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.
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The author thanks Edriss Titi for a helpful communication on the convergence of \(\alpha \)-models of turbulence.
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Communicated by G.P. Galdi
This work has been funded by University Politehnica of Bucharest, through the Excellence Research Grants Program, UPB GEX. No. 26/2016, Internal No. MA 52.16.01.
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Dunca, A.A. Estimates of the Modeling Error of the \(\alpha \)-Models of Turbulence in Two and Three Space Dimensions. J. Math. Fluid Mech. 20, 1123–1135 (2018). https://doi.org/10.1007/s00021-017-0357-y
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DOI: https://doi.org/10.1007/s00021-017-0357-y
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