The piston effect in supercritical fluids investigated via a reversible-irreversible vector field splitting-based explicit time integration scheme

TL;DR

The piston effect in near-critical fluids involves strong thermoacoustic coupling that rapidly converts boundary heating into pressure and temperature changes. The authors develop a fully explicit, second-order time integration method based on reversible-irreversible splitting of the semi-discretized linear thermoacoustic equations, yielding a decoupled hyperbolic wave equation with isentropic speed $a_s^0$ and a parabolic heat equation with speed $a_T^0$. They demonstrate second-order accuracy and dispersion-free wave propagation in 1D $CO_2$ across near-critical and nearly ideal-gas states and analyze piston-effect dynamics, including comparisons to Boukari's homogeneous-pressure model. The method is efficient, structure-preserving, and extensible to nonlinear, 3D, and GPU-accelerated settings for robust thermoacoustic predictions in supercritical fluids.

Abstract

In the vicinity of the liquid--vapor critical point, supercritical fluids behave strongly compressibly and, in parallel, thermophysical properties have strong state dependence. These lead to various peculiar phenomena, one of which being the piston effect where a sudden heating induces a mechanical pulse. The coupling between thermal and mechanical processes, in the linear approximation, yields a non-trivially rich thermoacoustics. The numerous applications of supercritical fluids raise the need for reliable yet fast and efficient numerical solution for thermoacoustic time and space dependence in this sensitive domain. Here, we present a second-order accurate, fully explicit staggered space-time grid finite difference method for such coupled linear thermoacoustic problems. Time integration is based on the splitting of the state space vector field representing the interactions that affect the dynamics into reversible and irreversible parts, which splitting procedure leads to decoupled wave and heat equations. The former is a hyperbolic partial differential equation, while the latter is a parabolic one, therefore, different time integration algorithms must be amalgamated to obtain a reliable, dispersion error-free, and dissipation error-free numerical solution. Finally, the thermoacoustic approximation of the supercritical piston effect is investigated via the developed method.

Figures (19)

• Figure 1: From left to right: Temperature dependence of density, isobaric specific heat capacity, isobaric volume thermal expansion coefficient, isothermal compressibility, shear viscosity, and thermal conductivity for carbon dioxide along the isobaric curves $p/p_{\rm c} = 0.8 , \ 1.05 , \ 1.2 , \ 1.35 , \ 1.5$ (from dark blue to red, respectively). Critical pressure and temperature of carbon dioxide are $p_{\rm c} = 7.377 \ {\rm MPa}$ and $T_{\rm c} = 304.128 \ {\rm K}$. Below the critical pressure (i.e., in the figures $p/p_{\rm c} = 0.8$) jump occurs in the thermophysical properties, where left and right sides belong to the liquid and gas phase, respectively, this jump disappears for pressures above the critical pressure. Numerical data are taken from the NIST database lemmon2024thermophysical.
• Figure 2: Temperature dependence of thermal diffusivity for carbon dioxide along the isobaric curves $p/p_{\rm c} = 0.8 , \ 1.05 , \ 1.2 , \ 1.35 , \ 1.5$ (from dark blue to red, respectively). Numerical data are calculated from the NIST database lemmon2024thermophysical.
• Figure 3: Spacetime plot of the temperature field.
• Figure 4: Calculated time evolution of temperature at the front side and at the rear side of the sample.
• Figure 5: Time evolution of temperature, density, and pressure at the front side and at the rear side of the sample at the beginning of the process.