of 18
11
FLOWS WITH GAS DYNAMICS
11.1 INTRODUCTION
This chapter addresses the class of compressible flows in which a gaseous
continuous phase is seeded with droplets or particles and in which it is nec-
essary to evaluate the relative motion between the disperse and continuous
phases for a variety of possible reasons. In many such flows, the motivation
is the erosion of the flow boundaries by particles or drops and this is directly
related to the relative motion. In other cases, the purpose is to evaluate the
change in the performance of the system or device. Still another motivation
is the desire to evaluate changes in the instability bo
undaries caused by the
presence of the disperse phase.
Examples include the potential for serious damage to steam turbine blades
by impacting water droplets (e.g. Gardner 1963, Smith
et al.
1967). In the
context of aircraft engines, desert sand storms or clouds of volcanic dust can
not only cause serious erosion to the gas turbine compressor (Tabakoff and
Hussein 1971, Smialek
et al.
1994, Dunn
et al.
1996, Tabakoff and Hamed
1986) but can also deleteriously effect the stall margin and cause engine
shutdown (Batcho
et al.
1987). Other examples include the consequences of
seeding the fuel of a solid-propelled rocket with metal particles in order to
enhance its performance. This is a particularly complicated example because
the particles may also melt and oxidize in the flow (Shorr and Zaehringer
1967).
In recent years considerable advancements have been made in the numer-
ical models and methods available for the solution of dilute particle-laden
flows. In this text, we present a survey of the analytical methods and the
physical understanding that they generate; for a valuable survey of the nu-
merical methods the reader is referred to Crowe (1982).
267
11.2 EQUATIONS FOR A DUSTY GAS
11.2.1 Basic equations
First we review the fundamental equations governing the flow of the indi-
vidual phases or components in a dusty gas flow. The continuity equations
(equations 1.21) may be written as
∂
∂t
(
ρ
N
α
N
)+
∂
(
ρ
N
α
N
u
Ni
)
∂x
i
=
I
N
(11.1)
where
N
=
C
and
N
=
D
refer to the continuous and disperse phases re-
spectively. We shall see that it is convenient to define a
loading
parameter,
ξ
,as
ξ
=
ρ
D
α
D
ρ
C
α
C
(11.2)
and that the continuity equations have an important bearing on the varia-
tions in the value of
ξ
within the flow. Note that the mixture density,
ρ
,is
then
ρ
=
ρ
C
α
C
+
ρ
D
α
D
=(1+
ξ
)
ρ
C
α
C
(11.3)
The momentum and energy equations for the individual phases (equations
1.45 and 1.69) are respectively
ρ
N
α
N

∂u
Nk
∂t
+
u
Ni
∂u
Nk
∂x
i

=
α
N
ρ
N
g
k
+
F
Nk
−I
N
u
Nk
−
δ
N

∂p
∂x
k
−
∂σ
D
Cki
∂x
i

(11.4)
ρ
N
α
N
c
vN

∂T
N
∂t
+
u
Ni
∂T
N
∂x
i

=
δ
N
σ
Cij
∂u
Ci
∂x
j
+
Q
N
+
W
N
+
QI
N
+
F
Ni
(
u
Di
−
u
Ni
)
−
(
e
∗
N
−
u
Ni
u
Ni
)
I
N
(11.5)
and, when summed over all the phases, these lead to the following combined
continuity, momentum and energy equations (equations 1.24, 1.46 and 1.70):
∂ρ
∂t
+
∂
∂x
i

N
ρ
N
α
N
u
Ni
= 0
(11.6)
268
∂
∂t

N
ρ
N
α
N
u
Nk
+
∂
∂x
i

N
ρ
N
α
N
u
Ni
u
Nk
=
ρg
k
−
∂p
∂x
k
+
∂σ
D
Cki
∂x
i
(11.7)

