  We present a novel interface-capturing scheme, THINC-scaling, to unify the
VOF (volume of fluid) and the level set methods, which have been developed as
two different approaches widely used in various applications. The key to
success is to maintain a high-quality THINC reconstruction function using the
level set field to accurately retrieve geometrical information and the VOF
field to fulfill numerical conservativeness. The interface is well defined as a
surface in form of a high-order polynomial, so-called the polynomial surface of
interface
  (PSI). The THINC reconstruction function is then used to update the VOF field
via a finite volume method, and the level set field via a semi-Lagrangian
method. Seeing the VOF field and the level set field as two different aspects
of the THINC reconstruction function, the THINC-scaling scheme preserves at the
same time the advantages of both VOF and level set methods, i.e. the
mass/volume conservation of the VOF method and the geometrical faithfulness of
the level set method, through a straightforward solution procedure. The
THINC-scaling scheme allows to represent an interface with high-order
polynomials and has algorithmic simplicity which largely eases its
implementation in unstructured grids. Two and three dimensional algorithms in
both structured and unstructured grids have been developed and verified. The
numerical results reveal that the THINC-scaling scheme, as an interface
capturing method, is able to provide high-fidelity solution comparable to other
most advanced methods, and more profoundly it can resolve sub-grid filament
structures if the interface is represented by a polynomial higher than second
order.
