A NUMERICAL AND EXPERIMENTAL STUDY OF THE
SCALAR TRANSPORT AT A TURBULENT LIQUID
FREE SURFACE
by
Boo Cheong Khoo
B.A., University
of Cambridge
(1980)
M.Eng., National University of Singapore (1984)
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment
for
of the Requirements
the Degree
of
DOCTOR OF PHILOSOPHY
at
the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
November 1988
@ 1988 Massachusetts Institute of Technology
Signature of Author
Department of Mechanical Engineering
November 1988
Certified by
Professor Ain A. Sonin
Thesis Supervisor
Accepted by
Professor Ain A. Sonin
Chairman, Department Committee on Graduate Studies
MA8SSWCr1S
MAR 1
8TITJ
1989
URlW
A Numerical and Experimental Study of the
Scalar Transport at a Turbulent Liquid Free Surface
by
Boo Cheong Khoo
Submitted to the Department of Mechanical Engineering in
partial fulfillment of the requirements for the degree of
Doctor of Philosopy
Abstract
An
study
experimental
was
carried
out
on
the
relationship between the gas absorption rate at
free surface and the parameters which define the
fundamental
a turbulent
turbulence on the liquid side. Measurements were carried out
in a system
similar to
(1986) to study
that
vapour
used by Sonin, Shimko and Chun
condensation
on surfaces with zero
shear. In this apparatus
the turbulence
intensity can be
varied over a broad range, and the macroscale can be changed
by using test
cells
of different
size. The present study
focused on transport at high Schmidt numbers (230-1600) and
was conducted with C02 as the transfer agent and water or
water-glycerol mixture as the liquid. The results revealed
that the rate equation has
the other
for high
eddy
Reynolds
numbers,
two asymptotes, one for low and
Reynolds
number.
At
the lower
the
transport
linearly with
V'
and
is
higher
Reynolds
numbers,
coefficientl/
ncreases
1
proportional
to Sc
the rate
of increase
; at the
of the
coefficient with V' is sufficiently higher, and the Schmidt
number dependence subsides. The break point between the two
regions
is characterised
by
a critical
eddy Reynolde
number
which is Schmidt number dependent.
None of the features
of the experimental correlation
are explained by any of the numerous simple models which
have been postulated for free surface transport. This led us
to
attempt
turbulence
a
numerical
rigorous
distribution
region near the
and passive
liquid
interface.
analysis
condition
by
represented
applied
at
the
a
bottom
both
the
in the
Computational were made
for the layer between the free
surface
macroscale. The effect of the turbulence
liquid was
of
scalar transport
and a depth of one
in the bulk of the
Dirichlet-type
of
this
boundary
layer: at that
plane, an unsteady
velocity
field was imposed which
with
specified V' and
simulated isotropic turbulence
macroscale. The
Navier-Stokes
and scalar conservation
equations were solved until both the turbulence and the
temperature
distributions
in
the
domain
attained
statistically steady states. Two eddy Reynolds number cases
were
computed,
1000 and 100,
with
the Schmidt number
in the
range 1 - 50 (higher values of Schmidt number would have
limited the Reynolds number to uncomfortably low values for
simulating turbulence). The solutions showed that some
fundamental
difficulties
arise when
2
one tries to replace an
outer
turbulence
condition
at
some
field
with
plane
near
physically unrealistic (but
a
Dirichlet-type
the
transfer
boundary
surface:
a
computational correct) boundary
layer forms near that plane, and the turbulence
distribution
in the computational domain shows unrealistic features.
Nevertheless, the computed transport coefficient agreed
surprisingly well with the experimental data of Brown (1989)
for the Schmidt number range 1.5-5.
Thesis Supervisor: Professor Ain A. Sonin
Thesis Committee: Professor Antony T. Patera
Professor Joseph H. Haritonidis
3
Acknowledgement
My deepest
Professor Ain
appreciation
A.
Sonin
who
goes
provided
guidance and yet afforded me
important
aspect
face the frontier
I would
of
my
education
like
to
Hartonidis: to Professor
rich
intricacies
of
implementation, and
The present
enjoyable.
to
express
my appreciation
to my
T. Patera and Joseph H.
Patera for providing me with the
computational
techniques
and
Professor Hartonidis for offering
to
into the experiments.
and former
Laboratory have given me
more
here in M.I.T.,
of science with confidence.
also
insights
with invaluable
His presence has been the
thesis committee, Professors Antony
helphful
me
plenty of academic freedom to
explore and satisfy my curiosity.
most
to my thesis advisor,
Special
members
of the Fluid Mechanics
much
help
and
thanks
goes
to Dick Fenner for his
seemingly unlimited ideas
also to Claire Sasahara
made my stay here
for design simplification. Thanks
and Leslie Regan who selflessly
helped me with the administrative chores.
I
would
also
like
Government for funding my
to
acknowledge
study
here
the
Singapore
at M.I.T., and NASA
Lewis Research Centre for supporting part of my experimental
expenses
(under Grant
No.
NAG3-731)
and for providing
time
on their Cray for the numerical
simulation.
Finally, I would like to
provided me with a supportive
profess my belief in God who
family, my wife Peck-Ha and
son Ivan who
have been a
source
occasions.
4
of joy and comfort
in many
Table of Contents
ae
Title
page
Abstract
................................................
2
Acknowledgements
........................................
4
Nomenclature
...........................................
7
List of Tables .........................................
List of Figures
13
.........................................
14
1. INTRODUCTION ........................................
22
2. MODELS OF AND CORRELATIONS FOR SURFACE TRANSPORT .....
25
Part
I
3. EXPERIMENT ...........................................
3.1 Apparatus for Gas Absorption Rate ................
3.2 LDA System for Velocity Measurement ..............
31
31
34
3.3 Velocity Measurement with High Speed Video Camera
System
...........................................
4. RESULTS AND DISCUSSIONS ..............................
4.1 Turbulence Characteristics .......................
4.1.1 RMS Velocity Correlation ...................
36
40
40
40
4.1.2 An attempt to measure Surface RMS Velocity
on the Free Surface
........................
4.1.3 The Turbulence Macroscale ..................
4.1.4 Eulerian Time Spectrum ....................
4.2 Mass Transfer Rate ...............................
4.3 Comparison with Other Results ....................
4.3.1 Turbulence Imposed from Wi thin the Liquid ..
S
41
42
43
44
51
52
4.3.2 Turbulence Imposed via Shear at the
Interface
.....................
5.
. .
. .. .
61
. .
. ..
.
68
5.1 LDA Calibration of Turbulence in Test Cell . . .
.
68
5.2
.
69
CONCLUSIONS FOR EXPERIMENTS .............
Gas Absorption Rate Correlation .....
.
.
.
. .
. ..
Part II
6. NUMERICAL SIMULATION .................................
71
6.1 Background and Overview ..........................
71
6.2 Discretized Equations ............................
6.3 Boundary Conditions ..............................
6.4 Mesh Size Considerations .........................
72
7. RESULTS AND DISCUSSIONS II ...........................
75
81
84
7.1 Approach to Steady State .........................
7.1.1 Time Evolution of Stanton number ...........
84
7.1.2 Energy Transfer Across Each Plane ..........
7.2 Velocity Characteristics .........................
85
7.2.1 Ensemble Averaging of the RMS Velocity in
the Steady State ...........................
7.2.2 RMS Velocity Profile .......................
7.2.3 Energy Spectra
............................
84
87
87
88
90
7.3 Temperature Profiles .............................
91
7.4 Turbulent Diffusivity ............................
93
7.5 The Condensation Stanton number and Comparison
with Experiment ..................................
94
8. CONCLUDING REMARKS ON COMPUTATION ....................
98
References
.......................................
100
Tables
........................................
110
Figures
................................
120
Appendix ................................
212
6
General Nomenclature, Part I
Note
: SI units
are used throughout,
unless
otherwise
stated
A
gas-liquid interfacial area
C
species concentration
Cs
saturated condition
Cp
specific heat of liquid in constant
d
nozzle diameter
D
diameter
of test cell
D
molecular
diffusivity
k
average turbulent kinetic energy per unit mass
K*
mass/heat transfer coefficient corresponding to the
'break' velocity as defined in equation 4.2.5
KL
mass/heat transfer coefficient (equation 1.1)
Lk e
length scale in k-e model analysis
P
pressure
AP
pressure difference across the nozzle
Pr,Prb
of the gas in the liquid
pressure
of gas in the liquid
bulk Prandtl number
Q
volumetric flow rate circulating through system
r
radial coordinate measured
from the cylinder
axis
R
<
R
(t)
>
u(t'+t)u(t')
Eulerian
/
<
u
2
as
>
autocorrelation, defined as
>
< w(t'+t)w(t')
Re
defined
autocorrelation,
(t) Eulerian
/
< w
2
>
system Reynolds number, (Q/Dd)(D/V), where D - test
cell diameter
ReA
eddy Reynolds number, V'A/v
(ReA)* break Reynolds number, V/v
Sc
Schmidt number, v/D
St
Stanton number, KL/V'
t
time
U*w
friction velocity based on shear stress at interface
and water density
u(t)
local horizontal
component
of the fluctuating
velocity (measured as azimuthal)
v'(t)
local vertical ar horizontal component of fluctuating
velocity
w(t)
local vertical component of the fluctuating velocity
V'
rms value of the local horizontal
component
of turbulent velocity
rate correlations,
or vertical
in the system; in
V' is used to indicate
the rms
value of either component at the "interface" obtained
by extrapolating from the bulk, disregarding surface
damping.
8
V,*
rms velocity
in the 'break' region, as defined
in
equation 4.2.5
V
volume of water in the system
z
coordinate measure vertically upwards from the nozzle
exit in the test section
z
height of' interface above nozzle
Greek symbols
a
thermal/molecular diffusivity of liquid
C
viscous dissipation
A
integral
rate per unit mass
length scale of turbulent
eddies as defined
in equation 4.1.6
liquid dynamic viscosity
v
liquid kinematic viscosity
p
liquid density
a
surface tension
T
turbulence macroscale
0(Re
s)
4I(f)
uu
function
time
used in figures 3.3.1 and 3.3.2
Eulerian time spectrum in the horizontal direction
(f) Eulerian
time spectrum
in the vertical
Subscript
9
direction
b
bulk liquid condition
s
evaluated at interfacial condition
*
conditions
at the 'break' region (see section 4.2)
Additional nomenclature for Part II
1
s
smallest length scale in the velocity field
Lx ,Ly
size of computational domain in the horizontal
directions
Pe
Peclet number
Q(z)
total heat flux across plane at elevation
dimensionless quantity,
z; used as
non-dimensionalised w.r.t.
U(Ts-Tb) in figures 7.1.6 - 7.1.9, 7.1.14 - 7.1.16
Reb
eddy Reynolds
velocity
S
uu
number, UA/v, which
at the lower surface
dimensionle
ss
longitudinal
horizontal plane
in
the
dimensional ised w.r.t. U 2
S
vv
is based on the rms
power spectrum
on
a
x-direction, non-
dimensionle ss longitudinal
power spe ctrum on a
horizontal plane
in
the
y-direct ion, nondimensionalised w.r.t. U2
S
dimensionless transverse power spectrum on a
horizontal
plane
in th.e z-direction, nondimensionalised w.r.t. U[2
T
dimensionless temperature, (T"-Tb)/(Ts-Tb )
T"
dimensional temperature
10
VVW
TsT
1
saturation temperature at the interface
Tb T 2
bulk temperature within the liquid
u
dimensionless
vector
velocity
(u,v,w)
non-dimensionalised w.r.t. to U
u,v
velocity component in the horizontal directions,
non-dimensionalised w.r.t. to U
urms
dimensionless rms velocity in the x-directions,
non-dimensionalised w.r.t. to U
vrms
rms
dimensionless rms velocity in the y-directions,
non-dimensionalised w.r.t. to U
w rms
dimensionless
rms velocity in the z-directions,
nrm
s
U
non-dimensionalisedw.r.t.
rms velocity
to U
at the lower surface of computational
domain, serve as reference velocity used for nondimensionalisation
w
dimensionless velocity in the vertical direction,
non-dimensionalised w.r.t. to U
x,y
horizontal coordinates
z
vertical coordinate
Greek symbols
a'
defined as 2/L
aart
artificial diffusivity replacing the molecular
diffusivity in the energy equation for region near
x
the lower boundary
(see section 6.3)
11
aT
turbulent
diffusivity;
used
as
dimensionless
quantity, non-dimensionalised w.r.t. U
7.4.1
- 7.4.4,
in figures
7.4.6 - 7.4.8
P
defined as 2/L
17
dimensionless dynamic pressure, P/pU2 + O.51u.ul
A
grid spacing
At
dimensionless time increment, non-dimensionalised
Y
in the computational
domain
w.r.t. A/U
'
time constant
dimensionless
A/U
in figure
as defined
in equation
7.4.2; used as
quantity, non-dimensionalised w.r.t.
7.4.5
12
List of Tables
Title
Table
1
Major conceptual models for transport across a free
surface
2
LDA measurement of turbulence macroscale time (r)
and length (A=V'r) in the system with
diameter
of
0.152m at 0.03m beneath the interface
3
Operating conditions of the mass transfer experiments
4
Conditions
at the "break" point as defined
in
equation 4.2.1
5
Mass
transfer
result for ReA < (ReA)*
6
Summary of comparison with experiments where
turbulence is imposed
7
Summary of comparison with experiments where
turbulence is induced
8
form within the liquid
by shear at the interface
Parameters associated with the numerical simulations
for case A and case B
9
Computed
St
vs
Pr
for case A and comparison with
vs
Pr
for case B and comparison with
experiment
10
Computed
St
experiment
13
List of Figures
Figure
Title
3.1.1
Schematic
diagram
3.1.2
System flow rate calibration curve, Q/Dd vs Ap1/2
3.2.1
Schematic
diagram
of the test system
of the test section used in
velocity measurement by the LDA
3.3.1
The function
Shimko
3.3.2
(1985)
The function
(Re s) in the correlation
for V' by
used by Sonin et al (1986)
(Re ) in the correlation for V' by
Brown (1989)
4.1.1
Horizontal rms velocity profile for Re -29140
4.1.2
Horizontal rms velocity profile for Re-574020
4.1.3
Horizontal rms velocity profile for Res -57690
4.1.4
Vertical
4.1.5
Vertical rms velocity profile for Re
4..6
Vertical rms velocity profile for Re-57690
4.1.7
Azimuthal rms and mean velocity at Smm beneath
s
rms velocity
profile
for Re s -29140
574020
the interface as a function of radius for
Re -40120
4.1.8
Surface rms velocity vs Q/Dd at height z/D-3.67
14
4.1.9
Surface rms velocity vs Q/Dd at height z/D=4.17
4.1.10
Surface rms velocity vs Q/Dd at height z/D-4.50
4.1.11
Eulerian autocorrelation for the horizontal
velocity
interface
4.1.12
at the centreline
30mm beneath
the
for Re -25000
Eulerian autocorrelation for the horizontal
velocity
at the centreline
30mm beneath
the
interface for Re -35200
4.1.13
Eulerian autocorrelation for the horizontal
velocity
at the centreline
30mm beneath
the
interface for Re -50300
4.1.14
Eulerian autocorrelation for the horizontal
velocity
at the centreline
30mm beneath
the
interface for Re -70500
4.1.15
Eulerian autocorrelation for the vertical
velocity
at the centreline
30mm beneath
the
interface for Re -25000
s
4.1.16
Eulerian autocorrelation for the vertical
velocity
at the centreline
30mm beneath
the
interface for Re -35200
4.1.17
Eulerian autocorrelation for the vertical
velocity
at the centreline
30mm beneath
the
interface for Re -50300
4.1.18
Eulerian autocorrelation for the vertical
velocity at the centreline 30mm beneath the
interface
4.1.19
for Re -70500
Eulerian time spectrum uu(f) for the horizontal
uu
15
velocity
at
the
centreline
30nmn beneath
the
interface for Re =25000
s
4.1.20
Eulerian
time spectrum
(f) for the vertical
velocity at the centreline 30mm beneath the
interface for Re -25000
s
4.1.21
Eulerian
time spectrum W Uu (f) for the horizontal
velocity at the centreline 30mm beneath the
interface
4.1.22
for Re -50300
S
Eulerian time spectrum
velocity
interface
WW(f)
for the vertical
30mm beneath
at the centreline
the
for Re =50300
s
4.2.1
Mass transfer coefficient K L vs V' for Sc=230
4.2.2
Mass transfer coefficient KL v
4.2.3
Mass transfer coefficient KL vs V' for Sc=525
4.2.4
Mass transfer coefficient K L vs V' for Sc=1600
4.2.5
Mass transfer coefficient KL vs V' for Sc-525
wi th smaller test-section
4.2.6
Mass transfer coefficient KL v's V' for Sc=525
with both test-sections
4.2.7
Dimensionless
plot of StSc
4.2.8
Dimensionless
plot of (KL-K,)/(V'-V
nV
J
V'
for Sc=377
vs ReA
)
vs Re A for
ReA < (ReA)*
4.2.9
Plot of St vs ReA/(ReA)*
and 4.2.10(b)
16
for equations
4.2.10(a)
4.2.10
Plot of KL vs V'
for Sc-525 with prediction
given by equations 4.2.8 and 4.2.9
4.2.11
Plot of K L vs V'
for Sc-1600 with prediction
given by equations 4.2.8 and 4.2.9
4.3.1
Dimensionless plot of StSc
of submerged oscillating
generating
4.3.2
Dimensionless
5
ve Re I
for
grid as means
system
of
turbulence (Isenogle (1985))
plot of StSc
ve Re H
for system
of submerged oscillating grid as means of
generating
turbulence
(Ho (1987))
turbulence
cube
6.3.1
Plot of f(r) vs r for isotropic
6.3.2
Profile of at
art /a
for case A
6.3.3
Profile
of a
art /a
for case B
6.4.1
Diagram
showing the grid arrangement
for case A
6.4.2
Diagram
showing the grid arrangement
for case B
7.1.1
Time evolution of computed St for case A (Pr-l.0)
7.1.2
Time evolution of computed St for case A (Pr=1.5)
7.1.3
Time evolution of computed St for case A (Pr-2.2)
7.1.4
Time evolution of computed St for case A
(Pr-3. 15)
7.1.5
Time evolution of computed St for case B
(Pr-3.15)
7.1.6
Total heat flux profile
17
in the steady state for
case A (Pr-1.0)
7.1.7
Total heat flux profile
in the steady state for
case A (Pr-1.5)
7.1.8
Total heat flux profile
in the steady state for
case A (Pr-2.2)
7.1.9
Total heat flux profile
in the steady state for
case A (Pr-3.15)
7.1.10
Turbulent
heat flux profile
in the steady state
for case A (Pr-l.0)
7.1.11
Turbulent heat flux profile in the steady state
for case A (Pr-1.5)
7.1.12
Turbulent heat flux profile in the steady state
for case A (Pr-2.2)
7.1.13
Turbulent
heat flux profile
in the steady state
for case A (Pr-3.15)
7.1.14
Total heat flux profile
in the steady state for
case B (Pr-3.15)
7.1.15
Total
heat flux profile
in the steady state for
case B (Pr-10.0)
7.1.16
Total heat flux profile in the steady state for
case B (Pr-50.0)
7.1.17
Turbulent
heat flux profile
in the steady state
for case B (Pr-3.15)
7.1.18
Turbulent
heat flux profile
for case B (Pr-10.0)
18
in the steady state
7.1.19
Turbulent heat flux profile
in the steady
state
for case B (Pr=-50.0)
7.2.1
Computed rms velocity profile in the x-direction
for case A
7.2.2
Computed
rms velocity profile in the y-direction
for case A
7.2.3
Computed
rms velocity profile in the z-direction
for case A
7.2.4
Computed rms velocity profile in the x-direction
for case B
7.2.5
Computed rms velocity profile in the y-direction
for case B
7.2.6
Computed rms velocity profile in the z-direction
for case B
7.2.7
Longitudinal power spectrum S uu for case A
7.2.8
Longitudinal pow er spectrum Svv for case A
7.2.9
Transverse power spectrum SX~V for case A
7.2.10
Longitudinal power spectrum S uu for case B
7.2.11
Longitudinal power spectrum
7.2.12
Transverse
7.3.1
Mean temperature profile in the steady state for
case A (Pr-l.0)
7.3.2
Mean temperature profile in the steady state for
19
S
vv
for case B
for case B
power spectrum S
ww
case A (Pr-1.5)
7.3.3
Mean temperature profile in the steady state for
case A (Pr=2.2)
7.3.4
Mean temperature profile in the steady state for
case A (Pr-3.15)
7.3.5
Mean temperature profile in the steady state for
case B (Pr-3.15)
7.3.6
Mean temperature profile in the steady state for
case B (Pr-10.0)
7.3.7
Mean temperature
case B (Pr-50.0)
7.4.1
Turbulent diffusivity profile in the steady state
for case A (Pr-l.0)
7.4.2
Turbulent diffusivity profile in the steady state
profile
in the steady state for
for case A (Pr-1.5)
7.4.3
Turbulent diffusivity profile in the steady state
for case A (Pr-2.2)
7.4.4
Turbulent diffusivity profile in the steady state
for case A (Pr-3.15)
7.4.5
Plot of time constant
' vs Pr for both case A
and case B
7.4.6
Turbulent diffusivity profile in the steady state
for case B (Pr-3.15)
7.4.7
Turbulent diffusivity profile in the steady state
for case B (Pr-10.0)
20
7.4.8
Turbulent diffusivity profile in the steady state
for case B (Pr=50.0)
7.5.1
Plot of St vs Pr for both case A and case B
21
1.
Transport
of
INTRODUCT ION
heat
surface of a turbulent
or
plays
liquid
a central
(Higbie,
liquid solvents
engineer
sanitation
of
oxidation
the
(Downing, 1960;
is concerned
(Streeter
Ratbun
interested
sludge
in
with
the
spread
Ratbun,
bodies of water
al, 1982), and with
droplets
pass
dioxide
(Matteson
the
through
et
in
from
increasing
treatment
the
plant
1979). The environmentalist
reaeration
through
role in many
al, 1984). The
et
sewage
the
Tchobanoglous,
et al, 1925;
pollutants
1935;
is
the free
to or desorption
the rates of gas absorption
maximising
across
engineers are interested in
fields of engineering. Chemical
turbulent
across
mass
of
streams
and rivers
1977), with how the gaseous
absorption
and
desorption
(Brtko et al, 1976 & 1978; Munz
al,
of
et
of acid rain when water
formation
clouds
by
like sulphur
pollutants
1984).
Mechanical
and nuclear
engineers are concerned with steam condensation rates on
process which is particular
cold, turbulent water, a
important in certain nuclear power plant accident scenarios.
Direct
condensation
heat transfer
which
of vapour on
process,
since
the latent heat can
it
cold liquid is basically
a
limited by the rate at
is
be removed from the surface. More
recently, NASA has focused on direct condensation as a
central problem in pressure control during the handling and
transfer of cryogenic
(such as hydrogen
liquids
under zero-gravity conditions (Merino
and oxygen)
et al, 1980; Aydelott
et al, 1985).
The transport rate is usually limited by the turbulent
transport
on the liquid side,
and
is defined
in terms of a
transport coefficient, KL:
KL
-
j/AC
(mass transfer)
1.1
q/pc AT (heat transfer or condensation)
P
22
where
flux ,
j is the molar
difference
between
or equivalently,
AC
the surface
is
q
the
is the molar
and
the bulk of the liquid,
thermal
temperature difference between the
the liquid, and
heat
capacity
negligible
p
and
of
the
of liquid properties
Lewis
surface
is the
and the bulk of
liquid,
respectively.
Assuming
and
the
parameters which control the
on the liquid side.
in
this
(1924)
consensus
AT
are the density and specific
Despite the significant
published
flux,
in the gaseous phase, KL is a function
resistance
turbulence
cp
concentration
and
area
body
since
Higbie
of work
the
(1935)
that has been
early contributions
to
gas
absorption,
exists as yet on a single universal model,
a single empirical
see the review
in
correlation,
Sonin
for the transport
et
al,
1986).
of
no
or even
rate (eg.
Our work
in this
laboratory has been aimed at developing a better fundamental
understanding
interface,
of the transport
in
the
hope
process
that
this
at a free, turbulent
will
lead
to
a more
universal model for the transport rate. We have developed an
experimental apparatus in which an easily controlled, steady
turbulence can be imposed from below on a shear-free liquid
surface,
and
transport
terms of turbulence
rates
measured
intensity,
and
turbulence
liquid-side Prandtl or Schmidt
number.
careful
dependence
investigation
of
the
correlated
in
length scale, and
Our hope was that a
of the transport
coefficient on these parameters would guide us toward a more
general and accurate model for the transport process.
Our earlier work with this system dealt with vapour
condensation
on a liquid (Sonin
1988; see also Brown,
1989)
coefficient was directly
the fluctuating
velocity
et al, 1986; Helmick
and
et al,
showed that the transport
proportional
to
the rms value of
on the liquid side (at least at low
Richardson number) and to the negative one-third power of
the bulk liquid Prandtl number, approximately. These results
are not explained by any of the available models which might
have been thought to be applicable.
23
(Each of these models
is
based on some
particular
assumptions about the
simplfying
turbulent transport process - see Sonin et al, 1986.)
The objective
advance
most
of
the
art
the state of the
of the available
work
present
in
simple
was
three ways.
models
to
try to
First,
since
for the transport
rate
were based on simplifications appropriate for high Schmidt
number, we embarked on an experimental study of high Schmidt
number
gas
turbulence
in
absorption
conditions
similar
experimental
to
system
at
those of the condensation
this investigation
studies. The results of
are described
in
undertook a more detailed study of
Second, we
section 4.2.
our
the turbulence structure in our experimental system, using a
laser doppler velocimeter.
in large measure
by the
region very close
to
transport
The
of the turbulence
structure
the
rate is controlled
surface,
in the
where measurements
are
very difficult. Our new measurements offer some insight into
the larger-scale
structure
system, however,
and
correlation
of
the
give
data
of the
a
turbulence
in our type of
rational
basis for the
more
(see
4.1).
section
Third, we
attempted a numerical simulation of the turbulence field and
passive scalar distribution near a shear-free surface, with
turbulence imposed from
below.
The
full, unsteady Navier-
Stokes equations and the energy equation are solved using a
spectral Fourier expansion in the two horizontal directions,
and a Legendre
spectral
direction. The
computational
full
flow,
is
neighbourhood
of
representing
element
technique
the
free
the
"extracted" from the
domain,
to
restricted
interface,
surface
in the vertical
a
with
single-macroscale
the
conditions,
top
boundary
and the bottom
boundary simulating the "far-field" isotropic turbulence.
Two statistically steady calculations at eddy Reynolds
number of 100 and 1000 and Prandtl numbers
50 are
presented.
compared to the
Brown (1989).
The
computed
experimental
24
ranging from 1 to
transport
rates are then
steam condensation results of
2.
