A Particle-Based Method Coupled with Finite Volume Solver for Interfacial Flows

A hybrid Lagrangian-Eulerian (hLE) scheme, combining a particle-based, mesh-free technique with a finitevolume flow solver, is developed for direct simulations of two-phase flows. The approach uses marker points around the interface and advects the signed distance to the interface in a Lagrangian frame. The kernel–based derivative calculations typical of particle methods are used to extract the interface normal and curvature from unordered marker points. Connectivity between the marker points is not necessary. The fluid flow equations are solved on a background, fixed mesh using a co-located grid finite volume solver together with balanced force algorithm (Francois et al. JCP, 2006, Herrmann JCP, 2007) for surface tension force. The numerical scheme is applied to standard test cases to show promising results: (i) parasitic currents in a stationary spherical drop, (ii) small amplitude damped surface waves, (iii) capillary waves on droplet surface, (iv) Rayleigh-Taylor instability, and (v) gravity-driven bubble/droplet in a stationary fluid. Introduction Particle-based Lagrangian, mesh-free algorithms such as moving particle-methods [1], vortex-in cell methods [2, 3], and smoothed-particle hydrodynamics [4] have been popular for large-scale free-surface flows. These pure Lagrangian methods are promising as they avoid enormous memory requirements for a three-dimensional mesh. These methods automatically provide adaptive resolution in the high-curvature region [2] and have been applied successfully to many two-phase flow problems [5, 6, 7]. However, they exhibit other difficulties such as high cost of finding nearest neighbors in the zone of influence of a Lagrangian point, true enforcement of continuity (or incompressibility) conditions, and problems associated with accurate one-sided interpolations near boundaries [2]. In the present work, we develop a hybrid approach, wherein the Lagrangian nature of the interface motion is captured by particle-based method, and the fluid flow is computed using a finite-volume solver using variable density, single-fluid model. The basic idea is to merge the locally ‘adaptive’ mesh-free particle-based methods with the relative ‘ease’ of Eulerian finite-volume formulation in order to inherit the advantages offered by individual approaches. The interface between two fluids is represented and tracked using Lagrangian points or fictitious particles [7]. Unlike particle level set method [8] or the semi-Lagrangian methods [9], in the present approach the interface is represented by Lagrangian points (LPs) (or particles) that are advanced in a Lagrangian frame. The motion of the interface is determined by a velocity field (interpolated to the particle locations) obtained by solving the Navier-Stokes equations on a fixed background mesh in an Eulerian frame. The interface location, once determined, identifies the region of the mesh to apply jump-conditions in fluid properties. In this sense, it is in the realm of Arbitrary Lagrangian-Eulerian (ALE) schemes, wherein the computational grid deforms to conform to the shape of the dispersed phase. The potential advantage of the present hybrid method is that the background mesh could be of any kind: structured, body-fitted, or arbitrary shaped unstructured (hex, pyramids, tetrahedrons, prisms) and may be stationary or changing in time (adaptive refinement). Here, we use a co-located grid, incompressible flow solver based on the energy conserving finite-volume algorithm [10]. The Lagrangian points (LPs) in our interface calculations, are particles distributed in a narrow band around the interface [11]. These LPs are initially uniformly spaced and carry information such as the signed distance to the interface (SDF) along the characteristic paths. Variations in flow velocities leads to an irregular distribution of the initially uniform LPs. Regularization of the particles are performed by mapping the particles on a uniformly spaced lattice [7]. Values for particle properties at new LP locations are obtained through kernel mollification as done in Smoothed Particle Hydrodynamics [4] and remeshed-SPH [6]. The novelty in our approach is that this mesh-free interface representation is integrated with a finite-volume solver where the governing equations for flow ∗Corresponding Author: [email protected] 1In this paper, the term ‘particles’ means Lagrangian points (LPs) that are used to represent the interface. evolution are solved. The Lagrangian points provide sub-grid resolution and in this respect the method is similar to the Refined Level Set Grid (RLSG) approach [12, 13]. However, here the LPs move in space with the flow velocity and different discretizations are necessary and we use high-order schemes based on mollification kernels. Hybrid Lagrangian-Eulerian (hLE) Scheme Following Hieber & Koumoutsakos [7], the interface between two fluids is represented using uniformly spaced Lagrangian points (LPs) or fictitious particles in a narrow band around the interface. Each LP is associated with position xp, velocity up, volume Vp and a scalar function Φp which represents the signed distance to the interface. The average spacing (h between the uniformly spaced LPs is related to the volume Vp. In this work, we use cubic elements (h = V p ). As the LPs move, they carry the SDF value along the characteristic paths and implicitly represent the motion of the interface. The evolution of the interface is calculated by solving level set equations in the Lagrangian form: