A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows
Authors
Abstract
We consider the problem of finding an energy-based formulation of the Navier–Stokes equations for reactive flows. These equations occur in various applications, e. g., in combustion engines or chemical reactors. After modeling, discretization, and model reduction, important system properties as the energy conservation are usually lost which may lead to unphysical simulation results. In this paper, we introduce a port-Hamiltonian formulation of the one-dimensional Navier–Stokes equations for reactive flows. The port-Hamiltonian structure is directly associated with an energy balance, which ensures that a temporal change of the total energy is only due to energy flows through the boundary. Furthermore, the boundary ports may be used for control purposes.
Keywords
Reactive flow; Port-Hamiltonian formulation; Navier–Stokes equations; Hamiltonian formulation; Energy-Based modeling
Citation
- Journal: Systems & Control Letters
- Year: 2017
- Volume: 100
- Issue:
- Pages: 51–55
- Publisher: Elsevier BV
- DOI: 10.1016/j.sysconle.2016.12.005
BibTeX
@article{Altmann_2017,
title={{A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows}},
volume={100},
ISSN={0167-6911},
DOI={10.1016/j.sysconle.2016.12.005},
journal={Systems & Control Letters},
publisher={Elsevier BV},
author={Altmann, R. and Schulze, P.},
year={2017},
pages={51--55}
}
References
- Borggaard, Model reduction for DAEs with an application to flow control. (2015)
- Nagarajan, U. & Yamane, K. Automatic task-specific model reduction for humanoid robots. 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems 2578–2585 (2013) doi:10.1109/iros.2013.6696720 – 10.1109/iros.2013.6696720
- van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
- Olver, (1993)
- Marsden, (1999)
- Ennsbrunner, H. & Schlacher, K. On the geometrical representation and interconnection of infinite dimensional port controlled Hamiltonian systems. Proceedings of the 44th IEEE Conference on Decision and Control 5263–5268 doi:10.1109/cdc.2005.1582998 – 10.1109/cdc.2005.1582998
- Schöberl, M., Ennsbrunner, H. & Schlacher, K. Modelling of piezoelectric structures–a Hamiltonian approach. Mathematical and Computer Modelling of Dynamical Systems vol. 14 179–193 (2008) – 10.1080/13873950701844824
- Van der Schaft, A. J. & Maschke, B. M. Fluid dynamical systems as Hamiltonian boundary control systems. Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228) vol. 5 4497–4502 – 10.1109/cdc.2001.980911
- Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Port Hamiltonian formulation of infinite dimensional systems I. Modeling. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3762-3767 Vol.4 (2004) doi:10.1109/cdc.2004.1429324 – 10.1109/cdc.2004.1429324
- Kanatchikov, I. V. Canonical structure of classical field theory in the polymomentum phase space. Reports on Mathematical Physics vol. 41 49–90 (1998) – 10.1016/s0034-4877(98)80182-1
- BRIDGES, T. J. Multi-symplectic structures and wave propagation. Mathematical Proceedings of the Cambridge Philosophical Society vol. 121 147–190 (1997) – 10.1017/s0305004196001429
- Nishida, G. Hamiltonian Representation of Higher Order Partial Differential Equations with Boundary Energy Flows. Journal of Applied Mathematics and Physics vol. 03 1472–1490 (2015) – 10.4236/jamp.2015.311174
- Ramirez, H., Maschke, B. & Sbarbaro, D. Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR. Chemical Engineering Science vol. 89 223–234 (2013) – 10.1016/j.ces.2012.12.002
- Morrison, P. J. & Greene, J. M. Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics. Physical Review Letters vol. 45 790–794 (1980) – 10.1103/physrevlett.45.790
- Morrison, (1984)
- Siuka, (2011)
- Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
- 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
- Beattie, C. & Gugercin, S. Structure-preserving model reduction for nonlinear port-Hamiltonian systems. IEEE Conference on Decision and Control and European Control Conference 6564–6569 (2011) doi:10.1109/cdc.2011.6161504 – 10.1109/cdc.2011.6161504
- Polyuga, R. V. & van der Schaft, A. Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos. IEEE Transactions on Automatic Control vol. 56 1458–1462 (2011) – 10.1109/tac.2011.2128650
- Warnatz, (2006)
- Zhou, W., Hamroun, B., Gorrec, Y. L. & Couenne, F. Infinite Dimensional Port Hamiltonian Representation of reaction diffusion processes. IFAC-PapersOnLine vol. 48 476–481 (2015) – 10.1016/j.ifacol.2015.05.119
- Tartar, (2007)
- Sutherland, J. C. & Kennedy, C. A. Improved boundary conditions for viscous, reacting, compressible flows. Journal of Computational Physics vol. 191 502–524 (2003) – 10.1016/s0021-9991(03)00328-0