Sara RINALDI, Ph.D. Candidate 40th cycle, University of Trento, DICAM

My research focuses on developing high-accuracy, conservative, and structure-preserving numerical schemes for the equations governing nonlinear continuum mechanics with phase transitions. Specifically, I will work with the unified and thermodynamically compatible model proposed by Godunov, Peshkov, and Romenski, which describes both the dynamics of elastoplastic solids undergoing large deformations and fluid flows. These types of flows can be of particular interest for the dynamics of industrial flows, as well as for the description of environmental and geophysical flows. 
My work began with the study of the Zeldovich–von Neumann–Döring (ZND) model to simulate combustion phenomena. Detonation can initially be described as an infinitesimally thin shock wave that compresses the explosive material to a high-pressure state known as the von Neumann spike, where the explosive remains unreacted. This spike marks the beginning of the exothermic chemical reaction zone, which concludes at the Chapman–Jouguet condition. Beyond this point, the detonation products expand backward.I analyzed steady planar detonation waves, where the reactive flow is governed by the compressible Navier–Stokes equations with chemical reactions. It is possible to reduce the associated partial differential equation (PDE) system to a single ordinary differential equation (ODE) for any fixed time t. Then I computed the temporal evolution of the detonation wave.

Currently, I am working on solving the compressible Navier–Stokes equations with chemical reactions, incorporating viscous forces. This addition introduces a stiff problem, making the numerical solution both challenging and computationally demanding. My ongoing efforts aim to develop a robust numerical approach to address this complexity.
Building on this foundation, my future research will focus on extending the chemical reaction relation of the ZND model to multiphase flows with phase transitions. The ultimate goal is to create numerical schemes for multiphase flows with phase transitions that preserve essential properties of the continuous equations, such as conservation laws and thermodynamic consistency. These properties are critical for accurately capturing the intricate interactions within multiphase systems involving compressible and quasi-incompressible flows.

Bibliography

[1] Menikoff R. Detonation Wave Profile 2017.
[2] Hidalgo A, Dumbser M. ADER Schemes for Nonlinear Systems of Stiff Advection–Diffusion–Reaction Equations. Journal of Scientific Computing. 2010;48(1-3):173-189.
[3] Romenski E, Drikakis D, Toro E. Conservative Models and Numerical Methods for Compressible Two-Phase Flow. Journal of Scientific Computing. 2009;42(1):68-95.
[4] Romenski, Resnyansky, Toro. Conservative Hyperbolic Formulation for compressible two-phase flow with different phase pressures and temperatures. 2007;Quarterly of Applied Mathematics(2).
[5] Thein F, Romenski E, Dumbser M. Exact and Numerical Solutions of the Riemann Problem for a Conservative Model of Compressible Two-Phase Flows. Journal of Scientific Computing. 2022;93(3). 
[6] Ferrari, Peshkov, Romenski, Dumbser A unified SHTC multiphase model of continuum mechanics. arXiv (Cornell University). doi:https://doi.org/10.48550/arxiv.2403.19298