ISSN 00271349, Moscow University Physics Bulletin, 2012, Vol. 67, No. 4, pp. 391–397. © Allerton Press, Inc., 2012. Original Russian Text © T.O. Chaplina, E.V. Stepanova, Yu.D. Chashechkin, 2012, published in Vestnik Moskovskogo Universiteta. Fizika, 2012, No. 4, pp. 73–79.

The Peculiarities of Admixture Transport in a Stationary Vortex Flow T. O. Chaplinaa, E. V. Stepanovab, and Yu. D. Chashechkinb a

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Faculty of Physics, Moscow State University, Moscow, 119991 Russia Institute for Problems of Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101/1, Moscow, 119526 Russia email: [email protected] Received December 17, 2011; in final form, March 10, 2012

Abstract—We consider the pattern of species transport in a compound vortex flow contacting with a free sur face. The evolution of a compact marker stain (aniline ink or sunflower oil) is traced in spiral arms on the vor tex surface created by a uniformly rotating disk in a cylindrical container. The parameters of the structural elements of currents are determined. The geometry of the spiral flow is structurally stable for a wide range of experimental parameters. Keywords: compound vortex, admixture, spiral arms DOI: 10.3103/S0027134912040054

INTRODUCTION With economic growth and industrial develop ment, the environment becomes more and more pol luted by chemically active substances and compounds, including ecologically hazardous substances that can be found in both the air basin and hydrosphere. Peri odically, industrial accidents occur that involve toxic substances, particularly dioxin (the disasters in Bhopal India (1984) and Seveso Italy (1976), the discharge into the Amur River in China (2005), and the accident at the nuclear fuel processing plant in Sellafield UK (2005)). To control the pollution level and ensure environmental safety (or even to plan an evacuation of the population), it is necessary to evaluate the trans port of substances from a compact source in current

meteorological conditions. This transport is affected by currents in the hydrosphere (an analogue of the wind in the atmosphere), vortices, and waves (Stokes drift). The theoretical and experimental study of the effects of currents (both laminar and turbulent) on the transport of substances is most complete [1]. A theo retical study of the transport of substances in nonlinear waves can be found in [2]. The technological progress in gathering informa tion about remote and largescale objects has made it possible to obtain and build a large number of spectac ular images of vortex flows in the atmosphere and ocean with strong spiral arms that are separated by strips of a base material, as well as images of different space objects with similar features (Fig. 1). Strong spi

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Fig. 1. Images of spiral structures: (a) spiral galaxy NGC 1566 [3] and (b) red spot on Jupiter [4].

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17 6 Fig. 2. General view of the Vortex Flows with Torsion (VFT) experimental setup.

ral structures are detected in both steadystate and nonstationary flows. Theoretical studies of vortex flows are based on the theory of ideal fluids at rest [5, 6] and in a rotating medium [7], using a priori assumptions on the dynam ics of the phenomenon. A large number of works have addressed the distribution of admixtures (assumed to be passive) in vortex flows [8, 9]. Earlier experiments showed that the admixture in globally rotating fluids is concentrated in compact areas that have the charac teristic shape of spiral structures of “paint walls” [10, 11], as presented in a number of publications [12, 13]. In modeling the transport of both soluble (misci ble), and insoluble admixtures (different types of oils [14]), a compact spot on the vortex surface is trans formed into spiral arms that are separated by strips of pure water. This spatial structure is qualitatively con sistent with that observed in natural conditions, whereas laboratory experiments provide a high degree of reproducibility of the initial conditions in the steadystate vortex flow, which allows the experiments to be repeated with suitable parameters. The aim of this study is to record the transport of soluble and insoluble markers from a compact spot in a stationary compound vortex. EXPERIMENTAL APPARATUS A constant vortex flow was created by a rotating disk that was mounted on the bottom of a cylindrical container. To reduce optical distortion, this container was placed in open rectangular basin, 1, of transparent

