A numerical investigation using in-house code is performed to encapsulate the effect of solid boundaries on two-phase laminar flow. The specific problem of plume rise is tackled using the Level Set method. The plume is considered to have negligible surface tension effects and is solely driven by density differences, such that the density ratio is fixed to two. The Navier-Stokes solver is coupled to the Level Set advection subroutine explicitly. A high-resolution WENO-5 scheme is used with the Runge Kutta (RK) 4^th order multi-step method for time marching of the Level Set function. Re-initialization and Level Set advection are implemented using the same spatial discretization subroutine but re-initialization procedure leverages TVD RK-3 scheme for temporal discretization. The presence of walls of confinement profoundly impacts the rising plume and its breaking up in other fluid, providing non-intuitive results as the lighter fluid rises up differently from the way it would have in an unbounded case. Plume diameter is varied as fraction of cavity width for detailed understanding of underlying effects.