Taylor-Green vortex

 

Taylor-Green vortex (1937) is a simple perioid initial value problem for the 3D Euler equation:

v = (v_1,v_2,v_3) with

v_1 =\sin x \cos y \cos z, v_2 = - \cos x \sin y \cos z, v_3 = 0

which leads to a complicated turbulent flow, and has become a standard example in fluid dynamics.

Taylor and Green conjectured that their vortex would develop a finite-time blowup. However, numerical simulations didn’t show a very high increase of maximal vorticity in time. (Only a factor of 4 ?). Nevertheless, the dynamics of the Taylor-Green vortex seems to be extremely complicated.

Some pictures of the Taylor-Green vortex from the internet:

Initial value:

The vortex after some time:

A video:

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