On a multi-scale approach to analyze the joint statistics of longitudinal and transverse increments experimentally in small scale turbulence

We analyze the relationship of longitudinal and transverse increment statistics measured in isotropic small-scale turbulence. This is done by means of the theory of Markov processes leading to a phenomenological Fokker-Planck equation for the two increments from which a generalized Karman equation is derived. We discuss in detail the analysis and show that the estimated equation can describe the statistics of the turbulent cascade. A remarkably result is that the main differences between longitudinal and transverse increments can be explained by a simple rescaling-symmetry, namely the cascade speed of the transverse increments is 1.5 times faster than that of the longitudinal increments. Small differences can be found in the skewness and in a higher order intermittency term. The rescaling symmetry is compatible with the Kolmogorov constants and the Karman equation and give new insight into the use of extended self similarity (ESS) for transverse increments. Based on the results we propose an extended self similarity for the transverse increments (ESST).