An implicit algorithm within the arbitrary Lagrangian-Eulerian formulation for solving incompressible fluid flow with large boundary motions

The objective of the paper is to present an implicit algorithm for incompressible fluid flow solution using the arbitrary Lagrangian-Eulerian (ALE) formulation and to investigate solution accuracy and stability of the algorithm. The governing equations of the implicit procedure are derived using isoparametric interpolations for the fluid velocities and pressure. The details suitable for general use are presented in our derivations of the fundamental equations and of the basic finite element balance equations. The penalty method is utilized to eliminate the pressure on the element level. Accuracy and stability of the solutions are demonstrated in three examples for which the analytical solutions are known. In the first example, the Burger's equation analogue to 1-D fluid flows is solved without and with FE mesh motion, to show that the mesh motion practically does not affect the solutions. In the second example, the solitary wave motion with large displacements of the free boundary is solved as a benchmark problem. In the third example, the fluid flow in an infinite contractible and expandable pipe with prescribed large radial wall motion is solved. This example is specifically attractive for biological flows as in blood vessels and lung airways. All solutions presented show that the proposed algorithm is sufficiently accurate and stable. Since the algorithm is implicit, high accuracy of results can be achieved with a relatively large time step.