MODELS OF AND CORRELATIONS FOR SURFACE TRANSPORT
liquid
A turbulent
continually replaced with fresh
bulk. Higbie
direct exposure
into
0 5
(D/r)
reduced to that of expressing
'
2.1
in terms of the properties
r
on the liquid side.
of the turbulence
The above expression
also be derived differently.
can
turbulent
The Reynolds-averaged
equation can be
diffusion
as
/az
aC/at =
where
the liquid-side
is the molecular diffusivity. The problem is thus
D
written
If a typical
elements.
fluid
of
can be expressed as
coefficient
K
L
where
the periods
occur by transient
at the surface,
time
eddy has a residence
transfer
the
as being
of fluid from the
during
transport
to the surface,
diffusion
molecular
that
viewed
be
elements
postulated
(1935)
can
surface
z
is
the liquid,
the
((D + DT)C/iaz)
distance measured
from the interface
is the turbulent
DT(z)
the Reynolds averaged species
2.2
diffusivity
concentration.
into
C is
and
The turbulent
diffusivity is defined by the equation
<
where w'
is
component,
w'C'
the
>
part
fluctuating
C' is the
near
-DTaC/ay
fluctuating
and the symbol <
region very
=
>
the
represents
free
2.3
of
the normal
velocity
part of the concentration
ensemble
surface,
average.
where
For the
the transport
resistance is greatest, Levich (1962) has argued that
DT
z 2 /r'
2.4
25
where
is
T'
a
quantity
statistically steady
with
dimension
solution
of
of
equation
time.
2.2
The
is (King,
1966)
KL - (2/)(D/r')05
Equation 2.1 (with an
2.5
equality sign) is
exact
identical to
equation 2.5 if we make the identification r-rr'/2.
Numerous models have been proposed for relating r to
the properties
of the turbulence on the liquid side. Table 1
(see also Sonin
models. We
et
al,
1986)
model which
assumes
responsible
for
that
summary of five major
proposed the large-eddy
large (macroscale)
fresh
'enriched'
fluid
liquid
to
eddies are
the surface and
into
the
bulk.
The
time scale of these large eddies is given by
r
L
(1967)
the
bringing
the
characteristic
where
a
shall give a brief outline of each below.
Fortescue and Pearson
removing
is
is an
-
L/V'
integral
2.6
length
turbulence intensity near the
scale
and
interface.
V'
is
the
Based on equation
2.1 or 2.5, KL can be expressed as
KL ~ (DV'/L)
0 5
'
2.7
Fortescue and Pearson obtained some experimental support for
their model in an
by a grid in
an
experiment where turbulence was produced
open
channel
across the free surface was
out, however,
'raw' mass
that they
made
transfer data
account for the
entrance
(of
flow, and the mass
transfer
measured. It should be pointed
numerous
the
effect
corrections
order
of
to their
10% - 20%)
to
immediately downstream of
the grid and the end region of the channel, where
positioned to maintain a constant water height.
26
a weir was
Lamont
controlled
and Scott
not
by
(1971) suggested
the
large
that the transfer
eddies,
Kolmogorov microscale size, which
but
by
eddies of
'piggy-back' on the large
eddies that sweep by the surface. The appropriate
r
is then the Kolmogorov
r
0
25
(vD2/L)0
measured
could be correlated with
suggested
the
transport
process.
comparison
of their
where
0
2.8
75
2.9
absorption
turbulent
the
importance
They
of
also
model
rates from small
pipe flow. Their data
energy dissipation
with
Pearson and agreement was off
Subsequently,
V
the
gas bubbles transported in
thus
'
5
as
KL
Lamont and Scott
time scale
time scale, that is
(L/V'3)
and KL is obtained
small
made
the
by
rate
eddies
some
was
used
enclosed vessel, provided
eddy model.
Meanwhile,
sets of available
to
data
Theofanous
models
in
the
data of Fortescue
and
a factor of at least two.
generate
which
experiment
turbulence
in an
supported the small-
et al (1976) made use of several
experimental
large and small eddy
, and
quantitative
Prasher et al (1973), in a separate
an impeller
is
data
to try and combine
into a more generalised
the
model.
They suggested that there was a critical turbulence Reynolds
number
(based
on
the
turbulence
intensity
and
the
macroscale) of 500, below which the large-eddy model is more
suitable, and above which the small-eddy model appeared to
be preferable.
data with
However,
respect
to
considering
their
the large scatter
suggested
correlation
in the
and the
numerous assumptions made to relate the experimental flow
parameters to V' and
L, their conclusions should be taken
cautiously.
27
Earlier,
argued
Levich
(1962) had taken a different
that the length scale
is controlled
surface
that the
of the eddies near the surface
by surface tension and curvature. Equating
tension force with
length
interfacial
tack, and
scale
transport
the
of
the
inertia force, he suggested
the
eddies
which
control the
rate is
L - a/pV'
2
2.10
where
is the surface
follows
that the time scale L/V' of these eddies is
tension
of the liquid, from which
a/(PV'3 ).
r
it
2.11
Levich 's argument yields K L as
J. T. Davies
transfer
and
data for
00.5 5
-(p
3
(pDV'3/or)
KL
collaborators
flows
down
2.12
(1972)
an
have obtained mass
inclined plane, with
and
without roughened surfaces. They also studied gas absorption
between
a free turbulent
jet
of
liquid and a gas. Davies
claims approximate agreement with
Levich.
the model
put forth by
A still different correlation has emerged from studies
of transport at surfaces where turbulence is imposed by
interfacial
shear.
Based
on
Ueda et al (1977) suggested
near such interfaces
their
data,
that the length scale of eddies
is dominated
scaling law, just as it is
own experimental
by the viscous
inner layer
near a solid wall. This suggests
a time scale r of the form
~r
v/V'2
2.13
Upon substitution into 2.1, KL becomes
KL
(D/v)O 5 V'
28
2.14
Finally, an entirely different model was proposed very
early by Kishinevsky
completely
right at
(1955) who
dominated
the
by
surface.
turbulent
In
this
diffusivity,
model,
the time scale r
equation 2.1, and with
that transport is
argued
DT
DT, even
replaces D in
L/V', yields K L as
KL~ V'
2.15
which is entirely independent of the molecular diffusivity.
Kishinevsky and Serebryansky (1956) supported this model
with data from an experiment where turbulence was produced
by a mechanical stirrer operating at very high speed (1700
r.p.m.) below the liquid interface. They acknowledged,
however,
that at lower turbulence
intensities,
the effect of
molecular diffusivity would play an ever increasing role.
Unfortunately, the threshold of turbulence intensity above
which equation .15 is applicable was not established.
The five models
ones that have
given above
been
proposed.
are
They
by no means
are
the only
provided
here to
illustrate the diverse concepts which previous investigators
have proposed
for
claimed some
support
the
transport
from
that other experiments
means
Prominent
examples
oscillating
taken
is created by mechanical
grid (see Isenogle,
flows (see Won
in systems where
stirrers
operating
speed than in Kishinevsky
and Serebryansky
not agree well with
of
experiments
of Davies
any
's
by
1985,
et al, 1982,
the turbulence
at a much
lower
tests, also do
these proposed models
(see the
et al, 1979, and Luk, 1986). For flows
in which turbulence is
above the interface,
are (a) the class
in the liquid is generated
and Ho, 1987) and (b) thin-film
and Bin, 1983). Data
has
experiments. We note, however,
the turbulence
of a submerged
Each model
have yielded data that do not conform
to any of these models.
of flows where
mechanism.
the
generated
data
regimes,
as found for example
Broecker
et al (1984). None
by
a strong wind blowing
show two different
in
Merlivat
et al (1983) and
of the model discussed
29
transport
above is
capable of predicting these regimes. Further development of
analytical models has been held back in large measure
because of the lack
turbulence
of
understanding
of,
or data on, the
structure near a free surface.
From table
1,
we
see
can
that
the various models
suggest Reynolds number exponent from -0.5 to 0.5, while the
Schmidt number appears always to be -0.5 power except for
the fully turbulent model of Kishinevsky. It should be
mentioned that most of these models were proposed for fairly
large Schmidt or Prandtl numbers
were
carried out with gas
as
and most
of the experiments
the transfer
agent to achieve
the high Schmidt number.
Our hope in this work was
if any, of these models
to shed some light on which,
are correct,
and to better
define KL
in terms of easily measured parameters, general enough for
wide application, irrespective of how the turbulence is
generated.
30
EXPERIMENT
3.
3.1 Apparatus for Gas Absorption Rate Measured
Sonin et
al
(1986)
have
shown
that an axisymmetric
confined jet in a cylinder partially filled with water gives
a bulk-flow-free
and
reasonably
free surface, provided
the
than 3D from the
nozzle,
jet
uniform
surface
D
is at distances
turbulence
decays with
nozzle and is
relatively
isotropic
constant
has
the
integral
added
elevation
greater
of the
from the
in a horizontal plane.
characteristic
length scale
at the
being the diameter
cylinder. The
The system
turbulence
that
of possessing
appears
a
to be 'locked'
to D.
Our apparatus was
al.
It consisted
of
a
similar
to
vertical,
the
system of Sonin et
axisymmetric
nozzle
in a
pyrex tube partially filled with water, as shown in figure
3.1.1. A centrifugal pump circulated the water through a
closed-loop
system. The total volume
was kept to a minimum
mass
transfer
experiment,
to minimise
experiment.
the
amount
in the system
the time required
(As
of
of water
in
dissolved
any
mass
C0 2
in
for each
transfer
the system
reached a readily measurable value only after a fairly long
time.) A heat exchanger was used to maintain the bulk water
temperature constant.
The turbulence intensity in the water was controlled
by varying
water
the
momentum
flux
through
the
nozzle
and the
level in the test section (see Sonin et al, 1986). Two
different test section diameters were used, 0.152m and
0.038m. The respective nozzles had diameters of 0.0064m and
0.0016m.
For
the
larger
system,
calibrated against the pressure drop
AP was measured with
a
Kistler
31
the
flow
rate
Q
was
P across the nozzle.
model RP15 pressure
transducer
linked
to
a
Dynascience
Demodulator whose output
voltage
model 3800B multimeter. The
model
can
be
calibration
CD10
Carrier
read from a Dana
is shown in figure
, and is given as
3.1.2
(Q/Dd)/AP0 ' 5 - 0.171 + 5%
where Q
pressure
section
flow rate through nozzle,
is the volumetric
the pressure
drop
in
and
d
smaller system,
D
is
the
the
as
voltage
,
transducer
3.1.1
is
registered
the
flow
of
the
by
nozzle.
determined
rates were
the
of the test
diameter
diameter
AP is
In the
using a
calibrated Fisher Rotameter inserted in the flow loop.
CO2
,which
measurements,
was
was
introduced
exhausted at a higher
build up of any
used
just
noncondensable
out
flushed
CO 2 coming
into
test
turbulence
at the interface,
reduce inflow velocity.
and after
still before
order to ensure
section
The
the
CO 2 , would
tend in
CO2 and maintained
In
a
air, next to
the test section. The gas
with
pressure.
and
aim was to prevent the
than
of
transfer
the interface
gases, notably
psi above atmospheric
the
the mass
above
lighter
any case to be exhausted
continually
all
station. Our
the interface. Air, being
space was
in
did
at 1
that the
not induce
'extra'
large inlet tube was used to
interface
remained perfectly
valve controlling
the supply of
CO2 was turned off. Also,
the incoming CO2 was preheated by
passing the
a
gas
water-glycerol
temperature
through
mixture
bath
as the bulk water
copper
so
tubing,
that
in
it
immersed
was
in a
at the same
the test section.
In this
way, we were able to maintain a gas temperature within
+0.5°C of the water temperature, thereby ensuring a known
gas temperature at the interface, which was important since
the
saturated
dependent
concentration
on temperature
of
(which has
CO2
in
water
a variation
for every degree Celsius change in temperature).
32
is highly
of about 3%
For each test, the system was
+
I
h
I() mil+*a
e
±L VI
I
LJ
±
tf'
r-L t A.L1
16&
LLLL
steady operating condition.
each) of the bulk
initial average
water
desired
being
the inlet nozzle. The
system
CO2 .
The water
to the
interface
of
the
above
and run for a
restarted
to 12 hours, depending on the turbulence
time ranging from
imposed. At the
taken for
th e height
s
(50 ml
taken to determine
reduced p recisely
the n
wa
samples
dissolved
of
was
zs/D level, z
intensity
t hen
were
cell
different
Five
run for
to
to reach the desired
i nflow
I
concentration
level in the test
first allowed
determinating
The amount of dissolved
run, 7 samples were
end
of
the
a verage final concentration.
C02
the
the samples was determined
in
using an Orion Research
02)
which
has
an
carbon d ioxide elect rode (model 95accuracy
of +5%. The electrode was
calibrated against two known samples of
(supplied by Orion Research); the output (in
measured with a
electrometer. In
through
very
high
the
hydrogen
ions
electrode
acc ording
in
CO
and
the
internal
filling
to
the reaction
+
H20
H
+
+H
The pH in the internal
the hydrogen ion
E
O
affects
diffuses
the level of
solution
of the
3.1.2
solution
is then related to
3.1.3
electrode
and
S
calibration mentioned
response
E = E
potential,
E
o
is the
hydrogen ion
is the el e ctrode slope obtained by
above.
By virtue of 3.1.2, the
reference potential (a constant),
electrode
CO 2
+ S ln[H + ]
is the measured
concentration
Ke i thl ey 602
HC3-
filling
wa s
concentration by the Nernst equation
E = E
where
dissolved
membrane
C02
llivolts)
inLput-impedanc
electrode,
a gas -perme a ble
dissolved
[H
+]
is the
is al s o Nernstian,
to the C02
+ S ln[CO 2]
33
3.1.4
where [C 2 ] is the
electrode were
CO 2
Calibration of the
concentration.
twice; first for the measurement
carried out
of the initial samples and second for the measurement
of the
final samples.
The governing equation
the liquid volume
transfer of C02 into
the
for
of the test system is
VdC/dt - KLA(Cs-C)
where V
is the volume of water
area of the interface,
in water,
C
is
the
in
the system, A
is the bulk concentration
C
saturated concentration
t is the time in seconds.
Upon
integrating
the
of C0 2
of CO2 at the
and
3.1.5, we have,
is constant,
KL - V/(At) ln((Cs-Ci)/(Cs-Cf))
where
is
the mass transfer coefficient (m/s),
interface, KL is
if K L
3.1.5
C i is the
initial
final concentration
the run. With
C i and
concentration
of CO2 and
Cf
t
measured,
of
3.1.6
CO,2
Cf is the
is the time duration
equation
of
3.1.6 gives the
mass transfer coefficient KL . After each run, the system was
flushed thoroughly with tap water before a new run is
initiated.
3.2 LDASystem for Velocity
Measurement
The turbulence in the system was investigated using a
back-scattered two-color
LDA.
Both
the vertical and
horizontal components of velocity were measured. The LDA
consisted
of a Lexel model
95
ion
laser as the source and
DISA-made optics, namely, 55X20/21 cover and retarder, 55X22
beam splitter, 55X28 beam
beam waist adjuster, 55X25
displacer, 55X29 Bragg cell, 55X32 beam translator, two
focal length achromatic
310mmnn
55X12 expanders, and a 55X57
34
front
lens.
The
receiving
optics
consisted
of
a 55X31
pinhole section, 55X34 PM optics, 55X30 backscatter section,
55X35 color separator,
55X36 and 55X37 interference
Signals from the PM and the Bragg cell were
frequency
shifter and 55L90a
counter was linked to
via
a
DEC
a
DRV11-J
Finally
the
portable DEC MINC-11 mini-computer
parallel-line
which
The internal oscillator
fed to the 55N10
counter processor.
interface. Data
collected by a series of assembly-code
the mini-computer,
filters.
in
routines
was
addressed
to
turn drove the interface-card.
of the computer was used to time the
arrival of each validated doppler signal for autocorrelation
purpose.
A window
of
specially machined
flat
plexiglass was
slot in
the cylindrical
shown in figure 3.2.1. The
was
slot
measured
had a radius of
test section,
53nim in width
as
and
76.2mm. Provided velocity measurements were
too far away
from the centreline,
of the cylindrical geometry was
beam expanders
into a
2mm from the axis of the test section, which
located at
not made
attached
in series
the perturbation
minimial. The usage of two
resulted
in
very small 40Am x
a
401Amx 600/m measuring volume.
The test section was mounted
on a
provided
traversing
orthogonal
plane.
table
which
linear directions
Instead of adjusting
and
movement
rotation
in three
on a horizontal
the laser beams for intersection
at the various position
in
the test section. This
left
the
flow field, we would
shift
the optical system stationary,
and avoided frequent optical realignment of the laser beams.
The centrifugal
specially made
absorbers
sound-proof
to minimise
LDA. It was connected
long stainless
pump for the test cell was
any
to
box
and
rested
transmission
the
housed
in a
on vibration
of vibration
to the
test section by an eight feet
steel flexible
hose.
Prelimary
tests of the
LDA system were carried out with a plexiglass disk filled
with water rotating at a known angular velocity, mounted on
the test cell stand. Laser reading of velocity were
without the pump running,
and
35
were
recorded
found to check against
the actual velocity
velocity
to
fluctuations
5% of the mean.
+
within
was also
Subsequently,
3%.
The rms value of the
noted
and found to be about
tests
were
carried out with
the centrifugal pump running. We found that the rms velocity
fluctuations did not differ significantly with or without
the pump running;
this implied that the precautions
taken to
minimise pump vibrations were successful.
Seedlings for the back-scattered LDA were provided in
the form of aluminium
signal-to-noise
particles
ratio
of
of size 3
about
5.
of size 1 Am
but
found
information
about
the
good. More
is
particles
We
that
m. This gave us a
also
tried
TiO 2
the SNR was not as
LDA system and its set-up
given in Appendix 1.
3.3 Velocity Measurement with High Speed Video Camera
System
Sonin et al (1986)
jet apparatus
to
were
study
liquid surface. They
the
transport
first to use a confined
across a free, turbulent
characterised
the interface by extrapolating
the
rms velocity V' at
the value of V' from the bulk
region to the surface.
The
using
spheres (which have a small negative
3nmm polypropelene
velocity measurements were made
buoyancy) as 'seedlings' for
system. The video
a grid-covered
tapes were
monitor
to
the
high
then replayed
determine
speed video camera
frame by frame on
the displacement,
and
vertical and horizontal components of particle velocity were
computed.
Sonin et al also
used the k-e model
to
an
analytical
give
turbulence
approximate
in the
region
the inlet nozzle) where
compared to the
turbulence
the
mean circulatory
energy k
reduced the high-Reynolds
0
>
fluctuating
kinetic
d/dz(C/O
k
k
description
z/D
(z is
of
the distance
the
above
flow is small
velocity. Assuming the average
is
number
SLk
3
of turbulence
a
of z only, they
form of the k-e model
dk/dz)
36
function
- kl
5/Lk
to
3.3.1
LkE
where
is
coefficients
model
k-e
= 0.09
are C
oo and to
the
length
scale,
ok = 1.0.
and
3.3.1, subject to the
equation
as z -
the
boundary
the
and
The solution
condition
of
0
of k -
that the length scale LkE
assumption
is 'locked' to the tube diameter, D,
Lk_E
is
where
where
k
o
is
a constant,
k 0.5 = k
05
o
3.3.2
=
1(a /6C)
exp(-
5
k
is the magnitude
at z=z0 .
(z/D
Using
z/D)
z0 /D)
3.3.3
arguments based on
dimensional analysis, Sonin et al further obtained for their
system,
V'
where
Re s is the
and
V'
vertical
is
3.3.4
= Q/Dd F(Res, z/D)
syst em
the
rms
direction
number, defined as Q/d,
Reynolds
velocity
s ince
it
in either the horizontal
was
found
that
flow
or
is
z is
plane,
provided
essentially isotropic in a horizontal
not too close to the interface. Equation 3.3.3 suggests that
the dependence
et al
were
on z/D should
able
fit
to
be an exponential
decay. Sonin
experimental
data to the
their
function
V' = Q/Dd
for
where
(Re ) -
(Re
S
) e -1 .2z/D
3.3.5
3.1 < z/D < 4.2
21.8
for
Re
as in figure 3.3.1
for
Re
> 25000
S
< 25000
and
Lkc
- 1.1D
37
3.3.6
The data used in the
25000
were
diameter
correlation
obtained
0.0381m.
the
smaller
Subsequently,
measurements
of
section
diameter
of
using
V'
for
Re
Brown
<
s
15000
0.038m
(Re )
function
and
<
for Re
test-section
(1989) made
with
of
more
the same test
essentially
the
same
experimental procedure as Sonin et al. However, he used much
smaller (0.1-1.mm diameter) polystyrene beads as seedlings,
compared to the 3mm polypropelene spheres used formerly, and
a nozzle with a smoother inlet region. Brown 's data is
summarised
in
smaller value
figure
3.3.2,
(Re )
of
and
than Sonin
shows
a
substantially
et al found in the lower
Reynolds number range. Brown 's data should be more reliable
because of the smaller size seedlings, which would follow
the fluid motions
size of 200
readings
more
readings
in Sonin
considerable
constant
al.
in
versus
the
the larger sample
sample
Nevertheless,
Brown
's
data.
size of 60
there
If
is still
we
a
assume a
obtained
by least square fit. This would
the LDA results (refer
to figure 3.3.2 and see section
4.1.1 for more details)
somewhat
LDA results, which
tend
data,
and
(Re s) through the whole range, 4000 < Re s < 70000,
a value of 23.5 is
make
each
et
scatter
faithfully,
show
0(Re8)-21.8,
we
better
which
range 30000 < Re
data shows that
The
to trust more than the video
agreement
with the correlation of
is the least square value for data in the
< 70000.
one
A closer examination
cannot
O(Re s) increases slightly as
are available,
lower than the average.
especially
rule
of Brown
out the possibility
's
that
Res decreases. Until more data
range 15000 < Re s < 30000,
in the
we shall adopt as our correlation
O(Re ) - 24.5
(
= 21.8
The value 24.5 is
ReS < 15000 )
( 30000 < Re
3.3.7
< 70000)
the least square fit in the lower Reynolds
number range.
38
It should be
noted
set-up) was only capable of
and
5mm
beneath
the
LDA (in our particular
the
that
providing measurement
interface
for
the
up to 2mmn
horizontal
and
vertical components of velocity respectively. To measure the
surface velocity directly, we mounted the video camera
vertically facing the interface from above. Air bubbles were
introduced into the system. They floated on the interface
as very bright
and appeared
the synchronised
synchronised with
conveniently
resolution.
lightings
the
addressed
Only
the
provided.
frame
speed
the
problem
at
bubbles
bright spots in contrast
due to the reflection
spots
to
the
lightings were
(The
of
the
of
camera.)
depth
interface
those beneath
of
of
This
field
appear
as
the interface. A
transparent 'Scriptel' digitizing tablet was mounted against
the monitor
during the
playbacks
and
the positions
of the
bubbles were recorded directly via the tablet through its
interface to the laboratory VAX 11/750 computer for further
processing. Each "data point" of rms velocity was based on
about 200 measurements.
These
results
next section.
39
are discussed
in the
4. RESULTS AND DISCUSSIONS I
4.1 Turbulence Characteristics
4.1.1 RMS Velocity Correlation
Both horizontal and
vertical velocity components
the
were measured with the LDA near the interface. Figures 4.1.1
the axis of the test section to
show data taken at
to 4.1.6
depths up to 30nmm. The
z/D-3.67, where
z is the
are shown
in
figures
of
and D is
the nozzle
above
height
diameter of the test-section.
kept at a height
was
interface
The horizontal rms velocities
4.1.1,
4.1.3 for three
and
4.1.2,
- Q/(dv) = 29140, 40120, and
system Reynolds numbers Re
57690 respectively. The corresponding plots for the vertical
rms velocities are given in figures 4.1.4, 4.1.5, and 4.1.6.
Each rms value is computed from at least 5 samples, with
each sample composed of 10,000 data points. Also shown as a
the correlation proposed by
broken line in each figure is
Sonin et al
(1986)
equation 3.3.5 with
on
based
(Re ) -
an
21.8 ) for points sufficiently
by it.
far below the interface to be unaffected
Overall
correlation
locations
our
of Sonin et
far from
the
show
data
LDA
al
( see
independent method
(equations
the
3.3.5 and 3.3.7) for
is to be expected
This
interface.
with
agreement
since that correlation was based on the data taken far below
the interface. Nearer the
rms velocity
to be damped and the horizontal
increase due to the
to the
horizontal
4.1.6,
the presence
rms velocity
interface, we expect the vertical
transfer
of momentum
Based
component.
on
rms velocity
to
from the vertical
figures
4.1.1 to
of the free surface starts to affect the
at a depth of about 0.13D,
irrespective
of Re .
We shall call this layer the 'damped layer'.
Figure 4.1.7
beneath
the
shows the
interface
azimuthal
as
a
40
rms velocity
function
of
at 5mm
radius
for
Re -4.01xl104. As expected,
as one approaches
the wall.
of the centreline
value
Also shown
as
a
in
(bounded by +0.01 m/s)
fluctuating
velocity.
4.1.7
of
and more dramatically
is
r/D.
As
the mean azimuthal
expected,
its value
is much smaller than the rms value of
This
is
also
and
the test points
4.1.1
the +0.01
to about 85%
at 0.SD.
velocities, both horizontal
in
value tends to decrease
r0.4D,
figure
function
rms
It drops gradually
at
between 0.4D and the wall
velocity
the
figures
true
for
the
mean
vertical, measured for all
to 4.1.6. All were
below
m/s bound (not shown). This supports the existence
of an essentially
bulk-free-flow
test region, as postulated
by Sonin et al (1986).
The mass transfer across
will
depend on the
side. We
assume
macroscale,
a turbulent liquid interface
turbulence
that
the
.
For
and
imposed on the liquid
is
latter
an rms velocity,
kinematic viscosity
state
characterised
the
by a
liquid density
and
the characteristic velocity, we
take the value of V' extrapolated
surface (see figure 4.1.1 to
from the bulk to the free
4.1.6). For the macroscale,
we
choose the quantity A defined in section 4.1.3 below.
4.1.2 An attempt
to measure Surface RMS Velocity on the
Free Surface directly
An attempt was made to determine the actual turbulent
velocity fluctuations on
the
free surface itself by
measuring the motion of air bubbles floating on it. Air was
introduced
into the
test
cell
at
a
depth near the inlet
nozzle. The bubbles were broken up into much smaller ones by
the
eddies
as
they
rose
towards
the
free
surface. We
recorded the movement of these bubbles using a video camera,
limiting the range of bubble
in diameter,
sizes
to between
to obtain the rms velocity
v"-[
< (v 2 + ve2) >
41
1 mm and 5mmn
v" defined
1/2
as
4.1.1
where
vr
is the radial
component
of
velocity,
component
tangential
ve
of velocity,
Figures 4.1.8, 4.1.9 and
4.1.10 show the plot of v" against Q/Dd for z/D and 4.5
respectively.
Each
point
average of 200 measurements.
solid line
representing
data are close
them
to
lie
to
Also
close
represents
3.67, 4.17
an ensemble
shown on each figure
equation
equation
>
<
and
represents the ensemble average.
is the
is a
3.3.5. Surprisingly, the
3.3.5,
though we would
expect
to
the
LDA-obtained
horizontal
velocity extrapolated to
the
surface
multiplied by a
/2- to
factor of
and
account for the two-dimensional
of the rms value
of
bubble
motion,
dimensional component measured
rms
measurement
as opposed to the one-
with
the
LDA. We have no
explanation for the low v" observed except the most obvious:
i.e., one may question whether the air bubbles truly follow
the fluid
motions
bubble motions
research
on
the
free
at the interface
topic and we will
surface.
The analysis
and its dynamics
of
is still a
have to defer the explanation
to
future work.
