The driving performance of the vehicle with automatic transmission is influenced by the performance of the wet clutch directly. But at present it is still a challenge to build a reliable predictable model for the torque of the engagement process of the wet clutch. Focusing on the wet clutch of vehicle, this paper starts from mechanism analysis, and a modified Reynolds equation with the consideration of the centrifugal force of the squeeze-film is established. In the model, we can consider the speeds of the friction and separator plates independently. At the same time, flow factors have been used to research the impacts of rough surface on the flow of the lubrication oil. In the three-dimensional solution domain, the circumferential pressure gradient of lubrication oil is considered. The model is solved with the finite volume method. The simulation of the torque of the asperity contact calculates the real contact area changed with the engagement process, and the microscopic texture direction of rough surface is considered. Subsequently, the squeeze-film flow model is combined with the asperity contact model to create an integrated clutch engagement model. Finally, the influence of applied force, viscosity of lubrication oil, friction material, the depth of grooves and the width of the grooves are investigated. Based on the comparison with the experimental data, the performance of the proposed model is found satisfactory. Because in this model more detail properties of material and geometric features of the friction plate are include, the wet clutch model developed in this research can become a baseline model for the prediction of the engagement behavior of a real wet clutch. The present model may become an efficient alternative to laboratory testing and lead to designs that can not be envisioned by other approaches.