ALE method to Solve Multiphase Flows

Maurya, Mithilesh Kumar and Dixit, Harish Nagaraj (2018) ALE method to Solve Multiphase Flows. Masters thesis, Indian Institute of Technology Hyderabad.

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Abstract

The problems of uid can be studied with any of the two approaches i.e. Eulerian (�xed position) and Lagrangian (�xed uid particle) method. In ALE (Arbitrary Lagrangian Eulerian) approach, we derive the bene�ts of both the methods. This approach is very helpful in tracking the boundary of uid. There are many methods which help to track the boundary of uids like Volume-of-Fluid (VOF), Level-Set (LS) and Front-Tracking (FT) methods. In all these methods, a �xed (or) Eulerian grid is used to solve the Navier-Stokes equations for the uid ow whereas the free surface/interface capturing/tracking technique di�ers in each method. In ALE method, the grid is not rigid but it moves. ALE method incorporated in FEM is used here to solve uid problems. The governing equations are Navier-Stokes and continuity equation which is discretized using Galerkin method. Finite element methods were introduced to uid mechanics with great expectations in 1970s. The signi�cant success of Galerkin method in structural dynamics and heat conduction problems would be replicated in uid dynamics did not realize. Convection operators present in the non-Lagrangian formulation (i.e. Eulerian) of the governing equations render the system of equation non-symmetric and the best approximation property in energy norm which made Galerkin a success in structural mechanics is lost. This problem has motivated the development of alternatives to the Galerkin which preclude oscillations without requiring mesh or time-step re�nement. One such method is Petrov-Galerkin, which is similar to upwind scheme in Finite volume method. The shape function is altered to give more weight for the direction of ow. Since here we deal with low Reynolds number problems, Galerkin approach gives good results. Using ALE, we here try to solve some of the problems of multiphase ow where we have to give additional jump conditions depending on the density and viscosity of the interacting uids. Surface tension plays a crucial role when it comes to problems of uids where dimensions are of order few mm. The ow of thesis is such that we have a brief introduction of FEM followed by a chapter on N-S equations discretization using FEM and how ALE method is used. Third chapter deals with some problems validating the code and �nally we have the summary as our last chapter.

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