Fig.1 (a) Two-dimensional water tunnel setup, $x$ axis is the gravity direction, $y$ axis is the horizontal direction, $z$ axis is the film thickness direction. The plane of soap film is consisted by two nylon ropes whose diameter are 0.32mm. To create turbulent flow, we insert a comb in the upstream. The laser illuminates the downstream regions below the comb, whose flow trajectories are recorded by two high speed cameras. (b) The real image of comb equipment. The upper cylinders make up a comb, the vertical bright lines are nylon ropes, and the soap film flows down between the ropes. The whole flow are in green background, owing the scatter from 532nm laser

Fig.2 (a) Using Taylor frozen flow hypothesis, we calculate the relationship between $D_{ii}$ and $r$ in local downstream location $x\pm 0.2$mm. Different colors correspond to different separation below the comb $x/M$. The capture window of two cameras is 100mm =20M in each time. To obtain a wider regions statistics, we set the relative separation between camera and comb as $S/M=12$, 20, 30 and 50. (b) The relationship between $-\langle \delta u_i \delta a_i \rangle$ and $r$, which shows $r^{1.5}$ power law in direct enstrophy range. The colors mean different downstream locations which are the same with (a). The vertical solid line lies in $r=0.35 M$, whose cross points with different colorful lines are $-\langle \delta u_i \delta a_i \rangle$ and $D_{ii}$ in $r=0.35 M$

Fig.3 (a) In scale $r/M=0.35$, the red circles are VASF which have been multiplied by $-1$. Green squares are inhomogeneous term which is the second term of VASF. Blue stars are the scale-to-scale transfer term which is the third term of VASF. We plot their values in different downstream location $tU/M$. (b) The ordinate is the logarithmic value of $-\langle \delta u_i \delta a_i \rangle$, the abscissa is the logarithmic value of $-\langle (U_n/2){\rm \partial} D_{ii} /{\rm \partial} X_n\rangle$, most of the data lies around the contour line

Fig.4 (a) In the region which lies in $20M\sim40 M$ below the comb, the dispersion relationship of two particles $\langle (R(t)-R_0)^2 \rangle $ with time. Different colors correspond to different initial separations $R_0$. The black line in sub-figure is $(t/\tau_{\rm F})^2$, which is not the fitting curve. (b) Considering the effect of VASF, the compensated dispersion relationships