N

ρ
N
α
N
c
vN
∂T
N
∂t
+
u
Ni
∂T
N
∂x
i

=
σ
Cij
∂u
Ci
∂x
j
−F
Di
(
u
Di
−
u
Ci
)
−I
D
(
e
∗
D
−
e
∗
C
)+

N
u
Ni
u
Ni
I
N
(11.8)
To these equations of motion, we must add equations of state for both phases.
Throughout this chapter it will be assumed that the continuous phase is an
ideal gas and that the disperse phase is an incompressible solid. Moreover,
temperature and velocity gradients in the vicinity of the interface will be
neglected.
11.2.2 Homogeneous flow with gas dynamics
Though the focus in this chapter is on the effect of relative motion, we must
begin by examining the simplest case in which both the relative motion
between the phases or components and the temperature differences between
the phases or components are sufficiently small that they can be neglected.
This will establish the base state that, through perturbation methods, can
be used to examine flows in which the relative motion and temperature
differences are small. As we established in chapter 9, a flow with no relative
motion or temperature differences is referred to as
homogeneous
. The effect
of mass exchange will also be neglected in the present discussion and, in such
a homogeneous flow, the governing equations, 11.6, 11.7 and 11.8 clearly
reduce to
∂ρ
∂t
+
∂
∂x
i
(
ρu
i
) = 0
(11.9)
ρ

∂u
k
∂t
+
u
i
∂u
k
∂x
i

=
ρg
k
−
∂p
∂x
k
+
∂σ
D
Cki
∂x
i
(11.10)


N
ρ
N
α
N
c
vN
∂T
∂t
+
u
i
∂T
∂x
i
=
σ
Cij
∂u
i
∂x
j
(11.11)
269
where
u
i
and
T
are the velocity and temperature common to all phases.
An important result that follows from the individual continuity equations
11.1 in the absence of exchange of mass (
I
N
=0)isthat
D
Dt
ρ
D
α
D
ρ
C
α
C
=
Dξ
Dt
= 0
(11.12)
Consequently, if the flow develops from a uniform stream in which the load-
ing
ξ
is constant and uniform, then
ξ
is uniform and constant everywhere
and becomes a simple constant for the flow. We shall confine the remarks in
this section to such flows.
At this point, one particular approximation is very advantageous. Since
in many applications the volume occupied by the particles is very small, it is
reasonable to set
α
C
≈
1 in equation 11.2 and elsewhere. This approximation
has the important consequence that equations 11.9, 11.10 and 11.11 are now
those of a single phase flow of an
effective
gas whose thermodynamic and
transport properties are as follows. The approximation allows the equation
of state of the
effective
gas to be written as
p
=
ρ
R
T
(11.13)
where
R
is the gas constant of the effective gas. Setting
α
C
≈
1, the ther-
modynamic properties of the effective gas are given by
ρ
=
ρ
C
(1 +
ξ
);
R
=
R
C
/
(1 +
ξ
)
c
v
=
c
vC
+
ξc
sD
1+
ξ
;
c
p
=
c
pC
+
ξc
sD
1+
ξ
;
γ
=
c
pC
+
ξc
sD
c
vC
+
ξc
sD
(11.14)
and the effective kinematic viscosity is
ν
=
μ
C
/ρ
C
(1 +
ξ
)=
ν
C
/
(1 +
ξ
)
(11.15)
Moreover, it follows from equations 11.14, that the relation between the
isentropic speed of sound,
c
,inthe
effective
gas and that in the continuous
phase,
c
C
,is
c
=
c
C

1+
ξc
sD
/c
pC
(1 +
ξc
sD
/c
vC
)(1 +
ξ
)

1
2
(11.16)
It also follows that the Reynolds, Mach and Prandtl numbers for the effective
gas flow,
Re
,
M
and
Pr
(based on a typical dimension,

, typical velocity,
U
, and typical temperature,
T
0
, of the flow) are related to the Reynolds,
Mach and Prandtl numbers for the flow of the continuous phase,
Re
C
,
M
C
270
and
Pr
C
,by
Re
=
U
ν
=
Re
C
(1 +
ξ
)
(11.17)
M
=
U
c
=
M
C

(1 +
ξc
sD
/c
vC
)(1 +
ξ
)
(1 +
ξc
sD
/c
pC
)

1
2
(11.18)
Pr
=
c
p
μ
k
=
Pr
C

(1 +
ξc
sD
/c
pC
)
(1 +
ξ
)

