This paper presents results from a flow visualization study of the wake
structures behind solid spheres rising or falling freely in liquids under the
action of gravity. These show remarkable differences to the wake structures
observed behind spheres held fixed. The two parameters controlling the rise or
fall velocity (i.e., the Reynolds number) are the density ratio between sphere
and liquid and the Galileo number.Comment: 9 pages, 8 figures. Higher resolution on demand. To appear in
  Nonlinearity January 200

Biesheuvel, A.

Lohse, Detlef

van Wijngaarden, L.

Veldhuis, C.H.J.

English

arXiv

This paper presents results from a flow visualization study of the wake structures behind solid spheres rising or falling freely in liquids, under the action of gravity. These show remarkable differences to the wake structures observed behind spheres held fixed. The two parameters controlling the rise or fall velocity (i.e. the Reynolds number) are the density ratio between the sphere and liquid and the Galileo number

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Motion and wake structure of spherical particles

University of Twente Research Information

INSTITUTE OF PHYSICS PUBLISHING NONLINEARITYNonlinearity 18 (2005) C1–C8 doi:10.1088/0951-7715/18/1/000COVER ILLUSTRATIONMotion and wake structure of spherical particlesChristian Veldhuis1, Arie Biesheuvel2, Leen van Wijngaarden1 andDetlef Lohse1,31 Department of Applied Physics, University of Twente, Enschede, The Netherlands2 Department of Mechanical Engineering, University of Twente, Enschede, The NetherlandsE-mail: d.lohse@utwente.nlReceived 21 October 2004Published 13 December 2004Online at stacks.iop.org/Non/18/C1AbstractThis paper presents results from a flow visualization study of the wake structuresbehind solid spheres rising or falling freely in liquids, under the action of gravity.These show remarkable differences to the wake structures observed behindspheres held fixed. The two parameters controlling the rise or fall velocity(i.e. the Reynolds number) are the density ratio between the sphere and liquidand the Galileo number.PACS numbers: 47.32.-y, 47.54.+r, 47.27.-i1. IntroductionThis year’s cover illustration shows the boundary layer separation from a light sphere rising inquiescent water at high Reynolds number. The wake structure behind the sphere is visualizedwith the Schlieren technique. The experiment is part of a project to study wake structuresbehind falling and rising spherical particles in a large range of Reynolds numbers.In the past years, extensive numerical investigations [1–5] have established how the wakeof a sphere held fixed in a uniform flow undergoes a series of transitions as the Reynoldsnumber Re = Ud/ν is increased. Here, U is the free stream velocity, d the diameter ofthe sphere, and ν is the kinematic viscosity of the water. It was found that the wake isaxially symmetric up to Re = 212. Above this value, a planar-symmetric wake is found thatconsists of two steady counter-rotating threads. At Re ≈ 270 there is a further transitionand the planar-symmetric flow becomes time-dependent: opposite-signed streamwise vorticesthen form a series of loops that resemble hairpin vortices. As the Reynolds number is furtherincreased, the flow gradually becomes more irregular and finally turbulent. The digital particleimage velocimetry (DPIV) measurements by Brücker [6] and the flow visualization studies bySchouveiler and Provansal [7] have confirmed most of these numerical results and have furtherelucidated the sequence of transitions.3 Author to whom any correspondence should be addressed.0951-7715/05/010001+08$30.00 © 2005 IOP Publishing Ltd and London Mathematical Society Printed in the UK C1C2 Cover Illustrationf =1000 mmf =1000 mmf =1000 mmf =1000 mmf =150 mmf =150 mmf =100 mmFigure 1. Top view of the Schlieren set-up used to visualize the sphere wakes.For freely moving spheres the Reynolds number is defined by the measured mean velocityof the rise or fall of the sphere UT and the corresponding ‘Reynolds number’ becomesReT = 〈UT 〉d/ν. The mean velocity is the time-averaged velocity of the sphere, not includingthe acceleration of the sphere from rest. The flow loses its axial symmetry at a criticalReynolds number, which is not significantly affected by the density ratio, between solid andliquid, ρs/ρ [8]: Recr = 211.9 for ρs/ρ → ∞ (i.e. sphere held fixed), Recr = 206.3 forρs/ρ = 0.5 and Recr = 205.8 for ρs/ρ = 0.0. This is in good agreement with, for example,the experimental results on solid spheres [9], on surface-contaminated gas bubbles [10], andon wake