Curvature statistics of streamlines in various turbulent flows

Abstract

Geometrical features of turbulent flows are analyzed by studying the curvature field of streamlines based on the instantaneous fluctuating velocity field of four different direct numerical simulations (DNS) at Taylor-based Reynolds numbers in the range Re _λ = 50–300. Among the flows are two decaying homogeneous isotropic flows, one isotropic forced flow and the other homogeneous shear flow. Two different measures of curvature are extracted, the first one is the geometrical curvature of streamlines based on Frenet’s formulas, and the second one is the Gaussian curvature describing the divergence of streamlines. The two curvature fields are found to be related to each other by the divergence of the directional tensor of streamlines, which is also related to the total acceleration along streamlines. The latter identity allows the identification of the convective term in the Navier–Stokes equations, as the term determining the curvature-related geometrical features of streamlines. Based on the random sweeping hypothesis, a scaling of the standard deviation of the two curvatures of streamlines in turbulent flows with the inverse of the Taylor microscale is obtained. The curvature fields are in a first step analyzed via their probability density functions (pdfs) of the fluctuating fields (including their sign), and a normalization with the standard deviation yields a good collapse of the curves for different Reynolds numbers and flow types. Both, the positive and negative tails of the pdf display a pronounced algebraic tail with a decay exponent of −4. To analyze the curvatures as geometrical features of streamlines, the pdf of their absolute values normalized with their respective mean values are examined, and the curves are found to collapse well for the different Reynolds numbers. All pdfs again display an algebraic tail for large values of the curvature with a scaling exponent of −4. This result is theoretically explained based on the expression for the Gaussian curvature, and turns out to be related to the regions in the flow close to stagnation points where streamlines get strongly diverted. It turns out that at the origin the pdf of the Gaussian curvature is dominated by regions where the gradient of the absolute value in streamline direction u_s vanishes. u_s =0, However, it defines an isosurface in which all local extreme points of the field of the absolute value of the velocity u as well as those of the turbulent kinetic energy k lie. As stagnation points are absolute minima of the kinetic energy field, these points too lie in the isosurface. The topology of the isosurface is analyzed in the vicinity of stagnation points, and it is found that independent of the type of stagnation point, the isosurface is locally always a degenerated quadric surface of cone type, a conjecture that is verified based on the comparison of the analytical expansion and the DNS data.

Keywords:

• streamlines
• turbulent flows
• curvature
• probability density functions
• scaling laws
• local geometry
• stagnation points

Acknowledgments

[Acknowledgements]

This work was funded by the Deutsche Forschungsgemeinschaft under Grant Pe 241/38-2 and the Gauss Center for Supercomputing. The author would like to thank Prof. N. Peters for his continuous support and all the valuable discussions and help he provided. In addition, the author would like to thank M. Gampert for extensive discussions on the topic. Finally, special thanks go to J.H. Goebbert for his continuous support concerning numerical procedures and in particular his support regarding the visualizations in Section 5.