The length distribution of streamline segments in homogeneous isotropic decaying turbulence

Abstract

Based on the profile of the absolute value u of the velocity field ui along streamlines, the latter are partitioned into segments at their extreme points as proposed by Wang [J. Fluid Mech. 648, 183–203 (2010)]10.1017/S0022112009993041. It is found that the boundaries of all streamline segments, i.e., points where the gradient projected in streamline direction ∂u/∂s vanishes, define a surface in space. This surface also contains all local extreme points of the scalar u-field, i.e., points where the gradient in all directions of the field of the absolute value of the velocity and thereby those of the turbulent kinetic energy (k = u2/2, where k is the instantaneous turbulent kinetic energy) vanishes. Such points also include stagnation points of the flow field, which are absolute minimum points of the turbulent kinetic energy. As local extreme points are the ending points of dissipation elements, an approach for space-filling geometries in turbulent scalar fields, such elements in the turbulent kinetic energy field also end and begin on the surface and the temporal evolution of dissipation elements and streamline segments are intimately related. Streamline segments by construction evolve both morphologically and topologically. A morphological evolution of a streamline segment corresponds to a continuous deformation when it is subject to stretching or compression and thus also implies a continuous evolution of the arclength l with time. Such an evolution does not change the number of the overall streamline segments and hence does not involve counting. On the other hand a topological evolution corresponds to either a cutting of a large segment into smaller ones or a connection of two smaller ones to form a larger segment. Such a process changes the number of segments and thus involves counting. This change of integer variables (i.e., counting) yields discrete jumps in the length of the streamline segment which are discontinuous in time. Following the terminology by Schaefer et al. [“Fast and slow changes of the length of gradient trajectories in homogenous shear turbulence,” in Advances in Turbulence XII, edited by B. Eckhardt (Springer-Verlag, Berlin, 2009), pp. 565–572] we will refer to the morphological part of the evolution of streamline segments as slow changes while the topological part of the evolution is referred to as fast changes. This separation yields a transport equation for the probability density function (pdf) P(l) of the arclength l of streamline segments in which the slow changes translate into a convection and a diffusion term when terms up to second order are included and the fast changes yield integral terms. The overall temporal evolution (morphological and topological) of the arclength l of streamline segments is analyzed and associated with the motion of the above isosurface. This motion is diffusion controlled for small segments, while large segments are mainly subject to strain and pressure fluctuations. The convection velocity corresponds to the first order jump moment, while the diffusion term includes the second order jump moment. It is concluded, both theoretically and from direct numerical simulations (DNS) data of homogeneous isotropic decaying turbulence at two different Reynolds numbers, 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 \documentclass[12pt]{minimal}\begin{document}$Re_{\lambda }^{-1/2}$\end{document}Reλ−1/2. Its 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. The interplay of viscous drift acting on small segments and linear strain acting on large segments yield, as it has already been concluded for dissipation elements, that the mean length of streamline segments should scale with Taylor microscale.