The dynamics of a boiling bubble before and after detachment ...

Heat Mass Transfer (2009) 45:831–846 DOI 10.1007/s00231-007-0254-7

SPECIAL ISSUE

The dynamics of a boiling bubble before and after detachment C. W. M. van der Geld

Received: 18 August 2006 / Accepted: 18 December 2006 / Published online: 3 April 2007 Springer-Verlag 2007

Abstract An overview is given of prediction methods for motion and deformation of a bubble that is created by boiling at a wall, at times before and after detachment, with a focus on added mass forces in the vicinity of the wall. The possibility to apply added mass coefficients derived in potential flows also to flows with vorticity is examined. An introduction to Lagrangian methods is given. Added mass tensors are derived for deforming bubbles at and away from a plane wall. Expressions for induced hydrodynamic lift forces are given, and validation experiments are briefly discussed.

acceleration of gravity (m2/s) distance of center to the wall (m) added mass tensor (–) mass (kg) normal (m) velocity (m/s) generalized coordinate Generalized velocity generalized force generalized drag force (N) radius (m) radius (m) dimensionless vorticity (–) time (s) kinetic energy (J) added mass coefficient (–) velocity (m/s) velocity parallel to the wall (m/s) velocity of a bubble (m/s) velocity of the liquid (m/s) velocity of object relative to fluid (m/s) volume (m3) coordinate (m) position vector (m) position of the center of mass (m) coordinate (m) velocity component in x-direction (m/s)

List of symbols Roman letters A area (m2) A11 added mass coefficient (–) A~rs added mass coefficient (–) Ac acceleration number (–) bn generalized coordinate n (–) c coefficient (–) cAM added mass coefficient (–) db diameter of a bubble (m) dS element of area (m2) dV element of volume (m3) F force (N)

g h I M n U q q_ Q QW r R Sr t T tr(b) v V vb vL vrel V x x xCM y x_

C. W. M. van der Geld (&) Faculty of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands e-mail: [email protected]

Greek letters a added mass coefficient to U2 (–) a2 added mass coefficient to V2 (–) @V boundary of volume (m2) cm added mass coefficient m (–) h contact angle (o)

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l q r u U_ w wij x

Heat Mass Transfer (2009) 45:831–846

dynamic viscosity (kg/m/s) mass density (kg/m3) surface tension coefficient (N/m) velocity potential (m2/s) energy dissipation rate (W) added mass coefficient (–) added mass coefficient i,j (–) vorticity (1/s)

1 Introduction Boiling heat transfer takes place in various industrial processes. Among the physical parameters that determine heat transfer in convective flow boiling are the bubble volume at detachment and the frequency of bubble production. Prediction tools for them are usually based on a force balance and an assumption about the shape at detachment. When a bubble has been created at the wall of, for example, an evaporator tube and has actually detached from it, it should preferably migrate swiftly to the center of the tube in order to expedite the mixing of cold and hot fluid. If not, high heat fluxes might even lead to the built-up of an insulating layer of vapor covering a substantial part of the wall. The tube might be designed such that the interaction of the fluid flow with the detached bubble makes it gravitate to the tube center, as in rifled tubes. Prediction of migration of bubbles requires knowledge of the forces involved. Knowledge of the forces acting on a bubble that is close to a wall or that is growing at a wall, and of hydrodynamic forces in particular, is therefore important. This paper summarizes some of our capacity to predict the motion and deformation of a bubble, both at and close to a wall. The focus is on prediction methods that are applicable to bubbles created in flow boiling are highlighted. Because force prediction is critical, much work was expended in the decoupling and derivation of the basic force terms governing motion of the center point of a bubble [41]. Although decoupling is not always possible, there are reasons to believe that it is possible for the case of the added or virtual mass forces. As a consequence, added mass coefficients derived for potential flow components would retain their value when vorticity is added to these flow components. Since this is a crucial result, an analytical proof is given and both analytical and numerical proofs are discussed in Sect. 2.1. Suppose now that the velocity field that approaches a growing bubble at a plane wall can be meaningfully decomposed in potential flow components and vortical ones, as in Fig. 1. The inertia forces corresponding to the added mass can be derived with the approach of Sect. 3. This approach is based on the Euler–Lagrange–Kirchhoff method. The forces related to the vorticity in the flow can

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subsequently be added to the added mass forces, as argued in Sect. 2.1. These vorticity-related forces are discussed in sect. 2.2 that summarizes results applicable in the pointforce approach. This approach is common in the computation of trajectories of objects away from the wall. An important example of vorticity related forces is the lift force that results from vorticity in the approaching flow. Section 2.2.3 discusses its relation to added mass coefficients. Numerical prediction of trajectories of freely moving particles and bubbles usually employs point force equations of motion and correlations for coefficients in these equations. Point forces expressions were also found for, and applied to, bubbles in the vicinity of a wall, but usually knowledge in this area is far more limited. Section 2.3 gives some results for the effect of the wall on the drag force coefficient. The presence of a wall leads to inhomogeneous flow and to vortical components in the approaching flow of a bubble. Some of the effects of these components on coefficients in force expressions can be estimated with findings summarized in Sect. 2.2. The spatial acceleration that can be accounted for by pointforce results of Sect. 2.2.2 can be important in boiling, in particular in confined boiling [9]. In the last decade, interest in confined boiling applications has been growing rapidly [45] because of applications such as the cooling of components in consumer electronics. Nearly all point force expressions are for spheres; deformation of the bubble–fluid interface is usually not considered. In addition, correlations for the drag force coefficient are usually for steady flow situations when a fully developed wake is present. Knowledge like this might be applicable to so-called sliding bubbles that exist in the flow for quite some time [50], but not to a quickly growing boiling bubble. A typical boiling bubble in forced convection grows in 3 ms from a radius of zero to one of 0.5 mm. The wake is practically non-existent in this case. Vorticity generated at the interface is confined to a thin layer at this surface. A growing bubble that is being generated by boiling at a solid boundary and that experiences a homogeneous approaching flow (component) parallel to the wall therefore yields a typical example of inviscid flow. This example is therefore analyzed in detail in Sect. 3, using Lagrangian methods.

h

Fig. 1 Schematic of a way to decompose the flow that approaches a bubble; a uniform flow plus a simple shear component are shown


Point force equations typically require the prediction of vector quantities, the forces imparted to the vapor–liquid interface, and yield predictions of a vector quantity, the acceleration of the center point of the bubble. In the classical framework of inviscid potential flow theory it is easy to obtain equations that govern the motion and deformation of a bubble. This merely requires the assessment of a scalar quantity, the kinetic energy of the fluid–bubble system and the application of general principles of classical mechanics, the so-called Lagrangian approach. As discussed above, uniform flow over a rapidly growing boiling bubble at a wall can be modeled as inviscid. A model for this inviscid flow is presented in Sect. 3.3 for bubbles with the shape of a truncated sphere. Spherical expansion of a spherical bubble is governed by an equation that far away from the wall reduces to the well-known Rayleigh–Plesset equation that is often applied in the literature. Lagrangian methods easily provide this equation but are nevertheless seldom used in boiling studies. Section 3 therefore gives an introduction to Lagrangian methods. There is no ambiguity in the generalized forces that have to be applied to a boiling bubble at a wall. They can be derived from the mechanical energy equation, as will be shown in Sect. 3.3. The Lagrangian method readily applies to bubbles that experience more complex deformation of the interface. Some examples will be given in Sect. 3.4. To validate the computations of Sect. 3.3 it would be best to have a uniform flow approaching a growing boiling bubble. A dedicated experiment to this end, that is currently being carried out, is briefly discussed (Sect. 3.5). Predictions for an upper bound for the drag force on rapidly growing boiling bubbles of Sect. 3.3 will be compared with lift forces on the bubble (Sect. 3.5).

