High resolution TVD schemes for interface tracking

Abstract

A first order upwind difference scheme (UDS) is routinely adopted for representing convection terms in a discretised space. UDS provides stable solutions. However it also introduces false diffusion in situations in which the flow direction is oblique relative to the numerical grid or when the cell-Peclet number is large. In order to predict sharp interface, higher order upwind schemes are preferred because of they reduce numerical dissipation. In interfacial flows, density and viscosity vary sharply in space. Representation of convective terms by Total variation diminishing (TVD) schemes ensures reduced smearing without impairing convergence property. TVD schemes develop formulae for interpolation of a cell-face value of the transported variable. If the interpolated value is bounded by the neighbouring nodal values then the scheme is ‘Bounded’. However, not all TVD schemes possess this property of ‘Boundedness’. The Normalised Variable Diagram (NVD) defines a domain within which the TVD scheme is bounded. Thus by combining the features of both TVD schemes and ensuring that they fall with the defined area of NVD, the convergence as well as the boundedness of a computational scheme can be ensured. In this paper, six different higher order schemes are considered some which are TVD bounded or unbounded, to solve the well known interface tracking problem of Rayleigh-Taylor Instability. To the best of our knowledge, a comparison of combined TVD/NVD principles in the case of interface tracking problems has not been reported in published literature.