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Eulerian modeling of dispersed droplet inertia: Internal circulation transition
Journal of Colloid and Interface Science 291 (2005) 577–584 www.elsevier.com/locate/jcis
On the Lagrangian/Eulerian modeling of dispersed droplet inertia: Internal circulation transition G.F. Naterer a , M. Milanez a,∗ , G. Venn b a Department of Mechanical and Manufacturing Engineering, University of Manitoba, 15 Gillson Street, Winnipeg, Manitoba, Canada, R3T 2N2 b GKN Westland Helicopters Ltd., Yeovil, Somerset, England, BA20 2YB, UK
Received 15 October 2004; accepted 8 May 2005 Available online 24 June 2005
Abstract This article addresses a limitation of Lagrangian methods for droplet tracking, when approaching the transition point of internal circulation within droplets. Laminar multiphase flow with dispersed droplets in a co-flowing airstream is considered. Analytical and numerical formulations of droplet motion are developed based on a Lagrangian finite difference method of droplet tracking. Cases of both high and low relative Reynolds numbers are formulated. The role of interfacial drag in cross-phase momentum exchange increases at higher relative Reynolds numbers. A new transition criterion is developed to characterize conditions leading to shear-driven non-uniformities of velocity within a droplet. This criterion entails a momentum Biot number, in analogy with the Biot number criterion for conjugate heat transfer problems involving conduction and convection. At sufficiently high momentum Biot numbers, appreciable changes of velocity within a droplet imply that Lagrangian methods become unsuitable and transition to Eulerian volume averaging is needed. Predicted results of Lagrangian modeling of droplet motion in a co-flowing airstream are presented and discussed. 2005 Elsevier Inc. All rights reserved. Keywords: Multiphase flow; Lagrangian/Eulerian tracking; Droplet inertia; Droplet/air interface; Shear driven velocity non-uniformities; Transition to internal circulation; Momentum Biot number criterion
1. Introduction Accurate numerical modeling of multiphase flows with droplets is a technological goal of considerable interest. Detailed understanding of multiphase flows is often more comprehensive than single phase flows, due to complex transport mechanisms involving momentum exchange, interfacial interactions and phase change heat transfer (Crowe [1]; Naterer [2]). This article investigates convection modeling of multiphase flows with droplets, particularly relating to impinging droplets in icing applications. Icing of helicopter surfaces involves momentum exchange between incoming droplets and the co-flowing air stream.
* Corresponding author.
E-mail addresses: [email protected], [email protected] (M. Milanez). 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.05.025
Droplet momentum displacement in multiphase flow with droplets near a flat plate entails cross-phase interactions with air stream (Milanez et al. [3]). Convective acceleration of droplets depends on the volume fraction of the dispersed phase [4]. In regions where the multiphase flow is not largely affected by droplet/droplet interactions, a dilute flow assumption may be adopted [4]. Such situations occur when the dispersed (droplet) phase is predominantly controlled by the carrier (air) phase. For example, when a helicopter travels through clouds, the impinging supercooled droplets on the helicopter surfaces are immersed in moving air. Their movement is not dominated by droplet/droplet interactions, although distinct interfacial and bulk pressures may arise in such dilute flows (Milanez et al. [5]). This article considers convection modeling with a Lagrangian formulation of dilute multiphase flow with droplets. Past Lagrangian methods generally entail spatial tracking of droplets, which are approximated as discrete particles
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without internal circulation. But shear-driven deformation or rotation of droplets can lead to internal circulation with resulting upstream pressure disturbances. In contrast, Eulerian methods have been developed successfully for multiphase flows (i.e., Banerjee and Chan [6], Milanez et al. [3,5], Hetsroni [4]), but generally without volume averaging for spatial effects of internal circulation on upstream pressure disturbances. This article attempts to address the transition from Lagrangian tracking of discrete particles to Eulerian averaging of droplets, when local pressure disturbances arise with internal droplet circulation. Internal circulation affects the interfacial pressure interactions between dispersed and continuous (carrier) phases in a diffusive type manner (Gidaspow and Solbring [7]). Compression of the gas phase depends on the predicted interfacial pressure term from both kinetic energy and continuity equations (Sha and Soo [8]). Cross-phase interactions arising from internal circulation affect the momentum exchange within the dispersed phase. Pressure disturbances induced by fluctuations in the dispersed phase may affect the carrier phase, particularly if the dispersed phase is moved suddenly from a region to another region with different bulk pressures. Prosperetti and Jones [9] have presented a detailed formulation for such interfacial pressure effects. LeClair et al. [10] reported that boundary layer theory overestimates the internal circulation within a droplet. Klett [11] discusses the limitations of boundary layer equations to internal circulation within droplets. Earlier studies by Taylor [12,13] provided fundamental insight regarding emulsions of droplets and viscosity of fluids containing droplets of other liquids. Past studies by Marchioro et al. [14] have successfully applied methods of ensemble averaging to mixture pressure and viscous stress modeling of dispersed two-phase flows. Similar advances involving two-fluid models have been reported by Park et al. [15], but effects of internal circulation on the averaged pressure in the bulk flow were not outlined. In the particle tracking procedure of conventional Lagrangian methods, spatial variations of temperature and velocity within a droplet are lost through a lumped capacitance assumption. In classical heat transfer analysis, the lumped capacitance assumption for temperature is known to be valid in cases when the Biot number is less than 0.1 [2]. Beyond this value, spatial variations of temperature within the object become significant and the internal temperature cannot be assumed to be uniformly isothermal. This article attempts to derive a similar lumped capacitance analogy and iso-velocity criterion for dispersed droplets in multiphase flows. Chen et al. [16] develop Eulerian modeling of the turbulent carrier phase and Lagrangian tracking of particle dispersion for turbulent two-phase flows. This approach avoids empirical parameters required for the particulate dispersion width. Particle dispersion is affected by surface tension (Polat et al. [17]). Chen et al. [16] show that smoother droplet profiles are obtained with stochastic/probabilistic modeling, as compared with an eddy interaction model. Laminar mul-
tiphase will be investigated and solutions will be limited to cases with spherical, non-deforming droplets and negligible droplet/droplet interactions.
