Brennen Ch.E. Fundamentals of Multiphase Flow

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lutions are numerous. Experimentally, it is found that a recirculating zone

(or vortex ring) develops close to the rear stagnation point at about Re =30

(see Taneda 1956 and ﬁgure 2.4). With further increase in the Reynolds

number this recirculating zone or wake expands. Deﬁning locations on the

surface by the angle from the front stagnation point, the separation point

moves forward from about 130

◦

at Re = 100 to about 115

◦

at Re = 300. In

the process the wake reaches a diameter comparable to that of the sphere

when Re ≈ 130. At this point the ﬂow becomes unstable and the ring vor-

tex that makes up the wake begins to oscillate (Taneda 1956). However, it

continues to be attached to the sphere until about Re = 500 (Torobin and

Gauvin 1959).

At Reynolds numbers above about 500, vortices begin to be shed and

then convected downstream. The frequency of vortex shedding has not been

studied as extensively as in the case of a circular cylinder and seems to vary

more with Reynolds number. In terms of the conventional Strouhal number,

Str, deﬁned as

Str =2fR/W (2.22)

the vortex shedding frequencies, f, that Moller (1938) observed correspond

to a range of Str varying from 0.3atRe = 1000 to about 1.8atRe = 5000.

Furthermore, as Re increases above 500 the ﬂow develops a fairly steady

near-wake behind which vortex shedding forms an unsteady and increasingly

turbulent far-wake. This process continues until, at a value of Re of the order

of 1000, the ﬂow around the sphere and in the near-wake again becomes

quite steady. A recognizable boundary layer has developed on the front of

the sphere and separation settles down to a position about 84

◦

from the

front stagnation point. Transition to turbulence occurs on the free shear

layer (which deﬁnes the boundary of the near-wake) and moves progressively

forward as the Reynolds number increases. The ﬂow is similar to that of the

top picture in ﬁgure 2.2. Then the events described in the previous section

occur with further increase in the Reynolds number.

Since the Reynolds number range between 0.5 and several hundred can

often pertain in multiphase ﬂows, one must resort to an empirical formula

for the drag force in this regime. A number of empirical results are available;

for example, Klyachko (1934) recommends

= −6πρ





(2.23)

which ﬁts the data fairly well up to Re ≈ 1000. At Re = 1 the factor in the

Re =9.15 Re =37.7

Re =17.9 Re =73.6

Re =25.5 Re = 118

Re =26.8 Re = 133

Figure 2.4. Streamlines of steady ﬂow (from left to right) past a sphere at

various Reynolds numbers (from Taneda 1956, reproduced by permission

of the author).

square brackets is 1.167, whereas the same factor in equation 2.20 is 1.187.

On the other hand, at Re = 1000, the two factors are respectively 17.7and

188.5.

2.2.3 Molecular eﬀects

When the mean free path of the molecules in the surrounding ﬂuid, λ,be-

comes comparable with the size of the particles, the ﬂow will clearly deviate

from the continuum models, that are only relevant when λ  R. The Knud-

sen number, Kn = λ/2R, is used to characterize these circumstances, and

Cunningham (1910) showed that the ﬁrst-order correction for small but ﬁnite

Knudsen number leads to an additional factor, (1 + 2AKn), in the Stokes

drag for a spherical particle. The numerical factor, A, is roughly a constant

of order unity (see, for example, Green and Lane 1964).

When the impulse generated by the collision of a single ﬂuid molecule

with the particle is large enough to cause signiﬁcant change in the particle

velocity, the resulting random motions of the particle are called Brownian

motion (Einstein 1956). This leads to diﬀusion of solid particles suspended

in a ﬂuid. Einstein showed that the diﬀusivity, D, of this process is given by

D = kT/6πµ

R (2.24)

where k is Boltzmann’s constant. It follows that the typical rms displace-

ment of the particle in a time, t,isgivenby(kTt/3πµ

.Brownian

motion is usually only signiﬁcant for micron- and sub-micron-sized parti-

cles. The example quoted by Einstein is that of a 1 µm diameter particle

in water at 17

◦

C for which the typical displacement during one second is

0.8 µm.

A third, related phenomenon is the response of a particle to the collisions

of molecules when there is a signiﬁcant temperature gradient in the ﬂuid.

