A fully conservative and shift-invariant formulation for Galerkin discretizations of incompressible variable density flow

TL;DR

The paper tackles conservation challenges in incompressible variable-density flow by introducing a shift-invariant, fully conservative Galerkin formulation that extends the EMAC concept to variable density. It develops an inviscid and a viscous framework with a modified pressure and a density shift parameter $ar{\rho}$, ensuring mass, squared density, momentum, angular momentum, and kinetic energy conservation under specific discretization choices, both semi- and fully-discretized. It also analyzes viscous regularizations, revealing trade-offs (no single flux preserves energy and angular momentum with diffusion) and identifies a momentum/angular-momentum preserving flux within the Guermond–Popov family. Numerical validations on manufactured solutions, the Gresho problem, and lock-exchange demonstrate improved accuracy and robustness, with shift-invariant formulations outperforming non-shift-invariant counterparts. The approach is readily integrable into existing solvers and offers a pathway to more complex multiphase models and stabilization strategies while maintaining key structural properties.

Abstract

This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to full discrete conservation of mass, squared density, momentum, angular momentum and kinetic energy without the divergence-free constraint being strongly enforced. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.

Figures (8)

• Figure 1: Manufactured solution. Relative $L^2$-error versus degrees of freedom. Convergence study for $T=1$, $\mu=0.01$, CFL = 0.025. All formulations used $(\rho, {\boldsymbol u}, P)\in\textcolor{black}{ {\mathbb P}_3 {\times} {\mathbb P}_3 {\times} {\mathbb P}_2 }$, except SI-MEDMAC which used $(\rho, {\boldsymbol u}, P)\in\textcolor{black}{ {\mathbb P}_2 {\times} {\mathbb P}_3 {\times} {\mathbb P}_2 }$.
• Figure 2: Gresho problem with variable density and periodic boundary conditions using $\text{CFL} = 0.5$ and 4705 ${\mathbb P}_3$ nodes. $\kappa = \nu = \mu = 0$. No stabilization is used. All formulations used $(\rho, {\boldsymbol u}, P)\in\textcolor{black}{ {\mathbb P}_3 {\times} {\mathbb P}_3 {\times} {\mathbb P}_2 }$, except SI-MEDMAC which used $(\rho, {\boldsymbol u}, P)\in\textcolor{black}{ {\mathbb P}_2 {\times} {\mathbb P}_3 {\times} {\mathbb P}_2 }$.
• Figure 3: Comparison of viscous regularizations \ref{['eq:guermond_popov_flux']}-\ref{['eq:GP_S']} using $\mu = \nu = 0$, $\kappa = 0.01$ for a smooth variant of the Gresho problem with variable density and periodic boundary conditions. The solution was obtained using $\text{CFL} = 0.2$, 73728 ${\mathbb P}_3$ nodes and no stabilization was used.
• Figure 4: Momentum and angular momentum conserving viscous flux: ${\boldsymbol f}_{0}$. Lock-exchange problem at $Re = 4000$, Schmidt number 1 and density ratio $1/0.7$. Density $\rho$ at times $t = 1, 3, 5, 7, 10$.
• Figure 5: Momentum and angular momentum conserving viscous flux: ${\boldsymbol f}_{KS}$. Lock-exchange problem at $Re = 4000$, Schmidt number 1 and density ratio $1/0.7$. Density $\rho$ at times $t = 1, 3, 5, 7, 10$.

Theorems & Definitions (7)

• Remark 2.1
• Remark 3.1
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Proposition 5.1