metal cable tray thermally insulated on the inside for the remainder. The isothermal zone box was lined inside with 0.8 mm copper sheets to conduct away local ...
NASA Contractor Report 3510
Turbulent Boundary Layer Heat Transfer Experiments: Convex Curvature Effects Including Introduction and Recovery
T. W. Simon, R. J. J. P. Johnston,
M&at,
and W. M. Kays
GRANTS NSG312Li and NAG 3-3 FEBRUARY 1982
TECH LIBRARY
KAFB,
NM
IInIlll~llll~lilllnlll llllb2227
NASA
Contractor
Report 3510
Turbulent Boundary Layer Heat Transfer Experiments: Convex Curvature Effects Including Introduction and Recovery
T. W. Simon, R. J. Moffat, J. P. Johnston, and W. M. Kays Stanford Stanford,
University CaZifornia
Prepared for Lewis Research Center under Grants NSG-3124 and NAG
National Aeronautics and Space Administration Scientific and Technial Information Branch
1982 $ 4
d-
3-3
TABLE OF CONTENTS Page Nomenclature
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
V
Chapter 1.
INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 10 11
THE EXPERIMENTALAPPARATUS. . . . . . . . . . . . . . . . . . .
19
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
. . . . . . . .
19 20 21 24 25 28 28 31
THE EXPERIMENTALRESULTS . . . . . . . . . . . . . . . . . . . .
39
1.1 1.2 1.3 2.
3.
3.1 3.2 3.3 3.4 3.5 3.6 4.
Previous Research .................. ..................... Objectives ................... The Experiment
General Description . . . . . . . . . . . . . Hydrodynamic Aspects of the Facility . . . . Heat Transfer Aspects of the Facility . . . . Instrumentation for hydrodynamic Measurements Instrumentation for Heat Transfer Measurements Qualifications of the Facility: Hydrodynamic Qualifications of the Facility: HeatTransfer The Energy Balance . . . . . . . . . . . . .
. . . .
. . . . . . . . . .
. . . . . . . .
The Baseline Case .................. ... The Effect of Initial Boundary Layer Thickness The Effect of Upw .................. ....... The Effect of Free-Stream Acceleration ....... The Effect of Unheated Starting Length The Effect of the Maturity of the Momentum Boundary Layer ........................
. . . . . . . .
. . . . . . . .
. . . . .
. . . . . . . .
. . . . .
. . . . .
39 43 46 48 49 51
HEAT-TRANSFERPREDICTIONS . . . . . . . . . . . . . . . . . . .
67
. . . .
67
. . . .
72
. . . . . . . . . . . .
74 74 75
. . . .
75
CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.1 5.2
85
4.1 4.2 4.3 4.4 4.5 4.6 5.
1
State of Art . . . . . . . . . . . . . . . . . . . Predicitng the Effect of Curvature for the Baseline Case . . . . . . . . . . . . . . . . . . . . . . . Predictions of Cases with Differing Initial Boundary Layer Thickness . . . . . . . . . . . . . . . . . . . . . . . . Prediction of Cases with Differing U Prediction of Cases with Free-StreamPxcceleration . Prediction of Cases with Differing Unheated Starting Length . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . Recommendations for Future Work on the Convex . . . Problem.......................
87
-._...
.
_ .-.. ..--.-
-~
Page References..............................
89
Appendices A.
The Uncertainty
B.
Curved-Wall
Analysis
Integral
Parameters
and Energy Integral C.
Calibration
D.
Simple Model for
............
Equations
Boundary Layer Cases E.
Listing
of Profile
F.
Listing
of the Stanton
G.
The Reduced Data
H.
Modifications Predictions
......... .......
Program
...........
Program
.........
................
to STAR5 for
iV
. . . . .
99
. . . . .
102
. . . . .
105
. . . . .
108
. . * . .
120
. . . . .
137
. . . . .
203
the Transitional .............
..................
94
and the Momentum
of the Heat Flux Meters Ordering
. . . . .
Convex Cunrature
Nomenclature A+
Effective
sublayer
Cf/2
Skin-friction
C P
Static
pressure heat.
F
Blowing
fraction,
H
Shape factor,
HP
Heat flux
i
Enthalpy.
K
Acceleration
Ki
Heat flux
k
Inverse
R
Mixing
RO
in the Van Driest
coefficient,
Specific
=P
thickness
model.
rw/pIJ2 Pw'
coefficient,
es-P s,ref
1
$ PU2 Pw
I
l
pwVw/p,Uao. 61/62.
meter
signal.
parameter, meter
radius
v/U;wWpw/ds)
calibration
constant.
of curvature,
l/R.
l
length.
Flat-wall
mixing
length.
M
Mach number.
n
Distance
n+
Nondimensional
distance
P'
An independent
parameter.
p, ps
Static
Pr
Prandtl
-t
Turbulent
%
Total
PSW
kll
l mm 9
Heat flux.
q2
Turbulent
R
Radius
Reff
Effective
normal
to the test
wall.
normal
to the test
wall,
pressure. number. Prandtl
number.
pressure. static
pressure.
kinetic
'+,Tz.7 u'
energy,
of curvature. radius
of curvature. V
.
nu,/v.
Rex(Res) Recs
2
ReA2 RI
x(s)-Reynolds
number,
oJpws/v>
l
Momentrrm thickness
Reynolds
number,
upw62'v'
Enthalpy
Reynolds
number,
UpwA2/v.
thickness
Richardson
number.
S
Streamwise
distance
S'
Sensitivity
Si
Streamwise
S
Stability
St
Stanton
T
Temperature.
T+
Dimensionless
TO
Up&v
referenced
coefficient
for
parameter,
Stagnation
of curvature.
(see App. A).
conductance
number,
to start
plate-plate
heat flux.
LJ/R/(aU/an).
;“/PC~U~~(T~-T,)
l
temperature,
(Tw-T)u,/($/Pc~)
l
temperature.
U
Mean streamwise
velocity.
U+
Nondimensional
U'
Fluctuating
5
Shear velocity,
V
Mean velocity
V’
Fluctuating
velocity
W’
Fluctuating
cross-stream
Y
Distance
normal
Ycrit
Distance
from wall
YSl
Distance zero.
2
Distance
streamwise streamwise
U/u,.
velocity.
dv. normal
from
velocity,
to test normal
to test
wall.
wall. S = Scrit.
where
in spanwise
to the test
velocity.
where
wall
wall.
extrapolated
shear
stress
direction.
Greek Lztters 6
Empirical
constant
in curvature
mixing-length
8'
Empirical
constant
in turbulent
Prandtl
Vi
correlation.
number model.
equals
The value
Y 6
sl
6*sl
of the
Shear layer
turbulent
thickness
Displacement Thickness potential
6995
Thickness of the potential
number for
plane
integrated
of boundary layer, for mean velocity.
to
6,1*
where mean velocity
62
boundary layer, where mean velocity flow mean velocity. OD Displacement thickness, f Momentum thickness, (l+kn)U(Up-U)/UEwdn.
a( >
Uncertainty
A2
Enthalpy
EH
Turbulent
diffusivity
of thermal
EM
Turbulent
diffusivity
of momentum.
8
Angle of turn
K
Rarman constant,
x
Mixing
P'
An independent
P
Density.
T
Total
TW
Surface
61
(
OD
/
in
shear
(see App. B).
0.41. factor.
parameter.
stress. stress.
Subscripts P
Of the potential
flow.
Pw
Of the potential
flow
ref
Reference.
W
At the wall.
P' OD
Of an independent In the free
energy.
of curvature.
proportionality
shear
dn
0
from start
length
Wp-U)/Upw)
0
( >.
thickness
flow.
(see Sec. 1.1).
thickness
699
Prandtl
at the wall.
parameter
stream.
Vii
P'.
l
is 99% of the is
99.5% of
Chapter 1 INTRODUC\TION Turbulent in
boundary
many engineering
leading
edges of air
wings,
and rocket
fer
rates
tion
of
passages
lent
part
that
initial
affects
to
the
the
mechanisms
turbine
the
thermal
energy
dictions
it
trans-
where accurate are
predic-
critical
to
engines.
strain
There is
the structure rates
the efwith
of turbu-
[13,32].
experiment was to measure the effect transfer rate over a representative
conditions. projects
The work was undertaken
at
Stanford
problem
predict
project
model it
is
began,
University
as
sponsored
was the need to understand
the
heat
transfer
the
state
of
computer
differential dependent
uses. a trusted
stress
rates
program
acceleration Presently,
however,
The purpose of the to its useful domain.
it
its
empirical
on gas
and/or
curvature
1
for
of
code
transport
of
of the prethe base,
input
quite because
from twelve
transpiration
program is
Reynolds
STAN5 is
data
for
design
This
STAN5 accurately
does not account
Stanford
of
modeling
at Stanford.
or deceleration
govern
applicability
with
art
'Ihe quality
use within
supported
experimentation
which
stress
for
the
STAN5 [l].
layers.
upon the
Though the
model is
code
equations
and momentum in boundary it
of careful
or suction.
was the
makes is
streamwise
vature.
heat
associated
transfer
The motivating
partial
Reynolds
years
high
of
rates
and the heat
calculations
solves
its
aircraft
Some of this
[2,35]
and accurately
body,
transfer.
affect
of
a blunt
blades.
heat-load
simple,
heat
significantly
khen the curvature
stress
of
with
loads
which
series
Labs.
layers
heat
extra
are encountered
of turbomachinery,
blades,
blade
and boundary
of an ongoing
part
of modern high-performance
curvature
layers
by NASA-Lewis
ture
turbine
The primary objective of this convex curvature on the heat
domain of
for
on gas turbine
for
curvature
boundary
blade
walls
forward
Curved boundary
has been attributed
streamwise
of
intakes,
and efficiency
ample evidence
curved
the
nozzles.
and design
on convexly
applications:
are encountered
reliability fect
layers
accounts blowing
streamwise
cur-
to add curva-
of the work , numerous
During
the course
contributions.
Dr. J.
qualification
Kokichi
Furuhama,
the qualification
reduction many helpful
vature
is about
layer,
eddy-viscosity
to the existing
survey
strain
Ludwig
Wflcken
apassist-
of the dataProfessor,
made
was to test
He pointed
(2).
as strong
on the
Prandtl.
Prandtl
on curvature out
as one would adding
that
effects,
prior
the effect
predict
of cur-
from a thin
the extra
rate
shear
of strain
W/ax
field.
work
was the
time,
a Visiting
and
reliable
and the updating Hinami,
model by simply
effects
One early
first
was designed
that
in the design
in a very
of the literature
by Bradshaw
ten times
Experimental
facility
aided
and
Research
to 1972, was given
Prandtl,
Shinji
the eonstivction
and Vanessa MacLaren
of the facility
A comprehensive
of
Glass,
made significant
suggestions.
Previous
days
in
resulting
Michael
Professor
programs.
role
Robin A. Birch
of much of the facility,
paratus.
1.1
had a major
of the facility.
construction ed with
Gillis
colleagues
documented
of study
study
to keep secondary had put
forth
curvature
dates
by kilcken
[3],
back
to
the
a student
of
of curvature
effects
flow
to a minimum.
effects
a stability
argument
for
where the At
curvature,
it.
"Because the various parts of the boundary layer are variously affected by centrifugal force in the presence of a curved surface, a concave surface produces a tendency to force the fast parts of the flow toward the surface and the slow parts away from it. This tendency favors the exchange of the slow layers next to the surface with the faster ones on the inside of the flow. Thus, it reinforces the already existing turbulent exchange procedure. The contrary is the case for the convex faces. Here the centrifugal force has a stabilizing effect, reducing the turbulence. Prandtl
also
called
"free
secondary
had developed
flow
path,"
for
influence,
the early a flat found
form of a mixing
plate. that 2
WLlcken, curvature
length
though
model,
admitting
significantly
then some
affected
the
"free
path
surfaces
Hz stated,
lengths."
should
be ascribed
case up to the present." the Kaiser [4]
kttlhelm
The next could
although
that the decreases concave
his
that
They
critical
Reynolds
found
that
number
critical
the
of
concave wall In 1955,
that
the
than for
the flat
Frank
showed quite
wall
was considerably
flow,
he concluded
vortex
parameter.
Kreith
similar
[9]
more than that
was the
had been
applied
curvature
observations
[8]
that
the
curvature
to very
is
curved
increased
convex
of
This to
Their
time,
effect
might
extended
the
weak curvature
transition
was not
was affected
by concave
curvature.
transition
was different
for
performed that
of a fully Kreith
the
from
curvature
the
flow.
was found
of
No measurements to
of curvature
the
or convex wall.
conclusively
The hydrodynamics flow
process
also
measurements, LI+ = cy +1/n , for stronger
profile, increases
on transition
but
by
kttendorf
Hans Liepmann
range it
curvature
that
nel
number.
influenced
to turbulent
transition. the
1934
the
and delayed of
in
investigated
convex
In this
by convex
indicated
anemometry
curvature
[63
concave curvature.
[7] laminar
airfoils
where the flow
pressures
stabilizing
Reynolds
effect
for
weak
affected.
by Wttendorf
static
at
of Wandt
even for
channel,
of the strength
Clauser
hot-wire
(0 < 62/R < 0.001).
taken.
was that,
drop was not.
and wall
from
on typical
showed that,
This
and
the studies
was significantly
pressure
transition
flows.
affected
flow
under Prandtl
are significantly
study
R was positive
Clauser
time
the
where
,
on the
of
developed
His descriptor
1937,
first
study
with
conclusion
This
mean velocity
vhu,
curvature
double
been the
was in a curved
the overall
curvature.
In
than has generally continued
hydrodynamics
study
layer
in the power-law velocity power, n, with stronger convex curvature and
parameter
the
fully
on curved
Flow Research
developed.
the
from
layer
events
research
The general
documented
showed that curvature,
for
[5].
become fully
found,
Curved flow
the boundary
curvature,
more importance
Institute
and Schmidbauer
"Boundary
a clever heat
of
from
wall.
For his
scaled local
on
heat
developed, facility
transfer
transport
a convex
effects
heat
were
U/r,
studied
a concave channel
the
transfer
curved,
test
forced
rates
turbulent in
detail
were chanby
Eskinazi
and Yeh [lo].
downstream and,
for
and
u'v'.
Using
development the fully
of
hot-wire
profiles
developed
anemometry,
of
streamwise
measured
flow,
they
velocity
the profiles
measured
the
fluctuations 22 of u' , v'
,
curvature
They stated that one of the most important influences 2 is on the v' -production term -u'v'(U/r). They noted
near
convex
the
(negative
u'v'
production)
showed that
VI2
in
wall of
flux
50% of
they
was
in
indicating
a supession 2 measurements of u' and
spectral
turbulent
heat
measured
mixing
transfer
local
model
heat
activity
with
was largest
of
Kreith
wall-measured
With
fluxes
values
so considerable
test
and Wade [ll].
flat-wall
by the
1.0,
Their
of Schneider
the predicted
be predicted
.
positive,
that
range.
curved-flow
data was that
transducers
v'
the decrease
the low-wave-number The first
was 2
of
plug-type
on a convex
and considerably [9].
Their
contamination
heat flow
wall less
tunnel
heat
that
were
than would
aspect
by secondary
flow
ratio
was prob-
able. V.
C. Pate1
90" curved
duct
except
study, separate
[12] similar
that
change in
influenced
by secondary
shape factor
of
and, in
flows.
therefore,
1968,
lhomann
Sweden, made detailed
that study 2.5,
were
was performed
in
static
local a wind
pressure of curvature
effect
was found
of case
= 20% with
for
the concave
respect
to
the
through
was made to
about
effects
whether
aspect
He noted
that
the
or the local
Pate1 measured only
increased
a
and Wade
from the other
the bend.
ratio,
mean
may have been
curvature
affects
the
of entrainment.
at the
[13],
thickness-to-radius of curvature
No attempt
was due to curvature
the rate
heat
and convexly
straight
uniform
the
flow
Schneider
was some confusion
within
even with
the
used in the was 5.0.
profiles
and deceleration which,
of
and deceleration
and there
quantities,
Also
ratio
acceleration
the mean velocity
hydrodynamics
facility
aspect
curvature,
acceleration
the
to the
the
streamwise
of streamwise
studied
Aeronautical
transfer
Research
measurements
and concavely with
a freestream
on the
test
wall,
on surfaces The Ihomann
curved.
tunnel
Institute
Mach number of
and a boundary
layer
The resultant "99'R = 0.02. to be an increase in convective heat flux ratio
of
case and a decrease flat-wall
case 4
(see
of Fig.
= 15% for l-l).
the convex Since
the
Thomann study
was done in a supersonic
sible
were present.
effects
or dilation
produces
layer
may alter
which
fashion
1969,
curvature of
curved
the
flows
is
effects
transport
the
in turbulent
compression
in the boundary
process
in much the
meteorological layers.
It
drodynamic was
"99'R imaginative
acceptably
same
in
On the
convex
"turned
off"
in
which
found
tices,
analogous
stress
was inferred stress
them to check
it
of
results
wall
the
and negathe
mixing
-
was
the
length
value
stress
and has been verified
by later
[19] in
of
of
technique,
5
flows
layers.of
ratio an
stress
was
Over the concave longitudinal but
inferring e.g., of
experi-
were measured.
the
turbulent to enable plot
tech-
wall
shear
Gillis
Melbourne
a curved
vor-
Mall shear
The Qauser for
hy-
were kept
enough to the wall
University
boundary
detailed
cylinders.
experimenters,
the
boundary
shear
layer.
system
method
from
Because of
8.0.
stresses
by extrapolation.
become an accepted
the
success.
of curvature
secondary
of the boundary
plot
with
model
In their
turbulent
a stationary
Monin-
by analogy
layers.
jets,
that
for
stability
that
from a very
the Reynolds
a Clauser
has
mean velocity
number
stratified
to radius
ratio
curved
10.
formed between rotating
nique
of
to
wall
[14,15]
and unstably
was not measured close
and Joubert
is
be inferred
thickness
aspect
half
from
the wall
since
layer
was found
to those
could
boundary
of all
evidence
profile
S
the convex
to be the order
employed
the outer
they
B
on curved-wall
Profiles
wall,
work
of the apparent
published
and the
wall
for
a modifi-
Richardson
proposed
stably
agreed
of boundary
design
and introduced
= 1 - f3Ri could be used to This approach met with considerable
[16,17,18]
(J 0.07,
streamwise
a/a;
is generally
small.
Ellis
He then
experiments
the ratio
between
where
positive
the correlation
experiment
flow
Ri = 2S(l+S)
of the constant
So and Mellor
analogy
The gradient
wall.
for
the value
shear
computation.
concave
the
number used in meteorological
buoyancy effects of weak curvature.
In fact,
discussed
(u/R)/(au/an>,
Obouhkov formula small effects
[14]
written:
S =
for
profiles
compres-
by Rradshaw [2],
rate-of-strain
turbulent
Richardson flow
parameter,
ment,
extra
the
Rradshaw
rotating
shear
strong
and buoyancy
cation
tive
As discussed
significant
as does curvature.
In
and
freestream,
[35]. measured
duct.
aey
noted
that
the
width
Convex curvature lower
value
but
could
defect channel the
not
find,
similarity
flow
or
for the
the
197Os,
for
to become wake-like
fully
regions and
Joubert
curved,
a
for,
as represented
developed,
of convex
at
They searched
concave.
the mean flow,
the
Ihe
results
a curved
duct
of
through
by a
turbulent
wall
boundary
layers.
found
evidence
of
On
Taylor-
[23]).
activity of
ratio
"impulse"
of
at
local
the
curvature
in
and in an axisymmetric
flare
Gijrtler
and
recovery
cells
were
axisymmetric studies,
indicating
mean data
free-mixing
Effects
of mild
shad [25,26,27,28]. mann and
Castro
curvature
ratio
of 2.5.
