E.V. Stepanova, T.O. Chaplina, and Yu.D. ChashechkinReceived February 10, 2010
Abstract—The oil redistribution pattern in a compound vortex created by a uniformly rotating disklocated on the bottom of a cylindrical tank is investigated. At the initial instant, a round spot of a light-weight oil is deposited on the free surface of the liquid at the center of the tank. In the course of theformation of the compound vortex, the oil is partially entrained into the interior of the liquid and formsa body of revolution. On the free surface, the spot loses the round shape, spiral branches stretch outfrom the appeared protrusions and cusps. The orientation of these branches is opposite to the directionof fluid rotation in the tank. Geometrical parameters of the structures for different flow regimes arestudied.
In recent years, the interest in studying the vortex structure and dynamics, the traditional subject offluid dynamics, increased due to the environmental problems and the development of new visualizationtechniques which make it possible to register the fine structure of the velocity field in the laboratory andfull-scale experimental conditions [1].
The remote-control methods make it possible not only to visualize the flow but also to detect the leak-ages and dispersions of hydrocarbons, which create the increasing hazards to the ecology of the entirehydrosphere and, particularly, the World Ocean due to the increasing volumes of offshore oil productionand transportation. The scales of accidents increase with increase in the tonnage of the tankers, the wellproduction rates, and the dimensions of the pipe systems.
The observations demonstrated that the oil leaking out of compact sources of natural (deposits under theocean bottom) [2] or technological (tankers, oil platforms) [3, 4] nature is accumulated on the ocean surfacein thin long bands which may have both irregular and regular arc shape. Of interest is the modeling of theflows of immiscible liquids and the research of the mechanisms of the formation of fairly narrow oil bands(50–100 m long) in the form of regular arrays, which were observed in the Gulf of Mexico [2].
In laboratory conditions, the transformation of a color spot into spiral structures in a globally rotatingliquid was observed in the experiments [5] with a soluble dye at the beginning of the XXth century. Theseexperiments were repeated in [6]. The extrusion of spiral branches from a compact spot of a soluble dye ormarker admixture mixing with the bulk flow was observed on the surface of a compound vortex created in acylindrical tank by a uniformly rotating disk [7]. Moreover, the compact spot of an immiscible liquid (castoror sunflower oil) with a smooth contour was transformed into a polygon, from the corners of which spiralbranches stretched out [8]. The transport of floating oil spots on the surface of vortex flows has not beensystematically investigated so far. The aim of the present study is to visualize the oil redistribution patternon the surface of a compound vortex induced by a disk rotating on the bottom of a cylindrical tank.
1. PARAMETRIZATION OF THE FLOW
The vortex flow in the tank deforms the free surface whose shape reflects the pressure distribution in thefluid. On the deformed free surface, unsteady disturbances may appear, which are attributable to the coupledaction of the forces of different nature: the capillary, gravity, and centrifugal forces. Their contributions
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Table 1
Physical parameter at T = 18∘C Water Sunflower oil Castor oil
depend on the kind of the liquid and the values of all parameters of the problem, such as the depth of theliquid layer H , the tank radius R0, the radius of the disk R and its angular velocity Ω, the gravity forceacceleration g, the density ρ , the kinematic viscosity ν , and the surface tension coefficient σ of the fluid.
The ratios of these parameters form a set of nondimensional parameters, which includes the Reynoldsnumber Re = (R2Ω2)/ν , the Froude number Fr = (R2Ω2)/gH , and (for a two-layer medium) the Bondnumber Bo = gH2(ρ1 − ρ2)/σ , where ρ1 and ρ2 are the densities of the liquids.
2. EXPERIMENTAL TECHNIQUE
In our experiments, we studied the transport of light-weight oil on the surface of water entrained in thevortex flow. The main working medium was a tap water, preliminary degasified to decrease the number ofgas bubbles deposited on the set-up walls and making the visualization more difficult.
