A model equation for the joint distribution of the length and velocity difference of streamline segments in turbulent flows

Abstract

Streamlines recently received attention as natural geometries of turbulent flow fields. Similar to dissipation elements in scalar fields, streamlines are segmented into smaller subunits based on local extreme points of the absolute value of the velocity field u along the streamline coordinate s, i.e., points where the projected gradient in streamline direction us = 0. Then, streamline segments are parameterized using their arclength l between two neighboring extrema and the velocity difference Δ at the extrema. Both parameters are statistical variables and streamline segments are characterized by the joint probability density function (jpdf) P(l, Δ). Based on a previously formulated model for the marginal pdf of the arclength, P(l), which contains terms that account for slow changes as well as fast changes of streamline segments, a model for the jpdf is formulated. The jpdf's, when normalized with the mean length, lm, and the standard deviation of the velocity difference σ, obtained from two different direct numerical simulations (DNS) cases of homogeneous isotropic decaying and forced turbulence at Taylor based Reynolds number of Reλ = 116 and Reλ = 206, respectively, turn out to be almost Reynolds number independent. The steady model solution is compared with the normalized jpdf's obtained from DNS and it is found to be in good agreement. Special attention is paid to the intrinsic asymmetry of the jpdf with respect to the mean length of positive and negative streamline segments, where due to the kinematic stretching of positive segments and compression of negative ones, the mean length of positive segments turns out to be larger than the mean length of negative ones. This feature is reproduced by the model and the ratio of the two length scales, which turns out to be an almost Reynolds number independent, dimensionless quantity, is well reproduced. Finally, a relation between the kinetic asymmetry of streamline segments and the dynamic asymmetry of the pdf of longitudinal velocity gradients in turbulent flows, which manifests itself in a negative velocity gradient skewness, is established and it is theoretically shown that negative streamline segments are only smaller than positive ones, if the gradient is negatively skewed.