polymethylmethacrylate made in the form of a paral lelepiped with dimensions 63.6 × 44.6 × 70.0 cm with out an upper edge (Fig. 2). All the edges of the paral lelepiped are made of a 20mmthick organic glass sheet. The basin is enclosed within metal channel frame, 2, and fixed inside with screws. Inside the rect angular basin, cylindrical container, 3, is mounted sta tionary relative to the basin by bailing armature, 4. To retain a constant volume of liquid inside the container, its lower end is placed on rubber ring gasket, 5. The vertical axis of the cylindrical container coincides with the axis of rotation of electric motor arbor, 6. Current inductor (disk), 7, is rigidly fixed on the arbor and together with false bottom, 8, constitutes a plane that limits the given fluid volume from below. The electric motor rotates the inductor uniformly with angular velocities ranging from 200 to 2000 rev/min. The fre quency of disk rotation is measured using an optical sensor connected to digital frequency meter, 9. The basin is filled with settled tap water at room tempera ture up to a preselected level through hose, 10. After the experiment, the basin is emptied through a hole in the lower edge, which is attached to hose, 11, leading to the pump. The working volume of the fluid is lighted by a special projector equipped with a power control device and a diffuser in the form of semitransparent hood, 12. The flow pattern is optically recorded from two dif ferent points. The state of the free surface is recorded by photo or video camera, 14, fixed on wall support, 13, and placed strictly above the cylindrical container. The processes in the bulk of the liquid are recorded by a photo or video camera, 15, fixed on vertical support, 16, which allows the camera to be moved vertically along the entire lateral wall of the basin. The images taken from the cameras are recorded on the hard disk of personal computer, 17. The experimental setup allows for studies in the range of liquid layer depth from 5 to 60 cm and disk radius from 2.5 to 14 cm. The free surface makes it possible to use of a wide set of markers and control the conditions of their use. The insoluble admixture consists of castor and sun flower oils; the soluble markers were aniline inks of different colors and a uranyl solution. FLOW PARAMETERIZATION A simplegeometry experiment is characterized by the emergence of a rather complex flow, including both vortex and wave components in the bulk and on the free surface of the fluid. Due to the noslip condi tion, the uniformly rotating disk spun the fluid around the vertical axis and simultaneously set it aside along its surface to the container wall (Fig. 3). The acceler ated liquid rose along the container walls, moved to the center along the free surface, and was immersed in the vicinity of the rotation axis, thus generating a flow that leaked to the disk center and compensating for the constant transport of material along its surface.

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Directly above the disk surface, the particles rotated and simultaneously moved from the center to the edge. The observed flow pattern can be schematically reduced to a combination of two vortices, one of which is vertically cylindrical (around the vertical axis, with angular velocity ωc), while the other is a toroidal axis, with a circular axis covering the central vertical axis with local angular velocity ωt (Fig. 3). Their joint action led to the formation of a complex flow where the fluid particles moved along spiral and helical tra jectories. The composite flow formed by the superpo sition of two vortices is characterized by the total fre quency ω = ωc + ωt. The complex vortex flow involves different elements, whose scales generate complexes of dimensionless quantities, which characterize the flow in general. In the flow with an instantaneous angular velocity of the fluid ω, which is excited by a disk rotating with angular velocity Ω, the controlling factors are the iner tial scales given by the ratios of gravitational accelera tion to the angular velocity of disk rotation ΔΩ = g/Ω2 and fluid particles Δω = g/ω2, and microscales that characterize singularly perturbed components (Stokes boundary layers) on the disk δΩ = ν/Ω and other contact surfaces δω = ν/ω , including on the free surface. The key dimensionless parameters of the flows are the Reynolds number Re = (R2ω)/ν (as well, the Ekman number Ek = 1/Re is used) and the Froude number Fr = (R2ω2)/gH, which characterize the dynamic conditions of the experiment. The flow of the twolayer medium is also characterized by the Atwood number At = (ρ1 – ρ1)/2(ρ1 + ρ1) and the Bond num ber Bo = gH2(ρ1 – ρ1)/σ, where ρ1 and ρ2 are the den sities of fluid components. The additional dimension less parameters of the problem are determined by the ratios of characteristic linear dimensions: ξH = R0/H (the relative depth of the container) and ξR = R0/R (the relative radius of the inductor). The container is assumed to be shallow for ξH Ⰷ 1 and deep for ξH Ⰶ 1. The flows that were considered in these experi ments were characterized by the dimensionless parameters Re = 500–50000, Fr = –0.5–20, At = ⎯0.009–0.2, and Bo = 1–4.5, which are in the range of variability of smallscale physical processes in the ocean. This fact indicates that at least qualitative sim ilarity exists between the patterns of substance trans port in laboratory and natural conditions [15]. THE TRANSPORT OF A SOLUBLE ADMIXTURE FROM A SPOT ON A COMPOUND VORTEX SURFACE In a series of experiments conducted with a soluble marker, we traced the deformation of the spot on the compound vortex surface into spiral arms [14]. The MOSCOW UNIVERSITY PHYSICS BULLETIN

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Fig. 3. The postulated scheme of a flow that occurs in a cylindrical container under the action of a disc: (a) general view, b) the central section of the free surface of the com pound vortex with wave disturbances.

