## Abstract

In many dispersed multiphase flows bubbles, droplets, and particles move and disappear due to a phase change. Practical examples include vapour bubbles condensing in subcooled liquids, fuel droplets evaporating in a hot gas and ice crystals melting in water. After these ‘bodies’ have disappeared, they leave behind a remnant ‘ghost’ vortex as an expression of momentum conservation.

A general framework is developed to analyse why and how a ghost vortex is generated. A study of these processes is incomplete without a detailed discussion of the concept of momentum for unbounded flows. We show how momentum can be defined unambiguously for unbounded flows and show its connection with other expressions, particularly that of Lighthill. We apply our analysis to interpret new observations of condensing vapour bubbles and discuss droplet evaporation. We show that the use of integral invariants, widely applied in turbulence, introduces a new perspective to dispersed multiphase flows.

## 1. Introduction

Phase change occurs in many common multiphase flows, such as during nucleate boiling of subcooled liquids (e.g. heating water in a kettle), evaporating fuel sprays (e.g. inside the combustion chamber of an internal combustion engine) and melting ice particles in water. A common element in these problems is the movement and disappearance of discrete bodies. In the case of condensing vapour bubbles created during boiling, the bubbles and the surrounding fluid initially may carry momentum and buoyancy and added-mass forces are important later in driving the bubbles vertically upwards. When volatile fuel is sprayed into a gas, the droplets formed carry most of the momentum of the system. The consequence of these discrete bodies moving and disappearing in an unbounded flow is profound—a ghost vortex is left behind after these bodies have disappeared. In many examples, the vortex is sufficiently compact that it can be observed.

A whole range of diagnostic tools have been developed and applied to study turbulence and vortical flows; for instance integral constraints such as helicity, (angular/linear) impulse, circulation and kinetic energy (see Hunt *et al.* 2007) have been widely used to provide general insight into these flows. But in the field of dispersed multiphase flows, integral constraints have only been applied in a few instances to study the flow past bodies impulsively set into motion (Felderhof 2007) or interacting with vortices. Momentum is a fundamental quantity in fluid mechanics, but one reason that it has not been used for unbounded flows is that its heuristic definition, as the integral of density-weighted velocity over a flow domain, is ill-defined since the velocity decays so slowly in the far field that the integral depends on the shape of the domain at infinity (Theodorsen 1941). This deficiency is described in few textbooks on fluid mechanics (Batchelor (1967) calls this problem ‘frustrating’). This is perhaps why the fluid mechanics community has tended to use the concept of impulse, a concept which is only useful for fluids with a homogeneous density field (Saffman 1972).

Dispersed flows with phase change certainly represent one of the most challenging areas of multiphase flows because the chemical and thermal processes which accompany them are complex. One unexplored avenue is to consider the global momentum balance. The chemical and thermal processes associated with phase change are clearly important in setting the disappearance time scale of the bodies and the kinetic energy in the external flow, but in some cases these processes do not affect the momentum flux into the ambient fluid. In developing a general framework to study momentum globally in §2, we resolve the issue of how to define momentum unambiguously. We shall show in §3 that this expression is equivalent to Lighthill's (1986) formulation (for fluids of uniform density), but the effect of an inhomogeneous density field introduces additional new terms. We develop a general formulation and apply it to analyse vapour bubble collapse and droplet evaporation in §§4 and 5. Our conclusions are described in §6.

## 2. General formulation

### (a) Coupled motion

Consider a body1 of density *ρ*_{b}, volume *V*_{0} and mass *m*_{0}(=*ρ*_{b}*V*_{0}), which is introduced into an unbounded fluid at *t*=0 with velocity *v*_{0}, moves and disappears at *t*=*t*_{d}. For *t*>0, the mass of the body is *m* and moves with velocity * v*,(2.1)where

**is the centre of the body;**

*X***(0)=**

*X***0**; and

**(0)=**

*v*

*v*_{0}. The external fluid has density

*ρ*that tends to the uniform density

*ρ*

_{∞}far from the body. Phase change is assumed to convert the material within the body to a gas or liquid of density

*ρ*

_{S}at the surface of the body.

