Stratified flow boundaries

INTEGRATED STABILITY AND WELL-POSEDNESS CRITERIA, 350 Stability and Well-Posedness Map, 350 Constructing the Stratified Flow Boundaries, 352... 

STRATIFIED-FLOW BOUNDARIES COMPARISON WITH EXPERIMENTS, 354... 

It was shown by Brauner and Moalem Maron that linear stability analysis is insufficient to predict the stratified flow boundaries [40]. Parallel analyses on the stability as well as on the well-posedness of the (hyperbolic) equations which govern the stratified flow has been invoked. It has been shown that the departure from stratified configuration is associated with a buffer zone confined between the conditions derived from stability analysis (a lowerbound) and those obtained by requiring well-posedness of the transient governing equations (an upper-bound). These two bounds form a basis for the construction of the complete stratified/non-stratified transitional boundary to the various bounding flow patterns. 

The relative contributions of the gas and liquid destabilizing terms along the stratified flow boundaries are to be carefully considered according to the particular physical system (e.g., horizontal gas-liquid, inclined gas-liquid, or liquid-liquid flows) and the range of operational conditions. In the extremes of 1 the... 

Figure 10. Effect of liquid viscosity on the stratified flow boundaries comparison of theory with experiments (Andritsos and Manratty [39],...

Figure 13. Effect of pressure on the stratified-flow boundaries in steam water systems comparison of theory with experimental data (Nakamura et al. [82]).

Eq. (3.150) represents the wall law for turbulent flow, first formulated by Prandtl in 1925. The functions f(y+) and g(y+) are of a universal nature, because they are independent of external dimensions such as the height of a channel and are valid for all stratified flows independent of the boundary layer thickness. 

Ayotte, K.W., Finnigan, J.J., and Raupach, M.R. (1998) A second order closure for neutrally stratified vegetative canopy flows, Boundary-Layer Meteorol 90, 189-216. 

Stratified Flow. The gas and liquid flow are segregated in this case, as a result the liquid flows on the bottom and the gas flows on the top of the tube with a distinct boundary. 

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]). 

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 general implication of the stability boundary (ZNS or ZNS ) and the well-posedness boundary (ZRC) is in defining three zones the area within the stability boundary is well-understood to be the stable smooth stratified zone. Beyond the ZRC boundary, the complex characteristics indicate that the governing equations of the stratified flow configuration are ill-posed with respect to long wave modes in the wave spectra. In this sense, the ZRC boundary represents an upper bound... 

Brauner, N., and Moalem Maron, D., Analysis of Stratified/Non-stratified Transitional Boundaries in Inclined Gas-Liquid Flows, Int. J. Multiphase Flow, Vol. 18, pp. 541-557 (1992). 

Streefer, V. F., ed. 1961. Handbook of Fluid Dynamics. New York McGraw-Hill. A classic handbook on fluid dynamics wifh confributions from distinguished experts. Written for engineers and scientists in the field. Deals wifh bofh fundamenfal concepts and applications. Covers fluid flow (one-dimensional, ideal, laminar, compressible, two phase, open channel, stratified), turbulence, boundary layers, sedimentation, turbomachinery, fluid transients, and magnetohydrodynamics. Includes many formulas, equations, tables, graphs, and illustrations. Each chapter has a bibliography and the volume has subject and author indexes. 

Comparison of the boundaries of the observed flow patterns with the analytical criteria derived by Quandt showed that the bubble, dispersed, and annular flow patterns are subclasses of a pressure gradient-controlled flow. Similarly, flow patterns identified as slug, wave, stratified, and f ailing film are subclasses of a gravity-controlled situation. 

The mathematical similarity of the equations is not broken by the starved flow, but there is a physical and mathematical problem. Since the fluid flows radially outward near the rotating surface and flows radially inward further away from the surface, there is a problem in assigning physically meaningful boundary conditions. Where is the radially inward flow coming from and what is its temperature and composition Similarity requires that the velocities, temperature, and composition be functions of z alone. Thus it would be physically hard (probably impossible) to set up the stratified reservoir of fluid required by the similarity. 

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