  We show that every smooth Schubert variety of affine type $\tilde{A}$ is an
iterated fibre bundle of Grassmannians, extending an analogous result by Ryan
and Wolper for Schubert varieties of finite type $A$. As a consequence, we
finish a conjecture of Billey-Crites that a Schubert variety in affine type
$\tilde{A}$ is smooth if and only if the corresponding affine permutation
avoids the patterns $4231$ and $3412$. Using this iterated fibre bundle
structure, we compute the generating function for the number of smooth Schubert
varieties of affine type $\tilde{A}$.