A spot of the marker liquid of a given volume (30, 60, 90, and 120 ml) was preliminary deposited on thesurface of a quiescent fluid. As the immiscible marker, we used the unsaturated fatty acids, i.e. the castorand purified sunflower oils. The physical parameters of the oils are given in Table 1. The sunflower oil isthe most light-weight one and has the minimal value of the surface tension. The castor oil is more viscousand dense.
The scheme of the set-up is shown in Fig. 1a. The vortex flow was induced by a disk located on the bottomof open cylindrical tank 2 with diameter 29.4 cm. To decrease the optical disturbances of the recorded flowpattern, we put reservoir 2 in the interior of open basin 1 with plane walls, measuring 63.6× 44.6× 70.0cm. On the bottom of the tank, working disk 3 2 mm thick was located. In some experiments, for levelingthe bottom plane, on the level of the upper edge of the disk false bottom 4 was mounted. The disk was set inmotion by means of electric motor 6 with the frequency from 200 to 1200 rev/min, the work of which wascontrolled by block 7. On the axis of the electric motor, disk-mask 5 of the angular-frequency meter waslocated. The angular velocity was controlled by means of optical gage 8 and signal converter block 9.
The photo- and videorecording of the flow pattern were performed simultaneously from the top andside by apparatus 10. The control of the experiment and the recording of the data were performed usingcomputer 11. The oil spot was placed at the center of the quiescent fluid by means of a dosing pipet ormeasuring can 12.
The liquid in the tank was illuminated by white-light source 13 with scattering screen 14 or ultraviolet-light lamp 15. The tank was supplied with hydraulic system 16.
In all experiments, the water layer was 40 cm deep and the disk radius was 7.5 cm. The conditions ofthe illumination and the location of the cameras were chosen so that, in the subsequent processing of theimages, all details of the liquid free surface, in particular, the interfaces between the media (water, oil, andair) were clearly visible.
The rotation of the disk started after the suppression of all disturbances accompanying the depositionof the oil spot. The recording of the flow pattern began after the steady-state regime was attained andall visible transition processes, in particular, oil redistribution and the cavity surface formation, finished.A 20 min pause was also held out after the variation of the rotation frequency.
The complex flow in the tank, the schematic pattern of which is shown in Fig. 1b, contains both thevortex and wave components.
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Fig. 1. Block-scheme of the experimental set-up (a) and the scheme of the compound-vortex (b) induced by the rotatingdisk in the cylindrical tank.
The uniformly rotating disk swirls the liquid about the vertical axis and rejects it towards the tank wall.The accelerated fluid rises along the tank walls, is displaced towards the center along the free surface, andsinks near the rotation axis, forming an inflow which compensates the constant transport of the substancealong the disk surface (Fig. 1b). Due to the no-slip condition on the surface of the rotating disk, the liquidin the tank is entrained in the vortex flow about the vertical axis. As a result, a compound vortex is formedin the tank, which has a nonuniform angular velocity distribution and creates the surface cavity.
The observed flow pattern in the interior of the water is schematically reduced to a combination of twovortices, one of which is vertical and cylindrical with the angular frequency ωc, and the second vortex istoroidal, with a circular axis embracing the central axis. As a result of their coupled action, a compound vor-tex is formed, in which the liquid particles move along spiral and helical trajectories with the characteristicfrequency ω = ωc + ωt .
The location of the circular rotation axis depends on all parameters of the problem (in particular, theradii R, R0, the height H , and the frequency Ω). When a constant angular velocity of the disk is maintained,in the pure water a steady-state flow is observed, in which the periodic components, for example, inertial,gravitational, and capillary waves may be present. The calculations of the shape of the pure-water freesurface for different values of parameters are given in [9].
