New Insights into Energy Dissipation in Fluid Dynamics

A recent paper titled "Navier-Stokes bounds and scaling for compact trefoils in (2ℓπ)³ domains" by Robert M. Kerr explores the behavior of perturbed trefoil vortex knots across a range of viscosities. The study presents evidence that these knots generate Reynolds number independent finite-time energy dissipation, denoted as ΔEₑₘₑ, and suggests the presence of a transient Kolmogorov spectrum at high Reynolds numbers. This finding aligns with one definition of a dissipation anomaly in fluid dynamics.

The research indicates that to encompass the necessary range of Reynolds numbers, specific scaling factors must be considered. Notably, the largest scale L, which corresponds to the size of the periodic domain, increases as L ∼ ν⁻¹/⁴. The study also discusses the convergence of vorticity moments at early times, which scale as ν¹/⁴Ωₘ, for various moments m.

The paper further details how the scaling ceases at a specific time, tₓ, leading to the convergence of the volume-integrated enstrophy, Z(t), which is defined as Z = L³Ω₁². Following this, the growth of Z(t) accelerates until the dissipation rates ε = νZ approximate convergence at t ≈ tₑₘₑ ≈ 2tₓ.

Kerr's work also addresses the critical viscosities, νₛ, that bound Navier-Stokes solutions in (2π)³ periodic domains, showing that these solutions are constrained by their corresponding Euler solutions as ν approaches zero. By rescaling results from Sobolev space analysis into larger (2ℓπ)³ domains, the study demonstrates a reduction in the bounding critical viscosities, allowing for the growth of Z(t) in these expanded domains.

This research contributes to the understanding of fluid dynamics, particularly in the context of turbulence and energy dissipation, which has implications for various applications in engineering and physics. The findings may help refine models that predict fluid behavior in complex systems, enhancing our ability to manage and manipulate fluid flows in practical scenarios.

For further details, the paper can be accessed through arXiv: arXiv:2401.03578.