The effect of flow conditions on the "negative wake" generation (longitudinal velocity overshoot behind a cylinder in the viscoelastic fluid flow along the centerline) has been investigated. FENE-CR model that predicts constant shear viscosity and controlled extensional viscosity was considered as a constitutive equation. The discrete elastic viscous split stress-G/streamline upwind Petrov-Galerkin (DEVSS-G/SUPG) formulation was employed and the high-resolution solutions were obtained with an efficient iterative solver based on the incomplete LU(0)-type preconditioner and BiCGSTAB. We found that the "negative wake" generation was more obvious in uniform flow conditions than in Poiseuille flow, which suggests that the experimentally unrevealed "negative wake" generation of Boger fluids could be partially attributed to the geometrical effect of Poiseuille flow. The "negative wake" generation was more discernable at low extensibility and high value of viscosity ratio, which agrees well with the previous studies. In addition, we could observe an undershoot phenomenon in Poisseuille flow condition, which has never been reported.