4.1.3 The Turbulence
Since our
Macroscale
measurements
were
restricted
to a single
point at any given time, we were unable to determine the
more conventional macroscale which is based on the two-point
autocorrelation function. However, we can define an Eulerian
time macroscale
-
0
00
Ruu(t) dt
or
0
RW(t)
dt
4.1.2
where
R
(t)-
R!r(t)-
< u'(t'+t)u'(t')
> /
<
>
'(t'+t)w'(t')
42
< u2
/ < W' 2
>
4.1.3
are the Eulerian
w' being the
components,
autocorrelations
horizontal
and
respectively,
the ensemble
defined at a point, u' and
vertical fluctuating velocity
and the symbol
<
> represents
average. Based on this Eulerian
time macroscale
and the local rms velocity, we can define an Eulerian
length
macroscale
A
Figures
4.1.11,
V'r
4.1.12,
4.1.4
4.1.13
and
autocorrelation correlation
function
centreline
30mm
at
a
depth
of
4.1.14
R uu (t),
for
show
the
taken at the
four system Reynolds
number, Res -25000, 35200, 50300 and 70500. Figures 4.1.15,
4.1.16, 4.1.17 and 4.1.18 show the corresponding R (t) for
the vertical component
of velocity respectively. Each
autocorrelation plot was derived from four samples of 10 4
data
points
specifically
each,
collected
adjiusted the
'combined mode'
so
that
at
Doppler
each
about
100
frequency
counter
Doppler
frequency data after validation.
burst
Hz.
We
to the
yielded one
Table 2 sumnarises
all the
Ct) and R (t), r and A. The data imply
that the system possesses a Reynolds-number invariant length
results for both R
macroscale
(large eddy size),
A - V'r
In
what
follows,
characteristic
size
we
of
0.24D
4.1.5
adopt
this
the
turbulent
length
number based on A as the eddy Reynolds
scale
as
the
eddies and Reynolds
number.
4.1.4 ulerian TimeSectrum
Based
vertical
on our
velocities
measurement
with
to examine the behaviour
respect
of
both the horizontal
to time, it is interesting
of the corresponding
43
and
Eulerian
time
spectrum. Using the definition of the Eulerian time spectrum
(e.g. Tennekes
where
and Lumley,
u(f)
-
(f)
-.
f1O
1972),
exp(-i2rft) Ruu(t) dt
4.1.6
exp(-i27rft)
Rw(t) dt
f is the frequency
in Hertz, we plot the Eulerian
time
spectrum/energy spectrum for the horizontal and vertical
directions, in figures 4.1.19 and 4.1.20 respectively. The
system Reynolds number (Res ) is 25000. The corresponding
plots for the
Re s -
horizontal
50300
are
respectively.
and
depicted
All
vertical
in
energy spectrum
figures
4.1.21
and 4.1.22
the four figures show that at sufficiently
high frequency, the spectrum
decays approximately as f/
This follows
of
inertial
the
prediction
subrange,
as
turbulence
oscillating
away from
is
the
discussed
(1972). The experimental
whose
for
result
generated
3.
spectral decay in the
by
Tennekes
of
Hannoun
by
means
and Lumley
et al (1988),
of
a submerged
grid, gives similar decay rate in the region far
the
grid,
where
turbulence
is
expected
to be
homogeneous. Also, for each system Reynolds number, both
components of energy spectrum are quite identical, which
confirms
again
isotropic
in turbulence,
that
our
flow
is
provided
one
indeed
is
approximately
not too close to
the interface.
4.2 Mass Transfer Rate
All mass transfer data were
transfer
agent. Schmidt numbers
obtained using CO2 as the
of
230,
377, 525 and 1600
were used. The lower three Schmidt numbers were obtained
simply by operating at bulk water temperatures of 400 C, 290°C
and 23°C respectively. The molecular diffusivities of C0 2 in
water were
taken from Davidson
44
et al (1957) and Hayduk
et al
(1974),
while
the
condition
saturated
of
mixture
solvent. These
the
at
al (1943). To achieve Sc=1600,
et
and
glycerol
C02
3.1.6) were taken from Cox
interface (required in equation
et al (1962) and Harned
of
water
liquids were
used as the
250°C was
at
pre-mixed
a
in the proportion
of
glycerol. After each mixing, samples were
taken to determine the kinematic viscosity using a 'CannonFenske-Routine' viscometer placed in a precision temperature
21% by volume of
controlled
viscosity
74944, manufactured by
(model
bath
Precision Scientific Company) maintained at a temperature of
250C + 0.1°C. The density
the mixture was also measured
of
coefficient and the saturated
directly. The diffusion
condition of CO, at the mixture interface were obtained from
Brignole
et
al
(1981).
After
each
run,
the mixture was
discarded and the whole system flushed with tap water. A new
mixture was
repeated
then made
to obtained
for
the
the
data as described
relevant
The mean Schmidt number of
the parameters
next run, and the procedure
1600 has a
are tabulated
above.
10% uncertainty. All
in Table 3.
Results for KL, the mass transfer coefficient, in m/s,
versus V' are plotted
dimensionally in figure 4.2.1, 4.2.2,
for the four values
4.2.3 and 4.2.4 respectively,
most
features of
significant
of a 'break', which
has not been
previous
by
used a single power law
their dimensional
or
kind or another have,
(1976),
al
involving
McCready
the
plots is the appearance
these
separates two linear regions.
noted
non-dimensional
et
generation
function)
data.
shall defer
discussion
and
al
(1984)
of
turbulence
of
to fit all
Breaks
of one
been noted by Theofanous et
in
these
experiments
strong wind
by
blowing above the interface (eg. see Merlivat
We
This break
researchers, most of whom
(or polynomial
however,
of Sc. The
et al (1983)).
other data to section
4.3.
If V
is the
rms
velocity
corresponding
to
the
'break'
point, we expect that
V* = V(D,
p,
45
, A)
4.2.1
where
D
of
liquid,
the
is the
A
and
molecular
diffusivity,
is the kinematic
is
the
macroscale
p
is the density
viscosity
of the
length, as defined
liquid,
in 4.1.5.
Dimensional analysis then shows that a break Reynolds number
(ReA)*VsA/v must be a function only of the Schmidt number,
ie.
(Re A ) * = Vn/v
This
scaling
law
suggest
inversely proportional
is constant.
a
set
of
diameter
that
to the
the
'break' velocity
is
A if Sc
turbulence macroscale
In order to check this conclusion, we performed
mass
temperature
4.2.2
= f(Sc)
of
transfer
23 0 C
D -0.038m,
experiments
(Sc -
525)
four
the
associated
the smaller
with
a
bulk
water
in a smaller test cell with
times
system. Using
at
relation
smaller
4.1.5,
test
cell
than
the
the former
macroscale
A2
would be four times
smaller than the macroscale A1 (0.037m) associated with the
larger system. Equation 4.2.2 would imply that the break
veloci ty (V*)2 associated wi th the small system should be
four t imes larger than the corresponding break velocity of
the
large
system
Figure
at
4.2.5
the
same
shows
Sc hmidt number.
the
plot
D s -0.038m. The relationship is
the
break
velocity
of
linear,
large
the
of
KL
versus
V' for
and
passes
through
system,
V0.073
shown more
clearly
m/s,
without a break. This feature
is
following figure, 4.2.6.
scaling analysis thus appears
Our
to be confirmed. We
were
small system at
beyond
not
able
in the
to obtain data for the
it s
break velocity because the
interface became quite wavy before such high velocities
could be reached.
Throughout
all our experiment, we
deliberately kept the turbulence intensity below the value
where
or
the induced surface waves
a few mm high. The
of the small system
than
the
gradient
has
corresponding
a
had amplitudes
of more
of the linear profile,
somewhat
large
46
than
dKL/dV',
higher value (by 20%)
system.
By
using
similar
in
as employed
scaling arguments
Sonin et al (1986), we can
in simplest form, that
established,
A, v, D, p)
KL - KL(V',
L
4.2.3
whence, upon dimensional analysis, we obtain
St -
4.2.4
St( ReA, Sc )
St is a Stanton number,
where
St -= KL/V' = j/V'C,
and ReA
is the eddy Reynolds
number,
ReA - V'A/u.
concentratiion difference
AC is the molar
and
the
bulk
velocity,
our
implies that
liquiid.
the
of
data
St
KL/V'
indep endent
velociti,es
implies that, for
of
seen in
figure
uncertainty
4.2.6
involved
the surface
constant, which
This,
in turn,
dKL/dV'
for
should be the same as for
The apparent difference
be
possibly
can
and
below the break
the break,
4.2.3).
(figure
system
is
ReA.
below
the smaller system (figure 4.2.5)
the large
between
At velocities
that
shows
is
the interface
t:ransport across
flux
j is the molar
Here
attributed
to the
O(Re ) calibration (see section
in the
3.3).
With
try
to
tes ts
data from four
establish
empirically.
the
with
funct ion
f(Sc)
in
equation 4.2.2
Table 4) that the break
(see
The data suggest
diffe rent Sc, we can
point Reynolds number is given by
(ReA), * 3.5x104 Sc0for
230
<
Sc
47
<
4
1600
4.2.5
where
the break
intersection
4.2.1
point
point
(denoted
of
the
by
two
*)
is
defined as the
linear regions (see figures
to 4.2.4).
Although equation 4.2.5
is
derived for the indicated
range of high Schmidt number, we obtain some idea of whether
it
might
referring
apply
also
much
lower
Schmidt
number
to the data of Sonin et al (1986) and Brown
for condensation
range
at
heat
transfer
1.5 to 6.6. These
at
Prandtl
numbers
by
(1989)
in the
data show a linear relation between
KL and V', with no break
point, for rms velocities
as 0.15 m/s. This result
is at least consistent with 4.2.5,
which
is
predicts
1.4x104 ,
numbers
that at Sc-10,
which is
in the
much
as large
say, the break Reynolds
number
higher than the maximum Reynolds
experiments
described
by
Sonin
et al and
Brown.
In principle
observed
in
the
our
data
possibility
might
be
contamination with monol ayers or
might
then
signal
exists
the
result
of
surface
even dust. The break point
condition where
the
that a 'break'
the
turbulence
intensity becomes strong enough to actual ly b reak up the
surface contamination. Contamination migh t ar ise
in our
system from oil leakage from the pump, etc. J. T. Davies and
co-workers
have
done
extensive work on gas absorption
through monolayers and
cause a profound
a monolayer
transport
resistance"
effect. Davies
acts not
at the
that monolaye r res istance
found
only
interface,
et al
(1964)
post ulated
a direct physical barrier
as
but
can
that
to
also offers "hydrodynamic
by damping out the eddies
in the vicinity
of the
interface, and thereby causing a drastic reduction in the
transport rate compared to the clean surface condition.
Their experimental data showed that the greatest reduction
in transport rate occured when the turbulence intensity in
the liquid was fairly low, which suggested that the cause
was
the inability
of the surface-clearing
eddies
to break up
the surface film. Above a certain turbulence intensity, the
corresponding percentage reduction in K L became smaller. At
48
very high turbulence intensity, the effect of monolayers was
reduced to
the
transport.
layer's
resistance
(physical
barrier)
to
They found that the effect of even a small amount
of contamination,
of order
one
ppm
could be measured
with
suitable contaminants.
In view of this, we carried out a series of experiment
designed to check if there was any change in surface tension
before and after
generally
the
mass
transfer experiment
alter the surface tension). Care was
(monolayers
taken to test
only water taken from very near the interface. Our Cenco
tensiometer registered at most a + 0.0001 N/m variation in
the surface tension of the water, which was measured as
0.0728 N/m (200 C) irrespective of the test conditions.
Orridge
(1970) has studied the
sulfosuccinate as surfactant
plane. He registered
15%
a
with accompanying 0.0011
effect
on
the
of 0.1 ppm of sodium
flow down an inclined
in CO 2 transfer
reduction
N/m
decrease
rate
in surface tension.
Our maximum + 0.0001 N/m variation in surface tension thus
suggests
the
absence
of
significant surface-seeking
surfactants in our test. In addition, Orridge 's result also
shows a maximum
the
reduction
surfactants were
accompanying
Typically
of
50% in the transport
increased
decrease
in
to
surface
tension
the slope dKL/dV' increases
much greater factor -
at least
4
10
-
rate when
ppm,
of
with
an
0.009 N/m.
in our experiment
by a
as the break point is
traversed. This suggests that monolayer contamination is
most likely ruled out as the reason for the 'break' observed
in our data.
As discussed in chapter 2, most models for interfacial
transport suggest that Stanton number, St, is proportional
to the
-0.5
power
of Sc,
i.e.
St oc Sc -0.5
4.2.6
the only exception being Kishinevsky model which predicts an
independence
of Sc. This provides
49
us with
the motivation
to
those data with V' <
dKL/dV',
or slope
is
dependence
if Sc -. 5
ascertain
reasonable
at least for
V*. Table 5 gives the Stanton number,
for
<
ReA
(ReA)*.
The
data can be
correlated well with
4.2.7
< (ReA)* ]
[ Re
0.0092 + 0.0018
StSc 0.5
A non-dimensional plot of StSc 0 5 versus ReA (figure 4.2.7)
indicates the approximately constant regi n of about 0.0092.
region VI
For the 'post-break'
V,,
we plot (KL K,)/(V'-V*)) against ReA
in
data
in
tha t
region
disproportionately
results of figure
Stanton
number
constant,
Sc.
It
due
can
or at mo st
would
to
4.2. 8
approximated
weak
correlated
StSc05
summarise,
of
our
effect
as
V'
The
>> V,
the
roughly
being
function of both ReA and
and Ser ebryansky's (1956)
by their
1700 rpm to generate
about
da ta
felt
V'-V,.
region, as evidenced
data lies in the 'post-break'
very high stirring rate
turbulence.
f or
that
suggest
very
its
denomin ator,
that Kishinevsky
is likely
To
makes
the
be
a
those
excluding
region, as any slight scatter of
data very near the 'bre ak'
the
4.2. 8,
figure
on
mass
transfer
the
can
be
as
~ 0.010 + 0.0018
[ ReA
< 3.5x104Sc-04
1
4.2.8
and
(KL-K*)/(V'-V*
)
0.0020+0.0003
[ReA>3.5x10 4 Sc 0.4
4.2.9a
is
viewed as an approximation.
where equation 4.2.9
Equivalently, using 4.2.5 and 4.2.8, 4.2.9a can be expressed
as
St ~ 0.0020 + (0.010 Sc
for ReASc
0
- 0.0020)(ReA)*/ReA 4.2.9b
'
> 3.5x10
50
4
Note that
up
to
this
point,
the
velocity
define St is the nominal rms velocity, which
is taken at the
centreline
of the test-cell. The actual turbulence
decreases
as
the
radial
If we
use
a
4.1.7).
distance
simple
smooth
4.1.7, we find that the mean velocity
the centreline
St and ReCA on
velocity.
the
This
used to
increases
curve
intensity
(see figure
fit for figure
is about 25% less than
implies
that if we base both
local rms velocity,
equations
4.2.8 and
4.2.9b become
StSc0.5 ~ 0.013 + 0.002
St
4 [ ReA<2 . 7 x104Sc
0.0025 + (0.013 Sc 0
5
2.7x104 Sc-0
Equations
8000
4.2.10(a)
and
functional
230 < Sc
and (b) are
<
form of equations 4.2.10(a)
four Schmidt numbers as in
and 4.2.11 show respectively
Sc of 525 and
1600
with
as
4
4.2.11
applicable
1600. Plotted
4.2.10(b)
4
and the break Reynolds number, (ReA)*, is given
(Re),
4.2.10(a)
- 0.0025)(ReAd)/ReA
ReA > 2.7x104 Sc= 0
for
]
for 500 < Re A
<
in figure 4.2.9 is the
and (b) for the same
our experiment. Figures 4.2.10
the experimental data for the
the prediction
given by equations
4.2.8 and 4.2.9 for comparison.
4.3 Comparison with Other Results
One feature
correlation
is
the
that
fact
stands
that
out
there
in
our transport
are
two
rate
regimes of
behaviour, separated "break
point" characterised by a
Schmidt-number-dependent critical Reynolds number. Most of
the previously published work show no such break point. The
51
burden falls on us
explain the apparent discrepancy.
to
the literature,
for the data in
and
the parameters V'
of deducing
here is that
basic difficulty
so
The
that previous
data
the present data. The means
can be rationally compared with
: vertically
numerous
are
turbulence
generating
oscillating grid, flow down an inclined plane, wind driven
shear flow, falling film flow and many others. Comparison is
possible only with those data where empirical relationships
of
the
exist for converting
flow parameters to V'
respective
and A.
We
have fallen
investigations
previous
other of the two
of most
either one or the
by the break Reynolds
The discussion will be divided
number (see equation 4.2.3).
The
data
the
into
demarcated
regions
into two sections.
that
show
to
attempt
shall
will
first
cover those experiments
where turbulence is generated within the liquid and diffuses
towards the interface.
These
are
in in Table 6.
summarised
cases where turbulence is imposed
The second will deal with
from above the interface. These are sunmmarised in Table 7.
these
Before we embark on
worth
out that the two-regime behaviour
pointing
or
(1976)
that
with KL
(V')1/2
McCready
and
induced
postulated
found,
Hanratty
in
channel,
changed from a (V')
V' dependence
by
McCready
a
at Re >
50,
2
in
and Hanratty
at Re
break
(V')
at lower Re and KL
turbulence
coefficient
obtained
et al
(1984). Theofanous
in our data
that postulated by Theofanous
from
is distinctly different
et al
comparison, however, it is
3/ 4
500,
at higher Re.
experiments with wind-
that
the
dependence
mass
transfer
at Re < 50
to a
(ie. they found a very different
behaviour from that suggested by Theofanous et al). Our
break is different from both of these. It occurs at much
higher Reynolds number, and separates two linear regions of
KL versus V'.
4.3.1 Turbulence Imosed
from Within the Liuid
52
First
inclined
consider
the
channel flow
turbulence
is
channel flow without
similar
to
obtained
in
shear. This type of
ours
in
that
the
Uw
is
the
(1984)
have
indicated
that for a
surface shear,
au
u
u
surface
data
imposed from beneath the interface.
Plate and Friedrich
where
transfer
without
generation
turbulence is
mass
rms
Uw
4.3.1
0.1Uw
4.3.2
value
of the horizontal
turbulence
component at 1 cm beneath the
surface, Uw
is
velocity,
mean velocity
in the channel.
and
They managed
U
to get
data of Krenkel
which
the
an
et
empirical
al
(1963)
h
is
-
and
1.78x10-5 Uw
the height of water
the hydraulic radius.
and (Sc)
Taking
Eloubaidy
et al (1969),
h/R
in
the channel
and
R
is
- 2h for a wide channel
R
500, we have
(KL/V')Sc0
~ 5
which
fit for the experimental
can be expressed as
K
L
where
is
w
the friction
0.020
is identical to our 4.2.10(a)
differ by a factor of
some possible
about
1.5.
4.3.3
except
We
that the constant
have to keep
error in using the approximation
in mind
4.3.1 for the
comparison. Noting that the largest shear velocity and water
height were U,w0.038m/s
and h-0.145m respectively, we have,
on estimating
Appendix
the macroscale
2 for a detailed
Ah
in
a channel as
O.lh
(see
derivation),
[(V'Ah/V)Sc
4]ma
x
h
max
53
6600
4.3.4
which
is about
Reynolds
behaviour
a
factor
number
of
four
(4.2.11).
Hence
flow
channel as found in Fortescue
correlation
(4.2.10),
They
data
implies
made
numerous
data
effect
downstream
to
where they placed a weir
An estimate
of these
take
'single-regime'
our results.
a grid in an open
(1967). Unlike
our
to exhibit
a better
V' 0
(see also
~
5
corrections to their
the entrance
of the grid and the end region
to maintain constant water height.
that
is
responsible
for
and we would
was
not
correlation. Nevertheless, their
regime' behaviour
KL
than our break
into account
corrections
10% to 20%, a figure
part
seem
that
'raw' experimental
immediately
behind
and Pearson
their
a correlation
expression 2.7).
least in
the
of their data is consistent with
Next, we consider the
fit with
smaller
put at least between
too small and may
the
discrepancy
data
be at
with
our
only show a 'single-
like to test if their maximum
Reynolds number is below our (ReA)*. Fortescue et al assumed
that the turbulence induced by the grid is given by
v'/U - 0.086/(X/M -10) .5
where
v' is the local rms velocity,
in the channel,
grid
and
X
is
M
is
the distance
the mean
is the flow velocity
the ratio of mesh
is applicable througout the
can compute
U
4.3.5
size to diameter
from grid downstream,
channel
L
is
the
and
length. From 4.3.5, we
rms velocity V' as
V'/U - 0.172M(L/M -10)0/L
where
of
length of the channel. After
4.3.6
substituting
the appropiate quantities, one obtains
V'/U
V' obtained
in this way
the flow like that of an
0.02
is
most
4.3.7
likely too low. If we treat
inclined channel, we would
54
have V'
0 O.1U
(equation
validity of the
4.3.5
and
4.3.2).
the
raises
assumption made
its
throughout
This
by
application
channel
with
the
issue
respect to equation
Fortescue
length.
In
of the
fact,
and
Pearson
equation
4.3.5,
which was first proposed by Batchelor and Townsend (1948),
was clearly intended to be used only in the region close to
the grid, not further
and Pearson,
a channel depth, say. Fortescue
than
on the other
This may
larger than the channel depth.
conditions of Fortescue
used it for distances much
hand,
and
imply that the flow
Pearson were
more similar to
those of 'open channel flow' than to 'grid' flow, especially
in the
region
further
away
difference they observed in
the entrance
length
the
from
grid. However,
the
the gas absorption results when
between
the
grid and the open-channel
section was varied suggests that the grid-induced turbulence
should
not
turbulence
be
due to
Nevertheless,
any
discounted
the
shear
it is very
revelant
entirely
stress
favour
of
on the channel
difficult
(due
to
measurements)
in
relate
the
floor.
to the absence of
the
turbulence
intensity in their system to the mean flow parameters. If we
make the assumption that V' U , where
is some constant
whose
grid
value
is of
order
configuration,
4.2.10(a))
Since Fortescue
height
operating
then
but
dependent
our
on the actual
correlation
(equation
can be reduced to the form
KL
water
0.1,
and
0.013 Sc
Pearson
remaining
condition
5IOU
of
data
about
20 0 C,
were
constant
our
obtained with the
and
correlation
at
a single
implies the
following functional form
KL ~ const. ReFp
where ReFp
4Q/vw (Q and w being the volumetric flow rate
and the wetted perimeter respectively), for comparison. The
55
linear dimensions
of
ReFp
14.
in figure
logarithmically
in
their
Comparing
figure 12 and
these two figures,
the location of the origin for the linear plot
we can deduce
for
(which lacks a scale
this information,
plot can
vs
KL
Pearson were plotted in
and
experimental data of Fortescue
be
ordinate
the
in the paper). With
it turns out that the data in their linear
by
approximated
straight
for
through the
the two sets of data
origin. This observation
is
obtained with different
grid configurations, which suggests
rather good agreement,
true
lines
at least in the functional
our correlation (made with
above assumption about V').
the
as 0.2, these data also suggests
If we take
form, with
that, for Sc
570,
StSc 0.5
0.033
which agrees only very roughly with our correlation equation
4.2.10(a)
(within a factor
the largest ReFp of
10000
three for the constant).
of
in their experiment,
achieved
For
we
can calculate
]max ~ 6300
[(V'Dh/V)Sc
where
Dh is the
the
approximately
as
channel and we have taken
height
4.3.8
of water
in the
0.2. The value computed
in
4.3.8 is smaller than our break Reynolds number even though
we are using Dh instead of some macroscale compatible with
our
A (which
of Dh). In other words,
should be some fraction
their data lies in the pre-critical
region.
A submerged oscillating grid has been used by a number
of researchers to generate 'homogeneous' turbulence which
surface
decays as the free
following
up work
measurements
of
in such
is approached.
a
al
et
Dickey
system
of
Isenogle
(1984), made
(1985),
numerous
both the horizontal
and
vertical rms velocities at various depths beneath the
interface, under various operating conditions. He also
56
carried out measurement
of
the
integral
length scale, L,I
defined as
L I - 0I
where
f(r)
-
°
f(r) dr
< u(r )u(r
4.3.9
+ r) > / < u2(r)
u
is
the fluctuating
r0
is
the reference location
r
is
the mean
is
the
<
>
>
velocity
separation distance
ensemble
which is compatible with,
integral length scale A.
average
though
not exactly equal to, our
using the absolute turbulent
By
velocity Q
< (u + v + w2)0
> and L I extrapolated to the
interface, we were able to replot Isenogle's mass transfer
data as in figure 4.3.1. It is interesting
Re I (
QLI/v)
exceeds about
value of about 0.01.
If
energy does not change
next to the interface,
extrapolated
form V'
we
In
then
this
For Re <
we
0
5 /Q
levels off to a
that the total kinetic
in the 'damped'
layer
can relate our V' (which
to
the
case,
re-correlated as KLSc0 5/V'
which agrees rather well
4.2.10(a)).
assume
significantly
from the bulk
Q/V'.
KLSc
600,
to note that when
interface)
is
to Q in the
the data of Isenogle
can be
for Re(mV'LI/v) > 400,
our correlation (equation
0.017
with
400, KLSc0
5/V' shows some indication
of rising above 0.017. Our data
did not stretch to such low
Reynolds
that
numbers.
Note, however,
is quite different
from
(1984), who observed
that
a
the
rise at Re < 400
data of McCready
decrease
in
4 max
x
10400
KL/V'
at
and Hanratty
Re < 50. In
Isenogle's data
(RelSc
'
which is smaller than
regime behaviour
in
our
his
4.3.10
break Reynolds number. A single-
case
results.
57
is
thus consistent with
our
In an independent
finding, Ho (1987) made measurements
of the average mass
transfer
The turbulence was
induced
of 02
by
onto a turbulent
an oscillating
the interface. Ho's apparatus was
Brumley
is not
et al (1984) who
too
close
to
the
grid beneath
same as that used by
the
reportedly found that, provided
to be
interface
influenced
0.25 f S1 .5M
0
5 /z'
4.3.12
of the velocity field. Here,
is a fairly good approximation
macroscale
rms
horizontal/vertical
length scale,
is the stroke amplitude,
z'
its
4.3.11
LTH - 0.1 z'
is the
by
(1976)
V
V
one
consistent relationship of Toly
presence, the dimensionally
and Hopfinger
liquid.
is the distance
to the centre
of
LTH
velocity,
f
is the
is the grid frequency,
M
size of grid, and
is the mesh
from the virtual origin
oscillation).
