In this study, a numerical method based on solenoidal basis functions, for the simulation of incompressible flow through a circular-cylindrical pipe, is presented. The solenoidal bases utilized in the study are formulated using the Legendre polynomials. Legendre polynomials are favorable, both for the form of the basis functions and for the inner product integrals arising from the Galerkin-type projection used. The projection is performed onto the dual solenoidal bases, eliminating the pressure variable, simplifying the numerical approach to the problem. The success of the scheme in calculating turbulence statistics and its energy conserving properties is investigated. The generated numerical method is also tested by simulating the effect of drag reduction due to spanwise wall oscillations.