2. Formulation of dispersed droplet motion in an airstream The Lagrangian method is a well-known conventional technique for tracking droplet trajectories in multiphase flows. In this method, a force balance (including drag and gravitational forces) is applied to each discrete droplet throughout its spatial trajectory. From Newton’s Law of motion, the time-varying forces contribute to the droplet’s acceleration. Then, successive temporal integrations of this acceleration field yield the velocity and position of individual droplets within the flow field. In this section and the following section, analytical and numerical formulations of the Lagrangian method will be outlined for such droplet motion. A Lagrangian analytical formulation of droplet/air interactions over a range of relative Reynolds numbers (Rer ) will be presented. Consider a force balance with interfacial drag and gravity acting on dispersed droplets (subscript d) in an airstream (subscript a) in the low-Re regime (see Fig. 1). The laminar equations of motion in the x and y directions are d (vd,x ) = −FD,x , (1) dt d m (vd,y ) = −FD,y + mgy , (2) dt where FD,x , FD,y and gy are the Cartesian drag forces in the x and y directions and gravity in the y direction. For lowRer flow regimes, the linear drag force in each direction, i, is expressed as FD,i = 3µπDd (vd,i − va,i ). The quadratic law is used to model the drag force in high-Rer number flow regimes, i.e., FD,i = 1/2ρa (vd,i − va,i )2 × 0.3(πDd2 /4). Also, the relative Reynolds number between phases in contact, Rer,i , in each direction, i, is defined as m
Rer,i =
ρd (vd,i − va,i )Dd . µa
Fig. 1. Schematic of droplet trajectory.
(3)
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Solving Eqs. (1) and (2) in the low-Rer regime, subject to a known velocity at an initial time, t0 , yields the following analytical solutions of droplet velocity in the x and y directions, vd,x (t) = va,x + vd,x (t0 ) − va,x 18µa (t − t ) × exp − (4) 0 , ρd Dd2 ρd Dd2 ρd Dd2 vd,y (t) = va,y + g + vd,y (t0 ) − va,y − g 18µa 18µa 18µa (t0 − t) . × exp (5) ρd Dd2 Furthermore, integrating these equations over distance and time yields two separate equations for the x and y positions of the droplet (denoted by xd and yd , respectively). The initial time is set to zero value. These equations together represent the trajectory for droplet motion in the airstream. This analytical trajectory becomes ρd Dd2 va,x − vd,x (t0 ) 18µa 18µa × exp − t − 1 , ρd Dd2
xd (t) = xd (t0 ) + va,x t +
yd (t) = yd (t0 ) + va,y t +
(6)
ρd Dd2 gt 18µa
ρd Dd2 Dd2 ρd g va,y − vd,y (t0 ) + 18µa 18µa 18µa × exp − t −1 . ρd Dd2
+
(7)
In Section 3, numerical solutions of the previous governing equations with a finite difference method will be presented.
3. Numerical modeling of Lagrangian formulation Consider a specific case of droplet motion with a droplet diameter of 0.1 mm, water droplet density of 1000 kg/m3 , air density of 1 kg/m3 and a dynamic viscosity of 18.17 × 10−6 Ns/m2 for air. Also, interfacial drag correlations for the following two limiting cases will be used: (i) high relative Reynolds numbers (Case 1) and (ii) low relative Reynolds numbers (Case 2). Then, the x-component of the droplet velocity can be expressed in terms of a change of droplet position, xd , with time. Equation (1) can be rearranged in terms of the droplet displacement in the x direction as follows,
dxd 2 dxd d2 xd 2 v = −2.25 − 2 + v a,x a,x dt dt dt 2 (Case 1; high relative Reynolds numbers), d2 x
dxd = 32.71va,x dt (Case 2; low relative Reynolds numbers).
d dt 2
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(8)
+ 32.71
(9)
Equations (8) and (9) are separated individually, since correlations for interfacial drag forces are different in the lowRer and high-Rer flow regimes. Equation (8) was developed from Eq. (1) and equation FD,i = 1/2ρa (vd,i − va,i )2 × 0.3(πDd2 /4), whereas Eq. (9) was developed from Eq. (1) and equation FD,i = 3µπDd (vd,i − va,i ). Finite difference modeling of time derivatives in the previous equation (Case 1) yields 2 0.5625 x i+1 d + xdi+1 1 − 1.125xdi−1 − 2.25tva,x 2 = 2xdi − 0.5625 x i−1 d − xdi−1 1 + 2.25tva,x 2 − 2.25t 2 va,x ,
(10)
where the superscripts i + 1, i and i − 1 designate successive time levels (note: i + 1 refers to current time level). The nonlinear term will be approximated by i+1 2 x (11) = xdi+1 xdi . d Then, the finite difference equation can be solved explicitly for the droplet position at time i + 1. For Case 2 (low relative Reynolds numbers), the finite difference equation becomes i+1 xd − xdi−1 xdi+1 − 2xdi + xdi−1 = 32.71va,x , + 32.71 2t t 2 (12) which can be solved explicitly for the droplet position at the current time level as follows, 2xdi − xdi−1 (1 − 16.36t) + 32.71va,x t 2 (13) . 1 + 16.36t In the y-momentum equation of droplet motion, the gravitational effect is included for droplet tracking. For example, in the low-Rer flow regime, it follows that
xdi+1 =
d2 yd dyd (14) + 32.71 = 32.71va,y + g. 2 dt dt Using similar procedures of finite differencing, the y-com ponent of the droplet trajectory becomes ydi+1 2ydi − ydi−1 (1 − 16.36t) + t 2 (32.71va,y + 9.81) . 1 + 16.36t (15) These equations give the time-varying spatial trajectory of a particular droplet. These Lagrangian formulations assume that the droplets behave equivalently as spherically shaped solid particles. =
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This assumption neglects internal circulation within droplets, which may arise from shear-driven momentum exchange between droplets and the co-flowing airstream. Furthermore, the large density difference between the dispersed (droplet) and carrier (air) phases produces a “virtual-mass effect.” This effect creates a force that accelerates the mass of the surrounding carrier phase around each droplet, thereby leading to additional driving mechanisms for spatial velocity variations within a droplet. Section 4 establishes a new breakdown criterion of this lumped capacitance assumption in the Lagrangian numerical method, when such variations become sufficiently high. This criterion will be used as a transition point between Lagrangian and Eulerian formulations of multiphase flow with droplets. Then a hybrid method can use this criterion to evaluate when Lagrangian tracking of individual droplet trajectories should be transformed to Eulerian spatial averaging of droplet motion. For Eulerian methods, supplementary relations are needed to accommodate momentum exchange with droplets at the transition point.
Fig. 2. Schematic of spatially varying and uniform internal velocity within droplets.
