Fluid-phase velocity fluctuations in fixed beds and freely evolving suspensions are quantified using particle-resolved direct numerical simulation (PR-DNS). The flow regime corresponds to homogeneous gas-solid systems typically encountered in fluidized beds, with high solid to fluid density ratios and particle diameters being greater than dissipative length scales. The contribution of turbulent and pseudo-turbulent fluctuations to the level of fluid-phase velocity fluctuations is quantified in flow past fixed particle assemblies. The simulations are then extended to freely evolving suspensions with elastic and inelastic collisions. For the parameter ranges considered here (volume fraction 0.1 and 0.2, particle to fluid density ratio 100 and 1000, coefficient of restitution in the range 0.7-1.0), the level of gas-phase velocity fluctuations in freely evolving suspensions differs by only 10% from the value for a fixed bed at the same solid volume fraction and mean slip Reynolds number. Quantification of the Reynolds stress indicates that the second moments of the gas-phase velocity fluctuations are anisotropic, corresponding to unidirectional axisymmetric velocity fluctuations. The anisotropy increases with Reynolds number to a maximum in the range between 10 to 40, and then decreases. In addition, the anisotropy decreases with increasing solid volume fraction for all cases considered in this study. The Reynolds stress is decomposed into isotropic and deviatoric parts, and their dependence on solid volume fraction and Reynolds number is quantified and explained.