ILASS Americas 27th Annual Conference on Liquid Atomization and Spray Systems, Raleigh, NC, May 2015 Investigating Traditional Atomization Theories with Highly-Resolved Simulations S. Deshpande, S. Gurjar, and M.F. Trujillo * Department of Mechanical Engineering University of Wisconsin Madison, WI 53706 USA Abstract Atomization of a liquid sheet is studied using simulations based on a Volume of Fluid (VoF) method. Our aim is to evaluate the primary atomization models, which are often used in Lagrangian-Eulerian simulations–a prominent spray simulation method. The models assume that growth of sinuous unstable waves on the sheet causes its breakup and use linear theory to predict the wavelength [Dombrowski & Johns 1963; Senecal et al. 1999]. With respect to this, we address two points: (1) applicability of linear theory to instability prediction, and (2) relevance of this prediction to sheet breakup. To this end, a more general linear analysis considering capillary, viscous and boundary layer is performed using Orr-Sommerfeld (OS) theory. The VoF and OS simulations agree well for the 2-phase mixing layer problem. These disturbances, however, do not cause sheet breakup, and this contrasts prior linear theories. The structures which eventually do lead to breakup are shown to be two to three orders of magnitude greater than the ones predicted from linear instability analysis and emerge well beyond the initial linear regime. * Corresponding Author: [email protected]
Transcript
ILASS Americas 27th Annual Conference on Liquid Atomization and Spray Systems, Raleigh, NC, May 2015
Investigating Traditional Atomization Theories with Highly-ResolvedSimulations
S. Deshpande, S. Gurjar, and M.F. Trujillo∗
Department of Mechanical EngineeringUniversity of WisconsinMadison, WI 53706 USA
AbstractAtomization of a liquid sheet is studied using simulations based on a Volume of Fluid (VoF) method. Our aimis to evaluate the primary atomization models, which are often used in Lagrangian-Eulerian simulations–aprominent spray simulation method. The models assume that growth of sinuous unstable waves on the sheetcauses its breakup and use linear theory to predict the wavelength [Dombrowski & Johns 1963; Senecal et al.1999]. With respect to this, we address two points: (1) applicability of linear theory to instability prediction,and (2) relevance of this prediction to sheet breakup. To this end, a more general linear analysis consideringcapillary, viscous and boundary layer is performed using Orr-Sommerfeld (OS) theory. The VoF and OSsimulations agree well for the 2-phase mixing layer problem. These disturbances, however, do not causesheet breakup, and this contrasts prior linear theories. The structures which eventually do lead to breakupare shown to be two to three orders of magnitude greater than the ones predicted from linear instabilityanalysis and emerge well beyond the initial linear regime.
One of the major aspects of spray modeling isthe prediction of the primary atomization process,which is defined to be the complete severance of theliquid core [1–4]. Within the Lagrangian-Euleriancontext, which is the predominant method for simu-lating sprays [5–7], the breakup process has its rootsin interfacial (linear) instability analysis [2, 8–10].The Kelvin-Helmholtz (KH) instability is often con-sidered to be the key mechanism for the developmentof unstable interfacial waves. Different approxima-tions are used by various authors to arrive at a closedform expression for the dispersion relationship.
Over the last decade or so, increases in compu-tational resources, along with the development of in-terface capturing computational methodologies suchas the Volume of Fluid (VoF) and Level Set methods[11] have led to a number of detailed studies of theatomization process (see for instance [12–16] amongothers). As opposed to the Lagrangian-Eulerian ap-proach, where semi-empirical models must be usedto phenomenologically treat atomization, this ap-proach involves solving the Navier-Stokes equationswith an implicit interface description to simulate theentire liquid breakup process. As a consequence,well resolved computations can provide detailed in-formation regarding key physical processes under-lying atomization, in addition to quantitative re-sults pertaining to breakup lengths, droplet sizes,etc. Such simulations have been successfully em-ployed in the context of sheets and jets injected intoquiescent [12, 13, 16–18], crossflow [15, 19], and co-flowing environments [20, 21].
