Spectral response of a droplet in pulsating external flow fieldP. Deepu and Saptarshi Basu

Citation: Physics of Fluids (1994-present) 26, 022103 (2014); doi: 10.1063/1.4865550 View online: http://dx.doi.org/10.1063/1.4865550 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/26/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantitative analysis of the dripping and jetting regimes in co-flowing capillary jets Phys. Fluids 23, 094111 (2011); 10.1063/1.3634044 Investigation of the effect of external periodic flow pulsation on a cylinder wake using linear stability analysis Phys. Fluids 23, 094105 (2011); 10.1063/1.3625413 Oscillatory shear induced droplet deformation and breakup in immiscible polymer blends Phys. Fluids 21, 063102 (2009); 10.1063/1.3153304 Oscillatory convective structures and solutal jets originated from discrete distributions of droplets in organicalloys with a miscibility gap Phys. Fluids 18, 042105 (2006); 10.1063/1.2192531 Note on the memory force on a slightly eccentric fluid spheroid in unsteady creeping flows Phys. Fluids 18, 013301 (2006); 10.1063/1.2162468

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PHYSICS OF FLUIDS 26, 022103 (2014)

Spectral response of a droplet in pulsatingexternal flow field

P. Deepu and Saptarshi Basua)

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India

(Received 29 August 2013; accepted 30 January 2014; published online 12 February 2014)

A droplet introduced in an external convective flow field exhibits significant multi-modal shape oscillations depending upon the intensity of the aerodynamic forcing.In this paper, a theoretical model describing the temporal evolution of normal modesof the droplet shape is developed. The fluid is assumed to be weakly viscous andNewtonian. The convective flow velocity, which is assumed to be incompressibleand inviscid, is incorporated in the model through the normal stress condition at thedroplet surface and the equation of motion governing the dynamics of each mode isderived. The coupling between the external flow and the droplet is approximated tobe a one-way process, i.e., the external flow perturbations effect the droplet shapeoscillations and the droplet oscillation itself does not influence the external flowcharacteristics. The shape oscillations of the droplet with different fluid propertiesunder different unsteady flow fields were simulated. For a pulsatile external flow, thefrequency spectra of the normal modes of the droplet revealed a dominant responseat the resonant frequency, in addition to the driving frequency and the correspondingharmonics. At driving frequencies sufficiently different from the resonant frequencyof the prolate-oblate oscillation mode of the droplet, the oscillations are stable. Butat resonance the oscillation amplitude grows in time leading to breakup dependingupon the fluid viscosity. A line vortex advecting past the droplet, simulated as anisotropic jump in the far field velocity, leads to the resonant excitation of the dropletshape modes if and only if the time taken by the vortex to cross the droplet is lessthan the resonant period of the P2 mode of the droplet. A train of two vortices in-teracting with the droplet is also analysed. It shows clearly that the time instant ofintroduction of the second vortex with respect to the droplet shape oscillation cycleis crucial in determining the amplitude of oscillation. C© 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4865550]

I. INTRODUCTION

Various technological and environmental processes involve two-phase flows. Two phase liquid-gas flows (liquid droplets dispersed in a gaseous flow), in particular, occur in spray driers, gas turbineengines, industrial furnaces, cloud evolution, and aerosol generation. A fundamental understandingof the phenomena, therefore, is essential to optimize the industrial production and/or to predictthe parameters characterizing the processes. However, the interaction between the continuous anddispersed phase presents a challenging subject of research in fluid dynamics. The mass, momentum,and energy transfer between the two phases lead to highly nonlinear two-way coupling. Additionalphenomena like droplet collision, atomization, and chemical reaction further add to the complexityof the process.

The surrounding gas phase offers an unsteady chaotic environment for the droplets, be it the fueldroplets in a turbulent combustor or the cloud droplets exposed to winds. Under the aerodynamicloading, the droplets undergo considerable shape deformation. This shape deformation in turn

a)Electronic mail: sbasu@mecheng.iisc.ernet.in

1070-6631/2014/26(2)/022103/22/$30.00 C©2014 AIP Publishing LLC26, 022103-1

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022103-2 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

changes the aerodynamic pressure distribution around the droplet which can lead to multimodalshape oscillations. Since Rayleigh’s classical energy analysis1 of the natural oscillation of a fluidsphere suspended in a quiescent medium, a lot of research effort has been devoted to free and forcedoscillations of an isolated droplet placed in another fluid (see, e.g., Ref. 2 for a review). His modelwas later extended by Reid3 to include the effect of fluid viscosity.

In recent years, numerous experimental investigations have offered valuable further physicalinsight into the behavior of oscillating droplets. By employing optical deflection technique, Sharp4

examined the resonance modes of microlitre sessile droplet disturbed by a pulse of nitrogen gas. Hereported that the height of resonant peak decreases while the resonant width increases with increasingviscosity. Oh5 analyzed electrowetting driven shape oscillations of a sessile droplet. Shen6 studied thenon-axisymmetric deformations of acoustically levitated droplets through modulation of the acousticpower. Behavior of capillary waves on the surface of a sessile droplet under periodic forcing of thesubstrate was studied by Vukasinovic.7 They found out that when the amplitude of forcing is highenough, the surface perturbations result in interfacial breakup and droplet atomization. The dynamicsof sessile droplet under the combined effect of support vibration and an acoustic pressure wave wasrecently investigated by Deepu et al.8, 9 Hill and Eaves10 studied the oscillations of a magneticallylevitated uncharged droplet excited by a puff of air. Forced oscillations of droplets have been thesubject of many numerical studies as well.11–18

But all these studies on forced droplets focus on excitation of the capillary modes via electric,magnetic, acoustic or support excitation. It is surprising to note that, the oscillation dynamics ofthe droplet induced under the conditions that exist in practical liquid-gas systems, which werementioned earlier, has not received much attention from researchers. However, many numerical19–22

and experimental23, 24 studies have been reported investigating droplet deformation and breakupinduced in such conditions. Most widely used mathematical models for predicting the disintegrationof an isolated droplet under aerodynamic forces are, among others, Taylor Analogy Breakup (TAB)model25 and Droplet Deformation and Breakup (DDB) model.26 But, neither of these models iscapable of simulating the spatio-temporal modes of oscillations exhibited by the droplet. Otherhydrodynamic models that describe the characteristics of droplet gas interaction concentrates onlyon predicting the droplet motion and trajectory and thereby the droplet (or particle) dispersioncharacteristics of the flow.27

Gaseous flow induced droplet oscillation was studied in the recent experiments conducted byDeepu et al.28 Using high speed imaging and hot-wire anemometry, the shape oscillations exhibitedby a droplet suspended at the centre of an air jet was examined. Though the ambient flow has constantmean velocity and not pulsatile in nature, the droplet showed different oscillation levels dependingon the fluid properties and droplet Reynolds number. They found out that the inherent broadbandfluctuations of the incoming air jet drive the steady state oscillations of the droplet.