2 Point force equations of motion The ability to predict trajectories is largely dependent on the possibility to isolate and predict forces that are imparted to the bubble interface. In this section, point forces acting on the center of the bubble are summarized and the use of added mass coefficients in vortical flows is discussed. 2.1 The added mass in viscous flows Howe [28], using the concept of bound vorticity [48], showed that when a body with a constant volume moves in a viscous, incompressible flow at rest at infinity, the added mass force can be separated from the force due to the vorticity in the flow and that the added mass coefficients retain the values of inviscid theory. With the assumptions of constant volume, incompressibility and rest at infinity

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this result concerning added mass is alternatively obtained in the following way. Apart from the pressure contribution at the potential flow region at infinity, Fp,¥, the force on the body is given by minus the time rate of change of the momentum of the fluid. In the present case this momentum is finite and given by Z Z qL vL dV¼ qL xðvL nÞdS ð1Þ VL

@V

Here, qL is mass density of the liquid, @Vis the boundary of the liquid volume, VL ; and n is the normal inward in the body, outward to the fluid. Let ni be component i of n. The boundary comprises the body–fluid interface, Sb, and a contour at infinity. Define three functions u*i such that they satisfy the Laplace equation and ni ¼ n r/i for each i, and such that at large distances from the body, at S¥, the functions behave in a way to specified shortly. Consider the case that the body has instantaneous translation velocity U and is not rotating; the latter assumption is not essential. It is now straightforward to show that the fluid momentum is given by Z Z i qL U /i ndS þ qL x vL U i r/i n dS Sb

S1

where the Einstein summation convention is applied. The second integral is made constant in time by assumptions set regarding the velocity at S¥ and the behavior of functions u*i there. The velocity field far away of an expanding bubble behaves as 1/|x|2, and in this case a shape potential u* should be found such that r/ / v at S¥, making the second integral in the above expression equal to zero. This shape potential should be a source potential, but this case of a growing bubble is not further investigated here. In the analysis of Howe [28] the fluid is assumed to be at rest at infinity, the body volume is constant and an appendix is devoted to the velocities at large distances from the body. Let the time rate of change of the integral over S¥ now be zero. The force on the body is Fp,¥ minus the time rate of change of the integral over Sb in the above expression, i.e. Fp;1 dU i I b;i =dt with the virtual mass tensor {(Ib,i)j}i,j as defined by SaffR def man ([48], p. 81); qL ¼ 1 : I b;i ¼ qL Sb /i n dS: The added, or virtual, mass is not affected by the vorticity in the flow, and dependent on the shape potentials u*i . This is the desired result. For the case of flow in an unbounded fluid without restrictions at infinity, the fluid momentum may not be well defined defined ([48], p. 50), and the proof of the applicability of Ib has not been given yet. In the case of a growing,

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deforming bubble, Sect. 3 will utilize added mass tensors that can be seen as extensions of the above added mass tensor. Whether these tensors retain their values in vortical flows has not been proven yet. In order to have a zero velocity at infinite distance up- and downstream of the center of the bubble in the case of Fig. 1 an observer would need to be moving along with the fluid at the uniform approach velocity of the bubble center. The above components Ui are then components of the relative velocity. To guarantee zero velocity at infinite distance in other directions, it suffices to assume that the bubble is at the wall of in an infinitely long tube. Howe [28] used the same potentials u*i in a nice way to derive explicit expressions for the skin friction. The vorticity force can be split in one due to free vorticity and in one resulting from skin friction, or excess bound vorticity. In the case of a sphere with radius db/2 in inviscid, irrotational flow, Eq. (2.27) of Howe ([28], p. 409) applies and the skin friction corresponds to the vorticity bound to the sphere in a thin boundary layer, x ¼ r vL : This yields the drag force Z Fdrag ¼ lL ðx r/Þ n dS ð2Þ S

with lL the dynamic viscosity of the liquid and u the velocity potential that satisfies the condition of zero normal velocity at the interface1. As for other analytical proofs, Mougin and Magnaudet [43] used a kind of perturbation scheme to show that the short-time response of the body to a change in upstream flow conditions is predicted by potential flow theory. As for numerical proofs, Magnaudet et al. [39] showed that the added mass coefficient of a solitary sphere in a viscous flow is 12 if it is sufficiently remote from solid boundaries. Also other numerical studies [11, 12, 42, 47] show that the added mass term for finite Reynolds-number flows is the same as the one predicted by potential flow theory. Bagchi and Balachandar [6] used a pseudo-spectral DNS method to solve the flow around a solid sphere in a linearly varying approach velocity field for Reynolds numbers in the range 10–300. For a wide variety of straining flows, the added mass force arising from convective acceleration was found to be given by inviscid theory Bagchi and Balachandar ([5], p. 144). The above studies also proved that added-mass effects are independent on the type of boundary condition at the surface: noslip or free-slip.

Most of these numerical studies have efficiently considered a fixed body submitted to a prescribed flow. This type of generic research is appropriate for determining dependencies of coefficients of point forces, but does not account for adaptation of, for example, the bubble shape or bubble orientation to the dynamics of the flow. The Lagrangian method of Sect. 3 offers a convenient way for this adaptation which is also applicable if boundary layers are developing at the free surface since pressure does not change in a boundary layer in the direction normal to the interface. Another drawback of the fixation of the body in the above numerical studies is the fact that vortex shedding and wake instabilities depend on the possibility to move or not [25]. It is safe to conclude that the added mass tensor in some important applications retains the values of inviscid theory also in the presence of vorticity. 2.2 Point forces on spheres in an unbounded fluid 2.2.1 Governing equation The effect of spatial acceleration and vorticity of a carrier liquid on the motion of a freely moving sphere at bubble Reynolds numbers, Re, in the range 10 £ Re £ 300 was studied both analytically and numerically by quite some authors. Here Re = |vrel| qL db/lL, where db is the sphere diameter, and vrel = vb – vL the velocity of the sphere relative to the liquid. Acceleration and gravity are intimately connected. If the fluid is accelerated, the sphere experiences an additional buoyancy-type of force [7], which shall be named ‘‘apparent buoyancy’’. A suitable general form for the bubble equation of motion at moderate Re is qb Vb

dvb DvL DvL dvb ¼ Vb ðqb qL Þg þ qL Vb þ qL Vb cAM dt Dt Dt dt 1 cD pdb2 qL jvrel jvrel þ Flift þ Frest 8 ð3Þ

