This paper presents a triangular-to-quadrilateral mesh conversion method that can control the directionality of the output quadrilateral mesh according to a user-specified vector field. Given a triangular mesh and a vector field, the method first scores all possible quadrilaterals that can be formed by pairs of adjacent triangles, according to their shape and directionality. It then converts the pairs into quadrilateral elements in order of the scores to form a quadrilateral mesh. Engineering analyses with finite element methods occasionally require a quadrilateral mesh well aligned along the boundary geometry or the directionality of some physical phenomena, such as in the directions of a streamline, shock boundary, or force propagation vectors. The mesh conversion method can control the mesh directionality according to any desired vector fields, and the method can be used with any existing triangular mesh generators.