In the present work, the configuration chosencorresponds to a liquid sheet undergoing hydrody-namic breakup. The properties are given in Table 1.This configuration has received considerable atten-tion because its relevance to hollow cone [22–24]and flat fan sprays [25]. In analyzing this problem,prior studies [2, 8, 22, 26] have made a direct as-sociation between the most unstable mode and themode responsible for producing atomization. It isoften assumed that these two modes are the same,and this assumption is inherited in CFD atomiza-tion models of liquid sprays. This point is examinedin the present work employing highly-resolved simu-lations of the atomization process with an algebraicVoF technique [27]. As described below, the findingsshow a significant difference between the theory andthe computations on the order of two to three ordersof magnitude. To gain more insight into this discrep-ancy, a more generalized instability analysis is per-formed, which consists of a 2-phase Orr-Sommerfeld(OS) system. In the limit of vanishing boundary
Liquid density ρl = 666.7 kg·m−3
Liquid viscosity µl = 2.5× 10−4 N-s-m−2
Gas density ρg ∈ (7.84, 13.33, 19.61,26.67, 39.22) kg·m−3
Table 1: Physical properties and injection parame-ters.
layer thickness, the OS results agree with the earlierinstability analyses. However, the simulations revealthat the atomization event is well beyond the linearwindow of analysis, and hence, the equivalence madebetween the most unstable mode (linear instability)and the observed atomization mode from VoF doesnot hold in the conditions of current interest.
Orr-Sommerfeld Analysis
A significant limitation of the standard theoriesis the discontinuity in the velocity profile across theinterface. This essentially implies that both gas andliquid boundary layers are zero. Additionally, theviscous effects are neglected in either the gas phase,or in both phases. To overcome these limitations, anOS analysis [28–30] is adopted in the present workalong with the assignment of the appropriate inter-facial conditions at the gas-liquid interface.
The problem consists of a 2D flow of a vis-cous liquid sheet having an offset velocity of 200m·s−1 with a gaseous ambient. The offset velocity isdefined as the velocity difference between the liquidand the gas, where the location where the velocityis obtained in each respective phase is far removedfrom the interfacial region. In the y-direction, thedomain extends from −h, h, where the sheet half-thickness is much greater than either the liquid orgas boundary layer, i.e. h � δg and h � δl. Thelocation of the interface is identified by η(x, t), withη(x, t) = 0 corresponding to an undisturbed surfaceprofile. Following convention, the velocity and pres-sure field are decomposed in terms of a base (denotedby uppercase and subscript b) and a perturbed field(denoted by lower case), respectively given by
U(x, y, t) = Ub(y) + u(x, y, t) and (1a)
P (x, y, t) = Pb(y) + p(x, y, t). (1b)
The base fields are only a function of the (undis-turbed) surface normal coordinate (y) and the ve-
2
locity is given by the following error function
Ub(y) = (Ub(y), 0, 0) with (2)
Ub(y) =
{Ugerf(y/δg) for y ≥ 0,Ulerf(y/δl) for y < 0,
(3)
borrowing from the conditions considered in [30].Additionally, δl = δg in all OS calculations pre-sented here.
The perturbation velocity can be expressed by
u(x, y, t) = (u(x, y, t), v(x, y, t), 0) = ∇× (ψez),(4)
where the stream function takes the following con-ventional form [31]
ψ(x, y, t) = <{ϕ(y) exp (iα(x− ct))
}. (5)
The wavenumber is strictly a real quantity in ourcase, and c is its complex phase speed. Consider-ing a linearized version of the Navier-Stokes, takingthe curl of this equation, and noting that the vortic-ity is related to the stream function above throughωvort = −∇2(ψ(x, y, t)), yields the well known OSequation [31]
(Ub(y)− c)(D2y − α2)ϕ(y)− ϕ(y)(D2
yUb(y)) =
µ(y)
iαρ(y)
[D2
y − α2]2ϕ(y). (6)
Here the short hand notation Dy has been substi-tuted for ∂/∂y. Also, the thermophysical propertiesare given by
ρ(y) =
{ρg for y ≥ η(x, t),ρl for y < η(x, t),
and (7)
µ(y) =
{µg for y ≥ η(x, t),µl for y < η(x, t).