They did
wall,
contrary
to
admit flows.
that
this
not
facility
with
the
observe
findings
difference
were studied
test
facility, the
and Bradshaw [24]
lateral
that
but
Taylor-
not
in
the
In both
divergence. order
and
and Bradshaw
found
duct
through
Young,
4 were
were studying
taken.
the con-
layer.
Their
Bradshaw
(Smits,
recovery
of sustained
was
of lateral
data
and fluctuating
same time,
curved,
curved
they Later
the
Eaton,
effects
in
an effect
included
(Smits, It
the
due to weak
coefficient. duct
curvature.
established
flare,
this
from
quantities
on the concave wall;
a curved
was
The data
friction
College
and
No attempt
higher-order
skin-friction
Imperial
[20]
data were taken
6.0.
skin
of
on
weak curvature
effects.
important
The second case showed the combined
divegence
of
were observed
20% in
on curvature
Bradshaw [22])
and the
experiments
by Meroney
Mean and turbulence
Spanwise variations
variations
an
were presented
[21].
of
the weak acceleration/deceleration
cellular
experiments
results
a series
was one of very
of 30" bend and aspect
4.
Taylor-Gb'rtler measured
early
mean quantities
order
undertook
experiment
by Hoffmann
made to separate consisted
Bradshaw
The first
= 0.01).
the final
vexly
of
Ellis
effects.
(dgg5/R
lXlring
profile
vortices.
curvature
from
outer
zone was curvature-dependent.
opposite
either
surface,
During
in
velocity
and the
law
logarithmic
the
n+
type
Gijrtler
the
caused
of
concave
of
of
by Ramaprian
was very notable
may be attributed
6
similar
to the
exception
Taylor-Gb'rtler Boffmann
and Shivapra-
cells
of
an aspect
on the concave
and Bradshaw. to
Hoff-
significant
Ihe authors secondary
Detailed
measurements
of the hydrodynamics
were made by Hunt and Joubert were
taken
in
an apparatus
could
be straight
or
in
bent
allowing
the convex
40% of
the distance
around
the
nozzle,
curvature
so the
ture,
then
13.2
to 1.0,
the
Taylor-GErtler
grew
cave wall. the
around
the
to minimize
of
profiles
f
of
the
Profiles
at
quantities
Data
test
fully
Rrinich
developed
and
the
in
Eskinazi
one
transfer
an increase
wall
and a slight
ture
profiles
peak toward on the
than
in
inner
log-law
relationship.
effect
oped the rather
the
in
of
ratio
earlier
curvature
the con-
friction
across
profiles
simple
and
that
the
wall agreed technique. in the mean
quantities.
took
had an aspect rate
is
[16,17,18] turbulent
relationship:
7
work
with
temperature
gradient
profiles
were not
they
showed a
beyond the hydrodyl
an analytical
Prandtl
tempera-
by a skew of the
known whether
curvature
of 6.0, (concave)
Their
wall.
Temperature not
ratio
on the outer
(convex)
wall.
hydrodynamic Nevertheless,
was observed.
a much steeper
his
on the
of
off
developed
shape characterized
so it
continued
curva-
vortices
was found
and Yeh [lo]
inner
convex
of
was large,
just
skin
It
radius
start
Roll-cell
were fully
transfer
giving
coordinates,
discussed of
heat
on the
wall,
on the
R. M. C. So [31] the
the
to
mean velocity
influence
an asymptotic
the concave
concave
at
the channel
however,
flow
decrease
plotted
of
facility, in
reached
namic studies
which
secondary
noted
thickness
The aspect
in
at the exit
and ]301 measured mean velocity transfer rates in a curved channel
Graham
similar
they
layer
was
to merge about
started
in the turbulence
heat
and considerable
The curve
effects.
region
and wall
The heat
layers
stresses.
profiles
data.
The channel
curve.
boundary
included
temperature to
nozzle
= 0.005)
Reynolds
the end of the
1977,
the
shear stress extrapolated to the convex values found with the Clauser plot
and nearly
In
of
in variations
taken
important
measured Reynolds with skin friction
downstream
flow
resulted
4%.
a duct
were observed
These cells
span
study
bend.
secondary
type
this
boundary (dgg/R
rapidly
for
the 45" bend.
small
flow
The data
a large-radius
of
channel
1979.
and concave
ratio
was extremely
in
which
into
shallow, of
[29]
of a curved
number.
prediction He devel-
I I II
1 - ;
B’Ri(1
- -+
EM
1 -+
where ing
is
y
plane
the value
flow.
correlations cal
with
data
> 0
diffusivity ity.
Since
from
curved-flow
data,
Pr < 1. This
the
Reynolds
energy
Prandtl
analogy
number of a correspond-
has been found
gives
ratio
to give
data,
an increase
of
decrease
if,either
will
"M/&R
over
a convex
decreases
faster
than
the
for
air
(EM/CR = 1)
is
is
nearly
Ri
wall,
the
eddy diffusiv-
unity,
predicted
for
one of the
flowing
number
the best
and meteorologi-
For air
reversed.
thermal
Prandtl
swirling-flow
This equation is
of
turbulent B' = 6.0
and
two inequalities
the
f3'(4ivm
The value
[55,56,57].
(convex)
of
4Y2 - 1)(1 ++
the
deviation
by the above equation
to be small. Recently, fer
rates
study
Mayle,
in
a curved
was with
velocity
study of
ings the
of
m 33% tion
at
l-2),
heat
the
the concave
start
heat
those for
investigation
is
the first
introduction
of
and
hhen looking
at
applications
where
one realizes
that
the
time
problem curvature
the
dealt
trans-
4.25.
Their
to
high-
static
layer
the
pressure
thickness Their
to find-
showed a decrease
in
and an increase
of
wall
only
downstream
heat
with
flux
predic-
on heat
curved
region,
so the
of the introduction
recovery of curvature.
8
of both
been
investigated.
cooling
transfer
curvature
The present
transfer
have blade
on heat
strong
the
of the curve.
curvature
effects
of weak or no curvature,
wall
was - 0.01.
convex
of gas turbine of
of
heat
[33].
from
regions
test
of lhomann,
the effects
recovery
the
to the flat-wall
studies
of recovery
ratio
of boundary
the
local
as opposed
the curve
relative
transfer
results
nent as the.effects
of
Rays, and Kline
none reported
regions
The ratio
wall
measured
(M = 0.06 in which
IJ 20%
of
[32]
had an aspect
air
confirming
flux
of Reynolds, Previous
which
Thomann [13]),
curvature
for
duct
were uniform.
(see Fig. wall
and tipper
low-velocity
and temperature radius
Blair,
are
are often effects
the
or many other significant, followed are as perti-
by
The present the heat heat
heat
transfer is
input
hydrodynamic Their
to
study
present
heat
sented
herein.
1-3 through friction,
ratios
1-7.
Fig. of
near-asymptotic. 0.10
the
from
the
layer
thickness
two
for
the
within
the
at
6,
two
activity in
region
the
wall.
region
stress
for
a short
sign-reversal
behavior
was seemingly
(= 16 boundary for the - 30% less
region,
the
to
layer
dgg/R = than the
return
of
the
that, though the boundary differed by a factor of of
with
Cf/2
similar
of for
1-7
in the outer
distance. indicates
curved
for
the
This
self-similar turbulence
the
slow
shows that, portion
distance two cases.
profiles
9
in
at
beginning
the
the
l-
turbulence
to be by propagation of
production
1-5,
but especially
return
linked
of the
The recovery
of the boundary
term is
the
layer,
profile.
the
a negative
region,
the boundary
was slow and appears
growth Fig.
the
throughout
may be responsible
the shear
response
a value
variation
Within
to a near
The slow values.
the cases pre-
The initial
curvature
was surprisingly
quickly profile
flat-wall
thickness-to-
The effect of convex curvature on profiles 2 2 1 , (u' , q , and uv> is shown on Figs.
regions,
flat-wall
as
- 20 boundary layer thicknesses from the skin friction was still - ZO-25%
the
respectively.
declined
layer
region
recovery
[34,35].
plot of mean velocity profiles in wall coordinates, region shortens within the curved section but is
quantities
the outer
normal
After region,
experiments,
discernibile.
and 1-7,
In the
the
on skin
They also found beginning of curvature
the
curved
turbulence
reached
of
an
of convex curvature
value.
Fig. l-4, a typical shows that the log always
of
is
in Figs.
and the
start
study
are summarized
was fast, the
value.
flat-wall
results
the curved
Since
configuration
two of
plot.
friction
skin friction was slow. the start of the recovery below
0.05,
a Glauser
end of
skin
predicted
and
flow.
and Johnston
boundary
both
results
the same tunnel
and Johnston
curvature
curved
Detailed
by Gillis
and with
studying
hydrodynamic
study.
0.10
using
downstream
flat-wall
2-l) of
At the
case),
the
1-3 shows the effect
as calculated
thicknesses
transfer
Fig.
The Gillis
introduction
motion,
on essentially
(see
of curvature
of a convexly
have been reported
study
radius
the
the
are from a program
mechanics
by turbulent
work was performed
the
results
and fluid
transport
essential
transfer
the
skin
from
recovery
friction
to
of curvature,
layer
to the production of TKE in this
to a
changes sign of TKR, a region.
Gillis
and Johnston the n-distance
thickness," Fig.
l-8)
curved
The inner
layer
typical
boundary
remains
essentially
acterized
the
beginning ture outer tent
is
but
layer,
residue
of the recovery lifted
with
1.2
study
processes,
of Gillis
investigating
The outer
stress.
on
6s.
transfer
At the
layer.
due to curva-
what
and Johnston
the
but heated,
objective
to
the
turbulent
With this
goal
of
remains
of
the
study
are consis-
part
of an over-
is
heat
transfer
boundary
layers.
domain
of
the eventual upon
during
the
differing
layer
Stanford
curvature
the
processes
the objectives heat
be important.
of the present
study
rates
initial of
and Johnston
For the
e.g.,
recovery
controlled
understanding
initial of
programs,
transfer flat
this
study
through
curvature This will
on maximum physical
aid
the
10
over
minimal
of
to various
add
STAN5 [l]. were to: convexly
a large
enough
conditions
so
extrapolation. layer
and others 15 runs to
to
predicted
boundary
[35]
conditions
in developing
insight.
plate
curved
a series
effect
is
from a smooth,
and boundary
study
and boundary
program
can be appropriately
model can be used with
Gillis
1.1.
that prediction
and downstream carefully
Build
the
effects
measure wall
wall
sitivity
list
in mind,
curved
Section
of
boundary
Accurately
that 2.
is char-
Obiectives
curvature
1.
stress; a which 6s.,
layer
boundary
within
heat
1-9.
heat transfer rates on smooth, convexly curved experiments build upon their study of the hydro-
similar,
The final with
shear
upstream grows
(see
to visualize
to
The outer
zero
of the present
all program involving surfaces. The present structurally
layer
profile
model.
The hydrodynamic
dynamic
thickness
the restriction
inner
The results
this
in
the
stress
layer
TKE and shear
the curve.
of
region,
and the
layer.
within
"shear
model shown in Fig.
by non-zero
TKE but essentially
decaying
shear
the
a model by which
constrained
constant
called
a two-layer
layer:
characterized
by non-zero
is
They proposed
boundary
is
a parameter
of the extrapolated
Yisl ."
labeled
the strongly
layer
identified
learn
listed
in
were made with about
parameters a prediction
gained
the thought
sento
model based
3. 4.
Test
the preliminary
ston
1351, making changes where appropriate.
Construct to
a heat
accomplish
overall
transfer
the
program,
injection,
prediction
model proposed
above
specifically,
which
that
facility objectives
curvature
modern gas
and John-
sufficiently
and future
convex
simulates
is
by Gillis
turbine
flexible
objectives with
of
discrete
the jet
film-cooling
geom-
etries. 1.3
The Experiment In
layers
the
following
were grown on a flat
cm radius
of
curvature
wall
temperature
test
wall
thermal
experiment, preplate,
convex
was controlled
hydrodynamic
then were introduced followed
wall,
was maintained
and
to a prescribed
to a 90",
by a recovery
and the static
uniform,
function
boundary wall.
pressure
of downstram
45 The
on the
distance,
K (v/UL dUpw/ds). Detailed wall heat Pw as well as profiles of velocity and temperature were meaflux data sured in the developing, curved, and recovery regions. The initial and
either
uniform
boundary
or
constant
conditions
were
parameters. tivity
The 15 cases
of the effect Initial Magnitude
c>
Free-stream
d)
Location
in
layer
the
to
determine
present
study
sensitivity
to
investigated
the
on heat transfer
various sensi-
to:
thickness.
of free-stream
velocity.
acceleration. of the beginning
of curvature Maturity
varied
of convex curvature
boundary
4 b)
e>
l
of heating
or the beginning
with
respect
to the beginning
of recovery.
of the momentum boundary
layer
at the beginning
of curva-
ture. One other radius facility
parameter
of curvature with
which
of the test
such flexibility
should wall. precluded
11
be varied The cost this
systematically and complexity
entry.
is
the
of a test
+----+----
Fig.
l-l.
- -,j-=-G
i
Stanton number versus streamwise distance, To-T, = 77.5 K , from Thomann [13].
6(10)-3
1
d 6 3 \
I
I
I
l
CONVEX
fi ;
I-;-
-0
zi
t4= m
( 1or3
*
2(1015
I
I
I
I,
4
6
(IOP
2
REYNOLDS Fig.
l-2.
.@
l o
NUMBER,
Stanton number versus x-Reynolds Mayle, Blair, and K opper 1321.
12
Re, number, from
M = 2.5,
.003 - - - FLAT FLAT D DATA, A DATA, .002
WALL PREDICTION, FIRST EXPERIMENT WALL PREDICTION, SECOND EXPERIMENT FIRST EXPERIMENT SECOND EXPERIMENT
AA--m-
cr 2
m
--*--
------l A
0’
n
I
-75
A
n A
.OOl -
-50
I
-25
WA .A
3
A
A
.A
1
I
I
I
I
I
I
I
0
25
50
75
100
I25
I50
I75
8, cm
-%-FLAT-&-CURVED+-RECOVERY+-
Fig.
l-3.
Skin friction versus streamwise 0.10, from Gillis and Johnston
13
distance, [35].
b&R
=
LINE IS U+ = 2.44 LOG n+ +5.0
35
112 cm
30
97 cm
25
82 cm
20 15 IO
80”
64 cm
60’
51 cm
40”
34 cm
20”
16 cm
5 U+
0 0
-71 cm
0 0 0 0 0 I
IO Fig.
l-4.
11111111 .-
I -
100
I
lllllll
I
1000
n+ Mean velocity profiles in wall 0.10, from Gillis and Johnston
14
IIIU
lO$OO coordinates, [35].
6gg/R =
h
CURVATURE
Pig.
l-5.
Streamwise turbulence intensity profiles versus distance in the streamwise direction, 6gg/R = 0.10, from Gillis and Johnston [35].
15
q* 2 “PW
Fig.
l-6.
T.K.E. profiles direction, r351.
699’R
16
versus distance in the streamwise = 0.10, from Gillis and Johnston
16 I2 -uv x IO4 8 u2Pw
l
WALL SKIN FRICTION (CLAUSER PLOT) Fig.
l-7.
Reynolds shear stress profiles versus distance the streamwise direction, 6gg/R = 0.10, from Gillis and Johnston [35].
17
in
n STA. 5,40° + STA. 6,60° . STA. 7, 80”
8.0
d-
6.0
9 X ‘,a
4.0
72 -
2.0
I-
0
-2.0
0
.20
.60
.40
.80
1.00
n/6 Fig.
Reynolds shear stress Gillis and Johnston
l-8.
Fig.
l-9.
profile [35].
showing
Two-layer curved boundary layer Gillis and Johnston [35].
18
116slrr, from
structure,
from
Chapter
2
THE EXPERIMENTAL APPARATUS The test in
the
facility
hydrodynamic
used in study
of
the present Gillis
of the tunnel,
ized
Systems added to the
below.
operating
domain and systems
are discussed 2.1
General
that
layer (1)
heat
allows recovery.
from
of
facility
has
a smooth,
the hydrodynamic
transfer
measurements
the
main
The charging
been constructed
flat-wall,
heated
of curvature
circuits
loop:
discharge
fan,
then return
flow
to the tunnel
(Fig. boundary
followed
to the facility:
and back to the main fan via
through
the
return pack test
a plenum box.
to the room via via
the filter
louvres box and
blower.
The suction
loop:
suction
from the preplate,
the suction
reinjected
to
fan.
a 303R The cooling water loop for the heat exchanger: capacity water tank, water supply, and discharge lines to and from the tank,
(5)
35, are summar-
header, heat exchanger and screen nozzle combination, then into the
the plenum box via (4)
in Ref.
to the heat
There were five
me main loop: ducting, oblique and contraction
charging
transfer
development
and slots (3)
specific
The hydrody-
[35].
to broaden
of a 90" bend of 45 cm radius
regions (2)
facility
was used
Description
upstream
flat-wall
in detail
study
in detail.
A curved-wall 2-1)
transfer
and Johnston*
discussed
namic aspects
heat
for
circulation
and for
make-up.
loop: heated the preplate and recovery me hot water walls, using two temperature-controlled water heaters.
*Gases 2 and 3 of Ref.
35. 19
by a
2.2
Hvdrodvnamic
Asnects
The main flow
was driven
header
that
through
the
before
entering
nozzle
can be found
plenum
box,
controlled
turned
the
in
its
boundary
layer.
beginning
-
This and
wall
contraction
of the the
nozzle
screen
test
pack and
region
The tunnel
into
velocity
on the fan and motor,
a was
and could
system low
layer
adjusted
layer
pressure
acceleration
were set
was achieved
within
tained
slightly
2-la),
which
and -
by pivoting
could
be made nearly
the operating hhen just
region,
the
of
and
of
two-
curvature, khen the
layers,
the
outer
(concave)
wall
convex
test
trip
until
a nearly
pressure
the
adjustments. with
room via 20
the a filter
was
followed khen the
was no streamwise layer.
0.025-inch
wall
uniform.
there
in both
above
the
boundary
laminar
of the nozzle.
boundary
and error
distributed
from
beginning
usually
the
these
air
layer
turbulent,
boundary
on the
were used for took
the
flexible
Twenty-six
the curve.
number
fully
was uniform,
region
ambient
at
distance,
up by trial
tap holes
thick,
span
boundary
domain to thinner
thinner
pressure
inner
the
holes.
the
wall
across
allowing
Reynolds a
was constructed
spaced
downstream
to produce
test
curvature
2-la)
was desired
static
on the
of
directions
(Fig.
of streamwise
acceleration
pressure
fan
of
uniformly
of the suction
the
function
start
holes
was tripped
curved
so that
the desired
the
layers.
downstream the
pressure
momentum thickness
fan was operating
Within
of
extended
boundary
was located
section
and adjustable
the static
diameter
boundary
the boundary
cm by 56 cm in cross
was straight
edge so that
inch
very
dimensional
wise
and belts
to an auxiliary
transitional
static
fan.
passed
the growth of a normal flat-wall (non-accelerated) A 15 cm section of the test wall in the preplate
2000-l/16
suction.
static
main
was 16.5
84 cm upstream
and connected
suction
Details
to an oblique
The flow
and an 11:l
Flow exited
the
region
upstream allowing
layers
exchanger.
region. 44.
pulleys
The outer
uniform,
-
Ref.
air
from 3.5 m/s to 26 m/s.
200 cm long.
with
a heat pack,
test
supplied
The developing about
into
a screen
the tunnel which
by a fan which delivered
flow
exchanger,
by changing
be varied
of the Facilitv
(0.6
Cases with constant
K
mm) diameter
streamwise
and span-
The tunnel
was main-
charging box.
blower
(Fig.
Because the
separation of the boundary layer on the concave tunnel was pressurized, could be prevented by discharge wall, upon entering the curved region, of the boundary layer fluid through a series of seven spanwise louvres. Secondary off
flows
in
the sidewall
wall
slots
and downstream
boundary
within
the bend,
the convex surface span.
flow
region
cm and approximately
walls
control
boundary
The outer at five
for
walls
were
stations
and six
stations
within
positions
2.3
Aspects
Transfer
The test
wall
mented so that the convexly
the
curved
The preplate
constructed
for
wall
The
the
final
15 cm by 53
continued
down-
length.
region,
six
data
could
stations
within
Stations
region.
be the
typically
two-dimensionality.
of the Facility
heat
of copper
flux
could
and the recovery
was divided
the streamwise direction constantan thermocouple.
fences
profile
checking
was constructed local
of
of dimensions
on
of the
discussion
so that
the recovery
portion C. Gillis.
60 cm of recovery
in the developing
had seven spanwise &at
tunnel
side-
fences
by J.
Boundary layer
the first
layer
beyond the heated
was developed
was a straight
by peeling
to the bend, by using
and by installing
125 cm long.
stream of the curve
curve,
bend were reduced
of the evolution and a more thorough are presented in Ref. 35. The recovery
taken
the
upon entry
near the side
The secondary
details design
layers
of
into
be measured on the
preplate,
wall.
48 segments,
and each instrumented The last
and segmented and instru-
each with
2.5 cm long
in
an embedded iron-
24 segments were heated
with
circu-
lating hot water (Fig. 2-lb) and were instrumented for direct heat flux measurement. Each segment consisted of a 3 mm thick copper plate backed with 3-0.5 constantan
a silvermm thick bakelite sheets, the center one containing thermopile at the centerline location. The thermopile signal
was correlated with heat flux by calibration (see Appendix C for calibration details). The heated water circulated through a 1.2 x 2.5 cm copper waveguide channel behind the bakelite sheets. structed in 1960 and had been used in the studies 47. the
This wall of Refs.
Although the circulating water was isothermal, a film years prevented the test wall from being perfectly 21
was con-
43 through buildup over isothermal.
Small
segment-to-segment
smaller
than 0.2"C,
of approximately analysis.
2%.
This scatter
streamwise
wall
electrically
heated,
averaged
wall
included
correction
preplate/curve
by energy-balance
for:
plate-to-plate
uncertainty
for
small
stock
seg-
AT.
Gap-conductance
analysis;
for
they
phenolic
Data reduction
conduction,
losses
to the
gaps were measured during the gap under
heat flow
paths
uncertainties
are typically
the 90" arc with
was
were insulated
or
less
across
support
then
correc-
the gap with
were incorporated small
wall
the
cutting
the
aluminum
support
ribs
The center
used for
the machining
to the
were supported
adding
turning
a single-point
into
contributors
The ends of the ribs
ribs.
prelinr study
to each plate,
The curve was cut by first
additional
operation,
were removed.
techniques.
The 14 segments of the curved
aluminum frame. cutting
Each segment was
was presumed to be conducted
by ten circumferential
machining
lhese
measurement of the spanwise-
The power delivered
losses,
uncertainty.
2-2)
copper
direction.
drop across
and the other
to zero AT.
(see Fig.
meter calibration.
6 mm thick
heat
wall
temperature
large
other
of
flux
where the
the measured
a large
each gap within
and radiation losses. The plate-to-plate conductances Gap conductances at the ends of the plates and for the
made artificially
overall
for
program to correct
steady-state
and curve/recovery
tests
for
number data
in the uncertainty
conductance
in the streamwise
allowing
heat
support assembly, were calculated.
tions
of the Stanton
the heat flux
was constructed
5 cm lengths
controlled
typically
heat flows.
mented into
inary
existed,
was not considered
were then used in the data-reduction
The convex
while
differences
in scatter
was measured during
plate-to-plate
the
which resulted
The plate-to-plate
the preplate values
temperature
were held by aluminum ribs
entire tool.
and side operation
rail
assembly After the spacers
was part
of
the frame and was the center of rotation of an arm used for traversing the various boundary layer probes. The heating elements for each curved wall
segment were
into
parallel
two 46 cm lengths
heater
grooves
of
AK; #28 chrome1 wire
and epoxy-bonded.
embedded
At one end of
plate,
each
the two embedded wires were connected by a large copper bus bar and at the other end were connected to the output terminal of a variable transformer.
The overall
heater
resistance
22
was about 8 fi.
To have well-stabilized passed through two voltage ond regulator
was connected
This reduced powerstats
the voltage
that
terference
into
was a system the last
input
for
plate
plugs
before
study,
the surface
Each plate across the span temperature. nearly
facility
into but
transition transfer plates these
injection
cases,
the output
arrangement
flows
spanwise
used for
the
discrete
jets
film-cooling
of centerline
(see Section
2.5.~).
transition
flat-wall
studies
were filled
reducing this
the
correction
Appendix that error, tion
there
G.
data
of
air
angle. [48]
balsa
wood
the the
using
correction
beginning heat in
thermocouples
but
earlier
contact
length
the techniques
near the ends of the copper
plates
23
where
runs.
that
from the measured by comparison to heat
was small the
flux
into
meters
for
cases where
it
was used in
uncertainty
uncertainties
were laid [49].
For
transition.
stick-on
(L/D > 20)
of Wffat
cases
was
model was necessary
overall
The embedded thermocouples
for
of the curve,
transfer the
distributed variation of
the convective heat section, higher near the ends of the
promoted
flux
this
was included
was a long
following
heat at
all
of
[43,44,45].
with
of the three
heat-conduction
Although
was complete
present
experiments
could be used to estimate the centerline heat flux spanwise average heat flux. This model was tested measurements
in-
permit-
at a 30" injection
two-dimensionality,
secondary
a simple
accuracy.
was sanded and polished.
within the curved was significantly
corner
al-
to minimize
had three iron-constantan thermocouples for redundancy and to detect spanwise
occurred coefficient where
not
wall
holes
were
each circuit.
1 cm diameter
of the curved
For most cases, same, indicating
the
Next in line
conduit
A switching
wattmeter
injecting
the
transformer.
This arrangement
power.
These will be used in future curved-wall, which are a continuation of the Stanford For the present
AC.
inside
cables.
test
13 plates
a step-down
N 40 volts
were enclosed
the
of
the building power was The output of the sec-
power over a wide range with
of a precision
Constructed through
the
instrumentation
ted the insertion
study
of
power cables with
the
each test
control
electrical
to
to typically
controlled
lowed individual All
power to the plates, regulators in series.
milled
to minimize
of
given
in
slots
SO
conduction
The aluminum frame sec-
was heated with
hot circulating
water
to minimize
water
were crushed
in
Fig.
lhe copper
end losses. against
the aluminum with
The side-rail,
2-2.
high-conductivity
grease
heated
heaters
with
patch
tubes
frame,
before that
for
ducting
this
the side-rail
blocks
and tube assembly Ihe large
crushing. were controlled
heating shown
was filled support
with
drum was
to a specified
tempera-
ture. The recovery test
surface
wall
was symmetric
To minimize walls, with
the
commercial
radiation
copper
details
estimated
Mean velocity
profile
pressure
ducer
calibrated
static
pressure
probe was 0.7 mm. differences
so that
the
heated
sandpaper the
polishing, could
the
copper
then polished
surface
was shiny
be examined in the reflecpolished emissivity
to remove oxide was held to an
to assure
model 2401C using
a two-point Pressure read the
directly static
was then calculated
Measurements were taken
ports.
using
Ihe outside
The wall-static
a total
pres-
diameter
pressure
of the
and total-to-
were read from a Statham model PM-97 trans-
voltmeter
was read
linearity with
a 33-second
differences off
to within
than
Combist
pressure
integration
against less
the
f
a Hewlett-Packard
recalibration
was read
a Combist approximately
0.25% of integrating
digital Before
each
micromanometer
was
time. 0.25
mm of
water
the
curved
In
micromanometer. at the wall.
full-scale
The local
velocity
from the formula: U
local
finer
measurements
Output
where
from
was regularly the surface
Hydrodynamic
output.
region,
After
wall,
of the curve.
transfer
progressively
Ihe surface precautions,
for
pressure
were
heat
polish.
Instrumentation
static
made.
the center
preplate
0.05 to 0.15.
sure probe and wall
test,
same as the
of the surroundings
tion on the surface. buildup. Rith these
total
about
they were sanded with
enough that
2.4
was the
=
i
(Pt(n)
- Ps(n> 1
l/2
was the measured local total pressure Pt(n) static pressure, was calculated from Psw as
24
and
W 4 ,
the
PU2 =
Wn)
1
Pew+-
= 0.05,
the baseline and a third
case
(6gg/R
case with
The data upstream of the curve dif6gg/R (e=O> = 0.02 (see Fig. 3-6). have different virtual fer slightly for the three cases because they origins (Reynolds number effect). Within the curve, the 3-6) are similar for the three cases, in these coordinates, 43
data (Figshowing the
Table 3-l HEAT TRANSFERRUN DESCRIPTORS
U,,W~)
I.D.
Case I
(ref.) 100779
Extrapolated Re62 Re62 (S - -35 cm)
to S - 0 699fR
T.B.L. Virtual Origin (cm)
ReA2 (Beginning Curve
ReA2 (Beginning Recovery)
15.1
3712
4270
0.10
-217
1641
070280
6,,/R(S-0)
- 0.10
14.8
3613
4173
0.10
-214
1775
2845
022680
6gg/R('3-O) - 0.05
14.5
1937
2563
0.05
-119
1825
2892
060480
6gg/R(8-O)
- 0.02
24.2
464
1724
0.02
-39
1755
3572
112779
bgg/R(8-0)
- 0.05
26.4
3826
4795
0.06
-142
2476
4368
7.0
1365
1696
0.08
-147
998
1496
14.5
1937
2563
0.05
-119
0
1498
14.5
1937
2563
0.05
-119
0
0
7.0
689
1060
0.05
-81
697
1270
5.2
618
874
0.06
-86
480
953
Energy balance
- 26 m/e
U 011380
2759
6gg;~(8-0,
- 0.08
U - 7 m/s, K - 0.0 PW Turbulent 113079
agg/R(e-O)
030280
bgg/~(810)
- 0.05
A,/R(8-90")
- 0.0
A2/R(9-0)
010680 012480
U - 5.2 m/e PW Transitional Early
062580
Very early
042280 051080
- 0
U - 7 m/e PW Late transitional
062880 060980
- 0.05
transitional transitional
- 3.5 m/e PW Transitional
u
6gg/R(e-O)
- 0.05
U - 13 m/m, K - .57E-6 PW agg/R(e-o) - 0.05 u - 9 m/s, K - 1.253-6 pr
12.4
322
724
0.02
-30
600
1109
7.0
238
275
0.02
-15
472
746
3.5
145
222
0.02
-23
323
653
13.3
1719
2313
0.05
-113
1560
2869
9.0
883
1349
0.05
-85
1181
2038
first
evidence
of the way in which
The parameter
699/R,
as a curved-wall range is
labeled
flow
0.02 ( "qq/R
only
convex curvature
the "strength
descriptor
[2].
a weak dependence
on
of curvature,"
Figure
(6 = 0) C 0.10
and for
"gg/R
organizes
the data.
has been used
3-6 shows that, the conditions as viewed
(6 = 0)
within
the
tested,
there
in these coordi-
nates. If
the
or shear
streamwise
layer
distance
thickness,
6sl,
of
699 would again hold. the boundary layer thickness
beginning If
the
of
the
curve),
abscissa
were
abscissa
compressed
conclude
that,
is
than
more
St left
to
choose
enthalpy
3-7
three
The data
of
sloped
= 0.10
to those
and
curvature further expect zero
6gg/R, 7u'v
"strong"
the is
the
coordinates
-1
the
slope
0.05
Stanton
in
cases
these
are
versus
coordinates.
rapidly
approach
Because the cases of
the asymptotic It
parameter
state,
appears
that,
can be
once
the
is of minimal 699'R rSsl/R = 0.03 found that
and
0.05).
Therefore,
parameter
were
changed
from
would have their
R, whereas
the cSgg/R
they
= 0.10
0.03
data
number data
(6gg/R
to
of
no strong
St
[35]
6gg/R > 0.03
slope
so the
and Johnston
cases with compressed
in
curvature. the
then
of curvature
there
others,
have the
One would
Gillis no other
cases
with
its
one could test
region 6gg/R
value of non< 0.03
&99/R = 0.03. of the case is quite different from the above "99/R = 0.02 The asymptotic state is reached much more curvature cases.
show continued
The data
enter
would
because point,
at the
conclusion.
the effect
the
of curvature.
strong,
cases if
of and
strong
sufficiently
their that,
would
slowly,
is
represent
6.
Eventually
a line
immediately
significance.
both
but
0.05
smaller
on
"S".
recast
number.
at the start
to
quantity
dgg/R = 0.10
B
699(8=0)/K
over
were scaled
or the value
larger
At this
the effect
different
Bgg(B=O)/R,
data
follow
a quite
with
of curvature
about
value
case with
smaller
Reynolds
the
considered
for
of
shows these cases
line
reach
the
radius
distance
the local
(f)=O) is steeper. one scaling length
thickness
all
streamwise
(either
s'699 relative
on the
above conclusion
the
in terms of the dimensional Figure
-1
If
one would
cases
S/&g9
the
cases of larger
for
versus
reason
for
for
were scaled
presumably
growth
because
of
the
the
boundary
initial 45
layer
boundary
until
layer
thickness
is
less
than the curvature
eventual results
weak curvature
shear layer thickness. in an immediate decrease
case similar
to that
The minimum Stanton
however.
Hthin returns
the
to within
return
of
the
correlation Stanton
Stanton
The Effect
number
Stanton
the strong
for
the
in
curvature
three
cases of
this cases,
Fig.
3.7
length,
the
Stanton
data
for
each case
The rate of correlation values. ReA 2 number to the flat-wall turbulent boundary layer
number and the
Figures of
recovery
lo-15%
Reynolds
in
of
correlation.
of the
seems to be proportional
thickness 3.3
60 cm
introduction
seen for
numbers
are 30-40% below the flat-wall
The abrupt
to the difference
flat-plate
Stanton
between
number for
the local
the
same enthalpy
of a study
of the effect
number. *
of
uPw
3-8 through
3-10 show the results
u
Figure 3-8 shows Stanton number versus distance in the streamPw' wise direction for the baseline case, = 15 m/s, and for two other uPw values of Upw: 7 m/s and 26 m/s. lhe two high-velocity cases respond nearly the same (when corrected for the Reynolds number effect), while
the low-velocity
state
in the last
nator
of Stanton
tion
on the curve
profiles for
show the state.
Stanton
data
this
increases
almost
number
data
data
velocity
these
coordinates;
baseline
data
(Fig.
with
Ihe shear U Pw' display an asymptotic
l-7)
velocity
rapidly
(112779, locking
60"
the curved
the data
(070380)
linearly
and about
30" of bend, one can find
a very
into
this
shows the data of
state 3-8)
curved
flow
approaching
the
needed for
the
the
curve
is
boundary
layer
condition.
weak
stress
070280 in Fig.
In the
Upw dependence as viewed
of the 26 m/s case (112779)
and the
to the same
appears in the denomiU Pw the wall heat flux at any posi-
case (011280)
to attain
closely
Since
means that
and Johnston
more slowly,
low-velocity last
number,
The cases of higher
The lower
other
30" of the curve.
from Gillis
8 > 60".
case seems to be approaching
in
is 8% below the
Upw = 7 m/s case (011380)
data
are 14%.
above.
*
The velocity boundary layer.
that
would
exist 46
at
the
test
wall,
were
there
no
Figure 3-10 shows a three-dimensinal plot third axis. This allows one to visualize
the
and provides
space for
diate
velocity
Table
3-l).
compared
but
thickness
at the beginning
within
the low-
and high-velocity
case of
nearly
of the curve,
paragraph
cases can each be
the
separating
exemplifies
The Stanton
coordinates.
the last
more easily
same boundary
layer
the effect
"99'R
of
U Pw'
of
The following these
surface
case which has the intermedgg/R = 0.05 (see thickness of
layer
an intermediate
from the effect
in
3-10,
the
Upw is used as
of a fourth
a boundary
In Fig. with
insertion
in which
number data
30" of bend are nearly
at any given
the same for
data
S-location
the three
cases,
the
cases yield
dicted
Stanton numbers. The flat-wall Stanton number plane 3-10 as a nearly horizontal plane. The lower-velocity
shown in cases St/St
Fig.
would
therefore
flat-plate
values
show a stronger
The minima
0’
ReX
in viewing
lower-velocity flat-wall
lower
the difficulty
within
correlations
the
for
and therefore
effect
curve
are
of
higher
curvature
in
pre-
terms
50%, 42%, and 39% below and
Upw = 7, 15,
is of the
respectively.
26 m/s,
in the asymptotic curved boundary layer, the Stanton number is deIf, pendent upon only local conditions, as now appears to be true, the history
of
the
development
of
Comparisons
not important.
the
boundary
to flat-wall
layer
upstream
of
values
in these
coordinates,
done above and commonly done with
curvature
ate.
must be separated
The Reynolds
number effect
studies,
the
are not from
bend is as
appropri-
the curvature
effect. Figure three
cases,
examination
correlation
the
In
of
approach
of
the
the
-1
correlation,
case, of
then approach
the this
-1 the
St
versus
ReA 2
47
for
all
with
in the first
data
quickly Fig.
correlation effect without
reach
3-7)
highest-velocity
curve
region,
slope study,
This seems to be the principal
the shape of the tual -1 slope.
curved
ReA . Careful 2 shows that, for the low-
curvature
in the previous
The data
curve.
the
relationship
this
the mid-velocity
ning
within
-1
a
introduction
(as observed
of curvature.
the data
follow
the data
case,
curve.
shows that
eventually of
velocity the
3-9
30-40% of the
at the begin-
case drop in the first of
-1
Upw: changing
below
20-30" to change its
even-
Figure 3-8 or 3-10 shows that the increase recovery region is faster in the lower-velocity this
becomes evident
3-9.
The rate
appears
of return
of Stanton
to be proportional
tion
(a first-order
3.4
Effect
vature
lag in these
is
regions.
discussion
[38]
point
boundary
the
momentum
layer, value
increase.
In the
for
the
K.
Therefore,
the
Three
state
ratio
for Fig.
region
than
important
seen
cases was about in
the
wall
boundary
layers
Figs.
acceleration
3-13
with
(e.g.,
of
[38]
to
Reynolds they
gave
parameter
was expected
to lock
into
of curvature. K = 0.0,
0.57
of curvature
K = 1.25
number trace or
in
an acceleration
x 10-6,
x lo+
within the
the
curved
of
milder
case
effect
case
which
is
as
effect.
walls,
Examples
a
number continues
amount of work has been done on accelerating
on flat
review:
reaches
acceleration
The
figures
is
number
thickness-to-radius
Stanton
layer,
accelerated
10% of the value the
where
and recov-
asymptotic
study: 0.05.
preceding
as the curvature
brief
this layer
There apparently
layers
shown in
for
region
momentum thickness
at the start
The boundary
A considerable ary
the
cur-
boundary
Reynolds
layer
are compared
only
the preplate
the
was within
shows a different
acceleration.
thickness
state
the
Reynolds
the momentum bnoundary
three
3-12
thickness study,
is for
for
to
cases
correla-
of streamwise
accelerating
corresponding
acceleration
all
the
that,
of curvature
1.25 x 10-t
and
out
following
start
region
of
the enthalpy
asymptotic
the asymptotic
of
while
region
the flat-plate
effects
K = 0.0
acceleration;
In their
the recovery
coordinates).
The curved
Kays and tiffat
number at
from
3-11 and 3-12 show the combined
freestream
constant
within
of Fig.
Acceleration
and acceleration.
there
in the coordinates
number
to the distance
of Freestream
Figures
ery
when the data are viewed
in Stanton number in the cases. The reason for
the for
heat
transfer
rates.
Figure
1.45
x 10-6
results
in
effect
38). of
The following
acceleration
K = 0.57 x lo+
and 3-14,
K = 0.57
Ref.
does not
a dramatic 48
dropoff
a very
on smooth,
Figure
3-13
significntly
shows that
3-14
is
flat-
and K = 1.45 x 10B6 are
respectively.
x lo6
bound-
influence
acceleration of
the
shows that with
Stanton
the K =
number.
This
is
predominantly
viscous
sublayer
or
inner-region
effect.
acceleration
cases
these
ary
Stanton
the
case of
10-6
(Fig.
3-12)
are significant x 10-6
(Fig.
results
review
wall,
an effect
apparently
3-14
for
K
a
wall
value
slope.
Figure
slope.
It
3-12
to
boundary
layer.
This is
and wall
suction
[38].
Effect Figures
ning
growth
of Unheated
with
momentum boundary
is
not
of
the
similar
to (1)
and (2)
comparison
is
060480 have heating
the
should
not of
slope
curvature Figure
in a
-0.5
to have a
-1.0
results
boundary
layer
than
on
x lO+
appears.
results
x lo+
of
show an
K = 0.57
x lo+
is
in
a
-2
layer
is
more
the
flat-wall
between acceleration
of the location
the virtual
the beginning
to show that
studies
beginning
to
Length
layer
influencing
appears
shows that,
introduction
curved
conduction
K = 0.57
of curvature.
would be required the
X
both effects
there
3-13
that
-1
K = 1.25
K = 1.25
shows an interaction
to the interaction
Starting
respect
the
The
number for
of
but
Figure x 10 -6
K = 1.45
3-15 and 3-16 show the effect
of heating
The first
at
that
therefore,
sensitive
3.5
the
2.5
Reynolds
The case of
shows
shows
of the bound-
shows that
study
expected
shows that
appears,
this
x 10 -6
in
on the outer.
at the beginning
3-12
as the
strong
3-12),
parts
slope,
K = 0.57
shows that
of
-1
acceleration
disappears
the flat
which
and acceleration.
of
(Fig.
are additive.
for
the
an
number curve
acceleration
-2.0,
Figure
for
Stanton
thickness
strong
with
predominantly
that,
and curvature
enthalpy
of the data
acceleration
that slope;
inner
the
-1.0.
acceleration
acceleration.
in
the
is
on different
shows an eventual
of curvature
of
act
with
lesser
it
thickening
and acceleration
which
on the
is,
of
and
is approximately
of this
the effects effect
slope
-0.25
curvature
3-12)
A careful flat
shows also
and the two effects
be an effect
the
3-14
number versus
strong
acceleration's
that
case of curvature
acceleration
of
the
layer; the
of two effects
layer,
slope
Figure is between
The combined
of
conduction studied,
coordinates
a combination
a result
already
the
49
origin
of the turbulent
and ending
of curvature.
unheated
discussed.
- 63 cm upstream
of the begin-
starting length Cases 070280 and
of the beginning
of cur-
vature.
Case 070280
virtual
origin
of
the
060480 has heating fore
quite
that,
has
heating
turbulent
beginning
different
in
case 070280,
the
of
effect
and is gone before all
starting
inside
in the vicinity
inner
In the
sonable
to
within
boundary
cases
not allowed ber drops
fast
coordinates
at
upstream
of
presumed
asymptotic
the
where
case
again
the
attain
this
sufficient ture
curve
return
sloped
amount of thermal to
Fig. that
of
3-16 cases
proportional
quickly
this
to the
momentum boundary
effect
within
line
region
energy
eventually
line
beginning
of
contained energy in
between
began the In
the
data
the above scenario, layer
received
an asymptotic
the
the
slope.
curvature,
inner
recovery
is
num-
reach
-1
of
Using
when the
rea-
and the Stanton
for
070280 and 060480 characterized difference
is
is
Fhen viewed
by the
the
it
two cases where heating
to establish
that
of the
energy
off.
as well.
reached
*en
the scale
so, .the thermal
is
curvature to the
the curve,
and 060480)
the
of
is confined
reducing
levels
of the
070280
shows
a thermal
of curvature.
added thermal
If
began at
be attrib-
starting
of turbulence
characterized
was eventually
profile.
similar
state -1
the
the data
(cases
heating
this
condition
3-16,
of
shows the
to the outer
then
from case to
cannot
turbulent
begins
region.
first,
of Fig.
effect
effectively
layer,
to mix effectively
number data are
number data
therefore,
quickly
section.
where heating
(active)
is most
disappears
began at the start
much of
case
effect.
the production
assume that
the inner
far,
and 3-16
layer
length
the Stanton
to show the
3-15
of the boundary
layer.
length
a well-developed
Figs.
is introduced,
part
length
of the curved
Case 113079 of where the thermal
thus
the
whereas
starting
in the Stanton
is
of
starting
introduced,
the cases discussed
layer
curvature
is
layer,
of
The two cases are there. length. Figure 3-16 shows
the unheated
unheated
The second comparison layer
starting
where
Differences
uted to the unheated
boundary
24 cm upstream.
curvature
on the correlation. case for
the
154 cm downstream
momentum boundary
unheated
even for
severe,
beginning
this
a
temperacase
is
by a rate
of
local
Stanton
number
layer
as 113079,
and the correlation. Case 030280 had the same hydrodymamic the
heating
began at
the
beginning
of 50
the
boundary recovery
section.
but
In this
the thermal
case, Gillis
boundary
and Johnston
data
finally
of Fig.
layer
model began to recover
attained
a trajectory
3-16 was at about
was slower
is
the same rate
evidence
of the shear
from
layer
not important
in
after
unheated
However, layers
the for
heating amount
distributions the
within
thermal
which heating
energy
thermal
fashion,
this
recovery
process.
energy residual
The Effect
of
momentum boundary through
sitional
boundary
Figure
3-17
(cases
the four
versus
disappears.
the
recovering
must not
in
.
Case 030280, in section, would have
recover
in a similar
be instrumental
in
the
of
the
of the Momentum Boundary Layer showing
the
the
start
effect
of
of
the maturity
curvature
to identify
are
shown in
Figs.
the separate
cases is the * of curvature. For this
number at the start
shows that,
070280,
if
011380, organizer
cases are radically
streamwise
which
trana fully
layer.
becomes an effective for
energy
they
A
have a considerable
"6,l"
Since
flow
similarly
varies from 222, indicating a laminar-to-early Res 2 (e=o) boundary layer, to the baseline case of 4173 indicating
turbulent
bulent,
thermal
Reynolds
would
layer. the
recover
within
and
process.
030280
of the recovery .
of
Case 070280,
outside
"6,l"
The descriptor
3-21.
momentum thickness study,
as residue
at
case
energy
region,
beyond
layer
they
different.
preplate
a study
of
thermal
quite
of the Maturity
The results 3-17
the
are
the
effect
began at the beginning
minimal
3.6
of
two cases
began of
length
state
the recovery
of cases 070280 and 030280 shows that starting
cases,
boundary
the
comparison the
the other
flat-wall
that
The
to the correlation
as seen for
study
of the
conditions.
return
in a normal
this
is
to flat-wall
where its
than would be expected
There outside
began to grow as the shear layer
distance
the boundary 010680, of
and 012480),
the
data.
different,
within
the
*
A simple model has been devised starting profile data to the beginning presented as Appendix D. 51
layer
is
Though the
curved
tur-
the convex curvature
the traces
for of
sufficiently
region
extrapolating the curve.
preplate
of Stanton (Fig.
data number
3-17)
are
the measured This model is
remarkably using the
similar. "shear
A plausible layer"
the
boundary
layer
the
original
boundary
well duced, the
the
turbulent
of
the asymptotic
the
boundary
layer
shear
layer
states
of
essentially
the
same.
apparently
asymptotic
curved
Figure tional turbulent
In contrast layer 3-19
is
flat-plate
the
not
sufficient
ing
curvature.
or early
small
the
then,
the
cases
are
outer
layer,
transitional
considered
for
and
to be the
cases of late to those
transi-
of the mature
the
increase
of Stanton
convex finally
near
curvature 3-20
number toward
stabilizing
the
begins,
Figure convex
cases
activity laminar
the
face
shows the
transition,
curvature
is
until
delayed
the bend,
it
is
of stabiliz-
correlation
has therefore
turbulent
062880 and
may be present
within
the
Figs.
to the laminar
(e.g.,
that in
the boundary
of curvature,
of the data
to continue
remain
in which
start
laminar
turbulent
transition
The data
by curvature.
at the
by the nearness
amount of
to force
retarded the
In
state.
similar
For the nearly
hhen transition
until
layer,
in
cases are those
transitional
characterized
has matured:
tion.
A mature
become adjus-
contained
data
are
to the above four
correlation.
062580),
layers
and
mixing.
turbulent
layer
intro-
region,
and fully
-1,
has
layer.
laminar
and 3-20,
layer
boundary
shear
30% of
is
importance.
These late
the recovery
(-
If
layer,
inner
the scales
the slope
boundary
shear
this
the
influence.
display
3-18 shows that boundary
is
layer
turbulent
layer,
Mithin
The history 3-18,
to
of minor
transitional
turbulent
or turbulent
confined
boundary
thin
[35].
hhen curvature
activity.
thickness.
both
Fig.
become the
have large-scale
has a minor
cases,
will
is
can be made by
and Johnston the
layer
would
this
turbulent,
is
turbulent
asymptotic
turbulent
mixing
of
by Gillis
which
production
curved
the
which
layer),
turbulent
boundary
to
sufficiently
turbulence
remainder
ted
model proposed
is
established
explanation
the
transi-
appears
to be
as seen by an
correlation,
to be slow
removed and then
to progress
quickly. A notable 060980, nolds of
difference
a case with numbers.
curvature,
In
from
extremely this
case,
particularly
at
the
two cases
low velocity although the
there
introduction, 52
discussed
above
is
and momentum thickness appears the
case Rey-
to be some effect curvature
does not
seem to retard
the
and transition
is nearly
The laminar curvature but
transition
to that
cussed
above,
in
mechanism.
data was as large
as f lo-13%
Figure
3-20
boundary boundary
layer
ture
was the delay 3-21
three-dimensional
and retardation shows the view
momentum thickness fied
curvature
is
only
in
of
increasing
transition returns response
the
trajectory
within
the curved
cases 060980 showed a trend curved
boundary
of the
The uncertainty (see App.
layers
boundary
emergence of G
for
(the
layer,
Stanton
a mild
process to "normal" of a mature
and,
plot
of
Figs.
number versus
number at
activity
these
transi-
details).
effect
regimes.
transition
retardant.
3-17
flat-
of curva-
and 3-19
as a
distance
and
streamwise
the beginning
distinct
curve,
-1
as dis-
turbulence for
simi-
of transition.
combined
shows three
by the most distant order
of
Reynolds
The figure
(e=o).
correla-
Ihe most significant
correlation.
the
of the stabiliz-
shows the recovery data for laminar-to-early-transiThe data return quickly to the turbulent layers.
wall
Figure
3-20 show that the laminar
the
transport
region.
toward
turbulent with
cases
return
dominated
conjunction
in the other
The introduction
may be due to the growth
as the dominant
tional
transition.
in fully
did
cases in Fig.
end of the curve,
observed
This
slope).
tional
transitional
effects
as it
by the end of the curved
in a momentary
At the very
region. lar
and early
resulted
as strongly
complete
did not prevent
ing curvature tion,
transition
of the
curve
In the first,
dominates
In the second,
the
exempli-
trajectory
the next
Reg
2
and
two curves
Reg2 v=w, curvature delays and retards the when curvature is removed, the boundary layer
quickly. turbulent
In the third boundary
53
layer
regime,
there
is the typical
to convex curvature.
0.0040
r
0 0
0.0030
2 is 2
0.0020
E 5 tii 0.0010
o.oQoo
-60
STREAMWISEDISTANCE(CM) Fig.
Stanton number versus streamwise distance for the baseline case (070280) - Turbulent 6gg/R(8=0) = 0.10, TJpw= 14.8 m/s , K = 0.0.
3-l.
,.OE-02
O.SE-02
__
-F
SC
--
\
0.0125 p,-o.s
-
--
\ \ \ 5
-\ \ \
2 2
q0.25
\
l.OE-03
\ \ \
5 b
\ 0.220
o.m-03
p,-1.33
Re
-1.00
\ \
I.OE-Q4
1000
10000
6000
6Qooo
ENTHALPYTHICKNESSREYNOLDSNUMBER Fig.
3-2.
Stanton number versus enthalpy thickness Reynolds number for the baseline case (070280) - Turbulent, 6g,/R@=O) = 0.10. uPW = 14.8 m/s , K = 0.0.
54
10
25 20 F 15
,S=O.Ocm
T+ IO 5 0
n+
Fig.
3-3.
Temperature profiles in inner coordinates for the baseline case (070280) - Turbulent, $,/R(e=O) = 0.10, upw = 14.8 m/s , K 2 0.0.
1.2 I.0 .8 prt .6 .4 CASE
.2
’
0 -40
r-
-20
0
cuRVE’-j
20
40
60
80
A
070280
o
022680
100 120
S (cm) Fig.
3-4.
Turbulent Prandtl number versus streamwise distance for the baseline case (070280) - Turbulent, &,,/R(e=O) = 0.10 , Upw = 14.8 m/s , K N 0.0.
oFROt PROFILES -FRoMuALL?fE.ASuRp[ENTs
0
o”/ / /
2000
/
/ -Re ref Re* .2 *;
1000
CURVE / 0
10
x -50 v
50
0
100
150
S(d
Fig.
3-5.
from the energy integral Comparison of ReA for equation and ReA 2 Erom the measured profiles the baseline case2(070280) - Turbulent, $g/R(8=0) 0.10, lJ = 14.8 m/s , K e 0.0. PW
57
=
CASE
6..IR
.
070280
0.10
0
0
022680
0.05
.O
A
060480
0.02
, . 0.0030
o.“,O
I
0.0000
1
I
-20
20
,
60
100
STREAMWISE DISTANCE(CM) Fig.
3-6.
The effect of initial Stanton number versus
boundary layer thickness streamwise distance.
-
l.OE-02
O.SE-02
---
St
-
0.0~25
pr-o-5
p.A2-o-25
F --\ . -%&seug a
5 9 2
l.OE-03
% !s 2
CASE O.SE-03
I
l.OE-M
I lmo
bW0
6,./R
.
070280
0.10
0
022680
0.05
Y
0604.30
0.02
bWO0
low0
ENTHALPYTHICKNESSREYNOLDSNUMBER Fig.
3-7.
The effect of initial Stanton number versus number.
58
-
boundary enthalpy
layer thickness thickness Reynolds
J
IO0000
0.0010
c
LI L
.
CASE
UPu (m/s)
070280
14.8
. . 0
o.m30
6 9 r’
. 0
0.0020
oz G E O.rnlO
Fig.
t- CmVE -I I
I
0.0000
3-8.
The effect distance
of
I
100
m
20
-20
-80
STREAMWISEDISTANCE(CM) UP - Stanton number versus
streamwise
l.OE-02
O.M-02
i
l.OE-03
E
O.bE-03
i.OE-01
Sr -
r
lmo
.-
-
6000
0.0125
Pr-0.5
Re
CASE
Upw (m/s)
.
070280
14.8
0
011380
7.0
0
112779
&2
-".25
26.4
10000
WOO.0
I 100000
ENTHALPYTHICKNESSREYNOLDSNUMBER Fig.
3-9.
The effect of U - Stanton thickness Reynol r s number.
59
number versus
enthalpy
STANTON NUMBER vs STREAMWISE DISTANCE
.0030 .0025 .0020 St .0015 .OOlO .0005
STREAMWISE DISTANCE, cm Fig.
3-10.
The effect of u - Stanton wise distance an 8" ‘Jpw
l
number versus
stream-
-_
0.0010
CASE
*o&J 00 00 ah 000
0.0030
00 oooo
a:% 6 !i 5 2
K
.
022680
0
042280
0.57x10-6
0
0
051080
1.25~10
-6
0
O* 0 06. ‘rd L .O
0.0020
E ii
00
00010
00 o” 0
CURVE
I0 WOI,
Fig.
/
3-11.
I
-20
20
100
80
STREAMWISEDISTANCE(CM) The effect of free-stream acceleration boundary layer - Stanton number versus distance.
on the curved streamwise
I.OE-02
1 J.st r.S,
0.6E-01
--
0.0125
p;O.S P;“.5
rkqO.25 ReA2-0.25
0 -
~-=@L#&\ 0 ooooo~~~-
-
“w”
‘Y-,
:I?: CAiE
, 0
I
5mo
,
K
022680
a
042280
0.57x10-6
Y
051080
1.25~10
0
,
lom0
-
T
.
Y
1 OE-04
-....
-6
I I
mm0
loo000
ENTHALPYTHICKNESSREYNOLDSNUMBER Fig.
3-12.
The effect of free-stream acceleration boundary layer - Stanton number versus thickness Reynolds number.
61
on the curved enthalpy
I
8 6
I I
a!
I
I
I11 III
I
I I I
----
0,
I
I
constant
xm
4
. . -- -- F=O.ObO
2 Stanton nu&er 1 x10-
I
III
g&o.;?2##
.8 .6
-~qg+o.o04
.4 .2
I I
102
2
Enthalpy
Fig.
3-13.
2
46103 thickness
4 Reynolds
6
10'
2
4
6
number,
The effect of mild free-stream acceleration (from Kays and Moffat [38]) - Stanton number versus enthalpy thickness Reynolds number.
4
Stanton number
K
2 A 0
-6 x 10 x 10 -6
1.45 1.99
m z.55 x lo
1x10-:
3
10'
Fig.
3-14.
-6
2
4
Enthalpy
thickness
6
10' Reynolds
2
4
6
number,
The effect of strong free-stream acceleration Kays and Moffat [38]) - Stanton number versus enthalpy thickness Reynolds number.
62
1
(from
0.0040
0
.
0.0030
E B 3 2
0.0020
E E O.WlO
o.oolm
STREaMWISEDISTANCE(CM) Fig.
3-15.
The effect of unheated starting length number versus streamwise distance.
- Stanton
l.OC-CQ
OAC-02
B 2
1.01-03
E E
om-03
l.OC-04
Fig.
I im.0
3-16.
5000
RtA2
Be62
070280
1770
4173
n
060180
1750
1724
0
113019
0
2563
o
030280
Aft.
2563
WO60
ENTHALPYTHICKNESSREYNOLDSNUMBER The effect of unheated starting length - Stanton number versus enthalpy thickness Reynolds number.
63
-
IWO0
CASE .
-
-
1OOW.O
0 06
0
.
Aa 01 .
0.0030
oo**Aaa
0
-a 0
.
070280
4173
a
011380
lb96
0
010680
1060
0
012480
674
CURVE
O.JOOO
-60
20
20
50
100
STREAMWISEDISTANCE(CM) Fig.
The effect of the maturity of the momentum boundary layer - Stanton number versus streamwise distance for the late-transitional and turbulent cases.
3-17.
l.oE-02
0
CASE
OAE-02
St
--\ E
\
\
-
0.0125
Pr-Om5
/ Paah
--iA%
%, 4173
.
070280
BeA2-op25
O
010580
1060
-
1
011380
1696
-0
012480
a74
\
E. 2
l.OE-03
E t;
om-03
.OG
\ \
1.OE-04
,000
5000
1000 0
3oooO
loo
ENTHALPYTHICKNESSREYNOLDSNUMBER Fig.
3-18.
The effect of the maturity of the momentum boundary layer - Stanton number versus enthalpy thickness Reynolds number for the late-transitional and turbulent cases.
64
0
0.0060 CASE
.
O.W30
0
060980
Reb2 222
-. D
g 0 E 2
0.0020
1
o.m10
CURVE
0.0000
Fig.
loo
60
20
-20
-bc 1
STREAYWISEDISTANCE(CM) The effect of the maturity of the momentum boundary layer - Stanton number versus streamwise distance Sor the Laminar and early-transitional cases.
3-19.
Irn-02
0.6X-02
g z
l.oc-03
\
om-03
m
0.220
,;1.33
P.
-l.OOA2
\ ca(Ic 0 . 0 I.OE-04
L 1’06.t I
\
w2
060980
222
062580
275
0628.90
724 so00
10060
WOO0
1oooO.
ENTHALPYTHICKNESSREYNOLDSNUHBER Fig.
3-20.
The effect of the maturity of the momentum boundary layer - Stanton number versus enthalpy thickness Reynolds number for the laminar and earlytransitional cases.
65
0.W CASE OMmo
o.oo(o
/
3-21.
STREAMWISE
DISTANCE
(CM)
The effect of the maturity of the momentum boundary thickness layer - Stanton number versus enthalpy Reynolds number and momentum thickness Reynolds number at the start of curvature.
66
Chapter 4 HEAT TRANSFERPREDICTIONS One of
the
objectives
of
the
curvature
project
turbulence model for curved-wall heat transfer used for practical engineering calculations. useful ments. 4.1
form in
which
to convey
State
of the Art
Prior
to the beginning
most prevalent
curvature
by
Bradshaw [14] Recently, [401-
the
results
predictions that lhis is, in fact, of
these
of the curved-wall
curvature
studies
model was one which
and was formulated Gillis and Johnston
was to develop
ing
length
across
below. the
boundary
layer
experi-
the
stemmed from a suggestion
and tested [35], with
They modified
could be the most
at Stanford,
by Johnston the benefit
recovery data, proposed a second model consistent with layer" interpretation of the curved turbulent boundary models are discussed
a
their
"shear
layer.
both
the distribution
to account
for
and Eide of their
of the mix-
streamwise
curva-
ture. models have been developed by launder et al. Other, higher-order, These models are quite [541, Irwin and Smith [39], and Gibson [53]. Although higher-order models give more detail good for mild curvature. of
the
and less
turbulence
structure,
useful
as engineering
the modeling
effort
keep the computation 'Ihe mixing-length
they
most useful
are generally
to industry,
hypothesis
-
layer
it
gradient
of mixing
of
was deemed desirable
to
approach.
prescribed
to the Reynolds
au anan I 1
is:
67
"R"
length
shear stress,
%2 au
length
to run
to make the results
uses an empirically
- uv = distribution
In order
tools.
scheme simple-- hence the mixing-length
scale to relate the mean velocity through the formula:
A typical
more difficult
(4-l) for
a flat-wall
boundary
R =
r
Kn
1
1 - exp(-n+/A+)
L
the modified layer, of
van Driest
where
the boundary
model [36]
for
the inner
the Karman constant,
K,
layer,
(4-2)
-I
the typical
is
portion
0.41.
flat-wall
of the boundary
For the outer
mixing
length
portion
distribution
has R = The smaller the
of
thickness
of
deceleration layer
with
(4-2)
(4-3)
the viscous
or
blowing/suction.
uniform
freestream
The curvature wall
or
is
for
is
gradient
Richardson
velocity
S,
the stability
The effective
is
8 radius
model proposed
is
vestigated tion
for
the wall
radius
in detail the value
2S(l
modified
A+ = 25. this
flat-
, length
+ S)
U R eff
and
Ri
is
the local
,
au an
I
parameter
10
6gg
1
generally
of the order
comes from a first-order
Reff' [41]:
and Eide
Ro -
1
and Eide
of the constant
f3 was 68
10. lag
[ 1 Reff
1 -
This mixing-length
of curvature. by Johnston
is
is
=-
=
boundary
by
of curvature,
dx W/Reff) Ro
mixing
an empirical
by Johnston
a flat-wall
to the relationship:
- f3Ri)
parameter, S
The parameter
llo(l
=
for
identifies
by acceleration/
and no transpiration
according
number defined
A+
affected
by Bradshaw [14]
flat-wall
Ri where
and is
The value
curvature
the local
The parameter
sublayer
11 = R0
6gg
used.
model suggested
distribution
where
0.085
[40,41]. 6.0.
model was inTheir
Figure
recommenda-
4-1 shows its
performance for the baseline case. The prediction was initialized with starting profiles taken 35 cm upstream of the beginning of the curve. No correction for The test wall was isothermal and Upw was uniform. the effect of curvature on turbulent Prandtl number was made. The response to the introduction after
Cf/2
the curve,
bulent
Prandtl
the curve
of curvature and
St
recovered
number were decreased
of Fig.
3-4,
was predicted far
process
well, but, If the tur-
too quickly.
in the recovery
the recovery
quite region
according
would be even faster.
model was devised merely an empirical
Since it before recovery data had been taken. fit of curved and rotating wall mixing lengths,
is not surprising was poor.
that
ing
the model's
performance
Gillis and Johnston [35] improved a model in which the main effect
turbulent
motions
large-scale
close
motions
the wall
to decay.
regrowth of the inner on a flat plate within The mixing
to
in
the recovery of curvature and allow
recovery
is it
region
prediction by deviswas to confine the
the
In the recovery
the
to This
previously
region,
existing
they modeled the
boundary layer as a normal boundary layer growing a velocity gradient. Their model follows.*
length
in
the outer
portion
of
the boundary
layer
was
modeled as: R 0: 6 where
6Sl
placement Layer
the width
thickness
thickness
0.10. wall
is
The shear was found
profile
to
Johnston
hsl. layer
thickness
by extrapolating
35 for
shear layer
the linear
S,
69
portion
l-8.
defined
more details.
and
layer integrated of proportionality
from Gibson [42],
parameter,
Sl
in the flat-wall
as shown in Fig.
use a proposal
See Kef.
of the active the boundary The constant
Uv = 0,
of the stability
*
of
4*
sl
Within that as:
there
* 6sl is the disout to the shear was found to be
preplate
or recovery
of the shear stress the curve,
Gillis
is a critical
and value
above which
the shear
this
critical
value
that
a critical
stress
is
value
essentially
exceptions
of
sustain
of 0.11 optimized the
the
Gibson
itself.
0.17.
was approximately
The model used in data
cannot
Gillis
and Johnston
the prediction
following
predictions
same model as presented
found that
of their
data.
of
the
heat
in
Ref.
35,
two changes made to the estimation
of
6sl
found
transfer with
the
within
the
curve. To describe
the first
it
change,
is necessary
to define
the follow-
ing two parameters: Ycrit
The n-distance
YSl
The n-distance away from the wall lated shear stress equals zero.
In the Gillis
away from the wall
and Johnston
6sl
Ycrit in switchover stress
= Ysl
in
the curved at
the
profile
the
0.10
profile,
Ysl
beginning
of
Y,l
Ycrit' This temperature profiles,
to increase within
asymptotic
curved
rebound causing
was small. lated profile
the
fraction
6sl
Ysl
affected
recovery stability
the extrapolated
values
=
Ycrit
when 70
shear But,
Ycrit.
the mean velocand finally
well
of
Cf/2
and
and
St
argument
that,
states
layer
thickness. But in
was found,
Ycrit.
< Ycrit
this when
a "controller"
in the Ysl, model, "99/R that
The new model calcu-
the curved-region
Y,l
the
the
though still
as: 6sl
that
=
from curvature, shear
to grow much above
bsl
the mean velocity
= 'crit' 'crit' it especially,
to be used in computing
of
up through
The change in the model was to construct
would not allow
altered
began to alter
passing
and
was found
quickly
the calculated
equals the n-distance where ' YSl could be well above Ycrit'
regions
model it
profile
eventually
The critical state,
their
down to a small
in a manner much like
the curve.
recovery
curvature
began to rebound,
above
where the extrapo-
- Cl) or
With
as the change in the shear stress ity
(6sl
developing
region.
driving
s = scrit.
model,
11 = where
where
mixing
length
6 when
sl
Ysl ' 'crit It
=
is a stiff
trying to force Note that when
(
'crit
but
Y p( crit
1.0 - 2.0
Gsl/Ycrit
that
drives
6sl
down when
for the asymptotic the new instruction
three
value for
cases
of
of
Scrit
cases of
differing
It "gg/R. of 6gg/R.
was a function 6gg/R = 0.10,
ary
layer
699/R
the cases
do equally R while
of curvature, thickens.
for
well
if
holding
runs
Fitting
0.02
with
to predict
that
the optimum
the optimum values a smooth curve gave:
4
Scrit
is
This
correlation
studied.
It
6gg/R sgg(e=O)
In the following transfer
was found
used in the model is the local
dgg/R
at the start
when trying
layer.
(~gg/R)l”
=
S crit Since the
and
0.05,
Ysl > Ycrit
curved boundary has no effect.
The second change to the model was developed the
)I
> 0.33.
"controller" Ysl = Ycrit Ysl < Ycrit
- 1.0
value
continually gives
and not the value
updated as the boundthe
correct
is not known, however,
trend that
it
with would
were changed from case to case by changing constant.
sections,
predictions are made of many of the heat earlier. In these predictions the turbulent
presented
Figure 3-4 Prandtl number was not changed as a result of curvature. shows that, within the curve, the change of turbulent Prandtl number is small
and
slightly
in
a
below
model could correction developed
direction
the
following
be made, then, for
the curved
by R.M.C.
Figure
3-4
the turbulent
So [31]
shows that
Prandtl
such correction,
that would predictions.
the Stanton numbers decrease A mild improvement to the
by incorporating boundary layer. discussed there
number.
are expected
a turbulent
Prandtl
One possibility
is
in Chapter
of recovery
predictions,
to show too slow a recovery.
ent, no model exists v' t' 127 across the
for making recovering
this correction. boundary layer
developing
accurately
accounts
a model that
71
for
the model
1.
seems to be an effect
The following
number
this
on
which make no At pres-
Profiles of local would be valuable in effect.
Y sl profile
After
to the wall.
some distance
curvature, the shear stress F to jump a considerable Ysl results
in variations
cillating very
slope
Wen
to run.
thickness
step
St
this
as an integral
no
would
is
osthe
goes irreparably frustrating
be to calculate
the
shear
thickness:
is a function
the n-distance
stabilize
usually
due to an inflection
solution
might
0
of
$dn s
where
Ysl
the
n Ysl
This
program can be quite
An improvement
can cause
or s), ReA 2 ( Occasionally value.
small
the
of
versus
happens,
this,
that
to step.
to be a reasonable
Because of
and expensive
of
of the beginning
inflections
from
to an extremely
the wall.
of control.
downstream
develops
distance
in the
extrapolate
near
layer
profile
about what appears
program will out
The calculation of of the shear stress
The present model can be quite temperamental. is done by extrapolation of a linear portion
,
W 0
when the shear
stress
goes to zero.
This
the model considerably.
The following
sections
demonstrate
the above changes predicts
the heat
how well
transfer
the
above model with
cases discussed
in
(;hapter
3. 4.2
Predicting
the Effect
Figures
of curvature
ton number.
For this for
within
some time
the curve
crease
St
rapidly
the Baseline
quite
slightly predicted.
predicted
St
the correction
in
near
the recovery
the
Prt
region.
increase
St,
To try
profile
data
plied
by the predicted
Cf/2
to get an estimated
Prt.
This
on
this
of
case from
corrected 72
and
curved number
would de-
The recovery
is
quite
would increase
to assess
from
correction
for
asymptotic
correction
Prt
correction
the ratio
The
and Stan-
and the Stanton
lated recovery
the
a
friction
the
the end of the curve.
Note that
would
friction
Note that
case.
enough to overshoot
down to
the skin
good.
the skin
strong
settling
of both
is
accurately
before
is
Case
of the baseline
decreases
case the effect
The prediction
state.
for
4-2 and 4-3 show the prediction
introduction oscillate
of Curvature
just
St/(Cf/2) Chapter Stanton
Stanton
the
how much was calcu-
3,
then multi-
number with
a
number is shown in
0 FROM PROFILES -FRoMu&LLMEASmmENTS
0
0 ,I / 0%
2000
/
0’ 1’
/ / -Re ReA 2 “;
ref
1000
1’ 1’c
CURVE -
0
/ A -50 LJ
1'
0
50
100
150
S(a)
Fig.
3-5.
Comparison of ReA from the energy integral equation and ReA 2 Erom the measured profiles for the baseline case2(070280) - Turbulent, &&R(8=0) 0.10, u = 14.8 m/s , K N 0.0. PW
57
=
0.0040
I
I
b
b..IR
CASE
0
be
.
070280
0.10
‘a
I
022640
0.05
I
a
060480
0.02
STREAMWISE DISTANCE(CM) Fig.
‘.OC-02
3-6.
The effect of initial Stanton number versus
boundary layer thickness streamwise distance.
-
F
0.x-02
SL-
r-
0.0125
Pr-Oe5
Bc
-Oez5 A2
E s i:
l.OE-03
% s 2
CASE
b,,lR
O.SE-03
,.OE-04
I IWO
6000
.
070280
0.10
0
022680
0.05
Y
060480
0.02
WOO0
10000
ENTHALPYTHICKNESSREYNOLDSNUMBER Fig.
3-7.
The effect of initial Stanton number versus number.
58
boundary enthalpy
layer thickness thickness Reynolds
I 1M
o.oobo
I
_ *
CASE
0.0030
WV
(m/s)
3
070280 14.8 011380 7.0
0
112779
.
26.4
g 2 z
0.0020
ii E 0.0010 CURVE
O.OWO
Fig.
3-8.
The effect distance
loo
40
20
-20
1
STREAMWISEDISTANCE(CM) of U,r- - Stanton number versus
streamwise 1-I
l.OC-02
OBZ-02
I
I.Ol?-03
E
OSE-03
CASE
upw 14.8
a
070280 oil380
0
112779
26.4
.
(m/s)
7.0
l.OE-04
ENTHALPYTHICKNESSREYNOLDSNUMBER Fig.
3-9.
The effect of U - Stanton Reynol El" s number. thickness
59
number versus
enthalpy
STANTON NUMBER vs STREAMWISE DISTANCE
STREAMWISE DISTANCE, cm Fia.
3-10.
The effect of LJpb7IT r wise distance and “pw
Stanton l
number versus
stream-
0.0040
CASE
0.0030
.
1 5 2.
.
, ,a %oo 00 0.3 0-a %O 00
022680
a 0
l a&, oooo0 0 l wco6‘.O.2 ’ ‘.a*L -0
K 0
042280
0.57x10-6
051080
1.25x10
-6
-0
0.0020
% k 2 0.0010
0 0001,
Fig.
-20
3-11.
20
40
100
STREAMWISEDISTANCE(CM) The effect of free-stream acceleration boundary layer - St.mton number versus distance.
on the curved streamwise
LOE-02
OS-02
m
0.0125P;"'5
Re "2
-".25
ri 4 z
I.OE-03
E t;
0.5E-03
C&E
K
.
022680
a
042280
o.57x1o-6
"
051080
1.25~10-~
0
J
LOE-04
0
1
I
IIIII.I 5000
10000
50000
lWW0
ENTHALPYTHICKNESSREYNOLDSNUMBER Fig.
3-12.
The effect of free-stream acceleration boundary layer - Stanton number versus thickness Reynolds number.
61
on the curved enthalpy
I
8 6
I
III
I
I
I
I
II
a!
I
III
----
I
K = 0.57
4
I
U, constant I
m
x lo -6
I,
t
2 Stanton number 1 x10.8 .6 .4
.2
102
2
4
6
Enthalpy
Fig.
3-13.
lo3
2
thickness
Reynolds
11
2
46104
6
number,
The effect of mild free-stream acceleration (from Kays and Moffat [38]) - Stanton number versus enthalpy thickness Reynolds number.
4
Stanton number
2
1 x10-
Fig.
3-14.
2
4
6
Enthalpy
thickness
10' Reynolds
2
4
6
number,
The effect of strong free-stream acceleration Kays and Moffat [38]) - Stanton number versus enthalpy thickness Reynolds number.
62
(from
o.owo
STREAMWISE DISTANCE(CM) 3-15.
Fig.
The effect of unheated starting length number versus streamwise distance.
- Stanton
l.oE-02
0.5C-02
/-- SL -
---
I!2 l.OE-03 i5 E E om-03
La!-M
Fig.
I
100.0
3-16.
WC.0
1ooO.O
0.0125
Pr-0.5
1e+-O.Z5
CASE
Ry2
%
.
070280
1770
4173
0
060480
1750
1724
0
113079
0
2563
0
030280
Aft.
2563
W.0
DlTHALPY THICKNESSREYNOLDSNUUBER The effect of unheated starting length - Stanton number versus enthalpy thickness Reynolds number.
63
IP
O.W40 0
0.0030
CASE
ne**
.
070280
417;
a
011380
1696
0
010680
1060
o
012480
074
ie-01
& B E
0.0020
E E 0.0010
t- CURVE 1 0.3000
STREAMWISE
Fig.
DISTANCE
(CM)
The effect of the maturity of the momentum boundary layer - Stanton number versus streamwise distance for the late-transitional and turbulent cases.
3-17.
I.ol!-02
100
60
20
20
-60
-I
-1
I
I
I-1111,
I
. 0.5E-02
St
-
0.0125
Pr-".5
p.A2-o.25
-\
I11111
CASE 070280
%az 4173
-
0
0106~30
1060
-
a
011380
1696
-
-0
012480
874
I-----, & ii i:
l.OE-03
8 b
0.6E-03
l.OE-04
\
1 1000
5000
10000
so000
4 100000
ENTHALPYTHICKNESSREYNOLDSNUMBER Fig.
3-18.
The effect of the maturity of the momentum boundary layer - Stanton number versus enthalpy thickness Reynolds number for the late-transitional and turbulent cases.
64
r
1 CASE
0
R-62
0
060980
222
.
062580 062880
275 724
0 0
o”*
“0
0 0 u” “0””
0”“”
””0
0
CURVE __I-.---
'.20
-20
Fig.
3-19.
'
loo
00
STREAMWISEDISTANCE (CM) The effect of the maturity of the momentum boundary layer - Stanton number versus streamwise distance for the laminar and early-transitional cases.
lS-co.
o.ec-OP.
I
0.0125
Pr-Oe5
PC 62
-Oez5
lg z
Lot-05
1
0.69-03
I.OE44
ENTHALPY
Fig.
3-20.
THICKNESS
REYNOLDS
NUHBER
The effect of the maturity of the momentum boundary layer - Stanton number versus enthalpy thickness Reynolds number for the laminar and earlytransitional cases.
65
dx c
Ro
is
vestigated tion
for
the wall
radius
in detail the value
This mixing-length
of curvature. by Johnston
and Eide
of the constant
f3 was 68
[40,41]. 6.0.
model was inTheir
Figure
4-l
recommendashows its
The prediction was initialized with performance for the baseline case. starting profiles taken 35 cm upstream of the beginning of the curve. No correction for The test wall was isothermal and Upw was uniform. the effect of curvature on turbulent Prandtl number was made. lhe response to the introduction after
Cf/2
the curve,
bulent
Prandtl
of curvature and
St
recovered
number were decreased
the curve of Fig.
