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Thermodynamics, Transport, and Fluid Mechanics

Vortex Formation and Subsequent Air Entrainment inside a Liquid Pool Parmod Kumar, Mihir Prajapati, Arup Kumar Das, and Sushanta K. Mitra Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00379 • Publication Date (Web): 11 Apr 2018 Downloaded from http://pubs.acs.org on April 11, 2018

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Industrial & Engineering Chemistry Research

Vortex Formation and Subsequent Air Entrainment inside a Liquid Pool Parmod Kumar1, Mihir Prajapati1, Arup K. Das1,* and Sushanta K. Mitra2 1

Department of Mechanical and Industrial Engineering, IIT Roorkee, Roorkee 247667, India

2

Waterloo Institute for Nanotechnology, University of Waterloo, Ontario, N2L 3G1, Canada

ACS Paragon Plus Environment

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ABSTRACT Genesis of free surface vortex and gas entrainment are investigated within a liquid pool using a rotating cylindrical disc having longitudinal axis normal to the gas-liquid interface with varying submergence from 1.40 to 2.85 times of disc radius (20 mm). A generalized vortex profile is obtained by suitable scaling for range of rotational Froude numbers (2.38-11.18). Transient evolution rate of the vortex tip followed logarithmic law and is correlated with experimental data of air and glycerin. The axial pull that holds the vortex against gravity is found to be linearly increasing with the rise of disc rotation. Asymmetric vortex profiles with increased extent of asymmetry are obtained upon increase of disc inclination (0-17.45°). Axial and radial circulations are revealed by following the trajectory of a solid particle in the flow field. Entrainment of discrete air volumes is observed from the vortex core at the corresponding Froude number and submergence ratio. KEYWORDS: Free surface vortex, Air core, Entrainment, Particle tracking, Air filament.

1.

INTRODUCTION

Entrainment of air bubbles at the free surfaces finds its enormous applications in civil, chemical, mechanical and nuclear engineering. Chanson1 reviewed occurrences of many such high velocity turbulent free surface flows and in particular, studied the open channel, plunging jet flows and discharging of turbulent water jets into air. Researchers continued making efforts in understanding the intriguing phenomenon of entrainment, which leads to successful prediction of entrainment location and its rate in situations like free surface bubbly flows2, air entrapment in a hydraulic jump3 and around the naval vessel Athena4. Majority of these entrainment occurrences lead to the formation of strong vortices within the liquid medium around the free surface. An amplified version of such vortices is often referred to as free surface vortex. The frequent


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occurrence of free surface vortex in the kitchen sink5, while emptying, or natural whirlpool in a river or ocean are amazing to visualize. In each of these scenarios, the air gets entrained within the vortex core and creates a nice pattern, which is indeed visually appealing but turns out to be detrimental for hydraulic machines6-9 and power systems such as sodium-cooled fast breeder reactors10-11. The loss in efficiency and cavitation in fluid machines along with safety issues in reactor vessel promote researchers to understand the fluidic stages and control of such air entrainment. Therefore, it is important to undertake a systematic study to understand such air entrainment effect inside a liquid pool. The first mathematical model for the tangential velocity of the vortex is proposed by Rankine12 considering a solid body rotation of inner core surrounded by free vortex. Schlichting13 have reported an exponential equation for tangential velocity considering dissipation of vorticity. Velocity correlation similar to that of Schlichting13 is obtained by Burgers14 and Odgaard6 using Navier-Stokes equation with the assumption that the radial velocity is proportional to the radial distance measured from the axis of the air core. An empirical formula for tangential velocity is also proposed by Rosenhead15 whose coefficients are further modified by Bennett16 and Mih17 to locate the maximum velocity at mathematically amenable radius, as obtained by Burgers14 and Odgaard6. These correlations are summarized in Table 1. These correlations13,15 have shown good agreement with the experimental data of Mih17 and Hite Jr18, obtained for air vortex near the water intake. In addition to the velocity of the vortex, the shape of the air core has also gained the due attention of the research community. Rouse19 proposed that maximum tangential velocity can be obtained at the outer radius of the vortex corresponding to the half depth of penetration. Hite Jr and Mih7 used the radial and the tangential components of Navier-Stokes equation to predict the



water surface profiles, depicting quadratic dependence between non-dimensional axial and radial coordinates. Their prediction agrees well with vortex shapes obtained from experimental investigations20-25. Baum26 proposed that entrainment inception is dependent on key nondimensional numbers such as Froude, Reynolds and Weber number under the laminar conditions. Experiments were also performed to study the influence of physical properties of fluid on the entrainment onset conditions27. The equivalent effort has been invested by nuclear scientists28-30 to notice the influence of fluid properties on entrainment profile. Researchers31-33 have also employed the volume of fluid approach based numerical techniques to study the gas entrainment inside reactor vessels. It can be observed that genesis of air entrainment is mainly studied as outflow induced vortex in the liquid pool, however, researchers have also investigated entrainment patterns, either in the form of vortex in unbaffled stirred tanks34-38 or as discontinuous entrainment of air entities in baffled stirred vessels39-41. Moreover, it is to be mentioned that all these efforts34-41 have utilized the impellers or mechanical agitators to impart the rotational field to the fluid medium and at the same time, the shaft of the prime mover holding impeller has interfered with the entrained vortex core. Such mechanical agitator induced swirling motion has led to the occurrences of various biasing features including excessive stress of shaft, loss of mixing power, imbibition of gases into liquid, vibrations, noise and genesis of large amplitude circumferential waves37. Therefore, the estimation of vortex tip penetration has remained the topic of prime interest for the chemical and process industry. Researchers37-42 over the last few decades have found the impeller diameter based Froude number as the controlling parameter for vortex penetration. Further, Rieger et al.34 have used Galileo number to consider the effect of liquid velocity and impeller size variation. Recently, Mahmud et al.43 have used a cylindrical magnetic agitator to study mixing and free surface vortex flows. They have also


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described the importance of unbaffled stirrers in precipitation, crystallization, fermentation, pharmaceutical, food and dairy applications. It can be observed from comprehensive literature survey that increasing stipulation of meeting the stringent quality of products in several mixing processes, batch reactors and mechanical separation devices, has led to the design of specific stirrer and vessel geometry based on the hydrodynamic understanding of the flow. Considering such process specific aspects, we have used a smooth cylindrical disc without any baffles/blades to impart the rotation to the fluid medium. Moreover, in the present situation the shaft of the prime mover is extended from the bottom face of vessel so that it should not interfere with the flow field around the vortex core. Gas entrainment, driven by external rotational fields using such cylindrical rollers are well established when axis of the disc/cylinder is parallel to the free surface44-51. Recently, Peters et al.52 have shown oil entrainment in the wake of a disc, translating inside water from the oil-water interface. However, to best of our knowledge, no comprehensive effort has been reported till date to investigate genesis of vortex and subsequent entrainment dynamics using submerged disc rotation when axis of the externally applied field is normal to the free surface, except the studies depicting polygon formation53-55 and fluidic circulations56-59 due to rotating fluid surfaces. Therefore, experiments of vortex genesis have been reported in the present study to identify the influencing parameters. Physical reasoning behind the occurrence of entrainment is also established using influencing parameters. A fundamental analysis of force balance has been used across the vortex core to evaluate the amount of axial pull imparted by disc rotation, causing the air volume to stay against gravity. Further, attempts have been made to reveal the flow field within the liquid medium around the vortex core using Lagrangian particle tracking. Generation of discrete air bubbles and their subsequent dynamics have been also reported at high rotation and low submergence of the disc.


