Wavy flow

Wavy flow As stratified but with a wavy interlace due to ... 

C. The Onset of Wavy Flow (Theoretical Stability Considerations). 163... 

A dimensional analysis of the problem of film flow (F7) has shown that in general the properties of a film flow may depend on the Reynolds, Weber, and Froude numbers of the flow, a dimensionless shear at the free surface of the film, and, for wavy flows, a Strouhal number formed from the frequency of the surface waves, and various geometrical ratios, e.g., the ratios of the wave amplitude and length to the mean film thickness. 

However, the flow regime of a film cannot be defined uniquely as laminar or turbulent, as in the case of pipe flow, due to the presence of the free surface. Depending on the values of AFr and JVw , the free surface may be smooth, or covered with gravity waves or capillary or mixed capillary-gravity waves of various types. Thus, under suitable conditions, it is possible to have smooth laminar flow, wavy laminar or turbulent flow, where the wavy flows may be subdivided into gravity or capillary... 

In the present section the general equations are set up, and the main results of the various treatments of smooth laminar flow, wavy flow, turbulent flow, and flow of films with an adjoining gas stream will be reviewed briefly. 

In addition to the general treatments of wavy flow, a number of theories concerning the stability of film flow have been published in these the flow conditions under which waves can appear are determined. The general method of dealing with the problem is to set up the main equations of flow (usually the Navier-Stokes equations or the simplified Nusselt equations), on which small perturbations are imposed, leading to an equation of the Orr-Sommerfeld type, which is then solved by various approximate means to determine the conditions for stability to exist. The various treatments are lengthy, and only the briefest summaiy of the results can be given here. 

In a simplified treatment for longer waves, Benjamin derived an amplification factor a, defined as the amplification experienced by the wave of maximum instability in traveling a distance of 10 cm. If neutral stability exists, a = 1, and if a > 1, wavy flow can be expected. For a vertical wall,... 

In the second approximation, the equation analogous to (56) contains terms to the second order in . The details of the calculation cannot be included here, but briefly, it is found that in this case the mean film thickness of the wavy flow depends on the wave amplitude a, or... 

It is seen from (65) that the wave velocity is considerably smaller than the value given by the first approximation, (58). From (63), the ratio of the mean film thickness in wavy flow to the thickness of a smooth film at the same flow rate is given by 4>1/3 or, from above, 0.93. The corresponding value obtained by Portalski (P3) was 0.94. It is thus seen that for wavy flow of the type assumed here, the mean thickness of the wavy film is 6-7% smaller than the corresponding smooth film. It is pointed out by Kapitsa that it does not follow that there may not be some other type of film surface configuration which would lead to a greater reduction in thickness and, therefore, to greater stability of flow. 

Semenov (S7) simplified the equations of wavy flow for the case of very thin films, and this approach has also been followed by others (K20, K21). These treatments refer mainly to the case of film flow with an adjoining gas stream and will be considered in Section III, F, 3. 

In addition to the theories reviewed above, there are many treatments in the literature which deal with the hydraulics of wavy flow in open channels. Most of these refer to very small channel slopes (less than 5°) and relatively large water depths. Under these conditions, surface tension plays a relatively minor part and is customarily neglected, so that only gravity waves are considered. For thin film flows, however, capillary forces play an important part (K7, H2). In addition, most of these treatments consider a turbulent main flow, while in thin films the wavy flow is often... 

Semenov (S7) simplified the wavy flow equations by omitting the inertia terms, which is permissible in the case of very thin films. Expressions are obtained for the wavelength, wave velocity, surface shape, stability, etc., with an adjoining gas stream the treatment refers mainly to the case of upward cocurrent flow of the gas and wavy film in a vertical tube. 

Feind (F2) measured the thickness of various films of kinematic viscosities 1 to 19.7 centistokes flowing in a vertical tube. An improved drainage technique was used. At the lowest values of Nr.,. (smooth laminar flow regime) the values of Nr fell along the line given by Eq. (97). Once wavy flow commenced, the values deviated towards the Kapitsa line,... 

Hence, the trend predicted by the Kapitsa theory is supported by the recent, more accurate, film thickness measurements. This does not indicate, however, that the Kapitsa theory will apply in detail over the whole wavy laminar regime of film flow, since Kapitsa (K7) pointed out that such a reduction in the mean thickness should result for other types of wavy flow besides the particular case considered in his theory. 