The objective of this paper is to develop a theoretical model describing the dynamics of theshape of a droplet placed in a convective flow field. To our knowledge, there have been no attemptsin the literature to develop such a model to predict the multimodal shape oscillations of a dropletunder aerodynamic loading. This model hence has relevant theoretical and applicative implications.For example, droplet deformation is the precursor to secondary breakup and affects the global sprayfeatures. Thus understanding the evolution of spatially resolved deformation of the droplet enablesone to predict the spray characteristics. Furthermore, in the existing droplet evaporation models,under the assumption of low Weber number, the drop is assumed to be always spherical. The presentmodel can be easily incorporated to include the effect of shape fluctuations of the droplet on the heatand mass transfer process. In Doppler method based diagnostics, the shape of the spray dropletsaffect the light scattering properties. By predicting the shape dynamics of the droplets, a correctionalgorithm can be devised to improve the accuracy of such measurements.

In the present study, discussion is restricted to axisymmetric oscillations of a non-evaporatingdroplet induced by the one-way coupling (the effect of the deformation of the droplet is assumed tobe not strong enough to alter the flow structure around it) with the external flow field. The dropletshape is described using a truncated series of the Legendre polynomials and the spatial distributionof pressure around the deforming droplet is assumed to be the same as that around a solid sphereplaced in a potential flow. The dynamical equations for each mode are derived and are numerically

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022103-3 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

solved for three different unsteady ambient flows: (i) pulsatile flow, (ii) advecting vortical flow,and (iii) velocity field with broadband energy spectrum. The reference fluid selected is water and aparametric study is carried out to investigate the effect of fluid properties on the spectral responseof the droplet. The semi-analytical model is also capable of predicting droplet breakup when theselected fluid and flow properties warrant the unstable growth of shape perturbation.

II. PROBLEM FORMULATION

The present study considers an initially spherical droplet of unperturbed radius of 1 mm sus-pended in an ambient uniform flow field as shown in Fig. 1. The direction of the free stream velocityis vertically upwards simulating the condition of a droplet placed in a vertical air jet. The droplet fluidproperties are taken to be that of water and the external fluid properties are taken to be that of air. Thesurface tension and viscosity of the droplet fluid are changed to study the effect of fluid properties ondroplet shape oscillations as explained later. A spherical coordinate system (r, θ , ψ) with the originat the centre of the unperturbed droplet is selected such that the polar angle, θ is measured from theforward stagnation point. As the translational mode of the droplet is not considered in this study,the origin always remains at the centre of mass of the deforming droplet and the relative effect of theexternal flow on the droplet will be analyzed from the droplet frame of reference. Since oscillationsare assumed to be axisymmetric, the azimuthal angle, ψ does not come into play in the followinganalysis.

The governing equations for the incompressible fluid flow inside the droplet can be written asContinuity equation

∇ · u = 0. (1)

Momentum conservation equation

ρ

[∂u∂t

+ (u · ∇)u]

= −∇ p + μ∇2u. (2)

Boundary conditions can be cast as follows.Kinematic boundary condition

∂ F

∂t+ (u · ∇)u = 0. (3)

Normal stress at the boundary

n · [(σ out − σ in) · n] = γ∇ · n. (4)

(iii) flow with broadband frequency spectrum

(ii) advecting vortices

(i) pulsatile flow (constant frequency)

mild oscillations

vigorous oscillations leading to droplet

breakup

oscillation level depends

upon fluid properties,

frequency and amplitude of fluctuating velocity and the crossing time of the

vortex.

r

FIG. 1. Schematic of problem configuration and solutions obtained.

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022103-4 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

Tangential stress at the boundary

n × [(σ out − σ in) · n] = 0. (5)

Here t denotes time, ρ denotes the density, μ denotes the dynamic viscosity, p denotes thepressure, γ denotes the surface tension, u denotes the velocity vector, and n denotes the outward unitnormal vector at the droplet surface. F (r, θ , t) = 0 denotes the instantaneous shape of the dropletinterface. If the droplet shape is decomposed into a series of the Legendre polynomials, F can bewritten as

F ≡ r − R − ε∑∞

n=2an(t)Pn(cos θ ), (6)

where r is the radial coordinate of the perturbed droplet interface at the polar angle θ , R is theunperturbed droplet radius, Pn is the nth order Legendre polynomial, an is the time-dependantamplitude of the nth Legendre mode and 0 < ε � 1. n = 0 is the mode associated with thevolumetric changes of the droplet and hence is discarded in the present analysis on account of massconservation of the droplet (non-evaporating). Likewise, n = 1 mode represents the translationalmotion of the droplet without any deformation. Hence this mode is also not considered.

σ denotes the hydrodynamic stress tensor:

σ = −pI + μ(∇u + ∇uT ). (7)

I is the second order unit tensor. σ in and σ out represent the stress tensor evaluated on the inner (liquid)and outer (gas) sides of the droplet surface, respectively. Following the formulation of Prosperetti,29

the velocity and pressure fields are decomposed into irrotational and rotational parts assuming thatthe viscous effects are relatively weak:

u = up + uv, p = pp + pv. (8)

Since up is the irrotational flow velocity it can be expressed as a gradient of velocity potential:

up = ∇ϕ. (9)

The mass and momentum equations along with the boundary conditions are also decomposed intothe corresponding potential and viscous parts, which are further linearized to solve for the respectivevelocity and pressure fields. The relevant results are given below.

