Analytic First Integrals and Asymptotic Behaviour as t → ∞ of Fourier Coefficients of Solutions of Two-Dimensional Navier-Stokes Equations

1988 ◽  
pp. 292-334
Author(s):  
M. J. Vishik ◽  
A. V. Fursikov
Keyword(s):  
Asymptotic Behaviour ◽  
Fourier Coefficients ◽  
Stokes Equations ◽  
First Integrals ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Analytic First Integrals

scholarly journals Asymptotic behaviour of solutions to the stationary Navier–Stokes equations in two-dimensional exterior domains with zero velocity at infinity

2014 ◽  
Vol 25 (02) ◽  
pp. 229-253 ◽  
Author(s):  
Julien Guillod ◽  
Peter Wittwer
Keyword(s):  
Asymptotic Behaviour ◽  
Stokes Equations ◽  
Invariant Solution ◽  
The Body ◽  
Dimensional Case ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Scale Invariant ◽  
Rigorous Results ◽  
Navier Stokes Equations

We investigate analytically and numerically the existence of stationary solutions converging to zero at infinity for the incompressible Navier–Stokes equations in a two-dimensional exterior domain. Physically, this corresponds for example to fixing a propeller by an external force at some point in a two-dimensional fluid filling the plane and to ask if the solution becomes steady with the velocity at infinity equal to zero. To answer this question, we find the asymptotic behaviour for such steady solutions in the case where the net force on the propeller is nonzero. In contrast to the three-dimensional case, where the asymptotic behaviour of the solution to this problem is given by a scale invariant solution, the asymptote in the two-dimensional case is not scale invariant and has a wake. We provide an asymptotic expansion for the velocity field at infinity, which shows that, within a wake of width |x|2/3, the velocity decays like |x|-1/3, whereas outside the wake, it decays like |x|-2/3. We check numerically that this behaviour is accurate at least up to second order and demonstrate how to use this information to significantly improve the numerical simulations. Finally, in order to check the compatibility of the present results with rigorous results for the case of zero net force, we consider a family of boundary conditions on the body which interpolate between the nonzero and the zero net force case.

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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