4. Criterion of internal circulation transition Fig. 3. Momentum exchange at liquid/air interface.
4.1. Thermal Biot number Appreciably large differences between air and droplet velocities may lead to shear-driven momentum exchange within a droplet (see Figs. 2 and 3). Consider a onedimensional approximation of Couette flow applied to a liquid layer subjected to differential edge velocities and internal diffusive momentum exchange in Fig. 3. This approximation attempts to represent an analogy to conduction heat transfer in a one-dimensional solid layer between two convective streams. In that case, the ratio between the temperature difference across the wall (Ts,1 − Ts,2 ) and fluid stream (Ts,2 − T∞ ) can be represented by the ratio of conduction to convection thermal resistances [2], i.e., Ts,1 − Ts,2 L/(kA) . = Ts,2 − T∞ 1/(hA)
(16)
The right side represents the non-dimensional Biot number (Bi = hL/k). When Bi < 0.1, temperature uniformity within the wall can be assumed (called the “lumped capacitance approximation”). Otherwise, spatial variations of temperature within the wall become sufficiently large that the wall is no longer isothermal. These conditions arise when the convective resistance is small compared to the thermal conduction resistance within the wall. 4.2. Momentum Biot number This section develops an analogous Biot number criterion for internal circulation within droplets. Conduction heat transfer within the wall is analogous to diffusive momentum
exchange within droplets arising from spatial velocity gradients across the airstream and droplet. Velocity gradients in the airstream produce spatial non-uniformities around the edge of a droplet, which will be represented by velocities u1 and u2 in Fig. 3. Considering the relative velocity difference across a droplet, the resulting diffusive momentum exchange may lead to spatial variations of velocity within a droplet. This section attempts to predict when these variations become significant, thereby requiring a transition from Lagrangian and Eulerian regimes. Although shear-driven momentum exchange may lead to droplet deformation/rotation, consider only internal circulation that is initially driven by cross-stream diffusion of momentum. A one-dimensional steady state approximation of the momentum equation becomes µ
∂ 2u = 0. ∂y 2
(17)
Solving this equation subject to specified velocities at both edges of the liquid layer, y u = u2,r . (18) L This result can be readily differentiated to give the shear stress within the liquid layer at the liquid/air interface. Equating this result to the shear stress computed from the other side of the phase interface (based on a local skin friction coefficient of external flow) constitutes the interfacial momentum balance in Fig. 3. This balance becomes analogous to the previously described thermal ratio between conduction and convective resistances of heat transfer within a
G.F. Naterer et al. / Journal of Colloid and Interface Science 291 (2005) 577–584
Fig. 4. One-dimensional velocity profiles at varying momentum Biot numbers.
one-dimensional wall. Using this thermal analogy, an equivalent lumped capacitance approximation for droplet velocity uniformity can be derived. On the air side of the interface, the local shear stress can be represented in terms of the friction coefficient, cf , and relative velocity as follows, 1 τ = cf ρ(u∞ − u2 )2 . (19) 2 Taking the ratio of this result to the previously differentiated velocity profile for the shear stress within the liquid layer, u2 − u1 ρcf (u∞ − u2 ) ≡ Bim . = u∞ − u2 2µ/L
(20)
This result for the momentum Biot number (Bim ) involves a local variation of skin friction coefficient due to boundary layer changes in the streamwise direction (i.e., direction traversed along the liquid/air interface). But for practical calculations in multiphase flows, it can be replaced by the drag coefficient for spherical droplets in a cross-flow, based on the relative Reynolds number. In this case, the momentum Biot number can be expressed as cd (µa /µ)Rer /2. The drag coefficient represents an averaged calculation over the droplet, rather than spatial changes at each circumferential position along the droplet surface. Various 1D approximations of the linear velocity profile within a liquid layer (or droplet) at varying momentum Biot numbers are shown in Fig. 4. These profiles confirm that spatial velocity variations are negligible at Biot numbers below about 0.1. But substantial variations arise at higher Biot numbers, thereby suggesting erroneous predictions of momentum exchange in standard Lagrangian (lumped capacitance) formulations. More substantial variations are observed at higher ratios of the freestream to droplet velocity (uf /u1 in Fig. 4), since this entails a higher relative Reynolds number between dispersed and carrier phases. Equation (20) appears analogous to the previous thermal Biot number correlation in Eq. (16). In particular, the convection coefficient is replaced by cf (u∞ − u2 )/2 and the thermal conductivity is replaced by the dynamic viscosity.
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A heat/momentum analogy becomes Bim < 0.1, when velocity uniformity within a droplet can be assumed (lumped capacitance approximation). Under these conditions, the Lagrangian method of non-deformable droplet tracking can be used; otherwise, velocity non-uniformities within a droplet may yield erroneous predictions of momentum exchange at the phase interface. Such errors due to internal circulation would arise at high relative velocities, large droplets or sufficiently small kinematic viscosities of the dispersed (droplet) phase. Although the current derivation of the momentum Biot number (Bim ) contains many simplifying approximations, it is worthwhile to compare it with the analogous thermal Biot number (Bi) criterion. The lumped capacitance approximation for Bi < 0.1 has been widely and successfully used in the past, including multiphase heat transfer with droplets, despite its simplifying assumptions. For example, Tsuruta and Tanaka [18] used a Biot number criterion successfully, despite spatial inhomogeneity of the surface heat flux along the droplet interface (analogous to varying shear stress in the case of interfacial momentum exchange). For liquid metal droplets sprayed through a convective gas stream, Chang and Chen [19] characterized the role of internal heat conduction within droplets by the Biot number, despite complex nucleation kinetics within a droplet. Wei and Yeh [20] reported that the Biot number can successfully characterize different regimes of heat transfer with sprayed droplets on a substrate. A 1D approximation of heat conduction was used successfully, similarly to the 1D approximation of momentum diffusion in this article. Dai et al. [21] showed that the Biot number characterizes temperature non-uniformity with droplets subjected to transient oscillations. The 1D Biot number approximation was successfully applied to the transient, 3D droplet oscillations. At large Biot numbers, the authors confirmed that effects of droplet oscillations on perturbed shapes became largest. This past success with the Biot number criterion suggests it can provide useful information regarding momentum exchange within droplets. The actual droplet dynamics is highly complex. For example, the surface layer exhibits complex motion due to droplet curvature. Furthermore, the convection coefficient and pressure gradient vary circumferentially around the droplet (not constant or zero). The aforementioned models have included such complexity by additional modeling within the droplet, while the Biot number provides a general transition criterion to internal temperature non-uniformity. Fully accommodating the complex mechanisms in a convenient Biot correlation would be infeasible, so a simplified criterion for Bi provides a valuable tool for predicting thermal transition within droplets. Thus, despite analogous approximations with momentum exchange in this article, it is viewed that Bim can also provide a useful tool for transition to internal circulation. It should be noted that the criterion Bim < 0.1 is intended to provide an approximate order of magnitude for this transition, rather than
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a specific value (due to the aforementioned simplifications and approximations). In general, the Biot number (Bi) represents a ratio between resistances associated with diffusive and convective transport. In this article, Bim refers to a ratio of internal diffusion within a droplet and convective exchange leading to friction between the droplet and surrounding airstream. This ratio is called the momentum Biot number, while the ratio between the conduction resistance within a droplet and convective heat transfer is represented by the thermal Biot number. A similar analogy can be constructed for a mass transfer Biot number. It should be noted that heat, mass and momentum transfer can occur simultaneously within the droplet, if each respective critical Biot number (0.1) is exceeded. In that case, coupled solutions of the energy, species concentration and momentum equations are needed. But this article considers isothermal droplets in air and neglects mass transfer associated with evaporation, so only Bim is analyzed. In other applications, such as sprays in internal combustion engines, each Biot number would become significant. When comparing each transport mode, a Reynolds analogy can characterize the ratio of convective exchange rates [1], while the Prandtl number (Pr) represents a ratio between momentum and thermal diffusivities. For oil droplets, Pr 1, whereas Pr 1 for liquid metal droplets. Thus, internal circulation would be noticed within liquid metal droplets earlier than oil droplets, provided the convective environment of surrounding air is equivalent in both cases. It is proposed that each Biot number could be used as a criterion for evaluating the effectiveness of Eulerian/Lagrangian formulations for each transport mode. If the thermal Biot number exceeds 0.1, but not the momentum Biot number, a mixed formulation could be adopted with Eulerian heat exchange and Lagrangian momentum modeling. However, this mixed formulation could become impractical from a computational perspective, so the lowest Biot number should be used as the criterion for coupled fluid and heat flow problems. In Section 5, predicted results from sensitivity studies, based on the previous Lagrangian formulation will be presented and discussed.
