Lagrangian Averaging for Compressible Fluids

Bhat, Harish S.; Fetecau, Razvan C.; Marsden, Jerrold E.; Mohseni, Kamran; West, Matthew

doi:10.1137/030601739

Cited by 27 publications

(41 citation statements)

References 26 publications

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“…In fact, alpha models can be derived from a standard fluid variational principle, by averaging over small scale fluctuations that are assumed to be advected by the larger scale flow 38,39 . Approaching the gyrokinetic variational principle in a similar way leads to the addition of extra, regularized terms into the gyrokinetic Lagrangian.…”

Section: Numerical Applicationsmentioning

confidence: 99%

“…38, averaging over perpendicular X-space fluctuations of scale length α, and ensuring gauge invariance, we were led to the regularized Lagrangian l = l GK + l α , where…”

Section: Numerical Applicationsmentioning

confidence: 99%

“…We note that in discretizing a variational principle it is obviously not desirable to be in an extended phase space, unless these extra dimensions can somehow be removed after a discretization. As well as integrators, other potential applications of the formulation presented here are the use in stability calculations with Casimir invariants 35 and the construction of regularized models for large-eddy simulation [36][37][38][39] .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

Squire¹,

Qin²,

Tang³

2012

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We present a new variational principle for the gyrokinetic system, similar to the MaxwellVlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with the Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincaré theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods.

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Section: Numerical Applicationsmentioning

confidence: 99%

“…38, averaging over perpendicular X-space fluctuations of scale length α, and ensuring gauge invariance, we were led to the regularized Lagrangian l = l GK + l α , where…”

Section: Numerical Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

Squire¹,

Qin²,

Tang³

2012

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“…5 where we show the temporal evolution of an initial bump. We 4 , ∆t 4 , µ 2 ∆x 2 , µ 2 ∆t 2 ) since, for example, y = z. We employ again periodic boundary conditions.…”

Section: Discussionmentioning

confidence: 99%

“…The nonlinear dispersive terms are of the same form as the nonlinear dispersive terms of the Camassa-Holm equation (4). The exclusive occurrence of dispersive terms is due to our scheme being centred in time and space.…”

Section: Backward Error Analysis and The Modified Equationmentioning

confidence: 95%

Dispersive regularizations and numerical discretizations for the inviscid Burgers equation

Gottwald

2007

J. Phys. A: Math. Theor.

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We study centred second-order in time and space discretizations of the inviscid Burgers equation. Although this equation in its continuum formulation supports non-smooth shock wave solutions, the discrete equation generically supports smooth solitary wave solutions. Using backward error analysis we derive the modified equation associated with the numerical scheme. We identify three different equations, the Korteweg-de Vries (KdV) equation, the Camassa-Holm (CH) equation and the b = 0 member of the b-family. Solutions of the first two equations are solitary waves and do not converge to the shock solutions of the Burgers equation. The third equation however supports solutions which strongly approximate weak solutions of the Burgers equation. We corroborate our analytical results with numerical simulations.

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Analysis of a General Family of Regularized Navier–Stokes and MHD Models

2010

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We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n ≥ 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-α model, the Leray-α model, the modified Leray-α model, the simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-α-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. This family of models has become particularly important in the development of mathematical and computational models of turbulence. We give a unified analysis of the entire three-parameter family of models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α → 0 limit in α models. Next, we show existence of a global attractor for the general model, and then give estimates for the dimension of the global attractor and the number of degrees of freedom in terms of a generalized Grashof number. We then establish some results on determining operators for the two distinct subfamilies of dissipative and non-dissipative models. We finish by deriving some new length-scale estimates in terms of the Reynolds number, which allows for recasting the Grashof number-based results into analogous statements involving the Reynolds number. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, and determining operators for the well-known specific members of this family of regularized Navier-Stokes and MHD models, the framework we develop also makes possible a number of new results for all models in the general family, including some new results for several of the well-studied models. Analyzing the more abstract generalized model allows for a simpler analysis that helps bring out the core common structure of the various regularized Navier-Stokes and magnetohydrodynamics models, and also helps clarify the common features of many of the existing and new results. To make the paper reasonably self-contained, we include supporting material on spaces involving time, Sobolev spaces, and Grönwall-type inequalities.

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Lagrangian Averaging for Compressible Fluids

Cited by 27 publications

References 26 publications

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

Dispersive regularizations and numerical discretizations for the inviscid Burgers equation

Analysis of a General Family of Regularized Navier–Stokes and MHD Models

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