Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow

International Journal of Multiphase Flow xxx (2014) xxx–xxx

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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w

Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow Sergey V. Alekseenko a,b, Andrey V. Cherdantsev a,b,⇑, Oksana M. Heinz a, Sergey M. Kharlamov a,b, Dmitriy M. Markovich a,b a b

Kutateladze Institute of Thermophysics, Novosibirsk, Russian Federation Novosibirsk State University, Novosibirsk, Russian Federation

a r t i c l e

i n f o

Article history: Received 2 March 2014 Received in revised form 16 July 2014 Accepted 18 July 2014 Available online xxxx Keywords: Annular flow Disturbance waves Ripples Space–time evolution Laser-induced fluorescence

a b s t r a c t Wavy structure of liquid film in downward annular gas–liquid flow is studied with high-speed laserinduced fluorescence technique. Film thickness measurements are resolved in both longitudinal distance and time with high spatial and temporal resolution. A method is developed to identify the characteristic lines of individual disturbance waves. Change of frequency of disturbance waves with downstream distance is modelled based on the obtained distributions of the disturbance waves by velocity and separation time. Using obtained characteristic lines further investigation of ripples’ properties is performed in reference system moving with the disturbance wave. As a result, velocities, amplitudes and frequencies of ripples are measured with relation to the distance from disturbance waves. Fast ripples travelling over crests of disturbance waves and slow ripples travelling over back slopes of disturbance waves and over the base film are studied separately. Besides, the average length of crest and back slope of disturbance waves were measured. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction In annular flow liquid film is flowing along pipe walls, sheared by high-velocity gas stream. At high enough gas and liquid flow rates liquid droplets are entrained from film surface into the core of gas stream. This flow pattern occurs in a wide range of industrial equipment, and prediction of its integral characteristics (in particular, entrainment rate and pressure drop) is of high practical importance. Appearance, evolution and interaction of waves on the film surface are the key processes, defining the mentioned characteristics. Any physical models of the flow should be based on the waves’ dynamics and their interaction to the gas shear. To create such models, experimental information on the wavy structure of liquid film is necessary. Wavy structure of liquid film in annular flow with liquid entrainment is represented by large-scale disturbance waves and small-scale ripple waves. Disturbance waves are separated by thin base film layer; ripples are travelling either over the base film surface or over the disturbance waves. Presence of disturbance

⇑ Corresponding author at: Kutateladze Institute of Thermophysics, Novosibirsk, Russian Federation. E-mail address: [email protected] (A.V. Cherdantsev).

waves is considered to be the necessary condition for the inception of entrainment (Azzopardi, 1997). They are distinguished from the ripples by large values of amplitude, velocity, longitudinal size and lifetime. In the last fifty years large number of experimental works was devoted to studying the properties of disturbance waves. Disturbance waves are created not far from the liquid inlet. Supposedly, they are formed from the high-frequency small-amplitude initial disturbances (e.g., Zhao et al., 2013). Properties of individual disturbance waves vary in a wide range of values. In particular, their velocity obeys normal distribution; its standard deviation is nearly constant at different gas velocities (Hall Taylor and Nedderman, 1968). Due to variation in velocity, multiple events of coalescence of individual disturbance waves occur. Because of coalescence, frequency of disturbance waves gradually decreases downstream. In most cases, when a faster wave overtakes a slower one, the resulting wave moves with the velocity of the faster wave (Hall Taylor et al., 1963). Under this scenario of coalescence, the average velocity of disturbance waves is expected to increase downstream (Azzopardi, 1997). Average amplitude of disturbance waves is also expected to increase downstream, since the velocity of individual disturbance waves is proportional to the waves’ amplitudes. Without coalescence, disturbance waves pass large downstream distances with nearly constant velocity.

http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009 0301-9322/Ó 2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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Properties of disturbance waves are affected by flow parameters as well. Passing frequency and velocity of disturbance waves grow linearly with superficial gas velocity, whereas the amplitudes of disturbance waves decrease with gas velocity. Increase of liquid flow rate leads to increase in all these properties (e.g., Chu and Dukler, 1975; Azzopardi, 1986; Han et al., 2006; Sawant et al., 2008). Longitudinal size of disturbance waves does not change much with gas velocity (Han et al., 2006). Frequency of disturbance waves is larger in small diameter pipes (Alamu and Azzopardi, 2011). Transverse size of disturbance waves is larger than their longitudinal size. The disturbance waves form full rings around the pipe circumference in small diameter pipes. In larger pipes circumferentially localized disturbance waves appear and the fraction of localized disturbance waves increases with pipe diameter (Azzopardi, 1997). Amplitudes of disturbance waves are not uniform along the circumferential coordinate (Belt et al., 2010; Alekseenko et al., 2012). Ripples are less studied; they were mostly studied independently of disturbance waves. Chu and Dukler (1974) measured amplitude, velocity and longitudinal size of ripples on the base film. They found that ripples have very short lifetime in comparison to that of the disturbance waves. The majority of mentioned works used either high-speed imaging visualization or conductance technique to study the disturbance waves. The former does not give any quantitative information on local film thickness; the latter has low spatial resolution and is unable to provide information on the properties of ripples and on structure of disturbance waves. In temporal records of film thickness obtained with conductance probes disturbance waves normally look like smooth high-amplitude waves with steep fronts, shallow rear slopes and pronounced crests. The shape is quite different when studied with high-resolution techniques such as planar laser-induced fluorescence (PLIF). In PLIF data, disturbance waves look like relatively flat ‘plateaus’ of higher thickness, covered with high-amplitude ripples (Hewitt et al., 1990; Schubring et al., 2010a; Farias et al., 2012; Zadrazil et al., 2014). Schubring et al. (2010a) performed separate study of film thickness behaviour in the base film zone and in the disturbance waves’ zone. They have shown that the average thickness in the waves’ zone is approximately twice larger than that of the base film. Standard deviation of film thickness (related to the amplitude

