Numerical investigation of a Navier-Stokes-Korteweg system through an extended Lagrangian formulation: nonclassical waves and phase transition problems
Abstract
The one-dimensional Navier-Stokes-Korteweg equations (a viscous-dispersive regularization of the one-dimensional Euler equations) are considered for modeling an elementary liquid-vapor phase transition. A cubic van der Waals equation of state is used to close the system. The viscosity is chosen so that the solutions of the Euler equations are recovered in the viscousdispersiveless limit, yielding classical shocks. Using an extended Lagrangian method, the dispersive Navier-Stokes-Korteweg system of equations is approximated by a first-order system of conservation laws. Despite the non-convex and non-monotone nature of the equation of state, the extended system can be made unconditionally hyperbolic through an adequate choice of the parameters of the extended system. Using a splitting method, the extended system is solved numerically by a second-order finite-volume numerical scheme. The convergence of the numerical scheme is tested on a stationary travelling wave. Finally, three numerical experiments are considered: a liquid-vapor shock tube, the cavitation of a liquid, and a physically motivated cavitation shock tube for the dodecane. These numerical experiments demonstrate that the extended system captures non-classical waves, such as undercompressive shocks and composite waves induced by the equation of state. Furthermore, for the cavitation of a liquid, multiple phase transition fronts are obtained and analyzed in terms of a characteristic length (related to the capillary coefficient) and of the expansion rate. A parallel is stressed with fracture dynamics in solid mechanics.
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BibTeX
@misc{didierlaurent2026numerical,
title = {{Numerical investigation of a Navier-Stokes-Korteweg system through an extended Lagrangian formulation: nonclassical waves and phase transition problems}},
author = {Quentin Didierlaurent and Nicolas Favrie and Bruno Lombard},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://hal.science/hal-05855842v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Numerical investigation of a Navier-Stokes-Korteweg system through an extended Lagrangian formulation: nonclassical waves and phase transition problems
AU - Quentin Didierlaurent
AU - Nicolas Favrie
AU - Bruno Lombard
PY - 2026
DA - 2026/11/07
PB - HAL
UR - https://hal.science/hal-05855842v1
AB - The one-dimensional Navier-Stokes-Korteweg equations (a viscous-dispersive regularization of the one-dimensional Euler equations) are considered for modeling an elementary liquid-vapor phase transition. A cubic van der Waals equation of state is used to close the system. The viscosity is chosen so that the solutions of the Euler equations are recovered in the viscousdispersiveless limit, yielding classical shocks. Using an extended Lagrangian method, the dispersive Navier-Stokes-Korteweg system of equations is approximated by a first-order system of conservation laws. Despite the non-convex and non-monotone nature of the equation of state, the extended system can be made unconditionally hyperbolic through an adequate choice of the parameters of the extended system. Using a splitting method, the extended system is solved numerically by a second-order finite-volume numerical scheme. The convergence of the numerical scheme is tested on a stationary travelling wave. Finally, three numerical experiments are considered: a liquid-vapor shock tube, the cavitation of a liquid, and a physically motivated cavitation shock tube for the dodecane. These numerical experiments demonstrate that the extended system captures non-classical waves, such as undercompressive shocks and composite waves induced by the equation of state. Furthermore, for the cavitation of a liquid, multiple phase transition fronts are obtained and analyzed in terms of a characteristic length (related to the capillary coefficient) and of the expansion rate. A parallel is stressed with fracture dynamics in solid mechanics.
ER -