Stratified-wavy transition boundary

The stability conditions derived from linear analysis represent the necessary conditions for maintaining a smooth interface configuration. Therefore, Equations 18 and 19 should predict the stratified-smooth to stratified-wavy transitional boundary. The SS/SW transition is one of the flow pattern transitions reported in experimental studies of horizontal and inclined two-phase flows, (e.g., Mandhane et al. [77], Taitel and Dukler [19], Shoham [78], Lin and Hanratty [37,75], Andritsos and Hanratty [39,81], Simpson et al. [79], Luninski [80], Nakamura et al. [82]).The various flow patterns boundaries are conventionally mapped in the two-phases flow rates coordinates, U. vs. U. ... 

The destabilizing term 1, in Equation 25 requires knowledge of the memory coefficient, C, as defined in Equation 23. The coefficient, C, is to be extracted from experimental findings which reflect the dynamic interfacial interactions. Observations of stratified-smooth/stratified-wavy transitional boundaries from various laboratories reported in the literature bear a potential of a data-base for correlating C. These are summarized in Table 1. 

Thus, the sheltering coefficient is determined by the liquid layer Froude number. However, for a given two-fluid system, the Froude number along the stratified-wavy transition boundary demonstrates a relatively small variation. Therefore,... 

Table 2 indicates the various controlling destabilizing terms along the stratified-smooth/wavy transitional boundary in some limited physical situations. 

Jeffreys stability condition (Equation 43.1) with a constant value for s was applied by Taitel and Dukler [19] in attempting to predict the stratified-smooth/wavy transitional boundary for air-water flows in closed conduits. The value needed for the sheltering coefficient, in order to fit transitional data in 2.5 and 5.1cm tubes, was s = 0.01, in reasonable agreement with Figure 21. However, the omission of the inertia destabilizing terms, Ja, Jb in Taitel and Dukler [19], while employing Equation 43.1 for small conduits, should be carefully considered. 

The data-base which has been found suitable for extracting the information on the dynamic interaction and correlating the dynamic interfacial shear stress component consists of the fluids flow rates along a stratified-smooth/wavy transitional boundary. More experimental data is needed to further substantiate the correlation for the dynamic coefficient. In particular, there is a need for additional data on this transition in systems of high liquid Reynolds numbers (e.g., high pressure steam/ water systems and large diameter tubes) and in two-fluid systems of either comparable phase velocities or faster lower turbulent layers (e.g., viscous-oil/water systems, downward inclined gas liquid systems). 

Consider Figure 7a for relatively large diameter tube, D = 9.5cm. For demonstration, maintaining the liquid flow rate at = 0.4cm/s, while increasing the gas rate, the latter becomes turbulent at point 1 (Re > 2,100). However Equation 33.1 is not yet fulfilled, and, therefore, at point 1, the smooth interface is still stable with turbulent flow in the upper phase. With further increase of the air flow rate. Equation 33.1 is met at point T , which represents a transitional point from stratified-smooth to stratified-wavy. The locus of all transitional points as obtained by Equation 33.1 (represented by the solid line) constitutes the predicted stratified-smooth boundary. Note that Equation 33.2 as represented by a dashed line is irrelevant in this case since it assumes laminar conditions. 

Thus, in the extreme of large or small diameter conduits, the stratified-smooth/ stratified-wavy boundary is predicted by Equations 33.1 or 33.2, respectively, while there exists an intermediate range of pipe diameters in between, where neither of these equations predict the locus of flow pattern transition. In these systems, transition from stratified smooth pattern coincides with the L-T transition of the gas phase and is predicted by the locus of operational conditions where Re = Re, (Brauner and Moalem Maron [105]). 

It is commonly believed that a correct mathematical presentation of physical situations ought to result in properly posed problems. In two-phase flow problems, however, the existence of an assumed physical situation, e.g., stratified wavy flow configuration, is not certain under all operational conditions. Therefore, ill-posedness in some domains of the parameters space does not necessarily imply that the formulation is globally incorrect. Moreover, the boundary of the well-posed domain may have physical significance since it signals the existence of additional physical features which the original model neglects. When these features become consequential, one expects a different physical behavior, such as transition to a different flow pattern, and a different model is required to simulate this transition. 

Thus, while the neutral stability boundary may represent preliminary transition from smooth-stratified flow to a wavy interfacial structure, the well-posedness boundary, which is within the wavy unstable region, represents an upper bound for the existence of a stratified wavy configuration. Beyond the well-posedness boundary transition to a different flow pattern takes place. In the ill-posed region, the model is no longer capable of describing the physical phenomena involved therefore, amplification rates predicted for ill-posed modes or numerical simulation of their growth is actually meaningless. 

The well-posedness boundary (ZRC) (included in Figures 10, 11, 13) represents the limit of operational conditions (U, U, ) for which the governing set of continuity and momentum equations is still well-posed with respect to all wave modes. Hence, it is considered as an upper bound for the stratified-wavy flow pattern. Indeed, the data of stratified-wavy/annular transition follows the ZRC curve in the region of H < 0.5. 

For the sake of clarity, the focus in Figure 18 is on the interplay and practical relevance of the stability boundaries (ZNS or ZNS) and the L-T laminar/turbulent flow regime transitional line in predicting the stratified-smooth/wavy flow pattern transition. Other transitional boundaries, which confine the stratified-smooth and stratified-wavy zones, are shown in Figure 19. For relatively low gas rates, the stratified-smooth zone is bounded by the slug or bubbly patterns while the stratified-wavy zone, at high gas rates, is bounded by the transition to annular pattern. 

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