Here Vb is the volume of the bubble, qb the mass density of the bubble, cD the drag coefficient, Flift the lift force that by definition comprises all force components that are acting in a direction perpendicular to vrel and Frest the sum of remaining forces that may comprise a history term. Near a wall, Flift will in Sect. 2.3 be seen to comprise a part of the drag force. The time rate of change of the bubble velocity is dvb /d t . and the other derivative in Eq. (3) is defined by D @ ¼ þ vL r Dt @t

ð4Þ

1

In the case of instantaneous motion with velocity U in the direction e1 ; / ¼ Ux1Rþ 12Uðdb =2Þ3 x1 r 3 and Eq. (2) yields the familiar result 2Ulð2=db Þ ðr/Þ2 dS ¼ 6plUdb :

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That the fluid acceleration in the added mass force, the term with the added mass coefficient, cAM, in Eq. (3), has


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to be evaluated in this way at the center of the sphere was shown by Auton et al. [2]. It is not obvious. Section 2.3 yields an added mass term with dU/dt, which can be interpreted as the time rate of change of the relative velocity, defined as d @ def d vrel ¼ vb vL vb rvL dt dt @t The difference of this added mass term with that of Eq. (3) is revealed by rewriting this Eq. (3) into

dvrel Vb ðqb þ cAM qL Þ þ ðvrel rÞvL þ bvrel dt DvL g þ Flift þ Frest ¼ Vb ðqL qb Þ Dt

ð5Þ

where the coefficient b accounts for drag: def b ¼ cD pdb2 18jvrel jqL : Equation (5) shows the equivalent L roles of fluid acceleration Dv Dt and the acceleration by gravity, g, although the fluid acceleration also occurs in the added mass force. Inertia terms in Eq. (3) are the body inertia term on the LHS and the added mass force, while the L term qL Vb Dv Dt in Eq. (3) is named the apparent buoyancy term. Vorticity effects occur in the drag, in the lift force and in Frest. Temporal acceleration or unsteadiness may increase viscous drag. This effect can be accounted for by history terms in Frest. Kim et al. [31] and Bagchi and Balachandar [5] show that the history force is not significant for freely translating particles. Things might be different if a small bubble is growing slowly, as will now be seen. Let the Jacob number, Ja, be defined by def

Ja ¼ qL cp jT1 Tv j=ðqv hvL Þ where T¥ is the temperature of the approaching liquid, Tv that of the bubble, hvL the specific enthalpy of evaporation and cp the specific heat of the liquid. If Ja is of order one, it _ might happen that RðtÞ=jv rel j and that Re ¼ Oð1ÞðRe\10Þ: In that case the history force on a freely growing spherical bubble might not be negligible [36]. Legendre et al. [36] also showed that recondensing bubbles have a thick thermal boundary layer and may also experience a history force that can not be neglected. In the present paper only fast growth of bubbles is considered.

eration that would be measured in the absence of the bubble. The acceleration a fluid particle experiences due to its motion relative to the bubble is of the order v2rel/db, while of the bubble the acceleration would be inDv absence L ¼ jvL rvL j in steady flows. The ratio is named the Dt acceleration number, Ac, that can be defined in the following, general way: def

Ac ¼

vrel ðvL rvL Þdb jvrel j3

ð6Þ

where the derivatives are evaluated at the center of the sphere. It is easily seen that the ratio of the apparent L buoyancy term of Eq. (3), qL Vb Dv Dt ; projected on def erel ¼ vrel =jvrel j; to the drag force, projected on –erel and divided by cD, equals 43Ac if @v@tL vL rvL : For the case of axisymmetric stationary straining flow around a stationary sphere, the definition (6) reduces to Ac ¼ adb =vrel

ð7Þ

if vL ¼ ðvrel þ azÞez 12 ar er in a cylindrical coordinate system centered at the sphere center. Such a flow was numerically studied by Magnaudet et al. [39], who defined Ac by Eq. (7). For 0.1 £ Re £ 300 they found for flow around a rigid sphere cD ðRe; Ac Þ ¼ ð1 þ 0:25 Ac ÞcD ðRe; 0Þ þ 0:55 Ac

ð8Þ

and no dependence of cD on Ac for an inviscid bubble. Because of the presence of surfactants tapwater bubbles often behave as solid spheres [16]. Experimental results of the effect of Re and Ac on the drag force coefficient of tapwater bubbles show an increase of cD with Ac that is somewhat bigger than the one given by Eq. (8) [23]. Kurose and Komori [33] numerically investigated the effect of linear shear on the drag of a spherical bubble with a fully mobile interface, using DNS. They found an increase in cD with increasing shear, especially for 50 £ Re £ 200. Legendre and Magnaudet [35] performed a numerical analysis of the effect of linear shear on cD as well. The dimensionless vorticity Sr was defined by Sr ¼ xdb =vrel

ð9Þ

if vL = (vrel + x y¢) ez¢ in a Cartesian frame of reference with its origin located at the center of the sphere. They correlated their results for the drag coefficient of a bubble with a fully mobile interface as

2.2.2 Coupling of acceleration and turbulence with drag cD ðRe; SrÞ ¼ cD ðRe; 0Þ ð1 þ 0:55Sr 2 Þ Let it now be assumed that the undisturbed flow is steady and that the effects of turbulence are negligible. The L acceleration Dv Dt is then the spatial acceleration vL rvL : DvL Note that Dt by definition is the unperturbed fluid accel-

ð10Þ

Equation (10) holds for Re £ 500 and Sr £ 1, and the increase of cD was found to stem from the contribution of the pressure.

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Bagchi and Balachandar [3] analyzed planar straining flows of a more general kind past a sphere and found similar complex dependencies of cD on strain. Spatial acceleration can be significant. When the relL ative velocity is negligible, Eq. (3) gives dvdtb 3Dv Dt : When a bubble experiences a constant acceleration given by Ac ~ 0.1 drag plays a role. If this bubble is moving along the center axis of a vertical venturi, Eqs. (3), (4), (6) yield

1 dvb qb þ qL Vb ¼ Vb ðqb qL Þg 2 dt 1 þ ð2Ac cD ðRe; Ac ÞÞpdb2 qL v2rel : 8

For clean (boiling) bubbles with cD(Re,0) ~ 0.5, an equivalent relative decrease of the drag coefficient due to acceleration effects, 2Ac/cD(Re,0), of 40% is found. For tapwater bubbles the effect of Ac on drag has to be accounted for, and Eq. (8) yields a lower relative decrease (28%). Bagchi and Balachandar [5] used DNS to show that for particles with diameters up to 10 times the Kolmogorov scale the freestream turbulence has a negligible effect on the mean drag. The standard drag correlation based on the mean relative velocity would yield a sufficiently accurate prediction. Burton and Eaton [10], on the contrary, claim that deterministic modifications as the one derived by Bagchi and Balchandar [6] would only be useful for relatively simples classes of flows. They used fully resolved DNS simulations to find that the drag term on a particle is dominant, and suggest that any useful correction to the drag term would need to be stochastic in nature. Probably the last word about the effect of turbulence on cD has not yet been told. 2.2.3 Lift due to vorticity in the approaching flow The lift force Flift by definition is the force component in the direction normal to the relative velocity. In the literature, often a lift coefficient cL,v, is defined analogously to the definition of the drag force coefficient: cL;v ¼ Flift

