An arbitrary Lagrangian-Eulerian formulation for the numerical simulation of flow patterns generated by the hydromedusa Aequorea Victoria.

*(English)*Zbl 1395.76062Summary: 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.

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

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\textit{M. Sahin} and \textit{K. Mohseni}, J. Comput. Phys. 228, No. 12, 4588--4605 (2009; Zbl 1395.76062)

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##### References:

[1] | Ahn, H.T.; Carey, G., An enhanced polygonal finite-volume method for unstructured hyrid-meshes, Int. J. numer. meth. fluids, 43, 29-46, (2007) · Zbl 1110.76034 |

[2] | Anderson, W.K.; Bonhaus, D.L., An implicit upwind algorithm for computing turbulent flows on unstructured grids, Comp. fluids, 23, 1-21, (1994) · Zbl 0806.76053 |

[3] | S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Users Manual. ANL-95/11, Mathematic and Computer Science Division, Argonne National Laboratory, 2004, <http://www-unix.mcs.anl.gov/petsc/petsc-2/>. |

[4] | T.J. Barth, A 3-D Upwind Euler Solver for Unstructured Meshes, AIAA Paper 91-1548-CP, 1991. |

[5] | Batina, J.T., Unsteady Euler airfoil solutions using unstructured dynamic meshes, Aiaa j., 28, 1381-1388, (1990) |

[6] | Benzi, M.; Golub, G.H.; Liesen, J., Numerical solution of saddle point problems, Acta numer., 14, 1-137, (2005) · Zbl 1115.65034 |

[7] | T.D. Blacker, S. Benzley, S. Jankovich, R. Kerr, J. Kraftcheck, R. Kerr, P. Knupp, R. Leland, D. Melander, R. Meyers, S. Mitchell, J. Shepard, T. Tautges, D. White, CUBIT Mesh Generation Enviroment Users Manual, vol. 1, Sandia National Laboratories, Albuquerque, NM, 1999. |

[8] | Chattopadhyay, S.; Moldovan, R.; Yeung, C.; Wu, X.L., Swimming efficiency of bacterium Escherichia coli, Proc. natl. acad. sci., 103, 13712-13717, (2006) |

[9] | Colin, S.P.; Costello, J.H., Reletionship between morpjology and hydrodynamics during swimming by the hyromedusae A. Victoria and aglantha didtale, Sci. mar., 60, 35-42, (1996) |

[10] | Colin, S.P.; Costello, J.H., Morphology, swimming performance and propulsive mode of six co-occuring hyromedusae, J. exp. biol., 205, 427-437, (2002) |

[11] | E. Cuthill, J. McKee, Reducing the bandwidth of sparce symmetric matrices, in: Twenty Fourth ACM National Conference, 1969, pp. 157-172. |

[12] | Dabiri, J.O.; Colin, S.P.; Costello, J.H.; Gharib, M., Flow patterns generated by oblate medusan: field measurements and laboratory analysis, J. exp. biol., 208, 1257-1265, (2005) |

[13] | Dai, M.; Schmidt, D.P., Adaptive tetrahedral meshing in free-surface flow, J. comput. phy., 208, 228-252, (2005) · Zbl 1114.76333 |

[14] | R.P. Dwight, Robust mesh deformation using the linear elasticity equations, in: H. Deconinck, E. Dick, (Eds.), Computational Fluid Dynamics, Springer, 2006. |

[15] | Ford, M.D.; Costello, J.H.; Heidelberg, K.B.; Purcell, J.E., Swimming and feeding by the scypomedusa chrysaora quinquecirrha, Mar. biol., 129, 355-362, (1997) |

[16] | Förster, Ch.; Wall, W.A.; Ramm, E., On the geometric conservation law in transient flow calculations on deforming domains, Int. J. numer. meth. fluids, 50, 1369-1379, (2005) · Zbl 1097.76049 |

[17] | Geuzaine, P.; Grandmont, C.; Farhat, C., Design and analysis of ALE schemes with provable second-order time-accuracy for inviscid and viscous flow simulations, J. comput. phys., 191, 206-227, (2003) · Zbl 1051.76038 |

[18] | Glowinskia, R.; Pana, T.-W.; Heslab, T.I.; Josephb, D.D., A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. multiphase flow, 25, 755-794, (1999) · Zbl 1137.76592 |

[19] | Guillard, H.; Farhat, C., On the significance of the geometric conservation law for floow computations on moving meshes, Comput. meth. appl. mech. eng., 190, 1467-1482, (2000) · Zbl 0993.76049 |

[20] | Hirt, C.W.; Amsden, A.A.; Cook, J.L., An arbitrary lagrangian – eulerian computing method for all flow speeds, J. comput. phys., 14, 227-253, (1974) · Zbl 0292.76018 |

[21] | M.J. Jensen, Numerical Simulation of Interface Dynamics in Microfluidics, Ph.D. Thesis, Technical University of Denmark, 2005. |

[22] | Johnson, A.; Tezduyar, T., Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces, Comput. meth. appl. mech. eng., 119, (1994) · Zbl 0848.76036 |

[23] | Johnson, A.; Tezduyar, T., Advanced mesh generation and update methods for 3D flow simulations, Comput. mech., 23, 130-143, (1999) · Zbl 0949.76049 |

[24] | Karypis, G.; Kumar, V., A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. sci. comput., 20, 359-392, (1998) · Zbl 0915.68129 |

[25] | Kobayashi, M.H.; Pereira, J.M.C.; Pereira, J.C.F., A conservative finite-volume second-order-accurate projection method on hybrid unstructured grids, J. comput. phys., 150, 40-75, (1999) · Zbl 0934.76049 |