(8)
Equation (6) represents a generalized eigenvalueproblem for the eigenpair (ϕ(y), c) and the relationis valid at all internal points of the physical domain,i.e. either in the liquid or gas phase. Additionally,both ϕ(y) and c are generally complex quantities,namely
ϕ(y) = ϕR + iϕI and (9a)
c = cR + icI . (9b)
These equations are solved in conjunction withthe interfacial conditions, namely: continuity ofnormal velocity, continuity of horizontal velocity,
continuity of tangential stress at the interface (noMarangoni effects), and balance of normal stress atthe interface. The 2-phase OS system is solved forthe eigenmodes ϕ(y) and associated eigenvalues (c).This provides a means of obtaining the perturbationvelocity and the growth rate as a function of pertur-bation wavenumber. The numerical solution is basedon a second order central finite difference differencescheme. In order to discretize interfacial conditionswhile maintaining symmetry of the finite differenceoperators, four ghost nodes are introduced, two oneither side of the interface.
Results pertaining to two configurations are con-sidered. These are identified by cases C and Din [30]. Periodic boundary conditions are imposed atthe boundaries X = 0 and X = 2λ = 4π/α, where(λ, α) correspond to the wavelength and wavenum-ber of interest in the OS solution. At the top andbottom surface, i.e. y = ±h, a zero Dirichlet con-dition is imposed on velocity and a zero gradientfor pressure. An initial perturbation, correspond-ing to a given α, is imposed on this base flow field.The characteristics of this perturbation are obtainedfrom an OS calculation, which generates predictionsfor c and ϕ(y). The resulting eigenfunction, ϕ(y), isemployed to construct the initial perturbed velocityfield given by
u(x, y, 0) = ε
{[DyϕR cos(αx)−DyϕI sin(αx)
]ex+[
αϕI cos(αx) + αϕR sin(αx)]ey
}. (10)
This is essentially the product of the perturbationvelocity, evaluated at t = 0, and a small constant ε =10−2. This small constant represents the magnitudeof the initial perturbation.
Regarding the initial interface displacement,η(x, t = 0), the interface kinematic condition can beemployed and linearized assuming |η(x, t)| is small.An analytic solution can thus be obtained and eval-
3
0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
αδg
ωδg/Ug
Algebraic VoF
Linear Theory
(a) Case C
0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
αδg
ωδg/Ug
Algebraic VoF
Linear Theory
(b) Case D
Figure 1: Comparison between VoF simulation re-sults (denoted by ×) and OS predictions (dashedlines) corresponding to the physical properties andparameter range of cases C and D, as reportedby [30].
uated at t = 0 yielding,
η(x, t = 0) = −ϕ(y = 0) exp(iαx)
Ub(y = 0)− c. (11)
This is the profile used to initialize the liquid frac-tion field for the VoF simulations. Overall goodagreement is achieved as shown in Fig. 1, where eachsymbol represents one VoF simulation.
Comparison among linear theories
As noted previously, one of the significant lim-itations in standard theories is the discontinuoustreatment of the velocity profile across the gas-liquidinterface. Within the framework of OS, a more re-alistic continuous profile can be employed. For in-stance in the present work the error function profiledescribed in Eq. (2) is used. A comparison of the
0 1 2 3 4 5x 107
105
106
107
108
109
_ (m−1)
t (s
−1)
Figure 2: Comparison between dispersion curvespredicted with OS and those originating from stan-dard theories.
wave growth based on the standard theories and OSpredictions is performed for values of δg = δl =(0.01,0.1, 1, and 10) µm. The results are included inFig. 2. The thermophysical property values usedare listed in Table 1. For the gas, a density is 19.609kg·m−3 is employed. All OS calculations are numer-ically converged with respect to spatial resolution.
