Compressive and extensive strain along gradient trajectories

Based on direct numerical simulations of forced turbulence, shear turbulence, decaying turbulence, a turbulent channel flow as well as a Kolmogorov flow with Taylor based Reynolds numbers Reλ between 69 and 295, the normalized probability density function of the length distribution P̃(l̃) of dissipation elements, the conditional mean scalar difference < Δk | l > at the extreme points as well as the scaling of the two-point velocity difference along gradient trajectories < Δun> are studied. Using the field of the instantanous turbulent kinetic energy k as a scalar, we find a good agreement between the model equation for P̃(l̃) as proposed by Wang and Peters (2008) and the results obtained in the different DNS cases. This confirms the independance of the model solution from both, the Reynolds number and the type of turbulent flow, so that it can be considered universally valid. In addition, we show a 2/3 scaling for the mean conditional scalar difference. In the second part of the paper, we examine the scaling of the conditional two-point velocity difference along gradient trajectories. In particular, we compare the linear s/τ scaling, where τ denotes an integral time scale and s the separation arclength along a gradient trajectory in the inertial range as derived by Wang (2009) with the s · a∞ scaling, where a∞ denotes the asymtotic value of the conditional mean strain rate of large dissipation elements.