ϕ = ε∑∞

n=2

rn

n Rn−1an(t)Pn(cos θ ), (10)

pp= − ε∑∞

n=2

rn

n Rn−1an(t)Pn(cos θ ), (11)

pv= − εμ∑∞

n=2

(n + 1)(n − 1)

n

rn

Rn−1an(t)Pn(cos θ ). (12)

Neglecting the gas phase viscosity, Eq. (4) can be written as

− pout= − pin+2μ∂2ϕ

∂n2+γ∇ · n, (13)

where 2μ∂2ϕ/∂n2 denotes the normal viscous term.The pressure inside the droplet can be written as

pin = p0 + pp + pv, (14)

where p0 is the static pressure. The spatial variation of the outside pressure pout is assumed to havethe same functional form of pressure around a sphere placed in an external potential flow. Thus30

pout = p∞+1

2ρ∞U 2

∞(1−9

4sin2θ ), (15)

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022103-5 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

where p∞ and U∞ are the ambient pressure and free stream uniform velocity at infinity (relative tothe droplet), respectively. This spatial dependence of the outside pressure at the droplet surface isassumed to be unaffected by the small deformation of the droplet. However, the unsteadiness of theambient flow is reflected in pout through the time dependence of U∞.

The classical derivation of Eq. (15) assumes the flow to be steady. For an unsteady irrotationalflow, as is in our case, one needs to use the unsteady from of Bernoulli’s equation to determinethe pressure distribution. In the following, we justify the use of the functional form of pressuredistribution obtained for a steady irrotational flow in our model describing an unsteady flow.

The unsteady Bernoulli’s equation applied on the external unsteady irrotational flow is given by

∇[∂ϕout

∂t+ 1

2ρ∞ |uout|2 + pout

ρ∞

]= 0. (16)

Here the subscript “out” stands for the gas phase close to the droplet interface. Let us compare theorder of magnitude of the first two terms on the left hand side of the above equation. For a uniformflow, ϕout ∼ U∞ R. Being in line with the analysis presented in Sec. III, the ambient velocity takesthe form:

U∞=〈U∞〉+κsin2π f t, (17)

where the angle brackets denote time average. Thus we get

∂ϕout

∂t∼Rκ2π f, (18)

whereas the second nonlinear term,

1

2ρ∞ |uout|2 ∼ 1

2ρ∞U 2

∞. (19)

For simulating the response of droplet, we chose the value for 〈U∞〉 as 7.8 m/s and droplet radius,R = 1 mm. The amplitude of the fluctuating velocity component, κ is calculated (as explained in detailin the subsequent sections) from the constraint that the ratio of energy of the fluctuating componentto that of the mean velocity component should be of the order of 0.001, i.e., κ2/〈U∞〉2 ∼ O(0.001).Under this constraint, it can be shown that U∞2 ≈ 〈U∞〉2. The values mentioned above were chosento match the experimental conditions used by Deepu et al.28 Taking ρ∞ to be the density of air(∼O(1)) and the maximum frequency component in the external velocity profiles used for currentsimulation to be 1000 Hz, it is found that the time gradient of velocity potential is about two ordersof magnitude less than the second nonlinear term. Neglecting this unsteady term, one gets the steadyform of Bernoulli’s equation, but with the difference that the nonlinear term is time-dependentthrough the unsteadiness of outside velocity. This further leads to the classical form of the pressuredistribution around a sphere in a potential flow (Eq. (15)) but with the difference that the free streamvelocity U∞ is a function of time.

This approach which is valid under the stated conditions leads to a much simplified model andsaves a lot of computational effort. The consequence of this simplification is that any disturbance inthe ambient flow is felt everywhere instantaneously. Hence the droplet sees a homogeneous uniformambient (at infinity) flow at every time instant and the pressure distribution around the droplet willbe that around a sphere in a potential flow corresponding to the free stream velocity at that instant.The propagation of a disturbance and its effect on the shape dynamics of the droplet through phaselag can be studied by retaining the time gradient of velocity potential in Eq. (16) and is the subjectof future work.

Upon substituting equations into Eqs. (14) and (15) into Eq. (13) and taking into account thatfor a droplet under equilibrium (in a quiescent surrounding fluid)

p0 − p∞ = 2γ

R, (20)

we get

− f (θ )1

2ρ∞U 2

∞ = −pp − pv + 2μ∂2ϕ

∂n2+ γ∇ · n − 2γ

R, (21)

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022103-6 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

where

f (θ ) =⎧⎨⎩

(1 − 94 sin2 θ ), if θ ≤ θsep

0, if θ > θsep

. (22)

This function generates an external pressure distribution, pout equal to that of potential flow arounda sphere till an angle of separation, θ sep and ambient pressure, p∞ beyond θ sep. This is to take intoaccount the presence of the wake behind the droplet at the Reynolds number considered.30 Discardingthe change in the location of the separation point of boundary layer due to the change in externalflow conditions and the droplet deformation itself, a constant θ sep = 150◦ is used throughout thesimulation presented in this paper. Since all other terms in Eq. (21) are expanded in terms of Legendrepolynomials, in order to derive the equation of motion describing the droplet shape oscillations weneed to expand the function f (θ ) in terms of Legendre polynomial as

f (θ ) =∑∞

n=2kn Pn(cos θ ). (23)

The coefficient kn can be determined using the orthogonality property of Legendre polynomialas

kn = 2n + 1

2∫π

0 f (θ )Pn(cos θ ) sin θdθ. (24)

The mean curvature appearing in Eq. (21) is expressed in terms of the Legendre polynomials tothe first order of ∈ as31

∇ · n =[

2

R+ ε

R2

∑∞n=2

(n − 1) (n + 2)an(t)Pn(cos θ )

]. (25)

Finally, upon substituting equations (10)–(12), (23), and (25) into Eq. (21), and evaluatingthe equation at the droplet interface, one arrives at the equation of motion governing the temporalevolution of each mode:

an + 2(n − 1)(2n + 1)μ

ρR2an + n(n − 1)(n + 2)

γ

ρR3an = − n

2ερRknρ∞U∞2. (26)