The vortex flow, which near the surface is directed from the walls towards the axis, transports the oil tothe cavity center and entrains it into the interior of the working liquid, where for a wide range of governing
Fig. 2. Composite vortex with a portion of castor oil (Vk = 30 ml, Ω = 640 rev/min): photograph (side view) (a) and theflow scheme (b).
parameters the oil takes the shape of an elongated body of revolution. A portion of the oil retains on the freesurface, whose shape also depends on all parameters of the problem, including the volume of the immiscibleliquid. For moderate angular velocities of the disk (Ω < 750 rev/min), the flow pattern is stabilized in10–14 min.
3. THE DISTRIBUTION OF THE OIL IN THE FLUID INTERIOR
The pattern of the oil distribution in the interior of the compound vortex and the notation for the maingeometrical parameters of the vortex are given in Fig. 2a, 2b. On the surface of the rotating water, a cavityis formed, with the maximal depth located at the cavity center. The main part of the oil is accumulated nearthe central vertical axis in a compact volume which has the shape of a body of revolution, adjacent to thecavity bottom. Here, H is the maximal thickness of the working liquid layer (measured near the wall), ht
is the difference in the heights of the free surface near the tank wall and the lower edge of the oil body, hk
is the height of the oil body of revolution at the center, h = ht − hk is the air-water interface deflection
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Fig. 3. Flow pattern with the compound vortex for a small rotation velocity of the activator (side view): (a) pure water,Ω = 377 rev/min; (b), (c) flow with the added castor oil, Vk = 30 ml and 60 ml, Ω = 250 rev/min.
Fig. 4. Central sections of the compound-vortex cavity: (1) pure water, Ω = 377 rev/min; (2), (3) oil body water interface,Ω = 240 rev/min, castor oil Vk = 30 and 60 ml.
indicate, and Rk is the radius of the contact line between the oil body and the water on the free surface.In the experiments, al contact lines, i.e. water-oil, oil-air, and water-air, were distinguished.
For the given conditions, the main portion of the castor oil from the lens-shaped spot on the surface ofthe quiescent water, with a characteristic transverse dimension 3.5 cm and about 4 cm thick, is accumulatedin an oil body with the height hk = 4.94 cm, rotating together with the ambient liquid. The volume of the oilbody (Fig. 2a) is Vk ≈ 29.5 ml, ht = 10.91 cm, Rk = 2.76 cm, and h = 5.97 cm. A small portion of the oil(Vk ≈ 0.5 ml) retains in a thin layer on the cavity surface and forms spiral branches which will be describedbelow.
On the surface of the fluid with the compound vortex, the central part of which is formed by the oilbody, there is a specific line, the water–oil interface. When this line is located inside the hollow of the freesurface, near this line the slopes of the tangents to the lateral surface of the oil body and the water surfaceare different.
For moderate angular frequencies of the disk (Ω = 377 rev/min), on the pure-liquid surface a cavity withthe height h = ht = 3.2 cm is formed, with the cavity walls remaining smooth (Fig. 3a). A small amount ofthe castor oil (Vk = 30 ml) is almost completely accumulated in this cavity with hk = 1.6 cm. The liquid-airinterface (the outer part is the water–air interface, and the central part the oil–air interface) remains almostflat (Fig. 3b).
When a larger volume of the castor oil (Vk = 60 ml, Fig. 3c) is added, the dimensions and the shape of theoil body of revolution (ht = 1.7 cm) change; however, its lateral surface and the lower edge remain smooth(symbols 1–3, Fig. 4). Here, symbol 1 corresponds to the liquid–air interface, a part of which may consistof two segments, and symbols 2–3 correspond to the oil–water interface in the fluid interior.
The shapes of the sections of the cavity for the pure liquid and for one or two added portions of the castoroil, 30 ml each, for small angular frequencies of the liquid are shown in Fig. 4. The comparison of the dataindicate that the cavity with smooth boundaries, formed in the pure water (Figs. 3a and 4a, symbol 1), turns
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Fig. 5. Flow pattern with the compound vortex for a moderate rotation velocity of the activator (Ω = 500 rev/min): (a) purewater; (b), (c) flow with the added castor oil, Vk = 30 ml and 60 ml.