Although the formulation presented for droplets and bubbles is very similar, it is important to highlight the different simplifying assumptions which we will apply later on in our analysis. For condensing vapour bubbles, the change in the density of the ambient fluid due to heating caused by vapour condensation is small, so that . We do not place any constraints on the shape of the bubble, which indeed may change topology. When a fuel droplet evaporates, *ρ*_{S} may be significantly different from *ρ*_{∞} (by a factor of 2) very close to the droplet. As is common in most studies of droplet evaporation, we will assume that droplets remain spherical and the loss of mass is symmetric. The symmetric loss of mass means that the evaporative flux does not explicitly contribute to the momentum flux into the gas.

The momentum and mass conservation equations for the flow ** u** outside the body, whose density is

*ρ*(

**,**

*x**t*), are(2.2)where

*p*is the pressure;

**is the stress tensor; and**

*τ**κ*is the diffusivity. The integral form of the conservation laws is studied in a large domain

*V*

_{∞}bounded by

*S*

_{∞}.

*V*

_{∞}is taken to be much larger than the displacement of the droplet so the flow at

*S*

_{∞}is irrotational. Note that

*S*

_{∞}is fixed but the surface of the body

*S*

_{b}is moving.

The surface of the body *S*_{b} moves outwards, relative to the centre of the body, at a rate *v*_{S}(<0). The kinematic condition applied to the fluid flow at the body surface is really an expression of mass conservation, i.e.(2.3)where is the unit normal out of *V*_{∞}. The subscripts ‘i’ and ‘o’ refer to the evaluation of ** u** just inside and outside the surface of the body, respectively. The mass of the body decreases due to the flux through its surface. Since the flow is incompressible, the volume flux through any surface encapsulating the body is(2.4)and the far-field flow is a moving source,(2.5)The strength of the source (2.4) is increased by the body losing mass and is reduced by shrinking. Evaporating droplets generate a source flow (since ), while condensing vapour bubbles generate a sink flow (since ).

The rate of change of the momentum of the body is(2.6)because there is a momentum flux through its surface. The hydrodynamic force on the droplet ** F** is (e.g. Crowe 1976, eqns (14) and (17))(2.7)The force on the body (minus the buoyancy term) can be calculated by examining the momentum flux in the far field. Jeffrey (1965) points out that there is an additional flux in the near field, because the surface of the body moves. The flow is irrotational on

*S*

_{∞}() where viscous stresses are negligible but the contribution from the pressure field, , is important. Using the momentum integral approach, where the force is expressed in terms of surface fluxes over

*V*

_{∞}and

*S*

_{∞}, the force on the body is(2.8)where(2.9)where

**is the momentum of the fluid flow. The first term in (2.9) is the integral of the density-weighted velocity over the flow domain. The second term is a far-field closure required for**

*M***to be defined unambiguously.**

*M*For condensing vapour bubbles, the velocity flux through the bubble surface is negligible and the second term on the right-hand side of (2.8) is negligible. For an evaporating droplet, (2.8) is similar to the rocket equations—for a symmetric loss of mass, the last term is also zero. The coupled set of equations describing the momentum of the flow and the particle equation of motion, and including buoyancy forces, is(2.10)where(2.11)where is directed vertically upwards. Golovin (1973) derived (2.10) but omitted the far-field closure in (2.9).

### (b) Momentum of the fluid

Using the identity (11) from Saffman (1992, p. 50), (2.9) can be written as(2.12)where . It is instructive to examine in more detail the momentum of the ambient fluid defined by (2.12) and show its equivalence to other representations of momentum.