Fig. 1. Scheme of the inlet section.

Fig. 2. Layout of camera and laser.

of ripples) is 0.3 of the average film thickness in the base film region and 0.2 in the waves region. All the mentioned ratios were shown to be the same for a wide range of gas and liquid flow rates. Pham et al. (2014) performed visualization of gas-sheared liquid film at the outer surface of a cylinder (part of a rod bundle). The film surface profile was obtained with high-speed video. In these data the fast ripples on disturbance waves can also be observed. They appear in the rear part of the disturbance wave and move toward its front. It was found that the fast ripples can be disrupted to droplets by the gas shear near the front of a disturbance wave. Alekseenko et al. (2008, 2009) used LIF technique in rather different approach, which could be called ‘brightness-based LIF’. They studied the spatio-temporal evolution of the film surface with good spatial and temporal resolution in one longitudinal section of the pipe with length 10 cm and duration of 2 s. In particular, it was found that all the ripples are generated at the rear slopes of the disturbance waves. Depending on the relative position of the point of origin, the ‘newborn’ ripples might travel either faster or slower than ‘parent’ disturbance waves. The slow ripples slide back to the base film behind the parent disturbance waves and travel over the base film with nearly constant velocity until the following disturbance wave absorbs them. The fast ripples travel over the parent disturbance wave toward its front and disappear there. This disappearance is supposed to occur due to disruption of the fast ripple by the gas shear into droplets, as it is was first observed by Woodmansee and Hanratty (1969).

Fig. 3. Example of spatial and temporal fragment of film surface.

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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In present paper, the quantitative study of spatio-temporal evolution of disturbance waves and both fast and slow ripples is presented. Automatic algorithm of data processing is developed, allowing to identify the characteristic lines of individual disturbance waves and to study the evolution of ripples in the reference system of disturbance waves. We acknowledge outstanding contribution of Professor G.F. Hewitt to the modern knowledge of multiphase flows. Our research would be much more difficult without his comprehensive review on annular flow (Hewitt and Hall Taylor, 1970), pioneering work on application of fluorescence technique to film thickness measurements (Hewitt and Nicholls, 1969), studies of spatiotemporal evolution of disturbance waves (Hall Taylor et al., 1963; Zhao et al., 2013), new notion on structure of disturbance waves (Hewitt et al., 1990), and many other papers, providing a solid base for future investigations. We wish Professor Hewitt health and happiness, and we are looking forward to read his new wonderful papers.

Experimental details Experiments were conducted for downward annular flow in vertical Plexiglas pipe with inner diameter d = 15 mm. The pipe was made with square outer section to decrease optical distortions. Gas entered the working section through a coaxial tube of slightly smaller diameter. Liquid was introduced as a film through a ringshaped slot between the inner wall of the pipe and the outer wall of gas-feeding tube. Slot thickness is 0.5 mm. In this configuration the annular flow regime took place from the liquid inlet (Fig. 1). Liquid was coming from a pressure tank placed 3 m above the inlet. Gas was coming from the centralized line of compressed air. Flow rates of gas and liquid were measured using orifice meter and float rotameter, respectively. Measurements were taken at equilibrium temperature which was reached after about an hour of running the rig in annular flow regime. This temperature was 2–3° lower than the ambient air temperature; viscosity of working liquids was measured at this temperature using capillary viscosimeter. Kinematic viscosity of water at this temperature was 1.15 * 106 m2/s. Water and two water– glycerol solutions with kinematic viscosities m = 1.5 * 106 m2/s (WGS1) and 1.9 * 106 m2/s (WGS2) were used as working liquids. The densities of WGS1 and WGS2 were 1030 and 1050 kg/m3, respectively. Surface tension was the same for all three working liquids. Range of superficial gas velocities was 27–58 m/s; liquid Reynolds numbers were 142, 220 and 350 (the last value was not reached for WGS2). The Reynolds number used in present work is constructed using film thickness and average film velocity. It is defined as Re = q/p dm, where q is volumetric liquid flow rate; it is four times smaller than the liquid Reynolds number constructed using liquid superficial velocity and pipe diameter. The used range of conditions corresponds to annular flow regime with liquid entrainment; presence of entrainment for this range of flow parameters was confirmed independently using sampling probe measurements. Area of measurements was located 50–62 cm (33–41 pipe diameters) below the inlet. According to Wolf et al. (2001), the flow is not fully developed at such distances. In particular, frequency and average velocity of disturbance waves are expected to change downstream. At the same time, this change may be explained by coalescence of disturbance waves. Such analysis will be performed in Section ‘Studying the disturbance waves’ of the present paper. Laser-induced fluorescence technique was used for field measurements of film thickness. Vertical laser sheet illuminated one longitudinal section of the pipe in order to excite the fluorescence of Rhodamine 6G, dissolved in liquid in low (20 mg/l)

