Area preservation in computational fluid dynamics

TL;DR

This paper addresses preserving area-related invariants in 2D incompressible flows by introducing two Eulerian schemes that enforce a discrete area preservation: a fully discrete cell rearrangement model and a smoother vorticity relabelling model. The cell rearrangement method uses a permutation-based update to enforce the preserved area distribution of vorticity, while the relabelling model computes a smooth area function $A_\omega(c)$ and projects the evolved field to maintain equal-area contours, yielding reduced spurious oscillations. Numerical tests on the Liouville equation show substantial suppression of nonphysical extrema, with the relabelling approach achieving smoother solutions at smaller remapping intervals. These methods offer a promising route to incorporate Casimir-like constraints into Eulerian schemes, with potential impact on multi-phase flows and level-set applications.

Abstract

Incompressible two-dimensional flows such as the advection (Liouville) equation and the Euler equations have a large family of conservation laws related to conservation of area. We present two Eulerian numerical methods which preserve a discrete analog of area. The first is a fully discrete model based on a rearrangement of cells; the second is more conventional, but still preserves the area within each contour of the vorticity field. Initial tests indicate that both methods suppress the formation of spurious oscillations in the field.

Figures (3)

• Figure 1: Initial condition for the test problem in Section 4. The contours show level sets of $\omega(0) = \exp(-45(x-{3\over4})^2 - 15(y-{1\over2})^2)$. The arrows show the vector field corresponding to the stream function $\psi = \sin(\pi x)\sin(\pi y)$ by which $\omega$ is advected.
• Figure 2: Numerical computation of the area enclosed within vorticity contours. (a): $C^0$ approximation to $A_{\omega(0)}$ using piecewise linear interpolation (Section 3). Here $\omega(0)$ is the initial condition shown in Figure 1. (b): Piecewise constant approximation to $A_{\omega(0)}$ by sorting the list of vorticity values (Section 2). (c): Area function $A_{\omega(t)}$ after evolving for time $t=1.2$ with no area preservation with an enstrophy-preserving scheme (Section 4). In the vorticity relabelling projection, vorticity values are mapped from this curve back to (a). (d): Finite difference approximation to $dA_{\omega(0)}(c)/dc$, showing that, although only $C^0$, for numerical purposes it can be regarded as being differentiable. The kinks in this derivative are due to $\omega$ being set to zero on the boundary.
• Figure 3: Results for the advection problem on Fig. 1 after 1.6 rotations about the center. (a): Arakawa differences with no area preservation. A large negative blob of vorticity forms and spawns a secondary positive blob. The dotted contour indicates the exact solution. (b): Arakawa differences with cell rearrangement applied every 20 time steps ($\Delta t = 0.001$). (c): Central differences with vorticity relabelling applied every 10 time steps. (d) Arakawa differences with vorticity relabelling applied every 10 time steps.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3