Steady “Attractors” for Forced Flow of Vapor Condensing Over a Horizontal Plate (Problem in Cess and Koh) and Their Unsteady Responses to Initial Disturbances and Noise

Abstract

Accurate steady and unsteady numerical solutions of the full 2-D governing equations – that model the film condensation of saturated vapor flowing over a horizontal plate (the problem of Cess [1] and Koh [2]) – are obtained and new results on the solutions’ unsteady response to disturbances are presented. The computations reveal important features of this classical condensing flow problem. The results highlight the scope and limitations of the well-known similarity solution given by Koh [2]. For the steady problem formulation, the paper discusses the similarities and differences between the solution obtained by solving the full 2-D governing equations and the one obtained semi-analytically by the similarity solution approach of Koh [2]. It is shown that the pressure variations in the vapor domain near the leading edge, though small, are important in deciding condensation dynamics (steady and unsteady) and cannot, in general, be neglected, as is the case with the similarity solution. For this shear driven flow, by considering the unsteady solutions, the paper finds that any initial guess leads to an unsteady solution which is attracted to a long-term steady solution (which is same as the solution as the steady problem). However, the attraction rates gradually diminish with increasing distance from the leading edge and decreasing inlet speed. The steady solutions for this external flow problem are generally found to be stable to initial disturbances at the interface or in the interior of the flow domain. However, since these flows can only be physically realized on suitable finite length portions of the plate, the issue of their stability and sensitivity to exit pressure disturbances and ever-present noise (through exit pressure or bottom plate) is also considered. For example, for the finite domain realization of this problem, it is found that the flows are stable to small initial disturbances to the nearly uniform value of exit pressure. These finite domain realizations of the flow are unique in the sense that they allow different non-uniform steady pressure prescriptions leading to different steady solutions – particularly near the exit zone. As a result, near the exit of a long plate, large unsteadiness is expected due to sensitivity to small exit pressure noise/fluctuations. The exit pressure noise for finite domain realization of these flows is expected because of practical difficulties in maintaining constant uniform pressures at downstream locations of the top and exit boundaries. It is shown that the transverse component of gravity does not affect the solution or its dynamic response except for the expected changes in the nature of hydrostatic pressure variations.

Computational and Ground‐Based Experimental Investigations of the Effects of Specified and Unspecified (Free) Pressure Conditions at the Condenser Exit for Condensing Flows in Terrestrial and Microgravity Environments

“…In fact, the strong versus weak attractor situations depicted, respectively, in Figures 2 and 3 for vertical (gravity‐driven) and 0g (shear driven) flows is quite typical and are also present, respectively, for their external flow counterparts, namely the Nusselt problem (see Phan and Narain 28 ) and the Koh problem (see Kulkarni et al 2–3 …”

“…The experiments reported here involve a single pure working fluid (FC‐72 by 3M Corp. (St. Paul, MN)) and focus on inlet mass flow rates that correspond to inlet vapor Reynolds numbers in the range of 10,000–40,000 and vapor to wall temperature differences of 3°–60°C (i.e., 0 ≤ Ja ≤ 0.4). Furthermore, computational results for external condensing flows that are obtained from this simulation tool have a proven record (see Phan and Narain 28 and Kulkarni et al 2,3 ) for making qualitatively and quantitatively correct predictions for steady flows. The steady flow predictions are in excellent agreement with the classical solutions of Nusselt 21 and Koh 29 .…”

“…This paper is a synthesis of results from a fundamental and novel experimental investigation 1 into the effects of exit conditions on internal condensing flows and computational studies 2–6 of internal and external condensing flows. Toward the end of the paper, we also present a new set of generalized results for these and other internal two‐phase flows.…”

“…This is not generally true. The attainment of natural steady flows under unspecified exit conditions requires special hardware arrangements and, even then, such flows occur more readily (over a larger‐parameter zone) for gravity‐driven condensate flows than they do for shear‐driven flows 2,3,6 .…”

“…In comparison, the representative 0g solutions (obtained by solving 2,3 the unsteady equations for the Koh problem 29 ) shown in Figure 5 are possible only if U is above a certain lower threshold (termed U ∞ | L‐cr in Kulkarni 3 ). Furthermore, the solution in Figure 5 shows the presence of an attractor whose attracting strength decreases with increasing x.…”