A numerical scheme is employed to solve this set of uncoupled equations to obtain the timeevolution of the shape modal amplitudes. A fourth order accurate Runge-Kutta method with adaptivetime step was used for time integration to solve the problem with the following initial conditions:

an(t = 0) = an(t = 0) = 0. (27)

As will be shown later, the energy carried by each mode decreases exponentially with the modenumber, n. Hence only first few modes are needed to describe the droplet shape with a high degreeof accuracy. In our analysis, we considered the modes from n = 2–6. The parameter ε was setequal to 0.001 during the simulations. This choice can be done arbitrarily because the role of ε isto merely scale up the solutions of Eq. (26) as it is a small number appearing in the denominator ofthe forcing term (see Eq. (24)). Finally, when the droplet shape is reconstructed using Eq. (6), themodal coefficients are scaled down by the same factor, εand hence its effect is nullified in our case.A numerical integration scheme was used to determine the value of kn for a given n. The equilibrium,radius of the droplet is fixed as R = 1 mm.

As a reference fluid, water is chosen as the droplet fluid (properties at a temperature of 25 ◦C:surface tension, γ = 0.072 N/m; viscosity, μ = 0.00089724 Ns/m2, and density, ρ = 996.85 kg/m3)and a parametric study of the influence of fluid properties on the shape dynamics of the dropletis performed by varying the fluid viscosity and surface tension about the properties of water. Thevalues of surface tension, γ ∗ and viscosity, μ∗ of the test fluids used relative to that of water are:γ ∗ = 0.3, 0.4, 0.5, 0.6, and 1; μ∗ = 0.5, 1, 3, and 10. The external fluid properties are set toair properties at a temperature of 25 ◦C: kinematic viscosity, υ∞ = 2.5 × 10−5 m2/s and density,ρ∞ = 0.946 kg/m3 (υ∞ is used only for estimating droplet Reynolds number and does not appearin the model due to the inviscid assumption of the external flow).

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022103-7 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

For analyzing the spectral response of the droplet, in addition to Fourier analysis, we use wavelettransform to reveal the time evolution of different frequency components. Wavelet transform usesa family of analyzing functions obtained by scaling and shifting a temporally localized function(called the mother-wavelet) as its basis functions,32 as opposed to Fourier transform which usescomplex exponentials as its basis functions. The continuous wavelet transform of a function g(t) isdefined as

g(s, l) = 1√s

∫ ∞

−∞g(t) ∗(

t − s

l)dt, (28)

where s and l are the translation parameter and dilation parameter, respectively, and ∗ representsthe complex conjugate of . We used the Morlet wavelet, defined as (t) = exp(−t2/2)cos(5t),as the mother wavelet. The dilation parameter is converted into an equivalent Fourier frequencyusing the formula

f (l) = fc

lTs, (29)

where fc is the central frequency of the mother wavelet and Ts is the sampling period of the signal.We impose three types of external flow field. First, to investigate the effect of fluctuations of

single, unique frequencies in the uniform ambient flow, a sinusoidally (at different preset frequencies)varying (non-zero mean flow) velocity field is imposed (see Fig. 1). Following this, we analyze theeffect of a convecting velocity pulse in the flow field which mimics the condition of a line vortex ofspecific strength (and transit time) interacting with the droplet. Finally, as a validation of the proposedmodel, the experimentally measured velocity signal19 is imposed as the free stream velocity profileof the model. The experimental velocity signal is found to exhibit broadband frequency spectralcharacteristics. The experimental and theoretical data related to the response of the droplet to suchan ambient flow fluctuations are compared to each other. In the following sections III A–III C, theresults of the simulations are presented.

III. RESULTS AND DISCUSSION

A. Pulsatile flow

The ambient flow velocity U∞ is temporally perturbed in a sinusoidal fashion with a forcingfrequency f as given by Eq. (17). But the spatial homogeneity of the ambient field (at infinity) is stillmaintained at every instant due to the reasons already explained. The different values of f selected are30, 90, 120, 180, and 200 Hz. The value of mean velocity, U∞, was taken to be 7.8 m/s correspondingto a Reynolds number, Re (= 2ρ∞ 〈U∞〉R/υ∞) of 650 thus justifying the inviscid assumptions usedin the analysis. In experiments19 it is observed that at this mean velocity, the energy contained in thefluctuating component of velocity is around 0.0027 times of the mean part of the velocity. Hencefor realistically mimicking the external flow characteristics, the above criterion is used to evaluatethe amplitude, κ of the fluctuating component of the velocity for each f. Furthermore, this conditionis required for the validity of Eq. (15) as previously mentioned. Mathematically, the criteria isexpressed as ∫ T

0 (U∞ − 〈U∞〉)2dt

〈U∞〉2T≈ 0.0027, (30)

where T is the simulation time period which is 2 s.In Fig. 2, the simulated droplet shapes in the case of water, obtained by superposing the

instantaneous modal shapes, are shown for driving frequencies, f = 120 and 180 Hz. From Eq. (26),the natural frequency of the nth mode of the isolated droplet can be written as (which is the sameresult obtained by Rayleigh1)

fn =√

γ n (n − 1) (n + 2)

2πρR3. (31)

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022103-8 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

f = 120 Hz f = 180 Hz0 69 0 69

0.25 69.25 0.25 69.25

0.5 69.5 0.5 69.5

0.75 69.75 0.75 69.75

1 70 1 70

FIG. 2. Instantaneous shapes (solid curve) of the interface of a water droplet for a pulsatile flow driven at frequency, f =120 and 180 Hz. The dashed line represents the equilibrium droplet shape. Normalized time (with respect to the time periodcorresponding to 120 Hz) is given in the corresponding panel.

The resonant frequency of different modes for the values of surface tension considered in this workis listed in Table I. For the n = 2 mode, the resonant frequency is 121 Hz. Hence the droplet exhibitsa resonance when the flow oscillates at a frequency close to this value. In fact, at f = 120 Hz, thedroplet oscillation amplitude grows in time as shown in Fig. 2. The normalized time is given in eachpanel. In Fig. 2, for normalizing the time, the time period corresponding to 120 Hz (= 1/120 s) isused for both the driving frequencies of 120 and 180 Hz. The reason for selecting this value forboth the cases will be clear in what follows. Compared to the first cycle of oscillation (first columnof panels), the amplitude of oscillation has increased considerably in a later cycle (second columnof panels) ultimately leading to a torus shape (see the second panel in the middle row of Fig. 2).

