Scaling of the two-point velocity difference along scalar gradient trajectories in fluid turbulence

In the context of dissipation element analysis of scalar fields in turbulence [L. Wang and N. Peters, J. Fluid Mech. 608, 113 (2008)], the elongation of elements by the velocity difference at the minimum and maximum points was found to increase linearly with the length of an element. This paper attempts to provide a theoretical basis for this finding by analyzing two-point properties along the gradient trajectories, of which dissipation elements consist. An equation of the two-point correlation function for the product of the scalar gradient along the same trajectory can be obtained. Similar to the idea of deriving Kolmogorov's 4/5 law, there exist a scaling in the inertial range for the velocity difference, however, not same as Kolmogorov's 1/3 scaling. Specifically, by conditioning on gradient trajectories we obtain a linear relation between the velocity difference and the arclength between two points on the same trajectory. Results from direct numerical simulation (DNS) show satisfactory agreement with the theoretical prediction. This result and the derivation thereof may generally be helpful for broad stream of similar statistical and scaling studies of turbulent flows.