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The wakes behind two-dimensional air bubbles
Chemical Engineering Science, 1967, Vol. 22, pp. 1517-1518. Pergamon Press Ltd., Oxford.
Printed in Great Britain.
The wakes bebind two-dimensional air bubbles (Received 2 March 1967; in revisedform 5 April 1967) INTIX~D~C~ON RECENTLYCOLLINS[1] has described experiments conducted in a 3 ft x 3 ft x) in. two-dimensional tank filled with water up which he passed single air bubbles. It was considered that the cylindrical cap bubble moved with a certain body of liquid, this liquid being present as a trailing vortex pair. The combined air and vortex pair system was approximately circular with there being no interchange between the liquid in the vortex pair and the surrounding water. Below the surface of the water a traceline in the water is drawn out into the characteristic drift profile first discussed by DARWIN [2]. When a bubble reached the surface each vortex of the vortex pair moved away from the line of rise of the bubble dissipating energy in the bulk liquid close to the surface. The present work has shown, however, that away from the surface a process of regular alternate vortex shedding occurs in the wake behind two-dimensional bubbles provided the Reynolds Number is sufficiently great. This is akin to the process described by Rowe and P&~~IDGE [3] and ROWE et al [4] for two- and three-dimensional bubbles in fluidixed beds-and very recently by MAXWORTHY[S] for a 2.5 cm3 three-dimensional bubble in water. In the present work the nature of the wake was found to change signiticantly at lower values of the Reynolds number. EWERIMENTALWORK Experiments were conducted in a two-dimensional tank having internal measurements of 8 ft x 3 ft x + in. made from perspex sheeting 3 in. thick. To prevent undue bowing of the tank, 2 in. x+ in. iron bars were clamped back and front spaced at 18 in. intervals up the tank. Air was injected from a graduated syringe through a small orifice into a cup, shaped as a half-cylinder of radius 4 cm, fitted in the liquid space. The air bubble was released by turning the cup with an external handle. Stable single bubbles without satellites of up to 35 cm3 could be readily obtained in this manner. The flow patterns of the liquids were studied using “Merlmaid”, a finely divided reflecting solid having a density of about 1 g/cm3 which remained in suspension forming a translucent liquid. Initial experiments showed the central 3 ft of the tank were free of injection and surface effects. This section of tank was used for all studies and was graduated with centimetre-scale tape. The central section of the tank was illuminated from the front by two top and two bottom photofloods. The movement of liquid following the passage of a bubble was filmed at 4 frames per second, the exposure time used being a millisecond. Experiments were conducted using five waterglycerol solutions having kinematic viscosities from 8.34 cm*/sec (98 % wt. glycerol) to 1.02 x lo-2 cm*/sec (water). For each liquid, air bubbles with volumes 10 cm3 to 35 ems by 5 ems increments were used.