Fig. 5. Case 1—droplets injected perpendicular to Co-flowing airstream.
Fig. 6. Effects of varying droplet diameters on droplet trajectories.
Fig. 7. Effects of varying droplet injection velocity.
5. Results and discussion Consider a sample case, Case 1, when droplet is injected into a horizontal airstream, as depicted in Fig. 5. The lumped capacitance assumption of the Lagrangian formulation can be used at low relative Reynolds numbers or small droplet diameters with low momentum Biot numbers. But increasing losses of momentum lead to erroneous predictions of droplet motion at higher Biot numbers. In those cases, an Eulerian method is needed. Figs. 6–8 show the sample Lagrangian predictions from sensitivity studies at varying droplet diameters and inlet velocities. Physical conditions associated with the Lagrangian/Eulerian transition will be analyzed.
Fig. 8. Case 2—gravitational field effects on droplet trajectories.
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In Figs. 6–8, the incoming horizontal air velocity is ua = 40 m/s. The elapsed simulation time is t = 10 s. The horizontal location of the injected droplet is x = 1 m. The constant density of air and droplet are ρa = 1.246 kg/m3 and ρd = 999.8 kg/m3 , respectively. In Fig. 6, the injected vertical droplet velocity is vd = 20 m/s. In Figs. 7 and 8, the droplet diameter is Dd = 0.05 mm. In Fig. 6, larger droplets penetrate further upwards in the co-flowing airstream, due to greater inertia of injected droplets with larger mass. But the Lagrangian formulation neglects droplet deformation, rotation and pressure interactions arising from internal circulation. Thus, larger errors arise for those predicted trajectories and an Eulerian formulation may be needed. When the droplet diameter becomes sufficiently large to produce a momentum Biot number exceeding 0.1, the lumped capacitance assumption becomes invalid. Similarly, high momentum Biot numbers occur for large injection velocities in Fig. 7. Unlike the Lagrangian results of Figs. 6–8, an Eulerian formulation can be successfully applied over the entire range of momentum Biot numbers. However, it requires supplementary relations for interfacial interactions arising from internal circulation and spatial averaging of momentum exchange within the dispersed phase of the control volume. In Fig. 8, droplets are injected at varying angles in the co-flowing airstream (Case 2). Consider droplets injected horizontally at a zero reference height and falling under the combined influence of gravity and interfacial drag. Droplets rise upwards over a certain distance due to varying vertical components of the injection velocity, before falling downwards under gravity. The curves in Fig. 8 represent predicted trajectories of these droplets from the Lagrangian formulation. As expected, the upward distance traveled is larger at higher vertical components of the injection velocity. In this example, internal circulation and droplet deformation are induced by directional differences between droplet motion and the co-flowing airstream, rather than differences of velocity magnitude. Internal droplet circulation generated at higher vertical injection velocities could be established with a twodimensional extension of the lumped capacitance analogy.
6. Conclusions Internal circulation within droplets, initiated by sheardriven momentum exchange in multiphase flow, leads to difficulties in conventional Lagrangian methods of droplet tracking, when droplets are modeled as discrete particles. Non-uniformities of velocity within dispersed droplets may appreciably affect the cross-phase momentum exchange. This article develops a new lumped capacitance criterion, which predicts transition to internal non-uniformities of velocity within dispersed droplets of a Lagrangian formulation. A new parameter (called the momentum Biot number) is shown to characterize this internal circulation, particularly
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arising from shear-driven momentum exchange between the dispersed (droplet) and carrier (air) phases. This parameter is developed in analogy with conjugate heat transfer problems involving conduction and convection. Transition to Eulerian volume averaging is needed, when sufficiently high momentum Biot numbers characterize the spatial variations of velocity within individual droplets. In this article, predicted results are shown for droplets injected into a co-flowing airstream at varying inlet velocities. For large droplets or high injection velocities, the trajectories of droplets reach further upwards into the airstream, due to higher droplet inertia.
Acknowledgments Financial support of this research from GKN Westland Helicopters Ltd. Canada Foundation for Innovation (CFI), Natural Sciences and Engineering Research Council of Canada (NSERC) and Western Economic Diversification (WED) is gratefully acknowledged.
Appendix A. Nomenclature A Bim cf Dd h k L T Re u, v
area (m2 ) momentum Biot number friction coefficient droplet diameter (m) overall heat transfer coefficient to air (W/(m2 K)) thermal conductivity of water (W/(m K)) thickness of liquid layer (m) temperature Reynolds number velocity components (m/s)
Subscripts a d f r s ∞ 1, 2
carrier phase (air) dispersed phase (droplet) fiction relative surface freestream inner/outer edges of liquid layer (plane wall or droplet)
Superscripts i, i + 1 previous, current time levels Greek µ τ µa
dynamic viscosity of water (kg/(m s)) shear stress (N/m2 ) dynamic viscosity of air (kg/(m s))
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References [1] C. Crowe, Modeling Fluid–Particle Flows: Current Status and Future Directions, AIAA Paper 99-3690, AIAA 30th Fluid Dynamics Conference, Norfolk, VA, 1999. [2] G.F. Naterer, Heat Transfer in Single and Multiphase Systems, CRC Press, Boca Raton, FL, 2002. [3] M. Milanez, G.F. Naterer, G. Venn, G. Richardson, Part. Part. Syst. Charact. 20 (1) (2003) 62–72. [4] G. Hetsroni, Handbook of Multiphase Systems, Hemisphere– McGraw–Hill, Washington, USA, 1982. [5] M. Milanez, G.F. Naterer, G. Venn, G. Richardson, AIAA J. 42 (5) (2004) 973–979. [6] S. Banerjee, A.M.C. Chan, Int. J. Multiphase Flow 6 (1980) 1–24. [7] D. Gidaspow, C.W. Solbring, Proceedings of the AIChE 81st National Meeting, Kansas City, 1976. [8] W.T. Sha, S.L. Soo, Int. J. Multiphase Flow 5 (1979) 153–158. [9] A. Prosperetti, A.V. Jones, Int. J. Multiphase Flow 10 (1984) 425–440.