3-4,
was predicted
the recovery
far
well, but, If the tur-
too quickly.
in the recovery process
quite region
according
would be even faster.
model was devised merely an empirical
Since it before recovery data had been taken. fit of curved and rotating wall mixing lengths,
is not surprising was poor.
that
ing
the model's
performance
Gillis and Johnston [35] improved a model in which the main effect
turbulent
motions
large-scale
close
motions
to decay.
regrowth of the inner on a flat plate within The mixing
to the wall
in
the recovery of curvature and allow
In the recovery
the
recovery
to This is it
region
prediction by deviswas to confine the
the
previously
region,
existing
they modeled the
boundary layer as a normal boundary layer growing a velocity gradient. Their model follows.*
length
in
the outer
portion
of
the boundary
layer
was
modeled as:
where
6sl
placement layer
thickness
thickness
0.10. wall
is the width
The shear was found
profile
to
Johnston
shear layer
and
the boundary
layer
hsl.
'Ihe constant
of proportionality
layer
thickness
the flat-wall
by extrapolating
in
the linear
uv = 0,
integrated
portion
* hsl out
is the disto the
shear
was found to be preplate
or recovery
of the shear stress
as shown in Fig. 1-8. Within the curve, Gillis and use a proposal from Gibson [42], that there is a critical value
of the stability
*
of
of the active
See Kef.
parameter,
S,
defined
35 for more details. 69
as:
above which
the shear
this
critical
value
that
a critical
stress
is
value
essentially
exceptions
sustain
of 0.11 optimized the
the
Gibson found that
itself.
0.17.
was approximately
The model used in data
cannot
Gillis
and Johnston
the prediction
following
predictions
of
same model as presented
the
of
data.
heat
in Ref.
two changes made to the estimation
of
of their
found
transfer
35, with
the
within
6sl
the
curve. To describe
the first
it
change,
is necessary
to define
the follow-
ing two parameters: Ycrit
The n-distance
Ysl
The n-distance away from the wall lated shear stress equals zero.
In the Gillis
away from the wall
and Johnston
6Sl
Ycrit in switchover stress
=
0 .lO
the
profile
beginning
driving
of
profile,
Y,l
Ycrit' This temperature profiles,
to increase within
asymptotic
rebound causing
curved
equals the n-distance where ' YSl could be well above Ycrit, was small.
profile
the
up through
stability
6sl
Ysl
values
=
Ycrit
when 70
But,
Ycrit.
the mean velocand finally
well
of
Cf/2
and
and
St
that,
states
layer
thickness. But in
was found,
Ycrit.
< Ycrit
this when
a "controller"
in the Ysl, model, &99/R that
The new model calcu-
the curved-region
Y,l
shear
argument
as: 6Sl
the
the
though still
= 'crit' 'crit' it especially,
to be used in computing
that
=
from curvature, shear
to grow much above
hsl
the mean velocity
The change in the model was to construct
would not allow lated
recovery
the extrapolated
state,
of
began to alter
the calculated
The critical
altered
fraction
affected
and
was found
quickly
passing
eventually
in a manner much like
the curve.
profile
regions
model it
curvature
began to rebound,
above
recovery
down to a small
Ysl
as the change in the shear stress ity
where the extrapo-
(As1 - Cl)
in the developing or = Ysl With their the curved region. at
S = Scrit.
model,
II where
where
mixing
length
6 Sl when
'sl It
Ycrit
=
but
> 'crit is a stiff
trying to force Note that when
1.0 - 2.0
(
Gsl/Ycrit
&-
1.0))
> 0.33.
"controller"
that
Ysl = Ycrit Ysl < Ycrit
(
drives
6sl
down when
for the asymptotic the new instruction
curved boundary has no effect.
The second change to the model was developed the
three
value for
cases
of
of
Scrit
cases of
differing
It "gg/R. of 6gg/R.
was a function 6gg/R = 0.10,
0.05,
and
Since the layer
bgg/R
well holding
if
the optimum
the optimum values a smooth curve
gave:
value and not the value updated as the bound-
This correlation gives the correct trend with studied. It is not known, however, that it would
the cases
do equally
to predict
4
used in the model is the local is continually of curvature, Scrit
thickens.
for
R while
with
6gg/R
at the start ary
Fitting
that
layer.
(sgg/R)1’3
=
S crit
when trying
was found
0.02
Ysl > Ycrit
hgg/K
were changed from case to case by changing
dgg(8=O)
constant.
In the following sections, predictions are made of many of the heat transfer runs presented earlier. In these predictions the turbulent Figure 3-4 Prandtl number was not changed as a result of curvature. shows that, within the curve, the change of turbulent Prandtl number is small and in a direction slightly below the following model could correction developed
be made, then, for
the curved
by R.M.C.
Figure
3-4
the turbulent
So [31]
shows that
Prandtl
such correction,
that would predictions.
decrease
by incorporating boundary
layer.
discussed there
number.
A mild
improvement
One possibility
is
the
number
the model
of recovery
predictions,
to show too slow a recovery.
for making recovering
this correction. boundary layer
developing
accurately
accounts
for
to
1.
seems to be an effect
71
numbers
Prandtl
ent, no model exists v't'fi?T across the a model that
Stanton
a turbulent
in Chapter
The following
are expected
the
this
on
which make no At pres-
Profiles of local would be valuable in effect.
Ysl profile
After
to the wall.
some distance
curvature, the shear stress e to jump a considerable Ysl results
in variations
cillating very
slope
the
of control.
hhen this
VerSUS
happens,
this,
the
as an integral
the
no
is
is a function
the n-distance
would stabilize
can cause
Or
usually S),
OS-
2 ( Occasionally
the
goes irreparably frustrating
be to calculate
the
shear
thickness: 0
of
gdn s
where
of
due to an inflection
solution
might
stress
This
program can be quite
n Y sl
that
ReA
Ysl
of
of the beginning
value.
small
An improvement
to run.
thickness
St
shear
to step.
to be a reasonable
Because of
and expensive
step
of
the
inflections
from
to an extremely
wall.
of
downstream
develops
distance
the
extrapolate
near
layer
in
profile
about what appears
program will out
The calculation
The present model can be quite temperamental. is done by extrapolation of a linear portion
when the
,
W
0
shear
stress
goes to zero.
This
the model considerably.
The following
sections
the above changes
demonstrate
predicts
the heat
how well
transfer
the
above model with
cases discussed
in
Chapter
3. 4.2
Predicting
the Effect
Figures
of curvature
ton number.
For this for
state.
some time
the curve
crease
is
accurately
predicted.
predicted
St
the correction
in
quite
the Baseline
is
the skin
strong
settling
friction
Note that
a
the recovery
the
Prt
region.
increase
St,
To try of
profile
data
plied
by the predicted
Cf/2
to get an estimated
Prt.
This
on
case from
corrected 72
and
curved number
would de-
The recovery
is
quite
would increase
to assess
from
correction
this
asymptotic
correction
Prt
lated recovery
for
The
and Stan-
and the Stanton
correction
the ratio
friction
the
the end of the curve.
Note that
case.
enough to overshoot
down to
the skin
Case
of the baseline
decreases
good.
near
would the
before of both
slightly
St
rapidly
case the effect
The prediction
within
for
4-2 and 4-3 show the prediction
introduction oscillate
of Curvature
just
St/(Cf/2) Chapter Stanton
Stanton
the
how much was calcu-
3,
then multi-
number with
a
number is shown in
4-2 and 4-3. With Figs. are quite close. Ideally, the
data
with
a
slope. shear
slope.
As the
and the
slope
decreases.
trend
curvature
show that
nearly
a
-1.0
is approximately portional
near
bsl, the in
correct,
to the distance
The following
layer trying
these
curve
the
Y,l it
curve.
a continued
results
continues to
than
a
to grow,
Ycrit,
the
seen as a
Ycrit,
decrease
in
sustained
in
St
with
The shape of the recovery
by a rate
from the flat-wall the
4-3 would follow
Runs made with
coordinates.
discuss
and the data
introduction
to drive
characterized
sections
of Fig.
becomes larger
Y,l
end of
prediction
thickens,
the model predicts
slope
the
The strong
hhen
decreases
downward
correction,
the prediction
-1
steeper "controller"
this
of return
roughly
pro-
correlation. predictions
of cases with
dif-
fering values of a single parameter. Table 4-l shows the case numbers and figure numbers that constitute each separate-effect study. Table 4-l Prediction f
Runs
Case
Parameter Value
Figure
070280
0.10
4-2,
4-3
022680
0.05
4-4,
4-5
060480
0.02
4-6,
4-7
112779
26.4 m/s
4-8,
4-9
070280
14.8 m/s
4-2,
4-3
011380
7.0 m/s
4-10,
022680
0
4-4,
051080
1.25 x 10-6
4-12,
Unheated
022680
Base
4-4,
4-5
Starting
113079
A2/R(9=0)
4-14,
4-15
Length
030280
A,/R(9=90“)
4-16,
4-17
Effect "99/R
u Pw
K
73
= 020 = 0.0
%'s
4-11 4-5 4-13
4.3
Prediction
of Cases with
Figures cases
of
4-2
through
differing
goods even for
4-7
boundary
the
thin
Differing
Initial
Boundary
show the
ability
of
layer
boundary
term "prediction"
might
values
were chosen that
of
culated
Scrit
through
Note that 5,
the preplate
and 4-7)
boundary fect.
is
In
cases.
momentum thickness
Reynolds
to be 1724 at the start early
turbulent
a fully plate to
boundary
turbulent data
for
improve
the thin
boundary
The predicted to be slightly 4-4,
4-S),
shows a faster
the data. a
for
Prt
4.4
this
correction
Prediction
differing
not modeled. vature. totic
of Cases with
the present it
4-7),
the
model was developed
for
underpredicts Ihe modeling
cases.
downstream.
thinner
Differing through
boundary
where
of
so that
model is
is not surprising
4-6,
turbulence
skin
of Stanton
and 4-8
number ef-
S = -35 cm and estimated
the
pre-
is expected The predicted
friction
in
Prt
a Stanton
74
cases appears = 0.05
than
number similar within
(Figs.
seen in
to that
the
seen in
the recovery
re-
number prediction
with
a recovery. Upw 4-11
show predictions
based mainly that
layer
6gg/R(8=0)
There seems to be a slight Upw. Ihis effect apparently begins at the
Since state,
4-3,
(Figs.
thinner
and
would show too strong
4-2,
= 0.02
thickens
recovery
however,
for
4-
a late-transitional
3-4 shows a decrease
case,
correlation
4-3,
Case 060480 was 3200.
the
and a recovery
Figure
Figures with
of
4-4
indicating
Case 022680,
too rapid.
measured values gion
recovery
for
to
sections
due to the Reynolds
layer
layer
of cal-
number (Figs.
and apparently
boundary
at the end of the curve
Re62
Ihe
layer
therefore,
the
number was 464 at
boundary
as the
than
dgg/R(6=0)
layer.
because
No adjustment
friction.
of Stanton
of curvature,
study,
are The
predictions.
probably
where
to model
the comparison
studies;
lower is
in this
would optimize
prediction
'Ihis
Case 060480,
especially
be called
progressively
layer
layer
subsequent
4-6 can, more honestly,
program
The "predictions" case where 699'R = 0.02.
number and skin
the model has been made for
lhickness
thickness.
be misleading,
and measured Stanton
the
Iayer
there
Upw
effect
introduction
on data
of
cases
that
is
of cur-
from the asymp-
are some deficiencies
near
the beginning of curvature. One might expect introduction of curvature might scale as
4.5
Prediction Figures
of strong the
-2
of Cases with
free-stream
acceleration
(Fig.
the shear
4-13) stress
extrapolation
technique
ferent
This
one.
recovery
the prediction
4.6
Prediction Figures begins layer.
case in which figures
is jumping
is
seen both
from one near
of the acceleration
of Cases with 4-14 at
and 4-15 the Figures
heating
show an excellent
the end of
the
curve
prediction
curvature
of inside
the
dif-
and in
the
good. Lengths case
a mature
of the recovery
of the two cases.
75
quite
in Section 4.1, where hsl is this problem, and it is expected
at the start
prediction
and the
to another
4-16 and 4-17 show the prediction
begins
Apparently,
an inflection,
Unheated Starting
show the of
6sl
region.
case would be quite
Differing
beginning
in the early
has developed
The model proposed by integration might correct
that
boundary
profile
section.
calculated
heating
well
of the
of the combined effects The prediction models
and curvature.
quite
the strength
Acceleration
4-12 and 4-13 show the prediction
slope
however,
Freestream
that
in
which
momentum
of a similar region.
The
0.0040
0
0.0030
0 00
2 z E 4 t;
00 00
FLAT PLATE 000
Ei 52
0
oo”oo 0.0020
CURVE
O0 9,-l\ 0 \ \ 0
1
0’0, o~;./---~~~~~~~ 00 --
0 OO 0
0.0010 CURVE
0.0000
,
-ZlJ
20
0”
1””
STREAMWISEDISTANCE(CM) Fig.
4-l.
Prediction of the baseline and Eide model.
76
case with
the Johnston
o St - BASELINE CASE a Cf12 - BASELINE CASE PtiDICTION - St --PREiXCTION - Cf/2 x yltb Prt brr.Ctim
L-60
-20
20 STREAMWISE
Fig.
4-2.
60 DISTANCE
Prediction of the baseline versus streaawise distance.
case:
101)
(CM)
Stanton
number
I.OL-62
O.M-02
6 B E
l.OE-63
E 2
o.m-on
l.OC-04
LOO.0
LOO.0 ENTHALPY
Fig.
4-3.
THICKNESS
1606.0 REYNOLDS
60060 NUMBER
Stanton mmber Prediction of the baseline case: versus enthalpy thickness Reynolds number.
77
1oooO.O
O.oo10 0 " --
St - CASE 022680 cf12 - CASE 022680 PREDICTION - St PREDICTION - Cf/2
0.6030
c s i-5 2
o.owo a
2
---
.
\
B -a-
z 2
-o--
-b---*b
0
---
__-a
6
O.WlO
0.6600
,
-
-20
20
106
60
STREAMWISEDISTANCE(CM) Fig.
$,/R(e=O) = 0.05: Prediction of Case 022680, Stanton number versus streamwise distance.
4-4.
l.OC-02
0.6C-02
& B ii
I.OC-w
E ii 2
0.6E-a3 0 -
1.6E-W
I11111
1
600.0
.O ENTHALPY
Fig.
CASE 022680 PREDICTION
4-5.
loon0
66600
THICKNESSREYNOLDSNUMBER
Prediction of Case 022680, $g/R(e=O) = 0.05: Stanton number versus enthalpy thickness Reynolds number.
78
101
o.omo o St - CASE - 060480 - PREDICTION - St --PREDICTION - Cf/Z 0.0030 c e f
o.OWo
i3 1
0.0010
CURVE
0
O.OOW
1
-2.0
20
-
2.0
100
STREAYWISEDISTANCE(CU) Fig.
4-6.
Prediction of Case 060480, 6gg/R(+O> = 0.02: Stanton number versus streamwise distance.
l.OC-02
7
O.fiC-02
.
---
0.~~2~ pr-o.5
Pe -0.25
c--Tz
- 0.02: Stanton number versus enthalpy thickness Reynolds number.
79
1WOO.O
O.oo10
o St - CASE 112779 PREDICTION - St --PREDICTION - Cfl2 0.0030
s u ii ;
0.0020
2 5 !s 2
0.0010 I
0.0000
--20
CURVE
-
20
80
100
STREWWISEDISTANCE(CM) Fig.
4-8.
Prediction of Case 112779, % = 26 m/s: number versus streamwise distance.
Stanton
l.OE-02
---~
OAE-02
St
-
0.0125
Pr-0.5
P.qO.25
--\ --_ uoo\ f!i B 2
0 l.OE-03
5 !2 2
o.ec-03 u -
St - CASE 112779 PREDICTION
l.OE-04
Fig.
4-9.
ENTHALPYTHICKNESSREYNOLDSNUUBER Prediction of Case 112779, UPw = 26 m/s: number versus enthalpy thickness Reynolds
80
Stanton number.
0 St - CASE 011380 PREDICTION - St --PREDICTION - Cfl2
--
I ?
-20
20
.STREAMWISE
Fig.
4-10.
0
---
_---
60 DISTANCE
100
(CM)
Prediction of Case 011380, UP = 7 m/s: number versus streamwise distance.
Stanton
1.0~~02
O&E-02
2
-o-25
/
St
0.0125
Pr-0’5
Pe 62
l.OC-03
z $ s
0.6E-03 0 -
l.OE-04
St - CASE 011380 PREDICTION
600 0
1000 ENTHALPY
Fig.
4-11.
THICKNESS
so00 0
10000 REYNOLDS
NUMBER
Prediction of Case 011380, Upw = 7 m/s: number versus enthalpy thickness Reynolds
81
Stanton number.
lOMO.0
0.0040
- St - CASE 051080 PREDlCTION - St --PREDICTION - Cf/2
0 "" O.CW30
s ” -.
6 g
Y -
O.MZO
,
1,--,--
/-
I
-0
2 !3 2 2 v-2 o.w10
O.WW
,
-20
STREAMWISEDISTANCE(CM) Fig.
Prediction of Case 051080, K = 1.25 x 10m6: Stanton number versus streamwise distance.
4-12.
r
I .OE-02
O.bd-02
--_
---/
/--- St -
0.0125
Pr-005 Ilc -".25 A2
" St - CASE 051080 - PREDICTION
,.Od-04
1
Fig.
)
4-13.
_-_
^
50” ”
.^^^
l”““”
^
_^^^^
XJLXJ”
._
IU
1
b.0
ENTHALPYTHICKNESSREYNOLDSNUMBER Prediction of Case 1351080, K = 1.25 x 106: Stanton number versus enthalpy thickness Reynolds number.
82
o St - CASE 113079 -PREDICTION - St --PREDICTION - Cff2
_----__ --__ ---__-I CURVE
-20
60
20
100
STREAYWISEDISTANCE(CM) Fig.
4-14.
Prediction of Case 113079, A2/R(B=O) = 0.0: Stanton number versus streamwise distance.
l.OE-02
O.bE-02
-
St
-
--
0.0125
P;"'5
Be
A2
-"'25
--a
0
--
-9
4
--P
“SC - CASE 113079 -PREDICTION
l.OE-0,
Fig.
5000
4-15.
5000 0
ENTHALPYTHICKNESSREYNOLDSNUMBER Prediction of Case 113079, A2/~(tl=O) = 0.0: Stanton number versus enthalpy thickness Reynolds number.
83
-
1000 0
10
30
cl
St - CASE 030280 PREDICTION - St PREDICTION - Cf/Z
--
c 3 2p o.w+?o 2 oz 5 it 0.0010
oc ---
\
\
\
.
.
--.__
O.OWO
--
-“’
-20
-_0
_/--
/
60
20
loo
STREAMWISEDISTANCE(CM) Fig.
Prediction of Case 030280, A2/R(B=900) = 0.0: Stanton number versus stream&se distance.
4-16.
.
I.oE-02
0.5E-02
_
--
_
_
.
. . . . .
.
cst
-
0.0125
p,-O*5
--
0
---_ ----od,
----
5 B 3 z
Re+-0.25
l.OE-03
0” 5 E
o.sc-03
-
LOC-04
LI SC - CASE 030280 PREDICTION
500 0
100 .o ENTHALPY
Fig.
4-17.
THICKNESS
wooo
LOW0 REYNOLDS
NUMBER
Prediction of Case 030280, &/R( =90"> = 0.0: Stanton number versus enthalpy thickness Reynolds number.