For wavy flow the experimental results of Konobeev et al. (K20, K19) are generally in quite good agreement with the theoretical predictions outlined in Section III, F, 3 for very low liquid flow rates. At larger flow rates there are serious deviations from the predictions. 

D. Surface Waves on Films 1. General Observations of Wavy Flow... 

The cause of this initial smooth zone and the subsequent fairly sudden transition to wavy flow is not completely clear. Working on a much larger scale, with mostly turbulent flow of the liquid layers on dam spillways, Bauer (Bl) has shown that the length of the smooth initial region is the same as the distance required for the turbulent boundary layer, which... 

In most cases, it is found that there is a considerable spread in the wave amplitudes, but that for given gas and liquid flow rates there is a certain wave height which occurs most frequently, and which can therefore be regarded as a characteristic of the wavy flow. The manner in which this characteristic frequency varies with the flow rates has been given in the literature, e.g. (B14, C4, H9). 

Grimley (G10, Gil) used an ultramicroscope technique to determine the velocities of colloidal particles suspended in a falling film of tap water. It was assumed that the particles moved with the local liquid velocity, so that, by observing the velocities of particles at different distances from the wall, a complete velocity profile could be obtained. These results indicated that the velocity did not follow the semiparabolic pattern predicted by Eq. (11) instead, the maximum velocity occurred a short distance below the free surface, while nearer the wall the experimental results were lower than those given by Eq. (11). It was found, however, that the velocity profile approached the theoretical shape when surface-active material was added and the waves were damped out, and, in the light of later results, it seems probable that the discrepancies in the presence of wavy flow are due to the inclusion of the fluctuating wavy velocities near the free surface. 

Kirkbride (K17), 1934 Flow of water and 4 oils outside tubes, JVro = 0.04-2000. Film thicknesses (maximum wave heights) measured by micrometer. Wavy flow is described, and corrections to Nusselt theory derived for heat transfer in laminar wavy film flow. 

Friedman and Miller (F5), 1941 Flow of films of water, oil, toluene, and kerosine inside - tubes, Aro = 0.02-115. Measurements of thickness, surface velocity, onset of wavy flow. 

Vedernikov (V2), 1946 Theoretical treatment of wavy flow in open channels. Wavy flow and turbulent flow clearly distinguished. 

Kapitsa (K7, K8), 1948 Theoretical treatment of wavy flow of thin films of viscous liquids, including capillary effects. Only regular waves considered. Wavy flow shown to be more stable than smooth film, and about 7% thinner than smooth film at same flow rate. Also calculates wave amplitudes, wavelengths, etc., onset of wavy flow, effects of countercurrent gas stream, heat transfer. Theory applicable only if wavelength exceeds 14 film thicknesses. Error in treatment pointed out by Levich (L9). 

Kapitsa and Kapitsa (K10), 1949 Wavy flow of water and alcohol films on outside of tube of diameter 2 cm., NRe < 100, studied photographically and stroboscopically. Experimental data at low flow rates in agreement with Kapitsa theory waves become random at large flow rates. 

Jackson et al. (J2), 1951 Experimental study of film flow inside tube with counter-flow of gas. Film surface velocity deduced from pressure drop readings in gas stream, neglecting wave roughness effects. Surface velocities appear to exceed the theoretical values in the wavy flow regime. 

Shibuya (S10), 1951 Mathematical treatment of onset of wavy flow in liquid films on vertical tubes. Waves appear for No, > 7. Wavelength of first waves = 3 film thicknesses. 

Ishihara et al. (12), 1961 Gives summary of recent Japanese work on wavy flow in open channels, and semitheoretical analysis of problem (wave velocities, frequencies, heights, lengths). Mostly small channel slopes considered. 

Mayer (M7), 1961 Experimental and theoretical study of wavy flow of water in open channel (slopes up to 5°). Data on growth of turbulent spots, local depths, surface velocity, length of entry zone, wave velocities, heights, frequencies, effect of surface-active materials. 

Brauner N, Maron DM, Dukler AE. Modeling of wavy flow in inclined thin films in the presence of interfacial shear. Chem Eng Sci 1985 40(6) 923-937. 

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