Angular-momentum conservation in discretization of Navier-Stokes equation for viscous fluids

TL;DR

This paper investigates how violating angular-momentum conservation in numerical simulations of viscous fluids leads to artificial effects, and proposes a method to preserve AMC by proper integration of viscous terms, validated through simulations.

Contribution

It reveals the cause of AMC violations in discretized Navier-Stokes equations and offers a correction method to ensure AMC in simulations of viscous fluids.

Findings

AMC violations cause artificial rotations in multi-component fluids.

Proper integration of viscous terms maintains AMC and matches analytical predictions.

Simulations confirm the importance of AMC for accurate fluid behavior modeling.

Abstract

Although the Navier-Stokes equation (NSE) is derived under angular-momentum conservation (AMC), numerical simulation methods often lack it. Here, we reveal that AMC violations result from implementation of the degenerated viscous terms of NSE. To maintain AMC, these degenerated terms must be separately integrated in accordance with their stress origins. As observed in particle-based hydrodynamics methods, the violation causes artificial rotations in multi-component fluids with different viscosities. At the interface between two fluids or with a mobile solid object, AMC must be satisfied, whereas AMC can be neglected in bulk fluids. We also clarify that the condition for constant fluid rotation as a rigid body in a container rotating at a constant speed is not the AMC of the stresses, but the invariance of the viscous forces under a global rotation. To confirm our theory, we simulated the circular laminar flows of single- and binary-component fluids using two-dimensional Lagrangian finite volume methods. The results show excellent agreement with the analytical predictions for fluids with and without AMC.