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Energy cascade for cross-shear length scales in free-shear three-dimensional incompressible viscous flows

Ricardo M. S. Rosa

Abstract

The phenomenon of energy cascade is addressed in the case of free-shear flows, modeled with the equations for incompressible Newtonian fluids with mixed periodic and free-slip boundary conditions driven by an imposed mean shear profile. The rigorous results are proved for ensemble averages with respect to stationary statistical solutions in the sense of Foias and Prodi. We obtain the energy-budget relations with an energy dissipation term, a shear-production term from the mean flow, and an energy flux term of the fluctuation field, based on a decomposition of the flow into high and low horizontal wavenumber components, corresponding to scales perpendicular to the mean shear gradient. We estimate the shear-production term exploiting the orthogonality of horizontal Fourier modes and look for an energy cascade of the fluctuation flux. For any given wavenumber, we define an associated horizontal Taylor wavenumber of the high-wavenumber band above the given wavenumber. We prove two energy cascade results, for the energy flux and for a restricted energy flux that discounts possible energy loss to flow singularities, both valid for wavenumbers whose associated horizontal Taylor wavenumber lies above the viscous shear wavenumber and whose low-wavenumber energy dissipation rate is negligible compared to the total energy dissipation rate. On heuristic grounds, these rigorous conditions correspond precisely to an energy cascade in the range from the classical Corrsin scale down to the Kolmogorov dissipation scale, as expected in the conventional theory of turbulent shear flows.

Energy cascade for cross-shear length scales in free-shear three-dimensional incompressible viscous flows

Abstract

The phenomenon of energy cascade is addressed in the case of free-shear flows, modeled with the equations for incompressible Newtonian fluids with mixed periodic and free-slip boundary conditions driven by an imposed mean shear profile. The rigorous results are proved for ensemble averages with respect to stationary statistical solutions in the sense of Foias and Prodi. We obtain the energy-budget relations with an energy dissipation term, a shear-production term from the mean flow, and an energy flux term of the fluctuation field, based on a decomposition of the flow into high and low horizontal wavenumber components, corresponding to scales perpendicular to the mean shear gradient. We estimate the shear-production term exploiting the orthogonality of horizontal Fourier modes and look for an energy cascade of the fluctuation flux. For any given wavenumber, we define an associated horizontal Taylor wavenumber of the high-wavenumber band above the given wavenumber. We prove two energy cascade results, for the energy flux and for a restricted energy flux that discounts possible energy loss to flow singularities, both valid for wavenumbers whose associated horizontal Taylor wavenumber lies above the viscous shear wavenumber and whose low-wavenumber energy dissipation rate is negligible compared to the total energy dissipation rate. On heuristic grounds, these rigorous conditions correspond precisely to an energy cascade in the range from the classical Corrsin scale down to the Kolmogorov dissipation scale, as expected in the conventional theory of turbulent shear flows.
Paper Structure (22 sections, 2 theorems, 109 equations, 1 table)

This paper contains 22 sections, 2 theorems, 109 equations, 1 table.

Table of Contents

  1. Introduction
  2. Framework
  3. Model derivation
  4. The flow domain
  5. Functional spaces
  6. Weak solutions and functional formulation
  7. The spectral decomposition of the Stokes operator
  8. Spectral representation of vector fields and horizontal spectral projections
  9. Stationary statistical solutions
  10. Ensemble averages and mean quantities
  11. Mean scalar physical quantities
  12. Energy-budget relations
  13. Mean energy-budget equations
  14. Estimating the production term
  15. Asymptotic behavior of the flux term as the wavenumber increases
  16. ...and 7 more sections

Key Result

Theorem 4.1

Let $\delta > 0$. Then, for any $\bar{\kappa}$ such that we have the following lower bound on the energy flux:

Theorems & Definitions (4)

  • Theorem 4.1
  • Remark 4.1
  • Theorem 4.2
  • Remark 4.2