[10] B.P. LeClair, A.E. Hamiliec, H.R. Pruppaher, W.D. Hall, Air. J. Atmos. Sci. 29 (1972) 728–740. [11] J.D. Klett, J. Atmos. Sci. 34 (1977) 551–552. [12] G.I. Taylor, Proc. R. Soc. London, Ser. A 138 (1932) 41–48. [13] G.I. Taylor, Proc. R. Soc. London, Ser. A 146 (1934) 501–523. [14] M. Marchioro, M. Tanksley, A. Prosperetti, Int. J. Multiphase Flow 25 (1999) 1395–1426. [15] J.W. Park, D.A. Drew, R.T. Lahey, Int. J. Multiphase Flow 24 (1998) 1205–1244. [16] X. Chen, J.A. Friedman, X. Li, M. Renksizbulut, Part. Part. Syst. Charact. 17 (4) (2000) 180–188. [17] M. Polat, H. Polat, S. Chander, R. Hogg, Part. Part. Syst. Charact. 19 (1) (2002) 38–46. [18] T. Tsuruta, H. Tanaka, Int. J. Heat Mass Transfer 34 (11) (1991) 2779– 2786. [19] K.C. Chang, C.M. Chen, Int. J. Heat Mass Transfer 44 (8) (2001) 1573–1583. [20] P.S. Wei, F.B. Yeh, ASME J. Heat Transfer 122 (4) (2000) 792–800. [21] M. Dai, J. Perot, D.P. Schmidt, Proceedings of the ASME Internal Combustion Engine Division, New Orleans, LA, September, 2002.
On the Lagrangian/Eulerian modeling of dispersed droplet inertia: Internal circulation transition G.F. Naterer a , M. Milanez a,∗ , G. Venn b a Department of Mechanical and Manufacturing Engineering, University of Manitoba, 15 Gillson Street, Winnipeg, Manitoba, Canada, R3T 2N2 b GKN Westland Helicopters Ltd., Yeovil, Somerset, England, BA20 2YB, UK
Received 15 October 2004; accepted 8 May 2005 Available online 24 June 2005
Abstract This article addresses a limitation of Lagrangian methods for droplet tracking, when approaching the transition point of internal circulation within droplets. Laminar multiphase flow with dispersed droplets in a co-flowing airstream is considered. Analytical and numerical formulations of droplet motion are developed based on a Lagrangian finite difference method of droplet tracking. Cases of both high and low relative Reynolds numbers are formulated. The role of interfacial drag in cross-phase momentum exchange increases at higher relative Reynolds numbers. A new transition criterion is developed to characterize conditions leading to shear-driven non-uniformities of velocity within a droplet. This criterion entails a momentum Biot number, in analogy with the Biot number criterion for conjugate heat transfer problems involving conduction and convection. At sufficiently high momentum Biot numbers, appreciable changes of velocity within a droplet imply that Lagrangian methods become unsuitable and transition to Eulerian volume averaging is needed. Predicted results of Lagrangian modeling of droplet motion in a co-flowing airstream are presented and discussed. 2005 Elsevier Inc. All rights reserved. Keywords: Multiphase flow; Lagrangian/Eulerian tracking; Droplet inertia; Droplet/air interface; Shear driven velocity non-uniformities; Transition to internal circulation; Momentum Biot number criterion
1. Introduction Accurate numerical modeling of multiphase flows with droplets is a technological goal of considerable interest. Detailed understanding of multiphase flows is often more comprehensive than single phase flows, due to complex transport mechanisms involving momentum exchange, interfacial interactions and phase change heat transfer (Crowe [1]; Naterer [2]). This article investigates convection modeling of multiphase flows with droplets, particularly relating to impinging droplets in icing applications. Icing of helicopter surfaces involves momentum exchange between incoming droplets and the co-flowing air stream.
* Corresponding author.
E-mail addresses: [email protected], [email protected] (M. Milanez). 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.05.025
Droplet momentum displacement in multiphase flow with droplets near a flat plate entails cross-phase interactions with air stream (Milanez et al. [3]). Convective acceleration of droplets depends on the volume fraction of the dispersed phase [4]. In regions where the multiphase flow is not largely affected by droplet/droplet interactions, a dilute flow assumption may be adopted [4]. Such situations occur when the dispersed (droplet) phase is predominantly controlled by the carrier (air) phase. For example, when a helicopter travels through clouds, the impinging supercooled droplets on the helicopter surfaces are immersed in moving air. Their movement is not dominated by droplet/droplet interactions, although distinct interfacial and bulk pressures may arise in such dilute flows (Milanez et al. [5]). This article considers convection modeling with a Lagrangian formulation of dilute multiphase flow with droplets. Past Lagrangian methods generally entail spatial tracking of droplets, which are approximated as discrete particles
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without internal circulation. But shear-driven deformation or rotation of droplets can lead to internal circulation with resulting upstream pressure disturbances. In contrast, Eulerian methods have been developed successfully for multiphase flows (i.e., Banerjee and Chan [6], Milanez et al. [3,5], Hetsroni [4]), but generally without volume averaging for spatial effects of internal circulation on upstream pressure disturbances. This article attempts to address the transition from Lagrangian tracking of discrete particles to Eulerian averaging of droplets, when local pressure disturbances arise with internal droplet circulation. Internal circulation affects the interfacial pressure interactions between dispersed and continuous (carrier) phases in a diffusive type manner (Gidaspow and Solbring [7]). Compression of the gas phase depends on the predicted interfacial pressure term from both kinetic energy and continuity equations (Sha and Soo [8]). Cross-phase interactions arising from internal circulation affect the momentum exchange within the dispersed phase. Pressure disturbances induced by fluctuations in the dispersed phase may affect the carrier phase, particularly if the dispersed phase is moved suddenly from a region to another region with different bulk pressures. Prosperetti and Jones [9] have presented a detailed formulation for such interfacial pressure effects. LeClair et al. [10] reported that boundary layer theory overestimates the internal circulation within a droplet. Klett [11] discusses the limitations of boundary layer equations to internal circulation within droplets. Earlier studies by Taylor [12,13] provided fundamental insight regarding emulsions of droplets and viscosity of fluids containing droplets of other liquids. Past studies by Marchioro et al. [14] have successfully applied methods of ensemble averaging to mixture pressure and viscous stress modeling of dispersed two-phase flows. Similar advances involving two-fluid models have been reported by Park et al. [15], but effects of internal circulation on the averaged pressure in the bulk flow were not outlined. In the particle tracking procedure of conventional Lagrangian methods, spatial variations of temperature and velocity within a droplet are lost through a lumped capacitance assumption. In classical heat transfer analysis, the lumped capacitance assumption for temperature is known to be valid in cases when the Biot number is less than 0.1 [2]. Beyond this value, spatial variations of temperature within the object become significant and the internal temperature cannot be assumed to be uniformly isothermal. This article attempts to derive a similar lumped capacitance analogy and iso-velocity criterion for dispersed droplets in multiphase flows. Chen et al. [16] develop Eulerian modeling of the turbulent carrier phase and Lagrangian tracking of particle dispersion for turbulent two-phase flows. This approach avoids empirical parameters required for the particulate dispersion width. Particle dispersion is affected by surface tension (Polat et al. [17]). Chen et al. [16] show that smoother droplet profiles are obtained with stochastic/probabilistic modeling, as compared with an eddy interaction model. Laminar mul-
tiphase will be investigated and solutions will be limited to cases with spherical, non-deforming droplets and negligible droplet/droplet interactions.
