A Unified Constitutive Framework for All Flow Regimes in Pipe Flow Based on Generalized Effective Viscosity
杜斯克1
作者信息
1. 水利科普号老杜侃江山工作室
折叠
摘要
本文从流体运动的纳维-斯托克斯（N-S）方程的宏观平均化形式出发，通过引入广义有效运动粘度（νx）及广义有效粘度比（x=νx/ν）的概念，推导并论证了管道流动的全流态统一本构方程：2ρνxVR=2ρνxVR=gρSR3。本文系统论证了该方程与N-S方程的内在联系，并证明其能够兼容哈根-泊肃叶定律、达西-魏斯巴赫方程等经典流体力学方程，实现了层流与湍流描述的理论统一。文中明确提出：驱动功率因子 gρSR3 作为整体，是决定平均流速 V 的基础性驱动因子。基于此统一方程，建立了完备的理论体系，推导了核心关系式：达西摩擦因子 λ=64x/Re、驱动雷诺数 Rc=√(128gR³S)/ν、状态方程 Re/Rc=1/√λ，以及广义有效粘度比 x 的等价表达式 x=Rcλ/64=Rc²/(64Re)。进一步，给出了 x 基于流速 V、雷诺数 Re、摩擦因子 λ 及粗糙度 ε 的四种求解公式。基于状态方程与科尔布鲁克-怀特方程的结合，推导了平均流速 V 的非迭代显式解析解。为验证理论，对尼库拉兹粗糙管实验、普林斯顿全流态数据及阿贝尔光滑管实验总计657个数据点进行了回溯验证，证实状态方程的平均相对误差小于0.05%。本工作的学术价值在于：建立了连接微观N-S方程与宏观工程公式的统一理论桥梁；提出了以 x 为序参量、Rc 为控制参数的流动状态描述新范式；为所有经典管道流动公式提供了统一的理论基石；衍生出无需迭代的显式计算新方法。
Abstract
Starting from the macroscopic averaged form of the Navier-Stokes (N-S) equations and introducing the concepts of generalized effective kinematic viscosity (ν_x) and generalized effective viscosity ratio (x = ν_x/ν), this paper derives and demonstrates the unified constitutive equation for all flow regimes in pipe flow: 2ρν_x V R = 2ρν x V R = gρS R^3. This paper systematically demonstrates the intrinsic connection between this equation and the N-S equations, and proves its compatibility with classical fluid mechanics equations including the Hagen-Poiseuille law and Darcy-Weisbach equation, thereby achieving theoretical unification of laminar and turbulent flow descriptions. We explicitly propose that the driving power factor gρS R^3, as a whole, serves as the fundamental driving factor determining the mean velocity V. Based on this unified equation, we establish a complete theoretical system, deriving the core relations: Darcy friction factor λ = 64x/Re, driving Reynolds number Rc = √(128gR³S)/ν, equation of state Re/Rc = 1/√λ, and equivalent expressions for the generalized effective viscosity ratio x = Reλ/64 = Rc²/(64Re). Furthermore, this paper presents four distinct solution formulas for x based on velocity V, Reynolds number Re, friction factor λ, and roughness ε. Based on the combination of the equation of state and Colebrook-White equation, this paper strictly derives the non-iterative explicit analytical solution for mean velocity V. To validate the theory, a rigorous retrospective validation is conducted using a total of 657 data points from Nikuradse’s rough pipe experiments, Princeton’s full-regime data, and Abell’s smooth pipe experiments, confirming that the equation of state has an average relative error of less than 0.05%. The academic significance of this work lies in: establishing a unified theoretical bridge connecting microscopic N-S equations with macroscopic engineering formulas; proposing a new paradigm for describing flow states using x as the order parameter and Rc as the control parameter; providing a unified theoretical foundation for all classical pipe flow formulas; and deriving a novel non-iterative explicit calculation method.
关键词
管道流动/统一本构方程/广义有效粘度比/驱动雷诺数/纳维-斯托克斯方程/功率平衡/全流态/序参量
Key words
Pipe flow/ Unified constitutive equation/ Generalized effective viscosity ratio/ Driving Reynolds number/ Navier-Stokes equations/ Power balance/ All flow regimes/ Order parameter
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