A computational approach for increasing the efficiency of multigrid solvers in computational fluid dynamics (CFD) problems is presented. An adaptive-smoothing multigrid algorithm (AS-MG) is developed in conjunction with a full-multigrid, full-approximation-storage (FMG-FAS) method and an incompressible three-dimensional Navier-Stokes solver. The AS-MG algorithm reduces the overall computational cost by performing smoothings in subsets of the computational grid. The latter are defined adaptively at each multigrid sweep according to a prescribed criterion. Three different adaptivity criteria are investigated in connection with the computation of external and internal flows. Several numerical experiments are presented in order to demonstrate the advantages of the new algorithm.