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Table 1: Studies mentioning characteristics of free surface vortex available in literature. Vortex formation due to the flow through hydraulic intakes Reference Tangential velocity equation Vortex depth 𝜔𝑟 ∀ 𝑟 < 𝑟𝑚 12 𝑢𝑡 = 𝛤 Rankine ∀ 𝑟 > 𝑟𝑚 2𝜋𝑟 𝑢𝑡 =

Odgaard6

𝛤 1𝑎 2 1 − 𝑒𝑥𝑝 − 𝑟 2𝜋𝑟 2 𝜗𝑒 𝑣𝑎𝑥𝑖𝑎𝑙 𝑎= 2𝐻𝑐𝑟𝑖𝑡

𝜗𝑒 𝑟 𝑟𝑚2 𝛤 𝑢𝑡,𝑚𝑎𝑥 = 0.71 2𝜋𝑟𝑚 𝑌 16 𝑢𝑎𝑥𝑖𝑎𝑙 = 𝜗𝑒 2 𝑟𝑚 1 + 2𝑟 2

𝐻𝑐𝑟𝑖𝑡 48 = 𝑑 𝐹𝑟𝑕

;

Rieger et al.34

Turbine impellers: 𝑕

𝐷

2

= 𝐵𝐺𝑎𝑖0.069

𝛿 𝐷

= 𝐵 𝐺𝑎𝑖0.33

Anchor agitator: 𝐷 = 2.82𝐹𝑟

1.07

𝑔𝑑 3

𝛿 = 1.98

Vortex formation due to the impeller rotations 𝜔𝑟 ∀ 𝑟 < 𝑟𝑐 ; 𝑟𝑐 < 𝐷 2 Deshpande et 𝑢𝑡 = 𝜔𝑟𝑐2 𝑟 ∀ 𝑟𝑐 < 𝑟 < 𝑇 2 al.37 𝛿

𝛤

2 𝑢𝑡,𝑚𝑎𝑥 𝑔 𝑌 − 𝑌𝑡𝑖𝑝 2𝑟 2 = ; 𝛿 1 + 2𝑟 2 𝑟 𝑟= 𝑟𝑚

𝑢𝑟𝑎𝑑𝑖𝑎𝑙 = −2.5

Hite Jr and Mih7

Key attributes  Considered inner core as forced vortex along with outer region as irrotational vortex  Velocity computed is too large than experimental data

𝛿 𝜔2 𝑟𝑐2 2𝑟𝑐 = 2− 𝐷 2𝑔 𝑇

𝑇 −0.38

1.14𝐺𝑎 𝑖−0.008 𝑇 𝐷 0.008

𝐷

𝐹𝑟𝑖

𝐷

𝐹𝑟𝑖

𝑇 −1.18

2

3.38𝐺𝑎 𝑖−0.074 𝑇 𝐷 0.14

∀ 2 × 107 < 𝐺𝑎𝑖 < 6 × 1010

 Velocity expression gives a forced vortex near the axis of rotation and irrotational vortex at large radius  Indicated critical depth of the submergence of the inlet at which vortex formation initiates  Presented the velocity equations for radial, tangential and axial velocity of the vortex core  Equations for vortex depression and water surface profile are also obtained  Proposed equations agrees well with the previous experimental measurements  Expression of tangential velocity profile in stirred tank  Presented an expression for depth of vortex tip using combined forced and free vortex concept  Vortex depth is correlated for different agitators using experimental measurements in unbaffled vessels  Coefficients B and 𝐵 have dependence on impeller type and Galileo number  Galileo number takes care of liquid viscosity and impeller size variation

𝜗𝑒 : Kinematic viscosity; 𝛤: Circulation; 𝛿: Vortex depth; d: Intake diameter; D: Impeller diameter; N: Impeller speed (rps); T: Vessel diameter; 𝐹𝑟𝑕 =

𝑣𝑎𝑥𝑖𝑎𝑙 𝑔𝑑

; 𝐹𝑟𝑖 =

𝑁2 𝐷 𝑔

; 𝐺𝑎𝑖 =

𝐷3 𝑔 𝜗 𝑒2


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2. EXPERIMENTAL METHODOLOGY An experimental facility is designed to create the free surface vortex induced gas entrainment using the externally applied rotational fields. Experimental setup mainly comprises a 0.3 m x 0.3 m x 0.3 m Perspex tank, a single phase induction motor (make: Crompton Greaves; rating: 0.5 hp; maximum speed: 3000 rpm), a variable frequency drive (make: Delta; series: L; range: 1-400 Hz) and a cylindrical disc as shown in Figure 1(a-b). This tank is placed on a metallic stand which houses the prime mover. The shaft of the vertical motor is extended in form of a metallic rod through a bearing fitted at the bottom surface of the tank. Perspex made a cylindrical disc of radius 20 mm and thickness 10 mm is fitted horizontally with the metallic shaft. An AC power of 230 V and 50 Hz, routed through a variable frequency drive, is supplied to the motor. A digital non-contact tachometer (range: 0-10000 rpm; least count: 0.1 rpm) is used to record the rpm of the disc. The phenomenon of the free surface vortex and gas entrainment is captured using high-resolution camera (make: Canon; model: EOS 5Ds; frame rate: 25fps). Glycerin and water are used as the working liquids at room temperature of 25 °C. Properties of both the liquids are measured accurately in experimental conditions and are provided in Table 2.

Table 2: Physical properties of the liquids used in experimentation at room temperature of 25°C. Fluid

Density, 𝜌 𝑘𝑔 𝑚3

Viscosity, 𝜇 𝑘𝑔 𝑚 − 𝑠

Surface tension, 𝜍 𝑁/𝑚

Glycerin

1258

0.934

0.0625

Water

1000

0.001

0.072



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Air Free surface Y H

Container

R X

Liquid

Disc Shaft

Disc

ω Bearing Shaft

Stand

Coupling Motor

230 V, 50 Hz Power source

VFD

Motor

a)

b)

Z

vr ω

Forced vortex Free vortex

Free vortex

vaxial

vt X

u=rxω

Y

u

d)

c)

Figure 1. a) 3D model of experimental facility, b) planer view of the test setup, c) schematic representation of different velocity components and d) velocity profile on the top plane of the disc depicting combined vortex field. It is found that the relevant non-dimensional numbers involving entrainment are Reynolds 𝑅𝑒 =

