Forced Flow of Vapor Condensing Over a Horizontal Plate (Problem of Cess and Koh)* - Steady and Unsteady Solutions of the Full 2-D Problem

Accurate steady and unsteady numerical solutions of the full 2-D governing equations – that model the forced film condensation flow of saturated vapor over a semi-infinite horizontal plate (the problem of Cess [1] and Koh [2]) are obtained over a range of flow parameters. The results presented here are used to better understand the limitations of the well-known similarity solutions given by Koh [2]. It is found that steady/quasi-steady film wise solution exists only if the inlet speed is above a certain threshold value. Above this threshold speed, steady/quasi-steady film condensation solutions exist and their film thickness variations are approximately the same as the similarity solution given by Koh [2]. However these steady solutions differ from the Koh solution [2] regarding pressure variations and associated effects in the leading part of the plate. Besides results based on the solutions of the full steady governing equations, this paper also presents unsteady solutions that characterize the steady solutions’ attainability, stability (response to initial disturbances), and their response to ever-present minuscule noise on the condensing surface. For this shear driven flow, the paper finds that if the uniform vapor speed is above a threshold value, an unsteady solution that begins with any reasonable initial guess, is attracted in time, to a steady solution. This long time limiting solution is the same – within computational errors – as the solution of the steady problem. The reported unsteady solutions that yield the steady solution in the long time limit also yield “attraction rates” for non-linear stability analysis of the steady solutions. The attraction rates are found to diminish gradually with increasing distance from the leading edge and with decreasing inlet vapor speed. These steady solutions are generally found to be stable to initial disturbances on the interface as well as in any flow variable in the interior of the flow domain. The results for low vapor speeds below the threshold value indicate that the unsteady solutions exhibit nonexistence of any steady limit of film wise flow in the aft portion of the solution. Even when a steady solution exists, the flow attainability is also shown to be difficult (because of waviness and other sensitivities) at large downstream distances.