Abstract:

[Objective] This study reveals the epistemological rationale underlying George Gabriel Stokes’ cautious and reserved stance in his review of Osborne Reynolds’ groundbreaking 1895 paper proposing the Reynolds-Averaged Navier-Stokes (RANS) equations. While Reynolds’ methodology has become a cornerstone of turbulence modeling in engineering practices, Stokes—co-developer of the Navier-Stokes (N-S) equations—notably refrained from offering an endorsement during its first-round review. Through an interdisciplinary investigation combining archival analysis, theoretical fluid dynamics, and philosophy of science, it is revealed that Stokes’ reservations stemmed not from technical negligence but from a profound understanding of the N-S equations’ physical completeness and the inherent epistemological limitations of turbulence closure models. [Methods] Three complementary approaches were employed: 1. Fundamental theory reconstruction: the N-S equations were reconstructed based on Stokes’ original axiomatic framework — Newton’s law of viscosity, the assumption of stress isotropy, and the law of mass conservation — confirming their physical completeness. However, introducing additional independent laws to close the unclosed terms derived from Reynolds averaging procedure would fracture the physical completeness of the N-S equations. 2. Theoretical framework comparison: Stokes’ derivation of viscous stress based on physical laws was juxtaposed with Reynolds’ empirical stress closure schemes, revealing a fundamental epistemological asymmetry between the physical law-based N-S equations and phenomenologically approximation methods. 3. Modern computational validation: Contemporary Direct Numerical Simulation (DNS) demonstrated that turbulent dynamics could naturally emerge from N-S equation solutions without auxiliary models, confirming Stokes’ intuition about the equations’ inherent prediction capability. [Results] 1. Closure paradox: unlike the viscous stress governed by Newtonian mechanics in the N-S equations, Reynolds stress lacks a definitive physical closure law. Any imposed closure model constitutes a departure from the N-S framework’s physical completeness. 2. Epistemological boundaries: Turbulence models essentially belong to engineering phenomenology rather than fundamental physics, with parameters relying on calibration and validation against domain-specific observational data rather than universal principles. 3. Computational confirmation: DNS technology validates Stokes’ foresight that turbulence is an inherent property of the Navier-Stokes equations, demonstrating that vortex dynamics and flow transition are natural solutions rather than modeling artifacts. [Conclusion] Stokes’ reserved position reflects a form of prescient scientific conservatism, recognizing that although RANS models have engineering utility, their operation has exceeded the epistemological boundary of first-principles physics. The physical completeness of the N-S equations essentially excludes the possibility of establishing an independent closure law for Reynolds stress, making turbulence models inherently approximate and limited in application. This study bridges historical insights with contemporary controversies in turbulence modeling, demonstrating that mathematical parameterization cannot compensate for the absence of physical laws. While RANS remains indispensable in engineering analysis, Stokes’ implicit critique continues to highlight the unresolved fundamental challenges in fluid mechanics.

Key words: Navier-Stokes equations, Reynolds stress, turbulence model, completeness