𝜌𝑢 𝑚 𝑅 𝜇

; 33.83 − 169.23 ,

Froude

𝐹𝑟 =

𝑢𝑚 𝑔𝐻

; 2.38 − 11.86


and

Weber

𝑊𝑒 =

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2 𝜌𝑢 𝑚𝑅

𝜍

; 635.70 − 15892.43 , respectively. The submergence (H) is characterized in terms of the

disc radius (R). Here, prime mover rpm decides the maximum tangential velocity at disc surface, 𝑢𝑚 as 𝑅 × 𝜔. A typical depiction of all the velocity components is given in Figure 1(c) with a schematic representation of the tangential velocity profile at the top plane of the disc in Figure 1(d), which is similar to the one proposed for combined Rankine vortex. The core of the fluid within the bound of the disc resembles the forced vortex and away from the disc surface there is a decrease of velocity due to the viscous decay. This external field imparted by the disc to the fluid is being propagated towards the free surface which results in the formation of vortices at the gas-liquid interface. Initially the genesis of free surface vortex is captured using high speed photography, followed by the establishment of effect of disc rotation on steady vortex behavior. Subsequently the effect of initial disc submergence is measured by keeping the disc rotation fixed. We also generated the asymmetric rotational field by orienting the disc at arbitrary inclinations with respect to the horizontal. In order to experimentally achieve such a situation, the whole experimental facility is tilted with reference to horizontally aligned base plate, using an indigenous mechanical assembly. Under the tilted condition, when the liquid is poured inside the vessel, the gas-liquid interface aligns with the horizontal due to the gravitational stratification and subjected to differential field because of unequal submergence of top surface of the disc. A digital inclinometer has been used to ensure the accurate measurement of disc inclination. In the later portion of the present investigation, efforts have also been extended to estimate the flow physics within the liquid medium around the vortex core. This has been achieved by tracking the path of a spherical buoyant tracer particle of approximately 3 mm diameter and ≈950.64 kg/m3 density. Size of the particle is selected so that it should not be either as big to affect the flow dynamics or as small to cause difficulty in visualization. Similarly density of the particle is kept



close to the liquid medium to make it neutrally buoyant. The procedure used for the estimation of flow dynamics using the Lagrangian framework has been detailed in section 3.6.

3. RESULTS AND DISCUSSION Present study elucidates the formation of the free surface vortex and subsequent air entrainment using submerged rotational fields. Efforts have been made to investigate the effect of rotational field, initial submergence and fluid properties on the vortex configuration. The motion of the surrounding liquid around the vortex core is tracked with the help of buoyant tracer particle. The pinch off of the vortex tip into discrete bubble entities and their subsequent dynamics is also studied in the later portion of this section.

3.1. Genesis of free surface vortex At first, we have illustrated the influence of rotational inertia of the disc keeping its submergence (H/R) at 2.16. Rotation of disc, maintaining Fr at 9.66 (3000 rpm) originates the vortex from the free surface and forces air to entrain inside the liquid. A series of experimental snaps depicting the axial growth of vortex profile at different time levels is shown in Figure 2(a). Inertial time 𝐻

scale 𝑡 ∗ = 𝑢

𝑚

is chosen for non-dimensionalization of the physical time instant. Initiation of air

penetration requires finite time (𝑡 𝑡 ∗ = 93.20 from undisturbed free surface) for the propagation of swirling motion till the free surface. In its development stage, triangular shaped air mass penetrates further down towards the source of swirling motion. At a certain height, when the strength of rotational inertia balances buoyancy pull, the vertical downward motion of the air core saturates. In Figure 2(a), it can be seen that at 𝑡 𝑡 ∗ = 1561.16, development of vortex stops, thereafter showing a steady behavior as revealed from experimental snapshots for 𝑡 𝑡 ∗ =


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1584.46 and 3215.52, respectively. The vortex profiles observed at different time instants are quantified with reference to the coordinate system located at the center of the top surface of the disc. Air

Air

Air

Air

Air

Glycerin

Glycerin

Glycerin

Glycerin

Glycerin

4

Y (cm)

3 2 1 0

t/t** == 00 t/t 4

* t/tt/t 93.20 * == 93.20

Air

Air

Glycerin

Glycerin

t/t*t/t=* 163.11 = 163.11

* t/tt/t 296.12 *== 296.12

* t/tt/t * = 279.61 = 279.61

Air

Air

Air

Glycerin

Glycerin

Y (cm)

3 2 1

Glycerin

0

t/t*t/t=* 629.12 = 629.12

* 862.13 t/tt/t * == 862.13

t/t*t/t=* 1561.16 = 1561.16

* 3215.52 t/tt/t *== 3215.52

t/tt/t * =* 1584.46 = 1584.46

a)

7

0.25

11

6

0.8 0.8

0.2

t/t* t/t* 00 93.20 93.20 163.11 163.11 279.61 279.61 396.11 396.11 629.12 629.12 862.13 862.13 1561.16 1561.16

0.4 0.4 0.2 0.2 00

5 3) X 10 6 V/V V (m cylinder

0.6 0.6 Y/H Y/H

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-1 -1

00 X/R X/R

11

3

0.1

2 0.05

1

-0.2 -0.2

-2 -2

0.15 4

2

0 0 0 0

b)

r

a b

𝑟

c d

δy 60010001200

1800 3000 2000 2400 3000 t/t*t/t*

c)

c)

b)

Figure 2. a) Experimental snaps depicting the evolution of vortices from free surface obtained at disc rotation based Fr = 9.66 and H/R = 2.16, b) dimensionless vortex profiles superimposed on the same plot at different time instants and c) non-dimensional volume of vortex core as function of dimensionless time. Time is given in non-dimensional form using the inertial time scale.



Interfacial boundary or any other feature is extracted by analysis of images, obtained during experiment using high speed photography. First, raw images, obtained from camera, are converted to grayscale and corresponding black and white snapshots are acquired. The interface is captured from the black and white images using Sobel technique60. One can refer Sanjay and Das60 for detailed elaboration of image analysis. It is also to be mentioned that due to the implication of the instinctive procedure for quantification of interface profiles, the errors encountered in measurements are negligible and range of coordinate variations obtained for different steady snapshots at same conditions is under 1%. The obtained ordinate and abscissa of the vortex boundary are scaled with respect to the initial submergence and radius of the disc, respectively. Temporal evolution of the non-dimensional symmetric vortex profiles is shown in Figure 2(b). The plot in Figure 2(c) shows the evolution of non-dimensional entrained air volume (V/Vcylinder) at a particular Fr (9.66) and H/R (2.16) combination. Initially the volume of the vortex core is obtained by applying Pappus theorem over the coordinate points of the 2D interface obtained using image analysis60. A representation for one half of the vortex profile is shown in inset of Figure 2(c). This vortex boundary is divided into large number of small surfaces along the axis of rotation with a constant step of 𝛿𝑦 (inset of Figure 2(c)). A magnified view of such an element abcd is also depicted in the inset figure. Centroid of the element abcd is estimated in radial direction using expression 𝑟 = 𝑟𝑎𝑐 + 𝑟𝑏𝑑 𝑑𝐴 = 2𝑟 × 𝛿𝑦 = 𝑟𝑎𝑐 + 𝑟𝑏𝑑

4 and its area is calculated as

2 × 𝛿𝑦. Finally the volume of the element is evaluated using the

expression 𝑑𝑉 = 𝑑𝐴 × 2𝜋𝑟 and total volume of the vortex core is obtained by summation of the small elemental volumes

𝑉=

𝑑𝑉 . This volume (V) of the vortex core is non-

dimensionalized with the volume of the hypothetical cylinder 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜋𝑅 2 𝐻 of radial span equivalent to the disc radius and length comparable to the initial submergence. The nature of the