We
S
(which is close
therefore
took the rms
velocity and length scale at the interface as being given by
4.3.11 and 4.3.12, respectively. We reduced Ho's mass
transfer data, which were collected at a fixed temperature
of 20°C, to the form StSc 5 versus ReH (- VLH/V). The
results are plotted
scatter
is
in figure
= 0.0156, which
in the plot, a mean value of StSc
indepedent
representation.
of
Re H,
is
there is some
4.3.2. Although
found
This compares
to
favourably
be
a
fairly
good
to our correlation
4.2.10(a). Next, we calculate that for Ho's data,
(ReHSc 0
)ma x
3100
4.3.13
which is very much smaller than our break Reynolds number. A
single-regime behaviour in Ho's data is thus consistent with
ours.
did experiments on steam
Finally, Thomas (1979)
condensation at a shear-free turbulent interface, using a
58
vertical
jet system
somewhat
like
ours,
but
with a much
shallower test cell with z/D
ranging from 0.3 to 1.3. His
system produced a bulk radial flow under the interface, with
the radial velocity
at the surface varying
i.e. the flow
distinctly
was
flow condition. Thomas
velocity
made
was
about
r,
different from our bulk-free
some measurements of the mean
near the interface and,
rms velocity V'
inversely with
on the assumption
one-quarter
that the
of the radial mean
velocity at the surface, suggested the following relations
V'
1.5U
V'
0.65U d/s for 0.3s < r < R
n
d/s
0 < r < 0.3s
for
4.3.14
n
and
Here U
0.039s
for
IT
0.045r
for
is the velocity
diameter,
1T
1T
s
at
the
0 < r < 0.3s
0.3s < r < R
nozzle,
d
is the depth of nozzle beneath
is the integral scale of
the radius of the test-cell.
the
4.3.15
turbulence,
Using
is the nozzle
the interface,
and
is
R
the above relations
to
obtain the integrated mean rms velocity, Thomas's heat
transfer data can be interpreted approximately as
StPr0
5
0.010
The corresponding correlation
used a very similar set-up, but with
nozzle beneath
and CO2 ,
the interface,
can be expressed
StSc0
for
4.3.16
of
Orridge (1970), who
even shallower
depth of
the gas absorption
of 02
as
5
0.004
4.3.17
We have implicitly assumed that the velocity and length
scale relations of Thomas (4.3.14 and 4.3.15) are applicable
59
to Orridge's
experiments,
though
smaller z /D than Thomas.
clearly
highest
's run is
associated
around
with Thomas
integral scale of
deduced
nozzle
data
are for
equations 4.3.16 and 4.3.17
bear strong a resemblance
The
Thomas
Both
Orridge's
to our 4.2.10(a).
Reynolds
80000.
number
Assuming
achieved in
the macroscale
AT
's system to be the mean value of the
eddies
(as
defined
in
4.3.15), we can
that
[(V'AT/v)Pr0 4]max ~ 14000
4.3.18
The equivalent result of Orridge experiment is
[(V'AT/V)Sc0 4]max
8200
4.3.19
The single-regime behaviour in both Thomas and Orridge tests
is thus expected.
All
the data we have
referred
'pre-critical' region. The
only
aware of which
our
those of
and
may
Kishinevsky
Serebryansky
turbulence
be
in
and
used
experiments
a
that we are
'post-critical'
Serebryansky
mechanical
in a cylindrical
kept constant at 1700
to thus far lie in the
region are
(1956). Kishinevsky
stirrer
to generate
test cell. The stirring
rpm
rate was
and mass transfer were collected
for 3 different gases, namely nitrogen, oxygen and hydrogen,
at 200 C. In order
the suggestion
to
relate
by McManamey
their
et
data ot ours, we follow
al (1973)
who
suggests
that
for turbulence generated by a stirrer,
V'
where
N
V'
0.55NL
4.3.20
is the turbulence fluctuation near the interface,
is the stirring
speed
(revolutions
per second), and
L
is the length (tip to tip) of the stirrer blade, and
AK
0.3L
60
4.3.21
where AK is the
characteristic eddy dimension (macroscale).
Note that subsequent hot wire measurement of the velocity
field in a stirrer vessel
the correlation
of
restate Kishinevsky
by
4.3.20.
Davies
In
et al (1979) supported
this
way,
we
are able to
et al result as
St - 0.0045
4.3.22
which is
less
than a factor of three higher than our
correlation (4.3.10(b)). As to whether their data belong to
the post-critical region, we compute
(V'A /)Sc0.4
54000, 66000, 71000
4.3.23
respectively for H 2 , 02 and N 2. These values are well above
our break Reynolds number which lends further credibility to
our postulation.
4.3.2 Turbulence
Imposed via Shear at the Interface
In this section,
we
will
look
at
data taken under
conditions where the liquid-side turbulence is generated by
imposeing a mean
shear
strictly speaking,
should apply in
whether
it does
these
that might
stress
at
not
cases,
the interface. Although
follow that our correlation
it
is
of
interest
to test
be the case.
Jensen and Yuen
channel of turbulence
(1982) made
intensity
measurements in a water
near
the interface
in the
presence of a cocurrent steam flow and found that
V' ~ 2.9U,w
4.3.24
where V' is
the longitudinal (flow-wise) rms velocity at
interface, and
U*w is the interfacial shear velocity based
on the water density.
Their experiment on cocurrent steam
condensation yielded an expression of the form
61
StPr0.5 ~ 0.048
where
Pr
is
the
Prandtl
number
temperature. Equation 4.3.25
(4.2.10(a))
only in that
the Prandtl number
4.3.25
based
on bulk
water
differs from our correlation
the
constant
dependence was
is larger. Note that
assumed, not determined
empirically. The largest Reynolds
number (based on the mean
water
water
velocity
reached
U*w
V
and
height
in Jensen and Yuen
O(V)
(which
is
of
's
an
data
in
the channel h)
was 6800. If we assume
overestimate)
and
make use of
4.3.24, we arrive at
[(V'h/v)Pr0
This value
number,
is
even
of
the
though
max
same
we
data
exhibited
proportional
a
order
have
macroscale length and taken
31000
as
used
Uw
h
~V
4.3.26
our break Reynolds
to
represent
the
(overestimates). Their
single-regime
behaviour,
with
KL
to V'.
the transfer of 02
Aisa et al (1981) measured
and C02
from a moving stream into a channel of constant water height
for a range of Schmidt number, 400 < Sc < 750. Using 4.3.24,
we can readily reduce their correlation function to the form
0
StSc
5
0.034
4.3.27
For their experiment,
[(V'M/v)Sc
where M is the height of
this estimate,
however,
macroscale
the
results
'pre-break'
water
their maximum
same order as our
mind,
max
break
(M is definitely
et al
Reynolds
used
number
number.
M
larger
would
4.3.28
in the channel. According
Reynolds
that we have
of Aisa
40000
be
is again of the
We must bear in
as an estimate
for the
than A), and that hence
expected
to fall
regime, as indeed they seem to have done.
62
to
in the
McCready
and
Hanratty
(1984)
to Aisa
et
experiment similar
similar to 4.3.27
for large Re
conducted
al.
a series of
They obtained results
> 50, which
(Re
is
based on
interfacial shear velocity and height of water). The largest
Reynolds number reached in their experiment was
U WM/v - 1000
4.3.29
w
where Uw is the
interfacial
shear velocity based on water
density. This corresponds to (with 4.3.24 and Sc-440)
[(V'M/v)Sc0.4]ax
which
33100
4.3.30
is of the same order as our break Reynolds
though we are
using
M
instead
scale (some fraction
of
only a single-regime
behaviour
50, their
data
the
(1972, pp
is
begins
made
inclined plane and found
measurements
that
for Re <
to show marked
correlation mentioned
194)
Evidently,
found. However,
KL
that
even
the appropriate length
for the computation.
indicates
variation from
Davies
M)
of
number
above (4.3.27).
of
flow down an
flow was essentially laminar
for Reynolds number (based on mean velocity u, and height of
water)
below 500. If we used the approximation
u
0( 10U*W)
which can be deduced from Mattingly 's (1977) wind imposed
shear flow experiments, then the departure from 4.3.27 for
McCready
et al data at Re < 50 can be attributed
turning laminar. Another
put forward by McCready
height became
boundary
explanation
thin,
increasing
allowing
In such
manifests
itselt
a
as
scenario,
the
the lower solid
influence on the turbulence
near the interface. The occurrence
uncommon.
for the deviation was
et al that at very low Re, the water
exceedingly
to exert an
to the flow
of capillary waves
the 'lower boundary
dependence
of
the mass
is not
effect'
transfer
coefficient on more parameters besides Re.
The only experimental data which have previously shown
the
existence
of
two
regimes
63
are
for
cases
where
the
turbulence in
blowing above
the
liquid
is
the interface
generated
e lg.
(
see
a strong wind
by
figure
11 of Broecker
et al (1978) which
show two diffierent linear regimes when KL
is plotted
the
against
wind
ve.locity). In all these cases,
speed, the li aear relation between K L and
speed appears to break down and K L increases more
at the higher wind
the wind
rapidly with further increases ii
n wind speed. This has been
attributed
by some to
effeict of small bubbles produced
the
by wave breaking and the presenc e of large gravity waves and
capillary waves (eg. see Merliva t et al (1983) and Broecker
et al
(1984)).
In order
result, we shall continue
(1982). However,
further comparison with our
to make
to
us e
4.3.24 of Jensen and Yuen
the determinati'on of a relevant
macroscale,
compatible with our A,
is more problematic. The macroscale
in a shear-imposed
i
mo st likely dependent on the
f low
shea r stress at both
the
i nter face and the channel floor,
Sinc e
and the water height.
ma c r oscale
th ere
determinat ion in this
make some order of ma gnitude
es
For flow condit ions where
wave s are relatively
breaking
small,
class
ti]mate
t he
a nd
are absent, Sivakumar
absorption
is
no reported work
on
of flow, we have to
for comparison.
ampli tude
capill ary
of
the
gravity
waves and wave
s (1984) correlation for 02
can be reduced to the form
StScO S ~
4.3.31
0.016
Here we have used equation 4.3.24 to obtain V' from U*w.
Sivakumar took measurement of the mean water velocity as a
function
of depth for a
the nominal wind
speed
constant water height of 0.3m, with
varying
blowing above the interface. We
from
can estimate
from the expression
au/l
du/yl.
u
s
interface
64
2.6
m/s to 10.2 m/s
a macroscale
1
s
4.3.32
where Au is the mean velocity
at the interface and y is the
vertical coordinate. Equation 4.3.32
about equal to the
the
'break'
expressed
Prandtl
condition
4.2.11)
is of the
two
Sivakumar
's
04
same
explanation
important
thus
bear in mind
' 25000
as
be
the
Secondly,
existence
by any
wave
this
of
bubbles. Perhaps with the further
both wave breaking and formation
case, the 'pre-critical'
break Reynolds
number
conditions where wave
as
breaking
a possible
to
region in our flowwas
nor the formation
of
increase in wind speed,
of bubbles become more
can
defined
in the
regimes. We have
complications.
region
number.
it shows that
provide
two
breaking
introducing more
One,
break lies essentially
that the 'post-critical'
accompanied
4.3.33
our break Reynolds
implications.
data before
for the
]break
order
'pre-critical' region.
prevalent,
can
for
as
This has
not
length. Our criteria
(equation
[(V'ls/A)Sc
which
mixing
defines a length scale
If
change
in
this is the
either at the
4.2.11,
occurs,
or, at the
depending on which
happens first.
Merlivat and Memery (1983)
Ar absorption
wind
in a wind-driven
obtained
data for N 20 and
channel with h=0.3m. At lower
speeds, their results can be interpreted
StSc
0.55
as
0.050
4.3.34
A break occurred when the wind velocity was about 10 m/s,
with KL incresing more rapidly with V' at the higher speeds.
This enable us to
of a macroscale
1
use
Sivakumar
according
to
data for the determination
4.3.32.
With V'
in their
break region, we calculate
[(V'lm/A)Sc
0
]break
65
16000
4.3.35
which
implies that the above
correlation
4.3.34 lies in the
'pre-critical' region.
Deacon (1977) also obtained correlation of the form
(KL/V')Sc0'67
for
the
region
of
the
0.027
lower
wind
interface is relatively unruffled.
much
higher
proportion
than
5
m/s,
he
gas
exchange
relation
in
a
where
the
the wind speed very
that
KL
is in rough
speed.
of limited fetch experienced
et al (1979) made measurements
circular
wind-water
tunnel
and
of
got
of the form,
5
(KL/V')Sc0
For this case, we are not
fair estimate
were
found
problem
in a linear tunnel, Jahne
velocity
For
to the square of the wind
To overcome the
4.3.36
~
able
of the macroscale
quite different.
If
we
0.007
4.3.37
to use Sivakumar
data for a
since the water height used
take
the
macroscale
1j to be
20%, say, of the water height of O.lm, we can compute
[(V'l/A)Sc
which
suggests
that the
4'
]break
15000
correlation
4.3.38
4.3.37
is in the 'pre-
critical' region.
In conclusion,
our postulation
of two regimes for the
mass transport correlation is
not inconsistent with the
previously published data. Furthermore, all the correlations
obtained by previous workers for the 'pre-critical region'
(equations 4.3.3, 4.3.16, 4.3.17, 4.3.25, 4.3.27, 4.3.31,
4.3.34, 4.3.36 and 4.3.37)
have forms similar to our
equation 4.2.10(a), though the constant in general differs
(at most
by
a
factor
of
four).
For
the 'post-critical
region', the only previously available data, equation 4.3.22
66
compares favourably with our
by Sonin et al (1986)
that
4.2.10(b).
the
way
It wa s pointed
out
in wh ich turbulence
is
being generated might have some
bearing on the value of the
constant. The important outcome
of the brief comparison (a
more extensive comparison is possible only if we could
relate the flow parameters in other exper iments to V' and A)
suggests
that St may be expressed as
StSc0 5
St
A
[ (V'A/v)Sc0.4
-0.5
B + (ASc '-B)(ReA)*/ReA
where A and
B
are
constants
to
< 27x10
4
]
[ ReA > (ReA)* ]
be determined.
4.3.39
4.3.40
Equations
4.3.39 and 4.3.40 may apply to a wider class of flow
conditions, irrespective of how turbulence is induced in the
liquid.
67
FOR EXPERIMENTS
5. CONCLUSIONS
5.1 LDA Calibration of Turbulence in Test Cell
(a)
The rms velocities measured
centreline agree fairly well
et al 's
turbulence
the
with
the correlation of Sonin
3.3.5 and 3.3.7), provided
(1986) (equations
not too close to
with the LDA at the system
interface. The interface
intensity
a
to
depth
of
about
one is
distorts
O.1D.
the
As one
approaches the interface from below, the horizontal rms
velocity increases and the vertical rms velocity decreases.
This
is
the
result
of
a
transfer
of
momentum
from the
vertical to the horizontal directions.
(b) With
nozzle,
the interface
at
the rms value
of
3.67
system diameters
the turbulence
above the
intensity near the
interface is overwhelmingly large compared with any vertical
bulk flow velocity
in the same region.
effects, the average value of V' is
(c) As a result of wall
about 25% below
(d)
The
integral
autocorrelation
yields
the centreline value.
and
the
respective
based
local
on
the
Eulerian
rms velocity V'
a length scale A,
A
which
scale
length
does not depend
postulation
of Sonin
-
V'r - 0.24D
on
et
V'.
al
This
(1986)
'locked' to the system diameter
5.1
is consistent with
that the macroscale
D at locations
the
is
greater than
about 3D away from the nozzle. Sonin et al also used the k-e
model
to
associated
1.5V'
5 /e,
fit
their
experimental
length scale
Lk _
being the dissipation
68
results
and obtained
an
1.1D, where Lk
=
rate per unit mass. The
fact that Lke
larger than A is consistent
is significantly
with what is found in most turbulent flows.
5.2 Gas Absorption Rate Correlation
Our mass
transfer
data
for
Sc
between 230 and 1600
suggest a 'two-regime' behaviour with
Reynolds
number,
the
two
regimes
being
"break" Reynolds number, (Red),, given
(ReA)*,
where
the
(average)
Reynolds
rms
separated
approximately
V'
is
of
based
one
on
by a
by
'
5.2
A
and the local
2.7x104 Sc 0 4
number
value
respect to the eddy
velocity
component.
At
Reynolds numbers below (Red),, the transfer coefficient KL
is directly proportional to V' and inversely proportional to
Sc
1/ 2
(see below), while at Reynolds
is proportional to V'
coefficient,
with
and essentially
04
Re CSe4
For flow with
found
for
the
<
4
2.7x10 ,
of Sc.
a correlation
5.3
interface with turbulence
KL/V' - j/V',C where j is the
flux transfer across the interface and
concentration
difference
constant
SC
is the molar
between the surface and the bulk of
the liquid. This expression
to
independent
shear-free
imposed from below. Here St
molar
a much higher proportionality
, 0.013 + 0.002
StSc05
is
number above (ReAd)* KL
on the right hand
is
side
similar in form (though the
differs
from case to case)
the
correlations which have been obtained by other
researchers for various other methods of producing liquidside turbulence, such as with flow down an inclined plane,
wind-induced interfacial shear and submerged oscillating
grids. It
is
shown
that,
with
previous correlations have been
69
a
few
exceptions,
these
obtained at Reynolds number
lower than our break Reynolds
number.
the Reynolds number exceeded
had in fact been observed.
(ReA),, a two-regime
St
This relation
behaviour
> 2.7x10 4 Sc-0.4, our correlation
For flow with Re
a shear-free
In the few cases where
for
interface has the form
0.0025
is in
+
(0.013 Sc -0
approximate
5
agreement with
Kishinevsky and Serebryansky (1956).
70
- O.0025)(ReA)*/ReA
the data of
6.
NUMERICAL SIMULATION
6.1 Background and Overview
The major
model
obstacle
to
the
formulation
for transport at a turbulent
lack of understanding
surface.
Very
of the
little
is
available
turbulence
the
liquid
on
free surface has been our
turbulence
information,
experimental,
for
structure
either
the
side,
of a general
at a free
theoretical
distribution
very
or
of the
near the interface,
where the major
resistance
to transport occurs. The
transport models which have been proposed to date have all
been based on simplified conjectures about what aspects of
the liquid-side
diversity
turbulence
and shortcomings
control
can
the transport,
be
traced
oversimplified input assumptions.
Detailed velocity measurements
the free
interface
are
not
easy
and their
directly
to the
sufficiently close to
to
obtain, because
the
region of interest in mass transfer problems is often less
than a millimetre from the interface. Usually, particularly
at the higher turbulence intensities, the surface moves with
an amplitude
larger than
the
thickness
of entire
region of
interest. (Even the resolution of a LDA with a typical
measuring volume of 40Am x 40/pmx 600/Amis barely adequate,
although the main difficulty is usually not resolution but
beam reflections
from
the moving
free surface). Recently
Brumley et
al
(1984) made
some
detailed velocity
measurement, using a split-film anemometer probe mounted in
front of
a
mechanical,
rotating
arm,
in
a system where
turbulence was generated by means of a submerged oscillating
grid. Even then, the
Reynolds
number
had to be limited to
fairly low values, since the surface turbulence had to be
low enough to ensure that the interface was essentially
flat).
71
In order to make
theoretical
further progress
decription
and
in general,
mass
particular, we undertook
the
turbulence
concentration
unsteady
from
some
initial
statistically
sc ala r
temp erature)
Navier-Stokes
steady
heat
transfer data in
n um.erical
direct
passive
and
or
a
and
ne ar
simulation of
transp ort
(spec ies
surface. The
free
a
a
process
of the free surface transport
our
of
towards deve loping
equations were nume r ical compu ted
unti 1 an
conditions
approximately
state
was
Because the
re ached.
computational domain ha d to be at least of the order of the
turbulence macroscale in size, and the computation had to
to
the
resolve features down
Kolmogorov microscale, the
Reynolds number based on rms
velocity andi macroscale
wa s
limited to values which
only
approach those of our
experiments . For similar reasons,
be limited to low values, Pr
refer
the
to
energy
the Prandtl number had to
equation
instead
concentration equation, where typically
To conserve
computational
solved for a number
of
field was utpdated from
continued
in
temperature
time
time,
the
rms val ues of the
velocity
fluctuations
both
reach
compared with
condensation
6.2
the
then
experimental
heat transfer
0
energy
the
(00
species
- 1000).
equation was
the velocity
velo c ity
approximately
vertical
were
Sc
the
time step. The c,
oImputation was
each
steady stat es. The Reynolds-averaged
the
of
numbers afte r
Prandtl
until
profiles
e:
X plains why we
0(10), which
and
the
statistically
heat tran s port rate and
and horizontal
computed,
data
of
components
of
and the results
Brown
(1989) on
at a free surface.
Discretized Euations
The
equation
dependent
incompressible Navier
in its rotational form is given by
time
(1/Re)V2 u - Vff- au/at + WXu
72
Stokes
6.2.1
V.u
where u is the
the
rms
non-dimensionalised w.r.t . U, U is
velocity
velocity
at
computational domain,
the
is
bottom
the
P/pU
number,
Temporal
implicit Crank-Nicolson
of
boundary
the
Vxu, H is the
Re is the Reynolds
vorticity,
+ 0.51u.ul
dynamic pressure,
UA/v.
6.2.2
= 0
N
and
discretisation
of
6. 1.1 by an
in time gives a semi -discrete
method
system of the form,
(1/2Re)V2 (un+l
+ un)
~
V
-
= (1/At)(un+l
- un) + wxu
_
V. n+l = 0
6.2.3
6.2.4
Similarly, the energy equation can be expressed as follows,
(1/2Pe)V
2 (Tn+l
+ Tn )
(/At)(T
n+ l
-Tn) + u.VT
6.2.5
where T is the non-dimensional temperature, defined as the
local-to-bulk temperature differential non-dimansionalised
with
the subcooling AT,
a is th e
Pe is the Peclet number, UA/a,
and
molecular diffusivity.
Us ing spectral Fourier expansions
in the horizontal x
and y di rections and Gauss-Lobatto Lagrangian interpolations
in the vertical
z direction, we
can
express u and T in the
form,
M-1
N-1
E
u(x,y,z) = -
-M -N
u (m,n,z)e
ia'mx iny
e
NN
u(m,n,z) =
E
O
M-1
T(x,y,z) =
E
-M
u11(m,n,j)hj(z)
N-i
i-a'mx iny
E
Tl(m,nz)e
e
TN
1
NN
T(m,n,z) =
where
i
=
a
= 2/L ,
o
O
T11 (m,n,j)h.(z)
,
L
x
is the
73
extent
of x domain,
This
a
= 2/L
hj(r)
is the elemental Lagrangian
effectively
is the extent of y domain, and
L
,
decouples
the
interpolants.
problem
into
a
one
dimensional problem to be solved for each wavenumber,
-M < m < M-1, and -N < n < N-1.
The next
is
step
the
variational discretisation of
both the Navier-Stokes and energy equations. This process is
described
by Maday
and
Patera
(1987),
who give the fully
discretised equations as
-(A + a 1 B)u i
n+l
H
+ DHH
n+1
D.u
1
where
i
A and B are stiffness
Maday
and Patera
the Hermittian
Bashforth
(1987),
treatment
(a2m 2
represents
6.2.7
fi
represents
explicit
)/Re + 2/At,
evaluated
followed
2/At.
From
is
symmetric
forms of the equations
of 'fast
direct
3
1
order Adam-
term
and
and
Due
a
wxu
finally
6.2.6
n+l .
u.
by
Poisson
in
H
rd
the
+
as defined
is the del operator, D
1
(a2m2 + p2n2)/Re
first
= 0
Di
of
+ p2n
6.2.6
and mass matrices
of D i ,
matrix
n
= (A + a2B)ui n + 2Bfi
is -
o2
6.2.7,
to
the
is
highly
involved, we use the method
as
solvers'
(1986), to solve for H. and u. n + 1
described
T.n+1
by Patera
is then evaluated
from the energy equation which is discretised as,
-(A 1 +
where A
1n (A1 + a21B1)T i
llB)Tn+ n+
and B 1 are stiffness
1
form to A and B,
ai2 2
a2A is -(Cam
3r
order
uxVT.
In
that the
account
a
2
+ p n )/Pe
Adam-Bashforth
all
the
appropriate
in the
and mass matrices
represents
211
above
+
2/At,
and
finally
of
the
representations,
conditions
matrices.
74
6.2.8
similar in
2n2)/Pe + 2/At,
(a2m2 +
treatment
boundary
respective
+ 2Blg
i
gi
is the
explicit
term
we have assumed
are
taken into
The implementation
of
the
boundary
condition
would
be
described
in
the next
section.
In the computation,
every
time
step
the
using
equations. At the end of
was "frozen", and the
each of the various
field was
then
velocity
the
field was
discretised
updated
at
Navier-Stokes
each time step, the velocity
field
temperature fields were evaluated for
specified Prandtl numbers. The velocity
obtained
for
procedure repeated until
an
the
next
time step, and the
approximately steady state was
reached for both the velocity and temperature fields.
6.3 Boundary Conditions
The problem we are addressing
is one which
involves an
essentially horizontal liquid interface with negligible
surface shear. The turbulence in the liquid is imposed from
below
(eg. by some mixing
or shear flow below the surface).
turbulence is
the source of
We consider the simplest case where the
isotropic in a horizontal plane, though turbulence
being
necessarily
elevation.
below
decreases
For
our
the
in
interface
the
A.
(A
being
defined
in
its
liquid with
computational
rectangular box with horizontal
-
domain,
intensity
increasing
we
take
a
and vertical sides equal to
4.1.5).
On
the
top horizontal
surface, we impose a stress-free condition
where
u and v
au/az - 0
6.3.1
av/az - 0
6.3.2
are
the
horizontal
velocities
distance measured vertically upwards,
interface remains horizontal,
w
0
and z is the
and assume that the
6.3.3
75
where w is the vertical velocity.
On the
sides
of
our
boundary condition,
which
with
Fourier
the
spectral
domain,
is
we
impose a periodic
automatically
expansion
taken care of
in
the
x
and
y
directions.
The specification
of
the
boundary
bottom boundary (henceforth
called
not so
must
straightforward.
dependent
velocity
We
condition
lower boundary) is
the
impose
field which
for the
a
time and space
simulates
at
least some
minimum statistical properties - which we take as V' and A of the real turbulence
This
is
done
as
-
we
computation
for
which
follows.
cube of the same
cube of
isotropic
a
method of Orszag (1969)
isotropic
and satisfying V.u=O, in a
is
based
from the
of
as our computational
turbulence
is aimed.
separately
"forgery"
flow
A
size
-
First
generate
turbulence, with zero mean
the computation
generated
domain. This
following
the
on a random number generator
and a knowledge of the turbulent energy spectrum, Ek). The
turbulence so generated has a zero mean flow and satisfies
the
incompressible
sides of
the
continuity
cube
take
on
equation
periodic
(ie. V.u-O). The
boundary
conditions,
consistent with our assumed boundary conditions.
The cube of turbulence
by a spatial
longitudinal
lateral velocity
definitions),
an
velocity
correlation
but is
appropriate
thus generated
time
correlation
g(r)
"frozen"
is characterised
(see
Hinze
f(r) and a
(1975) for
in time. In order to introduce
dependence,
we
identify
the
lower
boundary of our computational domain with a horizontal plane
which
moves
vertically
certain speed c. The
Eulerian
boundary
upward
speed
autocorrelation
ie. for a
R
point
the cube of isotropic
c
(t)
through
must
cube
chosen so
be
for
the
a
point
with a
that the
on the lower
moving vertically upward through
turbulence,
at
a
speed c)
matches
f(r). In other words,
RwsvW(ct) - f(r)
76
6.3.4
or equivalently
c
IO R (t')
= 0
dt'
The relation between c
6.3.12.
of the energy spectrum
As for the specification
cube, the
turbulence
Lumley,
Orszag's
in
which is required
is (Tennekes
and
1972),
where e is
the
k is the
wave
expressed
as
viscous
2/3 k-5/3
6.3.6
dissipation
rate per unit mass and
1.5
our
For
number.
e -
(1.V'
2) 1 5
where V' is the rms velocity
-
1.1D
(equation
s
can be further
6.3.7
Lk
is
the length scale
Sonin et al (1986) estimated
3.3.6) and from our experiment,
0.24D (equation 4.1.5), so that
=
case,
/Lk_
and
associated with the k-e model.
