New Immersed Boundary Method with Irrotational Discrete Delta Vector for Droplet Simulations with Large Density ratio

TL;DR

This paper introduces a novel immersed boundary method that uses an irrotational discrete delta vector to effectively reduce parasitic currents and improve interface reconstruction in droplet simulations with high density ratios.

Contribution

A new irrotational discrete delta function scheme is proposed to eliminate parasitic currents and enhance interface accuracy in immersed boundary simulations.

Findings

Significantly reduces unphysical parasitic currents.

Improves droplet surface stability and interface reconstruction.

Validated through standard test cases with positive results.

Abstract

The Immersed Boundary Method (IBM) is one of the popular one-fluid mixed Eulerian-Lagrangian methods to simulate motion of droplets. While the treatment of a moving complex boundary is an extremely time consuming and formidable task in a traditional boundary-fitted fluid solver, the one-fluid methods provide a relatively easier way to track moving interfaces on a fixed Cartesian grid since the regeneration of a mesh system that conforms to the interface at every time step can be avoided. In the IBM, a series of connected Lagrangian markers are used to represent a fluid-fluid interface and the boundary condition is enforced by adding a forcing term to the Navier-Stokes equations. It is known that the IBM suffers two problems. One is spontaneous generation of unphysical kinetic energy, which is known as parasitic currents, and the other is spurious reconstruction of interface. These two problems need to be solved for useful long-time-scale simulations of droplets with high density ratio and large surface tension. This work detects that the discrete delta function is the cause of unphysical parasitic currents. Specifically, the irrotational condition is not preserved when the common discrete delta function is used to spread the surface tension from Lagrangian markers to Cartesian grid cells. To solve this problem, a new scheme that preserves the irrotational condition is proposed to remove the spurious currents. Furthermore, for a smooth reconstruction of an interface, a B-spline fitting by least squares is adopted to relocate the Lagrangian markers. The conventional and new interpolation schemes are implemented in a multigrid finite volume Direct Numerical Simulation (DNS) solver and are subjected to standard test cases. It is confirmed that the unphysical parasitic currents are substantially reduced in the new scheme and droplet's surface fluctuation is eliminated.