## Abstract

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 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.

Original language | English |
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Pages (from-to) | 4588-4605 |

Number of pages | 18 |

Journal | Journal of Computational Physics |

Volume | 228 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1 Jul 2009 |

Externally published | Yes |

### Funding

This work was partially supported by the National Science Foundation and the Air Force Office of Scientific Research. The authors would like to thank Dr. Sean Colin at Roger Williams University for providing his original video recordings as well as his discussions on the hydromedusa A. victoria swimming. The authors would like to thank Dr. Greg Sheard at Monash University for computing the drag coefficient data in Table 2 which is not reported in [46] . The authors gratefully acknowledge the use of the IBM Machine Bluefire at NCAR.

Funders | Funder number |
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National Science Foundation | |

Air Force Office of Scientific Research |

## Keywords

- ALE methods
- Finite volume
- Geometric conservation law
- Jellyfish swimming