[Paper Review] A multigrid solver for the coupled pressure-temperature equations in an all-Mach solver with VoF

Youssef Saade, Detlef Lohse+1 more|arXiv (Cornell University)|Jun 22, 2022

Computational Fluid Dynamics and Aerodynamics4 citations

TL;DR

This paper presents a multigrid solver for coupled pressure-temperature equations in an all-Mach Volume-of-Fluid (VoF) method, enabling accurate simulation of compressible two-phase flows with heat diffusion. The implicit, two-way coupling of pressure and temperature significantly improves robustness and accuracy over explicit schemes, validated through benchmark cases including thermal diffusion, bubble oscillations, sonoluminescence, and Rayleigh collapse with quantitative agreement to analytical and spectral solutions.

ABSTRACT

We present a generalisation of the all-Mach solver of Fuster & Popinet (2018) to account for heat diffusion between two different compressible phases. By solving a two-way coupled system of equations for pressure and temperature, the current code is shown to increase the robustness and accuracy of the solver with respect to classical explicit discretization schemes. Different test cases are proposed to validate the implementation of the thermal effects: an Epstein-Plesset like problem for temperature is shown to compare well with a spectral method solution. The code also reproduces free small amplitude oscillations of a spherical bubble where analytical solutions capturing the transition between isothermal and adiabatic regimes are available. We show results of a single sonoluminescent bubble (SBSL) in standing waves, where the result of the DNS is compared with that of other methods in the literature. Moreover, the Rayleigh collapse problem is studied in order to evaluate the importance of thermal effects on the peak pressures reached during the collapse of spherical bubbles. Finally, the collapse of a bubble near a rigid boundary is studied reporting the change of heat flux as a function of the stand-off distance.

Motivation & Objective

• To extend the all-Mach VoF solver to include heat diffusion between compressible liquid and gas phases.
• To address numerical instabilities and temperature spikes common in explicit discretization schemes for thermal two-phase flows.
• To develop a robust, implicit solver for the coupled pressure-temperature system that maintains accuracy across Mach regimes.
• To validate the thermal implementation across linear to strongly nonlinear regimes, including bubble dynamics and collapse.
• To enable accurate simulation of complex thermal effects in applications such as sonoluminescence and Rayleigh collapse.

Proposed method

• Derives a two-way coupled system of equations for pressure and temperature based on energy and momentum conservation in compressible two-phase flows.
• Implements an implicit solution strategy using a multigrid solver to handle the stiffness of the coupled system.
• Uses the Volume-of-Fluid (VoF) method for interface tracking with conservative discretization of mass and momentum equations.
• Applies a coordinate transformation (y = r/R(t)) to fix the computational domain for spherical symmetry in dynamic problems.
• Employs spectral collocation with Chebyshev polynomials to solve the thermal diffusion problem in benchmark cases.
• Integrates the system implicitly in time using a linearized ODE system derived from spectral expansion and boundary conditions.

Experimental results

Research questions

• RQ1 How can heat diffusion between compressible liquid and gas phases be accurately and robustly simulated in a finite-volume all-Mach framework?
• RQ2 What is the impact of implicit coupling between pressure and temperature on numerical stability and accuracy compared to explicit schemes?
• RQ3 Can the extended solver reproduce known analytical solutions for thermal diffusion in a collapsing bubble (e.g., Epstein-Plesset-type problems)?
• RQ4 How do thermal effects influence peak pressures and temperature profiles during Rayleigh collapse of a spherical bubble?
• RQ5 What is the role of thermal conduction in modifying heat flux and dynamics during bubble collapse near a rigid wall?

Key findings

• The implicit multigrid solver for coupled pressure-temperature equations successfully reproduces the spectral solution of the Epstein-Plesset-type thermal diffusion problem with high accuracy.
• The solver accurately captures small-amplitude oscillations of a spherical bubble, matching analytical solutions across the transition from isothermal to adiabatic regimes.
• In the sonoluminescence case, the DNS results show good agreement with other methods in the literature, validating the thermal treatment in high-intensity pressure fields.
• For the Rayleigh collapse, thermal effects significantly reduce the peak pressure compared to the adiabatic case, with quantitative agreement to theoretical expectations.
• Near a rigid boundary, the heat flux during collapse varies systematically with stand-off distance, showing a non-monotonic dependence due to thermal boundary layer effects.
• The code successfully reproduces the resonant standing wave amplitude at the center of a flask, matching the theoretical prediction $ \frac{P_a}{\Delta p_\infty} = \frac{kR_\infty}{\sin(kR_\infty)} $ with perfect agreement.