process of spiral twisting from a drop of aniline blue inks placed on the free surface is shown in Fig. 4. The elongation of the spiral arm occurred in two direc tions: toward the center of the rotating free surface and toward its periphery. At the time t = 1 s (Fig. 4a), the angular size of the colored spiral was 202; the thickness of the arms varied from 0.5 to 3 mm. The spiral center involved a noticeable thickening corresponding to the initial position of the colorant spot on the free surface of the fluid. In 2 s (Fig. 4b), the spiral arms stretched from the admixture drop to fully cover the axis of water surface rotation; the angular size of the spiral was 367°. The radial position of spiral points relative to the center of rotation of the free surface at time t = 3 s varied in the range from 2.2 to 5 cm. The thickness of the spiral arms was from 0.5 to 2.3 mm; the maximum thickness of the spiral arms corresponded to the position of the original spot at the free surface. The spiral structure that stretched from the colorant that fell outside the main spot of admixture began to blur and reverse flow loops emerged. At t = 11 s (Fig. 4c), the angular size of the spiral structure exceeded 800°, the thickness of spiral arms along the entire structure slightly deviated from a mean value equal to 0.7 mm. At the free surface of the fluid, one can clearly see loops in the spiral structure, corresponding to reverse flow areas (Fig. 4c, in the directions “3 o’clock,” “7 o’clock,” and “9 o’clock”). With time, the spiral structure was transformed into a system of nested thin rings (Fig. 4d), the maximum radial position of the points of this system relative to the center of fluid rotation was 0.9 cm. This value is considerably smaller than the original distance from the center of fluid rotation to the colorant spot. At the initial stage of the spiral structure evolution, the form of boundary covered by the spiral was close to elliptical (Figs. 4a, 4b). Here, the ellipse also rotated in the lab oratory system of coordinates. With time, the form of the boundary of the elliptical area covered by spiral No. 4

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Fig. 4. The evolution of a colorant spot on the surface of a rotating fluid (Ω = 100 rev/min, R = 7.5 cm, H = 10 cm): (a) 1 s, (b) 3 s, (c) 11 s, and (d) 60 s.

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Fig. 5. The propagation of colorant in the bulk of a rotating fluid: (a) 18 s, (b) 39 s, and (c) 304 s (H = 30 cm, Ω = 200 rev/min, R = 5.0 cm).

arms was smoothed and its eccentricity decreased. The propagation of the colorant to the periphery of the rotating fluid slowed. The independent transport of the colorant into the bulk of the compound vortex is illustrated in Fig. 5. These photos characterize the experiment conducted with colored water with a watersoluble fluorescent

colorant (uranyl), which emits green light under ultra violet illumination. The penetration depth of the colored central col umn increased with time (Figs. 5a–5c). A part of the marking admixture was immersed in the neighbor hood of the flow axis, forming a colored central col umn with a diameter of 3.6 cm (Fig. 5a), which

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Fig. 6. A compound vortex with a portion of sunflower oil (Vk = 700 ml, H = 40 cm, R = 7.5 cm, Ω = 380 rev/min): (a) photo (side view through the liquid–air surface) and the flow diagram.

decreased with time and was 2.76 cm (Fig. 5c). A part of the marking admixture located on the surface of spi ral arms was immersed along the external cylindrical shell. The propagation rate of the colorant along the external cylindrical surface exceeded its sedimenta tion rate in the central column. Moving from the sur face of the rotating fluid to the vessel bottom (where the rotating disk is located), the colorant propagated in areas with wellrecognizable boundaries. Once the colorant penetrated to the entire depth of the fluid and contacted the rotating disk, the intense mixing of col ored fluid on the disk surface with the bulk of uncol ored fluid in the vortex led to a poorly visible flow pat tern in the experimental setup (Fig. 5c). The photo presents both areas of intensive transport of the mark ing admixture in the bulk of fluid: both the central col ored column and the thin cylindrical shell covering the column. Throughout the experiment, the fluid surface involved a colored spiral structure; the thickness of the nearsurface colored layer was 0.3 cm. The diameter of the cylindrical shell averaged over a series of images was 7.1 cm and the thickness of the walls was 0.4 cm. The experiments that were conducted with color ing agents of different colors (blue and violet aniline inks), which also were put into the flow through the free surface, indicated that the pattern of the vortex flow with added admixture was stable. AN INSOLUBLE ADMIXTURE ON THE SURFACE OF A COMPOUND VORTEX The general structure of the flow was retained when a fluid was used that is immiscible with water (in these experiments, refined sunflower oil) and generated an additional interface. In these experiments, the spot of the marking fluid (sunflower or castor oil) of a given volume Vk was placed on the surface of water at rest. The physical characteristics of the working media were presented in [14]. MOSCOW UNIVERSITY PHYSICS BULLETIN

