Asymptotic Exponents from Low-Reynolds-Number Flows

We present detailed studies of turbulence in the crossover region between the inertial and viscous ranges of turbulence. The work is based on two results: (1) the scaling exponents $\rho_n$ of the moments of velocity derivatives $\langle(\partial u/\partial x)^n\rangle$, with respect to the large scale Reynolds number $Re$, can be expressed in terms of the inertial-range scaling exponents $\xi_n$ of the longitudinal structure functions $S_n(r)=\langle(\delta_r u)^n\rangle$. (2) High-resolution direct numerical simulations of isotropic and homogeneous turbulence in a periodic box, in which sub-Kolmogorov grid has been employed to accurately resolve the analytic parts of the structure functions, are conducted. They show that the derivative moments for orders $0\le n\le 8$, obtained from relatively low Reynolds number flows with Taylor-microscale Reynolds number $10\leq R_{\lambda} \leq 63$, are represented well as powers of the Reynolds number. The exponents $\rho_{n}$ in these flows, though exhibiting no developed inertial range, agree closely with the exponents $\xi_n$ ($0\leq n\leq 17$) corresponding to the inertial range of high-Reynolds-number fully developed turbulence. The existence of a whole range of {\em local} dissipation scales, rather than a single Kolmogorov dissipation scale, is discussed.