(11.19)
Thus the first step in most investigations of this type of flow is to solve
for the effective gas flow using the appropriate tools from single phase gas
dynamics. Here, it is assumed that the reader is familiar with these basic
methods. Thus we focus on the phenomena that constitute departures from
single phase flow mechanics and, in particular, on the process and conse-
quences of relative motion or
slip
.
11.2.3 Velocity and temperature relaxation
While the homogeneous model with effective gas properties may constitute
a sufficiently accurate representation in some contexts, there are other tech-
nological problems in which the velocity and temperature differences be-
tween the phases are important either intrinsically or because of their con-
sequences. The rest of the chapter is devoted to these effects. But, in order to
proceed toward this end, it is necessary to stipulate particular forms for the
mass, momentum and energy exchange processes represented by
I
N
,
F
Nk
and
QI
N
in equations 11.1, 11.4 and 11.5. For simplicity, the remarks in
this chapter are confined to flows in which there is no external heat added
or work done so that
Q
N
=0and
W
N
= 0. Moreover, we shall assume that
there is negligible mass exchange so that
I
N
=0.Itremains,therefore,to
stipulate the force interaction,
F
Nk
and the heat transfer between the com-
ponents,
QI
N
. In the present context it is assumed that the relative motion
is at low Reynolds numbers so that the simple model of relative motion
defined by a relaxation time (see section 2.4.1) may be used. Then:
F
Ck
=
−F
Dk
=
ρ
D
α
D
t
u
(
u
Dk
−
u
Ck
)
(11.20)
where
t
u
is the velocity relaxation time given by equation 2.73 (neglecting
the added mass of the gas):
t
u
=
m
p
/
12
πRμ
C
(11.21)
271
It follows that the equation of motion for the disperse phase, equation 11.4,
becomes
Du
Dk
Dt
=
u
Ck
−
u
Dk
t
u
(11.22)
It is further assumed that the temperature relaxation may be modeled as
described in section 1.2.9 so that
QI
C
=
−QI
D
=
ρ
D
α
D
c
sD
Nu
t
T
(
T
D
−
T
C
)
(11.23)
where
t
T
is the temperature relaxation time given by equation 1.76:
t
T
=
ρ
D
c
sD
R
2
/
3
k
C
(11.24)
It follows that the energy equation for the disperse phase is equation 1.75
or
DT
D
Dt
=
Nu
2
(
T
C
−
T
D
)
t
T
(11.25)
In the context of droplet or particle laden gas flows these are commonly
assumed forms for the velocity and temperature relaxation processes (Mar-
ble 1970). In his review Rudinger (1969) includes some evaluation of the
sensitivity of the calculated results to the specifics of these assumptions.
11.3 NORMAL SHOCK WAVE
Normal shock waves not only constitute a flow of considerable practical
interest but also provide an illustrative example of the important role that
relative motion may play in particle or droplet laden gas flows. In a frame of
reference fixed in the shock, the fundamental equations for this steady flow
in one Cartesian direction (
x
with velocity
u
in that direction) are obtained
from equations 11.1 to 11.8 as follows. Neglecting any mass interaction (
I
N
=
0) and assuming that there is one continuous and one disperse phase, the
individual continuity equations 11.1 become
ρ
N
α
N
u
N
= ̇
m
N
= constant
(11.26)
where ̇
m
C
and ̇
m
D
are the mass flow rates per unit area. Since the gravi-
tational term and the deviatoric stresses are negligible, the combined phase
momentum equation 11.7 may be integrated to obtain
̇
m
C
u
C
+ ̇
m
D
u
D
+
p
= constant
(11.27)
272
Also, eliminating the external heat added (
Q
= 0) and the external work
done (
W
= 0) the combined phase energy equation 11.8 may be integrated
to obtain
̇
m
C
(
c
vC
T
C
+
1
2
u
2
C
)+ ̇
m
D
(
c
sD
T
D
+
1
2
u
2
D
)+
pu
C
= constant
(11.28)
and can be recast in the form
̇
m
C
(
c
pC
T
C
+
1
2
u
2
C
)+
pu
C
(1
−
α
C
)+ ̇
m
D
(
c
sD
T
D
+
1
2
u
2
D
)=constant
(11.29)
In lieu of the individual phase momentum and energy equations, we use the