[26] | Koobus, B.; Farhat, C., Second-order time-accurate and geometricaly conservative implicit schemes for flow computations on unstructured dynamic meshes, Comput. meth. appl. mech. eng., 170, 103-129, (1999) · Zbl 0943.76055 |

[27] | Krieg, M.; Mohseni, K., Trust characterization of a bio-inspired vortex ring generator for locomotion of underwater robots, IEEE J. oceanic eng., 33, 123-132, (2008) |

[28] | M.J. Lighthill, Mathematical Biofluid Dynamics, SIAM, Philadelphia, 1975. |

[29] | Mavriplis, D.J.; Yang, Z., Constraction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes, J. comput. phys., 213, 557-573, (2006) · Zbl 1136.76402 |

[30] | McHenry, M.J.; Jed, J., The ontogenetic scaling of hydrodynamics and swimming performance in jellyfish (aurelia aurita), J. exp. biol., 206, 4125-4137, (2003) |

[31] | Mei, R.; Xiong, J.; Tran-Son-Tay, R., Motion of a sphere oscillating at low Reynolds numbers in a viscoelastic-fluid-filled cylinderical tube, J. non-Newtonian fluid mech., 66, 169-192, (1996) |

[32] | Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. rev. fluid mech., 37, 239-261, (2005) · Zbl 1117.76049 |

[33] | Mohseni, K., Statistical equilibrium theory for axisymmetric flows: kelvin’s variational principle and an explanation for the vortex ring pinch-off process, Phys. fluids, 13, 1924-1931, (2001) · Zbl 1184.76372 |

[34] | Mohseni, K., Pulsatile vortex generators for low-speed maneuvering of small underwater vehicles, Ocean eng., 33, 2209-2223, (2006) |

[35] | Mohseni, K.; Colonius, T., Numerical treatment of polar coordinate singularities, J. comput. phy., 157, 787-795, (2000) · Zbl 0981.76075 |

[36] | Owens, R.G.; Phillips, T.N., Steady viscoelastic flow past a sphere using spectral elements, Int. J. num. meth. eng., 39, 1517-1534, (1996) · Zbl 0868.76071 |

[37] | Peskin, C.S., The fluid dynamics of heart valves: experimental, theoretical, and computational methods, Ann. rev. fluid mech., 14, 235-259, (1982) |

[38] | Rida, S.; McKenty, F.; Meng, F.L.; Reggio, M., A staggered control volume scheme for unstructured triangular grids, Int. J. numer. meth. fluids, 25, 697-717, (1997) · Zbl 0907.76049 |

[39] | M. Rozloznik, Saddle point problems, iterative solution and preconditioning: a short overview, in: I. Marek (Ed.), Proceedings of the XVth Summer School Software and Algorithms of Numerical Mathematics, University of West Bohemia, Pilsen, 2003, pp. 97-108. |

[40] | Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. statist. comput., 7, 856-869, (1986) · Zbl 0599.65018 |

[41] | Sahin, M., A preconditioned semi-staggered dilation-free finite volume method for the incompressible navier – stokes equations on all-hexahedral elements, Int. J. numer. meth. fluids, 49, 959-974, (2005) · Zbl 1077.76046 |

[42] | Sahin, M.; Owens, R.G., A novel fully-implicit finite volume method applied to the lid-driven cavity problem. part II. linear stability analysis, Int. J. numer. meth. fluids, 42, 79-88, (2003) · Zbl 1078.76047 |

[43] | Sahin, M.; Wilson, H.J., A semi-staggered dilation-free finite volume method for the numerical solution of viscoelastic fluid flows on all-hexahedral elements, J. non-Newtonian fluid mech., 147, 79-91, (2007) · Zbl 1195.76268 |

[44] | P.R. Schunk, M.A. Heroux, R.R. Rao, T.A. Baer, S.R. Subia, A.C. Sun, Iterative Solvers and Preconditioners for Fully-coupled Finite Element Formulations of Incompressible Fluid Mechanics and Related Transport Problems, SAND2001-3512J, Sandia National Laboratories Albuuquerque, New Mexico, 2001. |

[45] | Sfakiotakis, M.; Lane, D.M.; Davies, J.B.C., Review of fish swimming modes for aquatic locomotion, IEEE J. oceanic eng., 24, 237-252, (1999) |

[46] | Sheard, G.J.; Ryan, K., Pressure-driven flow past spheres moving in a circular cylinder, J. fluid mech., 592, 233-262, (2007), The value of Sheard & Ryan given in Table 2 is not reported in [46] · Zbl 1151.76423 |

[47] | Smith, R.W.; Wright, J.A., An implicit edge-based ALE method for the incompressible navier – stokes equations, Int. J. numer. meth. fluids, 43, 253-279, (2003) · Zbl 1032.76623 |

[48] | Thomas, P.D.; Lombard, C.K., Geometric conservation law and its application to flow computations on moving grids, Aiaa j., 17, 1030-1037, (1979) · Zbl 0436.76025 |

[49] | Triantafyllou, G.S.; Triantafyllou, M.S.; Grosenbaugh, M.A., Optimal thrust development in oscillating foils with application to fish propulsion, J. fluid struct., 7, 205-224, (1993) |

[50] | Wright, J.A.; Smith, R.W., An edge-based method for the incompressible navier – stokes equations on polygonal meshes, J. comput. phys., 169, 24-43, (2001) · Zbl 0989.76054 |

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