84
1,,OC
Chapter 5 CONCLUSIONSAND RECOMMENDATIONS This
study
curvature
results
on heat transfer For the baseline
layers. from
presents
other
effects
companies
of experiments
through turbulent case, the effect
such as streamwise
curvature
in
practice.
v/U2 dUpw/ds. Pw The results
of convex
and transitional boundary of curvature was separated
accleration,
hhere
they were carefully controlled, with curvature was to be tested,
on the effect
there
which
usually
were combined
ac-
effects,
i.e., where streamwise acceleration acceleration was with a constant K =
l
sight into transfer. previous [16],
the
investigations,
etc.
Reynolds
e.g.,
The studies and revealed
(e.g.,
the
-1
number
coordinates).
experiments
by subjecting
5.1
to
the
considerable
the
and Johnston effects
[35],
isolated
several
important
them to
modeler
and,
a broad data base with
a broad
range
eventually,
the
which to verify
of
robustness of in single--run
conditions.
designer,
future
Most
these
prediction
tests models.
CDnclusions
The effect
are as follows:
of convex
curvature
on heat
transfer
rates
is
cant for all the cases studied. Reductions of Stanton n~ber below the expected flat-wall values were observed. The effect curvature
is
ues to grow rate
in-
So and Mellor
Also, they checked the layer hypothesis," derived
"shear
The main conclusions 1.
here provide
some new characteristics of curved boundary layslope in Stanton number versus enthalpy thickness
like
important
Gillis
of separate
hypotheses,
provide
15 cases studied
the effects of curvature on turbulent boundary layer heat They complement the detailed hydrodynamic measurements of the
parameters ers
of
seen immediately throughout
on a downstream
ery length,
(15-20
flat
the
at the beginning curve.
wall
boundary-layer
of curvature
The recovery
is extremely thicknesses),
slow.
of
After the local
may still be 15-20X below what would be expected without ture, at the same enthalpy thickness Reynolds number.
85
the
signifiof 20-50% of convex
and continheat
transfer
60 cm of recovStanton
number
upstream
curva-
2.
The boundary
tion
characterized
significant. the wall
layer
eventually
by
It
may be indicat
accelerated
3.
effects
boundary
layer layer
the rate
at which
studies
affect state
4.
Studies
of
ture
effects
show that
to be additive. sensitivity
and wall
Studies
suction
of
tional
boundary
sition
from
the
near-laminar
recovery
is
Turbulent calculated region
7.
-1
the
of
the
of curvature
and
but not the even-
slope. and streamwise
can be significant
and that
seems to increase
the boundary
been observed
convex
curvature
on laminar
this
stabilizing
curvature
show that to
turbulent.
hhen the
similar
recovery
to that
Prandtl
curved
the effect
bhen the curved
data
curva-
they
seem
layer's
when accel-
layer
of a mature from
section,
the log-law
boundary
deduced
region
is
return
flat-wall
to
late-transitional, boundary
thermal
Prt
tran-
the curved
turbulent
increases slightly St/( Q/2) and decreases rapidly in the recovery
delays
is
is
the
and transilayer
rapidly
boundary
number deduced
of
as
of the recovery tinuing
of
and the
slow,
show that
acceleration
or early-transitional,
"forgotten," expected values.
layer
were combined.
layers
soon
from
boundary
and maturity
is approached,
as has previously
effect
laminar
velocity,
by the
Convex curvature
to be
transfer
surface
to the introduction
streamwise
to acceleration,
eration
one curved
state
both
considered
heat
-l
condi-
layer.
characterized
they
curved
as in a laminar
free-stream
combined
is
the overall
The
on this
the asymptotic
asymptotic
curved
l
the response
tual
or
ng that
“;
boundary
thickness,
boundary
6.
slope
has become conduction-dominated
Separate
an asymptotic
St Q: Re
or a strongly
5.
reaches
layer.
law-of-the-wall
(lo-15%) region.
within
At the end
N 0.50
is
the
and con-
to decrease.
The "shear
layer"
to be an effective
model proposed
model with
ture
on turbulent
8.
The curvature
prediction
1351
and modified
herein
reasonably
well,
boundary
although
by Gillis
and Johnston
which to view the effect
[35]
seems
of convex curva-
layers. model
predicts it
presented the
must still
86
present
by Gillis heat
be considered
and Johnston
transfer
studies
"developmental."
5.2
Recommendations This
program
for
Future
Lbrk on the Convex Problem
has answered
several
important
questions
about
the
curved wall convective heat transfer problem. It dealt with the scaling and sensitivity questions and provided a broad data base to build upon. An important the effects involves
next
step is
of curvature taking
of
shear
of
four-wire
anemometer probe at Stanford
ing
the
that
of
soon.
components
Prediction
velopment
to give
be available
three
sensor.
V't'
the
transport data should
of
of
experiment
important
assumed, but not
Prandtl
model,
of curvature
is
6&R. at which
complete,
fourth
Stanton
is
in
turn,
than the present
effect
study
of radius
not
a
measur-
a temperature require will
de-
require
data can reveal.
done in
of curvature.
the appropriate
of
and such
are for
number will
which,
turbulent
Development
three
wires,
'IhiS
parameter
for
the
present
It
has been
the strength
This study has indicated that 699/K affects the asymptotic state is approached. The ques-
only
the
tions held
that remain are: (1) If the radius is changed but 699/R(R=O) is constant, does the approach trajectory to the asymptotic state
change?
rate
shown, that
Prt.
and the of
between
and the
is now nearly
velocity
the effect
u'v'
of
Of the four
separate
would be of
stress
the recovery
a turbulent
the difference
of heat and momentum.
profiles
we know more about the processes Another
in detail
on the diffusivity
profiles
heat
to study
(2) Does the nature
Other
important
of the asymptotic
questions
remain
about
state
itself
the
curved
regions. These are: If the curved wall were allowed nitely (without influence of secondary flow), would the asymptotic the recovery tory
of
curvature tions
is
s
(Fig.
and transpiration,
about
or surface
the combined effects roughness.
A hydrodynamically rough when it becomes convexly
layer
suction
decreases. increases
It
or,
if
trajec-
3-4)?
to the designer.
boundary
to continue indefithe conclusion that
by
were much longer,
versus
and recovery
-1 remain true? St 0: Re *2 what would be the continued
represented
unanswered are questions
sitionally wall
wall
Prt
Still
state
change?
to 87
These are key ques-
smooth wall may become trancurved and the scale of the
has been shown that
sensitivity
of convex
streamwise
convex curvature acceleration.
or Row
sensitive tion
to acceleration
would
boundary
layer
with
wall
suc-
become? Another
important
question
does convex curvature influence * cooled blades? It most likely rate
a curved
to gas turbine the jet-freestream will
be learned
is curvature-dependent.
*
Research
activity
blade
now under way at Stanford. 88
designers
interaction that
is: for
Ebw film-
the optimum blowing
References 1.
Crawford, M. E. and W. M. Rays, "STAN5--A Program for Numerical Computation of Iwo-dimensional Internal and External Boundary Layer Flows ;' NASA CR-2742, Dec. 1976.
2.
Bradshaw, P., "Effects of Streamline AGARDograph No. 169, 1972.
3.
Wilcken, H., "Turbulente Grenzschichten an Gewoelbten Flaechen," Berlin, Vol. 1, No. 4, Sept. 1930, translated as NASA TT-F-11, 421, Dec. 1967.
4.
Is3ndt, F., "Turbulente StrSmungen zwischen zesi rotierenden ialen Zylindren," Ing.-Arch., Vol. 4, p. 577 (1933).
5.
Schmidbauer, Convex Wlls"
6.
GBttendorf, F. L., Developed Turbulent
7.
Clauser, M., and F. Qauser, "The Effect of Curvature on the Transition from Laminar to Turbulent Boundary Layer," NACA-TN No. 613 (1937).
8.
Liepmann, H. W., "Investigations on Iaminar Boundary-Layer Stability and Transition on Curved Boundaries," NACA Advance Confidential Report (1943).
9.
Rreith, F., "The Influence pressible Fluids," Trans.
Curvature
of Turbulent H., "Behavior (1936), NACA-TM No. 791.
on Turbulent
Boundary
"A Study of the Effect Flow," Proc. Royal Sot.,
Layers
Flow,"
honaxon Curved
of Curvature on Fully Vol. 148, 1935.
of Curvature on Heat Transfer ASME, Nov. 1955.
to Incom-
10.
Eskinazi, S., and Y. Yeh, "An Investigation Turbulent Flows in a Curved Channel," J. of No. 1 (1956).
11.
Schneider, W. G., and J. H. T. h&de, "Flow Phenomena and Heat Transfer Effects in a 90" Rend," Canadian Aeronautics and Space Journal, Feb. 1967.
12.
Patel, V. C., "The Effects of Layer ," Aeronautical Research Aug. 1968.
13.
Ihomann, II., "Effect of Streamwise kill Curvature on Heat Transfer in a Turbulent Boundary Layer," JFM., Vol. 33, Part 2, pp. 283-292 (1968).
14.
Bradshaw, P., "The Analogy Between Streamwise ancy in Turbulent Shear Flow," JFM, Vol. 36, (1969).
(Xlrvature Council,
89
of Fully Developed Aero Science, Vol. 23,
on the Turbulent Boundary A.R.C. 30 427, F.M. 3974,
Curvature and RuoyPart 1, pp. 177-191
15.
Bradshaw, P., "Review--Complex the ASME, June 1975.
Turbulent
Flows,"
Transactions
of
16.
"An Experimental Investigation of so , R.M.C., and G. L. Mellor, Turbulent Boundary Layers along Curved Surfaces," NASA-CR-1940, April 1972.
17.
"Experiment so , R.M.C., and G. L. Mellor, Layers on a Concave Wll," The Aeronautical Feb. 1975.
18.
so , R.M.C., and G. Effects in Turbulent 43-62 (1973).
19.
Ellis, L. B., and Joubert, Duct )I' JFM, Vol. 62, Part
20.
Meroney, R. N., and P. Bradshaw, "Turbulent Boundary Layer Growth over a Longitudinally Curved Surface," AIAA Journal, Vol. 13, No. 11, Nov. 1975.
21.
l-bffmann, P. H., and P. Bradshaw, "Turbulent Boundary Layers on Surfaces of Mild Longitudinal Curvature," Imperial College, Aero. Rept. 78-04 (1978) (submitted to JFM).
22.
Smits, A. J., S.T.B. Young, and P. Bradshaw, "The Effects of Short Regions of High Surface Curvature on Turbulent Boundary Layers," JFM, Vol. 94, Part 2, pp. 209-242, Dec. 1978.
23.
Smits, A. J., J. A. Eaton, and P. Bradshaw, "The Response of a Turbulent Boundary Layer to Lateral Divergence," JFM, Vol. 94, Part 2, pp. 243-268, Dec. 1978.
24.
Castro, Highly (1976).
25.
"Mean Flow Measurements in Ramaprian, B. R., and B. G. Shivaprasad, Turbulent Boundary Layers along Mildely (Xlrved Surfaces," AIAA Journal, Vol. 15, No. 2 (1977).
26.
Ramaprian, B. R., in Boundary layers Dec. 1977.
27.
"Some Effects Ramaprian, B. R., and B. G. Shivaprasad, inal kll Curvature on Turbulent Boundary Layers," First Turbulent Shear Flow Conference, April 1977.
28.
.Ramaprian, B. R., and B. G. Shivaprasad, "The Structure of Turbulent Boundary Layers along Mildly Curved Surfaces," JFM, Vol. 85, p. 273 (1978).
on Turbulent Boundary Quarterly, Vol. 16, 25,
L. Mellor, "Experiment on Boundary Layers," JFM, Vol.
Convex Curvature 60, Part 1, pp.
P. N., "Turbulent Shear Flow in a Curved 1, pp. 54-84 (1974).
I. R., and P. %adshaw, "The Turbulence Curved Mixing Layer," JFM, Vol. 73, Part
Structure of a 2, pp. 265-304
and B. G. Shivaprasad, "Turbulence Measurements along Mildly Curved Surfaces," ASME 77-HA/FE-6,
90
of Longitudpresented at
29.
Hunt, I. A., and Joubert, P. N., "Effects vature on Turbulent Duct Flow," JFM, Vol.
30.
Brinich, P. F., and R. W. Graham, "Flow Curved Channel," NASA TND-8464, May, 1977.
31.
of Streamline So, R.M.C., "The Effects ASME 77-WA/FE-17 (1977). WY 3"
32.
Mayle, R. E., M. F. Blair, and F. C. Kopper, "Turbulent Boundary Layer Heat Transfer on Curved Surfaces," J. of Heat Transfer, Vol. 101, No. 3, Aug. 1979.
33.
Reynolds, W. C., W. M. Kays, and S. J. Kline, "Heat Transfer in a Turbulent Incompressible Boundary Layer. I. Constant Wall TemperaNASA Memo 12-l-588, 1958. ture,"
34.
Gillis, J. C., and J. P. Johnston, "Experiments on the Boundary Layer over Convex Walls and Its Recovery to 2nd Symposium on Turbulent Shear Flows, July Conditions,"
35.
"Turbulent Boundary Layer on a Gillis, J. C., and J. P. Johnston, Convex, Curved Surface," draft of technical report for NASA-Lewis Research Center (Ph.D. thesis of J. C. Gillis, Dept. of Mech. Stanford University) (1980). NASA-CR 3391, March 1981. Engrg - ,
36.
Kays, W. M., and M. E. Crawford, Convective McGraw-Hill Publishing Co. (1980)
37.
Simon, T. W., and S. IJDnami, "Final Evaluation Report: Incompressible Flow Entry Test Cases Having Boundary Layers with Streamwise 1980-81 AFOSR-HTTM-Stanford Conference on Wall Curvature (0230)," Cmplex Turbulent Flows: Comparison of Computation and Experiment, Stanford University, 1980.
38.
Kays, W. M., and R. J. Moffat, "The Behavior lent Boundary Layers," HMT-20, Thermoscience of Mechanical Engineering, Stanford University,
39.
H.P.A.H., and P. A. Smith, Irwin, Streamline Curvature on lkrbulence," No. 6, June 1975.
40.
Boundary Layers in CenJohnston, J. P., and S. A. Eide, "Turbulent trifugal Compressor Blades: Prediction of the Effects of Surface and Rotation," J. of Fluids Engrg., 98(l), 374-381 Curvature (1976).
41.
Johnston, J. P., and S. A. Eide, "Prediction of the Effects of Longitudinal Wall Curvature and System Rotation on Turbulent Boundary Dept. of Mechanical Layers," Report PD-19, Thermosciences Division, Stanford University, Nov. 1974.
Engrg
l
,
91
of Small Streamline Cur91, Part 4, pp. 633-659. and Heat Transfer
Curvature
on Reynolds
in
a
Anal-
Turbulent Flat-Wall 1979.
Heat and Mass Transfer,
of Transpired IkbuDivision, Department April 1975.
"Prediction of the Physics of Fluids,
Effect Vol.
of 18,
42.
Gibson, M. M., "An Algebraic Stress and Heat-Flux Model for Turbulent Shear Flow with Streamline Curvature," Int'l. Jour. of Heat and Mass Transfer, Vol. 21, 1609-1617 (1978).
43.
Crawford, M. E., W. M. Kays, and R. J. Moffat, "Heat Transfer to a Full-Coverage Film-Cooled Surface with 30" Slant-Hole Injection," NASA Contractors' Report CR-2786, Dec. 1976.
44.
"Turbulent Boundary Layer aloe, H., W. M. Kays, and R. J. bffat, on a Full-Coverage, Film-Cooled Surface - An Experimental Heat Transfer Study with Normal Injection," NASA Contractors' Report CR-2642, Jan. 1976.
45.
Kim, H. K., R. J. tiffat, and W. M. Kays, "Heat Transfer to a FullSurface with Compound-Angle (30" and 45") Coverage, Film-Cooled HMT-28, Thermosciences Division, Mech. Engrg. Hole Injectin," 1978. Dept., Stanford University,
46.
MzCuen, P., "Heat Transfer with Laminar and Turbulent Flow between Parallel Planes with Constant and Variable h&l1 Temperature and Heat FLux," Ph.D. thesis, Ihermosciences Division, Mech. Engrg. 1962. Dept.3 Stanford University,
47.
mrretti, P. A., "Heat Transfer through an Incompressible Turbulent Boundary Iayer with Varying Free-Stream Velocity and Varying Surface Temperature," Ph.D. thesis, Thermosciences Division, Mech. Engrg. Dept., Stanford University, 1965.
48.
and J. P. Johnston, "Heat Transfer and Kays, W. M., R. J. hoffat, Hydrodynamic Studies on a Curved Surface with Full-Coverage Film Cooling," Proposal to NASA-Lewis Research Center, June 1976.
49.
Moffat, R. J., 68-514 (1968).
50.
Blackwell, B. F., and R. J. Moffat, "Design and Construction of a Iow-Velocity Boundary Layer Temperature Probe," Transactions of the ASME, Journal of Heat Transfer, May 1975.
51.
tiffat, R. J., "Gas Temperature Measurement," Temperature--Its Measurement and Control in Science and Industry, Vol. 3, Part 2, 553-571, Reinhold Publishing Corp., New York, N. Y. (1962).
52.
Pbnami, S., "New Definition of Integral thickness on the Curved Division, Dept. of Internal Lab. Report, Thermosciences Surface," (1980). Mech. Engrg., Stanford University
53.
Stress and Heat-Flux Model for TurbuGibson, M. M., "An Algebraic lent Shear Flow with Streamline Curvature," Int. Jour. of Heat and Mass Transfer, Vol. 21, 1609-1617 (1978).
54.
Launder, B. E., C. H. Priddin, and B. I. Sharma, "The Calculation of Turbulent Boundary Layers on Spinning and Curved Surfaces," ASME J. of Fluids Engrg., March 1977.
"Temperature
Measurement
92
in
Solids,"
ISA
Paper
55.
Mzllor, G. L., "Analytic Prediction of the Properties of Stratified Planetary Surface Layers," J. Atmos. Sci., 30, 1061-1069 (1973).
56.
So, R. M. C., "A Turbulence Velocity J. Fluid Mech., 70, 37-57 (1975).
57.
So, R. M. C., "Turbulence Velocity bulence in Internal Flows, edited Publishing Corp., kshington/I.ondon,
Scale
for
tirved
Shear Flows,"
Scales for Swirling Flows," Ikrby S. N. B. Murthy, Hemisphere 347-369 (1977).
Appendix
A
THE UNCERTAINTY ANALYSIS Processing quantity
the
and the
The uncertainty value
data
uncertainty
which
it
or 95% of the time. in
taker ular
Ihe
the
When many data tinuities, mean of
the
the
uncertainty
reduction sensitivity
the
reduction tial
calculated
in
reduced
Appendix
G).
the
body of
also
tells
the
that
applies
the
curved
each stated
value
report
designer
tends
or
in
to guide
the
recovery
than
3-5%,
wall,
any one data
the the
point.
are no discon-
the
decrease
data-
early
to any one data
wall,
typically
or
on-line
where there
to
can be
of any partic-
capacity
more certainty
is
odds,
this
was brought
preplate, averaging
2O:l
to the measurement
interval
known with
with
uncertainty.
uncertainty
of
is much smaller. analysis
built
was one that,
on the
Stanton
parameter, with
derivative,
sensitivity
each
into
the
as a first
Stanton
step,
number
calculated
all
datathe
coefficients,
influence
particular
(output would lie,
and used in
presented
uncertainty program
in
over a region
is
of
how closely
analysis
project
or
the mean of the data The
tells
are taken
data
integration
Though the
the value
be given
uncertainty
points
i.e.,
quantity
analysis
uncertainty
phase of this
One note:
point;
believed
should
calculation
as the band above or below the stated
G and discussed
how much effort quantity.
the
is
The uncertainty
the design design.
that
Ihe uncertainty
Appendix
believed.
of
the
can be interpreted
within
listed
included
that
This
P'. parameter
Kline
is
of the uncertainty
done by first
and McGlintock
the
overall
[A-l]:
m 3 2
=
Z&
i
94
6P'i
of a
the data-
l%, to get the par-
by the uncertainty
was calculated,
6St
rerunning
changed by, typically,
then multiplying
coefficients using
number uncertainty
of
P'.
