display and the input voltage range of the IFA200 digitizer is +5 volts. Most wire and film sensors will have .... FIFO is full, in which case ACK is held low until a word is read out of FIFO and into memory). ACK ... the waveform by digitizing at very.
i
NASA
Technical
Memorandum
105210
Experimental Investigation of Turbulent Through a Circular-to-Rectangular Transition Duct
David
Flow
O. Davis .......
Lewis Research Center Cleveland, Ohio _ " ..... - -
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i
0.70
i
0.80
.00
0.90
r/R
Fig.
4.9.
Shear
stress
correlation
I
0.50
0.40
distributions
I
I
•-
Rob=88,000
o-
R%=390,000
at Station
1.
I
.x 0.30 ¢'4 II o 0.20 0.15 0.10
0
-
0
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o
oo_OOO_8
0 •
• 0
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I
0.50
Fig.
4.10.
0
0.60
Shear-energy
I
0.70
ratio
parameter
I
0.80
distributions
I
0.90
1.00
at Station
1.
CHAPTER
5
AND
DISCUSSION
RESULTS 5.1
Introduction The sequence
in which
data were collected
in the transition
duct is as follows.
The first transition duct was installed and the peripheral distributions at each data station were obtained. Data were
wall static pressure then accumulated at
Station
of Reynolds
5 to check
dependence
flow symmetry
of the
flow.
The
and
determine
transition
duct
data were accumulated at Station 6. installed and measurements were made data
were
taken
from hot-wire however, structure
all stations,
were
3 and
where
cm
data
along
near
about
wall
were
spacing)
in each
cm in the 400
data
core
region were
a direct were spline
interpolated
onto
interpolant
no overshoot
5.2
Pressure
Wall coordinate,
static refer
an evenly
duct
of the
half.
as such,
3.2.d-3.2.f.
holes
spacing
than
made
(see Figs.
between
in both
region
data
of large
based the
of the
2.54
here
that
obtained
plane
of the
measurement
in
duct
is
techniques
results
at all stations
of a monotonic
performs
cm
Approximately
on data
x-y
the measured
of interpolation
from
be emphasized
6 are
by means
1 and (0.254
varied
gradients.
about
to 0.127 symmetry
grid
points
It should
data
Quadrants
x 19 rectangular
flow and mesh
The
core
5 and
purposes,
spaced
method
of the
in Figs.
access
vortex
and,
assuming
between
of symmetry
quality
plane
1.27 cm in the
were
for Stations level
deduced
quantities,
6, rather
on a 63
cm in regions
For plotting
[55]. This
guarantees Static
the
of the
in this study.
from 5 and
spacing
in the
results
so that
indication
employed
to 0.254
were
anticipated
x-z
probe
so indicated.
points
The
taken
contour
quadrants,
are
measurements
quadrant.
the
2 shown
of the
and
transition duct was Stations 3 and 4. No velocity
the
number
installed
6. Turbulence
6 since
1 and/or
varied
at selected
mean
then
in Quadrant 2. For presentation purposes, to show an entire duct half. All subsequent
At Stations
z=O,
taken
points
all subsequent both
traverses
surface.
midplane
2. Data
imaged
was
and
about
arrangement
taken only the y-axis
been
individual
the the
have
and 3,4,5
assumed
in Quadrants
to the
were about
pressure
at Stations
was
extent
Finally, the second at the intermediate
at Stations 5 and until Station 5.
only
4, due
3.2.d and 3.2.e), data the data were imaged points
Total
made
symmetry
accumulated
At Stations
plots
2.
were only measured was not well defined
At data
at Station
measurements
the
extension
derivative
no smoothing
and
data.
Distribution
pressure to Figs.
P,,A__INTENTION_LLy
distributions 3.2.d
through
BLANK
were 3.2.f)
measured of the
along lower
the half
PRECEDIb,'G
periphery of the
PAGE
duct
BLANK
(sat
NOT
FILMED
48
Stations
3,4,5,
shown
and 6. Peripheral
for Reb
= 88,000
is 1/4 of the duct pressure
laxity
at all stations
with
the
along
more
the duct
and
in static
flow
is the layer
centerline
pressure.
a positive
pressure
peak.
