Interface capturing schemes based on sigmoid functions

The non-polynomial THINC (tangent of hyperbola for interface capturing) scheme has been reported to show both numerical simplicity and high fidelity for resolving contact interfaces. In this paper, two types of smooth sigmoid functions are employed to construct the non-polynomial reconstructions for capturing interfaces (similarly, called SFINC schemes, sigmoid functions for interface capturing). One type is that the exact jump location (a parameter introduced in the reconstruction) can be analytically calculated, and another type cannot. The algebraic function and the Gudermannian function belong to the first and the second types, respectively, and are investigated in this paper. An approximate method for calculating the jump location of the Gudermannian function is proposed. The method avoids the iteration process of determining the jump location, and hence is efficient and practical in applications. The numerical validations and comparisons of SFINC schemes are presented to show their performance for simulating complex compressible flow fields.