One of the fundamental phenomena associated with the transport of rigid particles by the fluid flow in narrow ducts and tubes is the Segre–Silberberg effect. Experimental observations show that a spherical particle transported by the fluid flow in a long channel occupies a position of equilibrium between the wall and the centerline of the channel. In this study, this effect was numerically investigated using a novel semi-implicit immersed boundary method based on the discrete forcing approach. A uniform Cartesian mesh is used for the duct, whereas a moving Lagrangian mesh is used to track the position of the particle. Unlike previous studies, both cases of the duct geometry are considered: a round tube and a flat channel. Good agreement is shown to the available theoretical and numerical results of other studies. The problem is described by two dimensionless parameters, the channel Reynolds number, and the relative particle diameter. Parametric studies to these parameters were carried out, showing fundamental dependencies of equilibrium position on Reynolds number from 20 to 500 and on relative particle diameter from 0.2 to 0.7. It is demonstrated that the position of equilibrium becomes closer to the wall with the increase of Reynolds number, as well as with the decrease of particle diameter. In addition, the dependence of particle velocity on its diameter is investigated. The obtained results are of both theoretical and practical interest, with possible applications ranging from proppant transport to the design of microfluidic devices.