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concentration. Continuous 2 W laser with wavelength of 527 nm was used. High-speed camera equipped with orange filter was aimed at the illuminated section of the pipe. Top and side views of camera and laser layout are shown in Fig. 2. Single line of 600 pixels was used for data acquisition. Size of the area of measurements was 120 mm in longitudinal direction and 0.2 mm in circumferential direction. Spatial resolution was 0.2 mm per pixel in either direction. The camera frame rate was 10 kHz. Average amplitude of waves (and, consequently, the maximum brightness of emitted fluorescent light) depends on flow parameters. To provide the best correspondence between the brightness of fluorescent light and the dynamic range of the camera, the exposure time was varied between 30 and 70 ls depending on flow parameters. Two 2 s realizations were obtained for each regime point. Obtained values of brightness were recalculated into local film thickness. Details on calibration and recalculation are given in Alekseenko et al. (2012). In that paper, the film thickness measurements were resolved along both longitudinal and circumferential coordinates. In present paper, only the longitudinal coordinate was resolved, but the temporal resolution is much higher. Brief summary on the recalculation is given below. Local brightness of the fluorescent light J and local film thickness h obey the following relation:

JðxÞ ¼ CðxÞ  ½1  expðahðxÞÞ  ½1 þ K  expðahðxÞÞ þ DðxÞ:

ð1Þ

Here a is absorption coefficient; it is estimated by comparing levels of brightness of two liquid layers of different thickness. K is the reflection index of liquid–gas interface, which is 2%. D is dark level of camera matrix. Coefficient C is introduced to compensate the non-uniformity of the signal along longitudinal coordinate (which includes non-uniformity of the exciting light, difference in sensitivity of camera pixels, fouling of the wall, etc.). It is constructed based on the average value of fluorescence intensity registered by each pixel in long-time measurements of film of known average thickness (namely, the average thickness of thin film falling in absence of gas flow was considered to obey the Nusselt law). The random error of film thickness measurements was mainly defined by the camera noise. The amplitude of this noise was estimated as 2–3%. At low concentrations of the fluorescent dye, the dependence (1) is close to linear for the range of film thicknesses smaller than 1 m. Thus, the error, introduced by the camera noise, can be considered equal to the noise amplitude. The error caused by non-uniformity of individual pixels after the compensation procedure, was estimated as another 2–3%. Thus, the overall error of film thickness measurements can be estimated as 5–6% when studying slow ripples on the base film and on the rear slopes of the disturbance waves. Spatio-temporal behaviour of waves of different types In further analysis, we will present the obtained film thickness matrices in a graphical form. For this, fragment of matrix h(t, x) is transformed into an image with local brightness of the image directly proportional to the local value of film thickness. Example of such image is given in Fig. 3. Horizontal axis corresponds to longitudinal coordinate (total length is 120 mm); vertical axis corresponds to time. Two coalescing disturbance waves are seen as wide bright bands, moving along the diagonal lines. Slope of a wave’s trajectory to t-axis is proportional to wave’s velocity. The disturbance waves are covered by fast ripples (short bright bands), which normally appear at the rear slope of a disturbance wave, move faster than the disturbance wave and disappear near its front. The base film is covered by the slow ripples, which also appear at the rear slopes

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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Fig. 4. Example of instantaneous profile of a disturbance wave, obtained with LIF technique, with non-physically high values of the amplitude of the fast ripples (marked as 1).

of disturbance waves. Slow ripples decelerate and lag behind the disturbance waves; then they move with constant speed over the base film and are finally absorbed by the following disturbance wave. Such data enable one to perform quantitative study of evolution of disturbance waves and ripples in both space and time. Structure of disturbance waves and change of properties of each single disturbance wave can be investigated. Generation of fast and slow ripples and evolution of their properties with relative distance from the area of generation can be studied as well. Essential limitation of the technique is related to the optical distortions that appear under the fronts of fast ripples, yielding non-physically high amplitude values of order of several millimeters (example of film thickness record with four non-physical peaks is shown in Fig. 4). This is supposed to happen because of total internal reflection of exciting light under high-slope liquid– gas interface. On that reason, only frequency and velocity of fast ripples were studied. Amplitude characteristics of fast ripples are not considered, since they require additional methodic investigation on how to avoid, minimize or compensate the distortions, whether it is possible at all. At present, data of Schubring et al. (2010a) along with recent corrections by Kokomoor and

Fig. 5. Example of temporal record of average film thickness along the trajectories moving with cross-correlation velocity Vc. Water, Re = 220, Vg = 36 m/s.