Fig. 6. Central sections of the compound-vortex cavity (Ω = 500 rev/min): (1) pure water; (2), (3) water-castor oil inter-faces, Vk = 30 and 60 ml; (4) liquid-air interfaces, Vk = 60.
out to be completely filled with a single portion of the oil (Fig. 3b and 4, symbol 2). The addition of onemore portion of the oil results in the increase in the height of the oil body and the contraction of its upperpart (Figs. 3c and 4, symbol 3.
With increase in the angular velocity of the disk Ω), in the pure water small nonuniformities appear onthe cavity boundaries (Fig. 5a). They are manifested in the distortions of the free-surface shape, boundariesof the light spot at the cavity center, and the image of its spectacular reflection (a dark shadow above thecavity).
A small amount of the castor oil (Fig. 5b) is accumulated in the body of revolution near the central verticalaxis with the height hk = 3.4 cm, which is smaller than the cavity height in a pure liquid (ht = 4.1 cm). Withincrease in the oil volume (Fig. 5c), the deflection of the free surface almost completely restores (h = 3.2 cm,which is only by 0.9 cm smaller than for the cavity in Fig. 5a). The lower edge of the oil body is located onthe depth ht = 8.7 cm.
For a moderate frequency Ω, the central sections of the cavity in the pure liquid (Fig. 6, symbol 1) and thelocations of the oil–water interface (Vk = 30 ml of the castor oil (symbol 2)) almost coincide. With increasein the volume, the oil body is entrained into the interior of the water, its boundary is shown by dots 3. Theshape of the free surface (Fig. 6, symbol 4), the central part of which is formed by the oil body and the outerpart by the water, turns out to be similar to the cavity shape in the pure water (Fig. 6, symbol 1), but thecavity is more shallow (h = 3.2 and 4.1 cm, respectively).
For high frequencies Ω (Fig. 7a), in the uniform liquid two kinds of disturbances appear on the cavitysurface (h = 11.4 cm): large-scale (inertial waves) and small-scale (spiral disturbances). The liquid activelytravels along the free surface and entrains the gas bubbles towards the cavity center. In Fig. 7a, these bubblesare clear visible in the neighborhood of the rotation axis. The spiral waves are present on the entire cavitysurface, while the inertial waves are observed in its lower part.
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Fig. 7. Flow pattern with the compound vortex for a high rotation velocity of the activator: (a) pure water, Ω = 750 rev/min;(b), (c) flow with the added castor oil, Vk = 30 ml and 60 ml, Ω = 730 and 750 rev/min.
At a high frequency, a small amount of the castor oil reduces noticeably the free-surface deflectionindicator, which amounts to only 2.9 cm (Fig. 7b). A portion of the oil is accumulated in the central oil bodywith ht = 4.4 cm, and the other portion retains on the free surface in the form of individual droplets. In theliquid with the oil body at the center, the cavity surface remains smooth (Fig. 7b, 7c). The lateral surface ofthe oil body forms an acute angle with the free surface. On the specific line Rk = 1.56 cm, the angle betweenthe tangents to the oil body and the free surface is α = 27∘ (Fig. 7b).
After the addition of two portions of the castor oil, the free-surface deflection indicator decreases to2.9 cm, and the distance from the lower edge of the oil body to the marginal point of the free surface isht = 11.0 cm (Fig. 7c). The comparison of Figs. 3, 5, and 7 indicates that with increase in the rotationvelocity of the disk the effect of the oil on the free-surface shape increases.
For Ω = 750 rev/min, the cavity in the pure liquid has the maximal depth (Fig. 8, symbol 1, ht = 11.4 cm).The height of the oil body increases with increase in its volume, and under the conditions considered itremains smaller than the cavity height in the pure liquid (symbols 2 and 4). The free-surface deflectionindicator is almost independent of the oil volume (h = 2.8 and h = 2.9 cm, symbols 3 and 5).
