A Linear Hypergraph Extension of Turán's Theorem

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of hypergraphs $\mathcal{F}$, the Turán number ex$_r^{lin}(n,\mathcal{F})$ maximum on $n$ vertices that does not contain any member $\mathcal{F}$ as subgraph.
 Let $K_l$ be complete graph with $l$ and $r\geq 2$. The $r$-expansion $r$-graph $K_l^+$ obtained from by enlarging each edge $r-2$ new disjoint $V(K_l)$ such distinct are enlarged vertices. When $l\geq r \geq 3$ sufficiently large, we prove following extension Turán's Theorem $$ex_{r}^{lin}\left(n, K_{l+1}^{+}\right)\leq |TD_r(n,l)|,$$ equality achieved only design $TD_r(n,l)$, where $TD_r(n,l)$ an almost balanced $l$-partite pair parts contained exactly. Moreover, some results general configurations also presented.