=
E(k),
simulation of the isotropic
expression
simplest
E(k) -
that Lk-E
6.3.5
dr
given below in equation
is
V'
and
f(r)
A
can be rewritten as
(1.5V'2 )'5/4.583A
6.3.8
Mesh size consideration (see the following section for more
details) and the memory capacity of the Cray XMP/24 computer
allowed us to have
32
Fourier
modes
in the two horizontal
directions, thus giving minimum wave number kmin and maximun
wave number k
as
ma x
min
kmax
7/
*1x3-631
77
Upon
substituting
with
the limits imposed by 6.3.9 and 6.3.10, we find
6.3.8 into 6.3.6, and integration
1.5 V
of 6.3.6
f k max E(k) dk
>
6.3.11
kmin
which violates the principle of energy conservation. To
overcome this problem, we replace 6.3.6 with a modified
energy spectrum, E(k),
E(k) = (1.5 2/ 3k-5/3)(1 + B'e-(k-k/a)
where
k
. ,
o = kmin
determined
kmin
by
fkmax
'
= 2k
satisfying
E(k)
dk.
the lower wave
the higher wave
any small eddies that
in any
case
that
the
likely
be
reaching
eddies, which
contain the kinetic
of
damped out by viscosity
the top surface. Thus we
scales
the breakdown
from
to
at the lower boundary
turbulence
be the result
up
=
length scales). However,
specified
smaller
transferred
1.5 V'
length scales) relative
boundary will
of eddies
to be
constant
the energy distribution at
(shorter
inside the domain before
expect
bias
are
most
a
requirement
(longer
numbers
is
6.3.11
is found to be 2.784. The above
will
numbers
B'
the
B'
modification of E(k)
will
and
o
)
at
the
top
of the larger
energy, and not the result
the
lower surface where
the
turbulence is specified.
With
using the
of
isotropic turbulence
above-mentioned
calculate
shown
the cube
the
in
specification
longitudinal
figure
6.3.1.
spatial
Note
periodic boundary condition,
1.0 at r=0 and subsequently
In order
to evaluate
6.3.5 for the
0
computation
integral by truncating
is obtained
dr
of
since
we can
f(r), as
the
cube has
decreases initially from
increases
f(r)
E(k),
correlation,
that,
f(r)
of
thus generated
again to 1.0 at r=A.
as required
speed
at r=0.5A. With
in equation
c, we approximate
this approximation,
the
c
as
c = 0.
23 2
78
Vb
6.3.12
where Vb is the rms
the rms velocity
velocity
of the
in the cube, or equivalently,
lower boundary
of the computational
domain.
It should be noted that because we specify a Dirichlet
boundary
conditions
at
the
lower
surface,
irrespective
of
the flow conditions that are actually developed within the
computational domain, we expect an artificial boundary layer
to form at this boundary.
flow which
does not have
This boundary
simulate
layer
liquid via
a
a
is
the effect of
of a
lower boundary,
in contrast
boundary
the
an
boundary
grid
to the real
layer at this surface.
artifice
turbulence
boundary
normal
fine
is
simply
Dirichlet
this artificial
placement
This
in
of the way we
the bulk of the
condition. Resolution of
layer would require the
spacing
in
the vicinity
subject to the constraint,
A < v/W
6.3.13
max
where A
is the grid spacing in
kinematic
velocity
of the
viscosity
and
v
the z direction,
W max
is
is the
the maximum vertical
at the lower surface.
Expression 6.3.13, together with the requirement that
the mesh be
sufficiently
resolve the
thermal
fine
boundary
upper limit on the Reynolds
given computer. We
near
the upper boundary
layer
(see
number which
performed
layer was not
number
resolved.
number of 100,
both
lower
and
surface
the
the
case
B,
T 1 at
condensation,
saturation
the
thermal
where
top
the
by a
8). In case A, which
which
artificial
we
surface
interface
temperature). With
respect
79
boundary
has a Reynolds
boundary
boundary
interface were fully resolved.
In the
energy
equation,
temperature
can be handled
of 1000, the artificial
In
puts an
complete calculations for the
two cases, case A and case B (see Table
has a high Reynolds
below),
to
layer at the
layer
apply
a
at
the
constant
(this simulates vapour
is
to
at
a
constant
the sides of the
real
In the
< w'T' >
(which dominates the molecular heat flux aT/az),
< w'T'
where T
is
>
>>
6.3.14
aaT/az
a
temperature,
average
is the
is the molecular
component of temperature,
T' is the fluctuating
diffusivity,
w'
the fluctuating component of vertical velocity
symbol <
boundary
>
at
condition
average.
ensembl e
represents
lower
the
is
surface
boundary
temperature
provide
an
sure that
molecular diffusivity
const ant
diffusivity
artificial
surface, making
to represent
a
as
and the
Since the true
art
T2, but
at
the lower
is large compared with
art (z)
art
in that regi on.
Its
primary
the
the lower
temperature,
art (z)
of
part
solution, we simplify the problem by approximating
purpose
the
is
the turbulent heat flux,
tTaT/azllower surface
which
flux
heat
turbulent
due to the
distribution
temperature
has a time-varying
surface
lower
the
case,
periodic boundary condition.
takes
domain, the temperature
= -<w'T'>I
heat flux with
allows the passage of the
mean temperature gradient.
lower surface
a very small
typical turbulent
In
6.3.15
flows with
heat transfer,
a turb
b O.01uA
6.3.16
where U and A are the characteristic velocity and macroscale
of the flow. For our computation,
case B is given
figure 6.3.2 while
in
case A is plotted
in the following
art (z)/a takes the form
it approaches
for
A
a r/UA
case
art (z)/a
the counterpart
for
for
figure 6.3.3. In each case,
a parabolic
unity
the lower boundary,
art
of
the value of
from the interface
the function a
equation
smoothly at 0.7A
such that
and 0.8A
and case B respectively.
At
for case A,
= 0.2, 0.13, 0.09, 0.06
80
6.3.17
respectively
for Pr of
1.0,
1.5,
2.2
and 3.15, while
for
case B
a art /UA = 0.6, 0.20, 0.04
respectively
6.3.18
for Pr of 3.15, 10.0 and 50.0.
6.4 Mesh Size Considerations
A "rigorous"
solution
resolve the largest and
velocity
shown
as well
that
as
for
of
the
the
the
basic
smallest
equations must
length scales of the
temperature.
Townsend
flow,
of
turbulent
most
(1951) has
the
viscous
dissipation occurs at wave numbers below k
, where
max
0.3 < kmax/k
k
being
the
wavenumber
< 0.5
6.4.1
corresponding
to
the Kolmogorov
length,
k
where
is
= (/v3)1/4
6.4.2
the energy dissipation
per unit mass
0.2.
For
wavenumber kmax
our
0.4k.
=
flow,
is the
and U is the
k-e
adopt
a
maximum
6.4.3
lengthscale associated with the flow
velocity
which
is taken to be
at the lower boundary. We can, on using the
relation established earlier
and A - 0.24D
shall
) .5*/Lk
characteristic
the rms velocity
we
In equation 6.4.2, we set
=-- (15U2)1'
(1.5U
where LkC
is the
dissipation rate occurs at
kinematic viscosity. The maximum
k/k,
and
Lk
E
-
1.1D
(equation 3.3.6)
(equation 4.1.5), further express k
max
k
= 0.318 Re 0. 7 5 /A
max
b
81
as
6.4.4
where Re b is
the Reynolds
at the lower
boundary.
number
based on the rms velocity
Finally,
the smallest
associated with the flow, 1 s,
is given as
1
19.75A/Re 0.75
s
= 2r/k
-
max
length scale
6.4.5
b
(An expression similar to 6.4.5 can be derived independently
using
the relation
microscale,
resolve
1s
13~,
as proposed
the smallest
by
where
Pope
scales
and
is the Kolmogorov
1
Whitelaw
(1976)). To
of turbulent velocity,
the grid
spacing throughout the domain must satisfy
A<
Next, we
boundary
have
to
layer 6 in
appropriate
I
the
resolution
6.4.6
make
an
estimate
vicinity
can
be
of the thermal
of the interface
applied.
so
that
6 can be nominally
express as
o/K
~
where ac
turbulent
is
heat
the
6.4.7
L
molecular
transfer
diffusivity
coefficient.
and
It
KL is the
can be further
restated as follows
6
6/A
and ReA is based on
-
(oc/v)(v/V'A)(V'/KL)A
6.4.8
(1/Pr)(1/ReA)(1/St)
6.4.9
extrapolated
velocity
at the interface.
From the steam condensation
experiments by Brown (1989), in
the limit of zero Richardson
number,
St
0
0.0172 Pr
whence
82
33,
6.4.10
6/A
66
58.1/ReAPr
6.4.11
Since our computational domain is A, we can further express
6/A - 77.3/RebPr0
where Re b is the Reynolds
at the lower boundary.
rms velocity
number
We
correlation
of
diffusion
layer,
interface must be much
Finally,
accomodate
LD
6.4.12
based on the rms velocity
have
implicitly made use of the
3.3.5, which
experimentally, to relate ReA
thermal
66
to
the
was determined
Reb. Hence to resolve the
vertical
spacing
near the
smaller than 6.
the computational
LD
domain
the largest size eddy
A.
This
must
be able to
is done by making
A.
With
all
the
constraints mentioned
section 6.3, we performed
following
and
in
the complete calculations for the
two cases: case A with a higher Re b of 1000 and Pr
of 1.0, 1.5, 2.2 and 3.15,
100 and Pr of 3.15,
10.0
was chosen for case
B
Reb of 100 and Pr of
violates
above
the
and
case
and
-
implicit
artificial (turbulent) diffusivity
that 6 << A. The assumption
to be overly
influence
the lower surface.
above-mentioned
the
6.4.2 respectively, both
near the interface for
in
at
0(1) which
the
use
in
are
given
showing
the
the
a
art
of art(z)
z direction
in
figures
of
6.4.2. The horizontal
the
x
an
at
for the
6.4.1 and
very fine grid spacing
resolution
of
. Note that the
grid spacing is also finer at the inner boundary
the middle portion
of
the lower boundary -
specification
grids
two cases
6/A
is that 6/A is small enough not
by
The
range of Pr
from expression 6.4.5, a low
would have made
assumption
a lower Reb of
50.0. The higher
since,
0(1)
B with
compared
to
computational domain for figure
and
y
directions
have 32 Fourier
modes each. Table 8 shows all the parameters associated with
the implementation
of the two cases A and B.
83
7. RESULTS AND DISCUSSIONS II
The numerical
conditions,
simulations
identified
eddy Reynolds
were
as case A and
number Re b (based
lower boundary) was
1000 and
in figure 6.4.1. Re b for
on
carried
case B. In case A, the
the rms velocity
the grid mesh
case
B
out for two
was
spacing
at the
is shown
100; the grid mesh
is
shown in figure 6.4.2. For both cases, computations were
carried out with the NASA Lewis Research Centre Cray XMP/24
computer. The computations were started with the initial
velocity distribution in the computational domain equated to
that in the cube of
isotropic
turbulence which was used to
generate the lower boundary condition (see section 6.3). The
initial
temperature
function with
distribution
a value
of
unity
was
at
taken
as
a
step
the interface and zero
throughout the domain. Computations were allowed to run for
a period of at least 10
eddy turnovers
(an eddy turnover
defined as A/U, where U is the reference
lower surface)
to ensure
steady state had
temperature
been
mesh
to resolve
surface,
that a satisfactory
reached
distributions.
took over 10 hours of
the
required
7.1 Approach
rms velocity
CPU
In
for
terms
time.
artificial
both
of
is
at the
statistically-
the velocity and
Cray
time, case A
Case B, which had a finer
boundary
layer at the lower
close to 30 CPU hours.
to Steady State
7.1.1 Time Evolution of Stanton number
Since
we
started
initial states and allowed
a so-called
our
computations
the calculations
'steady' state was
track the time
evolution
of
reached,
a
from
arbitrary
to proceed until
it was necessary
quantity
which
to
is a good
representation of the flow. We selected the Stanton number,
St
KL/U,
based on
the
rms velocity
84
at the lower boundary
and on the average value
of its inherent
KL over the interface, because
of
dependence
on
both
temperature
fields. (In general,
temperature
field will not
time;
and
hence
quantity which
it
is
the
velocity
the velocity
reach
steady
convenient
to
and the
field and the
state at the same
use
as
a "marker"
a
depends on both).
Case A
The spatially-averaged
St
figures 7.1.1, 7.1.2, 7.1.3
and
2.2 and 3.15, respectively.
very
early
conditions
periods
is
(mentioned
shown
7.1.4
versus
not
shown
for St in the
because
artificial
characteristic
time in
for Pr of 1.0, 1.5,
The time evolution
above) are
value of St is not a
is
our
initial
and the initial
value of St. The dotted
lines in each plot indicate those regions where
the computed
data were not retrieved from NASA Lewis for post-processing.
Nevertheless,
we can see that
average after
about
7
St has stopped varying
eddy
turnovers,
implying
on the
that an
approximately steady state has been reached.
Case B
The average St (averaged
vs time is shown
dotted lines
data were
in
figure
represent
not retrieved
this case, St has reach
spatially over the interface)
7.1.5
those
from
for Pr=3.15.
Again,
the
sections where the computed
NASA. The plot reveals
an approximately
that in
steady level after
the 6 eddy turnover periods.
7.1.2 Energy Transfer Across Each Plane
Case A
If the deduction
that a
after the 7 eddy turnovers
time evolution
of
St,
steady state has been reached
can be made
then
should remains approximately
the
mean
invariant
85
from the plots of the
temperature
with
respect
profile
to time
the
heat
plane
of
and
(see section 7.3),
across each horizontal
should be the same
flux
The total heat flux, Q(z), can be defined
Q(z) - aT/az
where a is
the molecular
temperature
gradient
at
domain.
the computational
as
- < w'T' >
7.1.1
T/az
diffusivity,
z,
height
is the mean
is the fluctuating
T'
component of temperature and w' is the fluctuating component
The
velocity.
of the vertical
7.1.1 represents
the
term represents
due
flux
heat
flux, while
heat
the molecular
on the right in
term
first
the second
eddy
the turbulent
to
motion (turbulent heat flux).
7.1.6,
Figures
total heat
ensemble-averaged
spatially
over each plane
over the
last
numbers
3
of 1.0,
case, Q(z) is
reaffirms
7.1.8
7.1.7,
2.2
1.5,
and
approximately
the
time average at each z
then
turnover
eddy
show
is first averaged
(Q(z)
flux
and
7.1.9
and
periods)
for the Prandtl
In each
3.15, respectively.
invariant with distance, which
that a steady state exist.
>,
is
in figures 7.1.10, 7.1.11, 7.1.12 and 7.1.13.
It
The turbulent
shown next,
is important
part
of
heat
the
to note that as one moves
< w'T'
flux,
further away from the
interface, turbulent heat flux dominates progressively over
the molecular heat transfer. This is clear from a comparison
of
figures
7.1.6,
7.1.12
7.1.11,
magnitude
7.1.8
7.1.7,
and
7.1.13
at
transfer
interface. However,
the 'molecular'
in
the
heat
respectively;
transfer
note
7.1.10,
that
approaches
the
that of
away from the upper
locations
vicinity
with
7.1.9
heat transfer
of the turbulent
the total heat
and
of the lower surface,
predominates
again
at the
expense of the explicit turbulent heat transfer. This is
because we have artificially specified a variable molecular
diffusivity,
< w'T'
a
art (z),
to
> in that region
action was
conditions
necessary
represent
the turbulent
heat flux
(see section 6.3 for details).
because
for temperature
of
the
Dirichlet boundary
at the lower surface.
86
This
Case B
The ensemble-averaged heat
over each plane
last 3
eddy
first, then time
turnovers)
(averaged spatially
average at each z over the
across
Prandtl numbers 3.15, 10.0 and
flux
each
horizontal
50.0 are depicted
plane for
in figures
7.1.14, 7.1.15 and 7.1.16, respectively. These figures show
constant values across
each
plane,
implying that a
statistically steady state has been reached.
The
numbers
corresponding
3.15, 10.0 and
7.1.18 and
expected,
7.1.19,
turbulent
50.0
respectively. As
region where an
introduced.
far
flux
are plotted
the turbulent heat
flux sufficiently
heat
in figures
in
case
flux approaches
from
the
7.1.17,
A,
and as
the total heat
interface,
artificial molecular
for Prandtl
except
in the
diffusivity has been
7.2 Velocity Characteristics
7.2.1 Ensemble Averaging
of the RMS Velocity
in the Steady
State
For
a
stationary
flow,
ensemble
averaging is
equivalent to time averaging. The "ensemble-averaged" rms
velocity at a given plane was computed by first determining
the fluctuating component of velocity with respect to the
spatial mean at each horizontal plane, followed by taking
the rms
of
the
fluctuating
finally by time-averaging
velocity
(over
the rms velocity thus computed
average
is somewhat different
defined in an
velocity
at
determining
experiment.
a
particular
is no reason why
and
at each plane. This ensemble
a
point
the way
the rms value
LDA measurements,
is
calculated
the time series mean, followed
the rms quantity with
each plane,
the "steady state" period)
from
In
at
is
the rms
by first
by computation
of
respect to the time series mean. There
the two process of calculating
87
rms velocity
should be significantly different
use our sequence of
averaging
amount of memory
enormous
ensemble
because
it would
required
an
space for a computer
to store data
so
that the more
for every time step and for
conventional
in principle. We chose to
every location
can
average
be
evaluated.
To spot
check whether our ensemble averaging produced results in
agreement with the conventional ensemble averaging, we took
a limited
time series
on each plane,
profile
was
and
velocity
of
found
indeed
at a particular
the resulting
that
the
approximately
same
position
rms velocity
as
the
rms
velocity we computed (to within
about 309%;the differences
depended
and
on the point
examined,
could be explained
by
the relatively short time-wise averaging period).
7.2.2 RMS Velocity Profile
Case A
The
ensemble-averaged
rms
velocities
in
the
two
horizontal directions and the vertical direction are plotted
in
figures
ensemble
7.2.1,
7.2.2
averages were
and
mentioned
fact that the grid
spacing
resolve
probably
resulted
velocity
from about
the
in
elevation
of
drop in
rms
results
(see
0.4A
velocity
figures
attributable to the way
condition
the
by
not
boundary
increase
inner boundary
of the computational
rms
from
is
about
the
not
4.1.1
we
88
in
to about 1.5
domain. All
in
the experimental
It
is
fine
probably
the velocity boundary
of the computational
a
the rms
1.5 to about 0.2 at a
4.1.6).
imposed
having
layer has
lower surface. This sharp
found
to
not
then decrease with
velocity
at the lower boundary
and aggravated
The
section 6.3, the
the lower surface was
observed
from a high value of
distance of only
in
artificial
the
1.0 at
components
earlier
near
at the first grid location
the three
respectively.
taken over a period of the last three
eddy turnovers. As
fine enough to
7.2.3,
mesh
domain,
to resolve
the
normal boundary
However,
layer (see
as will
decline
be
the
seen
below,
in the rms velocity
boundary
layer was
therefore most
from
there
as
one
the
caused
likely
is
moves
However,
boundary.
that our imposed
turbulence
by
not
turbulence
the
In Orszag's
boundary
wavenumbers
is
but so is the fact
into wave
for
boundary
is
which
artificial
boundary
that
flow within
the
distance
than
the
each
present a
in
its
information
to the flow at the
the
result
are
by the fact that
prevents
domain
domain
for
the
between
might
compounded
The
of
between the wavenumbers
results
A
one
phase
condition
layer.
amplitudes
the
from the flow inside the
boundary,
isotropic
the specification of the
the
and
if the phase differences
Dirichlet
of generating
ignored. This
also important. This problem
lower
can be decomposed
only
completely
a
and phases between each wave
adopted
considered
of
application
Isotropic
scheme
(1969)
condition,
are
wavenumber
exchange
is
random phasing.
has
domain
in a spatial
turbulence, which we
the
from the lower
artificial boundary
our
is
only
numbers of different magnitude
problem
the artificial
interface. This decline
Dirichlet boundary condition artificial,
lower
still a large
at the lower surface. We simulate V' and A at that
condition
number.
in section 6.2).
in case B, where
resolved,
surface to about 0.6A
discussion
formation
of
of
all this may
requires
evolution
an
be
more spatial
into
a
more
"universal" turbulence.
It is interesting
rms velocity
experimental
profiles
to
bear
data of Brumley
submerged oscillating
grid
note, however,
a
and
to
strong
that our computed
resemblance
Jirka
generate
to the
(1984), who used a
their turbulence.
Their resultant rms velocity exhibits a sharp decline in
magnitude with distance measured away from the oscillating
grid, towards the
free
surface.
their oscillating
grid
is
condition,
in
It
fact
should
a
and that it takes some distance
the flow to develop
be noted that
Dirichlet
boundary
from the grid for
characteristics pertaining to isotropic
89
turbulence, with an accompanying
rate of decline
in
the
rms
large
drop in energy. The
velocity
subsequently
becomes
much more gradual, similar to our computed results. However,
there are some fundamental
induced by an
difference
oscillating
grid
between the turbulence
and
our case, notably
the
issue of length scales.
Case B
The use of
artificial
boundary
a
sufficiently
boundary
layer
in
eliminates
the
surge
fine
the
mesh
vicinity
in
rms
location next to the lower surface. This
the horizontal
and the
rms velocity
vertical
rms
Jirka. The rms
as
in
the
velocities
the lower
velocity
at
the
is clearly shown in
figure 7.2.6. However,
sharp decline
in
of
of figure 7.2.4 and 7.2.5
velocity
there is still a fairly
the lower surface,
plots
to resolve the
in magnitude
experiments
decrease
from
close to
of Brumley
and
about 1.0 at the
lower surface to about 0.15
at 0.4A from the inner surface.
As discussed above, this is
probably caused by our velocity
boundary
condition
It is
highest
at the inner surface.
interesting
note
that
the
ratio
of the
rms value near the inner surface to the rms value
the region close to
1.0/0.06
to
the
interface
16.6 for case
that the turbulence
are
1.5/0.1
A and B respectively.
intensity
decreases
in
- 15.0 and
This suggest
by roughly
the same
ratio irrespective of the imposed eddy Reynolds number Reb.
7.2.3Energy Sectra
Case A
To show
experimental
further
data
quantitative
of
calculations, we compare
Sv ) and transverse
Lumley,
1972, for
Brumley
et
similarity between the
al
(1984)
and
our
the
respective longitudinal (Suu,
(S W) power spectra (see Tennekes and
definitions).
90
Both
of
Brumley
et al 's
spectra exhibited
numbers
at
-5/3 power
locations
dependence
sufficiently
surface. The longitudinal
power
at the higher wave
remote
Suu
spectra,
u2
from
the free
and Svv,
are
w2
in
and
when expressed
proportional respectively to
terms of their Fourier-transformed wave number components,
and the transverse
power spectrum
its
when expressed in terms of
7.2.8 and 7.2.9 show the
to w2
wave numbers. Figures 7.2.7,
graphs
w (ensemble averaged over
plotted against wave number
is proportional
SVw
2
u , v
of our computed
2
and
the last three eddy turnovers)
(defined nominally as 2rk/A
k
where k is an integer) for different depths of 0.04A, 0.155A
and 0.5A beneath the interface. These figures show a -5/3
exponent
where
the
proximity
-5/3 power
Brumley
number region except for the case
in the high wave
depth
is
0.04A.
that
the the
of the interface has caused the deviation
from the
as
dependence,
et al
It
is
is
likely
also
seen
the data of
in
(1984).
Case B
The equivalent
S
for case B
power
are
7.2.12. These
are
period of the
last
spectral
plotted
ensemble
three
feature of these figures
it
is
Reynolds number
at
depth
evaluated)
are of
order
too
subrange where
a -5/3
and Lumley,
low
noted
0.09A,
10.
for
power
The
the
Case A
91
that
S
the local eddy
and
,
presence
-5/3
in figures
S
0.6A from the
and S
local Reynolds
exponent
7.3 Temperature Profiles
over a
interesting
profiles
0.SA
where
1972).
taken
they do not exhibit
velocity
be
locations
thus probably
Tennekes
to
values
turnovers. The
that
rms
7.2.4 to 7.2.6,
(ie. the
average
is
Suu, Svv and
figures 7.2.10, 7.2.11 and
eddy
exponent decays. From the
interface
in
densities
of
are
number
is
an inertial
can be expected
(see
The ensemble averaged ("mean")
temperature at a given
elevation was obtained by first averaging the temperature at
that elevation,
and then averaging
three eddy turnovers.
figures
7.3.1,
numbers
of
expected,
7.3.2,
1.0,
the
The
slopes
mean
7.3.3
1.5,
2.2
of
over a period of the last
and
and
the
is plotted in
temperature
7.3.4
for
3.15
the Prandtl
respectively.
temperature
profile
As
at the
interface are steeper for the higher Prandtl numbers.
One
presence
feature
out,
of a slight drop in
the interface.
last
stands
element.
be
The
attributed
needs
the profile
This happens also
spectral
however,
and
to
the
the
at about 0.7A from
be the location of the
'bump'
to
comment:
in
the
profile
artificially
can,
introduced
molecular diffusivity a art (z) which overwhelms the molecular
diffusivity a in the energy equation (see figure 6.3.2).
Recall
that
introduced
the
at
difficulties
temperature
artificial
the
with
lower
diffusivity
boundary
Dirichlet
to
boundary
in the energy equation
a art (z) was
overcome
condition
the
for
the
(see section 6.3 for more
details).
Case B
The corresponding mean temperature profile for Prandtl
numbers
of 3.15, 10.0 and
7.3.6
and
increasingly
increasing
50.0
are shown in figures
7.3.7
respectively.
steep
gradients
Prandtl number.
As
at
in
These
the
case
elements
located next
we have replace the molecular
artificial
diffusivity
a rt(z).
to
profiles
show
interface with
A, the profile
Pr-3.15 shows a slight drop at the beginning
spectral
7.3.5,
for
of the last two
the lower surface, where
diffusivity a with a varying
In
the
other figures,
the
temperature profile show smoother 'transition'.
Overall,
more
prominent
the slight drop in the temperature
in
case
A
probable causes. First, the
than
in
case
profile
is
B. There are two
distance over which a art (z) was
imposed near the lower surface was
92
smaller in case B than in
(
case A
0.2A
vs
0.3A).
This
allows
a
longer spatial
distance for the development of the temperature profile with
a
in case B.
finer
near
artificial
7.2.3),
Second,
the
the
lower
boundary
spacing
surface
for
layer;
the boundary
numerically-induced
grid
layer
in case B is much
resolution
of
the
in
case
was
not
resolved,
and a sharp
rms
velocity
takes place
increase
of
A
(figures
7.2.1 to
near the lower surface.
7.4 Turbulent Diffusivity aTZ)
Case A
The turbulent diffusivity aT(Z) is defined as
aT(Z) - -< w'T' >(z) /
where
the symbol <
average)
and T is the
expansion
apply the
au/az
> represent
mean
for the term, <
interfacial
T/az
7.4.1
ensemble
temperature.
average
(ie. time
If we use a Taylor
w'T' >(z), near the interface
boundary
conditions
and
w' = T' = 0,
av/az - 0, we can deduced that
aT(Z) Z2/
r'
where
is independent
'
+
+
7.4.2
-[(aT/az) / < w'/dz aT'/az >]
of z, and has the dimension
expansion in z, 7.4.2 with
only
of time. Being an
the first non-zero term is
valid only in the region very close to the interface.