Then the impulses imparted to the particle by molecular collisions on the

hot side of the particle will be larger than the impulses on the cold side. The

particle will therefore experience a net force driving it in the direction of the

colder ﬂuid. This phenomenon is known as thermophoresis (see, for example,

Davies 1966). A similar phenomenon known as photophoresis occurs when

a particle is subjected to nonuniform radiation. One could include in this

list the Bjerknes forces described in the section 3.4 since they constitute

sonophoresis, namely forces acting on a particle in a sound ﬁeld.

2.3 UNSTEADY EFFECTS

2.3.1 Unsteady particle motions

Having reviewed the steady motion of a particle relative to a ﬂuid, we must

now consider the consequences of unsteady relative motion in which either

the particle or the ﬂuid or both are accelerating. The complexities of ﬂuid

acceleration are delayed until the next section. First we shall consider the

simpler circumstance in which the ﬂuid is either at rest or has a steady

uniform streaming motion (U = constant) far from the particle. Clearly the

second case is readily reduced to the ﬁrst by a simple Galilean transformation

and it will be assumed that this has been accomplished.

In the ideal case of unsteady inviscid potential ﬂow, it can then be shown

by using the concept of the total kinetic energy of the ﬂuid that the force

on a rigid particle in an incompressible ﬂow is given by F

,where

= −M

(2.25)

where M

is called the added mass matrix (or tensor) though the name

induced inertia tensor used by Batchelor (1967) is, perhaps, more descrip-

tive. The reader is referred to Sarpkaya and Isaacson (1981), Yih (1969), or

Batchelor (1967) for detailed descriptions of such analyses. The above men-

tioned methods also show that M

for any ﬁnite particle can be obtained

from knowledge of several steady potential ﬂows. In fact,



volume

of f luid

d(volume) (2.26)

where the integration is performed over the entire volume of the ﬂuid. The

velocity ﬁeld, u

, is the ﬂuid velocity in the i direction caused by the steady

translation of the particle with unit velocity in the j direction. Note that

this means that M

is necessarily a symmetric matrix. Furthermore, it is

clear that particles with planes of symmetry will not experience a force

perpendicular to that plane when the direction of acceleration is parallel to

that plane. Hence if there is a plane of symmetry perpendicular to the k

direction, then for i = k, M

= M

= 0, and the only oﬀ-diagonal matrix

elements that can be nonzero are M

, j = k, i = k. In the special case of

the sphere all the oﬀ-diagonal terms will be zero.

Tables of some available values of the diagonal components of M

are

given by Sarpkaya and Isaacson (1981) who also summarize the experi-

mental results, particularly for planar ﬂows past cylinders. Other compila-

tions of added mass results can be found in Kennard (1967), Patton (1965),

Table 2.1. Added masses (diagonal terms in M

) for some three-dimensional

bodies (particles): (T) Potential ﬂow calculations, (E) Experimental data

from Patton (1965).

and Brennen (1982). Some typical values for three-dimensional particles are

listed in Table 2.1. The uniform diagonal value for a sphere (often referred to

simply as the added mass of a sphere) is 2ρ

πR

/3 or one-half the displaced

mass of ﬂuid. This value can readily be obtained from equation 2.26 using

the steady ﬂow results given in equations 2.7 to 2.10. In general, of course,

there is no special relation between the added mass and the displaced mass.

Consider, for example, the case of the inﬁnitely thin plate or disc with zero

displaced mass which has a ﬁnite added mass in the direction normal to the

surface. Finally, it should be noted that the literature contains little, if any,

information on oﬀ-diagonal components of added mass matrices.

Now consider the application of these potential ﬂow results to real viscous

ﬂows at high Reynolds numbers (the case of low Reynolds number ﬂows will

be discussed in section 2.3.4). Signiﬁcant doubts about the applicability of

the added masses calculated from potential ﬂow analysis would be justiﬁed

because of the experience of D’Alembert’s paradox for steady potential ﬂows

and the substantial diﬀerence between the streamlines of the potential and

actual ﬂows. Furthermore, analyses of experimental results will require the

separation of the added mass forces from the viscous drag forces. Usually

this is accomplished by heuristic summation of the two forces so that

= −M

−

(2.27)

where C

is a lift and drag coeﬃcient matrix and A is a typical cross-

sectional area for the body. This is known as Morison’s equation (see Morison

et al. 1950).