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volumetric evolution of the gaseous core has revealed the faster rate of air penetration inside liquid during the early stages of vortex formation, before obtaining steady profile. 3.2. Influence of rotational inertia on vortex profile After the study of genesis, efforts have been extended to investigate the effect of rotational inertia on steady vortex profile, keeping the disc submergence same. A sequence of snapshots capturing the steady free surface vortex profiles is shown in Figure 3 for a wide range of disc rotation based Froude numbers. Careful visualization of these vortices leads to their characterization in three different categories. Volume of air core in case of smaller disc inertia (Fr = 2.38) saturates at the initial stage. At this stage, the entrained profile can be characterized as a rounded shallow valley, accompanied by axial swirl. An increase of Fr has led to the decrease in radius of curvature of the valley base. At around Fr = 5.16, extent of the vortex to the free surface becomes commensurate with the disc diameter and the base appears to take an elongated, pointed profile. This can be considered as the most stable stage of the vortex. Further, penetration of the vortex tip inside the liquid pool faces the high inertial zone near the fast moving disc. Here, the vortex tip experiences centrifugal push and bulges in the radially outward direction (Fr = 5.56) following the disc. The more precise transition between the sharp vortex tip and radially bulging air core can be obtained by performing the experiments with lesser step of disc rpm than 100 for obtaining smaller variations between consecutive Froude number values. However, such estimation of exact transition limit has not been targeted in the present experiments due to the limitation of the motor characteristics. Moreover, radially bulging vortex stage can be considered as the boundary between the unified and bubble radiating entrainment zone, as a small increase of rotational inertia at this phase can lead to pinch off of air core into



the bubbles and their subsequent mixing with the liquid streams. The details of the free surface vortex driven air entrainment are discussed later.

Figure 3. Experimental photographs depicting the steady free surface vortex profiles obtained at a wide range of rotational speeds characterized in terms of Froude number at fixed submergence H/R = 1.42 of the disc. The rpm of the disc has been varied from 600 in steps of 100. The steady and non-dimensional vortex profiles shown in Figure 3 are quantified in left side of Figure 4(a) for different disc rotations. The shape of the steady vortex is found qualitatively similar to an inverted bell-shaped Gaussian distribution for lower rotational Froude numbers whereas for Fr = 5.56 vortex tip touches the surface of the disc. It is to be noted that vortex profile penetrates axially inside the liquid pool at different rates in the presence of dissimilar inertial pull depending upon the magnitude of the external force. Axial traversal rate of the vortex tip for the whole range of Froude numbers is shown in right side of Figure 4(a). It is evident from the temporal penetration history that vortex tip grows at faster rate upon increase of rotational inertia, however, the rate of its downward axial growth is not constant throughout its complete traversal inside the liquid, similar to Figure 2(b).


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1.05

1.05

1.05

0.7

1.05

0.7

Y/H

Fr = 5.16 0.7 Fr = 5.56 200 Fr = 2.78 Fr2.75 = 5.963.75 4.75 0.7 5.75 Fr Fr = 3.57 0.35

Fr = 3.18 0.35 3.97 Fr = 2.78

Fr = 4.37 0.35 Fr = 5.16 -1 X/R Fr = 5.96

4.77 Fr = 3.57

Y/H

Y/H

0

Fr = 2.78 0.7

-0.35 -3

-2 0

Fr = = 3.18 4.37 Fr 0.35 Fr = = 3.97 5.16 Fr

Fr = = 4.77 5.96 Fr 00 800 Fr 0 = 5.56

Fr 5.56 0 = 4.37 Fr = 5.16

Fr = 5.96 -0.35 -3

0 -0.35

1

Fr = 3.57

a) -0.35

-3

-2

-0.35 -3

-3-1

X/R

-2

Fr = 3.18 0.35 Fr = = 2.78 3.97 Fr

Fr = 2.78 0.7

Fr = 3.18

Fr = 3.57

Fr == 2.78 3.97 Fr

Free surface

Fr == 3.57 4.77

Fr = = 3.57 4.77 Fr

4.37 Fr = 3.18 0.35 5.16H Fr = 3.97

Fr = 4.37 5.56 Fr 0=

5.96 Fr = 4.77

Fr = 5.16

Fr == 4.37 5.56 Fr Ytip

0.4

0.2

0 -2

-1.5

-1

-0.5

0 X/R

Air

Fr = 3.97 0 Fr = 4.77

Liquid

Fr = 5.56 -0.35 0 3

-1

0 -1 1

X/R

02

1

0.5

1

1.5

2

2.5

b)

Figure 4. a) Non-dimensional steady vortex profiles superimposed on the same plot for different Froude number and quantification of vortex tip penetration rate inside the liquid pool at fixed submergence of the disc and b) proposition of combined parabolic (near core) and logarithmic fit (away) for generalization of vortex profiles. Only one half of the symmetric fit has been shown in the figure. Steady state occurs in lesser time for low disc rotations as maximum depth of penetration is less in these cases. This argument is valid until the Froude number has attained a magnitude around


13 2

Fr = 2.78 Fr = 3.18 Fr = 3.57 Fr = 3.97 Fr = 4.37 Fr = 4.77 Fr = 5.16 Fr = 5.56 Parabolic fit Logarithmic fit Hite Jr and Mih [3]

0.6

-2.5

Fr = 3.18

Fr = 5.16 Fr0 = 5.56 Fr = 5.96 600 1 400 2200 * t/t Fr = 5.96 -0.35 -3 -2 -0.35 X/R -2 -1 -3 -20 X/R -20 -11 X/R -1 0

X/R

0.8

0.35

Ytip/H

Fr = 4.77

Y/H

Fr = 4.37

0.7

Y/H

Fr = 3.971.05

Y/H

t/t*

Y/H

1.05

Fr = 3.57

500

0.35

0.7

1.05

800 Fr = 2.781.05 Fr = 3.18

(Y-Ytip)/δ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



5.16 and can also be elaborated with the help of the strength of the centrifugal pull 𝑓𝑐 = 𝜌𝑅𝜔2 = 4.66 × 105 𝑁/𝑚3 imparted by the disc rotation. At low disc rotations the liquid surrounding the disc periphery moves at lesser speed in radial outward direction, therefore, its replenishment from the free surface is also localized near the axis of rotation. Further, increase of the disc rotation or corresponding centrifugal pull increases both the radial extent and replenishment rate of the liquid in axial direction. Therefore, the axial velocity of the downward penetrating vortex core has increased but at the same time its penetration depth has also increased. Moreover, the subsequent growth of disc rotation has further led to the radial expansion of vortex tip along the surface of the disc. In such cases, free surface vortices cover the maximum possible axial penetration in lesser time for high rotational inertia 𝑓𝑐 = 6.21 × 105 𝑁/𝑚3 than weak vortex, as shown in Figure 4(a) for Fr = 5.96. It is due to the rapid replenishment and faster advancement of the vortex core inside liquid at higher disc rotations, whereas the maximum depth is same, which is limited by the initial submergence of the disc. An estimate for non-dimensional time taken by the vortex core to acquire the steady state is given in inset of Figure 4(a). It can also be observed from Figure 4(a) that vortex entrains at almost constant pace for higher Fr (5.96; 𝑓𝑐 = 6.21 × 105 𝑁/𝑚3 ) before touching the disc surface. Attempts have also been made to scale the ordinate of the steady vortex profile using maximum penetration of its tip (inset of Figure 4(b)) as length scale for the entire range of disc rotation. This has resulted in the collapse of all the steady interface profiles to similar curves irrespective of their Froude number (Figure 4(b)). The nature of these curves is considered as the generalized vortex shape and effort has been made to compare such experimental characteristics with the free surface profile 𝑌 − 𝑌𝑡𝑖𝑝

