  Understanding gas flows into and out of the most massive dark matter
structures in our Universe, galaxy clusters, is fundamental to understanding
their evolution. Gas in clusters is well studied in the hot ($>$ 10$^{6}$ K)
and cold ($<$ 10$^{4}$ K) regimes, but the warm gas component (10$^{4}$ -
10$^{6}$ K) is poorly constrained. It is challenging to observe directly, but
can be probed through Ly$\alpha$ absorption studies. We produce the first
systematic study of the warm gas content of galaxy clusters through synthetic
Ly$\alpha$ absorption studies using cosmological simulations of two galaxy
clusters produced with Enzo. We explore the spatial and kinematic properties of
our cluster absorbers, and show that the majority of the identified absorbers
are due to fast moving gas associated with cluster infall from IGM filaments.
Towards the center of the clusters, however, the warm IGM filaments are no
longer dominant and the absorbers tend to have higher column densities and
metallicities, representing stripped galaxy material. We predict that the
absorber velocity distribution should generally be bi-modal and discuss the
effects of cluster size, mass, and morphology on the properties of the
identified absorbers, and the overall cluster warm gas content. We find
tentative evidence for a change in the well known increasing N$_{HI}$ with
decreasing impact parameter for the most massive dark matter halos. Our results
are compared directly to observations of Ly$\alpha$ absorbers in the Virgo
cluster, and provide predictions for future Ly$\alpha$ absorption studies.
