An accurate THINC scheme for interface capturing based on homotopy analysis method

An accurate and robust Tangent Hyperbolic INterface Capturing (THINC) scheme is proposed in the framework of volume of fluid (VOF) method to capture the moving interface on the unstructured grids. In order to determine the interface position, a new iterative algorithm is put forward based on homotopy analysis method (HAM) to solve the high degree univariate polynomial equation, which hardly obtains converged solutions by the previous Newton-Raphson scheme. The so-called THINC/HAM significantly improves the convergence behaviour of interface reconstruction in terms of the maximum iterative residual and percentage of divergent cells, thus enhancing the local volume conservation of volume fraction. Different from Runge-Kutta schemes used in existing THINC methods, VOF equation is then solved by a direct time integral with varying velocity to update the volume fraction at each time step. It requires only one reconstruction step while preserving the numerical accuracy for higher Courant numbers, which substantially benefits the numerical efficiency. Numerical analysis is also carried out to investigate the appropriate values of some critical parameters in this method. As verified in the benchmark tests, the present scheme shows considerable improvements in numerical accuracy and robustness even if a larger time step and distorted unstructured grid are used. Despite of algorithmic simplicity, the solution quality of the present scheme is comparable to most geometric VOF methods, which is highly appealing for practical applications.