𝛿=2 𝑋 𝑅

2

1+2 𝑋 𝑅

2

given by Hite Jr and Mih7. Figure 4(b)

reveals that the qualitative nature of experimental interfaces is identical to the generalized profile


Page 16 of 47

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of Hite Jr and Mih7 but there is significant difference between the two in terms of specific coordinate values. This is due to the fact that the free surface equation of Hite Jr and Mih7 is obtained for outlet induced vortex, whereas the present configuration is different from their experimental conditions. Therefore, generalized experimental profile is compared with the parabolic fit

𝑌 − 𝑌𝑡𝑖𝑝

𝛿=2 𝑋 𝑅

2 61-62

, which is a fundamental vortex configuration in

fluid statics when the rotation is given to entire fluid body. The experimentally obtained profiles have shown good agreement with the parabolic fit near the vortex tip (shown in left portion of Figure 4(b)) and it has deviated from the parabolic nature close to the nominal gas-liquid interface. This is because of the forced vortex resemblance of the present situation near the vortex tip, which is a fundamental assumption for obtaining the parabolic nature of the interface, considering constant angular velocity of the liquid streams. However, in our experiments velocity is imparted to the fluid medium using the rotation of the disc and all the fluid streams are not having same tangential velocity due to viscous decay. Therefore, away from the vortex core, the interface profile has shown logarithmic nature (shown in right portion of Figure 4(b)), before its alignment with horizontal. Hence, using combined vortex understanding (Figure 1(c)), the shape of vortex boundary is depicted in equation (1), which has shown good agreement with experimentally obtained profiles.

𝑌 − 𝑌𝑡𝑖𝑝

𝛿=

2 𝑋 𝑅 2 ∀ 𝑋 𝑅 ≤ 0.5 0.3 ln 𝑋 𝑅 + 0.8 ∀ 0.5 ≤ 𝑋 𝑅 ≤ 𝑋 𝑅 ∈ 𝑌 − 𝑌𝑡𝑖𝑝

(1) 𝛿≈1

Present effort only targets study on shear induced vortex profiles whereas impeller driven vortex is resultant of shear and physical displacement of liquid between blades. One to one correspondence of both these means can be only made if present experiment is performed at a



Page 18 of 47

very high disc speed. Due to limitation of motor capacity of the developed setup, comparison between these two mechanisms of creating vortices is not targeted.

Figure 5. Variation of inertial pull in axial direction as function of rotational Froude number at fixed submergence of the disc, H/R =1.42. Efforts have been also extended to estimate the extent of axial pull imparted by the disc rotation, which holds the air volume against the gravitational field. When a volume of air entrains inside a liquid its motion is approximated by the balance of inertia against opposing gravitational field and viscous drag. However, drag force will not participate in the steady vortex profile, as air motion in axial direction seizes. Therefore, using the force balance in the axial direction (inset of Figure 5), one can relate the inertial pull of rotational field with the buoyancy force as: 𝐹𝐼 = 𝐹𝑏 = 𝜌 − 𝜌𝑎 𝑉𝑔

(2)

In equation (2), 𝜌 and 𝜌𝑎 are the densities of liquid and air, respectively, V is the volume of entrained air core and g is the gravitational acceleration. The volume of air core is obtained using Pappus theorem (inset of Figure 5) and amount of axial pull is calculated using equation (2). The obtained force is plotted as a function of Froude number in Figure 5 and depicts that axial pull or


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amount of entrained air core has increased linearly with an increase of externally applied rotational inertia.

Figure 6. Experimental photographs depicting the steady free surface vortex profiles obtained for range of disc submergence characterized in terms of H/R ratio and Froude number at fixed rotational inertia of the disc (3000 rpm).

3.3. Vortex configuration at different submergence for fixed disc rotation Apart from disc rotation, vortex profile is also influenced by the initial submergence. Experiments are performed to observe vortex profile for various free surface heights (characterize by H/R) at fixed disc rotation (ω). The experimental snapshots depicting the variations in steady vortex profile are shown in Figure 6 for H/R ratio ranging from 1.61 to 2.85. It can be observed that amount of entrained air core has decreased with an increase of the initial submergence of the disc, due to the increase of resultant gravitational pull. This is because in the present case the lighter phase is entraining inside the heavier one, therefore, net force due to gravity (resultant of the weight of the air core and weight of the water displaced by the vortex) is



in upward direction and is balanced by the axial pull acting in downward direction due to the disc inertia. Moreover, vortex profile has shown similar stages upon decrease of submergence as observed in the previous section about the increase of rotational Froude number. As in both the cases,the same fluid combination is used, the viscous effect has remained unchanged. Therefore, Froude number is considered as the most suitable dimensionless parameter to characterize vortex entrainment when the geometrical characteristics are varied, keeping fluid properties unchanged. This elucidates that extent of penetration of air core tip inside the liquid, can be well explained by the balance of inertial and gravitational pull. 1.05

1.05

H/R = 2.85; Fr = 8.41

0.7

0.7

H/R = 2.68; Fr = 8.66

H/R = 2.47; Fr = 9.02

0.35

0.35

H/R = 2.25; Fr = 9.46

Ytip/H

H/R = 2.58; Fr = 8.84 Y/H

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 47

H/R = 2.16; Fr = 9.66 H/R = 2.07; Fr = 9.86

0

0

H/R = 1.84; Fr = 10.45 H/R = 1.61; Fr = 11.18

-0.35 -3

-2

X/R

-1

0

1500

1 1000

t/t*

5002

0

-0.35 3

Figure 7. Non-dimensional steady vortex profiles shown for different Froude numbers, initial submergence of the disc and quantification of vortex tip penetration rate inside the liquid pool at fixed rotation of the disc. As the vortex profile is symmetric about the central axis, therefore, one-half of the steady vortex profiles are plotted in Figure 7 for all the H/R ratios. However, the axial penetration of vortex tip is plotted as a function of dimensionless time on the other side of the plot, shown in Figure 7, to characterize the penetration rates. Effect of initial submergence on penetration rate shows similar behavior, as observed for disc rotation (Figure 4). Moreover, the vortex tip has penetrated inside


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liquid at very fast rate upon decrease of submergence and its axial traversal rate is almost constant for H/R= 1.84.

Figure 8. Variation of axial pull induced by disc rotational inertia as function of Froude number and H/R ratio upon alteration in initial submergence of the disc at fixed external rotational field. Following the analogy from the previous section, here also, we evaluated the inertial pull using equation (2) that holds the air core inside the vortex against the gravity. The inertial pull is also plotted as a function of H/R ratio or Fr in Figure 8. The variation of the axial force has shown good agreement with the physical interpretation of the phenomenon, which depicts that inertial pull should increase with the increase in Froude number and decrease with H/R ratio, due to the dominance of disc rotational inertia over gravity. It has been already established that variation of vortex tip penetration is controlled by the geometrical parameters, gravitational field and rotations of the disc for a particular gas-liquid pair, as described here: 𝑦 = 𝑓 𝐻, 𝑅, 𝑡, 𝑢𝑚 , 𝑔

(3)

The application of dimensional homogeneity using Buckingham-theorem has resulted in a functional relationship of equation (3) in the form of dimensionless parameters as given in equation (4).



