Two-Dimensional "Poor Man's Navier-Stokes Equation" Model of Turbulent Flows

AIAA Journal ◽  
2003 ◽  
Vol 41 (9) ◽  
pp. 1690-1696 ◽  
Author(s):  
Tianliang Yang ◽  
J. M. McDonough ◽  
J. D. Jacob
Keyword(s):  
Turbulent Flows ◽  
Stokes Equation ◽  
Equation Model ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation

BASINS OF ATTRACTION OF THE TWO-DIMENSIONAL "POOR MAN'S NAVIER–STOKES EQUATION"

2004 ◽  
Vol 14 (07) ◽  
pp. 2381-2386 ◽  
Author(s):  
S. A. BIBLE ◽  
J. M. MCDONOUGH
Keyword(s):  
Turbulent Flows ◽  
Stokes Equation ◽  
Initial Conditions ◽  
Power Spectra ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Large Eddy

This research is part of an ongoing effort to construct "synthetic velocity" subgrid-scale (SGS) models using discrete dynamical systems (DDSs) for use in large-eddy simulations of turbulent flows. Here we will outline the derivation of the two-dimensional (2-D) "Poor Man's Navier–Stokes" (PMNS) equation from the 2-D, incompressible Navier–Stokes equation to be used in such models and report results from subsequent numerical investigations. In our results emphasis is placed on the effects of initial conditions on the dynamics of the 2-D PMNS equation, using such modes of investigation as regime maps, basins of attraction diagrams, phase portraits, time series and power spectra. The most important findings of this investigation concern applicable ranges of bifurcation parameters, causes and effects of symmetries seen in solutions of the PMNS equation, and the suitability of the methods of investigation used here.

scholarly journals GLOBAL EXISTENCE RESULTS FOR THE ANISOTROPIC BOUSSINESQ SYSTEM IN DIMENSION TWO

2011 ◽  
Vol 21 (03) ◽  
pp. 421-457 ◽  
Author(s):  
RAPHAËL DANCHIN ◽  
MARIUS PAICU
Keyword(s):  
Global Existence ◽  
Turbulent Flows ◽  
Vertical Direction ◽  
Boussinesq System ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Incompressible Euler Equation ◽  
Navier Stokes Equation ◽  
Transport Diffusion ◽  
Anisotropic Viscosity

Models with a vanishing anisotropic viscosity in the vertical direction are of relevance for the study of turbulent flows in geophysics. This motivates us to study the two-dimensional Boussinesq system with horizontal viscosity in only one equation. In this paper, we focus on the global existence issue for possibly large initial data. We first examine the case where the Navier–Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transport–diffusion equation with diffusion in the horizontal direction only. For both systems, we construct global weak solutions à la Leray and strong unique solutions for more regular data. Our results rest on the fact that the diffusion acts perpendicularly to the buoyancy force.

scholarly journals Solutions of Navier-Stokes Equation with Coriolis Force

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Sunggeun Lee ◽  
Shin-Kun Ryi ◽  
Hankwon Lim
Keyword(s):  
Diffusion Equation ◽  
Stokes Equation ◽  
Coriolis Force ◽  
Potential Flow ◽  
Convective Diffusion ◽  
Navier Stokes ◽  
Coriolis Effect ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Convective Diffusion Equation

We investigate the Navier-Stokes equation in the presence of Coriolis force in this article. First, the vortex equation with the Coriolis effect is discussed. It turns out that the vorticity can be generated due to a rotation coming from the Coriolis effect, Ω. In both steady state and two-dimensional flow, the vorticity vector ω gets shifted by the amount of -2Ω. Second, we consider the specific expression of the velocity vector of the Navier-Stokes equation in two dimensions. For the two-dimensional potential flow v→=∇→ϕ, the equation satisfied by ϕ is independent of Ω. The remaining Navier-Stokes equation reduces to the nonlinear partial differential equations with respect to the velocity and the corresponding exact solution is obtained. Finally, the steady convective diffusion equation is considered for the concentration c and can be solved with the help of Navier-Stokes equation for two-dimensional potential flow. The convective diffusion equation can be solved in three dimensions with a simple choice of c.

Numerical simulations using conserved wave absorption applied to Navier–Stokes equation model

2015 ◽  
Vol 99 ◽  
pp. 15-25 ◽  
Author(s):  
Zhe Hu ◽  
Wenyong Tang ◽  
Hongxiang Xue ◽  
Xiaoying Zhang ◽  
Jinting Guo
Keyword(s):  
Numerical Simulations ◽  
Stokes Equation ◽  
Equation Model ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Wave Absorption

Simulation of the dynamic two-dimensional Navier-Stokes equation applying the Galerkin methodology

2018 ◽  
Vol 12 (10) ◽  
pp. 467-475
Author(s):  
E.J. Canate-Gonzalez ◽  
W. Fong-Silva ◽  
C.A. Severiche-Sierra ◽  
Y.A. Marrugo-Ligardo ◽  
J. Jaimes-Morales
Keyword(s):  
Stokes Equation ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation

Casimir cascades in two-dimensional turbulence

2013 ◽  
Vol 729 ◽  
pp. 364-376 ◽  
Author(s):  
John C. Bowman
Keyword(s):  
Stokes Equation ◽  
Navier Stokes ◽  
Fourth Power ◽  
Two Dimensional ◽  
Vorticity Field ◽  
Navier Stokes Equation ◽  
Small Scales ◽  
Open Question ◽  
Incompressible Navier Stokes Equation

AbstractIn addition to conserving energy and enstrophy, the nonlinear terms of the two-dimensional incompressible Navier–Stokes equation are well known to conserve the global integral of any continuously differentiable function of the scalar vorticity field. However, the phenomenological role of these additional inviscid invariants, including the issue as to whether they cascade to large or small scales, is an open question. In this work, well-resolved implicitly dealiased pseudospectral simulations suggest that the fourth power of the vorticity cascades to small scales.

A two-dimensional vortex condensate at high Reynolds number

2013 ◽  
Vol 715 ◽  
pp. 359-388 ◽  
Author(s):  
Basile Gallet ◽  
William R. Young
Keyword(s):  
Stokes Equation ◽  
Numerical Solutions ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Finite Limit ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Quasilinear Theory ◽  
Zero Integral ◽  
High Reynolds

AbstractWe investigate solutions of the two-dimensional Navier–Stokes equation in a $\lrm{\pi} \ensuremath{\times} \lrm{\pi} $ square box with stress-free boundary conditions. The flow is steadily forced by the addition of a source $\sin nx\sin ny$ to the vorticity equation; attention is restricted to even $n$ so that the forcing has zero integral. Numerical solutions with $n= 2$ and $6$ show that at high Reynolds numbers the solution is a domain-scale vortex condensate with a strong projection on the gravest mode, $\sin x\sin y$. The sign of the vortex condensate is selected by a symmetry-breaking instability. We show that the amplitude of the vortex condensate has a finite limit as $\nu \ensuremath{\rightarrow} 0$. Using a quasilinear approximation we make an analytic prediction of the amplitude of the condensate and show that the amplitude is determined by viscous selection of a particular solution from a family of solutions to the forced two-dimensional Euler equation. This theory indicates that the condensate amplitude will depend sensitively on the form of the dissipation, even in the undamped limit. This prediction is verified by considering the addition of a drag term to the Navier–Stokes equation and comparing the quasilinear theory with numerical solution.

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