= Solution
Assume statistically steady, three-dimensional <turbulence> at large <Reynolds number>, with local <statistical homogeneity> and <statistical isotropy> at scales small compared with the forcing scale $L$. The <Kolmogorov 1941 theory> additionally assumes a local <energy cascade>, constant mean <energy flux> per unit mass $\epsilon$, and negligible <viscous dissipation> within the <inertial range>. Its dimensional form neglects corrections from <internal intermittency>.
At separation $l$, let $\delta u_l$ be a typical <velocity increment>. The <eddy turnover time> is $\tau_l\sim l/\delta u_l$. Constant transfer rate then gives $\epsilon\sim\delta u_l^2/\tau_l\sim\delta u_l^3/l$, hence
$$
\boxed{\delta u_l\sim(\epsilon l)^{1/3},\qquad E(k)=C_K\epsilon^{2/3}k^{-5/3}.}
$$
Here $E(k)$ is the one-dimensional <turbulent energy spectrum>, obtained from $\delta u_l^2\sim kE(k)$ with $k\sim l^{-1}$. The scaling holds only for
$$
\boxed{\eta\ll l\ll L,\qquad L^{-1}\ll k\ll\eta^{-1},\qquad \eta=(\nu^3/\epsilon)^{1/4}.}
$$
The <Kolmogorov length scale> follows by setting the scale-dependent <Reynolds number> $\delta u_l l/\nu$ to one. The associated <Kolmogorov microscales> give $\tau_\eta=(\nu/\epsilon)^{1/2}$. At larger scales the forcing matters; at smaller scales <kinematic viscosity> matters.
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