The bubble shape and wake produced by a 35 cm3 bubble in each liquid is shown in Figs. 1-5.t The camera is stationary and the bubbles are moving. The kinematic viscosity of the liquid has a great influence on the nature of the flow in the wake. The flow patterns obtained with a particular liquid but with smaller bubbles were of a closely similar kind to the 35 cm3 ones but of smaller scale, the Reynolds number of the flow not decreasing significantly with reduced bubble volume. The Reynolds number of the flow determines the immediate nature of the two-dimensional wake but then the front and back walls of the tank immediately start to retard the flow in the wake. For this reason detailed velocity measurements in the wakes were not undertaken. Note the change in bubble shape, the maximum to minimum dimension ratio being 1.7 for 98 ‘4 wt. glycerol but 4.0 for water. DIXUSSON In the most viscous liquid (Fig. 1) no vortex formation was observed. A darker line may be observed in the wake along the direct line of motion of the bubble. This is presumably the meeting and continuation of the boundary layer from around the bubble. Faint lines passing roughly round the edge of the bubble base were due to previous experiments and are not relevant to these discussions. As the liquid viscosity is reduced, the dark line becomes more prominent and moves away from the air base (Fig. 2) and eventually a weak systematic vortex pattern is obtained (Fig. 3). Further reduction of the liouid viscosity makes the vor&es yet stronger and larger (F&s-4,5). After-the passage of a bubble the shed vortices increase in radius and decay in strength at a rate dependent on the kinematic viscosity of the liquid [a. The vortex centres move slowly upwards and outwards in this process. With liquids of low viscosity, the decay time is longer and the radial growth greater with separate vortices ultimately merging in the experiments with water to form an apparently homogeneous wake. Following the discussions of ROSENHEAD[7] and GOLDSTEIN [8], the wake may be considered to be separated from the stationary liquid by layers of vorticity, termed vortex layers, originating in the bubble at the separation points. At low Reynolds numbers, these layers join together downstream and vorticity diffuses from this band into the bulk liquid. At higher Reynolds numbers, vorticity is further generated in the boundary layer around this system and it is feasible that a stable vortex pair may form directly behind the bubble. This may well be the situation in Fig. 2. Here the rates of generation and diffusion of vorticity are equal. With further increase in the Reynolds number there t The wake structure in Figs. 1,2 and 3 has been emphasixed with ink to ensure satisfactory reproduction.
1517
Shorter Communications is insufficient area for the generation-diffusion balance to be maintained, the vortices directly behind the bubble grow in size and ultimately one or the other is shed. As overall horizontal momemtum is conserved the liquid still directly associated with the bubble has a velocity in the direction opposite to that of the discharged vortex. Hence a periodic instability is set up with vortices discharging alternately from side to side. It is interesting to note that rotation of the bubble release cap clockwise caused the first vortex to be shed to the left and anticlockwise to the right. This was always reproducible, indicating that the method of bubble generation in liquids of low kinematic viscosity could be determining the wake structure. COLLINS' [l] bubbles, having a Reynolds number the same order of magnitude as that of Fig.. 5, were essentially symmetrically formed and showed a closed wake structure. However, in Collins’ experiments, a gap width of & in. was used (as against + in. in the present experiments). This could be a further stabilizing factor leading to a closed wake structure. At the surface the unequally sized vortex pair is deposited in a manner described by Collins. In passing it may be observed that should such a system of bubble with trailing vortex pair arise as described by Collins the drift profile observed would be much greater than
that of the DARWIN inviscid analysis {2] due to the momentum defect in the liquid following the passage of the bubble, i.e. the bubble experiences a finite drag. J. R. CRABTREE J. BRIDQWATER
Department of Chemical Engineering University qf Cambridge Pembroke Street Cambridge England NOTATION
de
equivalent diameter of a cylinder of air completely iilling the space between the plates, having a volume equal to that of the bubble (=(4V/‘lnt)9 t separation of plates u bubble velocity
Re
Reynolds Number
=$ C
> V bubble volume v kinematic viscosity of the liquid
REFERENCES COLLINSR., Chem. Engng Sci. 1965 20 851. ii; DARWIN C., Proc. Camb. Phil. Sot. 1953 49 342. B. A., Symposium on the Interaction between Fluids and Particles, [31 ROWE P. N. and PARTRIDGE
p, 135, Instn Chem. Engrs London 1962. B. A., CHENEYA. G., HENW~~D G. A. and LYALLE., Trans. Instn Chem. Engrs 1965 43 T271. 141 ROWEP. N., PARTRIDGE Dl MAXWORTHYT., J. Fluid Mech. 1967 27 367. SCHLICHTING H., Boundary Layer Theory, p. 71, McGraw-Hill 1960. L., Advances in Applied Mechanics, VON MISESR. and VON KARM~~N T. Eds., Vol. III, p. 185, Academic :t; ROSENHEAD Press 1953. S., Modern Developments in Fluid Dynamics, Vol. II, pp. 550-557, Oxford University Press 1938. WI GOLDSTEIN
1518
Printed in Great Britain.
