# A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method
###### Abstract
Some researchers employed the wake models to perform as the strong large vortices in the wakes. [PERSON] [43] developed a procedure based on the panel method for the calculation of the potential flow around bodies of a circular shape to investigate the flow around the train cross-section. [PERSON] [44; 45] proposed a quasi-three-dimensional approach using a 2-D panel method for train cross-section computation and then marching along the train length. He applied this model to the aerodynamic predictions of the train at low incidences. Discrete sources and discrete vortices were adopted to model the body and the wake. It was found that the model failed over the incidence of 50\({}^{\circ}\) due to the breakdown of the slender body analogy. [PERSON] [46; 47; 48] proposed a separated-flow model for both symmetrical and asymmetrical bluff bodies based on a 2-D second-order vortex panel method, and the computed pressure distribution around the train model and other 2-D bluff bodies agreed very well with the experiments. The model was then updated for the prediction of the aerodynamic loads on an idealised railway train model under crosswind at large yaw angles up to 90\({}^{\circ}\). This model involved a two- and a three-dimensional procedure and employed source and vortex panels to model the surface of the train and the wake on the leeside, respectively, and gave satisfactory results when the flow structure of the wake was mainly of a vortex-shedding type rather than a line-vortex type. [PERSON] [49] proposed a 2-D vortex panel method and utilised it in the multidisciplinary design optimisation of a magnetically levitated (MAGLEV) train. The model employs a continuous vortex sheet to model the solid surface and discrete vortices to model the shed wake, which better matches the mechanics of the flow and predicts the ground effect flow. Good agreement was shown by comparing the predictions with higher-order computational methods and experimental data. Therefore, only the flow in the nose region of the train has been accurately simulated using the potential flow theory. In order to simulate the flow and pressure distribution around the train, a wake model is needed to account for the vortices after the rear of the train. In addition, modelling of the wake requires the identification of the flow separation's locations in advance, which in turn relies on experimental or alternative numerical methods. The locations of the flow separation are determined by the boundary-layer development along the train body, which is a result of the viscous effect and can be predicted using integral boundary-layer methods. In this paper, the separated-flow model is developed to include the viscous and inviscid interaction for predicting the pressure distribution around the 2-D bluff body. In this method, a wake model is employed to simulate the separatedflow. The integral boundary-layer method is used to predict the separation position, which includes the approximation of both laminar and turbulent boundary layers as well as the transition in boundary layer. Moreover, an interpolative strategy is used to determine the \"correct\" wake model and to predict the base pressure. The general analysis procedure of this method consists of a potential flow calculation, a boundary-layer analysis, and a wake model for the separated flow. The entire calculation program is coded up in MATLAB and is validated using experimental data of the flow around a circular cylinder, which is a well-documented highly separated turbulent flow case. The method is then used to predict the pressure distribution along a 2-D train, with and without the ground configuration. The methodology of the present work is introduced in detail in Section 2, including the calculation strategy for potential flow, boundary-layer prediction, and modelling of the separated flow. The method is validated in Section 3 by predicting the separation and pressure distribution around the circular cylinder and comparing the results with the reference data.