Far-dissipation range of turbulence.

The very small scales of isotropic, Navier-Stokes turbulence at Reynolds number Rλ ≈ 15 are studied by high-resolution direct numerical simulation (DNS) and by integration of the direct-interaction (DIA) equations. The DNS follows the tail of the energy spectrum over more than thirty decades of magnitude. The energy spectrum in the far-dissipation range 5kd < k < 10kd is well-fitted by k α exp(−ck/kd), where kd is the Kolmogorov dissipation wavenumber, α ≈ 3.3 and c ≈ 7.1. For values of m that emphasize the fardissipation range, the fields (−∇2)mu exhibit strong spatial intermittency, associated with gentle spatial variations of the lower-k part of the velocity field. DIA analysis gives a prefactor k and an exponential decay more rapid than DNS. Averaging over an ensemble of DIA solutions, suggested by the observed intermittency, removes some of the discrepancy.