Flag-transitive block designs and unitary groups

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$G$isaflag-transitivealmostsimpleautomorphismgroupof$$\mathcal{D}$$whosesocleis$X=$\mathrm{P}\mathrm{S}\mathrm{U}$(n, q)$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 $(28,7,2)$.