[@BC16] and [@KBK19], to compute the Townsend-Perry constants, in the inertial (log) layer. These computations use the first structure function $S_1$ of turbulence and we explain how they are performed, see [@BC16] and [@KBK19] for more information. The computed Townsend-Perry constants are listed in Table I.

The first structure function of turbulence is, see [@KBBS17], $$\begin{aligned}
    &&E(\vert u(x,t)-u(y,t)\vert)=S_1(x,y,t){\nonumber}\\
     &&=\frac{2}{C}\sum_{k\in\mathbb{Z}^3\backslash\{0\}}\frac{\vert d_k\vert(1-e^{-\lambda_kt})}{\vert k\vert^{\zeta_1}+\frac{4\pi^2\nu}{C}\vert k\vert^{\zeta_1+\frac{4}{3}}}\vert \sin(\pi k\cdot(x-y))\vert,\end{aligned}$$ where the Reynolds number dependence enters through the viscosity $\nu$, and $E$ denotes the expectation (ensamble average). To get the Kolmogorov-Obukhov coefficients, $C_p$ in $$S_p(r, \infty) \sim C_p r^{\zeta_p},$$ for the lag variable $r$ small, and $\zeta_p$ the scaling exponents, we send $t$ to $\infty$ in the above formulas and project onto the longitudinal lag variable ${\bf r} = (r,0,0)$. For $p=1$ this becomes $$\begin{aligned}
&&S_1 \sim \frac{2\pi^{\zeta_1}}{C} \sum_{k\neq 0} \frac{|d_k |}{(1+\frac{4\pi^2\nu}{C}|k|^{4/3})} r^{\zeta_1}{\nonumber}\\
&=&\frac{4\pi^{\zeta_1}}{C} \sum_{k = 1}^\infty \frac{a}{(a^2+k^m)(1+\frac{4\pi^2\nu}{C}|k|^{4/3})} r^{\zeta_1},\end{aligned}$$ see [@KBBS17], where $\zeta_1 = 0.37$, see [@BB211]. Now we use the values for $\nu$ in Table 1 in [@KBK19], and the corresponding values for $a,\ m$ and $C$ from Table 3 in the same paper. The Reynolds numbers, 6430, 10,770, 15,740 and 19,670 are close enough to ours 6000, 10,000, 14,500, and 20,000, that we can 