For a punctured-disc homeomorphism given combinatorially, we give an algorithmic construction of the suspension flow in the corresponding mapping-torus. In particular one computes explicitly the embedding in the mapping-torus of any finite collection of periodic orbits for this flow. All these orbits are realized as closed braids carried by a branched surface which we construct in the algorithm. Our construction gives a combinatorial proof of the fact that the periodic orbits of such a suspension flow are carried by a same branched-surface.