An exact axisymmetric spiral solution of incompressible 3D Euler
  equations
  Spiral structure is one of the most common structures in the nature flows. A
general steady spiral solution of incompressible inviscid axisymmetric flow was
obtained analytically by applying separation of variables to the 3D Euler
equations. The solution, depending on 3 parameters, describes the spiral path
of the fluid material element on the Bernoulli surface, whereas some new exact
solutions were obtained to be bounded within the whole region. The first one is
a continued typhoon-like vortex solution, where there are two intrinsic length
scales. One is the radius of maximum circular velocity $r_m$, the other is the
radius of the vortex kernel $r_k=\sqrt{2}r_m$. The second one is a multi-planar
solution, periodically in $z$-coordinate. Within each layer, the solution is a
umbrella vortex similar to the first one. The third one is also a multi-planar
solution in $z$-coordinate. In each layer, it is a combination of two
independent solutions like the Rankine vortex, which is also finite but
discontinued for either vertical or horizontal velocity. The fourth one is a
multi-paraboloid vortex solution finite for $z$-coordinate but infinite for
$r$-coordinate. Besides, some classical simple solutions (Rankine vortex,
Batchelor vortex, Hill spherical vortex, etc.) were also obtained. The above
explicit solutions can be applied to study the radial structure of the typhoon,
tornados and mesoscale eddies. Both the solutions and approaches used here
could also be applied to other complex flows by the Navier-Stokes equations.