For a homogeneous fluid (where *ρ* is constant), the second term in (2.12) is zero. Any such flow can be expressed in terms of the sum of potential and vortical components,(2.13)The velocity potentials *ϕ*_{b} and *ϕ*_{i} correspond to the flow generated by the kinematic boundary condition on the body and the image vorticity. For an inviscid flow past a rigid body, the impulse of the body is (where *C*_{m} is the added-mass tensor) and the impulse of the free vorticity is(2.14)The total momentum of the flow is the sum of the impulse of the free, image and bound vorticity(2.15)Equation (2.15) has been derived by Howe (1995) and Shashikanth *et al.* (2002). Eames *et al*. (2007) used (2.15) and the conservation law (2.10) (see (2.17) for *F*_{B}=**0**) to study the coupled motion of vortices and rigid bodies in an inviscid fluid. When the irrotational flow does not contain vorticity or circulation, *I*_{v}=**0** and ** M**=

*I*_{b}(Saffman 1992).

When the density is piecewise constant or varies with time everywhere at the same rate (∇*ρ*=**0** or *ρ*=*ρ*(*t*)), the momentum is(2.16)which is identical to Lighthill's (1986) expression.

The momentum is equal to the sum of the impulse of the flow (defined as the integral of density-weighted moment of vorticity) and a contribution from the moving boundaries. This shows that impulse is still a useful concept for this class of flows. Eames & Hunt (2004) showed that the concept of impulse is still useful for irrotational flows where the density field is homogeneous but changes with time due to compression or expansion. When the density gradient is not zero, the second term in (2.12) may be important and the difference between the momentum and impulse of the flow may be large (as pointed out by Saffman 1972).

### (c) Conservation principles

From (2.10),(2.17)This conservation law applies to inertial flows (with finite or infinite Reynolds numbers) but not to Stokes flows. Immediately after the body has disappeared, the momentum of the ambient fluid is increased to(2.18)When the body forces acting on the flow are absent, *M*_{d} is conserved. In the examples that we will discuss relating to boiling and droplet evaporation, the body forces acting on the flow are weak and *M*_{d} changes slowly after the bubble/droplet have disappeared.

The flow generated after the body has disappeared can be evaluated from the vorticity field using the Biot–Savart rule,(2.19)Expanding (2.19), the far-field flow tends to(2.20)The first term is zero because the vorticity field is zero on *S*_{∞}, so that the far-field flow is dipolar (Saffman 1992, p. 49),(2.21)with a dipole moment(2.22)For condensing vapour bubbles, the relative change in the density of the ambient fluid is negligible, i.e. , so that(2.23)After evaporating droplets have disappeared, diffusion ensures that over much of the flow ∇*ρ* is small and the above relation is approximately satisfied. But, in general, for flows where *ρ* is not constant, the dipole moment is not a conserved quantity.

## 3. Vortex generation

The general analysis described in §2 provides a framework to understand how momentum is conserved and how to estimate the impulse/momentum of the vortex created. We describe a number of mechanisms that are important in transporting vorticity from the surface of the body into the ambient fluid and develop dimensionless groupings which determine their strength. We generalize our analysis by considering a body having a density *ρ*_{b}, moving with a characteristic speed *v*_{0} and having an initial radius *a*_{0}, with a collapse time *t*_{d} and acceleration d*v*/d*t*.

### (a) Inviscid mechanism: topological change

A collapsing vapour bubble can deform sufficiently for it to change from an ellipsoidal (genus 0) to a toroidal shape (a genus of 1; figure 1*a*). Circulation is created at the point where reconnection occurs because the integral of the velocity potential around the internal surface becomes non-zero (Lamb 1932). This does not contravene Kelvin's circulation theorem, which is associated with material surfaces. Benjamin & Ellis (1966) proposed and demonstrated that cavitating bubbles conserve impulse through a change in topology. This mechanism occurs widely in fluid mechanics and is responsible for the generation of a subsurface vortex by breaking waves (Melville *et al.* 2002). A suitable measure of this mechanism for dispersed flows is a Weber number defined as the ratio of pressure difference across the bubble height due to acceleration (which causes the bubble to deform) to the pressure change across the bubble surface due to surface tension (which stabilizes the bubble shape), expressed as(3.1)where *σ* is the surface tension. Acceleration leads to the bubble rear moving faster than its front.

