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A High-Order Analytical Extension of the Corrected Smagorinsky Model for Non-Equilibrium Turbulent Flow

Rômulo Damasclin Chaves dos Santos

Abstract

This study presents an extension of the corrected Smagorinsky model, incorporating advanced techniques for error estimation and regularity analysis of far-from-equilibrium turbulent flows. A new formulation that increases the model's ability to explain complex dissipative processes in turbulence is presented, using higher-order Sobolev spaces to address incompressible and compressible Navier-Stokes equations. Specifically, a refined energy dissipation mechanism that provides a more accurate representation of turbulence is introduced, particularly in the context of multifractal flow regimes. Furthermore, we derive new theoretical results on energy regularization in multifractal turbulence, contributing to the understanding of anomalous dissipation and vortex stretching in turbulent flows. The work also explores the numerical implementation of the model in the presence of challenging boundary conditions, particularly in dynamically evolving domains, where traditional methods struggle to maintain accuracy and stability. Theoretical demonstrations and analytical results are provided to validate the proposed framework, with implications for theoretical advances and practical applications in computational fluid dynamics. This approach provides a basis for more accurate simulations of turbulence, with potential applications ranging from atmospheric modeling to industrial fluid dynamics.

A High-Order Analytical Extension of the Corrected Smagorinsky Model for Non-Equilibrium Turbulent Flow

Abstract

This study presents an extension of the corrected Smagorinsky model, incorporating advanced techniques for error estimation and regularity analysis of far-from-equilibrium turbulent flows. A new formulation that increases the model's ability to explain complex dissipative processes in turbulence is presented, using higher-order Sobolev spaces to address incompressible and compressible Navier-Stokes equations. Specifically, a refined energy dissipation mechanism that provides a more accurate representation of turbulence is introduced, particularly in the context of multifractal flow regimes. Furthermore, we derive new theoretical results on energy regularization in multifractal turbulence, contributing to the understanding of anomalous dissipation and vortex stretching in turbulent flows. The work also explores the numerical implementation of the model in the presence of challenging boundary conditions, particularly in dynamically evolving domains, where traditional methods struggle to maintain accuracy and stability. Theoretical demonstrations and analytical results are provided to validate the proposed framework, with implications for theoretical advances and practical applications in computational fluid dynamics. This approach provides a basis for more accurate simulations of turbulence, with potential applications ranging from atmospheric modeling to industrial fluid dynamics.

Paper Structure

This paper contains 37 sections, 2 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Function Spaces and Norms
  4. Spaces for Velocity and Pressure
  5. Trilinear Forms and Relevant Inequalities
  6. Poincaré Inequality.
  7. Discrete Grönwall Inequality
  8. Strong Monotonicity and Local Lipschitz Continuity
  9. Reynolds Stress and Ensemble Averages
  10. Assumptions and EV Model for Turbulent Flows
  11. Mathematical Background and Extended Notations
  12. New Extensions and Functional Spaces
  13. Justification of Sobolev Regularity and Functional Space Selection
  14. Higher Regularity and Turbulence Modeling
  15. Theoretical Results and New Theorems
  16. ...and 22 more sections

Key Result

Theorem 1

Let $w \in H^s(\Omega)$ and $\partial \Omega(t)$ be a $C^k$-smooth evolving boundary. If $s > d/2$ and $w(0) \in H^s$, then $w \in C([0, T]; H^s(\Omega))$ for all $t \in [0, T]$ without singularities, given appropriate energy constraints:

Theorems & Definitions (2)

  • Theorem 1: Regularity for CSM with Dynamic Boundaries
  • Theorem 2: Error Estimation