The wakes bebind two-dimensional air bubbles (Received 2 March 1967; in revisedform 5 April 1967) INTIX~D~C~ON RECENTLYCOLLINS[1] has described experiments conducted in a 3 ft x 3 ft x) in. two-dimensional tank filled with water up which he passed single air bubbles. It was considered that the cylindrical cap bubble moved with a certain body of liquid, this liquid being present as a trailing vortex pair. The combined air and vortex pair system was approximately circular with there being no interchange between the liquid in the vortex pair and the surrounding water. Below the surface of the water a traceline in the water is drawn out into the characteristic drift profile first discussed by DARWIN [2]. When a bubble reached the surface each vortex of the vortex pair moved away from the line of rise of the bubble dissipating energy in the bulk liquid close to the surface. The present work has shown, however, that away from the surface a process of regular alternate vortex shedding occurs in the wake behind two-dimensional bubbles provided the Reynolds Number is sufficiently great. This is akin to the process described by Rowe and P&~~IDGE [3] and ROWE et al [4] for two- and three-dimensional bubbles in fluidixed beds-and very recently by MAXWORTHY[S] for a 2.5 cm3 three-dimensional bubble in water. In the present work the nature of the wake was found to change signiticantly at lower values of the Reynolds number. EWERIMENTALWORK Experiments were conducted in a two-dimensional tank having internal measurements of 8 ft x 3 ft x + in. made from perspex sheeting 3 in. thick. To prevent undue bowing of the tank, 2 in. x+ in. iron bars were clamped back and front spaced at 18 in. intervals up the tank. Air was injected from a graduated syringe through a small orifice into a cup, shaped as a half-cylinder of radius 4 cm, fitted in the liquid space. The air bubble was released by turning the cup with an external handle. Stable single bubbles without satellites of up to 35 cm3 could be readily obtained in this manner. The flow patterns of the liquids were studied using “Merlmaid”, a finely divided reflecting solid having a density of about 1 g/cm3 which remained in suspension forming a translucent liquid. Initial experiments showed the central 3 ft of the tank were free of injection and surface effects. This section of tank was used for all studies and was graduated with centimetre-scale tape. The central section of the tank was illuminated from the front by two top and two bottom photofloods. The movement of liquid following the passage of a bubble was filmed at 4 frames per second, the exposure time used being a millisecond. Experiments were conducted using five waterglycerol solutions having kinematic viscosities from 8.34 cm*/sec (98 % wt. glycerol) to 1.02 x lo-2 cm*/sec (water). For each liquid, air bubbles with volumes 10 cm3 to 35 ems by 5 ems increments were used.
