Laminar boundary layer flow of saturated vapor and its condensate over a horizontal tube


Smyrnaios D.N., Pelekasis N.A., Tsamopoulos J.A.


The steady two-dimensional laminar flow of a stream of saturated vapor flowing over a tube that is kept at a uniform temperature, below the saturation temperature, is examined. Owing to the temperature difference between the vapor stream and the solid surface a film of condensate is generated that flows along the surface due to shear, pressure-drop, and gravity. In the limit as the boundary layer and film thickness remain smaller than the radius of curvature of the surface a simplified lubrication-type formulation describes the temperature and flow fields in the film, whereas the usual boundary layer formulation is applied in the vapor boundary layer. The case of flow past a horizontal cylinder of radius R is investigated numerically with the oncoming stream aligned with gravity. The parameters that control momentum and heat transfer in this problem are the viscosity ratio, μcs/μss, the density ratio ρcs/ρss, the Prandtl number, Pr=Cpcsμcs/Kcs, the Froude number, Fr=U2∞/(gπR), and finally the thickness ratio between the condensate and the vapor boundary layer, ε, which is also a measure of the temperature difference between the vapor stream and the tube wall. Then, the Nusselt number and the skin friction coefficient, averaged over the upper half of the cylinder, are calculated for a wide parameter range. When Fr is very small and ε relatively large the flow remains attached until the trailing stagnation point of the cylinder. As the effect of adverse pressure drop becomes more pronounced (Fr increases or ε decreases) it is shown that the solution exhibits two different types of singularity in the rear part of the cylinder. The first one is a typical Goldstein singularity because it appears at the tube wall and it is associated with vanishing skin friction (wall shear) and rapidly increasing film thickness. The second one takes place near the interface between the vapor stream and the film of condensate in a region where very small velocities prevail in conjunction with vanishing shear rate. The latter has not been reported so far and it is expected to affect the flow locally, as opposed to the Goldstein singularity which is known to lead to massive separation in the case of a cylindrical surface. Upon proper rescaling of Fr and ε, Fr′= Fr ρss/ρcs, © 2002 American Institute of Physics.


DOI: 10.1063/1.1476674