Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local conservation laws of continuous Galerkin method for the incompressible Navier--Stokes equations in EMAC form

Maxim A. Olshanskii, Leo G. Rebholz

TL;DR

The paper addresses local conservation properties for continuous Galerkin discretizations of the incompressible Navier–Stokes equations by formulating diffuse-volume weak conservation laws and proving that the EMAC (Energy, Momentum, and Angular Momentum Conserving) nonlinear form yields exact discrete local balances for momentum and angular momentum in Eulerian form, with near-exact behavior in Lagrangian form. The authors derive these discrete balances and show their equivalence to the PDE-level weak forms for smooth solutions, then validate the theory through numerical tests using Taylor–Hood elements and BDF time stepping on Gresho, cylinder flow, and Kelvin–Helmholtz problems. Key contributions include a priori preservation of local momentum and angular-momentum balances by EMAC in the discrete setting, a clear separation between Eulerian and Lagrangian discrete forms, and demonstration that traditional nonlinear forms fail to retain these local balances. The results suggest EMAC improves long-time accuracy and physical fidelity by maintaining discrete invariants without extra residual terms, guiding the design of stable, structure-preserving finite element schemes for incompressible flow.

Abstract

We consider local balances of momentum and angular momentum for the incompressible Navier-Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier-Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory.

Local conservation laws of continuous Galerkin method for the incompressible Navier--Stokes equations in EMAC form

TL;DR

The paper addresses local conservation properties for continuous Galerkin discretizations of the incompressible Navier–Stokes equations by formulating diffuse-volume weak conservation laws and proving that the EMAC (Energy, Momentum, and Angular Momentum Conserving) nonlinear form yields exact discrete local balances for momentum and angular momentum in Eulerian form, with near-exact behavior in Lagrangian form. The authors derive these discrete balances and show their equivalence to the PDE-level weak forms for smooth solutions, then validate the theory through numerical tests using Taylor–Hood elements and BDF time stepping on Gresho, cylinder flow, and Kelvin–Helmholtz problems. Key contributions include a priori preservation of local momentum and angular-momentum balances by EMAC in the discrete setting, a clear separation between Eulerian and Lagrangian discrete forms, and demonstration that traditional nonlinear forms fail to retain these local balances. The results suggest EMAC improves long-time accuracy and physical fidelity by maintaining discrete invariants without extra residual terms, guiding the design of stable, structure-preserving finite element schemes for incompressible flow.

Abstract

We consider local balances of momentum and angular momentum for the incompressible Navier-Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier-Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory.
Paper Structure (16 sections, 2 theorems, 56 equations, 8 figures)

This paper contains 16 sections, 2 theorems, 56 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Eulerian and Lagrangian forms of momentum and angular momentum conservation
  3. Weak form of the conservation laws
  4. EMAC Galerkin formulation is locally conservative
  5. Local conservation in Eulerian form
  6. Local conservation in Lagrangian form
  7. Numerical Tests
  8. Discrete local conservation in Eulerian form
  9. Discrete local conservation in Lagrangian form
  10. Discrete local conservation in traditional form
  11. Gresho problem
  12. 2D flow past a cylinder
  13. Kelvin-Helmholtz flow
  14. Future directions
  15. Declaration of competing interest
  16. ...and 1 more sections

Key Result

Proposition 1

Assume $\mathbf u$ and $p$ are smooth and $\operatorname{div}\mathbf u=0$, then Laws2m (or Laws2a) holds for any subdomain $\omega\subset\Omega$ iff Laws2phim (or Laws2phia) holds for any $\phi\in W^{1,\infty}(\Omega)$ with $\hbox{supp}(\phi)\subset\Omega$.

Figures (8)

  • Figure 1: Initial velocity for the Gresho problem is shown above, as speed contours.
  • Figure 2: Shown above is the domain and $\omega$ (left), the mesh (center), and the mesh zoomed in near $\omega$ for the Gresho problem.
  • Figure 3: Shown above is error in discrete local Eulerian (left), Lagrangian (center), and traditional Eulerian (right) momentum and angular momentum conservation versus time in the (viscous) Gresho problem.
  • Figure 4: The domain for the channel flow past a cylinder numerical experiment.
  • Figure 5: Shown above are the t=1,2,3,4,5 solution plots of the Re=100 simulations of flow past a cylinder, as speed contours.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2