Characteristic Bending in Incompressible Flows

TL;DR

The paper addresses numerical advection under incompressible velocity fields, where discretization errors can introduce spurious compressibility and mass loss. It introduces Characteristic Bending (CB), a volume-preserving projection applied to near-identity reference-map steps within a semi-Lagrangian framework, enabling robust, conservative transport. CB can function as a drop-in replacement or augment existing reference-map methods (e.g., CLSRM, VPRM), including its long-time extension RMCB, and is demonstrated across 2D/3D advection and two-phase incompressible Navier–Stokes problems on adaptive grids. The results show CB improves robustness and accuracy in incompressible flow simulations, particularly in the presence of approximate divergence-free fields and deforming interfaces, with practical implications for multiphase and interfacial flows.

Abstract

We present the Characteristic Bending (CB) method, a general framework for advecting quantities under incompressible velocity fields. The method builds on standard semi-Lagrangian advection by interpreting the backward-in-time characteristic reconstruction as the construction of a reference map, a diffeomorphism between the current and initial geometries of the advected space. From this viewpoint, the CB method applies a volume-preserving projection to the map, systematically removing spurious compressible errors arising from time integration, interpolation, or from velocity fields that are only approximately divergence-free. This projection bends the characteristics toward the divergence-free space, preserving mass and geometric features of the advected fields, even in the presence of significant error. We demonstrate the method in both two and three dimensions using benchmark problems and for multiphase flows governed by the incompressible Navier-Stokes equations. The results show that the CB method serves as a drop-in replacement for traditional semi-Lagrangian schemes and as an augmentation of reference map formulations, offering improved robustness and accuracy in incompressible flow simulations.

Figures (30)

• Figure 1: Schematic of the semi-Lagrangian method where the characteristic passing through grid point $x_i$ at time $t^{n+1}$ is traced backward to the departure point $x_d$. Additionally, the value of the advected quantity at the departure point is computed through interpolation using the neighboring points, $x_{i-1}$ and $x_i$ at time $t^n$.
• Figure 1: Dimensionless parameters, Mo, Eo, and Re, for the Bhaga-Weber Cases $a$, $e$ and $f$.
• Figure 2: A schematic comparison of the semi-Lagrangian and Reference Map Advection methods using the reference map, $\mathbf{\xi}$. The semi-Lagrangian method can be seen as a single time step variation of the reference map method (e.g. using $\mathbf{\xi}_n^{n+1}, \; \mathbf{\xi}_{n+1}^{n+2}$), whereas the reference map method traces the characteristics over multiple time steps (e.g. using $\mathbf{\xi}_n^{n+2}$).
• Figure 3: The basic schematic of the reference map method. We consider the initial domain, $\mathcal{B}_0$ deformed by the velocity field $\mathbf{u}$. The motion map, $\chi(\cdot,t)$, transforms any point $\mathbf{x}_0$ in the domain, $\mathcal{B}_0$ to its associated image, $\mathbf{x}(t)$, in the deformed domain, $\mathcal{B}(t)$. The reference map, $\xi(\cdot,t)$, is the inverse of the motion map and transforms any point $\mathbf{x}(t)$ to its initial position, $\mathbf{x}_0$, in the domain $\mathcal{B}_0$.
• Figure 4: The Eulerian vs. Lagrangian perspective of incompressibility. In the Eulerian frame (a), incompressibility corresponds to a divergence-free velocity field, $\nabla \cdot \mathbf{u}=0$. In the Lagrangian frame (b), the equivalent statement is that a material control volume preserves its volume during motion, $V^n = V^{n+1}$.