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We experimentally study the temporal second-order structure functions for integer powers of turbulent fluid velocity fluctuations $S_2^m(N,\tau) \equiv \langle (\mathcal{U}^m(t+\tau) - \mathcal{U}^m(t))^2 \rangle$, in three dimensional (3D) and two dimensional (2D) turbulence. Here $\mathcal{U}^m(t) = (1/N) \Sigma_{i=1}^N u_i^m(t)$ is a composite time-series constructed by averaging the concurrent time-series ($u_i^m(t)$) sampled at $N$ spatially distributed Eulerian points. The $N = 1$ case has been extensively studied for velocity fluctuations ($m = 1$) and to a lesser extent for $m &gt; 1$. The averaging method in case of $N &gt; 1$ diverges from the Kolmogorov framework and has not been studied because fluctuations in $\mathcal{U}^m$ are expected to smooth with increasing $N$ leaving behind uninteresting large-scale mean flow information, but we find this is not so. We report the evolution of scaling exponents $\zeta_m(N)$ for $S_2^m(N, \tau) \sim \tau^{\zeta_m(N)}$ in going from a single ($N=1$) to a spatial average over several Eulerian points ($N &gt;&gt; 1$). Our 3D experiments in a tank with rotating jets at the floor show $\zeta_m(N=1) = 2/3$ for all $m$-values in agreement with prior results and evolves to an asymptotic value of $\zeta_m(N&gt;&gt;1) = 2m/3$. The evolution of $\zeta_m(N)$ follows the functional form $\zeta_m(N) = (2/3)[m - (m-1)\text{exp}(-N/N_0)]$, where $N_0 \sim 24 - 29$ points is the only fit parameter representing the convergence rate constant. Results for the 2D experiments conducted in a gravity assisted soap film in the enstrophy cascade regime are in sharp contrast with their 3D counterparts. Firstly $\zeta_m(N=1)$ varies polynomially with $m$ and asymptotes to a constant value at $m = 5$. Secondly, the evolution of $\zeta_m(N)$ is logarithmic $\zeta_m(N) = A + B \cdot \text{log}(N)$, where $A$ and $B$ are fit parameters and eventually deviates at large $N$ and asymptotes to $\zeta_m(N &gt;&gt; 1) = 2.0 \pm 0.1$ for all $m$. The starkly different convergence forms (exponential in 3D versus logarithmic in 2D) may be interpreted as a signature of inter-scale couplings in the respective turbulent flows by decomposing the two-point correlator for $\mathcal{U}^m$ into a self-correlation and cross-correlation term. In addition to aiding in the theoretical development, the results may also have implications for determination of resolution in 2D turbulence experiments and simulations, wind energy and atmospheric boundary layer turbulence.&#13.

Second order structure functions for higher powers of turbulent velocity