Stationary Solutions of Two-Dimensional Navier-Stokes Equations with Random Perturbation

1996 ◽  
pp. 237-245 ◽  
Author(s):  
P. L. Chow
Keyword(s):  
Stokes Equations ◽  
Random Perturbation ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations

ON STABILITY OF PHYSICALLY REASONABLE SOLUTIONS TO THE TWO-DIMENSIONAL NAVIER–STOKES EQUATIONS

2019 ◽  
pp. 1-52
Author(s):  
Yasunori Maekawa
Keyword(s):  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Domain Modeling ◽  
Asymptotically Stable ◽  
Two Dimensional ◽  
Spatial Infinity ◽  
Navier Stokes Equations ◽  
Unique Existence ◽  
Fundamental Object

The flow past an obstacle is a fundamental object in fluid mechanics. In 1967 Finn and Smith proved the unique existence of stationary solutions, called the physically reasonable solutions, to the Navier–Stokes equations in a two-dimensional exterior domain modeling this type of flows when the Reynolds number is sufficiently small. The asymptotic behavior of their solution at spatial infinity has been studied in detail and well understood by now, while its stability has remained open due to the difficulty specific to the two-dimensionality. In this paper, we prove that the physically reasonable solutions constructed by Finn and Smith are asymptotically stable with respect to small and well-localized initial perturbations.

scholarly journals Strong $L^2$ convergence of time numerical schemes for the stochastic two-dimensional Navier–Stokes equations

2018 ◽  
Vol 39 (4) ◽  
pp. 2135-2167 ◽  
Author(s):  
Hakima Bessaih ◽  
Annie Millet
Keyword(s):  
Stokes Equations ◽  
Random Perturbation ◽  
Time Discretization ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Exponential Moment ◽  
Navier Stokes Equations ◽  
Fully Implicit ◽  
Moment Estimates ◽  
Discretization Schemes

Abstract We prove that some time discretization schemes for the two-dimensional Navier–Stokes equations on the torus subject to a random perturbation converge in $L^2(\varOmega )$. This refines previous results that established the convergence only in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier–Stokes equations and convergence of a localized scheme we can prove strong convergence of fully implicit and semiimplicit temporal Euler discretizations and of a splitting scheme. The speed of the $L^2(\varOmega )$ convergence depends on the diffusion coefficient and on the viscosity parameter.

Direct simulation of transition in an oscillatory boundary layer

1998 ◽  
Vol 371 ◽  
pp. 207-232 ◽  
Author(s):  
G. VITTORI ◽  
R. VERZICCO
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Regime ◽  
Two Dimensional ◽  
Temporal Development ◽  
Direct Simulation ◽  
Navier Stokes Equations ◽  
Oscillatory Boundary Layer

Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

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