Schubring (2014) provide the best knowledge on the amplitude of fast ripples. Alekseenko et al. (2009) made preliminary quantitative investigation of generation of ripples using simple manual processing. To increase reliability of measurements and range of flow parameters, automatic algorithm of data processing is obviously required. The algorithm is described below. It identifies individual disturbance waves in x–t surface; it performs measurements of local characteristics of either fast or slow ripples depending on distance to the characteristic points of the disturbance waves; it finds the areas where fast and slow ripples exist and process the data in these areas in proper way. Studying the disturbance waves Identification of disturbance waves As it was noted above, for each combination of flow parameters (Re, Vg, m) two space–time records of film thickness were obtained. Each record is a matrix of film thickness values H(t, x) with dimension of 20,000 time moments (or 2 s) by 600 points in space (or 120 mm). In further description of the algorithm, t is given in time instant units, varying from 1 to T = 20,000; x is given in camera pixels units, varying from 1 to X = 600. First, the average velocity of disturbance waves Vc was estimated by cross-correlating the ‘temporal records’ (H(t) at constant x) separated with some distance. Not less than 20 pairs of such records with different position and separation distance were processed for each regime point in order to enhance the reliability of measurements. To identify the space–time trajectories of disturbance waves, new massive HV(t0) was constructed for each record. HV(t0) is equal to the average value of film thickness along the characteristic line with velocity V = Vc, beginning in the moment t0, namely:

HV ðt 0 Þ ¼

Fig. 6. Final correction of the trajectory of a disturbance wave. Points tm(x) are marked with black dots.

  X 1X x1 H t0 þ ; x : X x¼1 Vc

ð2Þ

Here t0 is varied from 1 to T  X/Vc. A portion of disturbance waves in the very end and in the very beginning of the record might be missed; though X/Vc is maximum 3% of T for the lowest observed Vc. To ‘catch’ the disturbance waves, threshold value of film thickness Htr was used, which was chosen equal to 1.5 of the base film thickness. The latter was supposed to be equal to the most

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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Fig. 7. Visual checking of the identification algorithm’s efficiency. Automatically obtained characteristic lines of disturbance waves are shown by straight lines.

Fig. 8. Passing frequency (a) and average velocity (b) of disturbance waves at x = 50 cm. Working liquids: water (1, 3, 5); WGS2 (2, 4). Re = 142 (1, 2): Re = 220 (3, 4); Re = 350 (5). Standard deviation of waves’ distribution by velocity is also shown by (6).

Fig. 9. Distribution of disturbance waves by separation time (a) and velocity (b). Re = 220, water.

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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For the whole set of new trajectories the one with maximum average film thickness along it was chosen. After that, additional correction was used. It is based on idea that each disturbance wave includes an area of existence of fast ripples. Though the amplitudes of fast ripples are overestimated by the optical distortions, the coordinates of the fast ripples are not significantly shifted due to the distortions. The ‘‘distorted’’ image of a fast ripple is still located on the disturbance wave’s crest. For temporal vicinity of the obtained line, time of maximum local film thickness tm(x) was obtained for each value of x. A linear approximation of tm(x) gave the best approximation of the space–time trajectories of disturbance waves (Fig. 6). The quality of approximation was checked visually for each neighbouring pair of detected disturbance waves. Examples of space–time images with two disturbance waves are shown in Fig. 7. The checking has shown that the number of waves, which were not detected by the algorithm, does not overcome 5% of the total number of the disturbance waves. Most part of such ‘lost’ waves was missed because of being overtaken by the following disturbance wave within the area of measurements (example of a missed disturbance wave is given in Fig. 7b). The overtaken waves might be missed by the algorithm on two main reasons. First, an overtaken wave exists over only a part of the length of the area of measurements, and continuation of its trajectory after its disappearance crosses the base film area. This leads to decrease of the average film thickness along its trajectory, which might not overcome the used threshold value. Second, local peak of HV, corresponding to the absorbed wave, might overlap with the peak, corresponding to the absorbing wave; in this case, the absorbed wave would not be identified. Number of disturbance waves that are overtaken in the area of measurements is of order of 5–10% of the total number of disturbance waves (part of the overtaken waves was detected by the algorithm). Scenario of coalescence always corresponds to the most likely scenario observed by Hall Taylor et al. (1963); namely, the resulting wave moves with the speed of the faster wave. Fig. 10. Evolution of disturbance waves’ passing frequency (a) and average velocity (b) with downstream distance, predicted based on distributions obtained experimentally at x = 50 cm below the inlet. Re = 220, water.

probable value of HV(t0). Example of HV(t0) along with used value of threshold Htr is shown in Fig. 5. We should note that the overestimated values of film thickness under the fronts of fast ripples, described in the previous section, also contribute to HV. Since the fast ripples are localized at the crests of disturbance waves, such values enhance the contrast in the values of HV between the disturbance waves and the base film. Thus, these values in a manner simplify identification of the disturbance waves. For each continuous area where HV(t0) > Htr, time instant t0 corresponding to maximum value of HV was found. This gave us a set of straight trajectories characterized by initial time t0i and velocity V (the latter is equal to Vc for all the waves at this stage). At the next stage, the trajectories were corrected by varying both V and t0. For each wave, velocity was varied within the range 0.5 * Vc < V < 1.5 * Vc with step 0.01 * Vc. Initial time varied respectively so that the new trajectory would always cross initial trajectory with V = Vc. Namely, for V < Vc, ti varied in the range:

t0  X

Vc  V 6 ti 6 t0 ; V  Vc

ð3Þ

and for V > Vc, ti varied in the range:

t0 6 ti 6 t0 þ X

V  Vc : V  Vc

ð4Þ

Dynamics of disturbance waves Obtained trajectories can be used to analyse the dynamics of the whole system of disturbance waves, since the distributions of disturbance waves by velocity and separation time are available. Fig. 8(a) shows passing frequency of disturbance waves at the distance of 50 cm below the inlet. The frequency linearly grows with gas velocity and also increases with liquid flow rate. Increase in liquid viscosity leads to small increase in the passing frequency of the disturbance waves at low gas velocities. Thus, the effect of liquid viscosity is not completely described by the liquid Reynolds number; nonetheless, Re yields better generalization than it would be obtained using the dimensional liquid flow rate as a parameter. The frequency of disturbance waves might be underestimated by approximately 5% on the reasons described in previous subsection; nonetheless, most part of missed waves disappear within the area of interrogation (namely, between 50 and 62 cm below the inlet). Fig. 8(b) shows the average velocity of the disturbance waves for the same data points. This quantity shows nearly the same behaviour with flow parameters as that of the passing frequency. Standard deviation of waves’ velocity distribution is marked by crosses for all regime points; its absolute value tends to be constant. This is consistent with the observations of Hall Taylor and Nedderman (1968); though, the absolute value of standard deviation is approximately 3 times larger in present experiments. The discrepancy might be explained by relatively small distance below the inlet in present experiments. Examples of distributions of disturbance waves by velocity and separation time are shown in Fig. 9. Average separation time

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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corresponds to the passing frequency of disturbance waves; standard deviation of the distributions decreases with growth of gas velocity. Distributions by velocity are nearly symmetric around the average value; the latter corresponds well to the crosscorrelation velocity Vc. Based on such distributions, it is possible to roughly predict the behaviour of the whole system of disturbance waves far downstream the area of measurements. As it was described in Section ‘Introduction’ and as it was confirmed in Section ‘Dynamics of disturbance waves’ for present experiments, downstream evolution of disturbance waves follows two simple rules: (1) without coalescence disturbance waves are considered to move with constant velocity over large distances; (2) coalescence occurs at the crossing of trajectories of two disturbance waves; resulting wave moves with the velocity of the faster disturbance wave (see Fig. 7b). Modelling of the initial state of the system of disturbance waves was performed in two different ways. In the first case, the experimentally obtained array of velocity Vi and initial time t0i of each individual disturbance wave was taken as the initial state. This array was repeated 100 times (giving a ‘temporal record’ with length of slightly less than 200 s) to minimize the effects of the edges of the array. In the second case, experimental distributions of disturbance waves by velocity and time separation were approximated as normal distributions with mean value and standard deviation taken from the experiments. Then the waves were placed

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randomly at the initial downstream position of x0 = 50 cm below the inlet over the time record of 200 s so that the waves obeyed the distributions by velocity and separation time. After that for both ways the evolution of the system of disturbance waves was modelled as follows. Let us consider a pair of disturbance waves with velocities Vi and Vj, which appear at the distance x0 in the time instants t0i and t0j, respectively. Their spatio-temporal trajectories are given by the equations:

x ¼ x0 þ V i ðt  t 0i Þ; x ¼ x0 þ V j ðt  t 0j Þ: These two lines will cross each other at the distance xcij, expressed as:

xcij ¼ x0 þ

V i V j ðt 0j  t0i Þ : Vj  Vi

ð5Þ

A disturbance wave with parameters {Vi, t0i} will be absorbed at the downstream distance xci, defined as the minimum value of xcij for all the waves with parameters {Vj, t0j}, satisfying to conditions Vj > Vi and t0j > t0i. Indeed, the assumption (2) above means that a disturbance wave disappears after the first coalescence with a following faster wave. After that, the number of disturbance waves which survived by particular downstream distance x was calculated as number of waves with xci > x. This quantity was calculated for the middle part

Fig. 11. Transformation of x–t-surface to the reference system of the disturbance wave: initial shape of the areas used in two different ways of transform (a); vicinity of the wave in ‘frozen space’ approach (b); vicinity of the wave in ‘frozen time’ approach (c). Water, Re = 142, Vg = 27 m/s.

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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able to develop into ‘real’ disturbance waves. The average velocity of disturbance waves grows with downstream distance (Fig. 10b), since the slowest waves are more likely to be absorbed. The standard deviation of disturbance waves’ distribution by velocity decreases as it should be expected, but its absolute value is still larger than that reported by Hall Taylor and Nedderman (1968).

