A direct time-integral THINC scheme for sharp interfaces

Highlights: • Direct time integration on hyperbolic tangent function. • Efficient and accurate compared to the RK3 THINC method. • Excellent boundedness propriety for large β values and large time steps. • Easy to implement on the current THINC schemes. -- Abstract: This paper describes an improved multi-dimensional Tangent of Hyperbola INterface Capturing (THINC) scheme that uses a direct time integration of the hyperbolic tangent function. The new time-integral THINC scheme is based on the assumption that the interface remains unchanged and moves with the velocity of the cell center during a computational time step. A time-varying hyperbolic tangent function representing the two-phase distribution is then constructed and directly integrated in time. Compared with existing THINC methods that generally use third-order total variation diminishing Runge–Kutta (RK3) schemes to update the volume fraction, the proposed method requires only one reconstruction step, thus reducing the computational cost. Several classical advection tests have been implemented, and the results indicate that the proposed direct time-integral THINC method: 1) achieves computational errors close to those of the RK3 scheme at a computational cost that is close to that of the first-order explicit scheme; and 2) preserves better boundedness of the volume fractions than the original RK3-based THINC methods when using a larger steepness parameter β and higher Courant numbers. The proposed approach is employed with the incompressible Navier–Stokes equations to solve a 2D dambreak problem, and the numerical results agree well with the corresponding experimental data, demonstrating the applicability of the direct time-integral THINC method.