Figures 7.4.1, 7.4.2,
ensemble-averaged aT(z)
spatially
7.4.3
(based
over each horizontal
over the last three
and
on
7.4.4 show plots of
first
averaging
w'T'
plane and then time averaging
eddy turnovers,
and separately
dividing
by the mean temperature gradient aT/az taken over each plane
before the time averaging)
versus
93
z
for Prandtl number
of
1.0,
1.5,
2.2
represent
the
indications
z2
the
figure, 7.4.5,
function
3.15
is
All
const.z
2
The
four
up
to
broken
plots
lines
provide
a depth of about
interface. Also
shown
' versus Pr
both case A and case B.
shows that
'
is
of Pr for case
between case A and
B.
of
at
most a slightly
increasing
independent
of Pr
' differs by roughly one order
It may be concluded therefore, that
interface
field, and insensitive
for
in the following
A and essentially
for case B. The magnitude
aT(z) near the
respectively.
dependence.
that aT(z)
0.3A beneath
The graph
and
is
a
property
of the turbulence
to the heat flux.
Case B
Figures
plots aT(Z)
7.4.6,
7.4.7
(evaluated
log-log scale for
and
7.4.8
in the same way as in case A) vs z in
Prandtl
numbers
respectively. The dotted line shows
the
spatial range. This
boundary
layer
T2
z
is
3.15, 10.0 and 50.0
a
z
dependence
not
increases
of
dependence. These
- const.z2 near the interface.
figures also show that a (z)
For the lower Pr,
show corresponding
extends over a wider
unexpected,
with
since the thermal
decreasing
Pr
for a given
Reynolds number flow.
7.5 The Condensation
Expe r iment
Stanton number and Comnarison with
The condensation Stanton number, St, is defined as
St - -
aT/aziinterface
7.5.1
where a is the molecular diffusivity, aT/z
is the mean
temperature gradient evaluated at the interface, V' is the
rms velocity
temperature
near the interface
between
(1986), Helmick
and AT is the difference
the interface and
et al
(1988)
the bulk. Sonin et al
and Brown
94
in
(1989) measured
the
Stanton number for
steam
condensation
in terms of the rms velocity
their Stanton number
Expressing
or horizontal) extrapolated from
component (either vertical
the bulk
to
the
surface,
The data of
interface.
correlated
damping near the
the
ignoring
which corrected
Brown,
the previous
number dependence,
Prandtl
the
data and established
on subcooled water.
can be
as
St - 0.0215 Pr -033
for
7.5.2
500 < Re A
< 12000
1.5 < Pr
< 4.7
provided that buoyancy effects are negligible.
This equation can, in
principle, be compared directly
with case A, which corresponds to ReA=1000 and, the flow
being assummed incompressible, negligible buoyancy effects.
An ambiguity arises, however, in determining the velocity V'
on which
St is to be based. Brown's experimental
the interface,
ignoring
(termed V' ) is not
the
V' from the bulk region to
of
is based on the extrapolation
correlation
layer. This value of V'
damped
different in the experiments from
very
(see figures 4.1.1 to 4.1.6).
the value of V' at z-A
In the
computation, however, the profile of V' is such that a value
extrapolated
to the surface is clearly very different
order of magnitude, approximately)
is
z-A. What
produces
more,
nonsensical
controlled
by the
rate
decay
of
layer rather than outside
what
since
results,
it,
the rms
velocity
imposed
at
the
of
as
follows, we base our St on
the value of V' at
at making
attempt
an
from
in
(by an
an extrapolation
extrapolation
V' within
is
the damped
the data of Brown.
In
the value of V' at z=A, ie.
the
lower
boundary
of the
computational domain.
Based
on the rms
velocity
at
the lower surface
ie.
the reference velocity used for normalisation), our computed
ensemble-averaged Stanton number (based on the time average
95
of the mean
temperature
gradient
last three eddy turnovers,
Re =
1000
values
agrees
divided
remarkably
in
Table
9
for
by the rms velocity)
with
well
(within about 15%). Results
are tabulated
the interface for the
at
the experimental
St v s
o f the computed
the values from
with
together
Pr
equation 7.5.2 for comparison.
This remarkable agreement is p robably in some measure
fortuitous, since we were not been able to simulate the rms
velocity profiles well (see figures 7.2. 1 to 7.2.3). The rms
velocity can be viewed as the sum of the energy cont ribu t ion
from eddies of
velocity
all
sizes.
That
at the lower surface,
the vicinity
of
the
interface,
across the free interface.
role
this is tru e,
If
rate
these large eddies.
large-eddy
Pearson
(1967)
Fortescue
clearly
and Pearson
a characteristic
value
of
(figure
does
7.4.5)
not
agree
is b ased
with
o n the
only
magn itude
in
the
of
and
data.
existence
of
T' wh ich has a constant
time
disagrees with
This
the
it still leaves
mode 1 of Fortescue
not
's model
well with
relate d to the motion
is
eddy renewal
0.20A/U.
so
in
perhaps only the large
in the transport process
unclear how the transport
The
based on the rms
the rms velocity
a grees
that
pivotal
St
not
and
experimental results suggests
scale eddies play a
the
our
but
computed
also
'
in the
functional dependence on Reynolds numb er.
From figure 7.5.1, we can see
that
the computed
results imply an exponent of Pr of -0.37, which is close to
Brown's experimental value of -0.33.
case B w'as computed was to check
One of the reason why
whether
the agreement
of the
Stanto n
experiment was a coincidence.
based again on the
tabulated
versus
computed
velocity
rms
Pr
The
in
Table
number
in cas e A with
St for case B,
at the lower surface,
10.
Also
shown
are
are
the
experimental quantities, evaluated by extrapolating equation
7.5.2 to Re 100. The
agreement between the computed and
the experiment
is within
about 20%.
96
Bearing
in mind
that the Prandtl number exponent of
established using data for Pr <
-0.33 in equation 7.5.2 was
5.0, and that our experimental mass transfer
suggests
an exponent
interesting
of
to see whether
Prandtl number exponent
corresponding
functional
-0.5
to
any
to
(equation
4.2.10(a)),
10.0
and
in
the 3 data points
Assumming
50.0.
the
form between St and Pr as
St - BPrP
the lower two Prandtl number
the higher
it is
trend exist for the negative
increase
Pr-3.15,
data of Sc >230
two imply Pr
1
-
values
-0.49.
97
7.5.3
imply
0
-
-0.40 while
8.
We
turbulence
CONCLUDING
have
EMARKS ON C(XMPUTATION
attempted
field
and
a
numerical
passive
scalar
shear-free surface, with turbulence
full,
unsteady
Navier-Stokes
equation are solved
the
two
using
horizontal
simulation
distribution
the
near a
imposed from below. The
equations
a
of
and
the
energy
spectral Fourier expansion in
directions,
and
a
Legendre
spectral
element
technique
in
the
vertical
direction.
The
computational domain, "extracted" from the full flow, is
restricted
to
interface,
with
a
single-macroscale
the
top
surface conditions, and
neighbourhood
boundary
representing
of
the
the free
the
bottom boundary simulating the
"far-field" isotropic turbulence. Two statistically steady
calculations at eddy Reynolds number of 100 and 1000 and
Prandtl numbers ranging from 1 to 50 are presented. Near the
free surface,
the
computed
turbulent
distance-squared dependence, in
series-expansion
result
the rms velocity
at
computed
the
lower
shows a
agreement with the Taylor-
appropriate
boundary conditions. The
diffusivity
for
the
free surface
Stanton number (based on
artificial
boundary
of the
extracted domain) is
found to agree with the experimental
steam condensation results (Brown, 1989) to within better
than 20%.
Despite
this
good
calculations are plagued
are, no doubt,
First there are
endemic
agreement with
by
several
experiment,
serious problems that
to the extracted-subdomain
numerical
our
approach.
difficulties associated with the
need to resolve the (physically unrealistic) boundary layer
which forms at "far-field" boundary, where we have applied
artificial Dirichlet boundary conditions. Second, the use of
random-phase simulated turbulent conditions at the "farfield" boundary can yield unreasonable (here, very low)
turbulent
intensities
in the interior. If these problems
can
be addressed, extracted-subdomain calculations can result in
significant computational savings, in that single-macroscale
98
domains can be used to investigate
characteristics.
99
(assumed
local) turbulent
References.
Aisa
L.,
(1981),'Exchanges
and
Masbernat
dissous en ecoulenments
de gaz
de gaz et de liquide',
J.
Goerge
B.,
Caussade
Int.
strati fies
Transfer,
J. Heat/Mass
L.
Vol
24,
No. 6, pp 1005.
Aydelott
J.
C.
R.
Rudland
and
S.
(1985),'Technology
by NASA Lewis Research Centre
requirements to be addressed
NASA Technical
Program',
Management
Fluid
Cryogenic
Memorandum 87048/AIAA paper 85-1229.
Aydelott J. C., Carney M. I. and Hobstein J. I. (1985),'NASA
management
fluid
low-gravity
Lewis
Research
Centre
technology program', NASA Technical Memorandum 87145/AIAA
GNOS Paper 85-002.
Barchilon M and Curtet R. (1964),'Some details of the flow
structure of an axisymmetric confined jet with backflow', J.
Basic Eng .,
Vol 86 , pp 777-787.
Batchelor G.
K.
Society, Vol.
193A, pp 539.
Battino
R.
and
Townsend
(1965),'The
A.
A.
solubility
(1948), Proc. Royal
of
gases
in liquid',
Chemical Reviews, Vol 66, pp 395-463.
Bin A. K. (1983),'Mass transfer into a turbulent liquid
flow', AIChE J., Vol 25, No.3, pp 405-415.
Boadway J.D.
readings for
information
Brignole E.
and Karahan E. (1981),'Correction of LDA
refraction at cylindrical interface', DISA
No. 26.
A.
and
Echarte
R.
(1981),'Mass Transfer in
laminar liquid jets', Chem. Eng. Sci., Vol
100
36, pp 695-703.
Broecker H., Petermann J. and Siems W. (1978),'The influence
of wind on
CO 2-exchange
in
a wind-wave tunnel, including
J . Mar. Res., Vol 36, pp 595-610.
effects of monolayers',
Broecker H. and Siems W. (1984),'The
transfer
fr om water
transfer
at water surfaces,
Gerhard,
19 84).
Brown
S.
J.
M. I. T. ;
to
air
at
role of bubbles for gas
higher windspeeds',
of 'Gas
pp 229-236'. (See Brutsaert and
(1989), S. M. Thesis, Dept. of
Mech.
Eng.,
in progress
Brtko W. H. and Kabel
R. L. (1976),'Pollution transfer into
water bodies', Water, Air and Soil Pollution, Vol 6, pp 71
95.
Brtko W. J.
natural
and
Kabel
R.
L.
air -water interfaces',
(1978),'Transfer
J. Phys. Oceanogr.,
of gas at
Vol
8, PP
543-556.
Brumley B. H. and
turbulence in a grid
Jirka
G. H.
stirred
tank',
(1987),'Near surface
JFM, Vol 183, pp 235-
263.
Brutsaert W. and Gerhard H. J. editors (1984),'Gas Transfer
at Water Surfaces', D. Reidel Publishing Company.
Chun J. H., Shimko M. A. and Sonin A. A. (1986),'Vapour
condensation onto a turbulent liquid II. Condensation
burst instability at high turbulence intensities', Int. J.
Heat Mass Transfer, Vol 29, No. 9, pp 1333-1338.
Cox J. D. and Head A. J. (1962),'Solubility of carbon
dioxide in hydrofluoric acid solutions', Trans. Faraday
Society, Vol 58, pp 1839-1845.
101
Danckwerts
coefficients
P.
V.
(1951),'Significance
in gas
absorption',
Ind.
of
liquid
film
Eng. Chem., Vol 43,
No.6, pp 1460-1467.
Davidson
J. F. and Cullen E. J. (1957),'The determination of
diffusion coefficients', Trans. Instn. Chem. Engrs., Vol 35,
pp 51.
Davies J. T. (1972),'Turbulence Phenomena', Academic Press.
Davies
J. T., Kilner A.
effect
of
A.
diffusivities
and Ratcliff G. A. (I964), 'The
and
surface
films
absorption',
Chem. Eng. Sci. , Vol 19, pp 583-590.
Davies
T.
and
ics
and mass
J.
characterist
AIChE
Lozano
F.
J.
on
gas
(1979),'Turbulence
transfer at air-water interfaces',
J., Vol 25, pp 405-415.
Deacon
E. L. (1977),'Gas
transfer
to and across
an air-water
interface', Tellus, Vol 29, pp 363-374.
Dobbins
W.
E.
(1962),'Mechanism
of gas absorption by
turbulent liquids', presented at International Conference on
Water Pollution Research, London, England, pp 61-96.
Downing A. L. (1960), 'Aeration in the activated sludge
the
Institute of Public Health
process', Journal of
Engineers.
Eloubaidy A. F., Plate E. J. and Gessler J. (1969),'wind
waves and the reaeration
coefficient in open channel flow',
Report No . 69.70 AFE
2, Dept. of
Univ., Ft . Collins, Colo ., U.S.A.
Civil Eng., Colo. State
Ferrell R. T. and
Himmelblau D. M. (1967),'Diffusion
coefficient of nitrogen and oxygen in water', J. Chem. Eng.
Data, Vol 12, pplll-115.
102
Fortescue
G.
E.
and
absorption into a
Pearson
turbulent
J.
R.
A.
liquid',
(1967),'On
gas
Chem. Eng. Sci., Vol
22, pp 1163-1176.
Hannoun
I.
A.,
Fernando
(1988),'Turbulence
H.
J.
structure
S.
near
and
a
List
sharp
E.
J.
density
interface', JFM, Vol 189, pp 189-209.
Harned H. S. and Davis R. (1943),'The ionization constant of
carbonic acid in water
in water
and aqueous
and
the solubility
salt
solution',
of carbon dioxide
J. Amer. Chem. Soc.,
Vol 65, pp 2030-2041.
Hayduk W.
and
coefficient
Laudie
for
solutions', AIChE
Helmick M.
liquid
R.
H.
(1974),'Prediction
non-electrolytes
J., Vol
for
environment', S. M.
dilute
aqueous
20, No. 3, pp 611-615.
(1988),'Vapour
interface
in
of diffusion
condensation
application
Thesis,
Dept.
in
on a turbulent
low
gravity
for Mech. and Nuclear
Eng., M. I. T.
Helmick M. R., Khoo
(1988),'Vapour
B.
C.,
Brown
condensation
rate
J.
at
B.
a
and Sonin A. A.
turbulent
liquid
interface, for application to cryogenic hydrogen', AIAA
2 6 th
Aerospace Science Meeting, Reno, Nevada.
Henstock W. H. and Hanratty
T. J. (1979),'Gas absorption by
a liquid layer flowing on a
wall
of a pipe', AIChE
J., Vol
25, No. 1, pp 122-131.
Higbie R. (1935),'The
a still liquid during
rate of
a
absorption
short
of a pure gas into
periods', Trans. AIChE, Vol
31, pp 365-389.
Hinze J. 0.
New York.
(1975),'Turbulence',
103
2 nd
Edition, McGraw Hill,
study
Ho A. W. K. (1987),'A
water
of
interface
a
agitated
grid
transfer at the air
oxygen
of
tank',
M Sc Thesis,
Cornell University, New York.
Isenogle S. S.
across
an
interface', M
air-water
Munnich
B.,
'Measurement
gas
of
of
Univ.
Geological Sciences,
Angeles, California.
Jahne
laboratory
(1985),'A
K.
Thesis,
Dept. of
Los
California,
Siegenthaler
and
exchange
Sc
Southern
and
0.
study of gas transfer
momentum
U.
(1979),
transfer
in a
circular wind-water tunnel', Tellus, Vol 31, pp 321-329.
Jensen R. J. and Yuen M. C. (1982),'Interphase transport in
a horizontal stratified concurrent flow', U. S. Nuclear
Regulatory Commission Report, NUREG/CR-2334.
Jobson H. E. and
Sayre
open channel flow', J.
Vol
W.
W. (1970),'Vertical transfer in
Hydraul. Div. Am. Soc. Civ. Engrs.,
96, pp 703-724.
liquid phase mass
King C. J. (1966),'Turbulent
free gas-liquid interface', I &
transfer
at a
EC Fundamentals, Vol 5, pp
1-8.
Kishinevsky M.
(1955),'Two approaches to the theoretical
absorption process', Jour. Appl. Chem. of
analysis
of
U.S.S.R.,
Vol 28, pp 881-886.
Kishinevsky M. and Serebryansky V. T. (1956),' The mechanism
of mass transfer at the gas-liquid interface with vigorous
stirring', Jour. Appl. Chem. of U.S.S.R., Vol 29, pp 29-33.
Komori
S.,
Ueda
H.,
(1982),'Turbulent structure and
free surface
Transfer,
in an
open
F.
Ogino
channel
Muzishina
T.
transport mechanism at the
flow',
Vol 25, No. 4, pp 513-521.
104
and
Int. J. Heat Mass
Krenkel P. A.
G. I. (1963),'Turbulent diffusion
coefficient,' Trans. ASCE, Vol 128, pp
and
Orlob
and the reaeration
293-323.
Lamont
mass
D.
J. C. and Scott
S. (1970),'An eddy cell model
into the surface of a turbulent
transfer
J., Vol
16, No. 4, pp 513-519.
Levich
V.
Hall Inc,
and Whitman
absorption',
L.
(1924),'Principles
G.
(1986),'Mass
of gas
16, pp 1215.
in eddies close to
transfer
interface', AIChE J., Vol 32, No. 9, pp 1546-1554
Maday Y. and Patera
the
W.
Ind. Engng. Chem., Vol
Luk S. Y. H.
for
liquid', AIChE
(1962),'Physiochemical Hydrodynamics', Prentice
Englewood Cliffs, NJ.
Lewis W.K.
air-water
of
A.
equations',
Navier-Stokes
surveys in
T. (1988),'Spectral element methods
computational mechanics
'State-of-the-arts
of
(A.
K. Noor editor)',
ASME, New York.
Matteson M. J., Flack W. H. and Woo Y. R. (1984),'Absorption
of gases at condensing
and
evaporating
'Gas transfer at water
surfaces,
water
pp 47-56'.
surfaces',
of
(See Brutsaert
and Gerhard).
Mattingly G. E. (1977),'Experimental studies of wind effects
on reaeration', ASCE, J. Hydr. Div., Vol 103, No. HY3, pp
311-323
McCready M.
J.
and
Hanratty
fluctuation close to a
T.
J. (1984),'Concentration
gas-liquid interface', AIChE J., Vol
30, No. 5, pp 122-131.
105
McCready M. J. and
turbulent mass
at
transfer
a
and Gerhard,
(See Brutsaert
and solid-liquid
gas-liquid
at water surface, pp 283-292'.
of 'Gas transfer
interface',
J. (1984),'A comparison of
T.
Hanratty
1984).
McManamey W. J., Davis J. T., Wollen
mole cular
(1973),'The influence of
transfer between turbulent
liquids',
M. and Coe J. R.
J.
diffusion
on
ma ss
Eng.
Sci
., Vol
Chem.
28, pp 1061-1069.
and Hill H. (1980),'Orbital refill
Merino F., Risberg J. A.
of propulsion vehicle tankage', NASA CR-159722.
Merlivat
L. and
Memery
L.
(1983), 'Gas
exchange
across an
air-water interface : Experimental
results and modelling
bubble contribution to
J. Geophysical Research,
transfer',
of
Vol 88, No. C1, pp 707-724.
Monin A.
S.
and
Yaglom
A. M.
(1975),'StatisticalFluid
Mechanics', Vol I & II, M. I. T. Press, Cambridge, MA.
Munz
C.
and
Roberts
P.
organics contaminants int
aeration', Technical Repc
(1982b),'Transfer
gas
a
No.
phase
262,
of volatile
during
Civil
bubble
Eng. Dept.,
Stanford Univ., Stanford C2
Orridge M. A. (1970),'Gas absorption into turbulent streams
of liquids', PhD Thesis in Chem. Eng., Univ. of Birmingham,
England.
Orszag S. A. (1971),'On the
finite difference schemes by
components',
elimination of
filtering high
aliasing in
wave number
J. Atm. Sci, Vol 28, pp 1074.
Orszag S. A. (1971),'Numerical integration of incompressible
flows with
simple boundaries:
accuracy',
112.
106
JFM, Vol 49, pp 75-
Orszag S. A. (1969),'Numerical methods for the simulation of
turbulence', Physics of Fluids Supplement II, pp 250-257.
Patera A. T. (1983),'Secondary instability
Orszag S. A. and
shear flows', JFM, Vol
of wall-bounded
128, pp 347-385.
Patera A. T. (1984),'A spectral element method for fluid
dynamics: Laminar flow in a channel expansion', J. Comp.
Physics, Vol 54, pp 468-488.
Patera A. T. (1986),'Fast
decomposable
domains',
Poisson solvers for highin
discretisations
element
finite
order
direct
Vol
Physics,
J. Comp.
rectangular
65, No. 2, pp
474-480.
J.
Plate E.
and
Friedrich
transfer
channel flow', of 'Gas
346'. (See Brutsaert
Pope S. B.
near wake
and
(1984),'Reaeration
of open
at water
pp 333-
surfaces,
and Gerhard).
Whitelaw
J.
H. (1976),'The
calculation
of
Vol 73, pp 9-32.
flows', JM,
Prasher B. D. and
R.
Wills
G.
B. (1973),'Mass
transfer
in an
agitated vessel', Ind. Eng. Chem. Process Des. Develop., Vol
12, No. 3, pp 351-354.
Ratbun R. E. (1977),'Reaeration coefficients of streamstate-of-art', J. Hydr. Div., ASCE, Vol 103, pp 409-424.
Ratbun
R.
E.
and
Tai
D.
Y.
(1984),'Volatilization
chlorinated hydrocarbons from water',
water
of
surfaces, pp 27-34'. (See Brutsaert
Shimko M. A. (1985),'Scalar
free surface',
of
'Gas transfer at
and Gerhard).
transport at a turbulent liquid
S. M. Thesis, Dept. of Mech. Eng., M.
107
I. T.
Sivakumar M. (1984),'Reaeration
shear
and wind induced turbulence
body', of 'Gas transfer at water
in a contained water
surface, pp 369-377'. (See Brutsaert and Gerhard).
Shimko M. A. and Chun J. H. (1986),'Vapour
onto a turbulent liquid - I. The steady
Sonin A. A.,
condensation
rate as a
condensation
function of liquid side turbulence',
Int. J. Heat Mass Transfer, Vol 29, pp 1319-1332.
Streeter H. W.
and
Phelps
B.
E.
in
concerned
the
study of the
the Ohio River. III.
of
pollution and natural purification
Factors
(1925),'A
of
phenomena
and
oxidation
reaeration', Public Health Bulletin No. 146, pp 75.
(1979),'Wastewater Engineering :
Treatment, Disposal and Reuse', McGrwa-Hill, New York.
G.
Tchobanoglous
Tennekes H.
and
editor
Lumley
J.
L.
(1972),'A
First Course in
Turbulence', M.I.T. Press, Cambridge, Massachusetts.
Theofanous
T.
Houze
G.,
R.
N.
Brumfield
and
L.
K.
free, gas-liquid
at
transfer
(1976),'Turbulent mass
interfaces, with applications to open-channel, bubble and
jet flows',
Thomas R.
Int. J. Heat Mass Transfer,
M.
(1979),'Condensation
turbulent motion',
Int. J. Multiphase
Townsend A. A. (1951), Proc.
of
Vol
19, pp 613-624.
steam
on water in
Flow, Vol 5, pp 1-15.
Roy. Soc. London, Vol 208A, pp
534.
Ueda H., Moller R., Komori
diffusivity
near the
free
S. and Mizushima T. (1977),'Eddy
surface
of
open channel
Int. J. Heat Mass Transfer, Vol 20, pp 1127-1136.
108
flow',
Won Y. S. and Mills A. F. (1982),'Correlation
of viscosity
and
surface
tension
on
gas absorption
into freely falling-turbulent liquid films',
Mass Transfer, Vol 25, No. 2, pp 223-229.
109
of the effects
Int.
rates
J. Heat
TABLE
1:
MAJOR
CONCEPTUAL
MODELS
FOR
TRANSPORT ACROSS SURFACE
ASSUMPT I ONS
MODEL
1.
diffusivity
time scale
a
Large-eddy
TRANSPORT COEFF.
k - c(a/7)
model
a
(Fortescue &
Pearson)
A/v
k
v
2. Small(Kolmogorov)
eddy model
(Lamont & Scott)
(v/v3
/ 1 2
-1
-1
-1
-1
2 Re2
Re
c
v
c 2 Pr
k
cPr2
Re
-1
3. Viscous
model
T
·.
4. Levich
5. Kishine
v/u* 2
inner layer
,
,
vsky
5*
*
~ ~ ~ . ~~
.
. .....
Pr - v/
Prandtl
Re
Reynolds
vA/v
(v2p/aA)
C1 to C6 :
surface tension
110
c ZPr
5
Re2
k
Z
v - kinematic viscosity
p - density
-a
v
vAv
integral length scale
of turbulent eddies
1
k
Kishinevs~ky|l/|k
a
molecular diffusivity
v - characteristic turbulent
fluctuating velocity
u* - shear velocity
A. -
__-1
a/pv
0t
3
1/ 2
no.
no.
Ohnesorge
constants
no.
Table 2 : LDA measurements of turbulence macroscale time
(r) and length (A = V'r) in system with diameter
of 0.152m at 0.03m beneath the interface
V' (m/s)T
-
- 0J R(t)dt
m (s)
A
- V'r (m)
Horizontal components:
0.062
0.58
0.0360
0.082
0.44
0.0363
0.116
0.34
0.0393
0.160
0.24
0.0283
0.062
0.58
0.0360
0.088
0.42
0.0370
0.131
0.31
0.0405
0.191
0.20
0.0382
Vertical components:
Average
A = 0.0365 m
= 0.24 D
111
Table 3:
Operating conditions of the experiment.
Bulk water
temperature
°C
40
23
29
25
10 6
vx
m2 /
0.6529
0.8148
0.9325
1.840
.
D x 10 9
C
m /s
2.8
2.18
Sc
230
2.407
sat
1.78
1.13
377
525
1600
3.120
3.640
x 10 2
mole/liter
V
is the kinematic
D
is the diffusivity
viscosity
2.630
of the bulk water
of CO2 in the liquid
Csat is saturation condition of C02 in the liquid
*
water mixed with 21% by volume of glycerol
112
Table 4:
Conditions
Sc
V,
(m/s)
at the 'break' point
1600
230
377
525
0.066
0.068
0.073
0.092
3851
3180
2983
1905
33900
34100
36500
36400
break velocity
(Re)*
break
Reynolds number
(Re)
0.4
Average
(ReA)*Sc04
113
3.5x104
Table 5:
St vs Sc for
ReA < (ReA)*
114
ka) .
0
a
5.4
0
44
0
A
0: m
u
a0
00U
Ad
490
;
-
a)
9
0
ao
U
.
