On conservation laws of Navier-Stokes Galerkin discretizations

Abstract

We study conservation properties of Galerkin methods for the incompressible Navier-Stokes equations, without the divergence constraint strongly enforced. In typical discretizations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier-Stokes equations dictate that they should. We aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). Several numerical experiments are performed, which verify the theory and test the new formulation.

References in corpus (1)

• A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations

Cited by in corpus (24)

• Longer time accuracy for incompressible Navier-Stokes simulations with the EMAC formulation
• Divergence-free $H$(div)-FEM for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics
• Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier-Stokes flows
• A Conservative Finite Element Method for the Incompressible Euler Equations with Variable Density
• An accurate finite element method for the numerical solution of isothermal and incompressible flow of viscous fluid
• Energy preserving model order reduction of the nonlinear Schrödinger equation
• Analysis of variable-step/non-autonomous artificial compression methods
• A unified framework of continuous and discontinuous Galerkin methods for solving the incompressible Navier--Stokes equation
• Dynamic resource allocation for efficient parallel CFD simulations
• Full and Reduced Order Model Consistency of the Nonlinearity Discretization in Incompressible Flows
• A new embedded variable stepsize, variable order family of low computational complexity
• Long time fate of two-dimensional incompressible high Reynolds number Navier-Stokes turbulence: A quantitative comparison between theory and simulation
• Local conservation laws of continuous Galerkin method for the incompressible Navier--Stokes equations in EMAC form
• Some continuous and discontinuous Galerkin methods and structure preservation for incompressible flows
• Artificial compressibility methods for the incompressible Navier-Stokes equations using lowest-order face-based schemes on polytopal meshes
• Reliable chaotic transition in incompressible fluid simulations
• Versatile Mixed Methods for the Incompressible Navier-Stokes Equations
• A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains
• Piecewise Divergence-Free Nonconforming Virtual Elements for Stokes Problem in Any Dimensions
• Ensemble algorithm for parametrized flow problems with energy stable open boundary conditions
• On well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations with no-slip boundary conditions
• On the Analysis of the Second Order Time Filtered Backward Euler Method for the EMAC formulation of Navier-Stokes Equations
• Iterative projection method for unsteady Navier-Stokes equations with high Reynolds numbers
• Intrusive and Non-Intrusive Polynomial Chaos Approximations for a Two-Dimensional Steady State Navier-Stokes System with Random Forcing