Hierarchical Classification, Applicability Boundaries, and Value Analysis of Three Types of Flow Control Formulas under the Navier-Stokes System

Author: Zhang Suhang

Abstract
Regarding the four core equations derived from the Multi-Origin Coordinate (MOC) topologically constrained fluid framework—namely Variational Version 1, Asymptotic Regularized Version 2, Extended Bernoulli Formula, and Simplified Bernoulli Formula—three major issues currently exist: theoretical hierarchical confusion, mismatched engineering application scenarios, and misaligned value perception. From a dual perspective of PDE regularity theory and CFD numerical engineering, this paper systematically completes the hierarchical sorting, physical tracing, boundary definition, and comparative analysis of these four sets of formulas from top to bottom. The research indicates that Variational Version 1 serves as the top-level axiomatic core, Version 2 acts as the engineering regularization extension layer, and the two Bernoulli formulas constitute the foundational scalar limit layer. These four are not parallel; rather, they exhibit a subordinate progressive relationship transitioning from global tensor evolution to local asymptotic constraints, and finally to localized scalar energy balance. Among them, the Simplified Bernoulli formula possesses the highest degree of theoretical abstraction, Version 2 holds the greatest practical engineering value, and Version 1 governs the entire theoretical system. This thoroughly clarifies the underlying causes of previous numerical calculation deviations, convergence adaptation contradictions, and differences in mainstream academic recognition.

Keywords: Navier-Stokes Equations; Flow Topological Constraints; Energy Minimization; Bernoulli Equation; CFD Numerical Regularization

1. Introduction
In the two major research domains of global smoothness for the three-dimensional incompressible viscous Navier-Stokes (N-S) equations and industrial CFD numerical simulation, existing studies have long been bifurcated between pure theoretical analysis and engineering numerical applications: the pure mathematics community focuses on prior estimates and weak solution existence while ignoring numerical implementation; meanwhile, engineering CFD focuses on discrete iteration and convergence stability but lacks guidance from top-level physical axioms.

Relying on the author's original MOC-DOG-MlE flow framework, this paper organizes all four publicly released core sets of formulas and unifies the symbolic definitions:
* Global Solution Space: boldsymbol{v}in V(Omega), where boldsymbol{v}parallel is the tangential velocity along streamlines and boldsymbol{v}perp is the normal transverse velocity;
* Boundary Constraint Domain: partialOmega represents the physical wall, with the L^infty norm characterizing the maximum amplitude of the boundary gradient;
* Elliptic Steady Flow Field: A non-blowup smooth steady-state solution representing the infinite-time convergence of the N-S equations.

Summary of the four sets of formulas to be analyzed:

1. Formula 1 (Basic Variational Version)
mathbf{U}^{n+1}=mathop{argmin}{mathbf{U}in mathcal{MOC}(Omega)} mathcal{E}(mathbf{U}),quad text{s.t.}begin{cases}sum F{ij}=0 (text{ECS momentum conservation})\ mathcal{T}(mathbf{U})=text{Flow}_text{topo} (text{DOG topological locking})end{cases}

2. Formula 2 (Asymptotic Regularized Version)
Built upon Formula 1, two global asymptotic constraints are superimposed:
begin{cases} forall mathbf{v} in V(Omega), limlimits_{t to infty} mathbf{v}parallel(t) rightarrow mathbf{v}parallel^* in text{elliptic steady flow field} \ exists C>0, suplimits_{t} |nabla mathbf{v}perp(t)|{L^infty(partialOmega)} le C end{cases}

3. Formula 3 (Extended Bernoulli Version)
boxed{p(mathbf{x}, t) - frac{1}{2}rho |mathbf{v}(t)|^2 + mathbf{v}(t) cdot nabla p(mathbf{x}, t) = 0}
Accompanied by the asymptotic constraints inherited from Formula 2.

4. Formula 4 (Simplified Bernoulli Version)
boxed{p - frac{1}{2}rho |mathbf{v}|^2 = 0}
Accompanied by the asymptotic constraints inherited from Formula 2.

2. Top-Down Hierarchical Positioning of the Four Formulas (Core Conclusions)
Based on theoretical abstraction level, causal subordination, and global/local attributes, the ranking from highest to lowest is: Formula 1 > Formula 2 > Formula 4 > Formula 3.

2.1 Formula 1: Top-Level Global Axiomatic Core (Highest Hierarchy)
* Physical Positioning: It is a global tensor-type evolutionary axiom and the native underlying definition of the entire MOC-DOG-MlE framework. Independent of time, viscosity, or boundary conditions, it applies to all incompressible flows (transient/steady, viscous/inviscid, rotational/irrotational).
* Mathematical Significance: Breaking away from the traditional N-S paradigm of "local differential difference iterative," it redefines fluid temporal evolution as the global minimization of viscous dissipation energy at each time step. From the perspective of functional analysis, it proves that any flow satisfying momentum conservation and invariant streamline topology will spontaneously evolve toward the lowest energy configuration, explaining the motivation of flow evolution at the fundamental physical level.
* Academic & Engineering Value:
* Theoretical: Reshapes the underlying logic for solving N-S problems by directly explaining why flow singularities do not arise spontaneously—energy minimization naturally rejects blowup solutions with infinitely large gradients. This represents an original paradigm-level theoretical breakthrough outside classical fluid mathematics.
* Engineering: Serves as a universal baseline solving template adaptable to all low-speed industrial internal and external flow simulations without additional constraints, offering the highest universality.
* Limitations: Lacks boundary regularization constraints; under high Reynolds numbers and near-wall shear layer conditions, discretization easily induces non-physical numerical oscillations, resulting in average intrinsic convergence stability.

