The <i>t</i>-Model as a Large Eddy Simulation Model for the Navier–Stokes Equations

Chandy, Abhilash J.; Frankel, Steven

doi:10.1137/090760787

Cited by 15 publications

(16 citation statements)

References 22 publications

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“…Applications of the Mori-Zwanzig approach to fluid dynamics can be found in Stinis [2007], Chandy and Frankel [2009], Hald and Stinis [2007], and Hou [2007]. A simple approximation to Mori-Zwanzig has been applied to jet formation on a beta plane in Tobias and Marston [2013].…”

Section: Projection Operator Techniquesmentioning

confidence: 99%

Mathematical and physical ideas for climate science

et al. 2014

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The climate is a forced and dissipative nonlinear system featuring nontrivial dynamics on a vast range of spatial and temporal scales. The understanding of the climate's structural and multiscale properties is crucial for the provision of a unifying picture of its dynamics and for the implementation of accurate and efficient numerical models. We present some recent developments at the intersection between climate science, mathematics, and physics, which may prove fruitful in the direction of constructing a more comprehensive account of climate dynamics. We describe the Nambu formulation of fluid dynamics and the potential of such a theory for constructing sophisticated numerical models of geophysical fluids. Then, we focus on the statistical mechanics of quasi-equilibrium flows in a rotating environment, which seems crucial for constructing a robust theory of geophysical turbulence. We then discuss ideas and methods suited for approaching directly the nonequilibrium nature of the climate system. First, we describe some recent findings on the thermodynamics of climate, characterize its energy and entropy budgets, and discuss related methods for intercomparing climate models and for studying tipping points. These ideas can also create a common ground between geophysics and astrophysics by suggesting general tools for studying exoplanetary atmospheres. We conclude by focusing on nonequilibrium statistical mechanics, which allows for a unified framing of problems as different as the climate response to forcings, the effect of altering the boundary conditions or the coupling between geophysical flows, and the derivation of parametrizations for numerical models.

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Section: Projection Operator Techniquesmentioning

confidence: 99%

Mathematical and physical ideas for climate science

et al. 2014

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“…The code is written in Fortran 90 and employs the P3DFFT parallel 3D fast Fourier transform library, which makes use of 2D, or pencil, decomposition to overcome an important scalability limitations associated with other 3D parallel FFT libraries using 1D or slab decomposition. For more details, see the authors' recent papers [26,30]. Computing time was provided by the Purdue University Supercomputing.…”

Section: Computational Detailsmentioning

confidence: 99%

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

Chandy

Frankel

2011

Journal of Turbulence

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A series of large eddy simulations is performed for decaying homogeneous magnetohydrodynamic (MHD) turbulence at low magnetic Reynolds number (Re m << 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.

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“…Its accuracy additionally requires that the flow invoked by the true dynamics e tL is slow to evolve in time. Despite its simplicity, the t-model has been successfully applied to Burgers equation and as a sub-grid model in Large Eddy Simulations [13].…”

Section: B Modeling Of the Memory Kernelmentioning

confidence: 99%

“…The t-model correctly preserved energy for two-dimensional simulations and provided results consistent with theory for three-dimensional simulations of the Taylor-Green Vortex. Chandy and Frankel [13] used the t-model in Large Eddy Simulations of homogeneous turbulence and the Taylor-Green vortex. The model was found to be in good agreement with Direct Numerical Simulation and experimental data for low Reynolds number cases, but discrepancies were seen for higher Reynolds numbers.…”

Section: Introductionmentioning

confidence: 99%

Non-Markovian closure models for large eddy simulations using the Mori-Zwanzig formalism

Parish

Duraisamy

2017

Phys. Rev. Fluids

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This work uses the Mori-Zwanzig (M-Z) formalism, a concept originating from non-equilibrium statistical mechanics, as a basis for the development of coarse-grained models of turbulence. The mechanics of the generalized Langevin equation (GLE) are considered and insight gained from the orthogonal dynamics equation is used as a starting point for model development. A class of sub-grid models is considered which represent non-local behavior via a finite memory approximation (Stinis, P., "Mori-Zwanzig reduced models for uncertainty quantification I: Parametric uncertainty," arXiv:1211.4285, 2012.), the length of which is determined using a heuristic that is related to the spectral radius of the Jacobian of the resolved variables. The resulting models are intimately tied to the underlying numerical resolution and are capable of approximating non-Markovian effects. Numerical experiments on the Burgers equation demonstrate that the M-Z-based models can accurately predict the temporal evolution of the total kinetic energy and the total dissipation rate at varying mesh resolutions. The trajectory of each resolved mode in phase-space is accurately predicted for cases where the coarse-graining is moderate. LES of homogeneous isotropic turbulence and the Taylor Green Vortex show that the M-Z-based models are able to provide excellent predictions, accurately capturing the sub-grid contribution to energy transfer. Lastly, LES of fully developed channel flow demonstrate the applicability of M-Z-based models to non-decaying problems. It is notable that the form of the closure is not imposed by the modeler, but is rather derived from the mathematics of the coarse-graining, highlighting the potential of M-Z-based techniques to define LES closures.

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The t-Model as a Large Eddy Simulation Model for the Navier–Stokes Equations

Cited by 15 publications

References 22 publications

Mathematical and physical ideas for climate science

Mathematical and physical ideas for climate science

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

Non-Markovian closure models for large eddy simulations using the Mori-Zwanzig formalism

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