The shape of the section of the oil body of revolution near the vertical axis of the tank is approximated bythe function h = ArB (Fig. 8). When one portion of the oil is added, A = 0.26 ± 0.06 and B = 1.91 ± 0.12(Fig. 7b); when two portions of the oil are added, A = 1.09 ± 0.49 and B = 2.55 ± 0.07 (Fig. 7c). Themaximal deflection depths of the free surface and the oil–water interface are given in Table 2.
The outer part of the cavity ia approximated by the curve h = D ln(r − C), where r is the radial coordinatemeasured from the rotation axis, C =−1.21 ± 0.43, D = 1.53 ± 0.07.
With the further increase in the angular velocity of the disk, the oil body is elongated in the verticaldirection and its lower edge oscillates irregularly. When the oil body touches upon the disk, a muddy water-oil suspension is created, which makes difficult the further observation, and the experiment finishes.
4. OIL DISTRIBUTION ON THE COMPOUND-VORTEX SURFACE
Same as in the case of a miscible liquid [10], a smooth oil spot in the center of the compound-vortex cav-ity is deformed into an asymmetric structure, from which spiral branches are stretched out. The dimensionsof the branches and their formation rate depend on all parameters of the experiment.
The successive photographs illustrating the evolution of the castor-oil spot for a low angular velocity ofthe disk are shown in Fig. 9. The oil distribution on the surface of a quiescent liquid is shown in Fig. 9a.
With the rotation, the oil spot loses the round shape, protrusions (Fig. 9b, in the angular direction “on9h”) and branches appear, which are separated from the main volume by the contact line (about 11 long and2.0 cm wide).
In this regime, the oil distribution pattern on the free surface is nonstationary, from the protrusion edgesthin spiral branches are shed episodically (Fig. 9c). In a short time, the branch formed may completely mergetogether with the central spot which becomes smoother. The end of the branch may be sharp or rounded, with
Fig. 8. Central sections of the compound-vortex cavity: (1) pure water, Ω = 750 rev/min; (2), (4) water-castor oil interfaces;(3), (5) liquid-air interfaces, Vk = 30 and 60 ml, Ω = 730 and 750 rev/min.
Table 2
Ω, rev/min Vk, ml h, cm ht , cm
377 0 0(2.51) 2.51
500 0 0(5.27) 5.27
750 0 0(12.30) 12.30
250 30 2.53 2.53
500 30 3.92 3.92
730 30 3.56 8.06
240 60 3.83 3.83
500 60 4.31 10.19
750 60 4.12 12.19
the liquid at the leading edge of the branch being accumulated in an elongated droplet (Fig. 9c). The lengthof the spiral branch increases with time and to the instant t = 143 s attains l3 = 12.83 cm, τ3 = 0.12 cm. Inthe region of contact of the branch and the central oil spot, a protrusion with the height a3 = 1.25 cm andthe width b3 = 4.75 cm is observed.
A thin long branch is located equidistantly to the spot edge and is separated from the spot by a pure-liquidband with average width d = 0.78 cm. The number, location and dimensions of individual branches varywith time. For the same flow regime, we observed simultaneously two and three branches (“triple spiralstructure”, Fig. 9e). No matter what the number of the branches, between the central spot and the branchesa pure-liquid layer retains.
With increase in Ω, on the compound-vortex surface the velocity of the flow directed from the wall to thetank center increases. Thus, the flow prompts the accumulation of the oil near the rotation axis (Fig. 10a).
The free surface becomes nonuniform. At the cavity center, the oil body surface is located, whichembraces the vertical rotation axis (Fig. 10b). On the surface of this body, a specific system of small-scale spiral waves with a characteristic step of 0.22 cm is visible (Fig. 10b). Near the oil body, individualoil droplets appear, which form spiral structures. At the outer edges of the branches, the castor oil isaccumulated in the droplets (with the size ranging from 0.25 to 1.15 cm) and elongated spiral branches(with the characteristic thickness from 0.01 to 0.3 cm and the length from 1 to 5.6 cm) (Fig. 10a).
