Abstarct: Free round jet results when fluid is issued with a given initial momentum, out of a circular nozzle into an infinite environment. Round jet flows, due to their simple geometry, are found in many engineering applications, e.g. propulsions, mixings and aeroacoustics. They are also important from theoretical point of view because they represent a convenient prototype of free flows. The equations in fluid mechanics are generally governed by the Navier-Stokes equations. Since, these equations are essentially non-linear, exact solutions are rarely found. The exact solutions can therefore only be obtained by numerically solving the Navier-Stokes equations. Here, we refer in particular to so-called direct numerical simulation (DNS) in which all flow characteristics are computed in detail. In this work, the direct numerical simulation of a spatially developing round jet flow is performed. The two dimensional incompressible Navier-Stokes equations in a cylindrical system are solved. These are solved in a domain which is finite in streamwise (x) direction and infinite in cross stream (r) direction. To compute the spatial derivatives, a compact finite difference scheme is used in x and a mapped compact finite difference method is employed in r. The compact third order Runge-Kutta method is used to advance the numerical simulation in time. The result of simulation in low Reynolds number is obtained and the power law relationship for the centerline deficit and the jet half-width as a function of the distance from the inflow is investigated. The velocity and vorticity profiles in different stations of the streamwise extent are depicted and the self-similarity phenomenon is studied. Also by imposing some perturbations at inflow, mean and fluctuating velocities are obtained and Reynolds stress distributions are studied.