After
uncertainty
the was
The following table tainty analysis and their
shows the parameters estimated uncertainty: Table
Input
to the Uncertainty
thermocouple
Calibration
for
halysis
thermocouples
10%
gap heat conductance meter
hkttmeter
reading
Heat flux
meter
20 pv
signal
0.05 w calibration
constant
3% 0.46 mm2
area
lhnbient
pressure
Relative
humidity
Free-stream
7.6 mm Hg 1% 3 PV
temperature
Dynamic pressure
0.13 mm H20
difference
0.02
Emissivity
Plate
in pre30%
Heat flux
Mttmeter
3 PV
output
and after-plate
Plate
the
Uncertainty
Embedded, calibrated
Plate-plate
in
A-1
Parameter
constant
included
calibration
heater
electrical
0.015 w
constant
0.015
resistance
95
R
uncer-
(Table
Plate
Heater and delivery
line
k-1 (cont.)
electrical 0.015
resistance Plate
heater
and delivery including
resistance, Resistance
electrical 0.015 n
transformer
0.005 R
of the ammeter circuitry
Tunnel static
section
Recovery factor Thermocouple
0.13 mm 30
pressure
Spanwise heat transfer Support
line
4 uv
of free-stream
thermocouple
pressure
Saturation
density
Inductance
of ammeter circuitry
Resistance
of voltmeter
Resistance
between wattmeter
0.14 atm
of water
0.0016 kg/m3
of water
section
Curved section/afterplate
0.04 0.10
constant
Saturation
Preplate-curved
20%
correction
temperature
calibration
n
0.003 D 100 n and transformer
gap conductance gap conductance
96
0.005 a 10% 10%
The uncertainty analysis reduction program (listed in
was written into both the profile dataApp. E) and the Stanton number-reduction
program
The total
(listed
given,
along
output
giving
within input tailed
in App. F). with
the
the
region,
the uncertainty
information
on the following
quantity
maxima and minima
the developing to
reduced
used in page for
uncertainty
curved
analysis the rig
in of
region, is
also
design.
the baseline
Appendix the
of each quantity G.
More detailed
sensitivity
or recovery available.
is
coefficients region This
AI example of this
is
for
each
the deis given
case.
References A-l.
Kline, S. J., and F. A. McClintock, "Describing Uncertainties Single-Sample Experiments," Mechanical Engineering, Jan. 1953.
97
in
RUN
070280
l
**
CURVATURE
RIG
l
**
NASA-NAG-3-3
SENSITIVITY
PREP 'LATE DIN
nAx
TO(I) , J= TTOPI I J= TCE:I( I J' TBOTI I J=
2 DELTA I I 3 DELTA I I 4 DELTA I J 5 DELTA
ARG=
0.00300
I
ARG-
0.00300
I
ARG=
0.00300
J
ARG=
0.00300
I
B(I) 1 J- 6 S(I) ( J= 7 HJ'llIl 1 J= 8
DELTA
ARG:
0.00039
J
DELTA
ARG:
0.07700
J
DELTA
ARG=
0.02000
1
DELTA
ARG:
0.05000
I
DELTA
ARG-
0.98160
)
DELTA
ARG'
0.00050
J
DELTA
ARG'
0.30000
J
DELTA
ARG:
1.00000
1 I
9111 I J= 9 KIII ( J=lO A( I I 1 J=ll PAna ( J=lZ RHUM I J-13 TRECOV I J=14 PDYN f J-15
DELTA
ARG:
0.00300
DELTA
ARG'
0.00500
I
DELTA
ARC:
0.02000
I
DELTA
ARG=
0.05000
J
EflIS 1 J=16 WCAL 1 J-17 RO1IJ 1 J-18 RBOI I I 1 J=19 RLODIII I J=20 RA 1 J-21 PSTAT 1 J=22 SPANI I J 1 J'23 TSUPPI 0 1 J=24 RFY 1 J=25 TCAL 1 J-26 PS 1 J'27 RHOS I J'28 XA I J=29 RV 1 J=30 RB 1 J=31 KUP I J'32 KDWN 1 J=33
DELTA
ARG'
0.01500
J
DELTA
ARG'
0.01500
I
DELTA
ARG-
0.01500
J
DELTA
ARG:
0.00500
J
DELTA
ARG=
0.00500
J
DELTA
ARG-
DELTA
ARG-
DELTA DELTA DELTA
-0.19815
J
0.00400
J
ARG'
0.04000
I
ARG=
O.lOOOO
J
ARG=
2.00000
J
DELTA
ARG'
0.00010
J
DELTA
ARG:
0.00300
J
DELTA DELTA
ARG=l00.00000 ARG=
0.00500
J J
DELTA
ARG=
0.00860
J
DELTA
ARG'
0.01790
J
0.26079E-04 III 1) 0.00000E (0 11 0.82705E-05 (U24J O.OOOOOE (II 11 0.45173E-05 IZ 0 O.l3696E-03 III IJ O.l0121E-04 (II 1) O.OOOOOE ICI 11 O.l3056E-03 (I) 11 0.37771E-05 IX 1) 0.22846E-06 (A 21 0.36lOSE-04 III 2) O.l3754E-04 IX 1) 0.30459E-05 1s 2) 0.74226E-05 1922) 0. OOOOOE 1s 11 O.OOOOOE IA 11 O.OOOOOE Ia 0 O.OOOOOE Itl 1) O.OOOOOE 11) 0 0.83559E-08 (tl 11 O.OOOOOE (# 11 O.OOOOOE 10 0 O.l0429E-05 (S lJ 0.67102E-06 IO lJ O.l662lE-07 I# 7) O.l4268E-05 I# 2) O.OOOOOE 1s 1) O.OOOOOE ltl 1) O.OOOOOE 11) 1) O.OOOOOE IC 1) O.OOOOOE f# 11
00
00
00
00 00 00 00 00
00 00
00 00 00 00 00
COEFFICIENTS TEST
O.l9330E-04 (t22J O.OOOOOE II 1) O.OOOOOE 111 11 O.OOOOOE (8 1) O.l0257E-05 1x24) 0.53416E-06 (#15J 0.3196lE-05 IU 5) O.OOOOOE Ill 1) 0.69306E-04 (%17J 0.22978E-08 (ti23J O.l6041E-06 (%24J 0.25454E-04 1024) 0.84155E-05 (P24J 0.21380E-05 (1)24J 0.72271E-05 Ill 41 O.OOOOOE Ill 1) O.OOOOOE (it 1, O.OOOOOE (0 11 O.OOOOOE (cl 0 O.OOOOOE (A 1) 0.26587E-08 Ill 91 O.OOOOOE (II 1) O.OOOOOE I* 11 0.63785E-06 (1124) O.l9418E-06 (11191 0.90028E-08 (1124) 0.99635E-06 (#24J O.OOOOOE (0 1) O.OOOOOE IO 1) O.OOOOOE III 11 O.OOOOOE (0 1, O.OOOOOE 1: 1)
00 00 00
00
00 00 00 00 00
00 00
00 00 00 00 00
98
IDST/LJARG
l
DELTA
S ECTION
tlAX
ttIN
0.38653E-05 (U25J 0.60197E-05 11125) 0.43886E-04 (11251 O.l4686E-04 lG25J O.OOOOOE 00 lU25J O.l7573E-05 I11381 O.OOOOOE 00 1325) 0.76188E-05 (U25J 0.34976E-06 III31 I 0.42677E-05 l#25J O.l3699E-06 (#25J 0.21862E-04 (8251 0.71244E-05 (#25J 0.182592-05 Ilt.25) 0.73668E-05 (a281 0.76684E-05 ((1253 0.46238E-05 (t25J 0.45330E-05 11125) 0.75734E-07 (11251 O.l2569E-05 IR251 0.41780E-08 (f25J O.OOOOOE 00 18251 0.96736E-06 (025) 0.5512lE-06 (0251 0.16857E-06 (U25J O.l1773E-07 (t125J 0.85607E-06 (#25J O.l2196E-07 1035) 0.5537OE-07 (825) O.l2995E-05 11125) 0.43539E-06 10251 0.20978E-06 (238)
O.OOOOOE 00 (126) 0.42149E-05 1026) 0.26972E-04 (tl38J 0.99133E-05 11137) O.OOOOOE 00 It251 O.l5367E-06 (11321 O.OOOOOE 00 IP25J 0.70733E-05 lt38J 0.28317E-07 1837) O.P4642E-05 I#381 0.82757E-07 (U38J O.l3102E-04 (U381 0.43376E-05 (3381 O.l1031E-05 11)38J 0.72550E-05 (1371 0.71334E-05 (0381 0.43014E-05 1038) 0.42166E-05 (338) 0.70482E-07 is381 0.78217E-06 (038) O.l1394E-08 (#38J O.OOOOOE 00 I8251 O.OOOOOE 00 (831 J 0.32596E-06 (S38J 0.94995E-07 (t137J 0.49477E-08 (838) 0.51332E-06 111301 0.55436E-08 11301 0.35375E-07 1:38J 0.8077lE-06 (0381 O.OOOOOE 00 IX261 O.OOOOOE 00 lt25J
ARGJ
AFTERPLATE UAX
0.20279E-04 1861 J O.OOOOOE (039) 0.44829E-05 (#39J O.OOOOOE (11391 0.75369E-05 W44J 0.46905E-04 (11621 0.7777lE-05 Is58J O.OOOOOE (11391 0.53652E-04 18623 O.l5659E-05 lit62J O.l1104E-06 1856) O.l7679E-04 111561 0.57944E-05 IPbOJ O.l480lE-05 (lt56J 0.74413E-05 (3411 O.OOOOOE (P39J O.OOOOOE (tJ39J O.OOOOOE (U39J O.OOOOOE ct391 O.OOOOOE (039) 0.34183E-08 (S39J O.OOOOOE (839) O.OOOOOE 111391 0.44075E-06 (lt6OJ 0.45029E-06 cc!441 0.96954E-08 (1156) 0.69187E-06 18561 O.OOOOOE 11139) O.OOOOOE (1139) O.OOOOOE 1139) O.OOOOOE 111391 O.OOOOOE (1391
HIN
00
00
00
00 00 00 00 00
00 00
00 00 00 00 00
0.171718-04 (a1461 O.OOOOOE 00 (139) 0. OOOOOE 00 (1401 O.OOOOOE 00 1839) O.l6503E-05 (1141 I 0.46892E-06 (11481 0.45852E-05 1155) O.OOOOOE 00 (839) 0.44104E-04 (lt401 O.l5342E-07 (#47J 0.93708E-07 (1162) O.l4917E-04 (0621 0.50328E-05 (045) O.l2493E-05 (A62J 0.73016E-05 It621 O.OOOOOE 00 (1139) O.OOOOOE 00 10391 O.OOOOOE 00 (P391 O.OOOOOE 00 111391 O.OOOOOE 00 (1139) O.l5193E-08 11148) O.OOOOOE 00 (X39) O.OOOOOE 00 11391 0.38336E-06 10451 O.P0955E-07 (1(43J 0.55402E-08 (1152) 0.58347E-06 (X62) O.OOOOOE 00 (1;39J O.OOOOOE 00 (0391 O.OOOOOE 00 1139) O.OOOOOE 00 1039) O.OOOOOE 00 (139)
Appendix Curved-Wall
Integral
B
Parameters
and Energy Integral &en
there
is
the definitions for
streamwise
the momentum and energy
wall
definitions
Displacement
integral
and momentum thickness
of Curved-I&11
of Curved-k&11 62
hhere it equation
is
to alter the form curved-
presented
=
s
Boundary layers
___ U Pw
dn
(B-1)
Boundary layers
m (l+kn) 0
assumed that
by
study:
h1=J m(Up-U) 0
Momentum Thickness
becomes necessary
somewhat and, hence, The following equations.
were used in the present Thickness
it
parameters
of the displacement
Honami [52,37]
Equations
curvature,
of the integral
and the Momentum
u (Up-U) u2 dn Pw Up = Upw/(l+kn), the integral
(B-2)
momentum
becomes 152,371: k
+ yJpw
w2pw”2)
dU *
- u'pw q,(s)
=
g
uf
(B-3)
where m nUp(Up-U) q,(s)
/
The definition following
of
m nLJ(Up-U) dn +
= the
0
curved-wall
study was derived
with
After
substituting
dn
0
the estimate
=
s
the basic
OD pU(i-ioD)
*2 /
dn v2 Pw thickness
0
enthalpy
by starting
pmUp(iw-im)
s
in
the
equation:
dn
0
Up = Upw/(l+kn)
99
used
and integrating:
(B-4)
-1 The simplest
form of the energy
integral
equation
(B-5)
is:
OD l
..
q.
=
WJ(i-i,)
&
f
Writing
ii
in terms of the Stanton
number defined
St = $1 b,upw(iw-i,> Eqn. (B-6)
(assuming
constant
(B-6)
dn)
0
free-stream
as
I
enthalpy
and density)
becomes:
) dn + k an(l+kA2) Taking
the derivative, -dA2 ds
Eqn. (B-5) =
(B-7)
becomes
4 dk -rds+
k
d OD pU( i-i,> + kds f o Po,Upw(iw-qJ Equating
the two terms that St
on the flat
=
wall,
(B-7)
and (B-8)
1 dA2 *2 1 + kA2 ds + C k(l+kA2) dU +$ Iln(l+kA2) -&*+A--Pw k = 0,
and (B-9)
dn have in common leads
- an(1+kA2) diw
study.
(B-9)
is
Note that
have accounted
for
the the
integral
reduces
energy
appropriate
changes in
and
100
(B-9)
to ti
di -d$ OD
equation
used
equation i,,,
dk ds
1-l ds w co
w Equation
1
to:
p,.
for
(B-10) in
high-speed
the flows
present would
In grated enthalpy
the Stanton number data-reduction over a wall streamwise distance thickness
as: 1+
611 2
=
program, Eqn. (B-9) is inte6s to determine the change in
(l+kA2)
St 6s -
kA2 k
1+kA2 En(l+kA2) k2
101
Iln(l+kA2)
-1
6k
+
6iW
6U + i _ i 00 pw pw w
Appendix C of the Heat Flux
Calibration Calibration that
would
asure
dimensional in Fig.
of
heat
flux
precisely
known
through
consisted
On the
and beyond that heating
trolled
of
outside
another
so that (typically
heater
adjacent
of
2.5
the
the
carefully
a heater
controlled This device
meter.
oneis shown
temperature
less
than
adjusted
to give
plates.
plates
on the
the
assembly
entire
the heating
hot-water
pad was equal
adjacent
patch
sheets
the
of
heater,
The power to was con-
two heaters
signal
was
from
the
to the control-
heat
to the copper
flux.
Ihe input
a transformer
circulating to the local
value
system. heat
flux
with
After
and the
was
power was regu-
The temperature
steady-state,
plates
and was measured
to a representative
reached
piece
any back-loss
The input
AC wattmeter.
was controlled
controller
between
heater
thermopile.
with
precision
two Styrofoam
preventing
to the heater
controlled
galvanometric-type
the
was another of Styrofoam.
difference
a representative
AC power
piece
O.l"C),
to the copper
The power supplied
Styrofoam
between
patch
backed by a 2.5 cm thick
cm thick
came from an iron-constantan
per
and
designing
a uniform-power-distribution cm high,
pad sandwiched
small
lated
required
the heat flux
was 15.2 cm wide by 45.7
the
meters
C-l.
Styrofoam.
ler
a
heat flow
The heater that
the
Meters
with
of the copthe
built-in
one-half
average
a
heat
hour, flux
near the center
of
of the
assembly. The heating that at
heat flux the
pad covered
meter
two center
2%, and usually Writing the plate
calibration
positions
less
six
copper
plate
constants,
agreed
with
segments.
K,
measured with
one another
to within
balance
for
the
ith
plate
area gives: l
.. =
was found the plate less
than
than 1%.
down the energy
91
It
Kimi
+
Si-1
(Ti-T+l)
+
where 102
Si+l
(Ti-TI+l)
and dividing
by
l 1. 9i
I-Pi Si-1'
si+l
the measured heating pad power divided by pad area, with small corrections for tiattmeter insertion and back-
=
conduction; the output
=
signal
of the heat flux
= the gap conductance between upstream or downstream plate.
hhen the test plate were either two or three plates were at small correction
meter; ith
the
plate
and the
was in one of the two center positions, there guard plates on each side, and the neighboring
nearly the same temperature. Therefore, only a very was needed in the above equation for streamwise conduc-
tion.
This correction was made using estimated values unknown, then, was the desired calibration constant, calibration constant was then input to the Stanton program along with a correction for temperature,
of Ki.
S.
The only
The measured
number data-reduction as suggested by the
manufacturer, Ki
=
Once the calibration side
of the heater
Kref , i (l constant
was found,
and the above test
hhen on the side of the heater, larger
- (Ti-80°F)/700"F) the plate
was rerun--then
plate-plate
temperature
was moved to one to the other differences
and the streamwise conduction terms became significant. could then be calculated from the two tests, since
S-values
two equations very accurate
side. were
The two there were
and two unknowns. The calibration device was designed for calibration of the heat flux meters but, unfortunately,
did not give accurate S-value measurements. The estimated uncertainty on the S-values was 10%. This term is not a large contributor to the overall
uncertainty,
however.
An uncertainty
analysis
on the heat
flux
meter
calibration
representative values was performed with the technique of It was found that the uncertainty on McClintock [C-l]. A value
2.7%.
of
3% was used in
the
Stanton
using
Kline and K was -
number data-reduction
program. References C-l.
Kline, S. J., and F. A. McClintock, "Describing Uncertainties Single-Sample Experiments," Mechanical Engineering, Jan. 1953. 103
in
r
CALIBRATION HEATER
COP&t PLATES
PRRPLATE OR RECOVERYWALL COPPERPLATES
Fig.
C-l.
Heat flux
meter calibration
heater.
Appendix D Simple Model for Ordering Transitional
Initializing
profiles
the
a fully
case of
correlations at
Boundary Iayer
In the study of the effect it was decided to order
layer,
of the momentum boundary according to Re62(e=o). were taken 35 cm upstream of curvature, and, in turbulent boundary layer at this point, standard
was laminar
more difficult.
Cases
of maturity the cases
were used to estimate
S=-35cm
the
Reg2(0=O).
khen the boundary layer
or transitional,
The following
simple
the extrapolation
model was used for
became
this
extrapo-
laminar
or fully
lation. The following turbulent
correlations
boundary
were used for
the fully
layers:
--Turbulent: 2 Cf/2
=
0.037 Reio8,
=
-0.2 0.0295 Res
Cf/2
z ds2/ds
Refi
leading
to: d8 '/ds
=
-0.25 0.0129 Re 62
and
(8
is the distance
from turbulent
boundary layer
Laminar: Re6 2 Cf/2
=
0.664 Re1'2
=
-81/2 0.664 Res
Cf I2
z
ds2/ds 105
virtual
origin.)
leading
to: -1 .o 0.441 Re6 2
=
db2/ds and
62
=
0.664
(s is distance It
from laminar
was also
boundary
assumed,
layer
where needed,
virtual that
origin.)
transition
was always
in
the interval < 800 Reg 2 the average was less than 400, 400
khen
(s = -35 cm)
Reg
s = -35 c2
and the beginning
iG6
=
of transition (Re& (S = -35 2
2 from which the average