Studying the fast and slow ripples New reference systems and local measurements of ripples properties

Fig. 12. (a) Local median film thickness; (b) local velocity of ripples (velocity of the disturbance wave is shown by the solid line); (c) local spatial frequency of ripples. Water, Re = 142, Vg = 27 m/s.

of the temporal record with length of 100 s. The velocity distribution of the survived disturbance waves was analyzed as well. The results did not change much depending on the way of introducing the initial state. In both cases the system of disturbance waves changes as it would be expected. The frequency gradually decreases with the downstream distance, and the rate of change of the frequency decreases as well (Fig. 10a). Though, even for the largest distances full stabilization was not achieved. It is possible that in real flow the stabilization might occur earlier due to appearance of ‘ephemeral disturbance waves’ described by Sekoguchi et al. (1985) and Wolf et al. (1996) in case if they are

To study the ripples, the obtained trajectories of disturbance waves were used as the base for further processing. The idea is to obtain the records of film thickness along the space–time trajectories, parallel to the characteristic lines of disturbance waves. Studying such records it is possible to investigate the ‘phase’ portraits of disturbance waves, and understand how the distance to the disturbance waves affects the local properties of ripples. Disturbance waves, separated by large enough distance (4 cm) from the neighbouring waves, were selected. Fraction of such ‘‘isolated’’ disturbance waves was over 90% for the most part of flow regimes, and it was never lower than 80%. The goal of this selection is to avoid ambiguities which might appear when overlapping or coalescing disturbance waves are processed. Possible drawback of this approach is related to the influence of small distance between the disturbance waves on the properties of the waves. E.g., in the paper of Schubring et al. (2010b) the waves with small time separation are characterized with lower velocity and large scatter in the longitudinal size. In our case, the difference between the average velocity of all disturbance waves and the average velocity of isolated disturbance waves does not overcome 1%. Though, it is difficult to estimate the effect of separation time on the other properties of disturbance waves and fast ripples. Thus, we cannot exclude that the selection of isolated waves might bias the sample and increase the error of measurements. Two ways of transformation can be used, which could be called ‘frozen space’ and ‘frozen time’ approaches. The first one is the common Galilean transformation, in which a disturbance wave is ‘standing’ at the same distance with time changing. In this case slow ripples move to the left (with negative velocity), and fast ripples move to the right (Fig. 11b). When the second transformation is used, the disturbance wave does not move in time (Fig. 11c). In this case, fast ripples move with negative velocity, whereas velocity of slow ripples is positive. Areas that are transformed into Fig. 11(b) and (c) are shown in Fig. 11(a) by parallelograms, marked by (1) and (2), respectively. In the first transform, spatial vicinity of disturbance wave with size of 4 cm is selected for processing. Borders of the second one are placed 2 cm in front of the current disturbance wave and 2 cm in front of the following disturbance wave. The main advantage of the ‘frozen space’ approach consists in its linearity and simplicity. But it can be applied only for quite a small fraction of H(t, x) matrix not far from each disturbance wave. It gives 200 temporal records of film thickness with length limited by 2/3 of time, required for a disturbance wave to pass the area of interrogation; this time decreases with gas velocity. On the contrast, with the ‘frozen time’ approach, nearly the whole H(t, x) matrix can be divided into the investigation areas, including the areas far behind each disturbance wave. Each area consists of a number of spatial records with length of 600 data points each. The number of records is defined by the separation time between the considered disturbance wave and the following one. On these reasons, only the ‘frozen time’ approach is used in subsequent processing. For each area surrounding a disturbance wave h(t0 , x), ‘local’ measurements of ripples properties are performed

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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The velocity of ripples in laboratory system is defined as:

Vr ¼

V 0r  V d : ðV 0r þ V d Þ

ð6Þ

Here Vd is the velocity of the disturbance wave, V 0r is the velocity of ripples, measured in transformed system. If the absolute value of Vr is negative or exceeds Vd more than 4 times, it is considered non-physical. Such values of velocity are discarded at the stage of investigation of CCF; the next peak of CCF is chosen for velocity measurements. (3) Local spatial frequency. Attempts to apply spectral analysis to measure the frequency were not successive due to large low-frequency component of spectrum. Direct counting of points where local film thickness record crosses threshold value of film thickness is used instead. Threshold values varied in the range hhi(t0 ) ± rh(t0 ); median value of halved number of crossings was used as a local frequency. To take into account the Doppler effect, measured frequency should be transformed to obtain the frequency in laboratory system: 0

fs ¼ fs 

Fig. 13. Velocity of fast and slow ripples on the base film, normalized to the velocities of the disturbance waves. (a) Influence of liquid flow rate for water. (1) Re = 142; (2) Re = 220; (3) Re = 350. (b) Influence of liquid viscosity at Re = 220. (1) Water; (2) WGS1; (3) WGS2.

for each moment of relative time t0 . The following quantities are defined: (1) Local average film thickness hhi(t0 ) and standard deviation of film thickness rh(t0 ) are defined, respectively, as the average and standard deviation of the spatial record with length of 600 points, obtained at relative time t0 . The standard deviation gives rough estimation of local amplitude of ripples. After tests, it was found that the median value of film thickness should be used instead of the average, since this value is less vulnerable to the optical distortions at the steep slopes of the interface. (2) Local velocity of ripples. The procedure involves crosscorrelating two spatial records, one before and one after the considered moment of t0 . The temporal delay between the correlated records was varied from 0.3 ms to 1 ms to increase reliability of the measurements. The velocity was calculated based on the maximum of cross-correlation function (CCF). To increase the resolution of measurements, vicinity of CCF’s maximum was approximated by cubic polynomial; interpolation of this approximation with 10 times smaller time step was used to obtain the position of maximum.

