Bounds for second order structure functions and energy spectrum in turbulence

In this paper we derive upper bounds for the second order structure function as well as for the Littlewood–Paley energy spectrum — an average of the usual energy spectrum E(k). While the upper bound results are consistent with a Kolmogorov type dependence on wave number k, the bounds do not involve the usual dissipation rate ε. Instead the bounds involve a dissipative quantity ε̂ similar to ε but based on the L3 average of ∇u. Numerical computations for a highly symmetric flows with Taylor microscale Reynolds numbers up to Rλ=155 are found to be consistent with the proposition that a relation in the inertial regime of the type E(k)∼Ĉε̂2/3k−5/3 holds with constant Ĉ.