"Beyond Inertial Torque: ΩFluidMAXRedefines Turbulence Singularities," which presents a fundamental resolution to the long-standing issue of hydrodynamic singularities in the Navier-Stokes equations.

The classical formulation of the Navier-Stokes equations permits unbounded vorticity amplification, leading to non-physical, infinite-energy-density states—the so-called turbulence singularities. This manuscript demonstrates that this divergence is averted by a previously overlooked physical constraint: a finite maximal rotational angular velocity for fluid elements, which we term ΩFluidMAX. This principle acts as a natural regulator for the energy cascade, replacing the unphysical mathematical blow-up with a physically realizable mechanism of intense energy dissipation.

Our key innovation is the identification and formalization of ΩFluidMAX as a universal physical limit. When the local rotation rate of a fluid element approaches this ceiling, further energy input cannot increase rotational kinetic energy. Instead, the excess energy is mandatorily converted into other forms—primarily internal energy, pressure fluctuations, and wave emission. This provides a concrete, physically-grounded mechanism for the extreme intermittent events and anomalous dissipation characteristic of fully developed turbulence. Crucially, this mechanism operates at the smallest scales, offering a definitive replacement for the singularities predicted by the classical equations.

This framework has profound implications across disciplines. By establishing a finite limit on rotational speed, ΩFluidMAX provides a new paradigm for understanding a wide range of extreme flow phenomena, from shock-turbulence interactions and supersonic boundary layers to astrophysical accretion disks and magnetohydrodynamic dynamos. Furthermore, its incorporation into models promises to revolutionize subgrid-scale closures in Large Eddy Simulation (LES) and Reynolds-Averaged Navier-Stokes (RANS) approaches, moving beyond heuristic eddy viscosities toward a physically derived model of small-scale dynamics.

A defining feature of this work is its experimental accessibility. Unlike abstract mathematical regularizations, ΩFluidMAX predicts measurable consequences: a sharp upper bound in the probability distribution function of local angular velocities, and characteristic signatures in thermal emission, acoustic noise, or electromagnetic radiation coincident with energy conversion events. Emerging capabilities in ultrafast imaging and quantum sensing now make direct verification feasible, offering a clear path to empirical validation.

In summary, this manuscript introduces ΩFluidMAX as a foundational physical law governing rotational motion in turbulent flows. We believe this work represents a critical advance in fluid dynamics, providing a complete and testable resolution to longstanding challenge in classical fluid dynamics.

researchgate.net/publication/395012380_Maximal_Rotational_Velocity_of_Fluid_Elements_OFluidMAX_A_New_Perspective_on_Turbulent_Singularities?_tp=eyJjb250ZXh0Ijp7InBhZ2UiOiJwcm9maWxlIiwicHJldmlvdXNQYWdlIjoicXVlc3Rpb24iLCJwb3NpdGlvbiI6InBhZ2VDb250ZW50In19

The classical Navier-Stokes equations, derived under the continuum hypothesis, extend Newton’s second law to fluid mechanics. Their standard formulation—based solely on mass and momentum conservation—neglects angular momentum principles and omits dynamical equations for fluid-element rotation. Consequently, these equations fail to capture inertial torques arising from moment of inertia and angular acceleration. While valid for translation-dominated flows, this omission becomes critical in turbulence where localized angular accelerations dominate. This limitation renders NS-based models (e.g., RANS, LES) ​​inherently incomplete for intensely vortical, nonequilibrium dissipative flows​​, forcing reliance on empirical closures that ​​limit fundamental understanding of turbulent dynamics​​.

This study proposes that fluid element rotation should be treated as an independent degree of freedom from translational motion. We introduce a testable hypothesis termed the Maximum Rotational Angular Velocity (ΩFluidMAX), which posits that angular velocity in fluid elements attains a universal physical upper bound determined by the fluid's speed of sound (a) and characteristic length scale (ℓ). A preliminary scaling relation is proposed as: 📷 , where 📷 is a dimensionless constant requiring experimental calibration. When external energy input drives angular velocity toward this limit, excess energy is forced to convert into thermal, pressure, or wave energy (including acoustic/shock waves), resulting in localized energy density spikes and nonlinear effects we term physical singularities. This mechanism provides a novel physical interpretation for turbulent singularity formation, addressing both mathematical limitations of unbounded rotation in the NS framework and offering a concrete picture of energy dissipation pathways and extreme events. In essence, ΩFluidMAX serves as a "safety valve" for rotational energy: when fluid element angular velocity nears this limit, excess input energy is forced into alternative pathways—thermal heating, pressure fluctuations, or wave radiation (including acoustic/shock waves). This energy conversion manifests as localized dissipation spikes, which we term physical singularities—nonlinear phenomena distinct from mathematical blow-up yet critical for understanding turbulence's extreme behavior.