U
a)
w
0
-43
a)
0
*-0
a)
4-O
a)
vw
bO
0
14
4
14
0O
,u
CI)
a)
a)
cd
.14
Q
A
o
0
>Q
k
Pr
0.'
bO
0
>a)= bo
u0
·r
43
U
112
cd
H
54
U ) 04
;j
'IO
.+
A
._7
0a)
.4a
10
v,
+3
w
0
a)
· v-.
0
a)
U
o
o
0E©
0
o
a)I-Q
r9
r-
cr)
1.
O
4
00ca
rc
0
o
._2
cJ
'4a)
0
Ita)a
0
o o
4
Q
U
t-
0
H
0)
Q
+3
St
44
X X
0
cF
·C(
a)
0
A
O
X
X
-4
e-"I
ubo
14
S
uu
o
c
%
z
0
0
V
o
00
to
0
.4
+3.
w
0
0co
v
m
0
E$
.43.
CD
o302
4
J
0
_I-
S3
oa)
V
1-4
0
00
01
-0
O0
cn
00
zd
%
0 cn
0
0
E
.'
1o oa
lo. o
oo
U
o
.2
o
-
a)
A~
H4.
*
UV
c cn
.l
co
An-, A
·c
.43
cd
0
O O
0
Ce
C.
Cs
-
A
0
o
CQ
V
Lo
M
14
V
o
Vd
NV
a)
0
0
V CD
C)
ed
tu
*
d
zi
"a
0
a)
a -
UiC UZ
)
_
C4' 3
_
__
1U
C'
0C
"
I-,
CD
Lo
a)
a)
a)
m0
'-4%-.
(3
I--,
0o
a)
t3
CO
a)
9l
a~Q
oO Wr
9 0 u
$-4 C,
V)
q1r
=i F-
frr
0
4
o~~~~~.
1.d
X1
rD-
'4-4
I
bO O
o
o0
cY X H O
115
a)
Zn
r)
C>I q(d
>%
r
.-
4
I
I'
C=
2
a
+
tko
4.
a
Co
U)
4-(
er,-
0a)
0
cl)
Ul
Is
B
a)
C'
I.
C
$-I E
.o
0 _
ed4)
_
04)
o
!-4
._
0
*Q:
4"
co 44
cdE
0
*~~~~~~~~~~~~
.
-'' _
0ua
u
.u
0 4a
. - +
I.-
0Q)
.-O
H
-4--
ce
;fl
Hd
0
r4 1r:
ada)
0'
0,
V
0
I i
P
0
v
3
XX
-- O
ed
4-4
D ceI
bO
O
X
'-4
(d
or
_
_-4
_g
_-
;.
0
C-
U,
.0
AO
-4
34
0
CO
1
x
bO
LtO
O C--4
H
-4
a)
V
8
N
V
3
u7
u
v
v
- -
D
1-4
0
0
~o o 00
o
ri v
1~
?4
-
D
0
s '3
-3
:.,
I?
ID
o
o
Qd
u CQ
0
7)
00
-)
-4
.4
a)
n
-4
a)
64a)
I-
C
_C,
00
g! i-fa
0
au
e
-
Y)
q
-4
) a)
'--
-4
I-
U
I)
42
L) II Q
4
116
0
U-
4 d
8
0
V
jo
°I
U
JCJ0 c
c;
zi
0
-
- -
0
?
_
zi
ZI
VQ CQ
'3
V)
D1
D Ct
0o
0
V
0
CeD
4
Od
*4
O o
C
u
9
0
0
'-4
x
V
a)
-4
a)I
'
d
.2
V
L)
.4
-4
F
Table
8
: Parameters
Reb
Pr
1 /A
associated with
case A and case B
Case A
Case B
1000.0
100.0
1.0,1.5,2.2,3.15
3.15,10.0,50.0
0.111
0.625
0.0362
0.0585
grid arrangement
fig. 6.4.1
fig. 6.4.2
aart(z)/a
fig. 6.3.2
fig. 6.3.3
A X A X A
A X A X A
(6/A)smallest
computational
domain
117
Table
9
: St vs
Pr
for Case
A
St=KL/V'
Pr
Computed
Extrapolated from
Experiment
1.0
0.0185
0.0215
1.5
0.0159
0.0188
2.2
0.01375
0.0165
3.15
0.01212
0.0146
118
Table 10
: St vs Pr for Case B
St=KL/V
Pr
Extrapolated
Computed
from
Exper iment
3.15
0.0158
0.0147
10.0
0.0101
0.0100
50.0
0.0046
0.0058
._
1~~
119
APPARATUS
*
CO,
FOR MASS TRANSFER
IN
DIA.
SYSTE
D -
NOZZ
d
COOLER
Figure 3.1.1
120
Q/Dd
VS
(A P)
.7
.6
.5
U)
N:
.4
-o
C3
.3
.2
.1
0
0
.5
1.0
1.5
2.0
1
2
(AP)
L
(VOLTS)
Figure 3.1.2
121
2.5
3.0
3.5
4.0
CYLINDRICAL
FOR
SECTION
LDA
-
k,
JB
5
0
3
3
L
bX
0
3o
ELEVATION
I
I
I
I
I
I1
I
I
3
3
I/
.C
INI .
'-1
PLAN
Lk_100
mm
w
Figure 3.2.1
122
i
,
0
N
N
C-
5(Re ) vs Re
-IU
I
I
I
I
I
I
I
I
0
35
q\0
I
I
I
F
Vertical component
Horizontal component
O
30
I
0
O
0
25
0
00)
0
gqo_ __
'" ~0
0
.
20
jmi
v*
.
15
10
5
0
_
0
10000
II
I
20000
30000
I
I
40000
Re
I
II
I
50000
60000
70000
s
Figure 3.3.1
123
I
I
80000
I
90000
O(Re ) vs Re S
_
40
__
I
__
I
I
'
'
I
I
35
average for lower Re
30
S
'
I
__
I
I
__
I
I
o
Vertical component
*
Horizontal component
data
__
overall average
0
0
O
25
O
0
*0To
0
P4
9
OF
,cA
0
O
J t- __
0
20
-- i___
A
00
.
0
C)
_
0
S
average for higher Res data
15
10
LDA
A
5
0
Vertical component
Horizontal component
A
I
0
I
10000
I
I
20000
I
I
I
30000
I
I
40000
I
50000
Re
S
Figure 3.3.2
124
I
I
60000
I
I
70000
I
I
80000
I
90000
RMS VELOCITY
PROFILE
.16
.14
.12
Cr)
.10
-J
uJ
.08
C)I
w
(n
2:
.06
.04
.02
0
0
.02
.04
.06
.08
.10
(Zs -Z)/D
Figure 4.1.1
125
.12
.14
.16
.18
.20
RMS VELOCITY
.16
I
.14
I
I
I
I
I
I
I'
correlation
I
I
I
I
I
I
I
of Sonin et al (1986)
-
.12
(0
I
PROFILE
-
t~~~~~
-
.10
t~~~~~~~~~~~~~~~~~~~~~~~~~~
t. .08
U-J
CO
a: .06
.04 _
.02 _
I
0
0
I
.02
I
I
.04
I
a
.06
I
.08
.10
(Zs -Z)/D
Figure 4.1.2
126
I
I
I
I
.12
I
.14
.
I
.16
I
.18
I
.20
RMS VELOCITY PROFILE
_
.16
I
I
I
I
I
I
_
-
__
I
I
I
I
I
I
correlation of Sonin et al (1986)
.12
.1
o()
t
{
.10
>-
o
±
.08
"J
(n
+
C-
I
.06
.04
.02
l
0
0
.02
I
I__
.04
I -~.......
I
.06
I
I I
.08
.10
(Z,-Z)/D
Figure 4.1.3
127
I
I
.12
.14
L
II
.16
,
L
i
I
IiI
.18
I
.20
RMS VELOCITY
.16
I
'
I
PROFILE
i
I
_--_
I
I
correlationof
I
I
Sonin et al (1986)
.1 4
7
7·
.12
.10
o
LU
.08
A
co
.06
-
-
- -
-4'
-
.04
.02
I
0
0
I
.02
I
I
.04
I
I
.06
I
I
_
.08
.10
(Z -Z)/D
Figu re
128
4.1.4
I
I
.12
I
I
1
. 14
I
~~~~~~~~~I
I
I
I
.16
.18
.20
RMS VELOCITY PROFILE
_
.16
__
I
·
I
I
i
I
I
·
I
·
i
__
I
i
I
I
correlation
-- _
I
I
I
of Sonin et al (1986)
.14
12
A
.10
{
o
tF
±
.08
-
--
-2 7
-T
F
iS
.
_
-
Lu
Cr
.06
.04
.02
0
B
0
I
.02
I
I
B
.04
I
B
I
B
.06
I
B
I
B
I
B
.08
I
.10
(s -Z) /D
Figure 4.1.5
129
I
I
.12
I
I
.14
I
I
.16
I
I
.18
.20
RMS VELOCITY PROFILE
.16
.12
.10
>.08
_J
LU
ct
.06
.02
0
U
.U2
.04
.06
.08
.10
(Z -)
.12
/D
Figure 4.1.6
130
.16
.18
.20
RMS VELOCITY
.11
I I I I
-
,I [ I
- --
--
-
-
-
II
-
I
ACROSS DIAMETER
-
-
- ~............
-
I
-
-
I II
-
II I
-
-
I
-
I
-
-
-
I
-
-
II I
-
-
-
-
-
I I I I I
-
-
-[
I I I I
.10
±
.09
-
.08
±
i
.07
.06
C!)
.05
,Of
.04
.03
.02
.01
E
E)
0
-. 01
e
-
0
I I I I I I I I I I I I I I I
.05
.10
.15
I
I
I
-
I
I
.25 --
.20
1( ) I I I I I I I I I I I I I I- I- - I- I- - - - - - - - - - - - - - -
.30
RRDOIRL DISTRNCE/DIRMETER
Figure 4.1.7
131
.35
.40
.45
.50
TURB INTENSITY ON INTERFACE,z,/D
3.67
.20
.18
.16
.14
,
.12
>.10
LU
z
Of
.08
.06
.02
0
0
.1
.2
.3
.4
Q/OD
Figure 4.1.8
132
(M/S)
.5
.6
7
TURB INTENSITY ON INTERFACE,z,/D=-4.17
.20
.18
.16
.14
r .12
IU)
.10
z
02
.08
.06
.04
.02
0
0
.1
.2
.3
.5
Q/Dd
( M/S )
Figure 4.1.9
133
.4
.6
.7
TURB INTENSITY ON INTERFACE,z/D=4.5
.20
.18
.16
.14
-n
.12
I-
z
.10
LIJ
Z:
n
.08
.06
.04
.02
0
0
.1
.2
.3
.5
Q/D ( M / S
Figure 4.1.10
134
)
.6
.7
Ruu (t)
vs
Time
(Re S -25000)
1.1
1.0
.9
.8
.7
.6
%.
:3
Zs
.5
.4
.3
.2
.1
-_
.
1
.d
0
.2
.4
.6
.8 1.0 1.2 1.4 1.6
TIME
1.8
(SECONDS)
Figure 4.1.11
135
2.0
2.2
2.4
2.6
2.8
3.0 3.2 3.4
RuuS(t)
vs
Time
(Re-35200)
1.1
1.0
.9
.8
.7
.6
:l:3
.5
:0
.4
.3
.2
.1
-.
1
0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
TIME (SECONDS)
Figure 4.1.12
136
1.8
2.0
2.2
2.4
2.6
2.8
Ruu (t)
vs
Time
(Re -50300)
S
1.1
1.0
.9
.8
.7
.6
4.
+A
a
.5
.4
.3
.2
.1
0
-.1
0
.2
.4
.6
.8
1.0
1.2
TIME (SECONDS)
Figure 4.1.13
137
1.4
1.6
1.8
2.0
2.2
R uu (t)
vs
Time
(Re $ =70500)
1.1
1.0
.9
.8
.7
.6
4.a
cs
.5
04
.4
.3
.2
.1
0
.1
0
.2
.4
.6
.8
TIME
(SECONDS)
Figure 4.1.14
138
1.0
1.2
1.4
1.6
R(.wwSt )
vs
Time
(Re -25000)
1.1
1.0
.9
.8
.7
.5
.4
.3
.2
.1
0
-(L
-.
1
O
.2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2
TIME (SECONDS)
Figure 4.1.15
139
R(t)
vs
Time
(Res-35200)
1.2
1.1
1.0
.9
.8
.7
.6
.5
.3
.2
.1
0
1
* I
0
.2
.4
.6
.8
1.0
1.2
TIME
1.4
1.6
(SECONDS)
Figure 4.1.16
140
1.8
2.0
2.2
2.4
2.6
2.8
Rw(t)
,ww
vs
Time
(Re S =50300)
1.1
1.0
.9
.8
.7
.6
04
ZI
.5
.4
.3
.2
.1
0
.,
0
.2
.4
.6
.8
1.0
1.2
TIME (SECONDS)
Figure 4.1.17
141
1.4
1.6
1.8
2.0
2.2
R.Ww'
(t)
vs
Time
(Re s =70500)
1.1
1.0
.9
.8
.7
.6
4
+A:
1$X
.5
.4
.3
.2
.1
0
-.1
0
.2
.4
.6
.8
TIME
(SECONDS)
Figure 4.1.18
142
1.0
1.2
1.4
1.6
U2 ENERGY SPECTRUM
10'
_.
I
I
.
I
.
I
.
I
1
..
II
.
I
I
.
I
.
I
.
I
I
.
..
I
I
I
__
.I
I
I
.1
I
I
.
1.1I
.
.
.
.
..
. I
..
I
I
I
.
I
AA
e_
A
A
10-
NX 3
Z
n
bL
0
LL
10 - 2
10-3
10 - 4
5
3
10-5
10 - 6
10 - 2
I
I
I
I I
Il1
I
I
I
I I
10-'
II I I
I
100
FREQUENCY
Figure 4.1.19
143
I
I I I II I I
10l
(Hz)
I
I
I
I I I I I
102
W2
ENERGY SPECTRUM
101
I
100
A
I
_
~~~~~~~
~I I I l ~
I
I
I
I
I___ I
I
I
I
11
1
I
I I I I
I___ I
I
I
I
I_ lii__
I I I I II
I
I
· I
I
I
I
I
I
I
I I I I
A
A
A LnAA
I
A AA
A ~ h~Crhg
A
1
A
A
10o-
A -
m9
I-
10 -2
A
A
n
L:
z
LLJ
10-3
a
N:x
Li
A]
10- 4
5
10-5
10 - 6
10 -2
3
I
I
I
II
IIlt
I
I
I
I
I I I I
10-'
I
I
100
FREQUENCY
Figure 4.1.20
144
I
I I I Il
10l
(Hz)
I
I
I
I
I I
II
102
U2 ENERGY SPECTRUM
10l
I
A
I
I
A
I
II
II
I I Il
AA
II
II
II IIIj
I I I
I
I
I
I
I
I
I
I
11111
I I I
I II IIII I C
I
I
I
I
I
I
I
I
I
I
I
I
A
^AA^~Aa
A
A
10°
Z~
Il
A
A
lt~:
A
10-1
Q1
10 -2
A
c
A
(c
LO
W
Z
W
N
A
10-3
10- 4
5
10-5
3
In- 6
I
-2
10'
I
I
I
I Ii,
I
I
I
I
,,111
I
I
100
10o-
FREQUENCY (Hz)
Figure 4.1.21
145
I
I, I1
10'
I I I I
102
W2
10'
ENERGY SPECTRUM
I
I
I
I
I
·
III
~~~~~~~~~
~ ~~~~~~~~~~~~
I
I
I
I
I I
I
I I
I
I
I-r-
I
I
117
1 1I1
lI
_III
I I I I I
~~~~~~~~~
I
I
I
I
I
II
r
A
A
A AAAP
A
A LA -
100
A
A
SA A
L;
0
I-
10- '
7\
10-2
U,
Qu
Z
L
10-3
N
3x
10 -4
5
10-5
3
_
10 -6
10 -2
L
II L
Il I
I
I
-
i lI
i
100
10o-
FREQUENCY
Figure 4.1.22
146
10l
(Hz)
L I
I II I II
102
KL VS RMS VELOCITY
(SC-230)
.00024
.00022
.00020
.00018
.00016
.00014
"r
.,*.c,
.00012
.00010
.00008
.00006
.00004
.00002
0
0
.02
.04
.06
.08
.10
.12
RMS VELOCITY (M/S)
Figure 4.2.1
147
.16
.18
.20
(SC-377)
KLVS RMS VELOCITY
.00024
.00022
.00020
.00018
.00016
.00014
O-
00012
.00010
.00008
.00006
.00004
. 00002
0
0
.02
.04
.06
.08
.10
.12
RMS VELOCITY
Figure 4.2.2
148
.14
.16
.18
.20
KLVS RMS VELOCITY (SC-525)
.00024
.00022
.00020
.00018
.00016
.00014
r)
.00012
-J
.00010
.00008
.00006
.00004
.00002
0
0
.02
.04
.06
.08
.10
RMS VELOCITY
Figure 4.2.3
149
.12
(M/S)
.14
.16
.18
.20
KLVS RMS VELOCITY
(SC-1 00)
.00024
.00022
.00020
.00018
.00016
.00014
z1-
.00012
J
b~
.00010
.00008
.00006
.00004
.00002
0
0
.02
.04
.06
.08
.10
RMS VELOCITY
Figure 4.2.4
150
.12
(M/S)
.14
.16
.18
.20
KLVS RMS VELOCITY
.00024
I
I
I
I
I
I'
I
(SC=525)
I
I
I
I
I
. 00022
.00020
.00018
Li
.00016
11
,-Cc
>t .00014
J
.00012
UCr
.00010
N-j
.00008
A
.00006
A
A
AA
.00004
A
.00002
0
A ,L
I
I
I
I
I
I
I
I
I
...
d
.02
.04
.06
.08
.10
.12
RMS VELOCITY (M/S)
Figure 4.2.5
151
.14
.
II1
.16
i
II
.18
.20
(SC-525)
K4VS RMS VELOCITY
.00024
I
'
I
I
I
I
I
I
I
Ir
I-
I
r
I
I
.00022
.00020
A
BIG
o
SMALL
.00018
.00016
I-
LU
.00014
IL
A
O3
.-
A
.00012
AAA
OL
I..1
A
.00010
A
A
Jc
A
.00008
A
.00006
IL
~h
.00004
0
0
O
;¢¢r~'!
.00002
IA A
0 10
-
I
I
·
.02
I
.04
I·
I·
.06
IL
I·
I·
.08
I·
.10
i·
I·
.12
RMS VELOCITY (M/S)
Figure 4.2.6
152
·I
I·
.14
1I
II
.16
I
II
.18
1
.20
KS /VI
VS
EDDY REYNOLDS NUMBER
.040
.035
.030
.025
4;-,~
U
.020
.015
.010
.005
0
1000
2000
3000
4000
5000
EDDY REYNOLDS NUMBER
Figure 4.2.7
153
6000
7000
8000
(Kc-[~)/(V'· 0050
I
II
VS EDDY REYNOLDS NUMBER
IM
II
II
I
I
I
I
I
I
I
I
I
I
'
I
E
I
I
I
Sc
0045
.0040
.0035
a
230
0
77
<
525
v
1600
.0030
Th
-I
IN
A
·0025
0
V
· 0020
V
0O
0
AA
VVVv
00 o',,
,~v
.0015
V
A-
I
OO0
o00 00
0
0
.0010
.0005
I
U
I
I
·
·
0
1UUU
I-
2000
II
I
II
'.,,,~~~~~~
-I
!
I
i
I
- -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
3000
4000
EDDY REYNOLDS
5000
NUMBER
Figure 4.2.8
154
a
I
I
I
6000
I
I
7000
I
8000
St
vs
ReA/(ReA)
.0026
.0024
.0022
.0020
.0018
.0016
.0014
u,
.0012
.0010
.0008
.0006
.0004
.0002
0
0
.5
1.0
1.5
2.0
2.5
ReA/(ReA)
Figure 4.2.9
155
3.0
3.5
4.0
4.5
5.0
KL
vs
RMS VELOCITY
(SC-
525)
.00024
.00022
.00020
.00018
.00016
.00014
(n
IN
r
.00012
-a
J--
.00010
.00008
.00006
.00004
.00002
0
0
.02
.04
.10
.08
.06
RMS VELOCITY
(M/S)
Figure 4.2. 10
156
.12
.14
.16
KL
vs
RMS VELOCITY
(SC=
1600)
.00024
.00022
.00020
.00018
.00016
.00014
co
rr- .00012
.00010
.00008
.00006
.00004
.00002
0
0
.02
.04
.06
.08
.10
RMS VELOCITY (M/S)
Figure 4.2.11
157
.12
.14
.16
KLSCz/Q
.050
·
I
I
VS
_
I
Re,
_
I
I
I
1_
I
·
I
·
I
·
·
I
I
·
·
I
I
I
GASES
.045
.040
A
02
0
Co0
N2
.035
.030
a
-CI
()
V
CH 4
o
CzH6
m
C;H8
.
R n
·
He
E
025
8~
J
.020
.
0
.015
90
*e9V
0
A
.
.010
0
.005
I
I
0
0
I
200
I
I
I
I
I
600
800
Re,
Figure 4.3.1
158
I
I
I
1000
t
I
1200
,
__
I
1400
KLSC/V
.040
I
.035
I
I
I
I
VS
I
,
I
R eH
'
I
i
I
I'
'
I
I
I
I
I
I
I
I
I
I-
I
I1
l
-
.030
-
.025 _
A
A
A
u
.o020
A
A
A
.015
A
A
A
A
A
A
.010
A
.005 _
n
0I
%J
-
II
-
20
I
-
I
-
40
I
-
I
-
60
I
-
I
-
80
I
-
I
-
-
I
-
I
-
I
-
-I
I
-
-
I-
I
-
I-
-,
I-
-,
I-
,-
I-
-,
I-
I
100 120 140 160 180 200 220 240 260 280 300
ReH
Figure 4.3.2
159
f(r)
I
I.U
IN
vs
r
A
2
.
_1IX,,,.,__
1
I
1
I
I
I
·
I
.!
I
I~~I
I
I
I
I
'
A
.9 _
A
I
I
I
I
'
A
A
A
-
A
.7
I\
E
A
A
.8
I
A
-
A
.6
-
A
A
%-. .5
A
.4
A
-
A
IL
A
.3
A
A
.2
A
.1
0
A
A
A
x, A
A LAA AAnS
A
-
I
I
I
I
I
I
I
I
I
I
!
f
I
m
I
1
I
0
.1
.2
.3
.4
.5
r
Figure 6.3.1
160
.6
.
I
I.
'.
m
.7
.8
.
I
I
.9
I
1.OA
a art (z)/a
vs
z for case A
200
180
160
140
120
N
"I
e-S
vN
100
.I
k
ed
80
As
60
20
0
0
.1
.2
.3
.4
.5
DISTRNCE
FROM INTERFACE
Figure 6.3.2
161
.6
.7
.8
.9
1.0
---
a art (z)/a
~
vs
z
for case B
LUU
180
160
140
120
N
-
100
80
60
40
20
n
0
.1
.2
.3
.4
.5
.6
DISTANCE FROM INTERFRCE
Figure 6.3.3
162
.7
.8
.9
1.0
Computational
grid arrangment for Case A
Interface
Lower Boundary
Figure 6.4.1
163
Computational
grid arrangment
Interface
Lower Boundary
Figure 6.4.2
164
for Case B
ST VS TIME (ReA-1000,
Pr-l.0)
.10
.09
.08
.07
wo
.06
C
z
z
Co
.05
z
cc
.04
.03
.02
.01
0
0
1
2
3
4
5
6
7
TIME (EDDY TURNOVERS)
Figure 7.1.1
165
8
9
10
11
12
ST VS TIME (Red-1000,
n
. IIU
I
I
,
Pr-1.5)
I
I
II
Ij I
I
I
I
II
.09
.08
.07
ac
LU
.06
z
z .05
Z
.0-
.03
.02
I-
.01
I/
I
n
0
I
1
I
I
2
I
I
I -- --I
3
4
I
I
5
TIME
I
6
(EDDY
7
TURNOVERS)
Figure 7.1.2
166
I
8
I
I
9
i
I
10
I
11
I
12
ST VS TIME (ReA-1000,
I A
.IU
'
II
II
'
Pr=2.2)
I
I
I
I
I
I
II
I
I
I
I
I
I
.09
.08
.07
a-
.06
02
z
z .05
z
I-o
ccd
O .04
.03
.02 _
"I
.01 _
I
I
1
2
I
I
I
I
I
I
0
0
3
5
4
TIME
6
(EDDY
7
TURNOVERS)
Figure 7.1.3
167
8
9
10
11
12
ST VS TIME
.10
I
I
(ReA-1000,
I '
I
I
I
Pr-3.15)
I
I
I
I I
I
I
I
I
I
I
I
I
I
I
.09
.08
.07
-
r, .06
L.J
z
zCo .05
z
CC
to
.04
.03
.02
.01
_
I
0
0
I
1
Ir
I
2
3
I
I
I
I
I
5
4
TIME
I
I
6
(EDDY
7
TURNOVERS)
Figure 7.1.4
168
I
I
8
I
9
10
I
I
11
II
12
ST VS TIME
(ReA-100, Pr-3.15)
.10
.09
.08
.07
L:J
zz
.06
.05
CD
I-
z
CC
c--
.04
.03
.02
.01
0
0
1
2
3
4
TIME
5
(EDDY
6
TURNOVERS)
Figure 7.1.5
169
7
8
9
10
TOTAL ENERGY FLUX ACROSS PLANE (PR=1.0)
1
.018
·
·
·
I
·
_
I
_ __ ___ _
I
_
I
_
_
I
_
I~ ~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~
I
i
I
I
I
1
I
I
I
.016
AA
AA04A A
~AAAALnAA
.0141
A LA
A
A
A AA
A
I
A
.012
x
_J
LL
>
.010
-0
z
_j .008
cr
0
.006
.004
.002
n
u
I
0
I
.1
I
I
.2
I
I
I
I
,
.3
I
I
.5
I
I
.6
DISTRNCE FROM INTERFRCE
Figure 7.1.6
170
I
.7
I
I
.8
I
I
.9
I
1.0
TOTAL ENERGY FLUX ACROSS PLANE (PR1 .5)
.014
.013
.012
.011
.010
.009
X
-J
.008
zLU
.007
_J
.006
(0
.005
.004
.003
.002
.001
n
u
0
.1
.2
.3
.4
DISTANCE
.5
.6
FROM INTERAFCE
Figure 7.1.7
171
.7
.8
.9
1.0
TOTAL ENERGY FLUX ACROSS PLANE (P R =2.2)
.012
.011
.010
.009
.008
.007
0
z
.006
LI
CE
C)
.005
I
.003
.002
.001
0
0
.
