scholarly journals Two-dimensional Navier-Stokes flow with measures as initial vorticity

1988 ◽  
Vol 104 (3) ◽  
pp. 223-250 ◽  
Author(s):  
Yoshikazu Giga ◽  
Tetsuro Miyakawa ◽  
Hirofumi Osada
Keyword(s):  
Stokes Flow ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Initial Vorticity

Final equilibrium state of a two-dimensional shear layer

1991 ◽  
Vol 233 ◽  
pp. 661-689 ◽  
Author(s):  
J. Sommeria ◽  
C. Staquet ◽  
R. Robert
Keyword(s):  
Equilibrium State ◽  
Shear Layer ◽  
Stream Function ◽  
Stokes Equations ◽  
Local Equilibrium ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Final Equilibrium State ◽  
Initial Vorticity ◽  
Pseudo Spectral Method

We test a new statistical theory of organized structures in two-dimensional turbulence by direct numerical stimulations of the Navier–Stokes equations, using a pseudo-spectral method. We apply the theory to the final equilibrium state of a shear layer evolving from a band of uniform vorticity: a relationship between vorticity and stream function is predicted by maximizing an entropy with the constraints due the constants of the motion. A partial differential equation for the stream function is then obtained. In the particular case of a very thin initial vorticity band, the Stuart's vortices appear to be a family of solutions for this equation. In more general cases we do not solve the equation, but we test the theory by inspecting the relationship between stream function and vorticity in the final equilibrium state of the numerical computation. An excellent agreement is obtained in regions with strong vorticity mixing. However, local equilibrium is obtained before a complete mixing can occur in the whole fluid domain.

Two-dimensional Navier-Stokes flow in unbounded domains

1993 ◽  
Vol 297 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Hideo Kozono ◽  
Takayoshi Ogawa
Keyword(s):  
Stokes Flow ◽  
Unbounded Domains ◽  
Navier Stokes ◽  
Two Dimensional

Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form

1989 ◽  
Vol 203 ◽  
pp. 1-22 ◽  
Author(s):  
S. Childress ◽  
G. R. Ierley ◽  
E. A. Spiegel ◽  
W. R. Young
Keyword(s):  
Stagnation Point ◽  
Blow Up ◽  
Stokes Equations ◽  
Point Vortex ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Infinite Domain ◽  
Initial Vorticity ◽  
Point Form

The time-dependent form of the classic, two-dimensional stagnation-point solution of the Navier-Stokes equations is considered. If the viscosity is zero, a class of solutions of the initial-value problem can be found in closed form using Lagrangian coordinates. These solutions exhibit singular behaviour in finite time, because of the infinite domain and unbounded initial vorticity. Thus, the blow-up found by Stuart in three dimensions using the stagnation-point form, also occurs in two. The singularity vanishes under a discrete, finite-dimensional ‘point vortex’ approximation, but is recovered as the number of vortices tends to infinity. We find that a small positive viscosity does not arrest the breakdown, but does strongly alter its form. Similar results are summarized for certain Boussinesq stratified flows.

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