The bubble shape and wake produced by a 35 cm3 bubble in each liquid is shown in Figs. 1-5.t The camera is stationary and the bubbles are moving. The kinematic viscosity of the liquid has a great influence on the nature of the flow in the wake. The flow patterns obtained with a particular liquid but with smaller bubbles were of a closely similar kind to the 35 cm3 ones but of smaller scale, the Reynolds number of the flow not decreasing significantly with reduced bubble volume. The Reynolds number of the flow determines the immediate nature of the two-dimensional wake but then the front and back walls of the tank immediately start to retard the flow in the wake. For this reason detailed velocity measurements in the wakes were not undertaken. Note the change in bubble shape, the maximum to minimum dimension ratio being 1.7 for 98 ‘4 wt. glycerol but 4.0 for water. DIXUSSON In the most viscous liquid (Fig. 1) no vortex formation was observed. A darker line may be observed in the wake along the direct line of motion of the bubble. This is presumably the meeting and continuation of the boundary layer from around the bubble. Faint lines passing roughly round the edge of the bubble base were due to previous experiments and are not relevant to these discussions. As the liquid viscosity is reduced, the dark line becomes more prominent and moves away from the air base (Fig. 2) and eventually a weak systematic vortex pattern is obtained (Fig. 3). Further reduction of the liouid viscosity makes the vor&es yet stronger and larger (F&s-4,5). After-the passage of a bubble the shed vortices increase in radius and decay in strength at a rate dependent on the kinematic viscosity of the liquid [a. The vortex centres move slowly upwards and outwards in this process. With liquids of low viscosity, the decay time is longer and the radial growth greater with separate vortices ultimately merging in the experiments with water to form an apparently homogeneous wake. Following the discussions of ROSENHEAD[7] and GOLDSTEIN [8], the wake may be considered to be separated from the stationary liquid by layers of vorticity, termed vortex layers, originating in the bubble at the separation points. At low Reynolds numbers, these layers join together downstream and vorticity diffuses from this band into the bulk liquid. At higher Reynolds numbers, vorticity is further generated in the boundary layer around this system and it is feasible that a stable vortex pair may form directly behind the bubble. This may well be the situation in Fig. 2. Here the rates of generation and diffusion of vorticity are equal. With further increase in the Reynolds number there t The wake structure in Figs. 1,2 and 3 has been emphasixed with ink to ensure satisfactory reproduction.
1517
Shorter Communications is insufficient area for the generation-diffusion balance to be maintained, the vortices directly behind the bubble grow in size and ultimately one or the other is shed. As overall horizontal momemtum is conserved the liquid still directly associated with the bubble has a velocity in the direction opposite to that of the discharged vortex. Hence a periodic instability is set up with vortices discharging alternately from side to side. It is interesting to note that rotation of the bubble release cap clockwise caused the first vortex to be shed to the left and anticlockwise to the right. This was always reproducible, indicating that the method of bubble generation in liquids of low kinematic viscosity could be determining the wake structure. COLLINS' [l] bubbles, having a Reynolds number the same order of magnitude as that of Fig.. 5, were essentially symmetrically formed and showed a closed wake structure. However, in Collins’ experiments, a gap width of & in. was used (as against + in. in the present experiments). This could be a further stabilizing factor leading to a closed wake structure. At the surface the unequally sized vortex pair is deposited in a manner described by Collins. In passing it may be observed that should such a system of bubble with trailing vortex pair arise as described by Collins the drift profile observed would be much greater than
that of the DARWIN inviscid analysis {2] due to the momentum defect in the liquid following the passage of the bubble, i.e. the bubble experiences a finite drag. J. R. CRABTREE J. BRIDQWATER
Department of Chemical Engineering University qf Cambridge Pembroke Street Cambridge England NOTATION
de
equivalent diameter of a cylinder of air completely iilling the space between the plates, having a volume equal to that of the bubble (=(4V/‘lnt)9 t separation of plates u bubble velocity
Re
Reynolds Number
=$ C
> V bubble volume v kinematic viscosity of the liquid
REFERENCES COLLINSR., Chem. Engng Sci. 1965 20 851. ii; DARWIN C., Proc. Camb. Phil. Sot. 1953 49 342. B. A., Symposium on the Interaction between Fluids and Particles, [31 ROWE P. N. and PARTRIDGE
p, 135, Instn Chem. Engrs London 1962. B. A., CHENEYA. G., HENW~~D G. A. and LYALLE., Trans. Instn Chem. Engrs 1965 43 T271. 141 ROWEP. N., PARTRIDGE Dl MAXWORTHYT., J. Fluid Mech. 1967 27 367. SCHLICHTING H., Boundary Layer Theory, p. 71, McGraw-Hill 1960. L., Advances in Applied Mechanics, VON MISESR. and VON KARM~~N T. Eds., Vol. III, p. 185, Academic :t; ROSENHEAD Press 1953. S., Modern Developments in Fluid Dynamics, Vol. II, pp. 550-557, Oxford University Press 1938. WI GOLDSTEIN
1518