2. Formulation of dispersed droplet motion in an airstream The Lagrangian method is a well-known conventional technique for tracking droplet trajectories in multiphase flows. In this method, a force balance (including drag and gravitational forces) is applied to each discrete droplet throughout its spatial trajectory. From Newton’s Law of motion, the time-varying forces contribute to the droplet’s acceleration. Then, successive temporal integrations of this acceleration field yield the velocity and position of individual droplets within the flow field. In this section and the following section, analytical and numerical formulations of the Lagrangian method will be outlined for such droplet motion. A Lagrangian analytical formulation of droplet/air interactions over a range of relative Reynolds numbers (Rer ) will be presented. Consider a force balance with interfacial drag and gravity acting on dispersed droplets (subscript d) in an airstream (subscript a) in the low-Re regime (see Fig. 1). The laminar equations of motion in the x and y directions are d (vd,x ) = −FD,x , (1) dt d m (vd,y ) = −FD,y + mgy , (2) dt where FD,x , FD,y and gy are the Cartesian drag forces in the x and y directions and gravity in the y direction. For lowRer flow regimes, the linear drag force in each direction, i, is expressed as FD,i = 3µπDd (vd,i − va,i ). The quadratic law is used to model the drag force in high-Rer number flow regimes, i.e., FD,i = 1/2ρa (vd,i − va,i )2 × 0.3(πDd2 /4). Also, the relative Reynolds number between phases in contact, Rer,i , in each direction, i, is defined as m
Rer,i =
ρd (vd,i − va,i )Dd . µa
Fig. 1. Schematic of droplet trajectory.
(3)
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Solving Eqs. (1) and (2) in the low-Rer regime, subject to a known velocity at an initial time, t0 , yields the following analytical solutions of droplet velocity in the x and y directions, vd,x (t) = va,x + vd,x (t0 ) − va,x 18µa (t − t ) × exp − (4) 0 , ρd Dd2 ρd Dd2 ρd Dd2 vd,y (t) = va,y + g + vd,y (t0 ) − va,y − g 18µa 18µa 18µa (t0 − t) . × exp (5) ρd Dd2 Furthermore, integrating these equations over distance and time yields two separate equations for the x and y positions of the droplet (denoted by xd and yd , respectively). The initial time is set to zero value. These equations together represent the trajectory for droplet motion in the airstream. This analytical trajectory becomes ρd Dd2 va,x − vd,x (t0 ) 18µa 18µa × exp − t − 1 , ρd Dd2
xd (t) = xd (t0 ) + va,x t +
yd (t) = yd (t0 ) + va,y t +
(6)
ρd Dd2 gt 18µa
ρd Dd2 Dd2 ρd g va,y − vd,y (t0 ) + 18µa 18µa 18µa × exp − t −1 . ρd Dd2
+
(7)
In Section 3, numerical solutions of the previous governing equations with a finite difference method will be presented.
3. Numerical modeling of Lagrangian formulation Consider a specific case of droplet motion with a droplet diameter of 0.1 mm, water droplet density of 1000 kg/m3 , air density of 1 kg/m3 and a dynamic viscosity of 18.17 × 10−6 Ns/m2 for air. Also, interfacial drag correlations for the following two limiting cases will be used: (i) high relative Reynolds numbers (Case 1) and (ii) low relative Reynolds numbers (Case 2). Then, the x-component of the droplet velocity can be expressed in terms of a change of droplet position, xd , with time. Equation (1) can be rearranged in terms of the droplet displacement in the x direction as follows,
dxd 2 dxd d2 xd 2 v = −2.25 − 2 + v a,x a,x dt dt dt 2 (Case 1; high relative Reynolds numbers), d2 x
dxd = 32.71va,x dt (Case 2; low relative Reynolds numbers).
d dt 2
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(8)
+ 32.71
(9)
Equations (8) and (9) are separated individually, since correlations for interfacial drag forces are different in the lowRer and high-Rer flow regimes. Equation (8) was developed from Eq. (1) and equation FD,i = 1/2ρa (vd,i − va,i )2 × 0.3(πDd2 /4), whereas Eq. (9) was developed from Eq. (1) and equation FD,i = 3µπDd (vd,i − va,i ). Finite difference modeling of time derivatives in the previous equation (Case 1) yields 2 0.5625 x i+1 d + xdi+1 1 − 1.125xdi−1 − 2.25tva,x 2 = 2xdi − 0.5625 x i−1 d − xdi−1 1 + 2.25tva,x 2 − 2.25t 2 va,x ,
(10)
where the superscripts i + 1, i and i − 1 designate successive time levels (note: i + 1 refers to current time level). The nonlinear term will be approximated by i+1 2 x (11) = xdi+1 xdi . d Then, the finite difference equation can be solved explicitly for the droplet position at time i + 1. For Case 2 (low relative Reynolds numbers), the finite difference equation becomes i+1 xd − xdi−1 xdi+1 − 2xdi + xdi−1 = 32.71va,x , + 32.71 2t t 2 (12) which can be solved explicitly for the droplet position at the current time level as follows, 2xdi − xdi−1 (1 − 16.36t) + 32.71va,x t 2 (13) . 1 + 16.36t In the y-momentum equation of droplet motion, the gravitational effect is included for droplet tracking. For example, in the low-Rer flow regime, it follows that
xdi+1 =
d2 yd dyd (14) + 32.71 = 32.71va,y + g. 2 dt dt Using similar procedures of finite differencing, the y-com ponent of the droplet trajectory becomes ydi+1 2ydi − ydi−1 (1 − 16.36t) + t 2 (32.71va,y + 9.81) . 1 + 16.36t (15) These equations give the time-varying spatial trajectory of a particular droplet. These Lagrangian formulations assume that the droplets behave equivalently as spherically shaped solid particles. =
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This assumption neglects internal circulation within droplets, which may arise from shear-driven momentum exchange between droplets and the co-flowing airstream. Furthermore, the large density difference between the dispersed (droplet) and carrier (air) phases produces a “virtual-mass effect.” This effect creates a force that accelerates the mass of the surrounding carrier phase around each droplet, thereby leading to additional driving mechanisms for spatial velocity variations within a droplet. Section 4 establishes a new breakdown criterion of this lumped capacitance assumption in the Lagrangian numerical method, when such variations become sufficiently high. This criterion will be used as a transition point between Lagrangian and Eulerian formulations of multiphase flow with droplets. Then a hybrid method can use this criterion to evaluate when Lagrangian tracking of individual droplet trajectories should be transformed to Eulerian spatial averaging of droplet motion. For Eulerian methods, supplementary relations are needed to accommodate momentum exchange with droplets at the transition point.