velocity and temperature relaxation relations 11.22 and 11.25:
Du
D
Dt
=
u
D
du
D
dx
=
u
C
−
u
D
t
u
(11.30)
DT
D
Dt
=
u
D
dT
D
dx
=
T
C
−
T
D
t
T
(11.31)
where, for simplicity, we confine the present analysis to the pure conduction
case,
Nu
=2.
Carrier (1958) was the first to use these equations to explore the struc-
ture of a normal shock wave for a gas containing solid particles,
adustygas
in which the volume fraction of particles is negligible. Under such circum-
stances, the initial shock wave in the gas is unaffected by the particles and
can have a thickness that is small compared to the particle size. We denote
the conditions upstream of this structure by the subscript 1 so that
u
C
1
=
u
D
1
=
u
1
;
T
C
1
=
T
D
1
=
T
1
(11.32)
The conditions immediately downstream of the initial shock wave in the
gas are denoted by the subscript 2. The normal single phase gas dynamic
relations allow ready evaluation of
u
C
2
,
T
C
2
and
p
2
from
u
C
1
,
T
C
1
and
p
1
.
Unlike the gas, the particles pass through this initial shock without sig-
nificant change in velocity or temperature so that
u
D
2
=
u
D
1
;
T
D
2
=
T
D
1
(11.33)
Consequently, at the location 2 there are now substantial velocity and tem-
perature differences,
u
C
2
−
u
D
2
and
T
C
2
−
T
D
2
, equal to the velocity and
temperature differences across the initial shock wave in the gas. These dif-
ferences take time to decay and do so according to equations 11.30 and
11.31. Thus the structure downstream of the gas dynamic shock consists of
a relaxation zone in which the particle velocity decreases and the particle
273
Figure 11.1.
Typical structure of the relaxation zone in a shock wave in
a dusty gas for
M
1
=1
.
6,
γ
=1
.
4,
ξ
=0
.
25 and
t
u
/t
T
=1
.
0. In the non-
dimensionalization,
c
1
is the upstream acoustic speed. Adapted from Mar-
ble (1970).
temperature increases, each asymptoting to a final downstream state that
is denoted by the subscript 3. In this final state
u
C
3
=
u
D
3
=
u
3
;
T
C
3
=
T
D
3
=
T
3
(11.34)
As in any similar shock wave analysis the relations between the initial (1) and
final (3) conditions, are independent of the structure and can be obtained
directly from the basic conservation equations listed above. Making the small
disperse phase volume approximation discussed in section 11.2.2 and using
the definitions 11.14, the relations that determine both the structure of the
relaxation zone and the asymptotic downstream conditions are
̇
m
C
=
ρ
C
u
C
= ̇
m
C
1
= ̇
m
C
2
= ̇
m
C
3
; ̇
m
D
=
ρ
D
u
D
=
ξ
̇
m
C
(11.35)
̇
m
C
(
u
C
+
ξu
D
)+
p
=(1+
ξ
) ̇
m
C
u
C
1
+
p
1
=(1+
ξ
) ̇
m
C
u
C
3
+
p
3
(11.36)
(
c
pC
T
C
+
1
2
u
2
C
)+
ξ
(
c
sD
T
D
+
1
2
u
2
D
)=(1+
ξ
)(
c
p
T
1
+
1
2
u
2
1
)=(1+
ξ
)(
c
p
T
3
+
1
2
u
2
3
)
(11.37)
and it is a straightforward matter to integrate equations 11.30, 11.31, 11.35,
11.36 and 11.37 to obtain
u
C
(
x
),
u
D
(
x
),
T
C
(
x
),
T
D
(
x
)and
p
(
x
)inthere-
laxation zone.
First, we comment on the typical structure of the shock and the relaxation
zone as revealed by this numerical integration. A typical example from the
274
review by Marble (1970) is included as figure 11.1. This shows the asymptotic
behavior of the velocities and temperatures in the case
t
u
/t
T
=1
.
0. The
nature of the relaxation processes is evident in this figure. Just downstream
of the shock the particle temperature and velocity are the same as upstream
of the shock; but the temperature and velocity of the gas has now changed
and, over the subsequent distance,
x/c
1
t
u
, downstream of the shock, the
particle temperature rises toward that of the gas and the particle velocity
decreases toward that of the gas. The relative motion also causes a pressure
rise in the gas, that, in turn, causes a temperature rise and a velocity decrease
in the gas.
Clearly, there will be significant differences when the velocity and temper-
ature relaxation times are not of the same order. When
t
u

