Joint multi-scale statistics of longitudinal and transversal increments in small-scale wake turbulence

Abstract

We analyse the relationship of longitudinal and transversal 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 Kármán equation is derived. We discuss in detail the analysis and show that the estimated equation can describe the statistics of the turbulent cascade. A remarkable result is that the main differences between longitudinal and transversal 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 Kármán equation and gives 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).