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Fig. 9. Evolution of the shape of the castor-oil spot on the compound-vortex surface (Ω = 250 rev/min, Vk = 120 ml):(a)–(d) t = 0, 1, 143, and 147 s.
Fig. 10. Shape of the castor-oil spot on the surface and in the interior of the compound vortex (Vk = 30 ml, Ω =510 rev/min), (a), (b) top and side view.
The qualitative features of the flow pattern in the compound vortex retain when the castor oil is replacedby the more light-weight and less viscous sunflower oil. In this case, the initially round spot on the surfaceof the compound vortex also acquires the angular shape (the corner points of the contour almost coincidewith the apexes of a right triangle, Fog. 11a). The center of the triangle is situated in the neighborhood ofthe rotation axis, the location of which is easily determined from the set of normals to thin light lines inFig. 11a, which are the trajectories of small gas bubbles fixed on the water-oil interface. The lengths of theleft, right, and bottom sides of the triangle are equal to 7.6, 7.7, and 8.8 cm, and the angles of the oppositeapexes are equal to 55.3, 61.2, and 63.5∘.
The long-term rotation results in the fact that some protrusions at the spot edge grow and are transformedinto thick spiral branches (Fig. 11b) (the dimensions are l4 = 3.48 cm, τ4 = 1.1 cm, and l5 = 2.45 cm),τ5 = 1.62 cm for the left and right branches, respectively). From the edges of the branches, the round (“on5h”, “on 10h”) and oval droplets separate, with the dimensions of 2.15 and 2.43 cm, respectively.
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Fig. 11. Shape of the sunflower-oil spot on the compound-vortex surface (Ω = 120 rev/min, (a), (b)) t = 1, 10 s, Vk = 30and 90 ml.
Fig. 12. Shape of the sunflower-oil spot on the compound-vortex surface (Ω = 120 rev/min, Vk = 120 ml: (a), (b))t = 1, 10 s.
In the initial stage of rotation, the oil spot of a larger volume acquires a more irregular shape, with irregu-lar protrusions, from which spiral bands 0.39 cm wide are stretched away. These bands end by droplets withthe sizes ranging from 0.22 to 1.35 cm (Fig. 12a). With time, the spot is transformed into a round structurewith two symmetric spiral branches. The droplets with diameters from 2.13 to 2.43 cm are separated fromthe central spot and slowly travel towards the tank wall (Fig. 12b). The motion of the droplets is irregular,they may merge together with the branches and the main spot.
Summary. The experiments performed indicate that, as in the case of a miscible admixture, the dynamicsof the transport of oil deposited on the surface of a compound vortex corresponds to the behavior of an activeadmixture. The structural properties of the distribution patterns of individual oil droplets and the orientationof spiral branches differ from the main flow scheme.
Due to the rotation of the disk, in the fluid interior the oil is accumulated in a body of revolution em-bracing the vertical axis of the tank. The location and the shape of the oil body are determined by thebalance between the buoyancy forces, ejecting the light-weight oil, and the drag force attributable to themain circulatory flow in the compound vortex.
The surface of the oil spot may be smooth or be perturbed by spiral waves, which are transformed intospiral branches on the flow periphery. The main flow direction and the orientation of the oil branches onthe free surface are opposite. The experiments demonstrated that the addition of even a small volume of oil
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(30 ml per the working volume of 54 l) significantly changes the vortex flow pattern and the shape of theentire free-surface.
The work was supported in part by the RF Ministry of Education and Science (State contractNo. 02-518.11.11.7157), the Russian Academy of Sciences (Program of the Presidium of RAS P-20 “Fun-damental problems of oceanology: physics, geology, biology, ecology”, Program of DMEMCP RAS “Me-chanics of nonuniform liquids in the fields of external forces”), and RFBR (No. 08-05-00473).
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