Local Dynamics and Statistics of Streamline Segments in Fluid Turbulence

Abstract

Based on local extreme points of the absolute value $u$ of the velocity field $u_{i}$, streamlines are partitioned into segments as proposed by Wang (J. Fluid. Mech. 648:183–203, 2010). The temporal evolution of the arc length l of streamline segments is analyzed and associated with the motion of the isosurface defined by all points on which the gradient in streamline direction $\partial u/\partial s$ vanishes. This motion is diffusion controlled for small segments, while large segments are mainly subject to strain and pressure influences. Due to the non-locality of streamline segments, their temporal evolution is not only a result of slow but also of fast changes, which differ by the magnitude of the jump $\varDelta l$ that occurs within a small time step $\varDelta t$. The separation of the dynamics into slow and fast changes allows the derivation of a transport equation for the probability density function (pdf) P(l) of the arc length l of streamline segments. While slow changes in the pdf transport equation translate into a convection and a diffusion term when terms up to second order are included, the dynamics of the fast changes yield integral terms. The convection velocity corresponds to the first order jump moment, while the diffusion term includes the second order jump moment. It is theoretically and from DNS data of homogeneous isotropic decaying turbulence at two different Reynolds numbers concluded that the normalized first order jump moment is quasi-universal, while the second order one is proportional to the inverse of the square root of the Taylor based Reynolds number $Re_{\lambda }^{ - 1/2}$. It’s inclusion thus represents a small correction in the limit of large Reynolds numbers. Numerical solutions of the pdf equation yield a good agreement with the pdf obtained from the DNS data. It is also concluded on theoretical grounds that the mean length of streamline segments scales with the Taylor microscale rather than with any other turbulent length scale, a finding that can be confirmed from the DNS.