### 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 u s = 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, l m , 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 Re λ = 116 and Re λ = 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.

Received 14 November 2012
Accepted 19 August 2013
Published online 22 November 2013

Acknowledgments:
This work was funded by the Deutsche Forschungsgemeinschaft (DFG) project PAK 213 and by the Gauss Center for Supercomputing in Jülich. In addition, the authors would like to thank J. H. Goebbert for his continuous support regarding numerics and visualization.

Article outline:

I. INTRODUCTION
II. DIRECT NUMERICAL SIMULATIONS
III. THE JOINT PDF MODEL EQUATION
IV. THE INNER STRUCTURE OF STREAMLINE SEGMENTS
V. DRIFT VELOCITIES IN PHASE SPACE
A. Drift in *l*-direction
B. Drift in Δ-direction
VI. MODEL VALIDATION
VII. KINEMATICS OF STREAMLINE SEGMENTS AND THE SKEWNESS OF VELOCITY GRADIENTS
A. The mean length of streamline segments and the velocity gradient skewness
B. Skewness of the mean gradient within streamline segments
VIII. CONCLUDING REMARKS

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