visualizations in experiments with drops of tetrachloride and chlorobenzene fallingin water [11–13]. As pointed out by Natarajan and Acrivos [14], these drops must haveeffectively behaved as solid spheres due to the presence of surface-active impurities, and thesevisualizations have, therefore, often served as a basis of comparison with numerical studies onfixed spheres.What happens for freely falling or rising spheres at higher Reynolds number, which ismore common in multiphase flow applications? How are the wake structures and transitionsobserved for the fixed sphere case modified? Is there a (clear) difference in wake structurebetween rising and falling spheres? In this article, we present flow visualizations of the wakesbehind freely moving solid spheres at large Reynolds numbers (Re = 450–4623) for whichthe density ratio ρs/ρ is in the range 0.50–2.63.2. Experimental detailsThe flow visualizations of the sphere wakes were carried out in a transparent tank (0.15×0.15×0.5 m3) filled with decalcified water. Smooth plastic spheres with diameters between 1.5 and10 mm and densities between 500 and 2781 kg m−3 were released from rest. By means of anoptical system consisting of two LED-lights, pinholes, lenses and mirrors, two perpendicularimages of the particle and its wake were created and recorded at 500 frames/s with a CCD-camera (figure 1). Hence, each image consists of two perpendicular views of the same sphere.The images are taken at a position in the transparent tank where the spheres do not accelerateCover Illustration C3Table 1. Parameter values in our visualizations; d in mm, ρs in kg m−3. G is defined by equation (1)and ReT is the mean Reynolds number (ReT = 〈UT 〉d/〈ν〉).Number d ρs ρs/〈ρ〉 G ReT FigureFalling 1 3.2 1028 1.03 121 205 82 4.0 1058 1.06 239 325 83 1.5 2781 2.79 304 450 84 6.0 1035 1.04 359 546 25 6.0 1043 1.05 394 608 26 4.0 2629 2.63 1261 1970 8Rising 7 3.2 965 0.97 121 210 68 5.0 950 0.95 297 450 39 5.0 947 0.95 306 475 310 4.0 873 0.88 334 565 611 10.0 988 0.99 350 576 412 8.0 982 0.99 331 602 613 6.0 958 0.96 355 647 214 6.0 950 0.95 390 656 215 6.4 925 0.93 534 920 616 6.4 864 0.87 728 1180 717 7.9 925 0.93 732 1350 718 6.4 650 0.65 1160 1965 719 9.5 500 0.50 2548 4623 7anymore. The wake was visualized using the Schlieren technique, which visualizes densitydifferences due to changes in the refraction index of the fluid. To this end a small verticaltemperature gradient in the water was maintained (1 K cm−1). The mean water temperatureat the measurement section was 302 K, with corresponding values of the density (〈ρ〉) andviscosity (〈ν〉) of 996 kg m−3 and 0.802 × 10−6 m2 s−1, respectively. Hence, the Reynoldsnumber is based on the mean viscosity and is defined as ReT = 〈UT 〉d/〈ν〉. It turned out tobe difficult to keep a constant temperature gradient. Therefore, the error in the mean watertemperature at the measurement section is about 3 K, leading to a relative error in the viscosityof 10%. As opposed to the fixed-sphere problem, the Reynolds number for freely movingspheres is not an independent parameter. Following Jenny et al [8] we choose as independentdimensionless variables the ratio ρs/〈ρ〉 of the densities and the Galileo numberG = (|(ρs/〈ρ〉 − 1)|g)1/2d3/2〈ν〉 . (1)Since (|(ρs/〈ρ〉 − 1)|gd)1/2 can be considered as a velocity scale, G plays a similar dynamicalrole as the free-stream Reynolds number in the case of a fixed sphere. The parameter valuesfor which we made the flow visualizations are summarized in table 1.3. ObservationsFigure 2 shows stereoscopic images of the wake structure behind falling spheres with densitiesapproximately 4% (figure 2(a)) and 5% (figure 2(b)) higher than that of the surrounding liquid,and, for comparison, that behind rising spheres with densities that are approximately 4%(figure 2(c)) and 5% (figure 2(d )) lower. In all these examples, the sphere diameter is 6 mm,so that the parameter G is roughly identical in cases (a) and (c), and in cases (b) and (d ). Thelighter spheres have a slightly higher vertical velocity than the heavier spheres, as indicatedC4 Cover Illustration(a) (b) (c) (d)Figure 2. Stereoscopic views of falling and rising spheres and their wakes. The left part of eachframe shows the xz-plane and the right part the yz-plane. In each case the sphere diameter is 6 mm.The values of the parameters ρs/〈ρ〉, G and ReT are, respectively: (a) 1.04, 359, 546; (b) 1.05,394, 608; (c) 0.96, 355, 647; (d ) 0.95, 390, 656.