gives the Townsend-Perry constant $$\label{eq:TP}
A_1=\frac{2C \rho^{1/2} y\langle |w_x|\rangle}{\sqrt{\langle \tau_0 \rangle}},$$ and the generalized Townsend-Perry constants $$\label{eq:GTP}
A_p=\frac{2C \rho^{1/2} y \langle |w_x|^{p}\rangle^{1/p}}{\sqrt{\langle \tau_0 \rangle}},$$ by use of Eq. (\[eq:wpder\]). This justifies the form of the stress tensor assumed in Ref. [@BC16] and used in Ref. [@KBK19]. Finally, we get the expressions $$A_1 = K \langle |w(x+y)-w(x-y)|\rangle$$ and $$A_p = K \langle |w(x+y)-w(x-y)|^{p} \rangle^{1/p},$$ where $K$ is a constant and this produces the relationship between the Townsend-Perry and the generalized Townsend-Perry constants and the structure function of turbulence, see Ref. [@BB211; @BB314; @KBBS17], used in Ref. [@BC16; @KBK19], $$\label{eq:TPstru1}
A_1 = K C_1|y^*|^{\zeta_1},$$ $$\label{eq:TPstru2}
A_2 = K C^{1/2}_2|y^*|^{\zeta_2/2},$$ and $$\label{eq:TPstrup}
A_p = K C^{1/p}_{p}|y^*|^{\zeta_{p}/p},$$ where $-y\leq y^* \leq y$. Considering the ratio, washes out the constant $K$, $$\label{eq:ratio}
\frac{A_p}{A_2}= \frac{C^{1/p}_{p}}{C^{1/2}_2}|y^*|^{\zeta_{p}/p-\zeta_2/2},$$ where the $C_p$s are the Kolmogorov-Obukhov coefficients of the structure functions from Ref. [@BB211; @BB314; @KBBS17]. The last ratio was used in Ref. [@KBK19] to get agreement between experimental data and theory.

![The average of the MVP as a function of $log(y)$, where $y$ is the distance from the wall. Comparison of experimental data with theory (black line). (a) Theoretical curve is given by an $I$-integral that interpolates between the $k_x^{-5/3}$ 