From UW-Math Wiki
Jump to navigation Jump to search

ACMS Abstracts: Fall 2020

Nick Ouellette (Stanford)

Title: Tensor Geometry in the Turbulent Cascade

Abstract: Perhaps the defining characteristic of turbulent flows is the directed flux of energy from the scales at which it is injected into the flow to the scales at which it is dissipated. Often, we think about this transfer of energy in a Fourier sense; but in doing so, we obscure its mechanistic origins and lose any connection to the spatial structure of the flow field. Alternatively, quite a bit of work has been done to try to tie the cascade process to flow structures; but such approaches lead to results that seem to be at odds with observations. Here, I will discuss what we can learn from a different way of thinking about the cascade, this time as a purely mechanical process where some scales do work on others and thereby transfer energy. This interpretation highlights the fundamental importance of the geometric alignment between the turbulent stress tensor and the scale-local rate of strain tensor, since if they are misaligned with each other, no work can be done and no energy will be transferred. We find that (perhaps surprisingly) these two tensors are in general quite poorly aligned, making the cascade a highly inefficient process. Our analysis indicates that although some aspects of this tensor alignment are dynamical, the quadratic nature of Navier-Stokes nonlinearity and the embedding dimension provide significant constraints, with potential implications for turbulence modeling.

Harry Lee (UW Madison)

Title: Recent extension of V.I. Arnold's and J.L. Synge's mathematical theory of shear flows

Abstract: A viscous extension of Arnold’s non-viscous theory ([1]) for 2D wall-bounded shear flows is established ([3]). One special form of our linearized viscous theory recaps the linear perturbation’s enstrophy (vorticity) identity derived by Synge in 1938 ([2]). For the first time in literature, we rigorously deduced the validity of Synge’s identity under nonlinear dynamics and relaxed wall conditions. Furthermore, we discovered a new ‘weighted’ enstrophy identity.

To illustrate the physical relevance of our identities, we quantitatively investigated mechanisms of linear instability/stability within the normal modal framework. We observed a subtle interaction between a critical layer and its adjacent boundary layer, which governs stability/instability of a flow. We also proposed a boundary control scheme that transitions wall settings from no-slip to free-slip, through which the 2D base flow was stabilized quickly at an early stage of the transition. Effectiveness of such boundary control scheme for 3D shear flows is yet to be tested by DNS/experiments.

Apart from physics, I shall also talk about the potential of using our nonlinear enstrophy identity to generate rigorous bounds on flow stability.


[1] V. I. Arnold. Conditions for the nonlinear stability of the stationary plane curvilinear flows of an ideal fluid. Doklady Akademii Nauk, 162:975–978, 1965. URL:

[2] F. Fraternale, L. Domenicale, G. Staffilani, and D. Tordella. Internal waves in sheared flows: Lower bound of the vorticity growth and propagation discontinuities in the parameter space. Physical Review E, 97:063102, 2018. URL:

[3] H. Lee and S. Wang. Extension of classical stability theory to viscous planar wall-bounded shear flows. Journal of Fluid Mechanics, 877:1134– 1162, 2019. URL: