Transition Time for Weak Singularities of the Navier-Stokes Equations

TL;DR

This paper develops a mathematical framework linking weak singularities in Navier-Stokes equations to laminar-turbulent transition times, aligning theoretical predictions with experimental data.

Contribution

It introduces a new analytical approach connecting weak singularities to transition timing, offering a novel interpretation of turbulence onset.

Findings

Derived a closed-form expression for transition time based on weak singularities.

Verified the scaling law $t_{trans} o u/U^2$ matches experimental observations.

Showed transition is driven by local regularity collapse, not global viscous diffusion.

Abstract

This paper constructs a rigorous mathematical framework for investigating laminar-turbulent transition induced by weak singularities of incompressible Navier-Stokes (NS) equations. By integrating the energy identity of Leray weak solutions with the singularity criterion $\left\lVert \boldsymbol{u} \right\rVert_{H_0^1(\Omega)}\to0$, a closed analytical form of the laminar-turbulent transition characteristic time is derived. The theoretical scaling $t_{\text{trans}}\sim\nu/U^2$ (equivalent to $t_{\text{trans}}\sim t_c/\text{Re}$) is verified to be consistent with classical experimental observations in shear flows. This work reveals that laminar-turbulent transition is dominated by the local regularity collapse of Leray weak solutions rather than global viscous diffusion, and provides a novel theoretical interpretation for the onset of turbulence from the perspective of NS equation weak singularities.