TABLE I. Resonant frequency (in Hz) of the nth mode as a function ofsurface tension, γ ∗ = γ /γ water, where γ water = 0.072 N/m.

nγ ∗ 2 3 4 5 6

1 121 234 363 506 6660.6 94 181 281 392 5130.5 86 166 257 358 4690.4 77 148 230 320 4190.3 66 128 199 277 363

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022103-9 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 3. Energy fraction versus mode number for n = 2–6.

However, this shape is not a stable one. When the droplet shape reaches such a configuration, theresulting abnormally high value of droplet aspect ratio (defined as the ratio of the droplet dimensionperpendicular to the axis of symmetry to that parallel to the axis of symmetry) is to be regardedas a sign of droplet breakup. When the viscosity of the droplet fluid is increased by a factor of 3(μ∗ = 3), the droplet breakup is observed to be suppressed. While this is intuitively plausible, thereader is reminded that the proposed model is not intended to accommodate nonlinear deformationsof the droplet because of the inherent assumption of small-amplitude deviation of the droplet fromthe equilibrium shape. Hence a quantitative evaluation of the criteria of droplet breakup cannotbe made in the light of the current results. Nevertheless, as a first cut the predictive capability ofthe model enables one to answer the question as to whether a droplet will exhibit symptoms that areprecursor to breakup under a given set of operating conditions.

An interesting point to note is that when the driving frequency matches with the resonantfrequency of any mode other than the P2 mode (e.g., f = 234 Hz, the resonant frequency of P3 modefor water), no droplet breakup occurs. This can be easily understood by looking at the distributionof energy among the different modes. In Fig. 3, the fraction of energy, χ (n) contained in each modeis plotted against the mode number, n. χ (n) is defined as

(n) = ∫T0 an

2dt∑6k=2 ∫T

0 ak2dt

, (32)

and represents the fraction of the total energy of the k = 2–6 modes that is contained in the nthmode. The graph is plotted for a non-breakup case (f = 180 Hz, γ ∗ = 0.6, μ∗ = 1) and shows anexponential decay in energy with increasing mode number, which is a typical energy distributionamong the shape modes of a forced droplet.5 This means that P2 mode is the mode dominating thedroplet shape oscillations explaining why the external flow pulsating at the resonant frequencies ofother modes (keeping the relative fluctuating energy content constant) does not lead to a dropletresonance.

Additionally, under the current assumption of low energy of the fluctuating component of thefree stream velocity, droplet breakup occurs only under resonant conditions for all the cases studiedhere. The oscillation levels are found to decrease with time when the driving frequency is fairlydifferent from the resonant frequency of P2 mode as shown in Fig. 2 (f = 180 Hz as a representativecase). Compared to the initial cycle of oscillation (Fig. 2: third column of panels) in the later cycle(Fig. 2: fourth column of panels) the oscillation level has diminished significantly. The droplet

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022103-10 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 4. (a) The temporal variation of droplet aspect ratio and the modal coefficient of n = 2 mode for a driving frequencyof f = 30 Hz. (b) Wavelet spectrogram of the droplet aspect ratio for a driving frequency of f = 30 Hz.

assumes a final oblate shape (because the aerodynamic pressure is max at the pole and minimum atthe equator due to the potential nature of the pressure distribution around the droplet) with relativelylow oscillation levels compared to the initial cycles.

In order to understand the dynamics of these oscillations in more details, we consider the wavelettransform of the aspect ratio of the droplet. The temporal evolution of the droplet aspect ratio, a2

and the wavelet transform of the droplet aspect ratio for water at f = 30 Hz is shown in Fig. 4 as thiscase depicts the causes for the observed behavior very distinctly. As can be seen from the waveletspectrogram of the droplet aspect ratio, initially, the frequency component of 120 Hz (the resonantfrequency of P2 mode) is excited, whereas the driving frequency of 30 Hz is present throughout theoscillation time history. The intensity of 120 Hz frequency is much higher than that of 30 Hz andit decays over the time due to the viscous dissipation included in the model. This initial resonant

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022103-11 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

n = 2 n = 3 n = 4 n = 5 n = 6

FIG. 5. Modal shapes of the droplet corresponding to n = 2–6 (solid curve). The dashed line represents the equilibriumdroplet shape. The value of the amplitude of each mode is taken to be arbitrarily high to demonstrate the modal shape. Top(bottom) row of panels correspond to a positive (negative) amplitude, i.e., an > 0 (an < 0).

excitation is due to the impulsive starting (at t = 0 s) of the velocity field around the droplet as perthe initial conditions provided to the model. In other words the droplet is instantaneously introducedinto the flow field, a condition that is very close to real applications such as fuel droplets beingsprayed from the nozzle into a turbulent combustion chamber. This sudden jump in the velocityfield is mathematically isomorphic to a step forcing function in the model which excites the dropletat the resonant frequency of the predominant mode of oscillation, namely P2 mode. The P2 modesubsequently gets dampened out due to the effect of fluid viscosity. The driving frequency component(30 Hz in Fig. 4) on the other hand is always present in the temporal history of the droplet oscillationbut is much smaller in amplitude compared to the resonant frequency component. The negative valueof a2 implies that the shape of the droplet is always oblate as shown in Fig. 5, where the modalshapes of corresponding to n = 2–6 are depicted. The nth mode represents a standing wave at thedroplet interface with n number of nodal lines. As can be seen in the first column of Fig. 5, a positive(negative) temporal coefficient for the 2nd mode denotes the droplet shape deformed to a prolate (anoblate) spheroid. The aerodynamic pressure associated with the mean steady state flow flattens thedroplet at the poles as explained earlier and hence the value of a2 in our case is always negative andthe steady state mean aspect ratio of the droplet is greater than 1 (Fig. 4(a)).

