On port-Hamiltonian formulations of 3-dimensional compressible Newtonian fluids

Authors

Luis A. Mora, Yann Le Gorrec, Denis Matignon, Hector Ramirez, Juan I. Yuz

Abstract

In this manuscript, a general formulation of 3-dimensional compressible fluids based on the port-Hamiltonian framework is presented, both for isentropic and non-isentropic assumptions, describing the energy flux between the mechanical, chemical, and thermal domains, with an explicit characterization of the first and the second law of thermodynamics. For isentropic fluids, the conversion of kinetic energy into heat by viscous friction is considered as energy dissipation associated with the rotation and compression of the fluid. A dissipative port-Hamiltonian formulation is derived for this class of fluids, including vorticity boundary conditions in the port variables. For non-isentropic fluids, we consider a fluid mixture with multiple chemical reactions. To describe the energy fluxes, we propose a pseudo port-Hamiltonian formulation, which includes the rate of irreversible entropy creation by heat flux, chemical reaction, diffusion of matter, and viscous friction.

Citation

• Journal: Physics of Fluids
• Year: 2021
• Volume: 33
• Issue: 11
• Pages:
• Publisher: AIP Publishing
• DOI: 10.1063/5.0067784

BibTeX

@article{Mora_2021,
  title={{On port-Hamiltonian formulations of 3-dimensional compressible Newtonian fluids}},
  volume={33},
  ISSN={1089-7666},
  DOI={10.1063/5.0067784},
  number={11},
  journal={Physics of Fluids},
  publisher={AIP Publishing},
  author={Mora, Luis A. and Le Gorrec, Yann and Matignon, Denis and Ramirez, Hector and Yuz, Juan I.},
  year={2021}
}