1 Afr qL v2rel 2

geometrical center of the bubble. The relation between the two definitions of lift coefficient is given by 4 cL;v ¼ Sr cL;i 3 Bagchi and Balachandar [4] performed DNS computations on a fixed, rigid sphere and found a linear dependence of cL,v on Sr, as in the above equation. For the rigid sphere, cL,i was found to be dependent on Reynolds number, but for a clean bubble with a stress-free condition at its surface and at larger Reynolds numbers the values given by inviscid theory are applicable [35, 38]. The value of cL,i is affected by flow unsteadiness [35] and is therefore for a growing bubble expected to be time-dependent. There is a distinct connection between cAM and cL,i. The lift coefficient for bodies axisymmetric around the relative velocity is equal to the added mass coefficient [37]. Apart from the above vorticity contribution, the lift force may also comprise components due to interaction with a uniform approaching flow if the body is at or near a wall (Sect. 3), due to a drag force component in the presence of a wall (Sect. 2.3), and due to turbulent fluctuations [15]. 2.3 Drag force on a sphere in the vicinity of solid boundaries The drag force not only depends on Re, Ac and the relative velocity vrel = vb – vL but also on the distance to a wall. The latter effect causes the drag force to be directed not exactly opposite to vrel, since the drag force component perpendicular to the wall exceeds the drag force component parallel to the wall, as will be shown in the following. For ‘‘clean’’ bubbles, i.e. in the absence of surface active agents, moving at Reynolds numbers around 100 in water, the relative flow can be modelled as potential flow. The following expressions in terms of the drag force coefficients far away from a single plane wall, cD,¥, were analytically derived [23]: ( cD;== ¼

( 2

)2 1 R 3 1 cD;1 16 h

)2 1 R 3 1 cD;1 8 h

ð12aÞ

with Afr the frontal area, p R in case of a sphere. Here, the expression derived from inviscid theory:

cD;? ¼

Flift ¼ cL;i qL Vb vrel x

Here cD;? is the drag force coefficient for the component of the relative velocity perpendicular to the wall, and cD,// the one for the component parallel to the wall. Although Eq. (12b) is an approximation, it is for distances of the center point of the bubble to the wall, h, exceeding twice

ð11Þ

is preferred because inviscid theory yields a constant 0.5 for cL,i in an unbounded simple shear flow, see [2]. Here x is the vorticity of the undisturbed ambient flow at the

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ð12bÞ


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Fig. 2 Decomposition of vrel; schematic of coordinate axis and numbering of walls

the bubble radius, R, sufficiently accurate for our purposes. A much stronger influence is, for creeping flow in the vicinity of a single plane wall, predicted by the correlation of Kim and Karrila [29]. For creeping flow in channels, other resistance functions exist, see for example Higdon and Muldoweney [27]. Let f//i denote the ratio (cD,///cD,¥) as computed for the distance to the wall i, i = 1,2,3. The numbering of walls is depicted in Fig. 2. Suppose that the bubble center is at point P, and that the relative velocity vrel is as in Fig. 2 with components only in the yz-plane. The normal to the wall is nw and tw is the tangent in the yz-plane normal to nw, see Fig. 2. By definition, def

ð13aÞ

def

ð13bÞ

vrel;nw ¼ ðvrel nw Þnw vrel;tw ¼ vrel vrel;nw

and f//3 is applied to vrel,tw and f3? ; being the ratio ðcD;? =cD;1 Þ corresponding to wall 3 at distance h3, is applied to vrel,nw. The drag force is now given by

1 == Fdrag ¼ qL pR2 cD;1 jvrel j f3 vrel;tw þ f3? vrel;nw 2

ð14Þ

==

and (–Fdrag) is not parallel to vrel unless f3 ¼ f3? or vrel,nw = 0. A stationary particle in a uniform flow parallel to a plane wall satisfies vrel,nw = 0, but its wake is deflected while the axisymmetry of the vorticity distribution is broken, leading to a vorticity-induced force that repels it away from the boundary [30]. This force is also a lift force since it acts normal to the relative velocity, see Sect. 2.2.1. For a particle in the corner of a channel with a rectangular cross-section, the next step would be to compensate for the presence of other walls (1 and 2 in Fig. 2. In first order approximation this could lead to a vector in Eq. 14) of the form [23]:

== == == == == f1 f2 f3 vrel;tw þ f1 f2 f3? vrel;nw :

Expressions like this one merely represent attempts to correct for the presence of more than one wall if a bubble is moving not too close to these walls. The main finding is

that in the vicinity of a wall the drag force might induce a force component perpendicular to vrel, i.e. might have a contribution to the lift force. The drag force coefficient in unbounded liquid, cD, ¥, can be quite small for spherical bubbles in pure, clean liquids with a low viscosity such as water [18]. Bubbles created by boiling in a low-viscous component at a wall are surely clean, while the time of gathering vorticity of the approaching flow downstream of the bubble is relatively short. If a wake is being formed behind the growing bubble, it will be small at the time of detachment and have little contribution to the drag of this bubble. In the following it will be shown that the drag force is negligible for boiling bubbles that are growing fast.

3 Lagrangian description As discussed in Sect. 1, and partly proven in Sect. 2.2, inviscid potential flow theory is useful for the modelling of forces on a boiling bubble at a wall even if vorticity is present in the approaching flow. This section presents a convenient framework for application of this theory, as well as applications to truncated spheres at a wall and to bubbles with more complex deformation near a wall, to further prove the usefulness of the approach. 3.1 Applicability, equivalence and advantages It is well known that functions are equivalently expressed by different types of series expansions. Similarly, different mathematical principles furnish equivalent expressions of the same fundamental laws of dynamics. For example, the dynamical equations of the theory of relativity, Einstein’s gravitational equations, Maxwell’s electromagnetic equations, and Schro¨dinger’s wave equation can also be expressed by extremum principles ([17], p. 264). Here we like to employ the dynamical equations that Euler and Lagrange established for bodies and for continuous fluids. These equations are usually associated with a variational principle, but are merely an alternative way of describing the dynamics of a mechanical system. They can be derived from the variational principle of Hamilton, of least action, but also from the principle of d’Alembert-Lagrange that states that the work of the constraining forces on any virtual variation is zero; these principles are equivalent ([1], p. 92). It is easily seen that Newton’s second law of motion for a particle with a constant mass M, Fr = M ar, is also obtained from Lagrange’s equation d @T @T r ¼ Fr dt @ x_ r @x