t
T
the velocity
equilibration zone will be much thinner than the thermal relaxation zone and
when
t
u

t
T
the opposite will be true. Marble (1970) uses a perturbation
analysis about the final downstream state to show that the two processes
of velocity and temperature relaxation are not closely coupled, at least up
to the second order in an expansion in
ξ
. Consequently, as a first approx-
imation, one can regard the velocity and temperature relaxation zones as
uncoupled. Marble also explores the effects of different particle sizes and the
collisions that may ensue as a result of relative motion between the different
sizes.
This normal shock wave analysis illustrates that the notions of velocity
and temperature relaxation can be applied as modifications to the basic gas
dynamic structure in order to synthesize, at least qualitatively, the structure
of the multiphase flow.
11.4 ACOUSTIC DAMPING
Another important consequence of relative motion is the effect it has on the
propagation of plane acoustic waves in a dusty gas. Here we will examine
both the propagation velocity and damping of such waves. To do so we
postulate a uniform dusty gas and denote the mean state of this mixture by
an overbar so that ̄
p
,
̄
T
, ̄
ρ
C
,
̄
ξ
are respectively the pressure, temperature,
gas density and mass loading of the uniform dusty gas. Moreover we chose
a frame of reference relative to the mean dusty gas so that ̄
u
C
= ̄
u
D
=
0. Then we investigate small, linearized perturbations to this mean state
denoted by ̃
p
,
̃
T
C
,
̃
T
D
, ̃
ρ
C
, ̃
α
D
, ̃
u
C
,and ̃
u
D
. Substituting into the basic
continuity, momentum and energy equations 11.1, 11.4 and 11.5, utilizing
the expressions and assumptions of section 11.2.3 and retaining only terms
275
linear in the perturbations, the equations governing the propagation of plane
acoustic waves become
∂
̃
u
C
∂x
+
1
̄
p
∂
̃
p
∂t
−
1
̄
T
∂
̃
T
C
∂t
= 0
(11.38)
ρ
D
∂
̃
α
D
∂t
+
∂
̃
u
D
∂x
= 0
(11.39)
∂
̃
u
C
∂t
+
ξ
̃
u
C
t
u
−
ξ
̃
u
D
t
u
+
1
γ
∂
̃
p
∂x
= 0
(11.40)
∂
̃
u
D
∂t
+
̃
u
D
t
u
−
̃
u
C
t
u
= 0
(11.41)
∂
̃
T
C
∂t
+
ξ
̃
T
C
t
T
−
ξ
̃
T
D
t
T
+
(
γ
−
1) ̄
p
γ
̄
T
∂
̃
p
∂t
= 0
(11.42)
∂
̃
T
D
∂t
+
c
pC
̃
T
D
c
sD
t
T
−
c
pC
̃
T
C
c
sD
t
T
= 0
(11.43)
where
γ
=
c
pC
/c
vC
. Note that the particle volume fraction perturbation only
occurs in one of these, equation 11.39; consequently this equation may be
set aside and used after the solution has been obtained in order to calculate
̃
α
D
and therefore the perturbations in the particle loading
̃
ξ
. The basic form
of a plane acoustic wave is
Q
(
x, t
)=
̄
Q
+
̃
Q
(
x, t
)=
̄
Q
+
Re
$
Q
(
ω
)
e
iκx
+
iωt
%
(11.44)
where
Q
(
x, t
) is a generic flow variable,
ω
is the acoustic frequency and
κ
is a complex function of
ω
; clearly the phase velocity of the wave,
c
κ
,is
given by
c
κ
=
Re
{−
ω/κ
}
and the non-dimensional attenuation is given by
Im
{−
κ
}
. Then substitution of the expressions 11.44 into the five equations
11.38, 11.40, 11.41, 11.42, and 11.43 yields the following dispersion relation
for
κ
:

