= Solution
The first similarity hypothesis of <Kolmogorov 1941 theory> states that at sufficiently large <Reynolds number>, the small-scale statistics in the <equilibrium range> are locally universal and determined by the mean <turbulent kinetic energy dissipation rate> $\epsilon$ and <kinematic viscosity> $\nu$, together with the separation or wavenumber being measured. Large-scale forcing details are excluded only at scales well below the <integral scale of turbulence>.
These parameters define the <Kolmogorov microscales>
$$
\eta=(\nu^3/\epsilon)^{1/4},\qquad
u_\eta=(\nu\epsilon)^{1/4},\qquad
\tau_\eta=(\nu/\epsilon)^{1/2}.
$$
<Dimensional analysis> gives universal functions $F$ and $G$ with
$$
\boxed{S_2(r)=u_\eta^2F(r/\eta)=(\nu\epsilon)^{1/2}F(r/\eta),}
$$
$$
\boxed{E(k)=u_\eta^2\eta\,G(k\eta)
=(\epsilon\nu^5)^{1/4}G(k\eta).}
$$
The dimensional factors follow because $S_2$ has units of <velocity> squared, whereas $E$ has units of <velocity> squared times length. The universal equilibrium forms include viscous effects; they do not impose a power law through the <dissipation range>.
The second similarity hypothesis asserts independence of <kinematic viscosity> at scales large compared with $\eta$ but still small compared with the integral scale. In this <inertial range>, $S_2$ can depend only on $\epsilon$ and $r$. Balancing time and length dimensions forces powers $2/3$ and $2/3$. The same calculation for $E$, using $k$ instead of $r$, forces powers $2/3$ and $-5/3$. Hence
$$
\boxed{S_2(r)=C_2\epsilon^{2/3}r^{2/3},\qquad\eta\ll r\ll\ell,}
$$
$$
\boxed{E(k)=C_K\epsilon^{2/3}k^{-5/3},\qquad\ell^{-1}\ll k\ll\eta^{-1}.}
$$
The constants $C_2$ and $C_K$ are universal within the hypotheses and the stated normalization. They are related by the exact <structure-function spectral filter>, rather than by assigning all energy to one sharp scale.
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