ð15Þ

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if T is the kinetic energy of the particle, 12Mv2 and if xr is a rectangular cartesian coordinate system. Lagrange’s equation retains the same form in a curvilinear coordinate system, although v2 is then given by x_m gmn x_ n (Einstein summation convention applied), with gmn the metric tensor, and the acceleration is then given by ar ¼ grs x€s þ Csmn x_m x_ n : with Gsmn the Christoffel symbols of the coordinate system ([46], p. 424). Lagrange’s equations keep the same form also when xr represents a system of so-called generalized coordinates [24]. Generalized coordinates do not need to be position coordinates, but may comprise all sorts of quantities. The amplitudes in a Fourier expansion or quantities with the dimension of energy can be used as generalized coordinates. The only requirement is that the generalized coordinates are independent degrees of freedom. Generalized coordinates are a way to remove dependencies between coordinates of particles that can be expressed in the form f(r1, r2, r3, ... ,t) = 0, so-called holonomic constraints ([24], p. 11). A general Lagrangian mechanical system is given by a configuration space and by the Lagrangian function. A Newtonian potential system is the particular case when the space is euclidean and when the Lagrangian function is the difference between the kinetic and potential energies. The system of n second-order differential equations of a Lagrangian system is equivalent to Hamilton’s system of 2n first-order equations. First integrals are readily obtained with Noether’s theorem ([1], p. 88). Two advantages of a Lagrangian description are immediately clear from Lagrange’s equations: the coordinates are independent and rather than dealing with vector quantities in the equations of motion only the scalar T has to be dealt with. It is noted that in this paper conservative forces, forces for which a potential exists, are treated in the same way as nonconservative forces. Another advantage of a Lagrangian analysis is the possibility to extend and apply it to a system with an infinite number of coordinates, as in field theories. The hydrodynamics of an ideal (incompressible and inviscid) fluid are an especially interesting example of a generalized Euler theory, in which the principle of least action implies that the motion of the fluid is described by the geodesics in the metric given by the kinetic energy ([1], p. 318). As a result, hydrodynamic theorems can be derived about the stability of flows from analogies in rigid body rotation. The case of N bodies, solids or bubbles, moving through an ideal fluid is conveniently described by a system of generalized coordinates


d @T @T r ¼ Qr r dt @ q_ @q

The coordinates of the fluid particles, an infinite number, are in a way ‘‘ignored† ([34], p. 201). As a consequence, the motion of the bodies can be predicted if the dependencies of the added mass coefficients on generalized coordinates are all known. To fully appreciate this, the example that Lamb [34] treated on p. 190, an example first given by Thomson and Tait, is presented in the next section. The added mass coefficients are the parameters A~rs ¼ A~sr in the following expression for the kinetic energy of the fluid, T: 1 1 1 1X ~ T =qL ¼ A~11 q_ 21 þ A~12 q_ 1 q_ 2 þ A~34 q_ 3 q_ 4 . . . ¼ Ars q_ r q_ s 2 2 2 2 r;s ð17Þ The A~rs only depend on the instantaneous configuration of the bodies, and are therefore functions of qi only. In the case of a single body, with mirror images to account for walls, the volume Vb of the body can be singled out: A~11 ¼ Vb A11 ; etc.. Although Lamb restricted his analysis to a finite number of coordinates, the number of generalized coordinates is here taken to be infinite but countable. 3.2 The example of Lamb: spherical bubble motion not close to a wall The following example, already discussed by Lamb [34], contains all main features of more complex applications of the Lagrangian approach, and in particular shows the importance of the added mass tensor. A sphere with mass m supposedly moves in ideal flow in a plane perpendicular to an infinite plane wall. Rectangular coordinates (x,y) in this plane are used y being the distance to the wall. The kinetic energy comprises the part of the sphere 1 1 Tb ¼ mx_2 þ my_ 2 2 2 and a part in the liquid, TL: TL

123

1 1 qL Vb ¼ A11 x_2 þ A22 y_ 2 : 2 2

A ‘‘mixed’’ term in x_ y_ can not occur because ‘‘the energy must remain unaltered when the sign of x_ is reversed’’. Not too close to the wall, if R/y 1, R being the radius, the added mass tensor is approximately given by

q 1 ; q2 ; q3 ; . . . and with generalized forces Qr, see ([34], p. 188) yielding Lagrange’s equations

ð16Þ

A11

! ! 1 3 R 3 1 3 R 3 1þ 1þ ’ A22 ’ 2 16 y 2 8 y

ð18Þ


839

Expressions valid in the proximity of the wall still depend on y, be it in a more complex manner [20]. If Bx, By are the components of an extraneous force, acting on the center of the sphere, the equations of motion follow from Eq. (16) and are Bx ¼

d ðA11 qL Vb þ mÞx_ dt

ð19aÞ

d 1 dA11 2 dA22 2 By ¼ ðA22 qL Vb þ mÞy_ qL Vb x_ þ y_ dt 2 dy dy ð19bÞ Without extraneous forces, a non-deforming bubble with mass density qb satisfies qb dA11 2 dA22 2 x_ y_ : 2 A11 þ y€ ¼ qL dy dy Moving only normal to the wall an acceleration away from the wall is experienced since x_ ¼ 0 and since A22 decreases with increasing distance to the wall. This situation is similar to the situation that occurs if a nonsliding boiling bubble is projected away from the wall directly after detachment. A little later, this bubble would be taken along with the vertical, upward flow and be moving in a plane parallel to the wall, making the y_ 2 -term negligible as compared to the x_ 2 -term. Without extraneous forces,the bubble then appears to be attracted by the wall. In the case of downward motion of the liquid, buoyancy reduces the velocity of the bubble with _ and the bubble would be less respect to the wall, jxj; attracted by the wall. In actual flow conditions, the lift force associated to vorticity plays a role, but all the main familiar features of bubble dynamics after detachment, both in up-flow and down-flow, are already represented in the above simple example of Lagrangian dynamics. The main feature, however, to be stressed here is the fact that the dependence of the added mass tensor on generalized coordinates is essential for the dynamics of bubble motion near a plane wall. This is equally true in the more complex applications that are treated below.

3.3 A spherically expanding bubble with a foot at a plane wall Bubbles created by boiling in forced convection along a heated, plane wall at high system pressure usually expand spherically. As discussed in Sect. 1, homogeneous flow is an important approaching flow component. This case is therefore modelled here.

