In this paper, we consider nonsingular flows on closed 3-manifolds which are transversely modeled on the real affine geometry of the plane. We obtain classification results for the following three types of flows: Flows whose developing maps are $mathbb{R}$-bundle maps over $mathbb{R}^2$; Flows whose holonomy groups are contained in $SL(2,mathbb{R}$; and Flows with homotopy lifting property whose holonomy groups are contained in $SL(2,mathbb{R})ltimes mathbb{R}$.