scholarly journals Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain

2012 ◽  
pp. 1-17 ◽  
Author(s):  
Thierry Gallay
Keyword(s):  
Stokes Equation ◽  
Exterior Domain ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Time Asymptotics ◽  
Long Time ◽  
Long Time Asymptotics

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.

scholarly journals Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity

2013 ◽  
Vol 6 (4) ◽  
pp. 973-991 ◽  
Author(s):  
Thierry Gallay ◽  
Yasunori Maekawa
Keyword(s):  
Two Dimensional ◽  
Time Asymptotics ◽  
Long Time ◽  
Long Time Asymptotics

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.

scholarly journals Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations

2013 ◽  
Vol 143 (5) ◽  
pp. 905-927 ◽  
Author(s):  
Margaret Beck ◽  
C. Eugene Wayne
Keyword(s):  
Decay Rate ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Long Time ◽  
High Reynolds

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows, where they often emerge on timescales much shorter than the viscous timescale, and then dominate the dynamics for very long time intervals. In this paper we propose a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier–Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. We also provide numerical evidence that the same result holds for the full linear operator, and that our theoretical results give the optimal decay rate in this setting.

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