The solving of any kind of problem in mechanical engineering in the Lagrangian way requires the determination of the generalized forces involved, including the body forces. In the case of the dynamics of bodies moving in a fluid and/or partly on a wall there is a straightforward and unambiguous way to do so. It is described below before applying the results in Lagrange’s equations. A convenient strategy to identify the generalized forces is the computation of the time rate R of change of the kinetic energy in the liquid, TL ¼ 12qL v2 dV; with the socalled mechanical energy balance [8, Sect. 3.3]. Consider a bubble footed at a plane wall and symmetrical about the axis perpendicular to the wall through the center of the bubble. Let AbL denote the area of the interface between liquid and bubble content, Abw the area between the wall and the bubble, rfoot the radius of the foot of the bubble, xCM the location of the center of mass of the bubble in a coordinate system with its origin in the center of the bubble foot and m_ int the local mass flux across the interface, positive if evaporation from the liquid into the bubble takes place. By application of the Leibniz theorem it is easy to show that Z d d 1 2 TL ¼ TL;m_ int ¼0 v m_ int dS ð20Þ dt dt 2 L AbL

where TL;m_ int ¼0 is the kinetic energy that would be in the fluid if the normal component of the interface would be equal to the normal component of the liquid velocity there. The integral in the above equation gives rise to a force that is related to evaporation and that is usually negligible as compared to hydrodynamic lift because of low values of m_ int : It is considered no further here in order to highlight hydrodynamic forces. The subscript m_ int ¼0 will be dropped from TL, henceforth. The volume integral over the contribution from the body force, gravity qL g ¼ qL rg x; can be converted into surface integrals yielding Z Z dTL ¼ f p qL g xgn vdS U_ þ n s vdS: dt where the normal n points into the fluid, U_ denotes the total energy dissipation rate and the last term on the RHS is the so-called traction term due to viscous stresses. The integration areas comprise all boundaries, including a free surface at infinity. The pressure at infinity is taken to be constant and equal to pw minus a constant hydrostatic pressure drop. Here pw denotes the hydrostatic pressure at the wall. Application of mass conservation in the incompressible fluid allows the integral at infinity to be written as a surface integral over AbL, (see [20], p. 104). This yields a surface integral of pw over AbL. The surface

123

840


with the aid of the surface divergence theorem2. The angle h is the instantaneous, dynamic contact angle and r is the surface tension coefficient between liquid and bubble. If the bubble foot is directly on the solid wall, the following extended Young-equation may be used to eliminate the contact angle: Fig. 3 Schematic of bubble, areas and dynamic contact angle

integral over the bubble–liquid interface of the contribution of gravity is equal to Z

dV qL v gdV¼ qL Vb g vCM qL g xCM b ; dt

Vb

which is zero if gravity is parallel to the wall and motion normal to the wall. The first term on the RHS corresponds to the well-known buoyancy force. The shape of the bubble is now assumed to be that of a truncated sphere footed on a plane wall, see Fig. 3. Two generalized parameters describe the interface: mean radius of curvature R and distance h of the center to the wall. The time rate of change of R is R_ and that of h is U. The pressure in the bubble, pb, is taken to be homogenous, constant in time and corresponding to the saturation temperature at the vapor–liquid interface. R If gravity is normal to the wall, the above integral qL v gdV yields Vb four terms as follows: Z Vb

d d V xCM;i ¼ gqL Vb xCM;z dt b dt @ @ Vb xCM;z þ R_ Vb xCM;z ¼ gqL U @h @R

qL v gdV¼ gi qL

@V @xCM;z ¼ gqL UxCM;z b gqL UVb ... @h @h ð21Þ _ Here xCM,z is the two in U and the following two in R: distance of the center of mass to the wall. Note that |xCM| is unequal to h. Two generalized forces corresponding to coordinate h are the above two terms divided by U. To find the other forces, the surface integrals of pL nv and of the traction term are converted with the so-called dynamic stress condition to a surface integral of the pressure inside the bubble, pb, see also [20]. This leads to a surface integral over j vn, where j denotes the local mean curvature. This surface integral is reduced to dAbL dAbw þ cos hr r dt dt

123

r cos h ¼ W ðrbw rwL Þ

106 rfoot =½m

ð22Þ

Here the last term accounts for the line tension, the equivalent of the surface tension for the contact line and W is the Wenzel surface roughness factor [44]. Coefficient rbw is the surface tension coefficient between bubble def content and wall. Let now Dr ¼ rbw rwL ; and let W ¼ 1 and line tension be negligible. Let further F3 be the hydrodynamic force component on the bubble normal to the wall, positive if pushing it away from the wall. Force F3 is minus the force on the liquid, Qh. The last step that is applied to identify the generalized forces is the substitution of all expressions found for dTdtL in dTL ¼ Qh U þ QR R_ dt

ð23Þ

and realizing that the two terms on the RHS are independent. If g is normal to the wall, this gives @V @AbL @Abw F3 ¼ pb b r Dr @h @h @h @V pw qL gxCM;z ðR; hÞ b @h @xCM;z þ QW;h : þ Vb gqL @h

ð24Þ

where the last term denotes the drag force. The rate of energy dissipation in the mechanical energy balance gives rise to drag force components that satisfy [20]

QW;j ¼

@ 12U_ @ q_ j

ð25Þ

if U_ is a second order polynomial in the generalized velocities, i.e. is twice a Rayleigh dissipation function, which usually requires the neglect of corrections for dissipation in boundary layers. Here, U_ is written as _ U_ ¼ W11 U 2 þ W22 R_ 2 þ 2W12 U R;

ð26Þ

2 It is noted that if a cylindrical coordinate system is employed, a contour has to be chosen that excludes the axis where r = 0 and the inverse mapping not defined. If a different type of contour would be used, an erroneous minus sign would appear. In the case of a truncated sphere this necessitates a limiting procedure in order to cover AbL entirely.


841

and each Wij only depends on the generalized coordinates, _ R and h. This yields, for example, QW;h ¼ W11 U þ W12 R: The drag functions Wij can be computed using inviscid potential flow theory. This yields an overestimation of the actual drag since a potential flow behaves like a flow of loose sand [19]: it may bend sharp corners without any problem. As a result, velocity gradients near a body may locally be extreme, whereas nature makes such gradients smooth by the action of viscosity. A thin layer of vorticity is created to either satisfy the no-slip condition (solid particle) or continuity of tangential stress (bubble). Smaller velocity gradients imply a smaller energy dissipation rate and hence a smaller drag force. @V Since @hb equals p r2foot it is easily seen that the sum of two terms containing this derivative in Eq. (24) corresponds to a force that was often denoted with ‘‘pressure correction force’’ or such [26]: 2 prfoot fpb

a ¼ 111:62137 844:131315k þ 2678:058461k2 4534:349913k3 þ 4311:889654k4 2180:345705k5 þ 457:591961k6 w ¼ 220:824854 þ 1639:114567k 5130:691427k2 þ 8625:857798k3 8169:91248k4 þ 4121:492877k5 863:784836k6 trðbÞ ¼ 104:601303 736:214699k þ 2293:784611k2

pw g ¼ Fpcorr :