With this understanding on the time evolution of the frequency components, we can now studythe effects of driving frequency and fluid properties on the frequency characteristics of the dropletoscillation. As can be seen from Figs. 3 and 4, the droplet shape oscillation and its frequencyspectrum are almost entirely dominated by P2 mode. Hence from now on focus will be on thetemporal coefficient a2 instead of the droplet aspect ratio. The power spectral density (PSD) ofwater droplet at the different driving frequencies is shown in Fig. 6. Note that the water droplet atf = 120 Hz undergoes breakup (resulting in the abnormal value of PSD at 120 Hz as indicated by theblack arrow in Fig. 6) and hence the corresponding PSD curve cannot be compared with the othercurves. However, the PSD plot is retained in the graph for completeness. Due to the reason alreadyexplained, irrespective of the driving frequency, the droplet always exhibits the resonant frequencyof the P2 mode, 120 Hz. In addition, as expected there is a spectral peak at the corresponding drivingfrequency. However, in all the PSD curves, there is an additional peak at the second multiple of thedriving frequency, which was not observed in the wavelet spectrum for f = 30 Hz because the PSDat this second multiple of the driving frequency is about three orders of magnitude less than thatat the driving frequency component itself (see Fig. 6). This mild droplet excitation at the secondharmonic of the second mode is the result of the quadratic dependence of pressure on velocity(Eq. (15)). When the ambient flow velocity is decomposed into its mean and fluctuating sinusoidalpart (Eq. (17)), the square of the instantaneous velocity leads to a term, (κsin2π ft)2 which fluctuatestwice as much as the instantaneous velocity itself. However, since this term scales as κ2(κ � 1), itseffect on the oscillation dynamics is expected to be very minimal, as can be seen in Fig. 6.

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022103-12 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 6. Power spectral density of a2, the modal coefficient of n = 2 mode, for various driving frequency as indicated in thelegend.

Figure 7 shows the effect of surface tension and viscosity on the spectral behavior of the dropletat a fixed driving frequency of 90 Hz. Besides the spectral peaks at 90 and 180 Hz, the resonantexcitation at the natural frequency of P2 mode can be noted. For γ ∗ = 0.3, this natural frequency is66 Hz (Table I) whereas the curves corresponding to γ ∗ = 1 show the resonant behavior at 120 Hz.The effect of viscosity is evident only in the free oscillation of the droplet; the PSD at the resonantfrequency is diminished substantially due to viscous dissipation. On the other hand, the PSD displaysalmost the same value at the driving frequency of 90 Hz irrespective of the fluid viscosity. This isexpected because the steady state oscillation phase (after the initial transient phase of oscillations,

FIG. 7. Power spectral density of a2, the modal coefficient of n = 2 mode, corresponding to various fluid properties (asindicated in the legend) at a driving frequency of f = 90 Hz.

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022103-13 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 8. A typical velocity jump imposed for simulating an advecting vortex. Dashed curve: tcross = 0.047 s; Solid curve:tcross = 0.0047 s.

dominated by the free oscillations) is governed by the balance between the restoring force (third termon the left hand side of Eq. (26)) and the external force and noting that the forcing function is keptconstant (due to the constraint, Eq. (30)). To explain this, assume that the steady state oscillation ofan is decomposed into its constant mean and fluctuating components and note that the fluctuatingpart is much lower compared to the mean. Hence the surface tension force (which is proportional toan itself) will be higher than the viscous force (which is proportional to an). Hence the amplitude ofsteady state oscillation is a strong function of surface tension but a weak function of viscosity.

B. Advecting line vortex

In this section we consider the free stream velocity profile as shown in Fig. 8 (dashed curve),which imposes a sudden disturbance in the external flow field. In physical terms, this velocity surgein the far field velocity can be interpreted to be caused when the droplet encounters a momentary puffof air that is convected with the ambient flow. Alternatively, this profile models the situation wherea sharp vortical structure moves past the droplet. However, since the propagation of disturbance inspace cannot be modeled because of the reasons mentioned earlier, the free stream velocity itself isapproximated to have the temporal variation as given in Fig. 8. The velocity profile given in Fig. 8is obtained by re-scaling the velocity profile used in Ref. 33 so as to satisfy Eq. (30). Mishraet al.33 used this profile to model the soot emission characteristics in a flame perturbed by anadvecting line vortex. This velocity profile was originally measured by Cetegen and Basu.34 Theundisturbed velocity in our case is taken to be 7.8 m/s; the ratio of the maximum velocity, U∞,max

to the undisturbed mean velocity, U∞,undisturbed is taken to be 1.4. The sudden rise in the free streamvelocity (which simulates the event of the high momentum fluid blob striking the droplet and hencetermed as vortex, hereafter) is introduced at t = 1 s, ensuring that the initial transient oscillationsof the droplet are dampened out, thus decoupling the effects of the free oscillations induced by theinitial jump in the velocity field (at t = 0 s) and the vortex itself. In Fig. 8 (dashed curve), the transittime of the vortex (the time during which the droplet interacts with the vortex) is denoted by tcross

which is taken to be the full width at half maximum (FWHM) of the velocity jump and is estimatedto be 0.047 s. In order to study the effect of this transit time, we also consider a velocity profile(solid curve in Fig. 8) obtained by scaling the template profile (dashed curve in Fig. 8) in the timedirection, thereby leading to a lower transit time of the vortex. The peak value of the velocity surge

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022103-14 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 9. (a) Effect of crossing time of an advecting single vortex on the temporal evolution of the droplet aspect ratio. Insetimages show the instantaneous droplet shapes. (b) Energy contained in each mode vs time for n = 2–4 for tcross = 0.0047 s.(c) PSD of a2, a3, and a4 for tcross = 0.0047 s.

is maintained constant (U∞,max/U∞,undisturbed = 1.4), while the FWHM is reduced to obtain a transittime of tcross = 0.0047 s (one-tenth of that of the template profile).