Download the bib file

References

• Macchelli, A., Le Gorrec, Y., Ramirez, H. & Zwart, H. On the Synthesis of Boundary Control Laws for Distributed Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 62 1700–1713 (2017) – 10.1109/tac.2016.2595263
• Trang VU, N. M., LEFÈVRE, L. & NOUAILLETAS, R. Distributed and backstepping boundary controls to achieve IDA-PBC design. IFAC-PapersOnLine vol. 48 482–487 (2015) – 10.1016/j.ifacol.2015.05.034
• Maschke, B. M. & van der Schaft, A. J. Port-Controlled Hamiltonian Systems: Modelling Origins and Systemtheoretic Properties. IFAC Proceedings Volumes vol. 25 359–365 (1992) – 10.1016/s1474-6670(17)52308-3
• Van Der Schaft, A. J. & Maschke, B. M. On the Hamiltonian formulation of nonholonomic mechanical systems. Reports on Mathematical Physics vol. 34 225–233 (1994) – 10.1016/0034-4877(94)90038-8
• 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
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Yamamoto Kyoto, Boundary control for a class of dissipative differential operators including diffusion systems.
• Brugnoli, A., Haine, G., Serhani, A. & Vasseur, X. Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control. Journal of Applied Mathematics and Physics vol. 09 1278–1321 (2021) – 10.4236/jamp.2021.96088
• Numerical Methods for Distributed Parameter Port-Hamiltonian Systems - Structure-Preserving Approaches for Simulation and Control (2019)
• Nielsen, A partitioned finite element method for the structure-preserving discretization of damped infinite-dimensional port-Hamiltonian systems with boundary control. Geometric Science of Information. GSI 2019 (2019)
• Serhani, A., Haine, G. & Matignon, D. Anisotropic heterogeneous n-D heat equation with boundary control and observation: II. Structure-preserving discretization. IFAC-PapersOnLine vol. 52 57–62 (2019) – 10.1016/j.ifacol.2019.07.010
• Vu, N. M. T., Lefèvre, L. & Maschke, B. A structured control model for the thermo-magneto-hydrodynamics of plasmas in tokamaks. Mathematical and Computer Modelling of Dynamical Systems vol. 22 181–206 (2016) – 10.1080/13873954.2016.1154874
• Boundary port Hamiltonian control of a class of nanotweezers. (2013)
• Guaranteed-passive simulation of an electro-mechanical piano: A port-Hamiltonian approach. (2015)
• Falaize, A. & Hélie, T. Passive simulation of the nonlinear port-Hamiltonian modeling of a Rhodes Piano. Journal of Sound and Vibration vol. 390 289–309 (2017) – 10.1016/j.jsv.2016.11.008
• Khurshid, S. & Donzis, D. A. Decaying compressible turbulence with thermal non-equilibrium. Physics of Fluids vol. 31 (2019) – 10.1063/1.5080369
• Beron-Vera, F. J. Nonlinear saturation of thermal instabilities. Physics of Fluids vol. 33 (2021) – 10.1063/5.0045191
• Wu, W. & Wang, J. Nonequilibrium thermodynamics of turbulence and stochastic fluid systems. New Journal of Physics vol. 22 113017 (2020) – 10.1088/1367-2630/abc7d2
• Zhan, N., Chen, R. & You, Y. Discrete gas-kinetic scheme-based arbitrary Lagrangian–Eulerian method for moving boundary problems. Physics of Fluids vol. 33 (2021) – 10.1063/5.0051299
• Xu, H. H. A. & Yang, X. I. A. Treatment of unphysical numerical oscillations via local grid refinement. Physics of Fluids vol. 33 (2021) – 10.1063/5.0054642
• Macchelli, A., Le Gorrec, Y. & Ramírez, H. Boundary Energy-Shaping Control of an Ideal Compressible Isentropic Fluid in 1-D. IFAC-PapersOnLine vol. 50 5598–5603 (2017) – 10.1016/j.ifacol.2017.08.1105
• Discretized models for networks of distributed parameter port-Hamiltonian systems. Proceedings of 8th International Workshop on Multidimensional Systems (nDS13) (2013)
• Matignon, D. & Hélie, T. A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems. European Journal of Control vol. 19 486–494 (2013) – 10.1016/j.ejcon.2013.10.003
• Mora, L. A., Gorrec, Y. L., Matignon, D., Ramirez, H. & Yuz, J. I. About Dissipative and Pseudo Port-Hamiltonian Formulations of Irreversible Newtonian Compressible Flows. IFAC-PapersOnLine vol. 53 11521–11526 (2020) – 10.1016/j.ifacol.2020.12.604
• 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
• 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
• Altmann, R. & Schulze, P. A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows. Systems & Control Letters vol. 100 51–55 (2017) – 10.1016/j.sysconle.2016.12.005
• Grmela, M. & Öttinger, H. C. Dynamics and thermodynamics of complex fluids.  I. Development of a general formalism. Physical Review E vol. 56 6620–6632 (1997) – 10.1103/physreve.56.6620
• Öttinger, H. C. & Grmela, M. Dynamics and thermodynamics of complex fluids.  II. Illustrations of a general formalism. Physical Review E vol. 56 6633–6655 (1997) – 10.1103/physreve.56.6633
• Öttinger, H. C. Nonequilibrium thermodynamics for open systems. Physical Review E vol. 73 (2006) – 10.1103/physreve.73.036126
• Grmela, M. GENERIC guide to the multiscale dynamics and thermodynamics. Journal of Physics Communications vol. 2 032001 (2018) – 10.1088/2399-6528/aab642
• Open physical systems: From GENERIC to port-Hamiltonian systems. (2018)
• Ramirez, H., Gorrec, Y. L. & Maschke, B. Boundary controlled irreversible port-Hamiltonian systems. Chemical Engineering Science vol. 248 117107 (2022) – 10.1016/j.ces.2021.117107
• Hadwin, P. J., Erath, B. D. & Peterson, S. D. The influence of flow model selection on finite element model parameter estimation using Bayesian inference. JASA Express Letters vol. 1 (2021) – 10.1121/10.0004260
• Thomson, S. L., Mongeau, L. & Frankel, S. H. Aerodynamic transfer of energy to the vocal folds. The Journal of the Acoustical Society of America vol. 118 1689–1700 (2005) – 10.1121/1.2000787
• Gassner, G. J. & Winters, A. R. A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where? Frontiers in Physics vol. 8 (2021) – 10.3389/fphy.2020.500690
• Guryanov, A. I., Piralishvili, Sh. A., Guryanova, M. M., Evdokimov, O. A. & Veretennikov, S. V. Counter-current hydrogen–oxygen vortex combustion chamber. Thermal physics of processing. Journal of the Energy Institute vol. 93 634–641 (2020) – 10.1016/j.joei.2019.06.002
• Ji, Y., Lin, C. & Luo, K. H. Three-dimensional multiple-relaxation-time discrete Boltzmann model of compressible reactive flows with nonequilibrium effects. AIP Advances vol. 11 (2021) – 10.1063/5.0047480
• Pandey, K., Chattopadhyay, K. & Basu, S. Combustion dynamics of low vapour pressure nanofuel droplets. Physics of Fluids vol. 29 (2017) – 10.1063/1.4991752
• Hoda, A., Rahman, T. M. R., Asrar, W. & Khan, S. A. A Comparative Study of Natural Gas and Biogas Combustion in A Swirling Flow Gas Turbine Combustor. Combustion Science and Technology vol. 194 2613–2640 (2021) – 10.1080/00102202.2021.1882441
• Lei, T. & Luo, K. H. Lattice Boltzmann Simulation of Multicomponent Porous Media Flows With Chemical Reaction. Frontiers in Physics vol. 9 (2021) – 10.3389/fphy.2021.715791
• Bhuvan, C. H., Hiranandani, K., Aravind, B., Nair, V. & Kumar, S. Novel flame dynamics in rich mixture of premixed propane–air in a planar microcombustor. Physics of Fluids vol. 32 (2020) – 10.1063/5.0020518
• Modeling and Control of Complex Physical Systems (2009)
• Transport Phenomena (2007)
• Cardoso-Ribeiro, F. L., Matignon, D. & Pommier-Budinger, V. Port-Hamiltonian model of two-dimensional shallow water equations in moving containers. IMA Journal of Mathematical Control and Information vol. 37 1348–1366 (2020) – 10.1093/imamci/dnaa016
• A Mathematical Introduction to Fluid Mechanics (1993)
• Least-Squares Finite Element Methods (2009)
• Mixed Finite Element Methods and Applications (2013)
• Recent Trends in Operator Theory and Partial Differential Equations. Operator Theory: Advances and Applications (Springer International Publishing, 2017). doi:10.1007/978-3-319-47079-5 – 10.1007/978-3-319-47079-5
• DOI not foun – 10.1002/(sici)1521-4001(199901)79:1<29::aid-zamm29>3.0.co;2-h
• Olshanskii, M. A., Heister, T., Rebholz, L. G. & Galvin, K. J. Natural vorticity boundary conditions on solid walls. Computer Methods in Applied Mechanics and Engineering vol. 297 18–37 (2015) – 10.1016/j.cma.2015.08.011
• Fluid Mechanics (1987)
• Beyond Equilibrium Thermodynamics (2005)
• Modern Thermodynamics (2014)
• Merk, H. J. The macroscopic equations for simultaneous heat and mass transfer in isotropic, continuous and closed systems. Applied Scientific Research vol. 8 73–99 (1959) – 10.1007/bf00411741
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling vol. 75 940–960 (2019) – 10.1016/j.apm.2019.04.035