3857:878559k3 þ 3659:955261k4

The sum of the surface tension terms equals 2 p rfoot r sin h, which is the usual expression for the component of the surface tension force that attracts the bubble to the wall, Fr. Force F3 is minus the force on the liquid, and follows from the LHS of Eq. (16) if the kinetic energy can be evaluated from a velocity potential. In the literature solutions for this potential were often found by matching series expansions, implying some arbitrariness in defining the type of expansion. There is a straightforward alternative way [20] that sets off with a suitable expansion in elementary functions. The boundary condition, in terms of the generalized velocities that are supposedly not all zero, is written as a matrix equation such that a Hilbert-Schmidt operator can be identified. An inverse exists because of the Fredholm alternative. The solution for the velocity potential is subsequently obtained by application of this inverse to a combination of generalized velocities, as prescribed by the velocity boundary condition at the interface of the bubble. A suitable expansion in elementary functions can for the case of a truncated sphere be obtained in various ways. The one based on Legendre polynomials of the first order have the drawback of satisfying velocity boundary conditions at places where it is not required, but naturally complement the solutions for a full sphere in the vicinity of the wall. They have been applied to obtain the added mass coefficients a, a2, tr(b) and w in the following expression for the kinetic energy: T

def

parameter k ¼ R=ð2hÞ: The dependencies of the added mass coefficients are shown in Fig. 4. For k < 0.5 (the full sphere case), explicit expressions for the added mass coefficients were given in an earlier paper [20]. For 0.5 < k < 1, the full expressions are similar but more complex. However, with sufficient accuracy polynomial fits represent the dependencies of the added mass coefficients on k in this case. These fits are the following:

1 1 1 _ 1 þ a2 V 2 ; qL Vo ¼ aU 2 þ trðbÞR_ 2 þ wRU 2 2 2 2

ð27Þ

where V is a uniform approach velocity parallel to the wall and Vo is 43pR3 : Note that Eq. (27) is analogous to Eq. (27). The added mass coefficients and the volume of the truncated sphere are only dependent on the geometrical

1849:854303k5 þ 388:412909k6 a2 ¼ 0:359528 þ 1:341274k 1:973813k2 þ 0:796613k3 Equations (27, 16) yield 1 qL Vo ¼ aU_ wR€ þ F~lift with 2 @a 1@a def 2 ~ _ þ a4pR =Vo U 2 Flift ¼ U R @R 2@h 1 _ 2 @w @trðbÞ 1@a2 þ w4pR2 =Vo þ V 2 R : 2 @R @h 2 @h

F3

ð29aÞ

ð29bÞ

Here F~lift of Eq. (29b) is a hydrodynamic lift force, divided by qL Vo ; that would be difficult to be captured in analytical form with an approach other than the Lagrangian approach. With the velocity potential also the drag force coefficients Wij are computed, but it suffices here to state that they depend on k and are usually less than 12p lL R. The derivatives occurring in Eq. (29b) can all be expressed as derivatives with respect to k since the added mass coefficients only depend on this parameter. Figure 4 shows that the derivatives that occur in Eq. (29b) change sign when the bubble shape changes from a truncated to a full sphere, i.e. when k ¼ 12: It can be shown that because of this change of sign in most practical cases also force F~lift changes sign. In practise, F~lift is promoting detachment of a bubble shaped as a truncated sphere, but it is driving the bubble back towards the wall once detached. Since two generalized parameters exits, h and R, there is a second equation of motion, associated with R. This is an extended Rayleigh-Plesset equation that can be circumvented if the bubble growth rate history is measured (as in

123

842

Heat Mass Transfer (2009) 45:831–846 We = 0.12 γ = 0.1 0.15

0.1

0.05

0 (h−h )/R 0

−0.05

t=0 −1

U/Rt=0 in s −0.1

Fig. 4 Added mass coefficients for spheres and truncated spheres. If the shape factor is less than 0.5, the center of a sphere with radius R has distance h to a plane wall

Sect. 3.5 below). The actual bubble dynamics before detachment is determined by the combination of Eqs. (24, 29a), and depends on the coupling of volume with pressure of the bubble content. For adiabatic processes, pVbk constant with k = cp/cv, and the relatively strong coupling results in high-frequency isotropic shape oscillations [22]. For nearly isothermal processes as in boiling, relatively low-frequency oscillations result from the interplay of surface energies [21]. The latter oscillations are only possible if the bubble foot is on the wall and if motion of the contact line is not hampered by artificial cavities or inhomogeneous substrate composition. 3.4 A bubble deforming near a plane wall More complex deformation than the one of Sect. 3.3 is readily modelled, taking full advantage of the possibilities that generalized coordinates offer. For sake of space only some typical results for a bubble in the vicinity of a plane wall are presented. Let the bubble be initially spherical, with radius Rt=0, while the shape at later times is supposedly star-shaped. The interface is described with a series of Legendre polynomials, Pn. The second term, b2 P1, of the expansion RðhÞ ¼ Rt¼0

1 X

bn ðtÞPn1 ðcos hÞ

ð30Þ

0

1

2

3

4

Dimensionless time, −db /dt t/R 3 t=0 t=0

Fig. 5 Histories of place and velocity of the center of a bubble deforming near a plane wall; mode 3 deformation

The kinetic energy of the fluid is, analogous to Eq. (17), written as the following function of the generalized velocities: 1 1 1 1 X 1X cm b_m Rt¼0 þ w b_i b_j R2t¼0 T=fqL Vb g ¼ aU 2 þ U 2 2 m¼1 2 i;j ij

ð31Þ Let k be 3, 4, 5, ..., and let excitation of mode k be defined as the initial condition, at t = 0, given by b1 ¼ Rt¼0 ; 0 ¼ b2 ¼ b3 ¼ b4 ¼ . . . ; U ¼ 0; b_j ¼ 0 for j „ k, and b_k 6¼ 0: Excitation of mode k in the absence of a wall leads to an oscillatory change of the bk-coefficient at a frequency known as the ‘‘natural frequency’’. In the presence of a plane wall this frequency is slightly reduced and whatever the direction of the initial velocity and whatever the value of k the bubble is eventually moved to the wall, see the example of Fig. 5. The initial distance is 1.1 Rt=0, the governing deformation parameter, def We ¼ Rt¼0 b_2k t¼0 =r; is 0.12, while the bubble content satisfies pVb0:1 constant. The added mass coefficients are not all equally important. Figure 6 shows typical histories of the dominant coefficients of this example. More details have been presented [22] and will be presented elsewhere. 3.5 Experimental validation

n¼1

is redundant since it is always possible to select the origin such that b2 is zero. This is done here, yielding a center that not necessarily coincides with the center of mass and has distance h to the wall. Distance h and all remaining coefficients bn serve as generalized coordinates.

