Finite dimensional aspects of turbulent flows

The time-asymptotic behavior of the solutions of the three-dimensional Navier-Stokes equations of turbulent flow is investigated analytically, summarizing the results obtained by Constantin et al. (1984). The intrinsically finite number of degrees of freedom (DOFs) is studied both in terms of determining modes and in terms of the fractal dimension of the attractors, and the relationship between the number of DOFs and the Reynolds number (R) of the flow is explored. It is found that the conventional estimate (varying as R to the 9/4ths) requires a priori knowledge of the spectrum of homogeneous isotropic turbulence and is applicable only at high R, and a more conservative estimate which varies as R cubed and is independent of spectrum knowledge is proposed.