Objective momentum barriers in wall turbulence

Abstract

We use the recent frame-indifferent theory of diffusive momentum transport to identify internal barriers in wall-bounded turbulence. Formed by the invariant manifolds of the Laplacian of the velocity field, the barriers block the viscous part of the instantaneous momentum flux in the flow. We employ the level sets of single-trajectory Lagrangian diagnostic tools, the trajectory rotation average and trajectory stretching exponent, to approximate both vortical and internal wall-parallel momentum transport barrier (MTB) interfaces. These interfaces provide frame-indifferent alternatives to classic velocity-gradient-based vortices and high-shear boundaries between uniform momentum zones (UMZs). Indeed, we find that these elliptic manifold approximations and MTBs outperform standard vortices and UMZ interfaces in blocking diffusive momentum transport, which suggests our momentum barriers are physical features that may be the cause of coherence signatures in statistical and non-objective diagnostics. We also introduce normalized trajectory metrics that provide unprecedented visualizations of objective coherent structures by avoiding strong turbulence biases.

Approximate streamsurfaces for flow visualization

“…We have additionally used the described algorithm to extract objectively defined instantaneous momentum transport barriers in a 3-D turbulent channel flow in an almost-automated fashion. Such barriers are streamsurfaces of an incompressible barrier equation defined by the Laplacian of the velocity field (Haller et al 2020;Aksamit & Haller 2022). We have extracted vortical momentum transport barriers without the need to advect arrays of trajectories and found such barriers across multiple spatial scales.…”

“…This provides a natural way out of the occlusion quandary that besets many visualization approaches while retaining the algorithm's capacity to capture the smallest structures, delineated in the aFTLE plots, by using finer partitions. In the next subsection, we will demonstrate this capacity by pinpointing structures that the active diagnostics-based methods in Haller et al (2020) and Aksamit & Haller (2022) would likely miss. We conclude this section with figure 19 depicting a well-formed spectral gap between the smallest and the second smallest eigenvalue of A for all the computational boxes.…”

“…To reconcile these two objectives, Haller et al (2020) developed a theory of objectively defined barriers to the transport of active vector fields, such as the vorticity and momentum. Aksamit & Haller (2022) used this theory to locate instantaneous (Eulerian) frame-indifferent momentum transport barriers in 3-D turbulent channel flows. Active barriers turn out to be distinguished streamsurfaces of appropriately defined steady, 3-D incompressible vector fields (Haller et al 2020).…”

“…For an incompressible velocity field, we further note that v = −∇ × ω holds. Haller et al (2020) and Aksamit & Haller (2022) detected distinguished streamsurfaces of the momentum barrier equation (4.1) using active versions of some of the passive hyperbolic and elliptic LCS diagnostics reviewed by Haller (2015). These Lagrangian calculations involve arrays of trajectories and return diagnostic scalar fields for visual inspection rather than explicit streamsurfaces families.…”

“…Specifically, the barrier vector fields are the velocity and vorticity Laplacian in the instantaneous (Eulerian) limit and the time-averaged pullbacks of these Laplacian fields in the Lagrangian case. The invariant manifolds of these barrier vector fields have been shown to highlight vortical features of the velocity field with increased accuracy in several 2-D and three-dimensional (3-D) flows (Haller et al 2020;Aksamit & Haller 2022).…”