ω
κc
C

2
=
(1 +
iωt
u
)(
c
pC
c
sD
+
ξ
+
iωt
T
)
(1 +
ξ
+
iωt
u
)(
c
pC
c
sD
γξ
+
iωt
T
)
(11.45)
where
c
C
=(
γ
R
C
̄
T
)
1
2
is the speed of sound in the gas alone. Consequently,
the phase velocity is readily obtained by taking the real part of the square
root of the right hand side of equation 11.45. It is a function of frequency,
ω
,aswellastherelaxationtimes,
t
u
and
t
T
, the loading,
ξ
, and the specific
276
Figure 11.2.
Non-dimensional attenuation,
Im
{−
κc
C
/ω
}
(dotted lines),
and phase velocity,
c
κ
/c
C
(solid lines), as functions of reduced frequency,
ωt
u
, for a dusty gas with various loadings,
ξ
,asshownand
γ
=1
.
4,
t
T
/t
u
=
1and
c
pC
/c
sD
=0
.
3.
Figure 11.3.
Non-dimensional attenuation,
Im
{−
κc
C
/ω
}
(dotted lines),
and phase velocity,
c
κ
/c
C
(solid lines), as functions of reduced frequency,
ωt
u
, for a dusty gas with various loadings,
ξ
,asshownand
γ
=1
.
4,
t
T
/t
u
=
30 and
c
pC
/c
sD
=0
.
3.
heat ratios,
γ
and
c
pC
/c
sD
. Typical results are shown in figures 11.2 and
11.3.
The mechanics of the variation in the phase velocity (acoustic speed) are
evident by inspection of equation 11.45 and figures 11.2 and 11.3. At very low
frequencies such that
ωt
u

1and
ωt
T

1, the velocity and temperature
relaxations are essentially instantaneous. Then the phase velocity is simply
obtained from the effective properties and is given by equation 11.16. These
are the phase velocity asymptotes on the left-hand side of figures 11.2 and
11.3. On the other hand, at very high frequencies such that
ωt
u

1and
277
ωt
T

1, there is negligible time for the particles to adjust and they simply
do not participate in the propagation of the wave; consequently, the phase
velocity is simply the acoustic velocity in the gas alone,
c
C
. Thus all phase
velocity lines asymptote to unity on the right in the figures. Other ranges of
frequency may also exist (for example
ωt
u

1and
ωt
T

1 or the reverse)
in which other asymptotic expressions for the acoustic speed can be readily
extracted from equation 11.45. One such intermediate asymptote can be
detected in figure 11.3. It is also clear that the acoustic speed decreases with
increased loading,
ξ
, though only weakly in some frequency ranges. For small
ξ
the expression 11.45 may be expanded to obtain the linear change in the
acoustic speed with loading,
ξ
, as follows:
c
κ
c
C
=1
−
ξ
2
⎡
⎣
(
γ
−
1)
c
pC
c
sD

(
c
pC
/c
sD
)
2
+(
ωt
T
)
2

+
1
{
1+(
ωt
T
)
2
}
⎤
⎦
+
....
(11.46)
This expression shows why, in figures 11.2 and 11.3, the effect of the loading,
ξ
, on the phase velocity is small at higher frequencies.
Now we examine the attenuation manifest in the dispersion relation 11.45.
The same expansion for small
ξ
that led to equation 11.46 also leads to the
following expression for the attenuation:
Im
{−
κ
}
=
ξω
2
c
C
⎡
⎣
(
γ
−
1)
ωt
T

