Aim of the present study is to investigate the effects of different numerical methods on the solution of laminar flow characteristics through an orifice plate inserted in a pipe. Ratio of the orifice diameter to the pipe diameter, beta=0.6 and dimensionless orifice plate thickness, L*=1/12, were kept constant throughout the study. The fluid flow was assumed to be two dimensional, axisymmetric, viscous, incompressible, steady and fully developed. Governing equations of the flow were solved with the aid of computer programs written in FORTRAN computer language by using two finite difference methods. These finite difference methods were alternating direction implicit method and upwind method. Additionally, finite volume method with the aid of Fluent package program was also employed to solve the flow for the purpose of comparison. Numerically obtained orifice discharge coefficient results were compared with experimental results obtained from literature. The best conformity with previous experimental results was obtained by using alternating direction implicit method. Vorticity contours, streamline and orifice discharge coefficient of the flow were presented in figures and discussed in details.