The Navier–Stokes-alpha model of fluid turbulence

By Ciprian Foias, Darryl D. Holm, Edriss S. Titi. 2001

468 citations recorded by the source corpus.

Abstract We review the properties of the nonlinearly dispersive Navier–Stokes-alpha (NS-α) model of incompressible fluid turbulence — also called the viscous Camassa–Holm equations in the literature. We first re-derive the NS-α model by filtering the velocity of the fluid loop in Kelvin’s circulation theorem for the Navier–Stokes equations. Then we show that this filtering causes the wavenumber spectrum of the translational kinetic energy for the NS-α model to roll off as k−3 for kα>1 in three dimensions, instead of continuing along the slower Kolmogorov scaling law, k−5/3, that it follows for kα

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/c66ef2c69b79ca76cfed8743a96133c9588203c5

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Grad–div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations (2009, background): “Note that the order of velocity and pressure finite elements was never taken into account in the considerations above.”
Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models (2006, methodology): “We will obtain our results about the decay of the energy spectrum for the filtered velocity u following similar techniques as those in the NS-α [17].”
Variational principles for stochastic fluid dynamics (2014, methodology): “To guarantee that the resulting stochastic partial differential equations (SPDEs) are well-posed, one may need to mollify the Lagrangian to ensure analytical control of these gradients along the lines followed for the NavierStokes-alpha turbulence models [11, 12].”
Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows (2004, background): “We recognize here the constitutive law of a rate-dependent incompressible homogeneous fluid of second grade in which the dissipation is slightly modified by composition with the Helmholtz operator (see [15] for more details).”
Regularization modeling for large-eddy simulation (2002, methodology): “Of particular interest is the Lagrangian averaged NS−α model [13] which arises in the Euler-Poincaré framework for smoothed flow dynamics.”
A modified-Leray-α subgrid scale model of turbulence (2006, methodology): “Following similar arguments to those presented in [14] and [16] (see also [3, 9, 17]) we will study in this section the energy spectra of the ML-α model.”
On questions of decay and existence for the viscous Camassa–Holm equations (2006, unknown): “In [9], [10] these equations were studied in relation to turbulence theory, this treatment includes existence and uniqueness theorems on the torus in three dimensions.”