Frequency susceptibility of rivulets flowing down vertical
Transcription
Frequency susceptibility of rivulets flowing down vertical
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 Frequency susceptibility of rivulets flowing down vertical plate Sergey V. Alekseenko 1,2, Aleksey V. Bobylev 1, Sergey M. Kharlamov 1, Dmitry M. Markovich1,2* 1: Institute of Thermophysics, Novosibirsk, Russia, [email protected] 2: Novosibirsk State University, Novosibirsk, Russia Abstract: Field measurements of the local thickness of wavy rivulets flowing down a vertical plate were conducted using LIF method. Cases with sufficiently different contact angles were studied. Detailed information on the wave structure of rivulets was obtained for different wave regimes and Reynolds numbers. For small Reynolds numbers, the cross-sectional form of smooth (waveless) rivulets is in good agreement with theoretical predictions. For wavy rivulets, contact boundaries as well as contact angle of rivulets are insensitive to the wave motion. For small contact angle, a significant increase in the area wetted by a rivulet at certain frequencies of wave excitation was found. For large contact angle, all experimentally registered waves were found to be two-humped. 1. Introduction Liquid film flow over inclined or vertical planes, especially with the presence of thermal flux, is often realized in the form of rivulets. For a number of practical applications (heat exchangers, absorbers, distillation columns, coating processes) as well as natural phenomena (lava flows, mudslides, glacier flows, etc) the knowledge about rivulet structure and possibility of its control is rather important. Similar to two-dimensional liquid films (Alekseenko et al. 1996b; Park and Nosoko 2003), the rivulet interface is unstable with wave formation that strongly affects heat and mass transfer intensity through a number of mechanisms. Irregularity and three-dimensional nature of liquid motion considerably complicate the theoretical and experimental analysis of that kind of flow, thus it is described mainly qualitative. Only few works exist on the study of rivulets either over flat or curved surfaces (Towel and Rothfeld, 1966, Alekseenko et al., 1996a, 2007, Holland et al., 2001, Wilson and Duffy, 2002, Carlos et al., 2004, Myers et al., 2004). The value of contact angle and its hysteresis strongly affect rivulet flow regimes (Kim et al., 2004). The aim of the present work is to explore the frequency susceptibility of straight rivulets flowing down vertical plate and to obtain quantitative characteristics of wavy rivulets at moderate Reynolds number of rivulet flow. 2. Experimental setup and measuring technique The test section was represented by a vertical glass plate with dimensions of 200×650 mm, Fig.1. Different polymer coatings on the plate were used to change contact angle of the rivulet flow for the same working liquid. The rivulets were formed by a slot distributor with variable width. Waterglycerol and water-alcohol solutions were used as working liquids. Physical properties of the solution were as follows: for water-glycerol solution - density ρ = 1.04·103 kg/m3, kinematic viscosity ν = 2.4·10-6 m2/s, and kinematic surface tension σ/ρ = 53.9·10-6 m3/s2; for water-alcohol solution - ρ = 0.925·103 kg/m3, ν = 2.65·10-6 m2/s, σ/ρ = 32.9·10-6 m3/s2. Liquid flow rates in the experiments ranged from 0.12 to 4.8 ml/s and Reynolds numbers of investigated rivulets were in the range Re = 7,2 ÷ 45,3 for water-glycerol and in the range Re = 25,5 ÷ 58,1 for water-alcohol. -1- 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 The LIF (Laser Induced Fluorescence) method was used to measure local thickness and wave characteristics of rivulets. The Rhodamine 6G was used as fluorescent dye. Hardware-based technique on standard PIV system “PIVIT” was used. The measuring technique provides spatial resolution of 0.1 mm per pixel and the thickness measurement accuracy of -2 approximately 10 mm. The regimes both with smooth stationary rivulets and with external periodical perturbations of liquid flow rate were investigated. The excitation frequency was varied in the range from 0.5 Hz to 50 Hz. In this range both the regimes with solitary Fig.1 Scheme of experiment and image of wavy rivulet waves of large amplitude and almost sinusoidal high-frequency waves with smallscale amplitude was observed at the same liquid flow rates. 3. Results and discussion The field measurement of rivulet local thickness allowed obtaining detailed patterns of rivulet wave forms for different wave regimes and Reynolds numbers. Wave structure of rivulets was found to be sufficiently different for small and large contact angles. 