In this work, we develop a numerical strategy for solving two‐phase immiscible incompressible fluid flows in a general arbitrary Lagrangian–Eulerian framework. As we use a conforming mesh moving with one of the fluids, there is no need to track or reconstruct the interface explicitly. A sharp interface, discontinuity in the fluid properties, and the jump in the pressure field are all accurately modeled using the dummy‐node technique. One of the challenges that this work addresses is to obtain a C0‐continuous approximation for the surface tension force term on the reference configuration, and to obtain its consistent linearization within the context of a Newton–Raphson strategy. It is shown by means of various numerical examples that the strategy is computationally efficient and robust, and circumvents the need for remeshing upto significantly large deformations.

Abstract:In this work, we develop a numerical strategy for analyzing the flows of an incompressible fluid in the gap between an arbitrarily shaped inner boundary that rotates inside a circular outer boundary. Such flows occur very commonly in turbomachinery applications. The numerical strategy is based on a noninertial frame of reference that is fixed to the rotating inner boundary so that Coriolis and angular acceleration effects have to be accounted for in its development. Since this strategy is based on a fixed mesh, it is much more economical and accurate than a general arbitrary Eulerian-Lagrangian strategy, which would typically require remeshing. In addition, we also conduct a numerical study for circular Couette flow with varying angular speed of the inner cylinder in an inertial frame of reference; such a study may prove useful in validating a theoretical stability analysis which currently seems to have been carried out only for the case of constant angular speed.

In this work, we develop a numerical strategy for solving two-phase immiscible compressible fluid flows in a general arbitrary Lagrangian-Eulerian framework.The interpolation functions for the field variables are chosen so as to produce a stable numerical formulation. We model one of the fluids in a Lagrangian setting and use a conforming mesh moving with it which circumvents the need of solving an additional diffusion equation to track or construct the interface. The discontinuity in the pressure field across the interface is accurately modeled using the dummy-node technique. This technique yields a sharp and accurate representation of the interface and in turn a correct computation of the surface tension forces. We present a variational form of the governing equations including the surface tension effect and their exact linearization. Various benchmark examples are presented to illustrate the good performance and good coarse-mesh accuracy of the proposed scheme.