.2
.3
.4
DISTANCE
.5
.6
FROM INTERFACE
Figure 7.1.8
172
.7
.8
.9
1.0
TOTAL ENERGY FLUX ACROSS PLANE (PR=3.1 5)
.016
I
'
I
I
I
I
I
*
I
I
I
'
I
A
A
.012
A
x
A
.010
A
A
_m,
>-It
LLJ
A
A
.008
A
Z-J
IC
I--
.006
A
.004
.002
I
I
I
0
0
.1
I
I
I·
.2
I·
I·
.3
·
·
·
.4
·
.5
·
·
·
.6
DISTANCE FROM INTERFACE
Figure 7.1.9
173
·
.7
·
L
.8
I·
I·
.9
I
·
1.0
TURB ENERGY FLUX ACROSS PLANE (PR=1.0)
1
.018
·
I
I
·
·
--·
·
·
I
I
I
I
I
-
1
I
-
_
I
_
_
I
I
Ir
_-
[
I
.016
A
A 'L, A A
.014
A
A
A
A
AA
.012
AAZCZ~ Z
A
A
A
X
J .010
A
LL
A
A
LO
.008
A
A
82
.006
A
.004
A
.002
,
0 1
0
,
I
I
I
B
U~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I
l
I
l
I
I
I
I
I
I
I
I
I
I
I
i
\
.1
.2
.3
.4
.5
.6
DISTRNCE FROM INTERFRCE
Figure 7.1.10
174
.7
.8
.9
1.0
TURB ENERGY FLUX ACROSS PLANE (PR=1 .5)
.014
T
I
'
I
~ ~
7
I
w
E
.
I
I
-r 1
7
-
-
.013
f
T
I
I
A
AAZA
.012
A
A
A
A
.011
A
AIL
.010
IL
A
A
IL
.009
A
X
LL
.008
I- .007
UL]
ILl
0h
A
.006
A
.005
.004
A
.003
.002
A
.001
0
.
L---------------------L----C
0
.1
I
i
I
.2
I
l
I
.3
l
,
IIl
Il
.4
l
II
l
Il
.5
I
l
II
.6
DISTANCE FROM INTERFACE
Figure 7.1.11
175
I
I
.7
.8
·
IL
.9
·
1.0
TURB ENERGY FLUX ACROSS PLANE (PR=2.2)
_
.012
'
·
I
_·
I
I
'
·
·
'
·
I
1
.011
_·
I
A
A
.010
A a
IL A
I
I
1
'
r
I
I
j
'2
·
·
I
A
A
i A 64n
A A
IL
·
I
a
A
. 009
A
A
- A
A
/\
.008
A
x
LI
.007
-A
>
L3
D-'
zIJ
Zr
IJ.
n-
.006
A
.005
A
A
.004
.003
.002
A
A
.001
0
I
..
O
.1
I
.
.2
I
·
I
.3
I
.I
I
I
I
.5
.4
I
I
.6
OISTANCE FROM INTERFRCE
Figure 7.1.12
176
I
I
I
.7
.8
.9
I
I
1.0
ENERGY FLUX ACROSS PLANE (PR=3.15)
TURB
.016
'
.014
I
'
I
I
I
I
·
I
I
I
__
'
I
l
__
__
I
I
I
-
A
.012
A
-
A
A
x
A
.010
ALr
AA A
LD
LU
z
A
.008
A
IU
ACb~
006
A
A
A
*006
A
.004
A
B
A
.002
ZN
A
-3
0
11
.-
0
I
I
.1
I
.2
I
.3
I
I
I
.4
I
,
.5
I
I
.6
DISTANCE FROM INTERFACE
Figure 7.1.13
177
I
.7
.8
I,
I
.9
I
#
L
1.0
TOTAL ENERGY FLUX ACROSS PLANE (PR=3.1
.020
-
I
-
|
I
|-
XX
'
I
s-
w
X
I
i
lw
w
|~~~~~~~T~
X
I
I
I
-
I
-
-
-
I
-
i
-
I|
-
F
5)
w
i
-
I
-
I -
.018
Ia
.016
A
6A
A
aA
A
A
A
A
A
.014
tAA
A
A
I
I
A At=
A
x
moaAA
-J .012.
I-
LO
zLU
.010
-J
cr
0
-
. 008
.006
.004
.002
I
0
0
.1
I
I
.2
I
I
.3
I
I
I
I
.4
I
.5
DISTANCE
I
I
.6
FROM INTERFACE
Figure 7.1.14
178
I
.7
I
.8
I
.9
I
1.0
TOTAL ENERGY FLUX ACROSS PLANE (PR=1 0.O)
.012
'
.011
I
I
I
'
I
I
I
I
I
i
I
I
I
I
I
A Aa
ha
Ah Aa'r.A
I
I
I
I
-
.010
-
A/ AA
.009
A4S A
A
A
A ALA
A
A
A
.008
A
waA
A
0
I
'L_ .007
z
,J .006
.J
--o
CD
.004
.003
.002
.001
-
-
-
-
I
0
0
I
.1
I
I
.2
I
I
.3
I
I
I
I
I
.5
.4
I
I
.6
.7
DISTANCE FROM INTERFACE
Figure 7.1.15
179
I
.8
I
.9
I
1.0
TOTAL ENERGY FLUX ACROSS PLANE (PR-50.0)
_
.011
x
_·
I
.010
w
.009
m
.008
m
.007
m
'
I
I
'
·
·
·
'
I
'
1
I
I
·
1
'
I
_·
'
I
_·
'
LL
, AA A
>- .006
A A
A
A
LU
zZ
·
.005
-J
A
·
I
-
A
A
AA
_J
·
A
A
AAN
A
A
A AAA A A
A
O
.004
m
A
.003
.002
.001
I
n
U
0
.1
I· _
I·
.2
·
I·
.3
I·
I·
I
I·
.4
I
.5
DISTANCE
I·
·
.6
FROM INTERFACE
Figure 7.1.16
180
IL
.7
I·
I·
.8
·
II
·
.9
·
1.0
TURB ENERGY FLUX ACROSS PLANE (PR-3.15)
_
.020
.018
·
·
I
I
I
__
·
I
I
i
·
·
I
II
·
·
I
·
·
·
I
I
·
_·
·
I
'
I
l
-
.016
-
A
A
.014
A
A
A a
A
A
A
A
A
· .012
LO
A
A
A
>-
A
j
A
008
008
A
A
A
.006
-
A
.004
A
A
A
.002
A
i
_
0
I
·
A
I
I
.1
.2
- I
I
.3
-
LI
-
.5
.4
I
I
.6
DISTANCE FROM INTERFRCE
Figure 7.1.17
181
I
I
Ik
... IIII
.7
.8
.9
L
1.0
TURB ENERGY FLUX ACROSS PLANE (PR-10.0)
.012
.011
.010
.009
I
I
I
I
I
I
I
I
I
I
-
-
-
.008
z
A
x
A
AA
A
AA
A
- .007
A
LL
A
LO
A
o~ .006
.U
o_
A
.005
.004
.003
A
A
m
-
A
A
.002
-A
.001
n
A
A
AA
4 .
I
I
*
-
I.
m
.1
0
To
1
I
I
I
I
.2
I
I
mm
I
.3
I
I
I
$
I
m
.4
I
w
I
m
.5
I
I
I
m
.6
OISTRNCE FROM INTERFRCE
Figure 7.1.18
182
m
\
I
m
.7
m
Im
m
I
m m
.8
I
I
I
zA L
d
.9
I\
1.0
TURB
ENERGY FLUX ACROSS PLANE (PR-50.0)
.011
'
I
'
I
I
I
'
I
'
I
I
I
'
I
I
I
I
.010
.009
.008
x
.007
-
AA
LAA
-j
UD
A
A A
iA
A
A
A
A
IL
Z
A
' .005
A
I-
.004
A
A
.003
'5
-
A-
.002 -A
A
.001
i a
0
j i
0
I
I
.1
I
I
.2
I
I
.3
I
I
I
.4
I
.5
I
I
I
.6
DISTANCE FROM INTERFACE
Figure 7.1.19
183
I
.7
I
I
.8
I
I
.9
I
I
1.0
RMS VELOCITY PROFILE (u,,,,, )
1.6
_
I
I
I
I
_
I
I
--
I
I
r
-
I
.
.
I
.
I
1.4
A_
1.2
A
1.0
x
-
.8
.6
.4
A
-
A
-
A
A
.2 _
p
A
A
I
I
I
A
Ai
,
n
J
I
Li
0
.1
A
I
__ I
.2
A
,
I
.3
I
.
I
.
.
.4
.
.5
.
I
.
I
.
.6
DISTANCE FROM INTERFRCE
Figure 7.2.1
184
I
.
.7
.
I
I
.8
I
.
I
.
.9
..
1.0
RMS VELOCITY PROFILE
1.6
-
I
I
I
· ··
·I
-
I
-r ·I ·
(
I
II
-1
·I
·
RM5 )
I
r~~~~~~
I
II
r
1
I
I
A
1.4
1.2
A
1.0
wA
A
.8
A
.6
AA A
A
.4
A
.2
A
A A
k4=4,6hocss
I
0
0
I
.1
/SZ
A
A
A
LA
I
I
.2
I
I
.3
I
I
1
.4
I
.5
I
I
i
.6
DISTRNCE FROM INTERFRCE
Figure 7.2.2
185
I
.7
I
I
.8
I
I
.9
1.0
RMS VELOCITY PROFILE
1.6
I
I
I
I
( WRm
I
I
I
)
1
I
I
I
1.4
A
1.2
z
1.0
A
.8
A
.6
A
AL
.4
A
A
.2
A
A
A
I
II
A
A
A
Ag~l
0
w
0
I
IAA
.1
I
.2
I
.3
I
I
I
.4
I
.5
.6
DISTANCE FROM INTERFRCE
Figure 7.2.3
186
I
.7
l
I
.8
I
.9
I
1.0
RMS VELOCITY
PROFILE (URMS)
4
1.L
1
I
I
I
I
I
_
I
'
I
I
I
I
I
I
1.1
1.0
.9
L
.8
.7
:D
A
A4
.6
1-
A~~
A
Li
.5
ZA
A
A
.3
A
A
.2
A
A
A
.1
A
0
I
m
0
I
m
I
I
.1
I
IA A
I
Im
I
.2
i
I
Im
.3
I
Im
I
.4
I
l
.5
I
I
m
I
.6
DISTANCE FROM INTERFRCE
Figure 7.2.4
187
I
.7
I
I
.8
I
I
.9
I
1.0
RMS VELOCITY
1.2
~.
I
'
I
PROFILE (VRMS)
.
.
I
I
.
.
.
.
I
.
.
-.
.
.
.
.
I
I
I
.
.
i
I
1.1
2
1.0
2
.9
A
.8
A
A -
A_
A
.7
A
>
.6
A
.5
A
.4
I
A
.3
IL
Z~'
.2
A
A
.1
AIA
r,
u'
I
O
I·
II
.
II
A
A
I
.2
I
I
.3
It
A
I
I
I
.4
A
A
AA
I
.5
I
I
l
.6
DISTRNCE FROM INTERFRCE
Figure 7.2.5
188
S
I
.7
I
!
.8
I
I
.9
1.0
PROFILE (WR,, )
RMS VELOCITY
1.2
I
I
I
I
I
I
I
I
I
I'
I
i
I
I
I
I
1.1
1.0
A
L
A
.9
A
-
A
.8
Ai
A
.7
IU
A
.6
A
A
A
A
.5
.4
A
A
.3
A
A
.2
A
A
A a
- ~A AA
.1
~AA#I
.
aI I. ,..
0
.1
.1
I
.2
I
I
.3
I
I
I
II
.4
.5
.6
DISTANCE FROM INTERFACE
Figure 7.2.6
189
I
.7
I
.8
I
I
.9
I
1.0
U2 ENERGY
10-2
_
SPECTRUM
I
I
I
I
Ir
I
I
I
I
0
A
A
10-3
I
I
I
I
I -
Depth
0.04A
o 0.155A
0
0
I
0 0.5A
0o
A
CI
L
¢\ 0
10 -4
I
u,
)
3
a
0
10-5
zIJ
L.
N
:::
0L,
10 -6
OA
0
A
A A
A0
10 - 7
A
10-8
I
I
I
I
I
i
I
t,
100
10l
WRVE
NUMBER
Figure 7.2.7
190
I
I, I , ,
·
I i I
I
II
I
I
I
I
I
102
V 2 ENERGY SPECTRUM
-
o10
I
I
_
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I-
10-2(
V
10-3
oO
0 °060
°o0f
U
0
LA
10 - 4
/A,
-
A
A
L-
I
10-5
10-6
I
0
A0
A
10- 7
n nA
nA
-
1n - 8
I
I
I
I
I
I
I
I
)O
1C
i
I
10'
WRVE
NUMBER
Figure 7.2.8
191
___
i
i
i
i
i
I
i.
102
Wz
10-l
ENERGY SPECTRUM
I
I
I·
I
I
I
1
I
I
I
I
I
· I_
I
I_
I
I
I
I
I
I
I
I
I
I
I
i
I
I
-
10-2
3UzC
0
10-4
A _
LU
A L-A
3
A
A
LIJ
Z
A
10-5
e3
0O
10 -6
At11
10 - 7
A
A
10 - 8
I
I
I
I
I
10'
100
WNVE
NUMBER
Figure 7.2.9
192
I
.
.
.
I
102
U2 ENERGY SPECTRUM
. --
lU -
i
I
I
I
I
t
I
I
I,
I
A
80
10-3
I
I
I
I
I
.
r
0.09A.
o 0.6A
2~~~~~~~i)
\
%og
a VO~~~~~~0
%
A,
10 - 5
I
Depth
~
10 - 2
10 - 4
I
n-
-0.5
_0
<X&
r
r
10- 6
-
=
A
I-
U
W 10 - 7
A
LLJ
-
_r4
_ff
10-9
`
-C:
.·'U.,-1r
-10
AA
10-11
10 -12
A
-
¶ o-
1
A
3
,
1 n-l4
,
I
I
I
IAt
100
I
10
WRVE
NUMBER
Figure 7.2.10
193
I
I
I
I
I I
102
VZ
ENERGY SPECTRUM
10'
I
I
I
I
I
I
Il
I
I
I
I
I
I
I
I=
Depth
10 - 2
0
10-3
2
A
o
O8
0.09A
0.6A
\o
00
10- 4
L(,)
-1..
0
10-5
2: 10-6
D
Ic-)
A
A
10-7
U
w
bJ
LO
ck:
w
(n
zu
3
0_0
A
10-8
10- 9
A
10o-0
A
10-l
10 -12
A
A
A
I I
I
I
I
I
i A~~~~i
100
I
10l
WRVE
NUMBER
Figure 7.2.11
194
I
I
_
I
I
_
I
I
!
_
102
W2 ENERGY SPECTRUM
10-1
10-2
O~~~
I-
I
I
I I
I
I
I
I
O0
°
I
I
r
-
~~~Depth
A 0.09 A
\o
o
Vc
.
10o-
II
i I-
oo
10-3
0.6Y-
0O.09A
o.5A
5
tS~~
A
?o
/\ O
'\ is),\
-- U
10-s
r
I
v.
A
10 - 6
D
U
IC-,
A
10 -7
LI-i
10 - 8
0
z
Lii
CA
X
A
10-9
A
800
10o10
10-11
A
A
10 -12
10o- 13
A
A
I
10-14
I
I
I
II
I
100
I
I
101
WRVE NUMBER
Figure 7.2.12
195
I
I
I
I
I
I
I
I
102
TEMPERATURE PROFILE (PRANDTL-1.0)
1.0
.9
.8
.7
A
Uj
A;
a:
LU
a:
a:
LU
.5
cr
W
cr-
A
.4
A
.3
A
A
A
.2
A4N
A
A
A A
6n
A
.1
A A AAA
A
A
I
0
.1
.2
.3
.4
.5
.6
DISTANCE FROM INTERFRCE
Figure 7.3.1
196
.7
.8
.9
1.0
TEMPERATURE PROFILE (PRANDTL-1.5)
4
,"
1.U
A
-
I
'I
1
I I
I
I
1
'
·
·
'
·
I
·
--1
I'
·
'
·
-r
I
I
·
·
I
·
I
I
.9
.8
.7
llJ
.6
A
0,
A
.5
-A
a:
c:
A
IL
C>
IL
A
.3
A
A
.2
A
A
.1
I
0
.1
I
l
.2
I
L
A
A A
I
.3
AA
nnnA
A AA
AA
I
.4
I
I
.5
.6
I|
l
DISTRNCE FROM INTERFRCE
Figure 7.3.2
197
I
l
.7
A
|
IA
|
.8
ll
A I
-
l
.9
A.M
{
--
nI\
A
wE~~~~~\
1.0
TEMPERATURE PROFILE (PRANDTL=2.2)
1.0
/
IA,
I
I
I
I
I
I
I
I
·1
I
I
I
1
r
I
-1~~~~~~~~~~~~~~~~~~
A"
.9
.8
.7
A
LIJ
D
.6
a:
LU
LU
.5
CD
C:
A
.4
.3
.2
-
A
A
A
Zn
/
A
A
ILA
.1
AA,
AA A
I
0
.1
,
I
I
.2
.3
I
I
A
I
I
.4
I
.5
DISTRNCE
FROM
I
I
.6
INTERFRCE
Figure 7.3.3
198
A A
z
I
.7
AI1
I
.8
.9
A
IA
I
1.0
TEMPERATURE PROFILE (PRANDTL=3.15)
1.0 /
I
im
.94
I
I
I
I
I
I
I
I
i
I
I
I
L
.8
.7
.
.6
'A
CL
cr
,i
LA
.S
A
C.3
*3
A
A
A
.2
_A
AA,
.1
4
I AAAA
A AA
I
0
0
I
.1
I
I
.2
I
I
.3
I
I
I
.4
I
.5
i
I
.6
DISTANCE FROM INTERFACE
Figure 7.3.4
199
I I - IA
.7
1A
.8
i
A I
.9
Al
A
1.0
TEMPERATURE PROFILE (PRANDTL-3.15)
nl l
I
,,
I
.
.
I
i
i
'
'
I
I
I
I
I
i
'
I
I
I
I
I
A
.9 -A
A
£
.8
aA
A
.7
A
uJ
0::
.6
A
LU
0(/
LJ
I-
.5
A
_
a
La
a:
LU
cr
A
.4
A
.3
A
_
-AM
A
.2 _
A
A
A
.1
A
_
I
0
0
I
X
.1
i
I
I
I
.i
.2
I
I
I
.3
n
I
I
I
I
,
I
,
I
a
.5
.4
I[-
I
A
I
.6
DISTRNCE FROM INTERFRCE
Figure 7.3.5
200
A
II
I
.7
I
I
A !~~
.8
NAAMAR
_ ____M
M1
.9
1.0
TEMPERATURE PROFILE (PRANDTL=1 0.0)
1.0
k
_
_·
I
.9
I
I
I
'
I
I
·
'
I
·
'
I
I
I
·
_·
'
·
I
·
_·
A
A
.8
A
.7
A
Z
.6
A
LL .5
A
CD
cc
cZ
A
_
.3
A
IL
A
.2 _
A~
A
A
_
.1
I_
0
0
I
.1
I
I
.2
I
I
.3
I
I
A
I·
.4
A
I
.5
.6
DISTANCE FROM INTERFACE
Figure 7.3.6
201
I,
I
.7
A
t
A.A N_ __Amur~
· ___··_ j\
II
-
.8
.9
i
1.0
TEMPERATURE PROFILE (PRANDTL=50.0)
1.0
A
I
I
I
r
I
I
I
I
'
'
I
I
.9
A
.8
.7
-A
LU
I-
.6
A
a:
LL
C
A
.5
LIJ
ILU
A
A
.4
A
.3
A
IL
A
.2
A
A
.1
I
0
0
.1
I
.2
I
I
.3
I
I
I
.4
AAA ,
I
.5
I
A
202
AAAA
" ~A4&wyL
I
.6
DISTANCE FROM INTERFRCE
Figure 7.3.7
A
.7
.8
.9
1.0
vs
aT(z)
100
I
z
_____
I
I
I
I
(Pr-l1.0)
I I
I
I
I
__
I
_
I
_
I
____
I I
I
__
I
I
I
_
I
I
_···
I
I I I i
7
10o-
AL
z
~c~n~n~n~_
~r
A
10-2
A!
A
A/
A
t%
EO 10
-3
A
A
10-4
A
./
10
-s
10 -6
10-3
A
I
·I
·I
_I
·I
_ I _I _I ·I ·
10
·I
·I
·I
·I
_I ·I _I ·I
·I
I
II
·I
I I
_I I··
I
-2
10o-
DISTRNCE FROM INTERFRCE
Figure 7.4.1
203
100
vs
aT(z)
10 °
I
I
.
.
.
.
z
I
.
.
.....
I
(Pr-1.5)
.
.
I
I
I
I
.
I
.
I
,,
.
~
.
,
I
.......... r
I I
r
I
.
I
~
s
I
I
.
.
.
r
I
.
I
]
I
Tz
.
I
I
I
I
I
I
I
=-
10o-
a -
A
A
a
10 - 2
A'
A
A
10-3
A
10-'
A
10- 5
10 -6
10-3
A
L
I·
I·
I·
I·
I_
I·
I· I·
·
·
10-2
_I
_I
I I
_1·_
I
10
DISTRNCE FROM INTERFRCE
Figure 7.4.2
204
I
-l
I
I
1I
_I
III
_··
100
vs
aT(Z)
100
_
I
I
z
I
I
1
I
(Pr-2.2)
·
I
--- ·
I
·
I
·
·
·
·
I
·I
I
·.
I
1
·
I
I
-·
I
1
·
I
.r
I
rr
I
I . I.
A
10o-
4$L
L~7
A /xA AA i
A%
A -
A
10-2
A
N
A
A
%P 10 - 3
v
A
10-4
A
10-
10 - 6
10- 3
A
A
A
I
I
I
I
I
I
I
I
I,
10
I
I
-2
I
I I I
I
10o-'
DISTANCE FROM INTERFACE
Figure 7.4.3
205
I
I
I
I
I
I II
100
vs
aT(Z)
_
I
100
I
z
_____
I
I
I
(Pr-3.15)
___
I I
I
I
I
______
I
I
I
I
I
I
I
I
_
I
I
____
I
I
I
I-
A
10o-
A ALA
A:
AA
-
AI~A
z~AZA
~
10-2
AA
4
AL
/\Y
a
A
t 10-3
A
10- 4
A
/
10-5
K
10 -6
10 -3
I
I
·
I
I
I
·
a
·
I
I
· I· ·I ·
·
I
I
I
I
-
10 2
·I
I
·
I
· I· II I·
10-1
OISTANCE FROM INTERFRCE
Figure 7.4.4
206
I
·
I
·
I
·
I
·
I
I I
· 1··.
I
-
100
T
102
I
vs
PRANDTL NUMBER
I
I
I
I
.
A
u-i
-
=
-
I
·
I
I
-
-
-
I
A
Re - 100
o
Re
1000
A
A
w
0z
r-
10l
00
L
F-
)
100
100
0 0
a
0
I
I
I
If
I
I
IX
I
m
II
II
II
I
101
PRRNDTL
NUMBER
Figure 7.4.5
207
I
I
I
I
I
I
I
102
vs
:T(z)
100
I
·
I
·
I
z
·
I
(Pr-3.15)
·--
·-rr
I
I
I
-- ·
I
II
-7-
I
·
I
-·
I,
··-·r
I,
-·
II
·
I
·
I
·
·
I
I
·
7r
I
I
I-
10o-
2ptA :
AL
10-2
7
A
7Z
i
A
/
N
vtr
10 - 3
A
A
A
A
A
10'
A
A
10- s
A
10 - 5
A
10 - 7
10 - 3
I
I
I
I
I
I
I
l
I
I
I
10 -2
I
II
I
I
I
10o-
DISTRNCE FROM INTERFRCE
Figure 7.4.6
208
I
I
I
I
I
I
I
I
100
vs
CT(Z)
100
I
_ ·I
I
I
z
I
(Pr-10.O)
I
I
I
I
I
I
I
- - -1I I· _·
II
I
1
·I!
_·I
1 r
~~~~~~~~~~~~~~
.
.
.
.
.-
/
/7
10-'
A
A
/
10 -2
z
/
A
4
7
AA
A1~A
A
10-3
t4
A
[-
A
10-4
AC
AN
-
A
/
10 - 5
A
10-6
10 - 7
10-3
A
I
I
I
I
I
I I tI
I
I
I
10-2
III
I
I
I
II
10o-
DISTRNCE FROM INTERFRCE
Figure 7.4.7
209
I
I
I
I
I
II
100
aT(z)
100
I
vs
I
z
...
I I
I
(Pr-50.0)
,,.
I
I
I
I I
I
.,..,.
I
I
I
I
I
l
I
I
I
I
II-
/
7
10'l
A
A
10 -2
z
i
A
A
A
A
10-3
A
A
/
10 -'
A
A
A
A
10-5
A
/
10-6
A
I -7
I
l
.
.
..
1,
o
_
10-3
I
II
I
I
I l
I I
I
I
I
10 -2
I
II
I
10o-
DISTRNCE FROM INTERFRCE
Figure 7.4.8
210
I
I
I
I
I
I
I
I
100
STANTON NUMBER
VS
PRANDTL NUMBER
lo-1
LLJ
z 10-2
Z
I-
10-3
10°
o10
PRRNDTL
NUMBER
Figure 7.5.1
211
102
1
Appendix
A schematic view of the laser set-up is given below.
MOdEI:
l
...
...
ar
I
I..W
I SfAMu I4TT
axe
m~n
uWa
n
~I IASnAoAT
IA4UIW*0
1.
am
t I-a
r*m
MS7
oA
waIrus1o
Ith0~
ws'
LS. &T
nLT
uTAATLII
U A
uIa)
.
atNC fIJTOA
fAIlOl
GAO)
It SATIahlla
front
--> lens
series of mirror
of
the expanders
reflections
to
reach the desired
of
equation
given nominally
location
through the 310mm focal length front lens.
before converging
The
go through a
coming out
The 3 beams
the
velocity
in a LDA
is
measurement
as
V
A.1
fDA/(2sinO/2)
where V is the fluid velocity
in m/s
fD is the doppler frequency in Hertz
A is the wavelength
of
laser
beams
in metres
0 is the angle between the intersecting
be ams.
212
laser
Boadway
and Karahan
found that equation A.1 is
beams
refracted through a flat
(1981)
still applicable when
have
the
surface into another medium,
wavelength
in
air
and
0
in
which
is
the
intersection.
213
A is taken
unrefracted
to be
the
angle
of
Appendix
In order to make
number, we need to
near the
constant water
better
have
interface
2
for
height.
a
comparison with our critical
fair estimate
the
flow
In the
in
of the macroscale
an
open channel of
absence of any data on direct
measurement of macroscale, we state that the macroscale
should not be very different from the Prandtl mixing length
L, given by
VT = L
u/Dy
A.2.1
where T is the turbulent eddy viscosity and au/ay is the
average velocity gradient. Jobson et al (1970) made some
detailed velocity measurements in a open channel flow and
their experimental
data
can
be
fitted
empirically
by the
expression,
A.2.2
VT=kuy((1-y/h)(1+0.4(y/h)2)where k is the von
velocity based
Karman
on water
constant,
density,
measured
from the base up and
separate
experiment
the velocity
h
by Ueda et
profile
fit
except for regions very
regions further away
the
close
from
the
u,
y
is
is
the friction
the distance
is the water height.
In a
al (1977), it was found that
logarithmic
to
law quite well
the interface. Hence for
interface,
velocity can be
taken to be
u/U=(ln(yu*/v))/k +
The expression
2
for L , not
then be expressed
5.75
A.2.3
very close to the interface,
can
as
L2=k2y2(1-y/h)/(1+0.4y2/h2)
214
A.2.4
The maximum
value
and decreases
length
of
toward
to be O.lh
the
should
the macroscale which
turbulence
L
is
obtained
is 0.174h at about y-0.6h
interface.
give
us a
So
taking the mixing
reasonable estimate of
responsible for characterising
near the interface.
215
the