3.1. Frequency susceptibility of the rivulets with small contact angles will be demonstrated below for the case of water-glycerol solution on a glass surface. In Figure 2, the reconstructed form of rivulet free surface is shown for the smooth rivulet with the contact angle α=5°. As seen in Fig. 2b, the contact angle is constant downstream and the crosssectional form of the rivulet agrees well with theoretical calculation (Carlos et al. 2004). In Fig. 3, correlations between Reynolds number of rivulet flow and volumetric flow rate as well as the diagram of frequency susceptibility of rivulet Fig.2 Reconstructed free surface of smooth rivulet a) and its cross-sections in different flow are shown. As parts of flow b). Black line is the calculation by Carlos et al. 2004. Qliq=1.25 ml/s, seen in Fig. 3а, with Re=22.2 the growth of the flow rate, the rivulet’s Re grows linearly up to the flow rate Q = 3 ml/s. In this range the relation between Q and Re is similar to that for film flow. For greater values of Q the linear dependency does not hold. Experiments with wavy rivulets were conducted only in the region of Q where the linear dependency exists. For such rivulets, susceptibility of the rivulet flow to the external perturbations exists only in the frequency range between bottom (blue) and upper (red) lines shown in Fig. 3b. In the vicinity of the low-frequency boundary, step-like waves with high amplitudes are realized. In the vicinity of high-frequency boundary, almost sinusoidal waves with small amplitudes are observed. In the middle part of the frequency range, there exist regular waves -2- 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 Fig.3 Reynolds number versus liquid flow rate a) and region of frequency susceptibility of rivulets versus Reynolds number b). Contact angle α=5º Fig.4 Reconstructed form of large solitary wave on the free surface of rivulet at excitation frequency F=15 Hz a) and cross-sections for different parts of the wave b). Qliq=3 ml/s, Re=42.6 Fig.5 Reconstructed form of the step-like wave at excitation frequency F=1 Hz. a) and cross-sections for different parts of the wave b). Qliq=0.2 ml/s, Re=7.2 -3- with a well-developed capillary precursor (Fig. 3b). It is interesting that the width and contact angle of rivulets are not sensitive to the phase of passing waves. As an example, in Fig. 4 the 3D form of the a large-amplitude solitary wave (Fig. 4a) and the crosssections of the rivulets for different wave phases (Fig. 4b) are shown. As clearly seen in Fig. 4b, the side walls of the rivulet and contact angle are constant for all phases. Additional peculiarity of the rivulet flow is the existence of stable step-like waves (Fig. 5). As seen in Fig. 5b, the amplitude of the “step” may exceed the thickness of the residue layer several times. The existence of step-like waves is a distinctive feature of rivulets. As know, in film flow excited step-like waves are unstable and quickly disintegrate with the formation of a number of solitary waves (Alekseenko et al., 1994). 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 Fig. 6 illustrates the case that is characteristic for all high-frequency regular wave regimes. For such regimes, the wave amplitude is small and its longitudinal crosssections are close to sinusoidal. With changes in Fig.6 Reconstructed form of sinusoidal regular waves at excitation frequency F=15 Hz excitation frequency a) and cross-sections for different parts of the wave b).. Qliq=0.2 ml/s, Re=7.2 in the region of existence of regular high-amplitude waves a change in the width of the rivulets downstream was observed. The 3D form and cross-sections in the upper and lower parts of the flow for one of such regimes are shown in Fig. 7a-c. It is seen that the boundary location and rivulet contact angle are insensitive to the phase of the passing wave. At the same time, the rivulet width downstream is sufficiently higher than that Fig.7 Reconstructed form of developed regular waves at excitation frequency F = 23 Гц a) and its cross sections for crest and trough of the waves in the upper b) and lover c) region of the flow. Qliq=1.25 ml/s, Re = 22.2 upstream (22 and 18 mm, respectively, for the above case). For the same regime, longitudinal crosssections are shown in Fig. 8 and the values of the wave’s amplitude downstream of the given regimes – in Fig. 8b. As seen in Fig. 8b, in this case the amplitude of waves grows nonmonotonically downstream, which is apparently due to the rivulet widening down-stream. In Fig. 9 the dependence of the rivulet width in the bottom region of rivulet flow on the excitation frequency is shown. The width Fig.8 Transversal cross sections of developed regular waves at excitation frequency of smooth rivulet was F = 23 Гц a) and amplitudes of the waves down stream b). Qliq=1.25 ml/s, Re = 22.2 taken for 100%. As -4- 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 Fig.9 Related width of rivulet in dependence of excitation frequency. The width of smooth rivulet accepted as 100%. Qliq=1.25 ml/s, Re = 22.2 follows from the Fig. 9, the rivulet width and consequently the area wetted by rivulet grow with the growth of frequency and reaches a maximum at F = 43 Hz. After that with further growth of excitation frequency the rivulet width sharply decreases and practically drops to the initial value of smooth rivulet width at F=50 Hz, which corresponds to the upper boundary of the rivulet frequency susceptibility range. 