The results show that as the boundary layerheight becomes small, the dispersion curve approxi-mates the predictions from standard theories. Thisis consistent with the notion that as δg → 0, thevelocity profile approaches a discontinuous charac-ter, and hence, it is expected to approximate thestandard theories. A second observation concernsthe enormous decrease in the maximum growth ratewith increasing boundary layer height. For instance,between δg = 0.01 µm and δg = 10 µm, the peakwave growth is diminished by approximately 3 or-ders of magnitude, i.e. inversely proportional to δg.This decrease in growth rate is also associated witha similarly drastic decrease in the wavenumber orequivalently an increase in the length scale of themost destructive mode.
Liquid Sheet VoF Simulations
The predictions from the standard theories arecompared to the results from VoF simulations. Theproperties and parameters for the simulations areagain described in Table 1. A visualization of a typi-cal simulation result is shown in Fig. 3, where the keystages during the atomization are outlined. The liq-uid section closest to the nozzle exit is characterizedby the initial development of interfacial instabilities.
4
9
Development of Initial Instability
Ligament Dynamics and Droplet Creation
Breakup of Intact Liquid Core
Development of Large Scale Liquid
Structures
Figure 3: Visualization of an injected liquid under-going deformation and breakup.
In this regime, which is governed by linear stabil-ity, the agreement with the current VoF predictionshas already been shown previously to be quite good.Further downstream, the disturbances become non-linear and evolve to large scale liquid structures,which subsequently disrupt and fragment the sheet.This behavior is further complicated by the emer-gence of spanwise structures marking a strong de-parture from the essentially 2D behavior of the lin-ear regime. The location where the liquid sheet isfragmented is indicated by a dotted line. This de-fines the atomization location. Its distance to thenozzle is given by Xliq.
The initial transient period is characterized bya linear growth in Xliq, which ends with the onset ofbreakup and the emergence of a quasi-steady stateperiod. The breakup characteristics during the ini-tial period are discussed in detail in the work of [16]regarding the disturbances produced by the jet tip.The demarcation between the initial transient andquasi-steady period, denoted as tbreak, can be es-timated by identifying the first break in the linearplot of Xliq. During the quasi-steady period varia-tions in Xliq continue; however, its mean value is nolonger changing. In the present work our interest isfocused on the length-scale characteristics of liquidstructures during this latter period.
To extract quantifiable length scale data at t >tbreak, a large number of computational probes areplaced in the interfacial region. These probes recordthe local instantaneous velocity and liquid fractionvalues and from a rather lengthy procedure (not in-cluded here for brevity) length scale data is obtained.The results, shown in Fig. 4, are given in terms ofa map of λ/2h as a function of density ratio cor-responding to standard theories, OS, and VoF cal-
culations in a linear-log format. Here λ pertains tothe optimal length scale. The range for λV oF showncorresponds to the span of the distribution betweenthe 10th and 90th percentile. From the results, it isclear that the standard theories underestimate thedominant length scales associated with atomizationby approximately 2 or 3 orders of magnitude, whichconstitutes a serious discrepancy. The OS calcula-tions show good agreement with standard theoriesonly for negligible values of δg, as previously elabo-rated. For relatively large values of δg, λOS valuesbegin to approach the smaller end of the correspond-ing VoF spectrum.
Conclusion
The 2-phase flow OS solution provides a meansto generalize some of the assumptions that were usedin the standard sheet breakup analyses. One of themost crucial ones concerns the base velocity profile.In the present OS treatment, the base velocity fieldcan take an arbitrary form relaxing the discontinu-ous character of profiles employed in standard theo-ries. Results show that this has a significant impacton the shape and magnitude of the resulting disper-sion curve. Specifically, the boundary layer height isfound to be inversely related to both the instabilitygrowth rate and its associated wavenumber.
The current OS predictions agree relatively wellwith standard theory in the limit of vanishingboundary layer thickness, i.e. as one approachesthe discontinuous profile. The length scales of themost unstable modes from standard theories andVoF simulations are found to be significantly differ-ent. The optimal modes from standard theory areroughly two to three orders of magnitude smallerthan the mode responsible for atomization. The OSpredictions span the range in between these two ex-tremes by varying the base profile boundary layerthickness.
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