Vd : jV d  V r j

ð7Þ

Besides, peak-to-peak amplitude of ripples A(t0 ) can be estimated this way as the difference between maximum and minimum values of film thickness in the areas between the crossing points. Fig. 12 shows an example of behaviour of these characteristics for a particular disturbance wave (the one shown in Fig. 11). Flow direction is from right to left. Local median film thickness profile is shown in Fig. 12(a). It consists of the main hump with steep front slope and shallow back slope behind the crest. This averaged shape is quite similar to the shapes obtained by conductivity technique (see, e.g., Han et al., 2006). Outside the disturbance wave (i.e., on the base film) local median film thickness shows stable behaviour with nearly the same average value in front of the disturbance wave and far behind it. The local velocity is presented in Fig. 12(b). Different plots correspond to different time delays (from 0.3 ms to 1 ms) between correlated spatial records. The plots show close results at the base film and at the back slope of the disturbance wave, where the velocity of ripples gradually decreases. At the crest, the scatter of the data is much larger, but the main amount of dots is lying in a relatively compact area above the velocity of the disturbance wave (which is shown by the solid line). Thus, the crest could be considered as the area of existence of fast ripples; their velocity can be roughly approximated by a single value, which was defined as the median value of all measurements within that area. The area of generation of fast and slow ripples is obviously located around the point t0 = 12 ms, where the sharp change in local velocity is observed. Local spatial frequency of ripples is shown in Fig. 12(c). It was transformed into laboratory system using formula (7); in the crest region a constant value of local velocity of ripples was used, which is equal 1.45 * Vd for the particular disturbance wave. The spatial frequency is nearly constant on the base film. It is notably smaller on the disturbance wave, reaching its minimum value approximately in the area of ripples’ generation mentioned above. The reason why the frequency of slow ripples increases with distance from the disturbance wave is clearly illustrated by Fig. 11(c). Quite large initial slow ripples are depleting into smaller waves simultaneously with the waves’ deceleration. Thus, regarding the properties of ripples, the whole film surface could be separated into three zones: (1) Crests of disturbance waves, covered by the fast ripples.

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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Fig. 14. Average number of fast ripples, generated by a single disturbance wave per unit time (a) and (b) and per unit length (c) and (d). Influence of liquid flow rate for water (a) and (c): (1) Re = 142; (2) Re = 220; (3) Re = 350. Influence of liquid viscosity at Re = 220 (b) and (d): (1) Water; (2) WGS1; (3) WGS2.

(2) Back slopes of disturbance waves, covered by the slow ripples with changing properties. (3) Base film, covered by the slow ripples with stable properties. Further, the algorithm is measuring ‘stationary’ values of ripples’ properties in zones (1) and (3), and estimates the average length of zones (1) and (2). Sum of these two values gives spatial length of the disturbance waves. Average characteristics of fast and slow ripples In present subsection, ‘stable’ values of properties of fast and slow ripples in zones (1) and (3) are investigated. Within the zone (1), the area of measurements is reduced to the short (1–2 mm) vicinity of the characteristic line of a disturbance wave in order to decrease the influence of the edges of the zone. In this zone, median values of velocity and frequency of fast ripples were measured. To obtain values of certain characteristics of slow ripples in the zone (3), the most probable values of each characteristic over all disturbance waves outside zone (1) are taken. Fig. 13 shows values of velocity of fast ripples (zone 1) and the most probable value of velocity of slow ripples (zone 3), normalized by the velocities of individual disturbance waves and averaged over all disturbance waves at fixed set of flow parameters. Influence of flow parameters on the normalized velocity of fast and slow ripples is rather weak. It is possible to say that the normalized

velocity of ripples tends to be slightly closer to unity at higher gas flow rates. On average, velocity of fast ripples is equal to 1.4– 1.5 * Vd, velocity of slow ripples – to 0.3–0.4 * Vd. The average number of fast ripples generated by a single disturbance wave per second is shown in Fig. 14(a) and (b). It grows approximately linearly with gas velocity and slightly increases with liquid Reynolds number. At the same time, the increase in frequency is mainly related to the increase in disturbance waves’ velocity: number of fast ripples generated by average disturbance wave over the distance of 1 m is nearly constant (Fig. 14c and d), except for the lowest Vg for the most viscous liquid. It means that the volume of liquid entrained to gas core from unit length of the pipe is mainly defined by the number of disturbance waves passing this distance per unit time and by the volume of liquid carried by a single fast ripple, whilst the number of generated ripples remains unchanged. Fig. 15 shows the average wavelength of the slow ripples on the base film. This value slightly reduces with increase of Vg and grows with both liquid Reynolds number and liquid viscosity. Fig. 16 shows the relative roughness of the base film, defined as the relation of standard deviation of film thickness record to the average film thickness. According to the measurements of Schubring et al. (2010a), this value does not depend on gas velocity and liquid flow rate, and is equal 0.3 for air–water flow. Our results on water are in rather good agreement with this conclusion (Fig. 16a). It should be noted that this ratio increases for more viscous liquids (Fig. 16b).

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

S.V. Alekseenko et al. / International Journal of Multiphase Flow xxx (2014) xxx–xxx

Fig. 15. Wavelength of slow ripples on the base film. (a) Influence of liquid flow rate for water. (1) Re = 142; (2) Re = 220; (3) Re = 350. (b) Influence of liquid viscosity at Re = 220. (1) Water; (2) WGS1; (3) WGS2.