Figure 9(a) compares the aspect ratio oscillations of a water droplet for these two values oftransit time. The initial droplet oscillations induced by the sudden initial impulse of velocity rise diedown to a negligible level at t = 1 s, when the droplet encounters the vortex. The sudden increase inthe velocity and thus the pressure field around the droplet further flattens the droplet thus increasing

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022103-15 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 10. Wavelet spectrum of (a) the velocity jump with a crossing time of tcross = 0.047 s and (b) corresponding a2. Themaximum deflected shape of the droplet in different time periods are shown as insets to (b).

the aspect ratio. For brevity, the vortex or the velocity profile with lower (higher) crossing time willbe referred to as V-low (V-high) hereafter. The droplet encountering V-high displays the expectedrise in the aspect ratio, followed by a decrease in the aspect ratio to the steady-state mean value.There is no overshoot resembling the response of a damped oscillator. On the other hand, the dropletinteracting with V-low exhibits very high level of shape oscillations for the same vortex strength(or velocity peak). The temporal variation of square of the temporal coefficients, a2, a3, and a4

(representing the instantaneous energy carried by each mode) after the V-low hits the water dropletare shown in Fig. 9(b). It can be seen that, as with the pulsatile flow, the dominant mode of oscillationin this case is also P2. The energy contained in P3 mode is two orders of magnitude lower than thatin P2 mode and that in P4 mode is relatively even lesser and so on. Hence it is sufficient to consideronly P2 mode in order to explain the observed behavior of the droplet interacting with the vorticesof different transit timescales. This argument is further corroborated by the much higher resonantpeaks in the PSD of a2, compared to that of a3 and a4 (see Fig. 9(c)).

Wavelet transform of the velocity profile for the different crossing times and the correspondinga2 are given in Figs. 10 and 11. In order to clearly illustrate the frequency contents of the velocityjump alone, the wavelet spectrum at the beginning and end of the velocity profile is not shown inFigs. 10(a) and 11(a). As can be seen in Fig. 10(b), after the initial resonant excitation of the droplet,

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022103-16 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 11. Wavelet spectrum of (a) the velocity jump with a crossing time of tcross = 0.0047 s and (b) corresponding a2.

the droplet exhibits very low frequency oscillations (<35 Hz) for V-high (tcross = 0.047 s). Butfor V-low (tcross = 0.0047 s), the droplet exhibits very high intensity oscillations at, the resonantfrequency of P2 mode, 120 Hz (ref. Fig. 11(b)). Hence it follows that the vortex with the lowercrossing time drives the droplet to the resonant condition, whereas the other vortex does not andthus resulting in very low level shape oscillations. The wavelet transform of the velocity profile forthe two vortices (Figs. 10(a) and 11(a)) reveals the reason for this. The lower crossing time leadsto a shaper velocity jump and the sharper the velocity jump is, the wider its frequency spectrumwill be. The vortex with lower crossing time has frequency contents less than only 35 Hz (seeFig. 10(a)). On the other hand the sharper velocity jump has its energy distributed over a wide rangeof frequencies extending up to about 200 Hz (see Fig. 11(a)). Note that these cutoff frequencies areof the order of 1/tcross. This indicates the paramount influence of the time of interaction between thedroplet and the vortex (≈ average relative velocity of the vortex with respect to the droplet dividedby the droplet diameter) on droplet deformation dynamics. For the same strength of vortex, a shorterinteraction time period results in higher droplet deformation. If tcross > 1/f2, the droplet deformationis less severe and no high frequency oscillations are to be expected. However, if the opposite is true,the droplet will be flattened to a higher extent followed by droplet resonance and can lead to breakup

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022103-17 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 12. A typical velocity profile imposed for simulating an advecting train of two sequential vortices.

at low enough fluid viscosity. The slight top-down asymmetry of the static droplet (see the last insetto Figs. 10(b) and 11(b)) can be attributed to the wake present in the behind of the droplet, since thestatic pressure is more at the forward stagnation point (in the wake region, the pressure at the dropletsurface is assumed to be the ambient pressure).

In the remainder of this section, the droplet oscillations induced by a train of two vortices(Fig. 12) are discussed. The separation between the peaks of the two velocity jumps is denoted byts. First let us consider the train of vortices with separation time equal to 5 times the crossing timeof each vortex, i.e., ts/tcross = 5. Figure 13 compares the effect of such a train of vortices with lowercrossing time (tcross = 0.0047 s, which corresponds to a frequency of about 210 Hz) and highercrossing time (tcross = 0.047 s, which corresponds to a frequency of about 21 Hz) on the evolutionof the droplet aspect ratio. For tcross = 0.0047 s, the response to the second vortex is noticeablydifferent from that to the first vortex. However, for tcross = 0.047 s the responses look exactly thesame. This shows that if the effect of the shape oscillations induced by a vortex is to last and couplewith the dynamics of the droplet during its interaction with a subsequent vortex, again the criteriontcross > 1/f2 is to be met for the first vortex. Otherwise, the response of the droplet to the secondvortex is not at all influenced by the first vortex; in other words, the droplet loses the memory of itsresponse to the former vortex while responding to the latter vortex. This is verified with two moreseparation times given by ts/tcross = 1 and 10. For higher tcross, even when the vortices are closer(ts/tcross = 1), the response of the droplet to the first vortex is exactly repeated while interacting withthe second vortex. For lower tcross, on the other hand, even when the vortices are further apart fromeach other (ts/tcross = 10), the effect of the first vortex is evident in the droplet response to the secondvortex (data not presented here).

Since a train made of vortices with only lower crossing time shows a coupled effect on thedroplet dynamics, from here on we focus only on such vortex trains (tcross = 0.0047 s). A moredetailed study is needed to clearly delineate the coupled effect of the two vortices. However, weaddress here a few main points that can be deduced from the results of the current work. In Fig. 14,the variation of the aspect ratio of water droplet with respect to time in response to vortex trains withts/tcross = 5 (referred to as V-train5 hereafter) and 10 (V-train10) is given. The data corresponding tothe single vortex (V-low) is also given for comparison. An interesting observation to be made fromthe figure is that the second vortex of V-train5 leads to an amplification of the oscillation (markedin the figure by the thicker arrow/black online) whereas that of V-train10 leads to a reduction in the

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022103-18 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 13. Temporal variation of the aspect ratio of a water droplet in response to a vortex train with separation time given byts/tcross = 5. Solid (dotted) curve corresponds to tcross = 0.0047 (0.047) s.

amplitude of the oscillation (marked by the thinner arrow/green online). This demonstrates that thetime after the first at which the second vortex interacts with the droplet relative to the first vortexplays a very important role in the subsequent droplet oscillation.

