Abstract
The subfilter-scale (SFS) physics of regularization models are investigated to understand the regularizations’ performance as SFS models. Suppression of spectrally local SFS interactions and conservation of small-scale circulation in the Lagrangian-averaged Navier-Stokes α-model (LANS-α) is found to lead to the formation of rigid bodies. These contaminate the superfilter-scale energy spectrum with a scaling that approaches k +1 as the SFS spectra is resolved. The Clark-α and Leray-α models, truncations of LANS-α, do not conserve small-scale circulation and do not develop rigid bodies. LANS-α, however, is closest to Navier-Stokes in intermittency properties. All three models are found to be stable at high Reynolds number. Differences between L 2 and H 1 norm models are clarified. For magnetohydrodynamics (MHD), the presence of the Lorentz force as a source (or sink) for circulation and as a facilitator of both spectrally nonlocal large to small scale interactions as well as local SFS interactions prevents the formation of rigid bodies in Lagrangian-averaged MHD (LAMHD-α). LAMHD-α performs well as a predictor of superfilter-scale energy spectra and of intermittent current sheets at high Reynolds numbers. It may prove generally applicable as a MHD-LES.
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Alexakis, A., Mininni, P.D., Pouquet, A.: Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence. Phys. Rev. E 72(4), 046301 (2005)
Aluie, H., Eyink, G.L.: Localness of energy cascade in hydrodynamic turbulence. II. Sharp spectral filter. Phys. Fluids 21(11), 115108 (2009)
Aluie, H., Eyink, G.L.: Scale-locality of magnetohydrodynamic turbulence. ArXiv e-prints (2009)
Baerenzung, J., Politano, H., Ponty, Y., Pouquet, A.: Spectral modeling of magnetohydrodynamic turbulent flows. Phys. Rev. E 78(2), 026310 (2008)
Berselli, L.C., Grisanti, C.R.: On the consistency of the rational large eddy simulation model. Comput. Vis. Sci. 6(2–3), 75–82 (2004)
Brachet, M.E., Mininni, P.D., Rosenberg, D.L., Pouquet, A.: High-order low-storage explicit Runge-Kutta schemes for equations with quadratic nonlinearities. ArXiv e-prints (2008)
Canuto, C., Yousuff Hussaini, M., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)
Cao, C., Holm, D.D., Titi, E.S.: On the Clark α model of turbulence: global regularity and long-time dynamics. J. Turbul. 6, N20 (2005)
Chandy, A., Frankel, S.: Regularization-based sub-grid scale (SGS) models for large eddy simulations (LES) of high-Re decaying isotropic turbulence. J. Turbul. 10, 25 (2009)
Chen, S., Foias, C., Holm, D.D., Olson, E., Titi, E.S., Wynne, S.: Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81, 5338–5341 (1998)
Chollet, J.-P., Lesieur, M.: Parameterization of small scales of three-dimensional isotropic turbulence utilizing spectral closures. J. Atmos. Sci. 38, 2747–2757 (1981)
Cichowlas, C., Bonaïti, P., Debbasch, F., Brachet, M.: Effective dissipation and turbulence in spectrally truncated Euler flows. Phys. Rev. Lett. 95(26), 264502 (2005)
Dahlburg, J.P., Montgomery, D., Doolen, G.D., Matthaeus, W.H.: Large-scale disruptions in a current-carrying magnetofluid. J. Plasma Phys. 35, 1–42 (1986)
Foias, C., Holm, D.D., Titi, E.S.: The Navier-Stokes-alpha model of fluid turbulence. Physica D, Nonlinear Phenom. 152–153, 505–519 (2001)
Galdi, G.P., Layton, W.J.: Approximation of the larger eddies in fluid motions. II: A model for space-filtered flow. Math. Models Methods Appl. Sci. 10, 343–350 (2000)
Galloway, D., Frisch, U.: Dynamo action in a family of flows with chaotic streamlines. Geophys. Astrophys. Fluid Dyn. 36, 53–83 (1986)
Geurts, B.J., Holm, D.D.: Regularization modeling for large-eddy simulation. Phys. Fluids 15, L13–L16 (2003)
Geurts, B.J., Holm, D.D.: Leray and LANS-α modelling of turbulent mixing. J. Turbul. 7(10), 1–33 (2006)
Goldreich, P., Sridhar, S.: Toward a theory of interstellar turbulence. 2: Strong Alfvénic turbulence. Astrophys. J. 438, 763–775 (1995)
Gómez, D.O., Mininni, P.D., Dmitruk, P.: MHD simulations and astrophysical applications. Adv. Space Res. 35, 899–907 (2005)
Gómez, D.O., Mininni, P.D., Dmitruk, P.: Parallel simulations in turbulent MHD. Phys. Scr. T 116, 123–127 (2005)
Guermond, J.-L.: On the use of the notion of suitable weak solutions in CFD. Int. J. Numer. Methods Fluids 57, 1153–1170 (2008)
Holm, D.D.: Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics. Physica D, Nonlinear Phenom. 170, 253–286 (2002)