123

Bubbles created by boiling at a wall in convective flow often have a so-called microlayer in-between the vapor dome and the wall [49]. Flow in this microlayer is lowReynolds-number flow governed by viscosity and is beyond the scope of the present paper. There are two ways to define a system boundary such that this area is circum-


843

We = 0.12 γ = 0.1 1

0.5 α 0 γ3 −0.5 γ

1

−1

−1.5

0

1

2

3

4

Dimensionless time, −db3/dtt=0 t/Rt=0

Fig. 6 Added mass coefficient histories of a bubble deforming near a plane wall; mode 3 deformation. Lines that are not labeled are for c4, c5, .... All these added mass coefficients have absolute values less than |c3|

vented while creating a system that is meaningfully related to the bubble. The first system comprises all the vapor bubble, the vapor-liquid interface, the vorticity attached to the bubble system and the microlayer between wall and vapor. The second system is the same except for the boundary at the wall, that is replaced by a flat surface that lies inside the bubble and intersects the interface in a way shown in Fig. 7. The problem with the first system is the need to evaluate the pressure at the wall; this would require an assessment of the flow inside the microlayer which is all but straightforward [40]. The second system does not comprise all of the vapor, but misses not much of it, while the pressure near the boundary close to the wall equals the pressure inside the bubble that is more easy to determine. This system is therefore selected here. Outside this system boundary, only the uniform approaching flow component is considered see Fig. 1. This flow, as well as the flow due to expansion of the bubble system, can be modeled as inviscid, while the shape in many cases can be approximated by that of a truncated sphere. The results of Sect. 3.3 regarding the hydrodynamic forces are therefore expected to be applicable, and if an experiment could be set up in

Fig. 7 Schematic of bubble system with a boundary not at the wall to exclude non-potential flow at the foot

which the approaching flow is nearly uniform its predictions could be validated. Experiments of the bubble foot are in boiling usually hampered by optical problems related to temperature gradients. It is therefore convenient to have a bubble system a little bit away from the wall, just outside the area where viscous flow occurs (Fig. 7). Angle h at its foot, see also Fig. 3, is a measurable quantity whereas the actual contact angle usually is not. Application of the Young equation, Eq. (22), is not allowed anymore, but the angles at the foot can be taken from measurements instead. If the bubble shape is close to being spherical, and the growth rate and shape histories are measured, each term in the force balance derived for h in Sect. 3.3, f.e. Eq. (29b), can be determined from measured parameters at each instant of time. Ideally, the approaching flow V would be nearly uniform to make the lift force due to vorticity negligible, and also turbulent fluctuations should have negligible infl\uence. The latter is likely to be the case, see also Sect. 2.2. The constraint regarding the velocity field, however, can only be satisfied in a dedicated experimental set-up. One way to arrange a nearly uniform approaching flow is by creating a bubble in the vicinity of a sharp-edged plate that is positioned in the center of a tube in which well-developed turbulent flow is present, see Fig. 8. An intrusive bubble generator of a proper shape possesses an artificial cavity that is locally heated to create a bubble in saturated flow in a pipe [32]. The cavity is located close to the upstream, sharp edge of the bubble generator. Typical measurement results, see Fig. 9, show that except for the initial and final stages of bubble growth shapes are close to that a truncated sphere. Sliding is not observed, probably due to the cavity used. This makes the force balance normal to the wall the most interesting one. This makes it possible to use for example Eq. (29b) to evaluate the hydrodynamic force components. Of course, the surface tension force needs to yield a net nonzero component parallel to the wall, and this surface tension force is therefore assessed using a 3D reconstruction of the actual bubble shape, see [(32]. In this article it is explained how Fr and Fp-corr, both plotted in Fig. 10, are evaluated. The way this is done is independent of the way the hydrodynamic forces are assessed. Because of experimental uncertainties

Fig. 8 Schematic of bubble generator in uniform approaching flow, V

123

844

Heat Mass Transfer (2009) 45:831–846 1.4 a b h c

1.2 1 [mm]

c 0.8 0.6 0.4 h

0.2 0

0

0.2

0.4 0.6 (t−t )/(t −t ) o

d

0.8

1

vorticity. This makes it important to assess the dependencies of the added mass coefficients on the governing geometrical parameters (generalized coordinates). This has been done for some categories of deforming bubbles at or near a plane, solid wall. The hydrodynamic lift forces related to the added mass of a growing truncated sphere in a uniform approaching flow have been quantified. The infinite, two-dimensional added mass tensor of an axisymmetric bubble near a wall has been investigated. The Lagrangian approach is particularly useful to determine added mass forces for a rapidly growing bubbles since

o

• Fig. 9 Typical histories of main dimensions of a bubble created in uniform approaching flow of 16 mm/s for a constant temperature boundary condition; td is time at detachment, to is time at bubble inception

• •

Fig. 10 Typical example of comparison of force components from measurement data of a bubble in nearly uniform approaching flow with a constant temperature boundary condition. Fr and Fp-corr are the dominant generalized forces and QW,h the drag force, see the text. Preliminary results

the force components not necessarily add up to zero. These and other experimental findings will be discussed more fully in another paper. The preliminary results of Fig. 10 show that the force balance of Sect. 3.3 is satisfied at times the shape is close to being a truncated sphere. This figure also shows that drag, overestimated in the way described in Sect. 3.3, is not contributing in the direction normal to the wall. Parallel to the wall, drag is compensated by surface tension and hydrodynamic forces and all these force components are probably small because of the slight deformation that is found in our measurements.

4 Closure There is quite some evidence that added mass forces can be decoupled from vorticity-related forces, and that the values computed in inviscid potential flow pertain to flows with

123

there are means to unambiguously determine the generalized forces involved, straightforward ways exist to solve for the velocity potential and vorticity generated at the vapor–liquid interface is weak and does not affect dynamics.

Lagrangian methods have been reviewed to expedite their use in boiling studies. In vortical flows, resulting predictions need to be combined with point force predictions, based on correlations, that have been summarized in Sect. 2. Both analysis and experiments suggest that the lift forces due to vorticity and added mass after detachment in upflow combine to keep a bubble close to the wall after detachment. The sum of added mass forces acting on truncated spheres footed at a plane wall usually has the opposite trend: it promotes detachment. This feature is for most force components apparent from the dependencies of the corresponding added mass coefficients on k = R/(2h), see Fig. 4. Because generated vorticity is negligible, the drag on a growing bubble can be overestimated with above-mentioned solutions of the velocity field. Results show that the drag force is negligible as compared to lift forces, the difference being one order of magnitude. It is noted that it is difficult to measure this drag force directly, for example because the growth rate needs to be large (2 m/s typically), the lifetime of the bubble must be short (order of magnitude 1 ms) and since the fluid must be clean, without surfactants. It is therefore not recommended to use the drag force coefficient as a fitting parameter to adjust predictions to experiments. The drag force on bubbles near a wall does not need to be antiparallel to the relative velocity. As discussed in Sect. 1, an important application of prediction methods of bubble dynamics of Sects. 3 and 2 is the prediction of the size of the bubble at detachment. One way to predict detachment is to model bubble growth evolution in time and to observe when the interface detaches from the wall [21]. It was often attempted in the literature to make an a priori assumption about the shape


at detachment, and derive a criterion from point forces acting on this shape. Alternatively, a family of possible shapes was predicted, and detachment defined as a condition when none of these shapes was possible [14]. A similar approach is also often used to predict drop detachment [13]. In the case of bubbles, usually a neck is formed shortly before detachment; this neck connects the vapor pocket with the wall. Detachment usually occurs so rapidly after the formation of a neck, that prediction of occurrence of a neck would be equivalent to prediction of detachment. In all these cases, comparison with experiments is necessary to validate the predictions and modelling assumptions, and to determine the range of application. Acknowledgments This research was funded by the EC project AD-700-2. I thank prof. J. Passos for providing a stimulating place to study, and Dr. J. Kuerten for useful comments to the manuscript.

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