3.2 The case of large contact angle For large contact angle, the rivulet wave structure is fundamentally different from that for small angles. In the region of rivulet frequency susceptibility, no step-like or sinusoidal waves were observed. For all frequencies of excitations within the region of susceptibility the developed waves have a distinct two-humped form. Characteristic properties of wave rivulets with large contact angle will be described below by the example of the rivulet with the contact angle 23º (in this case ethanol-water solution was the working liquid and fluorocarbon polymer coating on the glass plate was used). The form of smooth rivulet is still in good agreement with theoretical calculations. Similarly to the case of small contact angles, the side walls of the rivulet are also insensible to the phase of passing waves. The 3D form of the waves at different excitation frequencies is shown in Fig. 10 and longitudinal cross-sections for this flow regime at different excitation frequencies are shown in Fig. 11. As can be seen, at all frequencies of excitation the waves have a very similar twohumped form with a high-frequency front and low-frequency back humps. At low excitation frequencies, the amplitude of the front hump is sufficiently higher than that of the back one. With the growth of excitation frequency, the amplitude of the humps decreases at different rates and the humps become almost equal in amplitude near the upper frequency boundary of the existence of excited waves (Fig. 12). It is interesting that two-humped waves of similar shape were observed earlier for the rivulets flowing down the outer part of an inclined cylinder (Alekseenko et al., 1996a). The results of these authors obtained with the use of the shadow method are presented in Fig. 13. With the growth of Re, the form of the waves at all excitation frequencies is similar to that Fig.10 Reconstructed form of excited waves in the case of big contact angle at different excitation frequencies. Qliq=0.12 ml/s, Re = 25.5: a) F=7 Hz; b) F=17; c) F=28 Hz -5- 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 Fig.11 Profiles of waves at different excitation frequencies. Qliq=0.12 ml/s, Re = 25.5 described above (Figs. 14-16) with the only difference that the back hump at higher Re becomes less pronounced. 4. Conclusion Field measurements of the local thickness of wavy rivulets flowing down a vertical plate were conducted using LIF method. Cases with sufficiently different contact angles were studied. Detailed information on the wave structure of rivulets was obtained for different wave regimes and Reynolds numbers. It was found that general property of rivulet flow is insensitivity of contact angle of rivulets to the wavy motion at all contact angles. At the same time wave structure of rivulets was found to be sufficiently different for small and large contact angles. -6- 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 Fig.14 Profiles of waves on free surface of rivulet. Qliq=0.25 ml/s, Re=34.5 Fig.12 Amplitude versus frequency of excitation Qliq=0.12 ml/s, Re = 25.5. Fig.15 Profiles of waves on free surface of rivulet. Qliq=0.5 ml/s, Re=42.4 Fig.16 Profiles of waves on free surface of rivulet. Qliq=1 ml/s, Re=58.1 For small contact angles a wide variety of wave patterns in dependence on excitation frequency is observed. In this case the range of frequencies was observed when the amplitude of excited waves was maximal and comparable with the rivulet maximal thickness to all studied flow rates. For high frequencies of excitation, the small amplitude waves with sinus-like shape were observed. For frequencies in the vicinity of lover limit of the frequency susceptibility of rivulets step-like waves with high amplitude are formed. Also for the case of small contact angles existence of narrow range of excitation frequencies was found inside which considerable increase of rivulet width (without Fig.13 Profiles of stationary exited waves on the changing of contact angle), and consequently inclined cylinder at α=15º, Q=0.82 ml/s, ethanol. increase of wetting area occurs. Alekseenko et al., 1996a For large contact angles, the rivulet wave structure is fundamentally different from that for small angles. In this case, in the region of frequency susceptibility of rivulets, no step-like or sinusoidal waves were observed. Instead, for all frequencies of excitations within the region of susceptibility the developed waves are similar and have a distinct two-humped form. -7- 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 Acknowledgements 08-01501-a. This work is supported by RFBR Foundation, grants N 06-01-00762-a, 06- References Alekseenko SV, Antipin VA, Bobylev AV, Markovich DM (2007) Application of PIV to velocity measurements in a liquid film flowing down an inclined cylinder. Experiments in Fluids 43:197207 Alekseenko SV, Markovich DM, Shtork SI (1996a) Wave flow of rivulets on the outer surface of an inclined cylinder. Phys Fluids 8:3288–3299 Alekseenko SV, Nakoryakov VE, Pokusaev BG (1994) Wave Flow of Liquid Films. Begell House, New York Alekseenko SV, Nakoryakov VE, Pokusaev BG (1996b) Wave effect on the transfer processes in liquid films. Chem. Eng. Comm 141(142): 359-385 Carlos A, Perazzo A, Gratton J (2004) Navier-Stokes solutions for parallel flow in rivulets on an inclined plane. J Fluid Mech 507:367–379 Holland D, Duffy BR, Wilson SK (2001) Thermocapillary effects on a thin viscous rivulet draining steadily down a uniformly heated or cooled slowly varying substrate. 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