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Numerical Solution to the Riemann Problem for a Liquid-Gas Two-phase Isentropic Flow Model

Abdul Rab

TL;DR

This paper analyzes a one-dimensional, strictly hyperbolic $3\times3$ drift-flux model of liquid–gas two-phase isentropic flow to study Riemann problems and their wave structures. It combines a theoretical examination of the characteristic fields with a comparative numerical study of Lax–Friedrichs, Lax–Wendroff, and FORCE schemes, implemented in Python, to illustrate shocks, rarefactions, and contact discontinuities. The analysis reveals two genuinely nonlinear fields and one linearly degenerate field, aligning with the observed wave patterns in simulations where Lax–Wendroff best resolves individual waves while FORCE provides smoother, less oscillatory results. The work points to future adoption of higher-order methods such as Roe-type solvers, MUSCL reconstructions, and ENO/WENO schemes to achieve greater accuracy in two-phase flow simulations.

Abstract

A recently introduced two-phase flow model by Chun Shen is studied in this work. The model is derived to describe the dynamics of immersed water bubbles in liquid water as carrier. Several assumptions are made to obtain a reduced form of the mathematical model. The established model consists of nonlinear coupled PDEs in which the unknowns are the densities of the liquid and gas phases and the velocity of the liquid phase; these depend on space and time. For numerical purposes a one-dimensional space--time coordinate system $(x,t)$ is considered. Using the Python programming framework, several Riemann-type initial value problems for the two-phase flow model are solved numerically. A comparison of three finite-difference schemes is presented in order to examine their performance: the Lax--Friedrichs scheme, the Lax--Wendroff scheme, and the FORCE scheme. The FORCE scheme is total-variation diminishing (TVD) and monotone and does not create oscillations. As expected, the numerical solution of the Riemann problems consists of combinations of smooth profiles, shock waves, and rarefaction waves.

Numerical Solution to the Riemann Problem for a Liquid-Gas Two-phase Isentropic Flow Model

TL;DR

This paper analyzes a one-dimensional, strictly hyperbolic drift-flux model of liquid–gas two-phase isentropic flow to study Riemann problems and their wave structures. It combines a theoretical examination of the characteristic fields with a comparative numerical study of Lax–Friedrichs, Lax–Wendroff, and FORCE schemes, implemented in Python, to illustrate shocks, rarefactions, and contact discontinuities. The analysis reveals two genuinely nonlinear fields and one linearly degenerate field, aligning with the observed wave patterns in simulations where Lax–Wendroff best resolves individual waves while FORCE provides smoother, less oscillatory results. The work points to future adoption of higher-order methods such as Roe-type solvers, MUSCL reconstructions, and ENO/WENO schemes to achieve greater accuracy in two-phase flow simulations.

Abstract

A recently introduced two-phase flow model by Chun Shen is studied in this work. The model is derived to describe the dynamics of immersed water bubbles in liquid water as carrier. Several assumptions are made to obtain a reduced form of the mathematical model. The established model consists of nonlinear coupled PDEs in which the unknowns are the densities of the liquid and gas phases and the velocity of the liquid phase; these depend on space and time. For numerical purposes a one-dimensional space--time coordinate system is considered. Using the Python programming framework, several Riemann-type initial value problems for the two-phase flow model are solved numerically. A comparison of three finite-difference schemes is presented in order to examine their performance: the Lax--Friedrichs scheme, the Lax--Wendroff scheme, and the FORCE scheme. The FORCE scheme is total-variation diminishing (TVD) and monotone and does not create oscillations. As expected, the numerical solution of the Riemann problems consists of combinations of smooth profiles, shock waves, and rarefaction waves.

Paper Structure

This paper contains 27 sections, 3 theorems, 39 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Some theorems
  5. Numerical schemes
  6. Lax--Wendroff scheme (Richtmyer form)
  7. Lax--Friedrichs scheme
  8. FORCE scheme
  9. Theory of Hyperbolic Nonlinear PDEs
  10. Nonlinear scalar conservation laws
  11. Example (Burgers' equation).
  12. Self-similar rarefaction waves
  13. Shock waves and Rankine--Hugoniot condition
  14. Example (Burgers' equation).
  15. Systems of conservation laws
  16. ...and 12 more sections

Key Result

Theorem 2.11

Suppose a piecewise smooth solution of eq:conslaw contains a single discontinuity with left and right states $G_-$ and $G_+$. Then $G(x,t)$ is an entropy solution if and only if for all $G$ between $G_-$ and $G_+$.

Figures (6)

  • Figure 1: Lax--Friedrichs scheme at $t=0.17$ for initial data set 1.
  • Figure 2: Lax--Friedrichs scheme at $t=0.25$ for initial data set 2.
  • Figure 3: Lax--Wendroff scheme at $t=0.15$ for initial data set 1.
  • Figure 4: Lax--Wendroff scheme at $t=0.25$ for initial data set 2.
  • Figure 5: FORCE scheme at $t=0.15$ for initial data set 1.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition 2.1: Scalar conservation law
  • Definition 2.2: Riemann problem
  • Definition 2.3: Characteristic curves
  • Definition 2.4: Conservative numerical method
  • Definition 2.5: Consistency
  • Definition 2.6: Weak solution
  • Definition 2.7: Convergent scheme
  • Definition 2.8: Linear scheme
  • Definition 2.9: Monotone scheme
  • Definition 2.10: TVD scheme
  • ...and 3 more