Abstarct: Governing equations for the unsteady two dimensional plane mixing laxyer flow are Navier-Stokes and continuity equations. These equations in rotational form are solved using direct numerical simulation. The length scale and the velocity scale of the baxse flow at the inlet boundary of computational domain are used as two characteristics to define the flow Reynolds number. These two characteristics are the mixing laxyer half width and velocity difference of two streams. The governing equations are discretized in the streamwise direction and cross stream direction , using sixth order compact finite difference scheme and mapped compact finite difference scheme, respectively. A cotangent mapped of is used to relate the physical domain in the double infinity of to the computational domain of . The compact third order of Runge-Kutta method is used for the time-advancement of the computations. The numerical solution of the governing equations requires correct boundary condition implementations at suitable locations to produce well-posed problem. Most of numerical strategies exhibit weak performance and obtain inaccurate solutions since the outlet boundary conditions are not known accurately. To avoid this difficulty in this research, we have applied free boundary condition at the outlet boundary. This free boundary condition is equivalent to extend the validity of governing equations at the outflow instead of replacing them with unknown or natural boundary conditions. An invicid flow (Stuart flow) and a completely viscous solutions of the Navier-Stokes equations are used for verification of the numerical simulations. The numerical results show a good accuracy and agreement with exact solution of the Navier-Stokes equations. In this work, the velocity profiles in and direction and vorticity thickness are obtained and self-similarity characteristic for laminar and turbulent mixing laxyer flow is investigated.