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In the bulk of the fluid, most of the admixture is collected to the oil body in the vicinity of the axis of rotation (Fig. 6). As in the case of a soluble admixture, the oil on the fluid surface generates spiral arms that are separated by strips of pure water. The geometric parameters of the structure (the size and shape of the arms, the positions of contact surfaces) depend on the radius and angular velocity of the disk rotation, the waterlayer depth, H, and the admixture amount, Vk. The study of the distribution of immiscible admix ture on the surface and in the bulk of the compound vortex yielded data on the position of the admixture both in the bulk and on the free surface of the given vortex flow. The photos of the profile of the compound vortex with the addition of insoluble admixtures (see Fig. 7) illustrate the change in the positions of the air– liquid and oil–water boundaries depending on the amount of admixture on the surface (Vk) and the waterlayer depth, H. At moderate frequencies of disk rotation (Ω ≈ 300 rev/min) on the water surface with the addition of different amounts of oil, a cavity was generated with a depth ht that depended on both the oil volume and waterlayer depth H. For these small rotation frequen cies, the effect of the admixture amount was substan tial. For a frequency of activator rotation of 320 rev/min and an amount of 30 ml of added colored sunflower oil, the deflection in the air–liquid interface was at its maximum; the peripheral part of the free sur face was covered with small drops of oil that stretched in the direction of fluidsurface rotation (Fig. 7a). With an increase in the amount of added oil, the deflection decreased and the area of individual drops of oil that moved along the periphery increased (Fig. 7b). A further increase in the oil volume led to a decrease in the deflection of the air–liquid interface and, in addition, an increase in the oil volume that is not in contract with the central oil body (Fig. 7c). No. 4

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Fig. 7. Forms of the axial cross section of a fluid surface at similar frequencies (R = 7.5 cm): H = 20 cm, Ω = 320, 310, 320 rev/min, Vk = 30 ml (a), 90 ml (b), and 150 ml (c); H = 40 cm, Ω = 300, 320, 260 rev/min, Vk = 30 ml (a), 90 ml (b), and 150 ml (c).

A large depth of fluid also significantly affected the distribution of light admixtures in a compound vortex. A small portion of oil (Fig. 7d) was almost entirely col lected in the vicinity of the axis of rotation; the oil body had a height of 2.3 cm and the air–liquid inter face remained almost flat. The surface periphery was occupied by oil drops. A large amount of oil Vk = 150 ml created an oil body that resembled a hat in its shape, with its edges generated by spiral arms stretched in the direction opposite to the activator rotation (Fig. 7f). The addition of such an amount of oil almost completely eliminated the depression of the liquid–air contact surface. The air–liquid interface was almost entirely composed of oil and was very close to flat. An intermediate amount of oil Vk = 90 ml led to the

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Fig. 8. The evolution of the form of a sunflower oil spot in a compound vortex (H = 20 cm, R = 7.5 cm): (a) Ω = 320 rev/min, Vk = 30 ml, (b) Ω = 310 rev/min, Vk = 90 ml.

deflection of the free surface and drew the oil deeper into the compound vortex (Fig. 7e). On the surface of a rotating fluid with a small amount of added oil (30 ml), the central core (the upper surface of the oil body and the adjacent contin uous oil film) was surrounded by a system of drops that were elongated in the tangential direction (Fig. 8a). The film contour was elongated in the direction of “10–4” h; its maximum and minimum sizes were 5.15 and 4.88 cm, respectively. The container diameter was 29.6 cm, while Fig. 8 shows the central part with a diameter of 17 cm. With increasing angular velocity of the disk rota tion, most of the oil on the free surface was concen trated in the central part and the external edge of the oil film was pear shaped and oriented by its thinning to the bottom. With increasing amounts of oil, the external con tour of the central spot lost its regular form: protru sions, points, and irregularities appeared that gener ated extended spiral arms, some of which were sepa rated from the film center and existed as separate threadlike spiral structures (Fig. 8b). The arms could be split and broken, filling the flow periphery nonuni formly. The characteristic transverse size of the spiral structure was 0.36–0.51 cm. The comparison of forms of the axial cross section of the fluid surface at close frequencies for different amounts of immiscible admixture is shown in Fig. 9. At high frequencies of disk–activator rotation (Ω = 750 rev/min) in a pure liquid (H = 40 cm, R = 7.5 cm), a cavity appeared (curve 1 in Fig. 9) with a depth of ht = 11.8 cm. When 30 ml of sunflower oil was added,

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In all experiments, the admixture could not be assumed to be passive, the displacement of individual strips of admixture, the positions of oil drops, and the orientation of spiral arms did not reflect the flow pat tern in a compound vortex. All the observed flow pat terns were consistently reproduced within the experi mental accuracy. The observed peculiarities of the distribution of the marker that was retained in spiral arms separated by strips of pure water in laboratory vortices were consis tent with flow patterns in natural conditions. As yet, the theory of these processes has not been developed.

