Static structure function of turbulent flow from the Navier-Stokes equations

The velocity field of a fully developed isotropic turbulent flow is decomposed into a smoothed fieldu _i ^(r) and a strongly fluctuating part _i ^(r) depending on a lengthr which is varied through all scales. The eddies described byu _i ^(r) loose their energy either by direct dissipation or by energy transfer to the _i ^(r) -eddies. Both contributions can be traced back to the second order static structure functionD(r) and the Lagrangian time correlation function. The latter can also be evaluated in terms ofD(r). The energy balance then gives an integro-differential equation forD(r) which determinesD(r) uniquely. The solution is not only in agreement with the scaling behaviour in the viscous and inertial subranges as predicted by dimensional arguments, but it also gives the correct transition between the two regimes. A comparison with experimental data is offered.