Concave
in static decrease of the
convex
pressure
pressure occurs in cross-sectional walls
changes
relative
to the
inlet
is no curvature curvature
peaks
5.3
Effect Results
layer
at
the
ber
dependence
the
influence
lower
which
thus
far
operating
exists
over
and
some
limited
mean
velocity
conditions
results are shown here field behavior will be
notable
4 and
vector at the difference
upper
the
this
are
the
two operating is that
lower
associated
wall
sidewalls
in a net
is reflected
in the
(s/s,.eI=O)
induces
observed
pressure
in the
pressure
results
(S/Sre¢=:i:l)
minima.
induces
A net
decrease
is the result of a slight the radius of curvature
4 which
causes
maximum
only
to viscous
upstream
effects.
and
curvature
5) are still strongly and
Although
the
effects
present.
static
the
there
(the
wall
At Station
pressure
6,
is nominally
shown
that,
aside
number,
limited
range
no
turbulence numbers contours
shown
flow
data
were
of 88,000 and
in Figs.
and
transverse 5.2,
a thicker
appreciable
considered.
on downstream
from
5.3 and
largest
measured
conditions vortex
pair
are
vector observed
for the
lower
boundary
Reynolds
In order
num-
to determine
development,
mean
accumulated
at
390,000.
Total
velocity
vectors
in the
Station for
the
These
of the flowthe reference
plane.
to be very Reynolds
flow
pressure
5.4, respectively.
only for comparison; the physical significance discussed in the next section. In Fig. 5.4,
represents
the
Number
have
the
contours,
axial
plane.
numbers
for
losses
which
results
3 and
station,
number
Reynolds
flowfields
along
Reynolds
bulk
velocity
the
in the
is due
at this
5 for operating two operating
along
Stations
ReynOlds
of Reynolds
measurements
5.1,
(Cp=O)
> 0) which
curvature
Station
presented
regu-
z-z
Reynolds
is larger
viscous
to Fig.
value
change or curvature effects, the entire cross section.
of Varying
the
along the side and upper walls, respectively. At to the inlet value so that a net decrease in pressure
walls
between
there is no area constant across
inlet
between
occurs
of the
two
between Stations 3 and 4 which area and viscous losses. Also,
sign
spanwise
about
duct
8re!
represent
of cross-sectional area, wall curvature at an axial position where streamwise
resulting
minimum pressures to occur Station 5 the area has returned
dimension symbols
excellent
of increased
curvature
wall
the
are
growth.
(aP/Or
gradient
The to the
along
distributions
Solid
of symmetry
In reference
the
gradient
Conversely,
a negative
above
normalizing
The
station.
referred
result
is occurring.
pressure
5.1. data
drop
Local static pressure is a function viscous forces. Station 3 is located of the
coefficient
centerline.
results
boundary
pressure
assumption
coefficient
case
rapid
the
between
pressure
number
diffusion rise
net
in Fig.
at a given
supports
difference
the
Reynolds
and
measured
primary
is that
390,000
circumference
static The
and
wall static
The similar.
number
mean One is more
49 circular
and
observed both
centered
were
not
operating
significant
practical
applications,
duct
remaining
measurements
Mean
5.4.1
Flow
Mean Mean
and
6.
Figs. and
would
velocity
velocity
contours
results
5.6,
shown
duct.
induced
the
of the
At
relatively Between
boundary
layer
vicinity
thickening
pressure probe
pressure
the
variables at
Station
are
P, -
defined
5 was
are
divergence
station.
develops
centers
away
upper
boundary
by natural
wall are
vortices
further
respectively.
sidewall with
their
6, the
layer,
The
flow arises
of the
and
thickens
in the
which
vicinity
of the
sidewall
are
indicated
5.1) associated
centered
to the
from
(lower) layer
Between growth
have the wall causes
Stations and,
in the
convergence.
calculated
Pwhere
boundary
distribution and
and
of the
of hot-wires
secondary
Station
At
in the axial
similar
at that
in shape At
of the
everywhere
by lateral
to Fig.
duct.
a non-diffusing
flow pattern
oblong
5, lateral
convergence
layer
static
are
very
This
axial
vicinity
grows
vector
flow in the
(refer
sidewall.
circular
2 and of the
boundary sidewall,
static pressure butions:
duct
in the
observed
and
of the
by means
sidewalls.
