Leray-α Model Research Articles

Leray-α LES of magnetohydrodynamic turbulence at low magnetic Reynolds number

A series of large eddy simulations is performed for decaying homogeneous magnetohydrodynamic (MHD) turbulence at low magnetic Reynolds number (Rem <<1) with different strengths of magnetic field. The initial isotropic turbulence problem has the Taylor scale Reynolds number (Re λ) of 120. The regularization-based Leray-α model is used for sub-grid scale (SGS) closure for the first time and comparisons are made with our own direct numerical simulation (DNS) calculations conducted as part of this study. Analyses of turbulent kinetic energy decay rates, energy spectra, and vorticity fields are made between the varying magnetic field cases. The SGS model assessments are also made between the Leray-α model and the classic non-dynamic Smagorinsky with several model coefficients. Overall, the Leray-α model was unable to capture the anisotropy associated with MHD flows as well as the non-dynamic Smagorinsky model in this study or even the dynamic Smagorinsky model in previous studies.

On the Rate of Convergence of the Two-Dimensional α-Models of Turbulence to the Navier–Stokes Equations

Rates of convergence of solutions of various two-dimensional α-regularization models, subject to periodic boundary conditions, toward solutions of the exact Navier–Stokes equations are given in the L ∞-L 2 time-space norm, in terms of the regularization parameter α, when α approaches zero. Furthermore, as a paradigm, error estimates for the Galerkin approximation of the exact two-dimensional Leray-α model are presented in the L ∞-L 2 time-space norm. Simply by the triangle inequality, one can reach the error estimates of the solutions of Galerkin approximation of the α-regularization models toward the exact solutions of the Navier–Stokes equations in the two-dimensional periodic boundary conditions case.

Spectral scaling of the Leray-α model for two-dimensional turbulence

We present data from high-resolution numerical simulations of the Navier–Stokes-α and the Leray-α models for two-dimensional turbulence. It was shown previously (Lunasin et al 2007 J. Turbul. 8 30) that for wavenumbers k such that kα ≫ 1, the energy spectrum of the smoothed velocity field for the two-dimensional Navier–Stokes-α (NS-α) model scales as k−7. This result is in agreement with the scaling deduced by dimensional analysis of the flux of the conserved enstrophy using its characteristic time scale. We therefore hypothesize that the spectral scaling of any α-model in the sub-α spatial scales must depend only on the characteristic time scale and dynamics of the dominant cascading quantity in that regime of scales. The data presented here, from simulations of the two-dimensional Leray-α model, confirm our hypothesis. We show that for kα ≫ 1, the energy spectrum for the two-dimensional Leray-α scales as k−5, as expected by the characteristic time scale for the flux of the conserved enstrophy of the Leray-α model. These results lead to our conclusion that the dominant directly cascading quantity of the model equations must determine the scaling of the energy spectrum.

Three regularization models of the Navier–Stokes equations

We determine how the differences in the treatment of the subfilter-scale physics affect the properties of the flow for three closely related regularizations of Navier–Stokes. The consequences on the applicability of the regularizations as subgrid-scale (SGS) models are also shown by examining their effects on superfilter-scale properties. Numerical solutions of the Clark-α model are compared to two previously employed regularizations, the Lagrangian-averaged Navier–Stokes α-model (LANS-α) and Leray-α, albeit at significantly higher Reynolds number than previous studies, namely, Re≈3300, Taylor Reynolds number of Reλ≈790, and to a direct numerical simulation (DNS) of the Navier–Stokes equations. We derive the de Kármán–Howarth equation for both the Clark-α and Leray-α models. We confirm one of two possible scalings resulting from this equation for Clark-α as well as its associated k−1 energy spectrum. At subfilter scales, Clark-α possesses similar total dissipation and characteristic time to reach a statistical turbulent steady state as Navier–Stokes, but exhibits greater intermittency. As a SGS model, Clark-α reproduces the large-scale energy spectrum and intermittency properties of the DNS. For the Leray-α model, increasing the filter width α decreases the nonlinearity and, hence, the effective Reynolds number is substantially decreased. Therefore, even for the smallest value of α studied Leray-α was inadequate as a SGS model. The LANS-α energy spectrum ∼k1, consistent with its so-called “rigid bodies,” precludes a reproduction of the large-scale energy spectrum of the DNS at high Re while achieving a large reduction in numerical resolution. We find, however, that this same feature reduces its intermittency compared to Clark-α (which shares a similar de Kármán–Howarth equation). Clark-α is found to be the best approximation for reproducing the total dissipation rate and the energy spectrum at scales larger than α, whereas high-order intermittency properties for larger values of α are best reproduced by LANS-α.

Incompressibility of the Leray-α model for wall-bounded flows

This study shows that the Leray-α model does not explicitly enforce a divergence-free field for the filtered velocity. While this condition is automatically satisfied in the absence of boundaries, bounded domains require extra attention. It is shown, both analytically and through simulations of Rayleigh–Bénard convection, that incompressibility of the filtered velocity field cannot be guaranteed in the current formulation. Several suggestions are made to restore the incompressibility of the filtered velocity, and it is shown that free-slip boundary conditions for the filtered velocity do guarantee incompressibility for the domain under consideration.

On a Leray–α model of turbulence

In this paper we introduce and study a new model for three–dimensional turbulence, the Leray– α model. This model is inspired by the Lagrangian averaged Navier–Stokes– α model of turbulence (also known Navier–Stokes– α model or the viscous Camassa–Holm equations). As in the case of the Lagrangian averaged Navier–Stokes– α model, the Leray– α model compares successfully with empirical data from turbulent channel and pipe flows, for a wide range of Reynolds numbers. We establish here an upper bound for the dimension of the global attractor (the number of degrees of freedom) of the Leray– α model of the order of ( L / l d ) 12/7 , where L is the size of the domain and l d is the dissipation length–scale. This upper bound is much smaller than what one would expect for three–dimensional models, i.e. ( L / l d ) 3 . This remarkable result suggests that the Leray– α model has a great potential to become a good sub–grid–scale large–eddy simulation model of turbulence. We support this observation by studying, analytically and computationally, the energy spectrum and show that in addition to the usual k −5/3 Kolmogorov power law the inertial range has a steeper power–law spectrum for wavenumbers larger than 1/ α . Finally, we propose a Prandtl–like boundary–layer model, induced by the Leray– α model, and show a very good agreement of this model with empirical data for turbulent boundary layers.