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Viscous diffusion of multiple vortex system

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The following analytical solution satisfies the incompressible continuity and momentum
+
The following analytical solution satisfies the viscous, incompressible
-
equations in dimension-less form in the domain <math>0 \le x, y \le 2\pi</math>. The
+
continuity and momentum equations in dimension-less form in the domain <math>0
-
solution is periodic in both <math>x</math> and <math>y</math> coordinates.
+
\le x, y \le 2\pi</math>. The solution is periodic in both <math>x</math> and
 +
<math>y</math> coordinates.
:<math>
:<math>
-
u(x,y,t) = -\cos x \sin y e^{-2t/Re}
+
u(x,y,t) = -(\cos x \sin y) e^{-2t/Re}
</math>
</math>
:<math>
:<math>
-
v(x,y,t) =  \sin x \cos y e^{-2t/Re}
+
v(x,y,t) =  (\sin x \cos y) e^{-2t/Re}
</math>
</math>
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p(x,y,t) = -0.25( \cos 2x + \cos 2y) e^{-4t/Re}
p(x,y,t) = -0.25( \cos 2x + \cos 2y) e^{-4t/Re}
</math>
</math>
 +
 +
where <math>u,v</math> are the Cartesian velocity components, <math>p</math>
 +
is the pressure and <math>Re</math> is the [[Reynolds number]].

Revision as of 08:26, 16 September 2005

The following analytical solution satisfies the viscous, incompressible continuity and momentum equations in dimension-less form in the domain 0
\le x, y \le 2\pi. The solution is periodic in both x and y coordinates.


u(x,y,t) = -(\cos x \sin y) e^{-2t/Re}

v(x,y,t) =  (\sin x \cos y) e^{-2t/Re}

p(x,y,t) = -0.25( \cos 2x + \cos 2y) e^{-4t/Re}

where u,v are the Cartesian velocity components, p is the pressure and Re is the Reynolds number.

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