Fig. 2. Schematic of spatially varying and uniform internal velocity within droplets.
4. Criterion of internal circulation transition Fig. 3. Momentum exchange at liquid/air interface.
4.1. Thermal Biot number Appreciably large differences between air and droplet velocities may lead to shear-driven momentum exchange within a droplet (see Figs. 2 and 3). Consider a onedimensional approximation of Couette flow applied to a liquid layer subjected to differential edge velocities and internal diffusive momentum exchange in Fig. 3. This approximation attempts to represent an analogy to conduction heat transfer in a one-dimensional solid layer between two convective streams. In that case, the ratio between the temperature difference across the wall (Ts,1 − Ts,2 ) and fluid stream (Ts,2 − T∞ ) can be represented by the ratio of conduction to convection thermal resistances [2], i.e., Ts,1 − Ts,2 L/(kA) . = Ts,2 − T∞ 1/(hA)
(16)
The right side represents the non-dimensional Biot number (Bi = hL/k). When Bi < 0.1, temperature uniformity within the wall can be assumed (called the “lumped capacitance approximation”). Otherwise, spatial variations of temperature within the wall become sufficiently large that the wall is no longer isothermal. These conditions arise when the convective resistance is small compared to the thermal conduction resistance within the wall. 4.2. Momentum Biot number This section develops an analogous Biot number criterion for internal circulation within droplets. Conduction heat transfer within the wall is analogous to diffusive momentum
exchange within droplets arising from spatial velocity gradients across the airstream and droplet. Velocity gradients in the airstream produce spatial non-uniformities around the edge of a droplet, which will be represented by velocities u1 and u2 in Fig. 3. Considering the relative velocity difference across a droplet, the resulting diffusive momentum exchange may lead to spatial variations of velocity within a droplet. This section attempts to predict when these variations become significant, thereby requiring a transition from Lagrangian and Eulerian regimes. Although shear-driven momentum exchange may lead to droplet deformation/rotation, consider only internal circulation that is initially driven by cross-stream diffusion of momentum. A one-dimensional steady state approximation of the momentum equation becomes µ
∂ 2u = 0. ∂y 2
(17)
Solving this equation subject to specified velocities at both edges of the liquid layer, y u = u2,r . (18) L This result can be readily differentiated to give the shear stress within the liquid layer at the liquid/air interface. Equating this result to the shear stress computed from the other side of the phase interface (based on a local skin friction coefficient of external flow) constitutes the interfacial momentum balance in Fig. 3. This balance becomes analogous to the previously described thermal ratio between conduction and convective resistances of heat transfer within a
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Fig. 4. One-dimensional velocity profiles at varying momentum Biot numbers.
one-dimensional wall. Using this thermal analogy, an equivalent lumped capacitance approximation for droplet velocity uniformity can be derived. On the air side of the interface, the local shear stress can be represented in terms of the friction coefficient, cf , and relative velocity as follows, 1 τ = cf ρ(u∞ − u2 )2 . (19) 2 Taking the ratio of this result to the previously differentiated velocity profile for the shear stress within the liquid layer, u2 − u1 ρcf (u∞ − u2 ) ≡ Bim . = u∞ − u2 2µ/L
(20)
This result for the momentum Biot number (Bim ) involves a local variation of skin friction coefficient due to boundary layer changes in the streamwise direction (i.e., direction traversed along the liquid/air interface). But for practical calculations in multiphase flows, it can be replaced by the drag coefficient for spherical droplets in a cross-flow, based on the relative Reynolds number. In this case, the momentum Biot number can be expressed as cd (µa /µ)Rer /2. The drag coefficient represents an averaged calculation over the droplet, rather than spatial changes at each circumferential position along the droplet surface. Various 1D approximations of the linear velocity profile within a liquid layer (or droplet) at varying momentum Biot numbers are shown in Fig. 4. These profiles confirm that spatial velocity variations are negligible at Biot numbers below about 0.1. But substantial variations arise at higher Biot numbers, thereby suggesting erroneous predictions of momentum exchange in standard Lagrangian (lumped capacitance) formulations. More substantial variations are observed at higher ratios of the freestream to droplet velocity (uf /u1 in Fig. 4), since this entails a higher relative Reynolds number between dispersed and carrier phases. Equation (20) appears analogous to the previous thermal Biot number correlation in Eq. (16). In particular, the convection coefficient is replaced by cf (u∞ − u2 )/2 and the thermal conductivity is replaced by the dynamic viscosity.
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A heat/momentum analogy becomes Bim < 0.1, when velocity uniformity within a droplet can be assumed (lumped capacitance approximation). Under these conditions, the Lagrangian method of non-deformable droplet tracking can be used; otherwise, velocity non-uniformities within a droplet may yield erroneous predictions of momentum exchange at the phase interface. Such errors due to internal circulation would arise at high relative velocities, large droplets or sufficiently small kinematic viscosities of the dispersed (droplet) phase. Although the current derivation of the momentum Biot number (Bim ) contains many simplifying approximations, it is worthwhile to compare it with the analogous thermal Biot number (Bi) criterion. The lumped capacitance approximation for Bi < 0.1 has been widely and successfully used in the past, including multiphase heat transfer with droplets, despite its simplifying assumptions. For example, Tsuruta and Tanaka [18] used a Biot number criterion successfully, despite spatial inhomogeneity of the surface heat flux along the droplet interface (analogous to varying shear stress in the case of interfacial momentum exchange). For liquid metal droplets sprayed through a convective gas stream, Chang and Chen [19] characterized the role of internal heat conduction within droplets by the Biot number, despite complex nucleation kinetics within a droplet. Wei and Yeh [20] reported that the Biot number can successfully characterize different regimes of heat transfer with sprayed droplets on a substrate. A 1D approximation of heat conduction was used successfully, similarly to the 1D approximation of momentum diffusion in this article. Dai et al. [21] showed that the Biot number characterizes temperature non-uniformity with droplets subjected to transient oscillations. The 1D Biot number approximation was successfully applied to the transient, 3D droplet oscillations. At large Biot numbers, the authors confirmed that effects of droplet oscillations on perturbed shapes became largest. This past success with the Biot number criterion suggests it can provide useful information regarding momentum exchange within droplets. The actual droplet dynamics is highly complex. For example, the surface layer exhibits complex motion due to droplet curvature. Furthermore, the convection coefficient and pressure gradient vary circumferentially around the droplet (not constant or zero). The aforementioned models have included such complexity by additional modeling within the droplet, while the Biot number provides a general transition criterion to internal temperature non-uniformity. Fully accommodating the complex mechanisms in a convenient Biot correlation would be infeasible, so a simplified criterion for Bi provides a valuable tool for predicting thermal transition within droplets. Thus, despite analogous approximations with momentum exchange in this article, it is viewed that Bim can also provide a useful tool for transition to internal circulation. It should be noted that the criterion Bim < 0.1 is intended to provide an approximate order of magnitude for this transition, rather than