Holm, D.D.: Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics. Chaos 12, 518–530 (2002)
Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1–81 (1998)
Hughes, T.J.R., Mazzei, L., Oberai, A.A., Wray, A.A.: The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence. Phys. Fluids 13, 505–512 (2001)
Iroshnikov, P.S.: Turbulence of a conducting fluid in a strong magnetic field. Soviet Astron. 7, 566 (1964)
John, V.: An assessment of two models for the subgrid scale tensor in the rational LES model. J. Comput. Appl. Math. 173(1), 57–80 (2005)
Kim, T.-Y., Cassiani, M., Albertson, J.D., Dolbow, J.E., Fried, E., Gurtin, M.E.: Impact of the inherent separation of scales in the Navier-Stokes-α β equations. Phys. Rev. E 79(4), 045307 (2009)
Knaepen, B., Moin, P.: Large-eddy simulation of conductive flows at low magnetic Reynolds number. Phys. Fluids 16, 1255 (2004)
Kraichnan, R.H.: Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 1385–1387 (1965)
Labovschii, A., Trenchea, C.: Approximate deconvolution models for magnetohydrodynamics. Technical report, University of Pittsburgh (2010)
Larios, A., Titi, E.S.: On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models. ArXiv e-prints (2009)
Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9, 267–293 (1956)
Lee, E., Brachet, M.E., Pouquet, A., Mininni, P.D., Rosenberg, D.: Lack of universality in decaying magnetohydrodynamic turbulence. Phys. Rev. E 81(1), 016318 (2010)
Levant, B., Ramos, F., Titi, E.S.: On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model. ArXiv e-prints (2009)
Mason, J., Cattaneo, F., Boldyrev, S.: Numerical measurements of the spectrum in magnetohydrodynamic turbulence. Phys. Rev. E 77(3), 036403 (2008)
Meneveau, C., Katz, J.: Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech. 32, 1–32 (2000)
Mininni, P.D., Alexakis, A., Pouquet, A.: Nonlocal interactions in hydrodynamic turbulence at high Reynolds numbers: The slow emergence of scaling laws. Phys. Rev. E 77(3), 036306 (2008)
Mininni, P.D., Montgomery, D.C., Pouquet, A.: Numerical solutions of the three-dimensional magnetohydrodynamic α model. Phys. Rev. E 71(4), 046304 (2005)
Mininni, P.D., Montgomery, D.C., Pouquet, A.G.: A numerical study of the alpha model for two-dimensional magnetohydrodynamic turbulent flows. Phys. Fluids 17(3), 035112 (2005)
Mohseni, K., Kosović, B., Shkoller, S., Marsden, J.E.: Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous isotropic turbulence. Phys. Fluids 15, 524–544 (2003)
Montgomery, D.C., Pouquet, A.: An alternative interpretation for the Holm “alpha model”. Phys. Fluids 14(9), 3365–3366 (2002)
Müller, W.-C., Carati, D.: Dynamic gradient-diffusion subgrid models for incompressible magnetohydrodynamic turbulence. Phys. Plasmas 9, 824–834 (2002)
Pietarila Graham, J., Holm, D., Mininni, P., Pouquet, A.: Highly turbulent solutions of the Lagrangian-averaged Navier-Stokes alpha model and their large-eddy-simulation potential. Phys. Rev. E 76, 056310 (2007)
Pietarila Graham, J., Holm, D.D., Mininni, P., Pouquet, A.: Inertial range scaling, Kármán-Howarth theorem, and intermittency for forced and decaying Lagrangian averaged magnetohydrodynamic equations in two dimensions. Phys. Fluids 18, 045106 (2006)
Pietarila Graham, J., Holm, D.D., Mininni, P.D., Pouquet, A.: Three regularization models of the Navier-Stokes equations. Phys. Fluids 20(3), 035107 (2008)
Pietarila Graham, J., Mininni, P.D., Pouquet, A.: Lagrangian-averaged model for magnetohydrodynamic turbulence and the absence of bottlenecks. Phys. Rev. E 80(1), 016313 (2009)
Ponty, Y., Politano, H., Pinton, J.-F.: Simulation of induction at low magnetic Prandtl number. Phys. Rev. Lett. 92(14), 144503 (2004)
Ramos, F., Titi, E.S.: Invariant measures for the 3D Navier-Stokes-Voigt equations and their Navier-Stokes limit. ArXiv e-prints (2009)
Taylor, G.I., Green, A.E.: Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A 158, 499 (1937)
Theobald, M.L., Fox, P.A., Sofia, S.: A subgrid-scale resistivity for magnetohydrodynamics. Phys. Plasmas 1, 3016–3032 (1994)
van Reeuwijk, M., Jonker, H.J.J., Hanjalić, K.: Leray-α simulations of wall-bounded turbulent flows. Int. J. Heat Fluid Flow 30, 1044 (2009)
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Pietarila Graham, J., Holm, D.D., Mininni, P. et al. The Effect of Subfilter-Scale Physics on Regularization Models. J Sci Comput 49, 21–34 (2011). https://doi.org/10.1007/s10915-010-9428-4
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DOI: https://doi.org/10.1007/s10915-010-9428-4