  In this article, we study $2$-designs with $\gcd(r, \lambda)=1$ admitting a
flag-transitive automorphism group. The automorphism groups of these designs
are point-primitive of almost simple or affine type. We determine all pairs
$(\mathcal{D}, G)$, where $\mathcal{D}$ is a $2$-design with $\gcd(r,
\lambda)=1$ and $G$ is a flag-transitive almost simple automorphism group of
$\mathcal{D}$ whose socle is $X=\mathrm{PSU}(n, q)$ with $(n, q)\neq (3, 2)$
and prove that such a design belongs to one of the two infinite families of
Hermitian unitals and Witt-Bose-Shrikhande spaces, or it is isomorphic to a
design with parameters $(6, 3, 2)$, $(7, 3, 1)$, $(8, 4, 3)$, $(10, 6, 5)$,
$(11, 5, 2)$ or $(28, 7, 2)$.
