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Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 1123–1135 | Cite as

Estimates of the Modeling Error of the \(\alpha \)-Models of Turbulence in Two and Three Space Dimensions

  • Argus A. Dunca
Article

Abstract

This report investigates the convergence rate of the weak solutions \(\mathbf{w}^{\alpha }\) of the Leray-\(\alpha \), modified Leray-\(\alpha \), Navier–Stokes-\(\alpha \) and the zeroth ADM turbulence models to a weak solution \(\mathbf{u}\) of the Navier–Stokes equations. It is assumed that this weak solution \(\mathbf{u}\) of the NSE belongs to the space \(L^4(0, T; H^1)\). It is shown that under this regularity condition the error \(\mathbf{u}-\mathbf{w}^{\alpha }\) is \(\mathcal {O}(\alpha )\) in the norms \(L^2(0, T; H^1)\) and \(L^{\infty }(0, T; L^2)\), thus improving related known results. It is also shown that the averaged error \(\overline{\mathbf{u}}-\overline{\mathbf{w}^{\alpha }}\) is higher order, \(\mathcal {O}(\alpha ^{1.5})\), in the same norms, therefore the \(\alpha \)-regularizations considered herein approximate better filtered flow structures than the exact (unfiltered) flow velocities.

Keywords

Leray-\(\alpha \) Navier–Stokes equations Navier–Stokes-\(\alpha \) model Modified Leray-\(\alpha \) model Leray-deconvolution model Viscous Camassa–Holm Zeroth ADM model Simplified Bardina model Helmholtz filter 

Mathematics Subject Classification

35Q30 76D05 

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Notes

Acknowledgements

The author thanks Edriss Titi for a helpful communication on the convergence of \(\alpha \)-models of turbulence.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interests.

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© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsUniversity Politehnica of BucharestBucharestRomania

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