Exterior and vector calculus views of incompressible Navier-Stokes port-Hamiltonian models

Authors

Ramy Rashad, Federico Califano, Andrea Brugnoli, Frederic P. Schuller, Stefano Stramigioli

Abstract

In this paper we address the modeling of incompressible Navier-Stokes equations in the port-Hamiltonian framework. Such model not only allows describing the energy dissipation due to viscous effects but also incorporates the non-zero energy exchange through the boundary of the spatial domain for generic boundary conditions. We present in this work the coordinate-free representations of this port-Hamiltonian model using both exterior calculus and vector calculus as well as their corresponding coordinate-based descriptions.

Keywords

port-Hamiltonian; Navier-Stokes; exterior calculus; vector calculus

Citation

• Journal: IFAC-PapersOnLine
• Year: 2021
• Volume: 54
• Issue: 19
• Pages: 173–179
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2021.11.074
• Note: 7th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC 2021- Berlin, Germany, 11-13 October 2021

BibTeX

@article{Rashad_2021,
  title={{Exterior and vector calculus views of incompressible Navier-Stokes port-Hamiltonian models}},
  volume={54},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2021.11.074},
  number={19},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Rashad, Ramy and Califano, Federico and Brugnoli, Andrea and Schuller, Frederic P. and Stramigioli, Stefano},
  year={2021},
  pages={173--179}
}

Download the bib file

References

• Califano, F. et al. Decoding and realising flapping flight with port-Hamiltonian system theory. Annual Reviews in Control vol. 51 37–46 (2021) – 10.1016/j.arcontrol.2021.03.009
• Califano, F., Rashad, R., Schuller, F. P. & Stramigioli, S. Geometric and energy-aware decomposition of the Navier–Stokes equations: A port-Hamiltonian approach. Physics of Fluids vol. 33 (2021) – 10.1063/5.0048359
• Duindam, (2009)
• Frankel, (2004)
• Jagad, P., Abukhwejah, A., Mohamed, M. & Samtaney, R. A primitive variable discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes. Physics of Fluids vol. 33 (2021) – 10.1063/5.0035981
• Marsden, (1994)
• Marsden, J. E., Raţiu, T. & Weinstein, A. Semidirect products and reduction in mechanics. Transactions of the American Mathematical Society vol. 281 147–177 (1984) – 10.1090/s0002-9947-1984-0719663-1
• Mohamed, M. S., Hirani, A. N. & Samtaney, R. Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes. Journal of Computational Physics vol. 312 175–191 (2016) – 10.1016/j.jcp.2016.02.028
• Nitschke, Discrete exterior calculus (dec) for the surface navier-stokes equation. (2017)
• Rashad, R., Califano, F., Schuller, F. P. & Stramigioli, S. Port-Hamiltonian modeling of ideal fluid flow: Part I. Foundations and kinetic energy. Journal of Geometry and Physics vol. 164 104201 (2021) – 10.1016/j.geomphys.2021.104201
• Rashad, R., Califano, F., Schuller, F. P. & Stramigioli, S. Port-Hamiltonian modeling of ideal fluid flow: Part II. Compressible and incompressible flow. Journal of Geometry and Physics vol. 164 104199 (2021) – 10.1016/j.geomphys.2021.104199
• Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information vol. 37 1400–1422 (2020) – 10.1093/imamci/dnaa018
• Seslija, M., Scherpen, J. M. A. & van der Schaft, A. Explicit simplicial discretization of distributed-parameter port-Hamiltonian systems. Automatica vol. 50 369–377 (2014) – 10.1016/j.automatica.2013.11.020
• Seslija, M., van der Schaft, A. & Scherpen, J. M. A. Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics vol. 62 1509–1531 (2012) – 10.1016/j.geomphys.2012.02.006
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3