Actual unsteady high Reynolds number ﬂows are more complicated and

not necessarily compatible with such simple superposition. This is reﬂected

in the fact that the coeﬃcients, M

and C

, appear from the experimental

results to be not only functions of Re but also functions of the reduced

time or frequency of the unsteady motion. Typically experiments involve

either oscillation of a body in a ﬂuid or acceleration from rest. The most

extensively studied case involves planar ﬂow past a cylinder (for example,

Keulegan and Carpenter 1958), and a detailed review of this data is included

in Sarpkaya and Isaacson (1981). For oscillatory motion of the cylinder with

velocity amplitude, U

,andperiod,t

∗

, the coeﬃcients are functions of both

the Reynolds number, Re =2U

R/ν

, and the reduced period or Keulegan-

Carpenter number, Kc = U

∗

/2R. When the amplitude, U

∗

,islessthan

about 10R (Kc < 5), the inertial eﬀects dominate and M

is only a little less

than its potential ﬂow value over a wide range of Reynolds numbers (10

Re < 10

). However, for larger values of Kc, M

can be substantially smaller

than this and, in some range of Re and Kc, may actually be negative. The

values of C

(the drag coeﬃcient) that are deduced from experiments are

also a complicated function of Re and Kc. The behavior of the coeﬃcients

is particularly pathological when the reduced period, Kc, is close to that of

vortex shedding (Kc of the order of 10). Large transverse or lift forces can be

generated under these circumstances. To the author’s knowledge, detailed

investigations of this kind have not been made for a spherical body, but one

might expect the same qualitative phenomena to occur.

2.3.2 Eﬀect of concentration on added mass

Though most multiphase ﬂow eﬀects are delayed until later chapters it is

convenient at this point to address the issue of the eﬀect on the added mass

of the particles in the surrounding mixture. It is to be expected that the

added mass coeﬃcient for an individual particle would depend on the void

fraction of the surrounding medium. Zuber (1964) ﬁrst addressed this issue

using a cell method and found that the added mass, M

, for spherical bubbles

increased with volume fraction, α, like

(α)

(0)

(1 + 2α)

(1 − α)

=1+3α + O(α

) (2.28)

The simplistic geometry assumed in the cell method (a concentric spheri-

cal shell of ﬂuid surrounding each spherical particle) caused later researchers

to attempt improvements to Zuber’s analysis; for example, van Wijngaarden

(1976) used an improved geometry (and the assumption of potential ﬂow)

to study the O(α) term and found that

(α)

(0)

=1+2.76α + O(α

) (2.29)

which is close to Zuber’s result. However, even more accurate and more

recent analyses by Sangani et al. (1991) have shown that Zuber’s original

result is, in fact, remarkably accurate even up to volume fractions as large

as 50% (see also Zhang and Prosperetti 1994).

2.3.3 Unsteady potential ﬂow

In general, a particle moving in any ﬂow other than a steady uniform stream

will experience ﬂuid accelerations, and it is therefore necessary to consider

the structure of the equation governing the particle motion under these

circumstances. Of course, this will include the special case of acceleration of

a particle in a ﬂuid at rest (or with a steady streaming motion). As in the

earlier sections we shall conﬁne the detailed solutions to those for a spherical

particle or bubble. Furthermore, we consider only those circumstances in

which both the particle and ﬂuid acceleration are in one direction, chosen

for convenience to be the x

direction. The eﬀect of an external force ﬁeld

such as gravity will be omitted; it can readily be inserted into any of the

solutions that follow by the addition of the conventional buoyancy force.

All the solutions discussed are obtained in an accelerating frame of refer-

ence ﬁxed in the center of the ﬂuid particle. Therefore, if the velocity of the

particle in some original, noninertial coordinate system, x

∗

,wasV (t)inthe

∗

direction, the Navier-Stokes equations in the new frame, x

,ﬁxedinthe

particle center are

∂u

∂t

+ u

∂u

∂x

= −

∂P

∂x

+ ν

∂

∂x

(2.30)

where the pseudo-pressure, P , is related to the actual pressure, p,by

P = p + ρ

(2.31)

Here the conventional time derivative of V (t) is denoted by d/dt, but it

should be noted that in the original x

∗

frame it implies a Lagrangian deriva-

tive following the particle. As before, the ﬂuid is assumed incompressible

(so that continuity requires ∂u

/∂x

= 0) and Newtonian. The velocity that

the ﬂuid would have at the x

origin in the absence of the particle is then

W (t)inthex

direction. It is also convenient to deﬁne the quantities r, θ,

, u

as shown in ﬁgure 2.1 and the Stokes streamfunction as in equations

2.6. In some cases we shall also be able to consider the unsteady eﬀects due

to growth of the bubble so the radius is denoted by R(t).

