Phase-Field/Discontinuity Capturing operator for direct van der Waals simulation (DVS)

TL;DR

The paper tackles numerical stabilization for phase-transforming flows modeled by the isothermal Navier–Stokes–Korteweg equations within Direct van der Waals Simulation (DVS). It introduces a phase-field/discontinuity capturing (PF/DC) operator that honors the free-energy dissipation law at liquid–vapor interfaces while vanishing in the incompressible liquid bulk, addressing failures of classical DC operators. The PF/DC design combines phase-field nonlocal chemical potential concepts with targeted dissipative fluxes and is embedded in a dispersive-SUPG stabilized finite-element framework for robust, high-order simulations. Numerical results across 1D, 2D, and 3D cavitating flows show PF/DC yields superior accuracy, stability, and agreement with experimental data compared to classical DC operators, even on under-resolved interfaces. The work suggests PF/DC as a reliable stabilization strategy for DVS and related phase-transforming-flow models, with future extensions to thermal effects and multi-component mixtures.

Abstract

Discontinuity capturing (DC) operators are commonly employed to numerically solve problems involving sharp gradients in the solution. Despite their success, the application of DC operators to the direct van der Waals simulation (DVS) remains challenging. The DVS framework models non-equilibrium phase transitions by admitting interfacial regions in which the derivative of pressure with respect to density is negative. In these regions, we demonstrate that classical DC operators may violate the free energy dissipation law and produce unphysical wave structures. To address this limitation, we propose the phase-field/discontinuity capturing (PF/DC) operator. Numerical results show that PF/DC yields stable and accurate solutions in both bulk fluids and interfacial regions. Finally, we apply the proposed method to simulate cavitating flow over a three-dimensional bluff body, obtaining excellent agreement with experimental data and significant improvements over results produced using classical DC operators.

Figures (18)

• Figure 1: (a) Pressure and chemical potential as functions of density for a cubic equation of state (EoS) below the critical condition. (b) Pressure as a function of density at temperatures below, at, and above the critical temperature. The dashed and dash-dot lines in (b) mark the predicted phase envelope and the interfacial region, respectively.
• Figure 2: (a) Saturation pressure ($p^s$) as a function of temperature ($T$), and (b) corresponding phase envelopes for water predicted by various equations of state (EoS). Markers represent experimental data from Linstrom1997-pt. The results show that the GERG-2008 EoS provides the best agreement with experimental measurements.
• Figure 3: (a) Equilibrium liquid–vapor interface profiles along the spatial coordinate $x$ at temperatures $T = 350$, $450$, and $550$ K. (b) Corresponding profiles of the chemical potential $\mu$ and the non-local chemical potential $\mu_{\rm nl}$.
• Figure 4: Time evolution of (a) density and (b) velocity of the reference solution with $\rho^{\rm ref} = \rho^s_l + 50$ kg/m$^3$ on a one-dimensional periodic domain. Given the initial density distribution, the fluid remains entirely in the liquid phase, and the perturbation is insufficient to induce phase transformation. As a result, the solution remains single-phase and exhibits sinusoidal oscillations whose amplitude gradually decreases over time due to viscous dissipation.
• Figure 5: $L_2$ norm of the error in the numerical solution at $t = 0.076$ ns with $\rho^{\rm ref} = \rho^s_l + 50$ kg/m$^3$, evaluated against the reference solution for different DC operator choices in \ref{['eqn:DVS_Formulation']}. The DC (no scale) operator yields a suboptimal convergence rate, while both the DC (with scale) and PF/DC operators achieve optimal rate. These results provide numerical evidence supporting the effectiveness of scaling the DC operator based on fluid compressibility.

Theorems & Definitions (2)

• Remark 3.1
• Remark 4.1