The study of multi-phase flows is vital to industrial applications. In addition to analytical and experimental studies, an active attempt to find functional numerical solutions which accurately simulate the behavior of these flows is in progress. The jet breakup regimes have been studied analytically, experimentally and numerically. The front-tracking method is one of the most important numerical methods for Direct Numerical Simulation (DNS) of multiphase flows. This method is extremely desirable due to its high accuracy and suppression of numerical diffusion on the interface (front). The main drawback to the general form of this method in following jet breakup behavior is the application of topology change to the interface after a drop breakup. To tackle this problem, in this research, a new topology change algorithm is devised. In this algorithm, close elements are located at first and then, according to the direction of the perpendiculars to elements’ surfaces, a decision is made to either break or coalesce the shared interface. An important phenomenon that has been overlooked in previous works is a comprehensive study over the size of the droplets in various regimes and the influence of non-dimensional numbers on droplets’ size distribution. In this research, different breakup regimes are simulated and the influence of non-dimensional numbers on droplets’ size and distribution is evaluated. Hence, initially, the transition from dripping to jet regime and the size of drops formed are compared with experimental and other numerical results and they show consistency. Additionally, the size of the droplets formed in the jet regime of a co-flowing immiscible fluid system agrees well with the experimental findings. The jet’s configuration for various times in dripping regime of the same fluid system is also compared with the results simulated using ANSYS Fluent. After this validation of the method and the topology change algorithm, the effect of dimensionless numbers on drop size in a co-flowing immiscible fluid system in a dripping regime is investigated. Increasing the Weber number of the inner flow or the increase of the outer flow’s capillary number causes the reduction of drop diameter. On the other hand, drop enlarges the ratio of the velocities of the inner to the outer flow is raised. Drop diameter increases as either viscosity ratio decreases or density ratio is increased. The rate of diameter change decreases for cases where either viscosity or density ratio is greater than unity. Subsequently, in order to evaluate three-dimensional effects, the breakup of a jet emanating from an elliptic orifice is investigated. Axis-switching phenomenon is observed for the co-flowing immiscible fluid system and the effect of non-dimensional numbers on an elliptic jet’s axis-switching is examined. The initial wavelength is more affected by the density ratio change than by the jet’s maximum diameter. Increasing the viscosity ratio causes an increase in jet’s maximum diameter and a decrease in initial wavelength of axis-switching. Also, in dripping regime, the effect of non-dimensional numbers on drop size formed by an elliptic jet is investigated and is compared with the size of the drops formed by a circular jet. In dripping regime, the size of drops formed by an elliptic jet is less than that formed by a circular jet. In addition, the effect of elliptic nozzle’s aspect ratio on drop size is investigated and it is revealed that the increasing the aspect ratio decreases the size of drops. Keywords: Numerical, Front-tracking, Liquid jet breakup, Dripping, Jetting, Elliptic, Topology change