Numerical study of incompressible fluid dynamics with nonuniform density by the immersed boundary method

We apply the immersed boundary method to the dynamics of an incompressible fluid with a nonuniform density. In order to take into account both the inertial and gravitational effects of the variable density, the penalty immersed boundary (pIB) method is used [Y. Kim and C. S. Peskin, Phys. Fluids 19, 053103 (2007)]. Incompressible fluid motion with a nonuniform density has been extensively explored both experimentally and computationally. We show that the pIB method is a robust and efficient numerical tool for the simulation of fluids with variable density by conducting computations of some example problems: The falling of a heavier fluid surrounded by a lighter fluid and the Rayleigh-Taylor instabilities in two dimensions and three dimensions and the dynamic stabilization of the Rayleigh-Taylor instability.