𝑦 𝐻 = 𝑓 𝜏, 𝑅 𝐻, 𝐹𝑟 −2

Here, 𝜏 equals to

𝑢𝑚 𝑡 𝐻

Page 22 of 47

(4)

is the dimensionless time scale. Efforts are also extended to correlate these

dimensionless groups using the regression analysis applied to experimental measurements. In this regard the vortex tip entrainment rate curves (as shown in Figure 4 and 7) are fitted using the logarithmic variation, as a function of time (equation (5)) for a range of Froude numbers at H/Rratios of 1.417 and 1.57. It is to be noted that equation (5) is valid only prior to the occurrence of steady vortex profile, which depicts that it reflects the transient evolution of the vortex tip. 𝑦 𝐻 = 𝑎 ln 𝜏 + 𝑏

(5)

This unified temporal growth law even fails to portray steady penetration depth at long time interval. A representative case of such logarithmic fitted curve is shown in Figure 9(a) for Fr = 4.37 at H/R = 1.417. It is to be noted that in Figure 9(a), equation (5) is plotted only upto𝜏 ≈ 750 to avoid anomaly of proposed correlation for prediction of steady entrainment. The coefficients a and b of the logarithmic variation are plotted as the function of Froude number in Figure 9(b and c) for two different H/R ratios. These coefficients are correlated as a function of Fr and H/R ratio using the regression analysis and generalized form is given in equation (6) with the coefficient of determination equals to 0.968. 𝑎 = −0.06175 𝐻 𝑅 𝑏 = 0.9104 𝐻 𝑅

−7.52

−2.18

𝐹𝑟 2.45

𝐹𝑟 0.904

(6a) (6b)

It is to be noted that the correlation for the prediction of vortex tip location given in equations (5 and 6) is obtained from observation of experiments in the air-glycerin fluid pair. Developed correlation is validated with random data from experimental observation and a maximum error of


Page 23 of 47

±12% is obtained. It is also to be mentioned that the proposed correlation is applicable for Froude numbers within the range of 2.5 to 5.7 at submergence ratio of 1.4 to 1.57.

y/H = a ln(τ) + b

0.9

2

-0.07

1.5

a

0.5

-0.19

0.3

-0.25

0

200

400 600 τ = t/t* a)

800

H/R = 1.57

-0.13

b

Logarithmic Fit

H/R = 1.417

1.75

Experimental 0.7

y/H

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


H/R = 1.417

1.25

H/R = 1.57 2.5

3.5

Fr

1 4.5

5.5

b)

2.5

3.5

Fr

4.5

5.5

c)

Figure 9. a) Fitting of the vortex tip evolution using the logarithmic profile and b-c) variation of coefficients of the logarithmic equation as function of Froude number and H/R ratio.

3.4. Initiation of disc inclination for generation of asymmetric interface profile We established the free surface vortex formation in the previous section when the axis of the externally applied field is normal to the nominal interface profile. Here, we present the evolution of free surface vortex in a case when the axis of rotational field axis is oriented at an angle to the gas-liquid interface. Figure 10(a) shows the series of experimental photographs which depict the comparison of steady vortex profile between the horizontally aligned and inclined disc orientations. Here, 𝜃is the angle made by the surface of the disc with the horizontal. The photographs shown in Figure 10(a) are taken for same rotation of the disc with its fixed submergence about the point of inclination. It is evident from the figure that amount of entrained air core has remained almost same for all the cases of disc inclination, however, the vortex profile has shifted toward the upper surface of the disc, which is having least submergence. The horizontal shift of the vortex profile is clearly evident from Figure 10(b) due to its superimposition on the same plot for all the inclinations of the disc. It can also be observed that



vortex profile is symmetric about the axis of the disc for its horizontal alignment, whereas it has become asymmetric upon inclination. The extent of asymmetry in the interface profile has also increased upon increase of disc inclination (Figure 10(b)).

Figure 10. a) Experimental photographs depicting the horizontal shift of vortex profile from the axis of disc upon its inclination with the horizontal, b) quantification of entrained air core in terms of dimensionless parameters and c) measure of horizontal shift of the steady vortex tip and its inclination with vertical as function of disc angle.


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A representative case to compare the symmetric and asymmetric nature of vortex profile is shown in the inset of Figure 10(b) where the tip of both the vortices is positioned at the same location. The asymmetry in vortex boundary is observed due to the presence of differential rotational field around it. The portion of vortex more close to the disc surface experiences enhanced inertia than the remaining segment. Therefore, it draws more fluid and results in asymmetric interface profile. Efforts are also extended to measure the inclination 𝜑 and horizontal shift 𝑋𝑠 of the vortex tip with reference to the axis of symmetry. Both these parameters are represented in the inset of Figure 10(c). It is observed from Figure 10(c) that vortex tip inclination and its horizontal shift have increased together with an increase of disc inclination at an almost linear rate.

3.5.Comparison of vortex structures generated in water and glycerin The present study also aims at the characterization of vortex profiles upon variation of fluid parameters. Water is used as a substitute for glycerin and experiments are performed for both disc speed and its submergence variation, to elucidate the distinction in vortex profiles. A representative case to depict the relative comparison of vortex profiles between two different liquids is shown in Figure 11, where the disc rotation and its submergence, therefore, H/R ratio and Froude number are kept constant. Due to the variations of the liquid parameters, the Galileo number 𝐺𝑎 =

𝑅𝑒 2 𝐹𝑟

is varied, which signifies the relative comparison between the disc rotation

induced inertia and the viscous dissipation at fixed Froude number. Moreover, Figure 11(a-b) reveals that amount of air core penetration is significantly larger for water than its corresponding case of glycerin under the same set of parameters. This draws the important conclusion that apart from the rotational inertia of disc and gravitational pull, liquid viscosity also has significant



contribution in the origination of free surface vortex and subsequent gas accumulation. The higher viscosity of the glycerin has led to more viscous dissipation of the disc rotation and less rotational inertia in the surrounding medium. The difference in the inertial field is well reflected in the assessment of the downward axial pull for both the cases (Figure 11(c)). It can be observed from the figure that magnitude of inertial pull is higher for the case of the air-water medium 𝐺𝑎 ≈ 109 than the air-glycerin fluid pair 𝐺𝑎 ≈ 103 . Moreover, the present variations of vortex penetration are consistent with the previous studies utilizing impeller type agitators in stirred tanks.

Ga = 3250

Ga = 1.80 X 109

Ga = 2900

Ga = 1.60 X 109

0.25 Water

0.2

FI (N)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 47

Glycerin

0.15 0.1 0.05

0 Fr ≈ 8.80; H/R ≈ 2.6

c)

Fr ≈ 9.86; H/R ≈ 2.07

Figure 11. Comparison of the vortex profiles at same rotation (3000 rpm) and initial submergence of the disc for different fluid pairs: a) Fr = 8.80, b) Fr = 9.86 and c) assessment of inertial pull in axial direction for the two cases.


