zbMATH — the first resource for mathematics

Sigma transformation and ALE formulation for three-dimensional free surface flows. (English) Zbl 1154.76031
Summary: We establish a link between the sigma transformation approach and the arbitrary Lagrangian-Eulerian (ALE) approach. For that purpose we introduce the ALE-sigma (ALES) approach, which consists in an ALE interpretation of the sigma transformation. Taking advantage of this new approach, we propose a general ALES transformation, allowing for a great adaptability of the vertical discretization and therefore overcoming some drawbacks of the classical sigma transformation. Numerical results are presented, showing the advantages of this general coordinate system, as, for example, a better representation of horizontal stratifications.

76M10 Finite element methods applied to problems in fluid mechanics
76D33 Waves for incompressible viscous fluids
76D50 Stratification effects in viscous fluids
Full Text: DOI
[1] Hugues, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Computer Methods in Applied Mechanics and Engineering 29 pp 329– (1981)
[2] Soulaïmani, An arbitrary Lagrangian-Eulerian finite element method for solving three-dimensional free surface flows, Computer Methods in Applied Mechanics and Engineering 162 pp 79– (1998)
[3] Phillips, A coordinate system having some special advantages for numerical forecasting, Journal of Meteorology 14 pp 184– (1957)
[4] Blumberg, Three-dimensional Coastal Ocean Models 4 pp 1– (1987)
[5] Shchepetkin, The regional ocean model system (ROMS): a split-explicit, free-surface, topography-following coordinate ocean model, Ocean Modelling 9 pp 347– (2005)
[6] Song, A semi-implicit ocean circulation model using a generalized topography-following coordinate system, Journal of Computational Physics 115 pp 228– (1994) · Zbl 0853.76014
[7] Kocyigit, Three-dimensional numerical modeling of free surface flows with non-hydrostatic pressure, International Journal for Numerical Methods in Fluids 40 pp 1145– (2002)
[8] Zhou, An arbitrary Lagrangian-Eulerian sigma (ALES) model with non-hydrostatic pressure for shallow water, Computer Methods in Applied Mechanics and Engineering 178 pp 199– (1999) · Zbl 0967.76065
[9] Bryan, A numerical model for the study of the circulation of the world oceans, Journal of Computational Physics 4 pp 347– (1969) · Zbl 0195.55504
[10] Gerdes, A primitive equation ocean circulation model using a general vertical coordinate transformation, Journal of Geophysical Research 98 (14) pp 683– (1993)
[11] Mellor, Ocean Forecasting: Conceptual Basis and Applications pp 55– (2002)
[12] Hervouet, Hydrodynamique des Écoulements À Surface Libre. Modélisation Numérique Avec la Méthode des Éléments Finis (2003)
[13] Yuan, A two-dimensional vertical non-hydrostatic {\(\sigma\)} model with an implicit method for free-surface flows, International Journal for Numerical Methods in Fluids 44 pp 811– (2004) · Zbl 1094.76048
[14] Huwald, A multi-layer sigma-coordinate thermodynamic snow sea-ice model: validation against surface heat budget of the Arctic Ocean (SHEBA)/sea ice model intercomparison project (SIMIP2) data, Journal for Geophysical Research 110 (C5) (2005)
[15] Gary, Estimate of truncation error in transformed coordinate primitive equation atmospheric models, Journal of Atmospheric Sciences 30 pp 223– (1973)
[16] Janjic, Pressure gradient force and advection scheme used for forecasting with steep and small scale topography, Contributions to Atmospheric Physics 50 pp 186– (1977)
[17] Beckman, Numerical simulation of flow around a tall isolated seamount, Journal of Physical Oceanography 23 pp 1736– (1993)
[18] Chu, Hydrostatic correction for sigma coordinate ocean models, Journal of Geophysical Research 108 (C6) pp 3206– (2003)
[19] Haney, On the pressure gradient force over steep topography in sigma coordinate ocean models, Journal of Physical Oceanography 21 pp 610– (1991)
[20] Mellor, The pressure gradient conundrum of sigma coordinate ocean models, Journal of Atmospheric and Oceanic Technologies 11 pp 1126– (1994)
[21] Song, A general pressure gradient formulation for ocean models, Monthly Weather Review 126 pp 1749– (1998)
[22] Stelling, On the approximation of horizontal gradients in sigma-coordinates for bathymetry with steep bottom slopes, International Journal for Numerical Methods in Fluids 18 pp 915– (1994) · Zbl 0807.76062
[23] Mellor, Sigma coordinate pressure gradient and the seamount problem, Journal of Atmospheric and Oceanic Technologies 15 pp 1122– (1997)
[24] Malcherek A. Mathematische modellierung von StrÃd’ungen und Stofftransportprozessen in estuaren. Ph.D. Thesis, Institut für Strömungsmechanik und ERiB, Universität Hannover, 1995.
[25] Marcos F, Janin JM. Nouveaux développements dans l’étape de convection-diffusion de Telemac-3D. Rapport EDF R&D-LNHE, HE-42/94/025/A, 1997.
[26] Jankowski JA. A non-hydrostatic model for free surface flows. Ph.D. Thesis, Institut für Strömungsmechanik und ERiB, Universität Hannover, 1998.
[27] McCalpin, A comparison of second-order and fourth-order pressure gradient algorithm in a sigma-co-ordinate model, International Journal for Numerical Methods in Fluids 18 pp 361– (1994) · Zbl 0792.76054
[28] Slordal, The pressure gradient force in sigma-coordinate-ocean models, International Journal for Numerical Methods in Fluids 24 pp 987– (1997)
[29] Deleersnijder, On the use of the {\(\sigma\)} coordinate system in regions of large bathymetric variations, Journal of Marine Systems 3 pp 381– (1992)
[30] Li, A sigma coordinate 3D kappa-epsilon model for turbulent free surface flow over a submerged structure, Applied Mathematical Modeling 26 pp 1139– (2002)
[31] Lin, A multiple-layer sigma-coordinate model for simulation of wave-structure interaction, Computers and Fluids 35 pp 147– (2005)
[32] Deleersnijder, A generalized vertical coordinate for 3D marine models, Bulletin de la Société Royale des Sciences de Liège 61 (6) pp 498– (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.