This paper offers a procedure for the estimation of the numerical uncertainty of any integral or local flow quantity as a result of a fluid flow computation; the procedure requires solutions on systematically refined grids. The error is estimated with power series expansions as a function of the typical cell size. These expansions, of which four types are used, are fitted to the data in the least-squares sense. The selection of the best error estimate is based on the standard deviation of the fits. The error estimate is converted into an uncertainty with a safety factor that depends on the observed order of grid convergence and on the standard deviation of the fit. For well-behaved data sets, i.e. monotonic convergence with the expected observed order of grid convergence and no scatter in the data, the method reduces to the well known Grid Convergence Index. Examples of application of the procedure are included.