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a specific value (due to the aforementioned simplifications and approximations). In general, the Biot number (Bi) represents a ratio between resistances associated with diffusive and convective transport. In this article, Bim refers to a ratio of internal diffusion within a droplet and convective exchange leading to friction between the droplet and surrounding airstream. This ratio is called the momentum Biot number, while the ratio between the conduction resistance within a droplet and convective heat transfer is represented by the thermal Biot number. A similar analogy can be constructed for a mass transfer Biot number. It should be noted that heat, mass and momentum transfer can occur simultaneously within the droplet, if each respective critical Biot number (0.1) is exceeded. In that case, coupled solutions of the energy, species concentration and momentum equations are needed. But this article considers isothermal droplets in air and neglects mass transfer associated with evaporation, so only Bim is analyzed. In other applications, such as sprays in internal combustion engines, each Biot number would become significant. When comparing each transport mode, a Reynolds analogy can characterize the ratio of convective exchange rates [1], while the Prandtl number (Pr) represents a ratio between momentum and thermal diffusivities. For oil droplets, Pr 1, whereas Pr 1 for liquid metal droplets. Thus, internal circulation would be noticed within liquid metal droplets earlier than oil droplets, provided the convective environment of surrounding air is equivalent in both cases. It is proposed that each Biot number could be used as a criterion for evaluating the effectiveness of Eulerian/Lagrangian formulations for each transport mode. If the thermal Biot number exceeds 0.1, but not the momentum Biot number, a mixed formulation could be adopted with Eulerian heat exchange and Lagrangian momentum modeling. However, this mixed formulation could become impractical from a computational perspective, so the lowest Biot number should be used as the criterion for coupled fluid and heat flow problems. In Section 5, predicted results from sensitivity studies, based on the previous Lagrangian formulation will be presented and discussed.
Fig. 5. Case 1—droplets injected perpendicular to Co-flowing airstream.
Fig. 6. Effects of varying droplet diameters on droplet trajectories.
Fig. 7. Effects of varying droplet injection velocity.
5. Results and discussion Consider a sample case, Case 1, when droplet is injected into a horizontal airstream, as depicted in Fig. 5. The lumped capacitance assumption of the Lagrangian formulation can be used at low relative Reynolds numbers or small droplet diameters with low momentum Biot numbers. But increasing losses of momentum lead to erroneous predictions of droplet motion at higher Biot numbers. In those cases, an Eulerian method is needed. Figs. 6–8 show the sample Lagrangian predictions from sensitivity studies at varying droplet diameters and inlet velocities. Physical conditions associated with the Lagrangian/Eulerian transition will be analyzed.
Fig. 8. Case 2—gravitational field effects on droplet trajectories.
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In Figs. 6–8, the incoming horizontal air velocity is ua = 40 m/s. The elapsed simulation time is t = 10 s. The horizontal location of the injected droplet is x = 1 m. The constant density of air and droplet are ρa = 1.246 kg/m3 and ρd = 999.8 kg/m3 , respectively. In Fig. 6, the injected vertical droplet velocity is vd = 20 m/s. In Figs. 7 and 8, the droplet diameter is Dd = 0.05 mm. In Fig. 6, larger droplets penetrate further upwards in the co-flowing airstream, due to greater inertia of injected droplets with larger mass. But the Lagrangian formulation neglects droplet deformation, rotation and pressure interactions arising from internal circulation. Thus, larger errors arise for those predicted trajectories and an Eulerian formulation may be needed. When the droplet diameter becomes sufficiently large to produce a momentum Biot number exceeding 0.1, the lumped capacitance assumption becomes invalid. Similarly, high momentum Biot numbers occur for large injection velocities in Fig. 7. Unlike the Lagrangian results of Figs. 6–8, an Eulerian formulation can be successfully applied over the entire range of momentum Biot numbers. However, it requires supplementary relations for interfacial interactions arising from internal circulation and spatial averaging of momentum exchange within the dispersed phase of the control volume. In Fig. 8, droplets are injected at varying angles in the co-flowing airstream (Case 2). Consider droplets injected horizontally at a zero reference height and falling under the combined influence of gravity and interfacial drag. Droplets rise upwards over a certain distance due to varying vertical components of the injection velocity, before falling downwards under gravity. The curves in Fig. 8 represent predicted trajectories of these droplets from the Lagrangian formulation. As expected, the upward distance traveled is larger at higher vertical components of the injection velocity. In this example, internal circulation and droplet deformation are induced by directional differences between droplet motion and the co-flowing airstream, rather than differences of velocity magnitude. Internal droplet circulation generated at higher vertical injection velocities could be established with a twodimensional extension of the lumped capacitance analogy.
6. Conclusions Internal circulation within droplets, initiated by sheardriven momentum exchange in multiphase flow, leads to difficulties in conventional Lagrangian methods of droplet tracking, when droplets are modeled as discrete particles. Non-uniformities of velocity within dispersed droplets may appreciably affect the cross-phase momentum exchange. This article develops a new lumped capacitance criterion, which predicts transition to internal non-uniformities of velocity within dispersed droplets of a Lagrangian formulation. A new parameter (called the momentum Biot number) is shown to characterize this internal circulation, particularly
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arising from shear-driven momentum exchange between the dispersed (droplet) and carrier (air) phases. This parameter is developed in analogy with conjugate heat transfer problems involving conduction and convection. Transition to Eulerian volume averaging is needed, when sufficiently high momentum Biot numbers characterize the spatial variations of velocity within individual droplets. In this article, predicted results are shown for droplets injected into a co-flowing airstream at varying inlet velocities. For large droplets or high injection velocities, the trajectories of droplets reach further upwards into the airstream, due to higher droplet inertia.
Acknowledgments Financial support of this research from GKN Westland Helicopters Ltd. Canada Foundation for Innovation (CFI), Natural Sciences and Engineering Research Council of Canada (NSERC) and Western Economic Diversification (WED) is gratefully acknowledged.
Appendix A. Nomenclature A Bim cf Dd h k L T Re u, v
area (m2 ) momentum Biot number friction coefficient droplet diameter (m) overall heat transfer coefficient to air (W/(m2 K)) thermal conductivity of water (W/(m K)) thickness of liquid layer (m) temperature Reynolds number velocity components (m/s)
Subscripts a d f r s ∞ 1, 2
carrier phase (air) dispersed phase (droplet) fiction relative surface freestream inner/outer edges of liquid layer (plane wall or droplet)
Superscripts i, i + 1 previous, current time levels Greek µ τ µa
dynamic viscosity of water (kg/(m s)) shear stress (N/m2 ) dynamic viscosity of air (kg/(m s))
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