### (b) Inviscid mechanism: surface flux

The flow past a moving body may be interpreted as bound vorticity (moving with the flow), with the external flow described as free vorticity, with image vorticity required to satisfy kinematic boundary conditions (Saffman 1992). When a body ‘dissolves’, as in the case of Taylor's (1953) ‘dissolving disc’, bound vorticity is converted to free vorticity (figure 1*b*). This method is sometimes employed to generate dipolar vortices where an open-ended cylinder is translated forward in a fluid and then removed (Eames & Flor 1998). While a portion of the final vortex created is due to the vorticity in the wake, for short translation distances, the primary contribution is from the conversion of bound vorticity to free vorticity.

In the context of condensing bubbles and evaporating droplets, a flow through the body surface is generated by a phase change—in the combustion community, this through-surface flow is referred to as the blow speed. Denoting *ρ*_{S} as the density of the body following a phase change, the radial blow speed scales as , which convects bound vorticity into the ambient fluid. This mechanism is determined by the relative speed by which vorticity is created by the surface flux through the surface of the bubble or particle, estimated as(3.2)

### (c) Viscous mechanism

Vorticity can be generated on the surface of the body either through a shear-free condition or a non-slip condition. Vorticity can be communicated to the exterior flow by diffusion through the boundary layer. For high Reynolds numbers , the wake is turbulent and vorticity is deposited in the ambient fluid behind the body (see figure 1*c*). From a detailed analysis of this effect, a measure of this mechanism is the ratio of the disappearance time to the response time of the body (with a view of bringing together the results of bubbles and droplets, the added-mass contribution must be included),(3.3)For low Reynolds number flows , a measure of the strength of this mechanism is the ratio of collapse to viscous time scale,(3.4)Having set up a general framework to describe momentum conservation and for generating the vortices, we go on to apply these different measures (, and ) to understand how vorticity is created by condensing bubbles in nucleate boiling and evaporating droplets.

## 4. Nucleate boiling

The literature on subcooled nucleate boiling is vast, but few studies have dealt with the permanent flow generated by bubbles collapsing, the morphological changes which may accompany their collapse or, indeed, the vortices which are generated.

We undertook a series of laboratory tests using partially degassed water as the working fluid to examine these processes. The water was preheated in a beaker to a prescribed temperature. To visualize the wake and vortex generated by the bubble collapse, the boiling element was submerged within a layer of 2 mm of sediment whose diameter was 30–80 μm. The sediment was decanted many times to remove the fines. Image sequences were recorded using a high-speed camera (at 4000 frames s^{−1}) and interrogated digitally. The bubbles were characterized by their response time, determined by the initial radius *a*_{0} based on the bubble cross-sectional area (before detachment) and disappearance time *t*_{d} (after detachment). The heating element consisted of a metal rod encased in a ceramic plug. The bubbles were characterized in terms of their response time *t*_{R}=*v*_{T0}/*g*, where *v*_{T0} is a rise velocity based on the bubble radius *a*_{0}. Since vapour bubbles are so much less dense than the ambient fluid that , and the inviscid surface flux mechanism is clearly not important. Parameter (3.1) is estimated to be , where *σ*=75 dynes.

Figure 2*a* shows a time sequence following the emergence of a vapour bubble from the sediment layer for subcooling Δ*T*=30°. The bubble is initially oblate (with *a*_{0}=0.19 cm) and deforms during its collapse but does not change topology. The bubble collapses in a time *t*_{d}=14 ms after it has detached from the sediment layer. Following the collapse of the bubble, a vortex is observed which initially grows to a constant size and moves vertically at a constant speed. Dense sediment is essentially thrown out of the vortex core, further highlighting the presence of the vortex core. An air bubble is formed after the vapour bubble has condensed because the water was not completely degassed. At a later time, the sediment becomes detrained from the vortex as it settles. In this example, =1.1 and so while the bubble was expected to greatly deform, a topological change was not anticipated. Under these circumstances, the vortex is generated by the viscous mechanism.

