On velocity gradient dynamics and turbulent structure

Abstract

The statistics of the velocity gradient tensor A = ∇u, which embody the fine scales of turbulence, are influenced by turbulent 'structure'. Whilst velocity gradient statistics and dynamics have been well characterised, the connection between structure and dynamics has largely focused on rotation-dominated flow and relied upon data from numerical simulation alone. Using numerical and spatially resolved experimental datasets of homogeneous turbulence, the role of structure is examined for all local (incompressible) flow topologies characterisable by A. Structures are studied through the footprints they leave in conditional averages of the Q = −Tr(A 2 )/2 field, pertinent to non-local strain production, obtained using two complementary conditional averaging techniques. The first, stochastic estimation, approximates the Q field conditioned upon A and educes quantitatively similar structure in both datasets, dissimilar to that of random Gaussian velocity fields. Moreover, it strongly resembles a promising model for velocity gradient dynamics recently proposed by Wilczek & Meneveau (J. Fluid Mech., vol. 756, 2014, pp. 191-225), but is derived under a less restrictive premise, with explicitly determined closure coefficients. The second technique examines true conditional averages of the Q field, which is used to validate the stochastic estimation and provide insights towards the model's refinement. Jointly, these approaches confirm that vortex tubes are the predominant feature of rotation-dominated regions and additionally show that shear layer structures are active in strain-dominated regions. In both cases, kinematic features of these structures explain alignment statistics of the pressure Hessian eigenvectors and why local and non-local strain production act in opposition to each other.

New insights into the fine-scale structure of turbulence

“…With the increase of computational power, numerical simulations on the other hand now can forge into Reynolds-number ranges previously only accessible to experiments. The contribution by Lawson & Dawson (2015) is an inspiring example for these recent developments and highlights the synergetic potential of experimental, numerical and theoretical techniques that are now in the hands of the community. It appears likely that precise comparisons of numerical and experimental data will further evolve to become a standard procedure in the field, providing not only confidence in the characterization of the subtle details of turbulent flows, but also a testbed for novel theoretical approaches and reduced models.…”

“…The main question pursued by Lawson & Dawson (2015) is how the small-scale structure, dynamics and statistics of turbulence are interrelated, a long-standing and fundamental question in the field. To contribute to the answer, they acquire experimental data from a comparably large French washing machine set-up, consisting of a dodecagonal water tank of 2 m in height and 2 m in diameter equipped with two slowly turning, counter-rotating impellers at the top and at the bottom.…”

“…This so-called stochastic estimation put forward by Adrian (1994) and coworkers is extremely helpful to improve the statistical convergence of such complex statistical quantities. Lawson & Dawson (2015) extend this idea further by making contact to a recent closure theory suggested by Wilczek & Meneveau (2014), which quantifies non-local pressure Hessian effects on the velocity gradient statistics starting from the assumption of Gaussian velocity fields and subsequently taking into account non-Gaussian features by calibration against DNS data. When restricted to time-reversible terms, as dictated by the Gaussian closure assumption, their experimental data confirm the recent theoretical result.…”

“…With their recent work, Lawson & Dawson (2015) push the state of the art of this exciting subfield of turbulence research one step further in three respects. By a sophisticated experimental set-up, they were able to obtain high-fidelity, volumetric and timeresolved measurements of the velocity gradient and pressure Hessian fields which give novel insights into the statistical signatures of the small-scale structure of turbulence.…”

Synthetic velocity gradient tensors and the identification of statistically significant aspects of the structure of turbulence

“…This study makes use of velocity gradient tensors extracted from the Johns Hopkins numerical simulation of HIT at a Taylor-Reynolds number of 433 [29], which has become a popular resource for studying flow topologies [36,37]. We begin by looking at two basic quantities to derive from A given both the restricted Euler analysis of the velocity gradient tensor [20], and flow visualization studies [17]: Q and R as defined in (7).…”