A novel algorithm based on parameterization method for calculation of curvature of the free surface flows

In this paper, a new approach based on parameterization method is presented for calculation of curvature on the free surface flows. In some phenomena such as droplet and bubble, surface tension is prominent. Therefore in these cases, accurate estimation of the curvature is vital. Volume of fluid (VOF) is a surface capturing method for free surface modeling. In this method, free surface curvature is calculated based on gradient of scalar transport parameter which is regarded as original method in this paper. However, calculation of curvature for a circle and other known geometries based on this method is not accurate. For instance, in practice curvature of a circle in interface cells is constant, while this method predicts different curvatures for it. In this research a novel algorithm based on parameterization method for improvement of the curvature calculation is presented. To show the application of parameterization method, two methods are employed. In the first approach denoted by, three line method, a curve is fitted to the free surface so that the distance between curve and linear interface approximation is minimized. In the second approach namely four point method, a curve is fitted to intersect points with grid lines for central and two neighboring cells. These approaches are treated as calculus of variation problems. Then, using the parameterization method, these cases are converted into the sequences of time-varying nonlinear programming problems. With some treatments a conventional equivalent model is obtained. It is finally proved that the solution of these sequences in the models tends to the solution of the calculus of variation problems. For verification of the presented methods, curvature of some geometrical shapes such as circle, elliptic and sinusoidal profile is calculated and compared with original method used in VOF process and analytical solutions. Finally, as a more practical problem, spurious currents are studied. The results showed that more accurate curve prediction is obtained by these approaches than the original method in VOF approach.