On the Turán number of blow-ups of ${\mathcal{F}}_5$

Let ${\mathcal{F}}_5$ denote the $3$-uniform hypergraph on the vertex set $\{f_1,f_2,\dots ,f_5\}$ with hyperedges $\{f_1{f}_2{f}_3,f_1{f}_2{f}_4,f_3{f}_4{f}_5\}$. Recently, Balogh, Clemen and Luo determined the Turán number of a one-vertex blow-up of ${\mathcal{F}}_5$, more specifically, they blow up the vertex $f_5$ to $t$ vertices, the resulting hypergraph is denoted by ${\mathcal{F}}_5(f_5;t)$. They show that for infinitely many $t$, ${\mathcal{F}}_5(f_5;t)$ has exponentially many extremal constructions and positive Turán density. In this paper, we determine the exact Turán number of the hypergraph obtained by blowing up $f_3$ of ${\mathcal{F}}_5$ to $t$ vertices and show that it also has exponentially many extremal constructions. We also give a general upper bound and lower bound of the Turán number of every blow-up of ${\mathcal{F}}_5$. For some special blow-ups of ${\mathcal{F}}_5$, for example, $t$-disjoint copies of ${\mathcal{F}}_5$, we determine the exact Turán number. We construct a hypergraph ${\mathcal{F}}_{sim}(t)$ which is a subgraph of a blow-up of ${\mathcal{F}}_5$, and is contained in the hypergraph obtained by adding any new hyperedge to the Turán hypergraph (the balanced complete $3$-partite hypergraph), but its extremal construction is not the Turán hypergraph. We also determine the exact Turán number of ${\mathcal{F}}_{sim}(t)$.