An inexpensive non-intrusive polynomial chaos (NIPC) method for the propagation of input uncertainty in CFD simulations is presented. The method is straightforward to implement for any stochastic fluid dynamics problem and computationally less expensive than sampling or quadrature based non-intrusive methods. To validate the present NIPC approach, the method has been applied to: (1) an inviscid oblique shock wave problem with geometric uncertainty, (2) an inviscid expansion wave problem with geometric uncertainty, and (3) a subsonic, two-dimensional, laminar boundary layer flow over a flat plate with an uncertain free-stream dynamic viscosity. For all test cases, the statistics (mean and the standard deviation) obtained with the NIPC method were in good agreement with the results of the Monte Carlo simulations. A fourth order polynomial chaos expansion was sufficient to approximate the statistics and the shape of the output uncertainty distributions with the desired accuracy. Only in the shock region of the first test case a sixth order polynomial expansion was required to estimate the statistics of pressure within the 95% confidence intervals of the Monte Carlo results, since the shape of the distributions obtained with 3rd order spatially accurate Euler solutions were highly non-Gaussian in this region.