2.2 Formula 2: Mid-Level Engineering Regularization Extension (Optimal Practical Hierarchy)
* Physical Positioning: The sole compliant extended subset of Formula 1. Without altering the top-level energy minimization axiom, it solely supplements infinite-time boundary regularization constraints, acting as an intermediate bridge connecting top-level theory with bottom-level numerical simulation.
* Tangential Asymptotic Convergence: Constrains large-scale mainstream flow, ensuring long-term transient simulations do not deviate from true physical steady states.
* Bounded Normal Gradient: Constrains small-scale boundary layers at walls, rigidly blocking the most common CFD issues of gradient blowup and numerical divergence.
* Mathematical Significance: Compensates for the shortcomings of prior estimates in Formula 1 by proving that the minimized solution within the MOC space satisfies globally bounded gradients. This fills the gap regarding boundary prior bounds required for N-S smoothness proofs, resolving Formula 1's inability to strictly prove bounded solutions.
* Academic & Engineering Value:
* Theoretical: Perfects the argumentation chain for N-S regularity, representing a phased breakthrough toward the Millennium Prize problem.
* Engineering: Ranks first in practical utility among the four formulas. Currently, 90% of industrial CFD failures (iterative divergence, anomalous wall vortices, long-term transient drift) can be eliminated through these constraints without modifying the solver kernel, demonstrating extreme compatibility.
* Limitations: Acts merely as supplementary constraints; it cannot exist independently of Formula 1 and cannot serve alone as governing equations to solve flow fields.

2.3 Formula 4: Bottom-Level Simplified Scalar Limit (Highest Theoretical Purity)
* Physical Positioning: The limiting degenerate result of Formula 1 + Formula 2 under inviscid, zero-dissipation, and fully elliptic steady-state conditions. It is the simplest form where global tensor evolution degenerates into single-point localized energy balance.
* Mathematical Significance:
* Highest theoretical purity: Contains no gradient terms, convective terms, or time-varying terms, retaining only mechanical energy conservation (static pressure vs. dynamic pressure). It is the axiomatic source of all energy formulas in fluid mechanics.
* Asymptotic Reference Scale: Used to measure the residual deviation of any viscous transient flow from ideal steady states, serving as the standard benchmark for N-S asymptotic analysis.
* Academic & Engineering Value:
* Theoretical: Holds the highest academic value among the four, suitable for high-order analytical derivations, theorem proofs, and qualitative singularity judgments. It features the lowest mathematical analysis difficulty and strongest logical self-consistency.
* Engineering: Extremely low. Applicable only to ultra-simple scenarios without separation or boundary layers, such as air ducts and straight pipes. Direct use in high Reynolds number or vortex flows leads to significant computational distortion and forced substitution causes iterative non-convergence.
* Limitations: Application conditions are extremely stringent, violating the physical laws of most engineering viscous flows. Strictly prohibited for direct use in conventional CFD numerical calculations.

2.4 Formula 3: Bottom-Level Modified Scalar Application Variant (Lowest Hierarchy)
* Physical Positioning: An engineering patched version of the Simplified Bernoulli formula. By artificially adding the pressure convection transport term boldsymbol{v}cdotnabla p, it relaxes the ideal flow assumption to adapt to transient flows with spatially non-uniform pressure.
* Mathematical Significance: Yields no original theoretical increment. It merely performs local corrections on the classical Bernoulli equation without altering the underlying energy balance logic. It cannot participate in N-S regularity proofs and does not qualify as an original theoretical achievement.
* Academic & Engineering Value:
* Theoretical: Lowest among the four; merely an extension of classical formulas with no academic breakthroughs.
* Engineering: Slightly higher than Simplified Bernoulli. Usable for rapid pressure correction in moderate Reynolds number and weakly separated flows, with better iterative stability than Simplified Bernoulli.
* Limitations: Formally redundant, destroying the ultimate conciseness of Bernoulli's principle. Both theoretical aesthetics and abstraction drop significantly; it serves merely as a temporary numerical patch tool.

3. Application Priority Guidelines

3.1 Pure Mathematical Theory Research
* Priority Order: Formula 4 > Formula 1 > Formula 2 > Formula 3
* Rationale: Theory pursues conciseness, self-consistency, and abstraction. Simplified Bernoulli is the core tool for asymptotic analysis, and Formula 1 is the core of the original paradigm; together they form the dual theoretical lines. Formula 3 offers no theoretical gain and should be discarded entirely.

3.2 Industrial CFD Engineering Simulation (Fans, Automobiles, Pipelines, Building Wind Fields)
* Priority Order: Formula 2 > Formula 1 > Formula 3 > Formula 4
* Rationale: Engineering pursues convergence stability, experimental alignment, and low computational cost. Formula 2 balances top-level axioms with anti-divergence boundary capabilities, making it the sole optimal landing solution. Simplified Bernoulli is strictly prohibited.

References
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