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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 $SL(2,\mathbb{R})\ltimes \mathbb{R}$.