The hypothesis's core innovation lies in grounding singularity phenomena in observable physical processes rather than abstract mathematical formalisms. Validation will require high-resolution experimental techniques such as ultrafast particle image velocimetry (PIV) operating at megahertz frame rates, quantum sensing, or femtosecond laser spectroscopy. If confirmed, this work could transform turbulence modeling by enabling physically informed subgrid-scale closures in RANS/LES frameworks. Theoretically, it may provide a unified framework for understanding extreme phenomena across disciplines, including plasma turbulence in fusion reactors and angular momentum transport in black hole accretion disks. This would represent a paradigm shift from empirical fitting to physically-grounded modeling approaches in turbulence research.

researchgate.net/publication/395012380_Maximal_Rotational_Velocity_of_Fluid_Elements_OFluidMAX_A_New_Perspective_on_Turbulent_Singularities?_tp=eyJjb250ZXh0Ijp7InBhZ2UiOiJwcm9maWxlIiwicHJldmlvdXNQYWdlIjoicXVlc3Rpb24iLCJwb3NpdGlvbiI6InBhZ2VDb250ZW50In19

Core Limitation of the NS Equations: Absence of Angular Momentum Conservation

The NS equationsare based on the continuum assumption, where their momentum conservation equation solely accounts for the relationship between translational acceleration of fluid elements and external forces (e.g., pressure gradients and viscous stresses). A critical omission is the absence of explicit angular momentum conservation principles or terms representing inertial torque arising from fluid element moment of inertia and angular acceleration. This simplification implies three key implications:

1. Rotational effects are only indirectly represented through viscous stress gradients (e.g., diffusion terms in the vorticity transport equation), with inertial torque entirely neglected;

2. In laminar or weakly rotational flows, this approximation remains physically valid due to negligible rotational contributions;

3. In turbulent flows, however, intense deformation and vortex stretching induce large local angular accelerations, making inertial torque a dominant dynamical factor.

This limitation underscores both mathematical insufficiency and fundamental physical modeling gaps. Xie Xuegang emphasized that angular momentum balance must be explicitly incorporated into fluid dynamics; otherwise, non-conservation compromises accurate description of rotational structures . Joseph D. D. further demonstrated in multiphase flow studies that classical NS equationsfail to capture rotational degrees of freedom, requiring supplementary rotational dynamical equations .

Inherent Limitations of Mainstream Turbulence Models

Current mainstream turbulence models inherit this foundational deficiency from the NS framework, manifesting through three primary limitations:

1 Empirical Parameters as "Compensatory Fitting"

RANS models like the k-ε formulation introduce a "turbulent viscosity coefficient" to represent fluctuating effects as enhanced viscous stress. However, this empirically calibrated parameter lacks universal physical basis, rendering it effective only in specific flow configurations (e.g., boundary layers, pipe flows) while failing to generalize to complex turbulence or shock-interacting flows. This "compensatory fitting" strategy essentially sidesteps mechanistic description of energy transfer within rotational degrees of freedom.

2 LES and DNS: "Computational Bottlenecks" and "Physical Blind Spots"

Large Eddy Simulation (LES) employs spatial filtering to separate scales but relies on empirical subgrid-scale (SGS) models (e.g., Smagorinsky model) that ignore inertial torque effects from small-scale vortex rotation. Direct Numerical Simulation (DNS), despite aiming to solve NS equations directly, encounters physical breakdown at high Reynolds numbers when turbulent scales approach molecular mean free path, invalidating the continuum assumption. The NS equations' neglect of rotational degrees of freedom exacerbates discrepancies between numerical solutions and physical reality. Lesieur and Métais highlighted that NS equations require supplementary rotational terms to accurately describe vortex stretching and angular momentum transport in rotating turbulence [7], demonstrating that even high-resolution simulations may misrepresent physics if essential terms are missing.

