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 epsilon. Instead the bounds involve a dissipative quantity <(epsilon)over cap> similar to epsilon but based on the L-3 average of del u. Numerical computations for a highly symmetric flaws with Taylor microscale Reynolds numbers up to R-lambda = 155 are found to be consistent with the proposition that a relation in the inertial regime of the type E(k) similar to (C) over cap<(epsilon)over cap>(2/3)k(-5/3) holds with constant (C) over cap. (C) 1999 American Institute of Physics. [S1070-6631(99)00308-6].