(a) (b) (c)Figure 3. Stereoscopic views of rising zigzagging spheres and their wakes. The left part of eachframe shows the xz-plane and the right part the yz-plane. The views illustrate the crossing at thecentre-line of the zigzag path of the two counter-rotating threads of the wake (‘1’), the occurrenceof kinks (‘2’) at the extremes of the path, and the formation of hairpin-like vortices (‘3’) as twoneighbouring kinks connect. Values of the parameters d, ρs/〈ρ〉, G and ReT are, respectively:(a) 5 mm, 0.95, 297, 450; (b) 5 mm, 0.95, 306, 475. As shown in (c) the crossing of the vortexthreads results in a lift force L that is always directed towards the centre-line of the zigzag path.D shows the direction of the drag force and B the one of the buoyancy.by ReT in table 1. The wakes of the falling spheres appear to have a more ‘irregular’ structure,and the path followed by these spheres shows much larger deviations from a straight verticalline. These pictures illustrate that the density ratio ρs/〈ρ〉 matters, even at values close to 1.Figures 3(a) and (b) give examples of a phenomenon that we believe to be characteristicfor spheres following a zigzag path, namely that the two counter-rotating threads in the wakecross at the centreline of the zigzag (indicated by ‘1’ in figure 3(a)). The presence of thesethreads of opposite-signed streamwise vorticity implies that the sphere experiences a lift force.As a consequence of the periodic crossing of the threads this force is always directed towardsthe zigzag centre-line (see the sketch in figure 3(c)). A similar observation was made byDe Vries et al [15] on the wake behind zigzagging gas bubbles.Cover Illustration C5Figure 4. Sequence of stereoscopic views of a rising sphere and its wake. The left part of eachframe shows the xz-plane and the right part the yz-plane. The views illustrate the process offormation of a hairpin-like vortex (d = 10 mm, ρs/〈ρ〉 = 0.99, G = 350 and ReT = 576).Schouveiler and Provansal [7] remark that for a fixed sphere ‘the dynamics of the twoopposite-sign streamwise . . . vortices presents a striking similarity with the long-wavelength(or Crow) instability of a pair of counter-rotating parallel vortices’ and further ‘such a vortexpair instability could be responsible for the appearance of unsteadiness in the sphere wake’.Figure 3 suggests that the situation is slightly different for freely moving spheres. Here, itappears that close to the sphere each of the vortices first develop a ‘kink’ (indicated by ‘2’ infigure 3(a)), a process in which the curvature of the vortices presumably plays an importantrole [16]. As the kinks develop further downstream of the sphere they come near each otherand finally combine into what resembles a hairpin vortex (indicated by ‘3’). This sequence ofevents can be seen in the flow visualizations presented in figure 4; see also figure 6 of [11].As the kinks develop and hairpin-like vortices are formed further downstream, a patternresults. Lunde and Perkins [17] interpreted this pattern as a series of hairpin vortices ofalternating sign, shed periodically by the spheres at the extremes of the zigzag path. Ourvisualizations suggest instead that the streamwise vorticity produced at the surface of thesphere does not change sign; the legs of the like-signed hairpin vortices cross at the centrelineof the zigzag.Figure 2(b) is an example in which more than one kink develops in a half-period of thezigzag. We have not yet been able to determine the conditions (in terms of the parametersρs/〈ρ〉, G or ReT ) that select the number of kinks that are formed. What is remarkable isthat the development of the kinks and the subsequent formation of the hairpin vortices do notseem to affect the trajectory of the sphere. This corroborates the opinion that at high Reynoldsnumbers the details of the vorticity distribution very close to a body basically determine theforces that it experiences.C6 Cover IllustrationFigure 5. Phase diagram: density ratio ρs/〈ρ〉 versus Galileo number. The grey box is the regimeanalysed by Jenny et al [8]. They find in the leftmost region an axisymmetric wake. The symbols,directly taken from [8], denote: + steady and oblique, ∗ oblique and oscillating regime withlow frequency (0.045  f  0.068), × oblique and oscillating regime with high frequencies(f = 0.180), O zigzagging periodic regime (0.023  f  0.035) and  chaotic regime. Thenumbers denote the number of our experiment in table 1. Experiments 6 and 19 fall outside thediagram.We will now turn to experiments with density ratios more different from one. Recently,Jenny et al [18] reported on their numerical work on freely moving spheres in a Newtonianfluid. They focused on the frequencies in the wake and the path of the sphere in the parameterspace