  A new geometrically conservative arbitrary Lagrangian-Eulerian (ALE)
formulation is presented for the moving boundary problems in the swirl-free
cylindrical coordinates. The governing equations are multiplied with the radial
distance and integrated over arbitrary moving Lagrangian-Eulerian quadrilateral
elements. Therefore, the continuity and the geometric conservation equations
take very simple form similar to those of the Cartesian coordinates. The
continuity equation is satisfied exactly within each element and a special
attention is given to satisfy the geometric conservation law (GCL) at the
discrete level. The equation of motion of a deforming body is solved in
addition to the Navier-Stokes equations in a fully-coupled form. The mesh
deformation is achieved by solving the linear elasticity equation at each time
level while avoiding remeshing in order to enhance numerical robustness. The
resulting algebraic linear systems are solved using an ILU(k) preconditioned
GMRES method provided by the PETSc library. The present ALE method is validated
for the steady and oscillatory flow around a sphere in a cylindrical tube and
applied to the investigation of the flow patterns around a free-swimming
hydromedusa \textit{Aequorea victoria} (crystal jellyfish). The calculations
for the hydromedusa indicate the shed of the opposite signed vortex rings very
close to each other and the formation of large induced velocities along the
line of interaction while the ring vortices moving away from the hydromedusa.
In addition, the propulsion efficiency of the free-swimming hydromedusa is
computed and its value is compared with values from the literature for several
other species. The fluid dynamics video presented here shows the time variation
of the instantaneous three-dimensional vorticity isosurfaces around a
free-swimming hydromedusa \textit{Aequorea victoria}.