(
c
pC
/c
sD
)
2
+(
ωt
T
)
2

+
ωt
u
{
1+(
ωt
T
)
2
}
⎤
⎦
+
....
(11.47)
In figures 11.2 and 11.3, a dimensionless attenuation,
Im
{−
κc
C
/ω
}
,isplot-
ted against the reduced frequency. This particular non-dimensionalization
is somewhat misleading since, plotted without the
ω
in the denominator,
the attenuation increases monotonically with frequency. However, this pre-
sentation is commonly used to demonstrate the enhanced attenuations that
occur in the neighborhoods of
ω
=
t
−
1
u
and
ω
=
t
−
1
T
and which are manifest
in figures 11.2 and 11.3.
When the gas contains liquid droplets rather than solid particles, the
same basic approach is appropriate except for the change that might be
caused by the evaporation and condensation of the liquid during the passage
of the wave. Marble and Wooten (1970) present a variation of the above
analysis that includes the effect of phase change and show that an additional
maximum in the attenuation can result as illustrated in figure 11.4. This
additional peak results from another relaxation process embodied in the
phase change process. As Marble (1970) points out it is only really separate
278
Figure 11.4.
Non-dimensional attenuation,
Im
{−
κc
C
/ω
}
, as a function
of reduced frequency for a droplet-laden gas flow with
ξ
=0
.
01,
γ
=1
.
4,
t
T
/t
u
=1 and
c
pC
/c
sD
= 1. The dashed line is the result without phase
change; the solid line is an example of the alteration caused by phase
change. Adapted from Marble and Wooten (1970).
from the other relaxation times when the loading is small. At higher loadings
the effect merges with the velocity and temperature relaxation processes.
11.5 OTHER LINEAR PERTURBATION
ANALYSES
In the preceding section we examined the behavior of small perturbations
about a constant and uniform state of the mixture. The perturbation was
a plane acoustic wave but the reader will recognize that an essentially sim-
ilar methodology can be used (and has been) to study other types of flow
involving small linear perturbations. An example is steady flow in which
the deviation from a uniform stream is small. The equations governing the
small deviations in a steady planar flow in, say, the (
x, y
) plane are then
quite analogous to the equations in (
x, t
) derived in the preceding section.
11.5.1 Stability of laminar flow
An important example of this type of solution is the effect that dust might
have on the stability of a laminar flow (for instance a bo
undary layer flow)
and, therefore, on the transition to turbulence. Saffman (1962) explored
the effect of a small volume fraction of dust on the stability of a parallel
flow. As expected and as described in section 1.3.2, when the response times
279
of the particles are short compared with the typical times associated with
the fluid motion, the particles simply alter the effective properties of the
fluid, its effective density, viscosity and compressibility. It follows that un-
der these circumstances the stability is governed by the effective Reynolds
number and effective Mach number. Saffman considered dusty gases at low
volume concentrations,
α
, and low Mach numbers; under those conditions
the net effect of the dust is to change the density by (1 +
αρ
S
/ρ
G
)andthe
viscosity by (1 + 2
.
5
α
). The effective Reynolds number therefore varies like
(1 +
αρ
S
/ρ
G
)
/
(1 + 2
.
5
α
). Since
ρ
S

ρ
G
the effective Reynolds number is
increased and therefore, in the small relaxation time range, the dust is desta-
bilizing. Conversely for large relaxation times, the dust stabilizes the flow.
11.5.2 Flow over a wavy wall
A second example of this type of solution that was investigated by Zung
(1967) is steady particle-laden flow over a wavy wall of small amplitude
(figure 11.5) so that only the terms that are linear in the amplitude need be
retained. The solution takes the form
exp(
iκ
1
x
−
iκ
2
y
)
(11.48)
where 2
π/κ
1
is the wavelength of the wall whose mean direction corresponds
with the
x
axis and
κ
2
is a complex number whose real part determines
the inclination of the characteristics or Mach waves and whose imaginary
part determines the attenuation with distance from the wall. The value of
κ
2
is obtained in the solution from a dispersion relation that has many
similarities to equation 11.45. Typical computations of
κ
2
are presented in
figure 11.6. The asymptotic values for large
t
u
that occur on the right in
this figure correspond to cases in which the particle motion is constant and
Figure 11.5.
Schematic for flow over a wavy wall.
280
Figure 11.6.
Typical results from the wavy wall solution of Zung (1969).
Real and imaginary parts of
κ
2
/κ
1
are plotted against
t
u
U/κ
1
for various
mean Mach numbers,
M
=
U/c
C
, for the case of
t
T
/t
u
=1,
c
pC
/c
sD
=1,
γ
=1
.
4 and a particle loading,
ξ
=1.
unaffected by the waves. Consequently, in subsonic flows (
M
=
U/c
C
<
1)
in which there are no characteristics, the value of
Re
{
κ
2
/κ
1
}
asymptotes to
zero and the waves decay with distance from the wall such that
Im
{
κ
2
/κ
1
}
tends to (1
−
M
2
)
1
2
. On the other hand in supersonic flows (
M
=
U/c
C
>
1)
Re
{
κ
2
/κ
1
}
asymptotes to the tangent of the Mach wave angle in the gas
alone, namely (
M
2
−
1)
1
2
, and the decay along these characteristics is zero.
At the other extreme, the asymptotic values as
t
u
approaches zero corre-
spond to the case of the effective gas whose properties are given in section
11.2.2. Then the appropriate Mach number,
M
0
, is that based on the speed
of sound in the effective gas (equation 11.16). In the case of figure 11.6,
M
2
0
=2
.
4
M
2
. Consequently, in subsonic flows (
M
0
<
1), the real and imag-
inary parts of
κ
2
/κ
1
tend to zero and (1
−
M
2
0
)
1
2
respectively as
t
u
tends
to zero. In supersonic flows (
M
0
>
1), they tend to (
M
2
0
−
1)
1
2
and zero
respectively.
281
11.6 SMALL SLIP PERTURBATION
The analyses described in the preceding two sections, 11.4 and 11.5, used a
linearization about a uniform and constant mean state and assumed that the
perturbations in the variables were small compared with their mean values.
Another, different linearization known as the small slip approximation can be
advantageous in other contexts in which the mean state is more complicated.
It proceeds as follows. First recall that the solutions always asymptote to
those for a single effective gas when
t
u
and
t
T
tend to zero. Therefore, when
these quantities are small and the slip between the particles and the gas is
correspondingly small, we can consider constructing solutions in which the
flow variables are represented by power series expansions in one of these
small quantities, say
t
u
, and it is assumed that the other (
t
T
) is of similar
order. Then, generically,
Q
(
x
i
,t
)=
Q
(0)
(
x
i
,t
)+
t
u
Q
(1)
(
x
i
,t
)+
t
2
u
Q
(2)
(
x
i
,t
)+
....
(11.49)
where
Q
represents any of the flow quantities,
u
Ci
,
u
Di
,
T
C
,
T
D
,
p
,
ρ
C
,
α
C
,
α
D
, etc. In addition, it is assumed for the reasons given above that the slip
velocity and slip temperature, (
u
Ci
−
u
Di
)and(
T
C
−
T
D
), are of order
t
u
so that
u
(0)
Ci
=
u
(0)
Di
=
u
(0)
i
;
T
(0)
C
=
T
(0)
D
=
T
(0)
(11.50)
Substituting these expansions into the basic equations 11.6, 11.7 and 11.8
and gathering together the terms of like order in
t
u
we obtain the following
zeroth order continuity, momentum and energy relations (omitting gravity):
∂
∂x
i