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Fig. 16. Relative roughness of the base film. (a) Influence of liquid flow rate for water. (1) Re = 142; (2) Re = 220; (3) Re = 350. (b) Influence of liquid viscosity at Re = 220. (1) Water; (2) WGS1; (3) WGS2.

Zones of disturbance wave The most physical way to identify the borders of ‘crest’ area would be to find the area where the local velocity is larger than that of the disturbance wave. But, because of large scatter of local velocity, this method is not always reliable. So, more simple method based on threshold value of local median film thickness is used. Crest of a disturbance wave is defined as the area with local median thickness larger than 1.5 of the base film thickness. This gives slight overestimation of crest’s length, but the end of this zone is a good starting point for identifying the border between the back slope and the base film zones. For the latter, data on local median film thickness, velocity and frequency were analysed beginning from the obtained end of the crest zone for each disturbance wave. The moment of t0 , corresponding to stabilization of certain parameter, was defined as follows. For each value of t0 the temporal records to the left and to the right of it were approximated by a function and a constant, respectively. For local median film thickness and local velocity the function in form of a/(t0 + b) was used; for local spatial frequency it was used in form of kt0 + c. After that the value of t0 with minimal total error of approximation was selected. Example of the best approximation of local velocity is shown in Fig. 17. Each measured temporal length was recalculated into spatial length by multiplying it by individual velocity of a disturbance wave.

Fig. 17. Example of piecewise approximation of local velocity of ripples behind a disturbance wave (the same as in Figs. 11 and 12). Obtained border between the zones is marked by the dashed line.

As a result, the average distances between the beginning of the crest zone and the points of stabilization of film thickness, velocity and frequency behind the crests of disturbance waves (Lh, LV and Lf, respectively), were obtained. Fig. 18 shows dependence of these

Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009

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S.V. Alekseenko et al. / International Journal of Multiphase Flow xxx (2014) xxx–xxx

Velocity and separation time distributions were obtained for individual disturbance waves. Based on these data, a model for downstream evolution of the whole system of disturbance waves was developed. The ripples properties were studied in reference systems of individual disturbance waves. Dependence of ripples properties on the relative distance to the disturbance wave was studied. It was found that the film surface could be considered to consist of three zones: the crests of disturbance waves, where the fast ripples exist; the back slopes of disturbance waves, where the slow ripples are generated and their properties gradually change with increasing distance from the crests; and the base film zone, where the properties of slow ripples have stabilized values not affected by the disturbance waves. Properties of ripples in crest and base film zones were investigated. It was found that the velocities of both fast and slow ripples normalized by the velocities of disturbance waves are nearly constant for the whole investigated range of conditions. Number of fast ripples generated by a single disturbance wave while it passes 1 meter length was found to be constant as well. Wavelength of ripples on the base film and relative roughness of the base film were also investigated. Average length of crest and back slope of a disturbance wave were found to be decreasing with gas velocity. The distance required for stabilization of velocity and frequency of fast ripples was found to be larger than the length of disturbance wave. Acknowledgements The work was supported by Russian Foundation for Basic Research (Project 13-08-01400a) and Grant Council of President of Russian Federation (Project MK-5997.2014.1) Fig. 18. Average length of crest of a disturbance wave, Lc, (1, 5). Average distances from the beginning of a crest to the point of stabilization of: average film thickness, Lh, (2, 6), velocity of slow ripples, LV, (3, 7) and frequency of slow ripples, Lf, (4, 8). Re = 142, water (1–4 for both (a) and (b)); Re = 350, water (5–8 for (a)); Re = 142, WGS2 (5–8 for (b)).

values and the average length of disturbance waves’ crests, Lc, on gas velocity for different liquid Reynolds numbers and viscosities. All the quantities decrease with gas velocity and slightly grow with liquid viscosity and flow rate. Typical value of Lc changes in the range from 20 to 10 mm from low to high Vg. Lh, which is the most similar to the traditional definition of a disturbance wave’s length, is 2–2.5 times larger than Lc, varying in the range from 50 to 20 mm. LV is slightly larger than Lh, and Lf is larger than LV. Thus, behind the back slope of a disturbance wave, an area exists, in which the velocity and frequency of slow ripples are still different from that on the base film. Length of this ‘tail’ area, Lf  Lh, is smaller than the length of the back slope, Lh  Lc, for low Vg, and larger for high Vg. Length of the whole disturbance wave with its crest, back slope and ‘tail’ is equal Lf, and it varies from 70 to 40 mm from low to high gas velocities. Conclusions Quantitative study of waves of different types existing on liquid film in downward annular gas–liquid flow was performed. The experiments were performed in vertical pipe with inner diameter of 15 mm, for three working liquids with different viscosities, within a wide range of gas and liquid flow rates. High-speed LIF technique was used for field measurements of film thickness with high spatial and temporal resolution. The obtained matrices of film thickness were analysed using new data processing algorithm, which automatically identifies characteristic lines of individual disturbance waves.

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Please cite this article in press as: Alekseenko, S.V., et al. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas–liquid flow. Int. J. Multiphase Flow (2014), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.07.009