### A Conservative Volume of Fluid Method for General Grids in Three Dimensions

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** Christopher Ivey,
Stanford University **

The volume of fluid method is a widely used formulation for interface-tracking between two immiscible fluids. Although many variations of the volume of fluid method exist, the most popular schemes represent the interface by piecewise planar elements, a second-order approximation to the interface, and use geometric interface reconstruction and advection algorithms of the volume fraction for their conservation properties. For an incompressible fluid, local volume conservation ensures density conservation, an essential property for the simulation of multiphase flows due to the scaling of momentum, mass, and energy fluxes on density. For liquid fuel combustion, the fuel density is typically three orders of magnitude larger than that of the oxidizer, making local volume conservation paramount to simulation accuracy. Analytic geometric tools for interface reconstruction on general grids had been developed; however, conservative volume of fluid advection schemes were previously restricted to structured hexahedral grids. Typical combustion chambers are geometrically complex and require the use of unstructured polyhedral grids. To handle complex geometries, researchers previously used an advection scheme that conserves the global volume and an unphysical redistribution algorithm to move the volume around to keep the volume fraction bounded. Redistribution algorithms are a zeroth order correction to the volume fraction evolution equation and destroy the accuracy of the advection scheme. To maintain accuracy, a conservative scheme based on the use of edge-matched flux polyhedra to integrate the volume fraction advection equation on general grids is proposed. The algorithm prevents the formation of over/undershoots of the volume fraction by enforcing that the flux polyhedra do not over/underlap, removing the need for unphysical redistribution algorithms. The advection scheme is formally first order in the volume fraction; however, canonical advection tests, performed on a series of unstructured grids, show that the accuracy is between first and second order.

**Abstract Author(s):** Christopher Ivey, Parviz Moin