In this work, we are concerned with the flow arc-set polyhedron of the unsplittable capacitated network design problem. By analyzing the property of the nontrivial facet of the flow arc-set polyhedron, we design an exact algorithm to solve the separation problem over it such that the generated inequality always defines a facet. In order to make this approach practical, several techniques are proposed to speed up the exact separation algorithm. Finally, a comprehensive computational study is presented for the unsplittable capacitated network design problem, which demonstrates the effectiveness of the proposed algorithm.