First consider inviscid potential ﬂow for which equations 2.30 may be

integrated to obtain the Bernoulli equation

∂φ

∂t

+ u

)=constant (2.32)

where φ is a velocity potential (u

= ∂φ/∂x

)andψ must satisfy the equation

Lψ =0 where L ≡

∂

∂r

sin θ

∂

∂θ



sin θ

∂

∂θ



(2.33)

This is of course the same equation as in steady ﬂow and has harmonic

solutions, only ﬁve of which are necessary for present purposes:

ψ =sin



−



+cosθ sin



2Ar

−



+ E cos θ (2.34)

φ =cosθ



−Wr +



+(cos

θ −

)





(2.35)

=cosθ



−W −



+(cos

θ −

)



2Ar −



−

(2.36)

= − sin θ



−W +



− 2cosθ sin θ



Ar +



(2.37)

The ﬁrst part, which involves W and D, is identical to that for steady

translation. The second, involving A and B, will provide the ﬂuid velocity

gradient in the x

direction, and the third, involving E, permits a time-

dependent particle (bubble) radius. The W and A terms represent the ﬂuid

ﬂow in the absence of the particle, and the D, B, and E terms allow the

boundary condition

)

r=R

(2.38)

to be satisﬁed provided

D = −

,B=

2AR

,E= −R

(2.39)

In the absence of the particle the velocity of the ﬂuid at the origin, r =0,is

simply −W in the x

direction and the gradient of the velocity ∂u

/∂x

4A/3. Hence A is determined from the ﬂuid velocity gradient in the original

frame as

A =

∂U

∂x

∗

(2.40)

Now the force, F

, on the bubble in the x

direction is given by

= −2πR



p sin θ cos θdθ (2.41)

which upon using equations 2.31, 2.32, and 2.35 to 2.37 can be integrated

to yield

2πR

= −

(WR) −

RWA +

(2.42)

Reverting to the original coordinate system and using v as the sphere volume

for convenience (v =4πR

/3), one obtains

= −

∗

(U − V )

∗

(2.43)

where the two Lagrangian time derivatives are deﬁned by

∗

≡

∂

∂t

∗

+ U

∂

∂x

∗

(2.44)

∗

≡

∂

∂t

∗

+ V

∂

∂x

∗

(2.45)

Equation 2.43 is an important result, and care must be taken not to confuse

the diﬀerent time derivatives contained in it. Note that in the absence of

bubble growth, of viscous drag, and of body forces, the equation of motion

that results from setting F

=0is





∗

(2.46)

where m

is the mass of the particle. Thus for a massless bubble the accel-

eration of the bubble is three times the ﬂuid acceleration.

In a more comprehensive study of unsteady potential ﬂows Symington

(1978) has shown that the result for more general (i.e., noncolinear) accel-

erations of the ﬂuid and particle is merely the vector equivalent of equation

2.43:

= −

∗

− V

)

∗

(2.47)

where

∗

∂

∂t

∗

+ V

∂

∂x

∗

;

∗

∂

∂t

∗

+ U

∂

∂x

∗

(2.48)

The ﬁrst term in equation 2.47 represents the conventional added mass eﬀect

due to the particle acceleration. The factor 3/2 in the second term due to

the ﬂuid acceleration may initially seem surprising. However, it is made up

of two components:

/dt

∗

, which is the added mass eﬀect of the ﬂuid acceleration

2. ρ

vDU

/Dt

∗

,whichisabuoyancy-like force due to the pressure gradient asso-

ciated with the ﬂuid acceleration.

The last term in equation 2.47 is caused by particle (bubble) volumetric

growth, dv/dt

∗

, and is similar in form to the force on a source in a uniform

stream.

Now it is necessary to ask how this force given by equation 2.47 should

be used in the practical construction of an equation of motion for a particle.

Frequently, a viscous drag force F

, is quite arbitrarily added to F