Figure 2*b* shows a time sequence of a vapour bubble collapsing in a much shorter time, *t*_{d}=3 ms, where *a*_{0}=0.1 cm and =2.9. The bubble rear accelerates through its front generating an intense jet. The bubble becomes toroidal and finally, owing to the presence of dissolved air, two air bubbles are produced. In this example, is large enough so that the bubble changes from a topology genus 0 to a genus of 1, and the inviscid mechanism is important as confirmed by bubble jetting. When the vortex is created through a topological change, it is initially much smaller than the bubble and moves much faster (at 27 cm s^{−1}) than for the case when this does not occur.

Figure 3*a* shows a phase diagram which discriminates between when a collapsing bubble is observed to change topology and when it is observed not to change. For clarity, points corresponding to observations where we could not clearly determine whether a topological change did or did not occur are omitted. The parameters and (calculated from (3.3)) are plotted on the vertical and horizontal axes—the velocity scale chosen was the terminal rise speed based on the initial bubble radius on detachment. We observe that for >2, the collapsing bubbles undergo a topology change, while for <2, no topological change was observed. In the absence of a topological change, the vortex is generated by the viscous mechanism. When a topological change does occur, while most of the vorticity is generated by this mechanism, it is not possible to estimate the fraction.

We can apply the analysis of §2 to understand the impulse of the vortex generated. A massless vapour bubble condensing in a subcooled liquid generates a vortex of momentum(4.1)where is a constant which depends on the rate of bubble collapse and is estimated to be approximately 0.25–0.5. After the vortex has been created, the heat from the condensation process creates a buoyant packet of fluid or vortex. The maximum density difference between the hot fluid in the vortex and the cooler ambient fluid is Δ*ρ*∼*γρ*_{∞}, where *γ*∼0.02 (for a temperature contrast of 30°C) and is small. Buoyancy forces tend to increase the momentum of the vortex over a time scale , so over the period of our measurements, the vortex momentum is approximately constant. The vortex momentum can be estimated from its speed of propagation *U*_{ω} and volume *V*_{ω}, from (where for an approximately spherical vortex, *C*_{m}=1/2)(4.2)For a vapour bubble rising from rest and condensing, *M*_{0}=0 and . Figure 3*b* shows the variation in the ratio with *t*_{d}/*t*_{R}. The largest errors are associated with estimates of *t*_{d} and *a*_{0}, and probably leads to an error of 40% in *ρ*_{∞}*V*_{0}*gt*_{d}; *M*_{v} is estimated more accurately with an error less than 10%. For *t*_{d}/*t*_{R}>0.3, the results are consistent with the physical model based on the momentum being generated after the detachment process. For fast collapse times (*t*_{d}/*t*_{R}<0.3), the initial momentum of the flow prior to the bubble detaching (*M*_{0}) is clearly important, leading to a significant underestimation of vortex impulse, as shown in figure 3*b*.

## 5. Evaporating droplets

The literature on evaporating droplets has dealt to a great extent with rates of evaporation, the force they experience and the near-field component of the flow. In this section, we examine droplets evaporating in flows where gravitational forces are not important. The initial momentum of the external gas caused by the injection of a droplet is at least three orders of magnitude smaller than the droplet momentum, , and can be justifiably ignored. The momentum of the vortex created by the droplet disappearing is(5.1)the initial droplet momentum. This result is not a surprise, but what is interesting and important is the impact of the evaporation on the external flow, as we shall describe.