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3.6. Analysis of the fluid streams using particle tracking The present section extends our efforts towards characterization of some physical insights of the phenomenon using the particle tracking inside the flow field. A buoyant solid particle is seeded at the gas-liquid free surface sufficiently away from the axis of the disc and it is allowed to transverse along with the flow. The particle has started its motion from the rest with the initiation of disc rotation. Its motion is captured using a high-speed camera at 25 frames/sec. After obtaining the experimental photographs, image analysis, as described in section 3.1, has been conducted to obtain the particle positions at different time instants. A typical plot depicting the radial and axial motion of the particle around the axis of the disc is shown in Figure 12(a). The trajectory followed by particle inside the flow field is described in different stages (I-VI) using the particle position data at equal time intervals. The initial phase of particle traversal reveals its radial motion towards the axis of the externally applied field. However, one needs to note that particle motion is not purely radial during stage I, in spite, it has tangential component (as shown in Figure 13) as well which can be visualized from the top plane. Once the particle reaches close to the axis of rotation it continues its swirling motion, though, its radial traversal seizes. Axial penetration at the initial stage is slow, which enhances later on due to the closeness of the disc. When the particle has reached the surface of the disc its axial motion has seized and it started outward radial movement due to the centrifugal pull. After the particle has reached the periphery of the disc, it is thrown by the downward moving liquid streams and after attaining a certain depth, the particle has further raised towards the free surface (stage III). Subsequent follow up of particle history reveals that it has once again moved in the downward direction (stage IV) to reach towards the epicenter of the inertial field by following circulation and continuous decrease in the radial path.



Figure 12. a) Trajectory of the particle placed at the gas-liquid interface away from the axis of rotation obtained upon rotation of the disc and b) measure of its axial velocity as function of nondimensional time for Fr = 5.97 (1500 rpm) and H/R = 1.41.


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Figure 13. Polar plot depicting particle trajectories captured from the top plane for Fr = 5.97 (1500 rpm) and H/R = 1.41: a) initial motion of particle towards the axis of rotation, b) swirling motion of particle and subsequent circulation in radial direction, c) transition between two consecutive radial circulations and d) evolution of periodic circulation patterns. The scale of radial position is in m.



During the transition from stage IV to V particle drifts away from the disc and resulted in a clockwise axial circulation, which further shifts in counter clock direction (stage VI). It is also to be noted that particle has shown tangential motion as well, which is reflected as the sudden transfer of the trajectory in the plane of visualization. Stages I-IV is of transient nature whereas phase V-VI represents the periodic axial circulations, which continue during the subsequent rotations of the disc. Efforts are also extended to calculate the axial velocity of the particle to visualize the transition of trajectories. Figure 12(b) shows the variation of the axial velocity of the particle scaled with respect to the maximum tangential velocity of the disc, as a function of dimensionless time. In stage I and most portion of stage II axial velocity is almost negligible and it instantaneously picks up as it reaches the zone of strong inertial pull. Near the end of stage II when the particle reaches close to the disc surface it experiences sudden deceleration and velocity becomes zero. For the transition from stage II to III, initially the axial velocity is almost zero for few points and it suddenly accelerates from the periphery of the disc in downward direction followed by instantaneous transfer of field in the upward direction. Moreover, particle has shown a gradual increase in axial downward velocity when it again accelerates from the free surface. Further,the particle has shown a sudden transfer of velocity on the transition from stage IV-V. In both the stages V and VI the axial velocity of the particle has taken almost sinusoidal profile due to circulations and has shown periodic trend upon further rotation of the disc. The particle trajectory described above is captured from the front view, however, efforts have also been extended to measure its evolution from the top and to obtain radial velocity. A series of polar plots are given in Figure 13 to elucidate the particle dynamics starting from the transient phase to establishment of periodic circulations in the radial direction. Figure 13(a) represents the radial motion of the particle (stage I) towards the epicenter of the fields followed by its swirling


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motion (stage II). Near the axis of rotation particle swirls along with its axial penetration (Figure 12(a)) and thrown in radially outward direction by the centrifugal field at the disc surface. This led to the initiation of the circulatory motion of the particle in the radial direction (Figure 13(b)). During the circulation phase (stage III), the first particle moves in radially outward direction and then it is again pulled by the centripetal action towards the disc center once it reaches the gasliquid interface. This cycle repeats as depicted in Figure 13(c) for the transition between stage III and IV. The periodic cycle of radial circulations is shown in Figure 13(d) upon the subsequent progress of the time.

Figure 14. Variation of dimensionless radial velocity of the particle as function of time during its circulatory motion. We also obtained the radial velocity of the particle during its motion in Figure 14. The particle has accelerated towards the axis of the external field upon rotation of the disc. Particle radial velocity has exhibited sinusoidal nature before it reaches the axis of the disc (stage I), however, its magnitude has decreased progressively and becomes zero during the axial penetration of particle (stage II) inside the liquid. The particle has again accelerated and achieved very high radial velocity due to centrifugal action in stage III. Moreover, it has decelerated as it progressed



in radially outward direction and has attained zero velocity at the point of contra-flexure near the free surface at stage III. From this point, the particle starts its traversal again towards the radial inward direction. After the particle has once more reached the center of disc it accelerates and repeats the cycle (stage IV-VI). This results in periodic variation of particle radial velocity for subsequent rotation of the disc with almost same cycle time. The particle trajectories and its velocity patterns reveal the presence of axial and radial circulations in the surrounding liquid medium. The axial velocity variation of stages V and VI elucidates that liquid medium, thrown in by centrifugal action of the disc in radially outward direction, has risen toward the free surface by following a circulating trajectory. Also the liquid near the vortex core moves in the downward direction to replenish the radially outward moving fluid medium.The radial velocity of the particle signifies that liquid streams near the core have high rotational inertia and it decreases upon moving towards the radially outward direction. Moreover, it is also reflected that velocity of the liquid medium near the disc surface is comparable to that of its tangential velocity, whereas it has considerably decreased, at other peripheral locations. It is also to be mentioned that the present circulations can be easily obtained by initially placing the particles at nearly same locations. Though, the specific path of circulations may change, however, the generalized nature of the circulations will still remain the same. Moreover, in the present experiments the trajectory of the particle is obtained using image analysis, therefore, the measurement of coordinates is quite reliable and accurate (99%).

3.7. Discrete bubble pinch off at high Froude number In the preceding discussion, the genesis and dynamics of stable free surface vortex core is elucidated. However, it is to be noted that the stability of the vortex core can be distorted at very


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less submergence and sufficiently high rotation of the disc. Under such conditions, the tip of the stable vortex profile has grown in the radial direction over the surface of the disc. When the radially expanding air core has touched the periphery of the disc, it creates aninstability, which led to the pinch off of air in the form of discontinuous bubbles. The population density of these bubbles has increased upon continuation of the disc rotation. A series of experimental photographs are given in Figure 15 to characterize the bubble trajectories and their shape transformation in the region of differential velocity fields. Figure 15(a-b) depicts the bubble trajectory for two different bubbles, pinched off from the air core. The first bubble has shown circulation in the clockwise direction, which is almost identical to the circulation at stage V of particle trajectory shown in Figure 12(a). Second bubble (Figure 15(b)) has also shown the circulatory trajectory, however, it has shifted from the plane of observation due to its tangential motion. The trajectory of both the bubbles is plotted in terms of disc radius and submergence based spatial non-dimensionalization on the right portion of Figure 15(a-b). Here, snapshots of the bubble are also shown in the insets at different spatial locations along their trajectory. Next, we have shown a filament of air in the form of the thin lamella in the zone of high inertia near the periphery of the disc (Figure 15(c)). This filament of air has risen towards the free surface due to buoyancy effects and axial circulations in the surrounding liquid medium. Due to the decrease of inertial field, the collapse of the filament (stage II) has taken place and it has acquired the ellipsoidal shape (stage III-IV) during its upward axial journey. The bubble has subsequently taken the spherical shape (stage V) when it reaches close to the free surface, due to the dominance of surface tension over the inertia.



