A Differential Algebraic Closure Framework for Turbulence: Constructive Representation and Multiscale Analysis of Navier-Stokes Solutions
Description
This paper establishes a comprehensive differential algebraic framework for the constructive analysis and representation of turbulent flows governed by the incompressible Navier-Stokes equations. We define the turbulence differential closure KTurb through a systematic recursive adjunction process that incorporates fundamental flow structures, including exact vortex solutions, coherent structures, empirical eigenfunctions, and combinatorially corrected nonlinear interactions. Within this closure KTurb, we prove that a convergent sequence of approximants to the Navier-Stokes equations admits explicit representations through a constructive recursive procedure. This representation rigorously separates the dynamics into a spectrally expanded component and a combinatorially corrected nonlinear component, where the combinatorial corrections are derived through a recursive transport equation framework analogous to the parametrix method in geometric PDEs. The framework provides a novel algebraic perspective on turbulence closure, moving beyond traditional statistical or phenomenological models towards a structurally preserved, constructive solution theory. We provide detailed constructive proofs,derive explicit solution formulas with certified error bounds using interval arithmetic, and present algorithms for practical computation. Applications to canonical flows demonstrate the recovery of fundamental turbulent phenomena, including in termittency corrections and non-equilibrium dynamics, within our framework, establishing connections between differential algebra, geometric analysis, and turbulence theory while addressing modern challenges in turbulence modeling including non-equilibrium dynamics and control.
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turbenlence_tueory.pdf
Additional details
Additional titles
- Alternative title (English)
- A Differential Algebraic Closure Framework for Turbulence: Constructive Representation and Multiscale Analysis of Navier-Stokes Solutions
Dates
- Submitted
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2025-12-31
References
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