We consider the general features of the flow generated by a droplet moving and evaporating. The near-field flow consists of a source flow of strength *Q*_{S}, which combines the evaporative flux and wake volume flux, i.e.*Q*_{S}=*Q*_{E}+*Q*_{W} (figure 4*a*). The total force on the steadily moving droplet evaporating slowly can be interpreted as a thrust *ρ*_{∞}*Q*_{S}*v* due to the source flow (Lamb 1932) and a drag of 2*ρ*_{f}*Q*_{W}*v* generated by the volume flux deficit (see Hunt & Eames 2002), so that(5.2)This result was derived by Cliffe & Lever (1985), though the physical interpretation is different. From (5.2), the downstream volume flux is(5.3)where the effect of the source flow generated by the loss of mass from the body introduces the new term on the r.h.s. of (5.3). This shows that Betz's (1925) relationship between drag force and wake volume flux deficit (*Q*_{W}=*F*_{D}/*ρ*_{∞}*v*) is not valid when there is a flow through the droplet surface. The effect of the evaporative source flow separates the positive and negative components of vorticity in the wake of a two-dimensional body, or stretches the vorticity shed from a three-dimensional body. Because the positive and negative components of vorticity become separated for high blow velocities, they do not intermingle causing the maximum wake vorticity to decrease much more slowly downstream. Although the drag on a rigid body increases with *Q*_{E}, it is predominately the loss of mass—the second term on the right-hand side of (5.3)—which is responsible for increasing *Q*_{W}.

The far-field flow consists of a sink of strength *Q*_{W0} located at ** x**=

**0**, created by the particle starting from rest (Mei & Lawrence 1996). The decrease in the volume flux along the particle path appears as a series of source terms in the far-field flow, whose strength is(5.4)and

*X*=|

**| is the distance from the point of injection (see Hunt & Eames 2002; figure 4**

*X**b*). When the droplet has disappeared, only the free vorticity component is present and(5.5)where(5.6)(since ). From (2.3) and (2.10), . This result is exactly the same as (2.23) obtained from the global conservation of momentum. In general, when the vapour density is less than the gas (

*ρ*

_{S}<

*ρ*

_{∞}), the impulse will tend to be less than the momentum of the flow, |

*I*_{v}|<|

*M*_{d}|. Over much of the wake, |∇

*ρ*×

**| is small, so that we anticipate the second term is small (the third term is zero). For a long time, diffusive effects will ensure that |∇**

*u**ρ*|→0, the limit of

*I*_{v}→

*M*_{d}will be approached.

## 6. Concluding remarks

The disappearance of moving bubbles and droplets, through a phase change, must leave behind a ‘ghost’ vortex to conserve momentum. Three principal mechanisms largely determine how vorticity bound to the surface of the body is communicated into the ambient fluid to generate the vortex—a topological change in shape in an inviscid fluid; inviscid surface flux through the surface of the body; and a viscous mechanism. For the case of nucleate boiling, we report new experimental observations of a ghost vortex signature being generated by the disappearance of a single bubble. The degree of subcooling was sufficiently high in our experiments so that bubbles may go through a topological change as they collapse. While the observations are consistent with our physical model, the contribution to the momentum of the vortex from the bubble growing and rising is not included in our estimates. For evaporating droplets, we have shown that Betz's (1925) relationship between drag force and wake volume flux no longer holds. The evaporative flux separates the wake inhibiting vorticity cancellation and causing the wake velocity deficit to decay (in the near field) more slowly than for a rigid body.

The examples in this paper show that integral invariants are a powerful tool, providing a new perspective to understanding the global behaviour of dispersed multiphase flows. Other invariants are perhaps worth considering, for instance angular momentum, particularly since swirl is deliberately introduced into the internal combustion engine (in the air and fuel) to assist mixing and aid combustion.

## Acknowledgments

This work was supported by EPSRC (EP/E029302/1), a Philip Leverhulme Prize (2005) and a Global Research Award (2007) from the Royal Academy of Engineering. This paper is based on a talk given to the Royal Academy of Engineering on 25 September 2007, to summarize I.E.’s activity during their period of support. Dr Mark Landeryou is gratefully acknowledged for his help in processing the experimental data.

## Footnotes

One contribution of 11 to a Theme Issue ‘New perspectives on dispersed multiphase flows’.

↵The term ‘body’ is introduced for generality, but should be interpreted as a bubble, droplet and particle for boiling, evaporation and melting problems. The external fluid should, respectively, be interpreted as a liquid, gas and liquid. We deal explicitly with the kinematic boundary conditions on the surface of the body, and the imposition of shear-free or non-slip conditions depends on the particular problem and is taken as implicit.

- © 2008 The Royal Society