On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity

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.

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

Applied Mathematical Sciences ◽

10.12988/ams.2018.8242 ◽

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

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A CUDA Pseudo-spectral Solver for Two-Dimensional Navier-Stokes Equation

2012 11th International Symposium on Distributed Computing and Applications to Business, Engineering & Science ◽

10.1109/dcabes.2012.93 ◽

2012 ◽

Author(s):

Kaiyuan Lou ◽

Zhaohua Yin

Keyword(s):

Stokes Equation ◽

Navier Stokes ◽

Two Dimensional ◽

Navier Stokes Equation ◽

Pseudo Spectral

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Casimir cascades in two-dimensional turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.308 ◽

2013 ◽

Vol 729 ◽

pp. 364-376 ◽

Cited By ~ 5

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.

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A two-dimensional vortex condensate at high Reynolds number

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.524 ◽

2013 ◽

Vol 715 ◽

pp. 359-388 ◽

Cited By ~ 27

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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Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.278 ◽

2013 ◽

Vol 729 ◽

pp. 285-308 ◽

Cited By ~ 97

Author(s):

Maciej J. Balajewicz ◽

Earl H. Dowell ◽

Bernd R. Noack

Keyword(s):

Reynolds Number ◽

Galerkin Method ◽

Stokes Equation ◽

High Reynolds Number ◽

Transient Flow ◽

Navier Stokes ◽

Power Balance ◽

Two Dimensional ◽

Navier Stokes Equation ◽

High Reynolds

AbstractWe generalize the POD-based Galerkin method for post-transient flow data by incorporating Navier–Stokes equation constraints. In this method, the derived Galerkin expansion minimizes the residual like POD, but with the power balance equation for the resolved turbulent kinetic energy as an additional optimization constraint. Thus, the projection of the Navier–Stokes equation on to the expansion modes yields a Galerkin system that respects the power balance on the attractor. The resulting dynamical system requires no stabilizing eddy-viscosity term – contrary to other POD models of high-Reynolds-number flows. The proposed Galerkin method is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer. Generalizations for more Navier–Stokes constraints, e.g. Reynolds equations, can be achieved in straightforward variation of the presented results.

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