Figure 15. Experimental snapshots depicting the bubble trajectories after immediate occurrence inside the liquid as function of dimensionless time (t/t*) with a constant step of 4.39 (a-b) and c) transformation of an air filament to spherical shaped bubble due to its presence in the zone of differential fields (dimensionless time step between consecutive snapshots is 8.79). These snapshots are taken at Fr = 11.86 (3000 rpm) and H/R = 1.43.

4. CONCLUSIONS Present study depicts a comprehensive experimental investigation of vortex genesis and its fluidic understanding from the fundamental aspects. The key findings of the investigation are elucidated as:


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

Upon gradual increase of submerged disc rotation, it is observed that the interface

remains unperturbed at low inertia, however, there comes a balance of dominant forces when it forms a steady dip. Such interfacial depression of the free surface is significant at higher rotations and is termed as free surface vortex. The disc rotation characterized in terms of Froude number and its initial submergence ratio (H/R), is found to be the significant parameter controlling the vortex dynamics. 

Temporal measurements of vortex tip have revealed the logarithmic variation at moderate

dominance

of

inertia

over

gravity

𝐹𝑟 = 5.16; 𝑓𝑐 = 4.66 × 105 𝑁 𝑚3 ∀ 𝐻 𝑅 = 1.42 .

However, it became almost linear when the inertial pull is sufficiently high over the gravity 𝐹𝑟 = 5.96; 𝑓𝑐 = 6.21 × 105 𝑁/𝑚3 ∀ 𝐻 𝑅 = 1.42 . The centrifugal pull imparted by the disc rotation to the neighboring liquid medium and subsequent replenishment rate of liquid from the free surface are responsible for such a typical vortex dynamics. 

The experimentally obtained vortex profiles have shown self similarity when scaled with

respect to the maximum depth of the vortex tip. Therefore, a generalized equation based on parabolic and logarithmic fit in the regions of differential inertia is proposed to mimic the vortex profile. The axial pull imparted by disc rotation which holds the entrained air volume against gravity has shown a linear trend with variations of Fr and H/R ratio. 

Occurrence of asymmetric vortex profiles is also reported upon inclination of rotating

disc axis. Apex of the asymmetric vortex has shifted in almost linear manner from the axis, upon increase of disc inclination. Increase in the volume of steady entrained air core is observed upon decrease in viscosity of the liquid medium, characterized in terms of Galileo number variation from 𝐺𝑎 ≈ 103 to 𝐺𝑎 ≈ 109 , for fixed Froude number.





Utilization of the Langrangian particle tracing has resulted in the visualization of periodic

axial and radial circulations inside the liquid streams. In addition, using the temporal history of particle trajectories, the axial and the radial velocities are also quantified. At high disc rotation (Fr = 11.86) and low initial submergence ratio (H/R = 1.43), the vortex core apex touches the disc surface, grows in radial outward direction and sheds off discrete air entities. 

Trajectories of the discontinuous filaments pinched off from the main vortex core have

also shown similar circulations to that of the particle. Discontinuous air entities are like elongated filaments while pinched off and have transformed into spherical shaped bubbles via the route of ellipsoidal air volumes depending upon the interplay of inertia and surface tension.

AUTHOR INFORMATION Corresponding author * E-mail ID: [email protected] ACKNOWLEDGEMENTS MP wishes to acknowledge the partial financial support from IIT Roorkee, India via Summer Undergraduate Research Award (SURA)-2017. References (1)

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List of figure captions Figure 1. a) 3D model of experimental facility, b) planer view of the test setup, c) schematic representation of different velocity components and d) velocity profile on the top plane of the disc depicting combined vortex field. Figure 2. a) Experimental snaps depicting the evolution of vortices from free surface obtained at disc rotation based Fr = 9.66 and H/R = 2.16, b) dimensionless vortex profiles superimposed on the same plot at different time instants and c) non-dimensional volume of vortex core as function of dimensionless time. Time is given in non-dimensional form using the inertial time scale. Figure 3. Experimental photographs depicting the steady free surface vortex profiles obtained at wide range of rotational speeds characterized in terms of Froude number at fixed submergence H/R = 1.42 of the disc. The rpm of the disc has been varied from 600 in steps of 100. Figure 4. a) Non-dimensional steady vortex profiles superimposed on the same plot for different Froude number and quantification of vortex tip penetration rate inside the liquid pool at fixed submergence of the disc and b) proposition of combined parabolic (near core) and logarithmic fit (away) for generalization of vortex profiles. Only one half of the symmetric fit has been shown in the figure. Figure 5. Variation of inertial pull in axial direction as function of rotational Froude number at fixed submergence of the disc, H/R =1.42. Figure 6. Experimental photographs depicting the steady free surface vortex profiles obtained for range of disc submergence characterized in terms of H/R ratio and Froude number at fixed rotational inertia of the disc (3000 rpm).



Figure 7. Non-dimensional steady vortex profiles shown for different Froude numbers, initial submergence of the disc and quantification of vortex tip penetration rate inside the liquid pool at fixed rotation of the disc. Figure 8. Variation of axial pull induced by disc rotational inertia as function of Froude number and H/R ratio upon alteration in initial submergence of the disc at fixed external rotational field. Figure 9. a) Fitting of the vortex tip evolution using the logarithmic profile and b-c) variation of coefficients of the logarithmic equation as function of Froude number and H/R ratio. Figure 10. a) Experimental photographs depicting the horizontal shift of vortex profile from the axis of disc upon its inclination with the horizontal, b) quantification of entrained air core in terms of dimensionless parameters and c) measure of horizontal shift of the steady vortex tip and its inclination with vertical as function of disc angle. Figure 11. Comparison of the vortex profiles at same rotation (3000 rpm) and initial submergence of the disc for different fluid pairs: a) Fr = 8.80, b) Fr = 9.86 and c) assessment of inertial pull in axial direction for the two cases. Figure 12. a) Trajectory of the particle placed at the gas-liquid interface away from the axis of rotation obtained upon rotation of the disc and b) measure of its axial velocity as function of nondimensional time for Fr = 5.97 (1500 rpm) and H/R = 1.41. Figure 13. Polar plot depicting particle trajectories captured from the top plane for Fr = 5.97 (1500 rpm) and H/R = 1.41: a) initial motion of particle towards the axis of rotation, b) swirling motion of particle and subsequent circulation in radial direction, c) transition between two consecutive radial circulations and d) evolution of periodic circulation patterns. The scale of radial position is in m.


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Figure 14. Variation of dimensionless radial velocity of the particle as function of time during its circulatory motion. Figure 15. Experimental snapshots depicting the bubble trajectories after immediate occurrence inside the liquid as function of dimensionless time (t/t*) with a constant step of 4.39 (a-b) and c) transformation of an air filament to spherical shaped bubble due to its presence in the zone of differential fields (dimensionless time step between consecutive snapshots is 8.79). These snapshots are taken at Fr = 11.86 (3000 rpm) and H/R = 1.43.



List of Table title Table 1: Studies mentioning characteristics of free surface vortex available in literature. Table 2: Physical properties of the liquids used in experimentation at room temperature of 25°C.


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For table of contents only Air Glycerin

H/R

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Fr


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