(1 +
ξ
)
ρ
(0)
C
u
(0)
i

= 0
(11.51)
(1 +
ξ
)
ρ
(0)
C
u
(0)
k
∂u
(0)
i
∂x
k
=
−
∂p
(0)
∂x
i
+
∂σ
D
(0)
Cik
∂x
k
(11.52)
ρ
(0)
C
u
(0)
k
(
c
pC
+
ξc
sD
)
∂T
(0)
∂x
k
=
u
(0)
k
∂p
(0)
∂x
k
+
σ
D
(0)
Cik
∂u
(0)
i
∂x
k
(11.53)
Note that Marble (1970) also includes thermal co
nduction in the energy
equation. Clearly the above are just the equations for single phase flow of
the effective gas defined in section 11.2.2. Conventional single phase gas
dynamic methods can therefore be deployed to obtain their solution.
Next, the relaxation equations 11.22 and 11.25 that are first order in
t
u
282
Figure 11.7.
The dimensionless choked mass flow rate as a function of
loading,
ξ
,for
γ
C
=1
.
4 and various specific heat ratios,
c
pC
/c
sD
as shown.
yield:
u
(0)
k
∂u
(0)
i
∂x
k
=
u
(1)
Ci
−
u
(1)
Di
(11.54)
u
(0)
k
∂T
(0)
∂x
k
=

t
u
t
T

Nu
2

(
T
(1)
C
−
T
(1)
D
)
(11.55)
From these the slip velocity and slip temperature can be calculated once the
zeroth order solution is known.
The third step is to evaluate the modification to the effective gas solution
caused by the slip velocity and temperature; in other words, to evaluate
the first order terms,
u
(1)
Ci
,
T
(1)
C
, etc. The relations for these are derived
by extracting the
O
(
t
u
) terms from the continuity, momentum and energy
equations. For example, the continuity equation yields
∂
∂x
i

ξρ
(0)
C
(
u
(1)
Ci
−
u
(1)
Di
)+
u
(0)
i
(
ρ
D
α
(1)
D
−
ξρ
(1)
C
)

= 0
(11.56)
This and the corresponding first order momentum and energy equations can
then be solved to find the
O
(
t
u
) slip perturbations to the gas and particle
flow variables. For further details the reader is referred to Marble (1970).
A particular useful application of the slip perturbation method is to the
one-dimensional steady flow in a convergent/divergent nozzle. The zeroth
order, effective gas solution leads to pressure, velocity, temperature and
density profiles that are straightforward functions of the Mach number which
is, in turn, derived from the cross-sectional area. This area is used as a
283