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Mass conservation, positivity and energy identical-relation preserving scheme for the Navier-Stokes equations with variable density

Fan Yang, Haiyun Dong, Maojun Li, Kun Wang

TL;DR

The paper tackles the Navier-Stokes equations with variable density by designing a numerical scheme that preserves key physical properties: mass conservation, positivity of density, and the original energy identical-relation. Positivity is ensured via a square-density transform $\rho=\sigma^2$, while mass and energy identity are enforced through a recovery process and a transformed momentum formulation that preserves the energy structure. The authors provide a rigorous error analysis for a fully discrete, first-order finite element scheme, establishing convergence rates (notably $O(\tau^2+h^4)$ for density and velocity in $L^2$) under suitable regularity, and validate the method through numerical experiments including convergence studies and flows with variable density. The framework enables reliable long-time simulations of variable-density incompressible flows by maintaining essential invariants of the continuous model, with potential for extension to higher-order schemes and coupled multi-physics problems.

Abstract

In this paper, we consider a mass conservation, positivity and energy identical-relation preserving scheme for the Navier-Stokes equations with variable density. Utilizing the square transformation, we first ensure the positivity of the numerical fluid density, which is form-invariant and regardless of the discrete scheme. Then, by proposing a new recovery technique to eliminate the numerical dissipation of the energy and to balance the loss of the mass when approximating the reformation form, we preserve the original energy identical-relation and mass conservation of the proposed scheme. To the best of our knowledge, this is the first work that can preserve the original energy identical-relation for the Navier-Stokes equations with variable density. Moreover, the error estimates of the considered scheme are derived. Finally, we show some numerical examples to verify the correctness and efficiency.

Mass conservation, positivity and energy identical-relation preserving scheme for the Navier-Stokes equations with variable density

TL;DR

The paper tackles the Navier-Stokes equations with variable density by designing a numerical scheme that preserves key physical properties: mass conservation, positivity of density, and the original energy identical-relation. Positivity is ensured via a square-density transform , while mass and energy identity are enforced through a recovery process and a transformed momentum formulation that preserves the energy structure. The authors provide a rigorous error analysis for a fully discrete, first-order finite element scheme, establishing convergence rates (notably for density and velocity in ) under suitable regularity, and validate the method through numerical experiments including convergence studies and flows with variable density. The framework enables reliable long-time simulations of variable-density incompressible flows by maintaining essential invariants of the continuous model, with potential for extension to higher-order schemes and coupled multi-physics problems.

Abstract

In this paper, we consider a mass conservation, positivity and energy identical-relation preserving scheme for the Navier-Stokes equations with variable density. Utilizing the square transformation, we first ensure the positivity of the numerical fluid density, which is form-invariant and regardless of the discrete scheme. Then, by proposing a new recovery technique to eliminate the numerical dissipation of the energy and to balance the loss of the mass when approximating the reformation form, we preserve the original energy identical-relation and mass conservation of the proposed scheme. To the best of our knowledge, this is the first work that can preserve the original energy identical-relation for the Navier-Stokes equations with variable density. Moreover, the error estimates of the considered scheme are derived. Finally, we show some numerical examples to verify the correctness and efficiency.

Paper Structure

This paper contains 13 sections, 11 theorems, 139 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional spaces
  4. Some inequalities
  5. Some essential properties
  6. Property-preserving scheme
  7. Error estimate
  8. Numerical Results
  9. Convergence order
  10. Property-preserving test
  11. Back-step flow
  12. Flow around a circular cylinder
  13. Conclusions

Key Result

Lemma 2.1

(Gronwall inequality bib17) Let $B>0$ and $a_k, b_k, c_k$ be non-negative numbers such that If $\tau c_k< 1$ and $d_k=(1-\tau c_k)^{-1}$, then there holds

Figures (8)

  • Figure 1: Evolutions of the density, energy, mass without recovery, mass with recovery, $\lambda_h^{n+1}$, $\gamma_h^{n+1}$ and $D_E^n$ with $f=0$.
  • Figure 2: Evolutions of the density, energy, mass without recovery, mass with recovery, $\lambda_h^{n+1}$, $\gamma_h^{n+1}$ and $D_E^n$ with $f\neq 0$.
  • Figure 3: Analytical regions and boundary conditions.
  • Figure 7: Velocity $u_{1h}^n$ of the cylinder flow with a constant density at $t=3$ (top), $t=5$ (middle), $t=7$ (bottom).
  • Figure 10: Velocity $u_{1h}^n$ of the cylinder flow with a variable density at $t=0.5,~1,~2,~3,~5$ and $7$ (from top to bottom).
  • ...and 3 more figures

Theorems & Definitions (24)

  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 14 more