The Vortex Blob Method as a Second-Grade Non-Newtonian Fluid

Oliver, Marcel; Shkoller, Steve

doi:10.1081/pde-100001756

Cited by 54 publications

(87 citation statements)

References 21 publications

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“…This is a promising result for LES modeling, since the mathematical theorems available in [6] for existence, uniqueness and finite dimensional global attractor for the NS-α model will now be transferable to the continuous formulations of this class of generalized LES similarity models. As a basis for numerical schemes, the NS-α model also resembles vorticity methods, as observed in [20,40,41].…”

Section: Dns and Comparisons With Lesmentioning

confidence: 99%

The Navier–Stokes-alpha model of fluid turbulence

Foiaş

Holm

Titi

2001

Physica D: Nonlinear Phenomena

378

463

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We review the properties of the nonlinearly dispersive Navier-Stokes-alpha (NS-α) model of incompressible fluid turbulence -also called the viscous Camassa-Holm equations in the literature. We first re-derive the NS-α model by filtering the velocity of the fluid loop in Kelvin's circulation theorem for the Navier-Stokes equations. Then we show that this filtering causes the wavenumber spectrum of the translational kinetic energy for the NS-α model to roll off as k −3 for kα > 1 in three dimensions, instead of continuing along the slower Kolmogorov scaling law, k −5/3 , that it follows for kα < 1. This roll off at higher wavenumbers shortens the inertial range for the NS-α model and thereby makes it more computable. We also explain how the NS-α model is related to large eddy simulation (LES) turbulence modeling and to the stress tensor for second-grade fluids. We close by surveying recent results in the literature for the NS-α model and its inviscid limit (the Euler-α model).

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Section: Dns and Comparisons With Lesmentioning

confidence: 99%

The Navier–Stokes-alpha model of fluid turbulence

Foiaş

Holm

Titi

2001

Physica D: Nonlinear Phenomena

378

463

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“…If one discretizes the averaged Lagrangian Euler equations by the point vortex method, i.e. to approximate the initial vorticity by a collection of point vortices (Dirac delta functions), then the resulting numerical approximation is a vortex blob method with α being the vortex blob size [22,13].…”

Section: Introductionmentioning

confidence: 99%

On Global Well‐Posedness of the Lagrangian Averaged Euler Equations

Hou¹,

Li²

2006

SIAM J. Math. Anal.

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We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions. We show that a necessary and sufficient condition for the global existence is that the BMO norm of the stream function is integrable in time. We also derive a sufficient condition in terms of the total variation of certain level set functions, which guarantees the global existence. Furthermore, we obtain the global existence of the Lagrangian averaged 2D Boussinesq equations and the Lagrangian averaged 2D quasi-geostrophic equations in finite Sobolev space in the absence of viscosity or dissipation.

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“…The global regularity of the solution to LANS-α equations in a three-dimensional bounded domain at smooth initial velocity and zero volume forces, and zero boundary conditions is proved in [31] It is shown in [7], [35] that solutions to LANS-α equations converge to the solution of the Navier-Stokes equations as α goes to zero. Because of this, for any Reynolds number, the velocity profile of the LANS-α fluid in the circular tube is close to parabolic for small α.…”

Section: Figurementioning

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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The Vortex Blob Method as a Second-Grade Non-Newtonian Fluid

Cited by 54 publications

References 21 publications

The Navier–Stokes-alpha model of fluid turbulence

The Navier–Stokes-alpha model of fluid turbulence

On Global Well‐Posedness of the Lagrangian Averaged Euler Equations

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

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