3 "Passive Response" to Nonlinear Effects

Core turbulence phenomena—energy cascading, vortex fragmentation, and localized extreme gradients—are intrinsically tied to nonlinear evolution of rotational degrees of freedom. Existing models can only passively capture phenomenological manifestations (e.g., sudden vorticity increases) through numerical means but fail to explain underlying mechanisms. This stems from NS equations'lack of intrinsic constraints on rotational degrees of freedom, rendering current models more reactive recorders than active mechanism revealers.

3 Mathematical Evidence: Divergence of Rotational Degrees of Freedom

Mathematical analysis reveals structural deficiencies in NS equations' rotational modeling capabilities. Terence Tao noted that while total kinetic energy remains bounded, the nonlinear structure fails to control vorticity growth (a proxy for rotational degrees of freedom), leaving open the possibility of rotational degree blowup [8,9]. This mathematical phenomenon translates physically to the NS equations' failure to suppress unbounded angular velocity growth. Frank M. White confirmed that NS equations "lack sufficient mathematical structure to describe angular momentum balance" in rotational flows . Frisch et al. observed that NS nonlinear terms provide positive feedback during vortex stretching, enhancing vorticity without intrinsic limiting mechanisms—potentially leading to theoretically infinite local vorticity . This aligns with the physical argument that without rotational constraints (e.g., maximum angular velocity), energy continues injection into rotational modes, ultimately causing mathematical singularities.

3.4 The Gap Between Mathematical Solutions and Physical Reality

It is crucial to clarify that this analysis does not challenge the validity of NS equations in translation-dominated flows. Instead, it highlights potential decoupling between mathematical solutions and physical reality when rotational inertial effects are improperly accounted for:

Thus, current turbulence model limitations stem from NS equations' failure to properly model rotational degrees of freedom. To address this, theoretical frameworks must incorporate independent rotational dynamical equations—following rigid body dynamics' separation principle. This represents a physical completion rather than rejection of historical NS contributions, laying groundwork for next-generation physics-driven turbulence models.

The "singularity problem" in turbulence remains a dual challenge—both a mathematical regularity issue and a debate over physical mechanism completeness. Building on prior analysis of NS equations' deficiencies in modeling rotational degrees of freedom, this paper introduces a novel physical framework: the concept of a fluid element maximum rotational angular velocity (ΩFluidMAX) as a fundamental physical constraint. This hypothesis seeks to elucidate the formation of extreme nonlinear structures in turbulence while contrasting with traditional mathematical singularity paradigms.

1 The Principle of Maximum Angular Velocity and Rotational Constraints on Fluid Elements

In nature, the angular velocity of any finite-volume physical structure cannot increase indefinitely, constrained by fundamental physical laws akin to the speed of light as a universal linear velocity limit. This principle has been termed the Principle of Maximum Angular Velocity [10], which states that any volumetric entity possesses an upper bound Ωmax on its angular velocity, related to the vacuum speed of light c and the mathematical constant π. This principle arises from the observation that geometric points (zero volume) lack rotational attributes (angular velocity = 0), whereas real physical entities—including fluid elements—must adhere to governing physical mechanisms. Reference [10] formalizes rotational motion through the definition ω=dϕ/dτ (where ϕ is angular displacement and τis proper time), providing a foundation for analyzing fluid element rotational dynamics.

Inspired by this principle, we propose that fluid elements—as quasi-structured entities with physical volume—also exhibit an angular velocity upper bound, denoted ΩFluidMAX (in rad/s). This limit depends on fluid intrinsic properties and is preliminarily expressed as:

📷 (📷)

where 📷 is a dimensionless constant dependent on molecular structure, thermodynamic state, and other fluid-specific factors (to be experimentally calibrated), and 📷represents the characteristic length scale of the fluid element (e.g., Kolmogorov scale). We impose the conservative constraint:

ΩFluidMAX 📷

reflecting rotational degree saturation under extreme conditions and potentially limiting energy cascade in turbulence. The physical origin of ΩFluidMAX may stem from fundamental limits such as:

A. Maximal molecular collision frequency: At extreme angular velocities, fluid elements rotate faster than molecular collisions can redistribute momentum. This mismatch between acceleration timescale (Δt ≈ ℓ/Ω) and collision timescale creates a "kinetic bottleneck," where rotational energy saturates due to insufficient angular momentum transfer at the molecular scale.

B. Acoustic wave propagation limit: Rotational stresses cannot exceed the finite speed of sound (a), as excess energy must dissipate via thermalization or shock formation to maintain local mechanical equilibrium. This aligns with second-law thermodynamics, where entropy production scales with angular velocity gradients across acoustic wavelengths.