spanned by the density ratio and the Galileo number. Figure 5 reproduces their phasediagram. The numbers in the diagram refer to the numbers of the experiments given in table1. A lot of our experiments are outside their investigated region and new experiments shouldbe done in the interesting regions around a Galileo number of 200. Further experimentsshould focus more on the frequencies in the wake of the sphere and compare this to thefrequencies given by Jenny et al (see caption of figure 5). Furthermore, we must stress thatwake visualizations with the Schlieren method demand a temperature gradient in the water.Hence the density and viscosity of the water are not constant through the entire flow field and thelocal Galileo number will not be constant. The differences between the mean Galileo numberand the local Galileo number can reach 10% and must be taken into account when analysingfigure 5.A striking difference between our experimental data and the numerical data of Jenny et alis the behaviour of falling spheres with a density ratio close to one. From figures 2 and 8it can be seen that these falling spheres can also fall in a non-vertical path. This contradictsJenny et al who claim that only rising spheres can take a zigzagging path (the circles in thephase diagram figure 5).From our experiments one concludes that for increasing Reynolds number the wakebecomes more irregular (figures 6–8). The two-threaded wake structure is also present forhigher Reynolds numbers. Is the double-threaded wake structure also present in the case ofthe highest Reynolds numbers, where the wake structure has a turbulent structure? If so, doinstabilities in the wake cause kinking of the vortex threads, which leads to this turbulent wakestructure? Further research will address these questions in order to get a better understandingof the boundary layer separation from spheres at high Reynolds numbers as shown on thisyear’s cover.Cover Illustration C7(a) (b) (c) (d)Figure 6. Stereoscopic views of rising spheres and their wake structures observed at severalReynolds numbers (Reynolds number increases from (a) to (d ) and continues in figure 7). The leftpart of each frame shows the xz-plane and the right part the yz-plane. Values of the parametersd, ρs/〈ρ〉, G and ReT are, respectively: (a) 3.2 mm, 0.97, 121, 210; (b) 4.0 mm, 0.88, 334, 565;(c) 8.0 mm, 0.99, 331, 602; (d ) 6.4 mm, 0.93, 534, 920.(a) (b) (c) (d)Figure 7. Stereoscopic views of rising spheres and their wake structures observed at severalReynolds numbers (Reynolds number increases from (a) to (d )). The left part of each frame showsthe xz-plane and the right part the yz-plane. Values of the parameters d, ρs/〈ρ〉, G and ReT are,respectively: (a) 6.4 mm, 0.87, 728, 1180; (b) 7.9 mm, 0.93, 732, 1350; (c) 6.4 mm, 0.65, 1160,1965; (d ) 9.5 mm, 0.50, 2548, 4623.(a) (b) (c) (d)Figure 8. Stereoscopic views of falling spheres and their wake structures observed at severalReynolds numbers (Reynolds number increases from (a) to (d )). The left part of each frame showsthe xz-plane and the right part the yz-plane. Values of the parameters d, ρs/〈ρ〉, G and ReT are,respectively: (a) 3.2 mm, 1.03, 121, 205; (b) 4.0 mm, 1.06, 239, 325; (c) 1.5 mm, 2.79, 304, 450;(d ) 4.0 mm, 2.63, 1261, 1970.C8 Cover Illustration4. ConclusionsFlow visualizations of the wakes behind solid spheres moving under the action of gravity revealremarkable differences when compared to the wakes behind spheres held fixed: the crossing ofthreads of opposite-signed vorticity and the formation of kinks on these threads that developinto hairpin vortices. The ratio between the densities of the sphere and that of the surroundingfluid appears to be important. Our experiments clearly show the differences in path and wakestructure between rising and falling spheres with the same Galileo number. Furthermore, thedouble-threaded wake structure seems to be a basic feature, even for large Reynolds numbers.This should be investigated thoroughly in future research.AcknowledgmentsThe authors thank Andrea Prosperetti, M Versluis and C D Ohl for helpful discussions andsuggestions and G-W Bruggert and H Scholten for the technical support. This work is part of theresearch programme of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), whichis financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek(NWO).References[1] Johnson T A and Patel V C 1999 Flow past a sphere up to a Reynolds number of 300 J. 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Motion and wake structure of spherical particles

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