VELOCITY-ACCELERATION STRUCTURE FUNCTION IN TWO-DIMENSIONAL DECAYING TURBULENCE$^{\bf 1)}$

Second order velocity-acceleration structure function (VASF) is related to energy or enstrophy scale-to-scale transfer both in two-dimensional (2D) and three-dimensional (3D) turbulence, whose sign indicates the transfer direction. In 3D turbulence, energy transfers to smaller scale which results in negative VASF. In 2D turbulence, energy transfers to larger scale and enstrophy transfers to smaller scale, so the VASF are supposed to be positive in both inverse energy cascade range and direct enstrophy cascade range. However, comparing the abundant VASF researches in 3D turbulence, the sign of VASF in 2D turbulence is still lack of identification in neither experiment nor simulation. In this work, we give a general derivation which points out that apart from the scale-transfer term, the inhomogeneous term will also affect the VASF in spatial inhomogeneous turbulence. A commonly-used spatial inhomogeneous turbulence is that the turbulence below the turbulence generating device (such as comb) in wind tunnel or water tunnel. As flowing downstream, the turbulence intensity will decay, which brings the spatial inhomogeneity. We built a 2D decaying setup based on vertical soap-film flow, and performed particle tracking to measure VASF and its two components. Results show that the scale-transfer term is positive, but the inhomogeneous term is negative and dominates the VASF. As a result, the VASF is negative and lose its significance to identify the enstrophy transfers direction. Thus in similarly decaying turbulence, such as wind tunnel, water tunnel, sink et al., we shouldn't ignore the inhomogeneous term anymore. Finally, we discuss the dispersion process and find the slower dispersion is owing to the negative VASF.