Maximal Rotational Velocity of Fluid Elements (ΩFluidMAX): A New Perspective on Turbulent Singularities

The classical Navier-Stokes (NS) equations, rooted in the continuum hypothesis, essentially extend Newton’s second law to describe fluid motion. However, in their standard derivation, these equations rely solely on mass and momentum conservation and do not incorporate angular momentum conservation or dynamical equations for the rotation of fluid elements. As a result, they cannot explicitly describe the inertial torque arising from the fluid element’s own moment of inertia and angular acceleration. This simplification is reasonable and has been widely validated in translation-dominated flows. Nevertheless, in turbulent flows, local angular accelerations can become significant, and inertial torques may dominate the dynamics, rendering the NS equations physically incomplete in modeling strongly rotational, nonequilibrium flows. Consequently, turbulence models based on the NS equations—such as Reynolds-Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES)—often rely on empirical or semi-empirical closure assumptions to compensate for the missing physical mechanisms, thereby limiting their ability to reveal the intrinsic dynamical processes of turbulence.This paper argues that the rotational degree of freedom of fluid elements should be treated independently from translational motion. We propose a testable hypothesis: the Maximum Rotational Angular Velocity of Fluid Elements (​ΩFluidMAX). We posit that the angular velocity of fluid elements in a flow field is subject to a physical upper bound, the magnitude of which is determined by the fluid’s speed of sound a and the characteristic length scale ℓ of the fluid element. A preliminary expression is given by: ，where is a dimensionless constant to be calibrated experimentally. When external energy input drives the angular velocity of a fluid element toward this limit, excess energy can no longer be stored as rotational kinetic energy and is instead forced to convert into thermal energy, pressure fluctuations, or wave energy (e.g., acoustic or shock waves). This leads to a sharp rise in local energy density and strong nonlinear effects, resulting in what we define as a physical singularity. This mechanism offers a new physical interpretation for singularity formation in turbulence. It not only addresses the mathematical singularity problem arising from unbounded rotation in the NS framework but also provides a clear physical picture of energy dissipation and extreme events.The core value of this hypothesis lies in transforming the concept of physical singularity from an abstract mathematical notion into an observable physical state. Future validation requires high-resolution experimental techniques—such as ultra-high-speed particle image velocimetry (PIV), quantum sensing, or femtosecond laser spectroscopy—to verify the existence of ΩFluidMAX. If confirmed, this hypothesis could revolutionize turbulence modeling by enabling the reformulation of subgrid-scale dissipation mechanisms in RANS and LES. Theoretically, it may also provide a unified framework for understanding extreme phenomena such as plasma turbulence in fusion devices and angular momentum transport in black hole accretion disks, thereby advancing turbulence modeling from an empirically fitted paradigm toward a physically driven one.