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A Projection Method for Compressible Generic Two-Fluid Model

Po-Yi Wu

TL;DR

This work develops a projection-based, time-discrete–space-continuous scheme for a viscous compressible two-fluid model derived via volume averaging. By exploiting a single pressure determined from $φ_k ρ_k$, the method achieves unconditional energy stability at the time-discrete level and recovers stable mass transport through a carefully designed prediction–correction sequence and a projection step. Stability at the fully discrete level is fortified with artificial viscosity and a renormalization-like mechanism to prevent nonpositive masses and excessive compression, validated through convergence and benchmark simulations (channel flow, dam break, and bubble rise). The approach offers a computationally efficient framework by consolidating core iterations into the projection step and demonstrates robust performance on classical two-fluid test problems with notable agreement to reference data. Overall, the method provides a rigorous, stable, and scalable tool for simulating compressible two-fluid flows with strong density contrasts and complex phase interactions.

Abstract

A new projection method for a generic two-fluid model is presented in this work. Specifically, we extend the projection method, originally designed for single-phase variable density incompressible and compressible flows, to viscous compressible two-fluid flows. The key idea is that the single pressure $p$ can be uniquely determined by the products of volume fractions and densities $φ_k ρ_k$, of the two fluids. Additionally, the stability of the method is ensured by appropriately assigning intermediate step variables at the time-discrete level and incorporating a stabilizing term at the fully discrete level. We prove the energy stability of the proposed numerical scheme, and its validity is demonstrated through three numerical tests.

A Projection Method for Compressible Generic Two-Fluid Model

TL;DR

This work develops a projection-based, time-discrete–space-continuous scheme for a viscous compressible two-fluid model derived via volume averaging. By exploiting a single pressure determined from , the method achieves unconditional energy stability at the time-discrete level and recovers stable mass transport through a carefully designed prediction–correction sequence and a projection step. Stability at the fully discrete level is fortified with artificial viscosity and a renormalization-like mechanism to prevent nonpositive masses and excessive compression, validated through convergence and benchmark simulations (channel flow, dam break, and bubble rise). The approach offers a computationally efficient framework by consolidating core iterations into the projection step and demonstrates robust performance on classical two-fluid test problems with notable agreement to reference data. Overall, the method provides a rigorous, stable, and scalable tool for simulating compressible two-fluid flows with strong density contrasts and complex phase interactions.

Abstract

A new projection method for a generic two-fluid model is presented in this work. Specifically, we extend the projection method, originally designed for single-phase variable density incompressible and compressible flows, to viscous compressible two-fluid flows. The key idea is that the single pressure can be uniquely determined by the products of volume fractions and densities , of the two fluids. Additionally, the stability of the method is ensured by appropriately assigning intermediate step variables at the time-discrete level and incorporating a stabilizing term at the fully discrete level. We prove the energy stability of the proposed numerical scheme, and its validity is demonstrated through three numerical tests.
Paper Structure (13 sections, 2 theorems, 69 equations, 8 figures, 1 table, 2 algorithms)

This paper contains 13 sections, 2 theorems, 69 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Modeling equations
  3. Reformulation of the governing equations
  4. A priori estimates to \ref{['goveq2.1']}-\ref{['goveq2.3']}
  5. A time-discrete and space-continuous numerical scheme
  6. Stability analysis
  7. Finite element implementation
  8. Numerical test
  9. Channel flow with uniform inlet distribution of fluids
  10. Dam break
  11. Bubble rising
  12. Conclusion
  13. Derivation of Eq. \ref{['goveq1']}

Key Result

Proposition 4.1

If $\overline{\bm u}_k^{m+1}$, $k=g,l$ satisfy the boundary conditions $\overline{\bm u}_k^{m+1}\cdot\bm n |_{\Gamma} = 0$, then we have Proof. By scheme6 and scheme5, we have Taking the integration on both sides, the divergence theorem together with the assumption $\overline{\bm u}_k^{m+1}\cdot\bm n |_{\Gamma} = 0$ lead to the conclusion. Q.E.D.

Figures (8)

  • Figure 1: The physical domain and boundary conditions considered in the first test.
  • Figure 2: Plot of convergence of the first test.
  • Figure 3: Initial configuration of the dam break problem.
  • Figure 4: Comparison between numerical solution and experimental results for the dam break problem. Front position (top) and height of the column (bottom).
  • Figure 5: Isovalues of the volume fraction for the dam break problem.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Remark 3.1
  • Remark 3.2
  • Proposition 4.1
  • Theorem 4.2