  We study scaling of the superfluid density with respect to the film thickness
by simulating the $x-y$ model on films of size $L \times L \times H$ ($L >> H$)
using the cluster Monte Carlo. While periodic boundary conditions where used in
the planar ($L$) directions, Dirichlet boundary conditions where used along the
film thickness. We find that our results can be scaled on a universal curve by
introducing an effective thickness. In the limit of large $H$ our scaling
relations reduce to the conventional scaling forms. Using the same idea we find
scaling in the experimental results using the same value of $\nu = 0.6705$.
