Classical groups as flag-transitive automorphism groups of 2-designs with λ = 2

In this article, we study 2-designs with $\lambda =2$ admitting a flag-transitive and point-primitive almost simple automorphism group G with socle X a finite simple classical group of Lie type. We prove that such a design belongs to an infinite family of 2-designs with parameter set $\left(\right(3^n-1)/2,3,2)$ and $X={\mathrm{PSL}}_n\left(3\right)$ for some $n⩾3$, or $X={\mathrm{PSL}}_2\left(q\right)$ with point-stabiliser $D_{2(q+1)/\gcd (2,q-1)}$, or it is isomorphic to the 2-design with parameter set $(6,3,2)$, $(7,4,2)$, $(10,4,2)$, $(11,5,2)$, $(28,7,2)$, $(28,3,2)$, $(36,6,2)$ or $(126,6,2)$.