A Two-Level Discretization Method for the Smagorinsky Model

Borggaard, Jeff; Iliescu, Traian; Lee, Hyesuk; Roop, John Paul; Son, Hyungbin

doi:10.1137/070704812

Cited by 21 publications

(21 citation statements)

References 29 publications

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“…As will be seen from the following presentation, the Theorems 3.1-3.4 can be extended to this case. 4. Auxiliary results.…”

Section: Nonstationary Problemsmentioning

confidence: 99%

Model for laminar and turbulent flows of viscous and nonlinear viscous non-Newtonian fluids

Litvinov

2011

Journal of Mathematical Physics

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We develop a model that describes laminar and turbulent flows of viscous and nonlinear viscous fluids. A basis for the model is a characteristic, which we call a local Reynolds number, and which is calculated at each point of the domain of flow. In the subdomain wherein the local Reynolds number does not exceed some value, the flow is laminar; otherwise it is turbulent, i.e., the model identifies the areas of laminar and turbulent flows. The model predicts the existence of a laminar boundary layer at turbulent flows. It enables us to describe special features of turbulent flows such as a drastic increase in the resistance to flow and the variation of the velocity profile with the increase of the Reynolds number, and so on. The model is mathematically grounded. We prove the existence of global solutions to stationary and nonstationary flow problems with the nonhomogeneous Dirichlet and mixed boundary conditions, where velocities and surface forces are given on different parts of the boundary. Numerical methods for solving stationary and nonstationary flow problems are investigated.

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“…As will be seen from the following presentation, the Theorems 3.1-3.4 can be extended to this case. 4. Auxiliary results.…”

Section: Nonstationary Problemsmentioning

confidence: 99%

Model for laminar and turbulent flows of viscous and nonlinear viscous non-Newtonian fluids

Litvinov

2011

Journal of Mathematical Physics

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“…Proof The proof follows the same steps as that in [35], except for the fact that using the inverse estimate in [41], we have $|{\bf v}^{H}|_{1,r} \leq C H ^{{(2-r)d\over 2r}} |{\bf v}^{H}|_{1}$ , $\forall {\bf v}^{H} \in {\bf X}^{H}$ .…”

Section: Nonlinear Subgridscale Modelmentioning

confidence: 99%

“…The article [35], which examined error analysis and scalings associated with the Smagorinsky model, was the first application of the two‐level methodology in a large eddy simulation (LES) setting. The study in this article differs from that in [35] in several respects. The most important difference is that this study contains several 3D numerical tests, one of which is the flow past a backward‐facing step at a Reynolds number Re = 5100.…”

Section: Introductionmentioning

confidence: 99%

“…The only other 3D numerical investigation of the two‐level methodology was performed by Medjo and Temam in [37]; that investigation, however, was for a different two‐level approach and was applied to the primitive equations, not the Navier‐Stokes equations. Another departure from [35] is that the error analysis for the r ‐Laplacian introduces challenges that were not present for the Smagorinsky model; such a challenge and the way it is tackled are presented in Lemma 4.1.…”

Section: Introductionmentioning

confidence: 99%

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Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

Borggaard

Iliescu

Roop

2011

Numerical Methods Partial

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The r-Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two-level finite element approximation schemes applied to the Navier-Stokes equations with r-Laplacian subgridscale viscosity, where r is the order of the power-law artificial viscosity term. In the two-level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single-step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two-level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two-level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H , fine mesh size h, and filter radius δ. We also investigate the two-level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward-facing step at Reynolds number Re = 5100. In all numerical tests, the two-level algorithm was proven to achieve the same order of accuracy as the standard one-level algorithm, at a fraction of the computational cost.

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“…The fine structures are captured by solving one linear system (linearized about the coarse mesh approximation) on a fine mesh. On the two-level methods, Xu [30,31] gave a framework of two-level methods (sometimes named two-grid methods) for PDEs, Layton et al [32][33][34][35] developed the two-level method for the Navier-Stokes equations, and He, Li, and Hou [36][37][38] improved two-level method for the Navier-Stokes equations and developed one-step Newton method, Borggaard et al presented a two-level discretization method for the Smagorinsky model in LES in [39], and for the Navier-Stokes equations with r-Laplacian Subgridscale Viscosity very recently in [40], based on one-step Newton on the fine mesh, and so forth.…”

Section: Introductionmentioning

confidence: 99%

A two‐level variational multiscale method for incompressible flows based on two local Gauss integrations

Li¹,

Mei²,

Li³

et al. 2013

Numerical Methods Partial

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In this article, a two-level variational multiscale method for incompressible flows based on two local Gauss integrations is presented. We solve the Navier-Stokes problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations, then seek a fine grid solution by solving a linearized problem on a fine grid. In computation, we use the two local Gauss integrations to replace the projection operator without adding any variables. Stability analysis is performed, and error estimates of the method are derived. Finally, a series of numerical experiments are also given, which confirm the theoretical analysis and demonstrate the efficiency of the new method.

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A Two-Level Discretization Method for the Smagorinsky Model

Cited by 21 publications

References 29 publications

Model for laminar and turbulent flows of viscous and nonlinear viscous non-Newtonian fluids

Model for laminar and turbulent flows of viscous and nonlinear viscous non-Newtonian fluids

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

A two‐level variational multiscale method for incompressible flows based on two local Gauss integrations

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