Volume‐tracking methods for interfacial flow calculations

TL;DR: A new algorithm for volume tracking which is based on the concept of flux-corrected transport (FCT) is introduced, applicable to incompressible 2D flow simulations on finite volume and difference meshes and can be extended to 3D and orthogonal curvilinear meshes in a straightforward manner.

Abstract: SUMMARY A new algorithm for volume tracking which is based on the concept of flux-corrected transport (FCT) is introduced. It is applicable to incompressible 2D flow simulations on finite volume and difference meshes. The method requires no explicit interface reconstruction, is direction-split and can be extended to 3D and orthogonal curvilinear meshes in a straightforward manner. A comparison of the new scheme against well-known existing 2D finite volume techniques is undertaken. A series of progressively more difficult advection tests is used to test the accuracy of each scheme and it is seen that simple advection tests are inadequate indicators of the performance of volume-tracking methods. A straightforward methodology is presented that allows more rigorous estimates to be made of the error in volume advection and coupled volume and momentum advection in real flow situations. The volume advection schemes are put to a final test in the case of Rayleigh‐Taylor instability. 1997 by CSIRO. In the numerical computation of multifluid problems such as density currents or Rayleigh‐Taylor instability there is a need for an accurate representation of the interface separating two immiscible fluids. Free surface flows such as water waves and splashing droplets are an approximation to the multifluid problem in which one of the fluids (usually a gas) is neglected as having an insignificant influence on the dynamics of the system. In a general free surface flow problem, fluid coalescence and detachment may occur and deforming meshes cannot be used. In this case the need of an accurate and sharp interface is even greater than in true multifluid computations. Although a slightly diffuse interface may be acceptable in a problem where the continuity, momentum and energy equations are solved throughout the entire mesh, in a free surface simulation the location of the interface determines the size and shape of the computational domain and specifies where boundary conditions must be applied. In this case a diffuse interface cannot be tolerated. On finite volume (or difference) meshes, standard advection techniques can be used in multifluid problems to advect either the density or a material indicator function, however these methods are either diffusive (e.g. first order upwinding) or unstable (higher order schemes in which unphysical oscillations appear in the vicinity of the interface). Numerous techniques have been devised to limit the diffusiveness of low order schemes and to minimize the instability of high order schemes (see e.g.