On the scaling of the mean length of streamline segments in various turbulent flows

The geometrical properties of streamline segments (Wang, 2010 [1], ) and their bounding surface (Schaefer et al., 2012 [2]) in direct numerical simulations (DNS) of different types of turbulent flows at different Reynolds numbers are reviewed. Particular attention is paid to the geometrical relation of the bounding surface and local and global extrema of the instantaneous turbulent kinetic energy field. Also a previously derived model equation for the normalized probability density of the length of streamline segments is reviewed and compared with the new data. It is highlighted that the model is Reynolds number independent when normalized with the mean length of streamline segments yielding that the mean length lm plays a paramount role as the only relevant length scale in the pdf. Based on a local expansion of the field of the absolute value of the velocity u along the streamline coordinate a scaling of the mean size of extrema of u is derived which is then shown to scale with the mean length of streamline segments. It turns out that lm scales with the geometrical mean of the Kolmogorov scale η and the Taylor microscale λ so that lm∝(ηλ)1/2. The new scaling is confirmed based on the DNS cases over a range of Taylor based Reynolds numbers of Reλ=50–300.