Alexey Cheskidov, Charles R. Doering, Nikola P. Petrov
Energy dissipation in fractal-forced turbulence
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ABSTRACT.  The rate of energy dissipation in solutions of the body-forced 3-d 
incompressible Navier-Stokes equations is rigorously estimated with a focus 
on its dependence on the nature of the driving force. For square integrable 
body forces the high Reynolds number (low viscosity) upper bound on the 
dissipation is independent of the viscosity, consistent with the existence of a 
conventional turbulent energy cascade. On the other hand when the body force 
is not square integrable, i.e., when the Fourier spectrum of the force decays 
sufficiently slowly at high wavenumbers, there is significant direct driving 
at a broad range of spatial scales. Then the upper limit for the dissipation 
rate may diverge at high Reynolds numbers, consistent with recent experimental 
and computational studies of "fractal-forced'' turbulence.