavenumber bins are of nonuniform size (as is desirable for efficiency), 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 stiffness of the spectrally reduced equations. The prospect of developing an implicit nonlinear integrator for this highly stiffened convection problem is examined.