C. Quantum/relativistic constraints: While negligible in classical flows, quantum effects constrain angular momentum resolution (ΔL ≥ ħ) at molecular scales (ℓ ≲λmfp), potentially linking ΩFluidMAXto the classical-quantum transition. Relativistic considerations may further impose spacetime-based limits on rotational dynamics.

Collectively, these limits operate across scales—from molecular collisions (microscale) to acoustic propagation (mesoscale) and quantum/relativistic bounds (fundamental scale)—to constrain rotational angular velocity. These interdependent mechanisms jointly establish the physical upper bound of fluid rotation, rendering ΩFluidMAXa universal regulator of turbulent energy dissipation pathways.

Note: The inequality ΩFluidMAX 📷​ is conservative. Even if the universal 📷​ from [10] is invalid, ΩFluidMAX may still exist independently as a fluid-specific limit.

2 Formation Mechanism of Physical Singularities in Turbulence

Under the proposed hypothesis, turbulent singularities arise as follows: During energy cascading, large-scale vortices transfer energy to smaller scales via stretching/shearing, increasing fluid element angular velocity. When this velocity approaches ΩFluidMAX, the system enters rotational saturation. Subsequent energy injection (e.g., pressure gradient work or gravitational potential release) forces excess energy into alternative forms:

These phenomena collectively constitute a physical singularity in turbulence—an extreme state of energy dissipation and redistribution under rotational constraints, distinct from mathematical infinities. This interpretation aligns with experimental observations of turbulence intermittency, localized gradients, and multiscale dissipation.

3 Analogy with Shock Wave Phenomena

To contextualize ΩFluidMAX, we draw parallels with shock waves: When an object exceeds sound speed a in a medium, disturbances cannot propagate fast enough, leading to abrupt compression and shock wave formation. Across the shock front, discontinuous jumps in pressure, density, and temperature occur with significant energy dissipation (e.g., sonic boom). Similarly, when fluid element angular velocity approaches ΩFluidMAX, further energy input cannot sustain rotational kinetic energy. Excess energy instead converts into internal energy (local heating), pressure fluctuations, or wave energy (e.g., shock waves), triggering rapid energy density increases and nonlinear effects—forming a physical singularity. Both a (translational motion) and ΩFluidMAX (rotational motion) represent physical constraints mediating energy dissipation under extreme conditions. While acting on different degrees of freedom, they share similar energy transfer mechanisms, reinforcing our assumption that ΩFluidMAX is fundamentally linked to a.

4 Contrast with Traditional Views on Turbulent Singularities

Traditional studies focus on NS equation solutions' regularity—specifically whether velocity/vorticity can "blow up" in finite time (mathematical blow-up). Works by Leray, Beale-Kato-Majda, Tao, and others [1,8] center on this question but often neglect physical realism. Our hypothesis innovates in three ways:

1. A physics-based perspective: Singularities are viewed as forced energy conversion due to saturation, not mathematical divergence;

2. Mechanistic Clarity: Attributable to rotational energy upper bounds rather than uncontrolled nonlinear term growth in NS equations;

3. Bridging Mathematics-Physics Gap: Does not negate smooth NS solutions but argues such solutions may fail to describe real turbulence if rotational constraints are ignored.

Furthermore, traditional explanations—such as "molecular mean free path" or "chaotic self-organization"—can be integrated into this framework: the former describes microscopic energy transfer, while the latter reflects system responses to rotational-nonlinear coupling. Notably, this hypothesis complements rather than contradicts Kolmogorov scaling laws: while the latter describes statistical energy cascade averages, ΩFluidMAX characterizes energy saturation and forced dissipation at the smallest scales (inertial range vs. dissipation range).

Regarding turbulence problems, we prefer: 1. Viewing turbulence from beyond fluid mechanics. 2. Extreme complexity leads to simplicity.

"Maximal Rotational Velocity of Fluid Elements (ΩFluidMAX): A New Perspective on Turbulent Singularities" (DOI: 10.20944/preprints202508.2019.v1).

researchgate.net/publication/395012380_Maximal_Rotational_Velocity_of_Fluid_Elements_OFluidMAX_A_New_Perspective_on_Turbulent_Singularities?_tp=eyJjb250ZXh0Ijp7InBhZ2UiOiJwcm9maWxlIiwicHJldmlvdXNQYWdlIjoicXVlc3Rpb24iLCJwb3NpdGlvbiI6InBhZ2VDb250ZW50In19