To explain this phenomenon, in Fig. 15 the time evolution of the droplet aspect ratio in response toV-train5 and a vortex train with ts/tcross slightly less than 5 (4.28 to be precise: referred to as V-train4),is presented. As mentioned earlier, V-train5 results in an increase in the oscillation level, whereasV-train4 results in a reduction of the oscillation. At this point of the analysis only a speculative

FIG. 14. The imposed velocity profile and droplet response to single vortex (curve with square markers); train of vorticeswith ts / tcross = 5 (solid line) and train of vortices with ts/tcross = 10 (dashed line).

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022103-19 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 15. The imposed velocity profile and droplet response for single vortex (curve with square markers); train of vorticeswith ts/tcross = 5 (solid line) and train of vortices with ts/tcross = 4.28 (curve with circular markers).

explanation can be offered for this finding. The time instants at which the droplet starts to respondto the second rise in the vortex trains (marked by the arrows in the figure) is at the upper half of theoscillation cycle induced by the first vortex for V-train4 and at lower half of the oscillation cycle forV-train5. The second derivative of the droplet aspect ratio has opposite signs in the upper and lowerhalf of the oscillation cycle. Since the inertia term in the governing equation is proportional to thesecond derivative of the droplet oscillation level, the forcing imparted by the increase in the velocitydue to the second vortex and the inertia are competing with each other in the case of V-train4. Thisis because the sudden increase in the velocity and thus the pressure field around the droplet inducedby the second vortex tries to further flatten the droplet (increase the aspect ratio), whereas inertia istrying to reduce the aspect ratio at that instant. On the other hand, the two forces favor each otherin the case of V-train5. The combined action of the subsequently decaying forcing and restoring anddissipative forces leads to a reduction/magnification in the amplitude of oscillation. Although nocomplete explanation is offered, it has been shown that the separation between two sequential vorticesinteracting with the droplet plays a major role in determining the shape oscillation dynamics of thedroplet. The exact time at which the droplet interacts with the second vortex determines whether thealready induced droplet oscillation will be amplified or diminished.

C. Flow with broadband energy characteristics

The instantaneous velocity signal (given in Fig. 16(a)) measured experimentally28 at the centreof an air jet (at a mean velocity of 7.8 m/s) is imposed as the ambient flow velocity. The resultsof this simulation serve as an experimental validation of the proposed mathematical model. Thepower spectral density of the input velocity signal (see Fig. 16(b)) shows broadband characteristics.The frequency response is nearly flat up to about 100 Hz beyond which the spectral power densitydecreases monotonically. This energy spectrum is typical of a round jet.35

The model predicts a resonant behavior in the frequency spectrum of each of the modal coef-ficients. The PSD of computed a2 is given in Fig. 16(c). When the droplet is exposed to incomingvelocity fluctuations over a wide frequency range, the different modes of oscillation get excited,exhibiting response characteristics similar to the spectral power distribution of the incoming flow,but with the highest response around the resonant frequency of the corresponding mode. The ex-

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022103-20 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

FIG. 16. (a) The experimentally measured velocity signal. (b) Power spectral density of the experimentally measured velocitysignal. (c) Comparison of PSD’s of numerically and experimentally obtained a2.

perimentally observed PSD of a2 measured from the optical images of the oscillating droplet isalso shown in Fig. 16(c). The trends of the experimentally and numerically obtained PSD’s matchexceptionally well, although quantitatively there is a disagreement between the two. Furthermore,experimentally there is a band of frequency around the natural frequency which gets excited quite

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022103-21 P. Deepu and S. Basu Phys. Fluids 26, 022103 (2014)

prominently, whereas in the simulation, the spectrum shows a sharp peak at the resonant frequency.The possible reasons for these discrepancies are the various assumptions that were made in thetheoretical analysis such as asymmetric droplet oscillations, inviscid external potential flow and theabsence of nonlinear effects. Furthermore, the presence of the cross-wire used in the experiment tosuspend the droplet also may have some effect on the droplet oscillation dynamics. However, thisresult demonstrates the predictive capability of the model with regard to the spectral behavior of thedroplet in response to a fluctuating ambient flow field, which is the main objective of this study.

IV. CONCLUSIONS

The shape oscillation of a droplet subjected to a uniform external flow is numerically examined.The aim of the present work is to predict the spectral behavior of droplet shape oscillations inducedby unsteady flow fields. The fluctuating energy content in the external flow velocity is taken to bea value much lesser than unity. In all the cases analysed in the present study, P2 mode is found todominate the droplet oscillaiton. For a pulsatile external flow field, the droplet exhibits steady stateoscillations at the driving frequency and at twice that frequency. It is proved that the transient phaseof oscillation is dominated by natural oscillations and this is unavoidable for a droplet placed in animpulsively started velocity. This free oscillation dies down under non-resonant conditions. Dropletresonance and subsequent breakup result only if the driving frequency matches with the naturalfrequency of P2 mode. A higher fluid viscosity is found to reduce the intensity of free transientoscillations of the droplet and hence if the fluid viscosity is high enough the growth of amplitude oftransient oscillations under resonant condition is prevented, thereby the subsequent droplet breakupis impeded.

The effect of one-way interaction between an advecting line vortex and the droplet is simulated.For the same vortex strength, the vortex crossing time determines the oscillation intensity of thedroplet. A lower crossing time introduces more frequency components in the spectrum of the outsidevelocity field and hence below a threshold crossing time (∼1/f2), the droplet is driven to resonanceand exhibits high intensity oscillations. A train of two vortices is considered to study the couplingeffect in the droplet oscillations induced by sequential vortices. In this case also, only a train ofsharper vortices (lower crossing time) is found to produce any considerable coupling effect. At theinstant when the second vortex starts interacting with the droplet, the external forcing competeswith the inertial effect of the oscillations induced by the first vortex in a constructive or destructivemanner. The ensuing complicated dynamics of the different time-varying external and internal forcescan magnify or diminish the already induced droplet oscillation. The effect of fluctuating energycontent in the external flow field, droplet mass and the development of a stability criteria is thesubject of future work.

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