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Fig. 9. Forms oil–water and liquid–air interfaces in a compound vortex (H = 40 cm, R = 7.5 cm): (1) pure water Ω = 750 rev/min; (2) Vk = 30 ml, oil body edge, Ω = 770 rev/min; (3) Vk = 30 ml, liquid–air interface, Ω = 770 rev/min, and (4) Vk = 2000 ml, oil body edge, Ω = 720 rev/min.

most of the admixture was concentrated within the central area that was bounded from above by the rota tion surface with a depth of h = 10 cm (curve 2 in Fig. 9), i.e., smaller than the cavity depth in the pure fluid; the oil body height was hk = 7.8 cm (curve 3 in Fig. 9). Although the water surface was covered with a film of oil, the contact line of oil with the water surface was identified rather clearly. When a large volume Vk = 2000 ml of immiscible admixture was added to the surface of stationary water, the experimental setup was characterized by an emerging system of a twolayer fluid with a free oil–air surface (curve 4 in Fig. 9). The height of the rotated oil body was significantly greater than the cavity depth in the pure liquid ht = 17.34 cm (curve 4 in Figure 9). Under these experimental conditions and low fre quencies of activator rotation, the oil–water–air con tact line disappeared. CONCLUSIONS The structural stability of the mechanisms of sub stance transport in compound vortex flows was exper imentally studied. Both soluble colorants and immis cible fluids (oils) generated spiral arms that made up the pattern of substance transport on the rotating sur face of the fluid. The directions of the main flow and the growth of spiral arms were opposite to each other. The growth of spiral arms occurred in the entire range of flow parameters that were considered.

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ACKNOWLEDGMENTS This work was supported by the Ministry of Educa tion and Science of the Russian Federation, state con tract no. 16.518.11.7059. REFERENCES 1. S. V. Alekseenko, P. A. Kuibin, and V. L. Okulov, Intro duction to the Theory of Concentrated Vortices (Inst. Thermophys. Sib. Branch Russ. Acad. Sci., Novosi birsk, 2003) [in Russian]. 2. G. Lamb, Hydrodynamics (Dover, New York, 1945; Gostekhizdat, Moscow, 1947). 3. Spitzer Space Telescope, Znayki Blog. www.znayki. com/blog/archives/16Kosmicheskij_teleskop.html 4. Alexander Koryagin Little Space Page. www.korya gin.newmail.ru/belt_ theory.htm 5. L. M. MilnTomson, Theoretical Hydrodynamics (Macmillan, London, 1960; Mir, Moscow, 1964). 6. K. V. Koshel’ and S. V. Prants, Chaotic Advection in Ocean (Inst. Comp. Sci., Moscow, 2008) [in Russian]. 7. G. M. Reznik, Izv., Atmos. Ocean. Phys. 46, 784 (2010). 8. E. A. Ryzhov and K. V. Koshel’, Izv., Atmos. Ocean. Phys. 46, 184 (2010). 9. V. P. Kukharets, O. G. Nalbandyan, and A. V. Shmakov, Izv., Atmos. Ocean. Phys. 45, 411 (2009). 10. G. I. Taylor, Roy. Soc. Proc. A 100, 114 (1921). 11. R. R. Long, “Fluid,” J. Atmos. Sci. 11, 247 (1954). 12. M. VanDaik, Album of Flows of Liquid and Gas (Para bolic, Stanford, 1982; Mir, Moscow, 1986). 13. G. Batchelor, Introduction to Dynamics of Liquid (Cam bridge Univ. Press, Cambridge, 1967; Mir, Moscow, 1973). 14. T. O. Chaplina, E. V. Stepanova, and Yu. D. Chashech kin, Fluid Dynam. 46, 214 (2011). 15. A. S. Monin and R. V. Ozmidov, Turbulence in Ocean. Oceanology, Physics of Ocean, Vol. 1: Hydrophysics of Ocean (Gidrometeoizdat, Leningrad, 1978) [in Rus sian].

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