near-wall
vortices
are more lateral
duct
gradients
the
Stations and
magnitude
the
of the
5, the near
of the
The
along
shape
velocity
in
probe)
and 6 are shown
flow through
measured
3,4,5
shown
pressure
is qualitatively
vectors
are (Pitot
3,4,5 total
the distortion
of the reference
pressure
extent,
sidewall.
distortion
vector
skewing
Station
in lateral
and
for
of the
at Stations
stations
to develop
that
flow is due to a secondary
pair
of lateral
6, the
most
contours
sectional
is seen
6 show
velocity
primary vortex
positioned
5 and
of the
maximum
by transverse
thinning
5 and
data
3, the
cross-
contours
plane
at Stations
Station
the
The magnitude
curvature. grown
Since,
condition,
pressure
et al. [39] for turbulent
Transverse
a discrete
as a result
of the
by Taylor
represents
into
of the
case.
to theses
probes) At
follow
at Stations
in Fig. 5.7.
distortion
for
bulk
for a circular-to-rectangular
transverse
Total
(hot-wire
respectively.
development
presented
S-shaped
the
condition.
operating
in the
applicable
generally
The
operating
= 390,000
respectively.
contours
and
The data
direction.
plots
3.2.f,
4, a distortion
sidewall.
Res
were measured
coordinates
through
5.5
Station
axe for the
all measurements
to restrict
condition higher
the differences
contours
Reference
axial
to one
to the
though,
repeating
therefore,
stations
closer
flow variables
in Figs.
to justify
decided,
the operating
be
Overall,
Results
flow
3.2.d
the wall.
enough It was
at the remaining
transition
$.4
away from
conditions.
measurements most
further
at Station from
the
total
pressure
directly
and
velocity
1 2 _Pa,,_bU_/(R,_i,T,,mb)
as in equation
possible
5 was measured
since
the
(3.1). total
with
a
distri-
(5.1) Measurement flow
angle
of the is less
than
static 10 °.
5O The
measured
static
and
pressure
shown
in Fig.
agrees
well with
5.1.b.
The
calculated
pressure
field
but
doesn't
show
layer
mean
divergence
along
the
layer.
On
midplanes the
were
y=0
The
wall tap
agrees
5.9.
midplane
Along
part,:conflned ther illustrate
the
z=0
duct
z=0
y=0
(side
to estimate (negative
(upper
evaluated
upper
wall),
the
strength
divergence
wall),
the
midway
the
duct
divergence
the
parameter
2.8 and strain
rate
wall,
which
5.4.3
Mean Total
that
flow
is defined
profiles,
These
layer
substantially
This
is due
periphery Beyond layer
from
along
both
to converge a large
jump
most
To furof the
jump
in a manner
similar
to the
is observed
between
x/R
is due to a reduction
in the
primary
result
effects
results thickens
the
at
of the
will
Stations
wall) traverses Similarly, axial
to the secondary
Station
is, for the
vortex
pair.
The
results
= at
be reflected
in the
local
turbulence
profiles
pressure
5.12.b.
divergence
as
(_ 0.5 cm). is shown in
boundary layer continues to converge well into the Based on the strength of the divergence observed,
divergence
(sidewall) and y3 (upper and 5.11.b, respectively. and
This
is a direct
z/R = 8.0 shows that the transition duct extension. it is anticipated structure.
but then
4 and 5).
OU/O_/2
the
boundary
section, peaking at around 20%. on the upper and lower surfaces
the flow begins
upper
4.0 (Stations
surface,
same
parame-
surface oil flow results by Reichert et al. [4] obtained duct at an operating condition of Recl,i = 1.57 × 10 6.
sidewalls,
on the
the
of the
is the
divergence
through
the divergence
(lower)
to the actual transition the degree of divergence
duct, Fig. 5.10 reproduces in an identical transition Along
with
as clearly.
(OW/Oz)/(OU/Oy2) evaluated at the first data point from the wall An estimate of the axial distribution of these divergence parameters Fig.
measured
measurements
distributions
used
and
midplane
as (OV/Oy)/(OU/Oyl)
the
the
5.8.
qualitatively
saddle-shaped
distributions
On
is defined
the
in Fig.
divergence
velocity
as convergence). ter
are shown
wall region
Boundary The
distributions
in the near
measurements, 5.4.2
calculated
vicinity
show
that,
along
1,3,4,5
between
traverses
thickens.
measured
Stations
the Y2 traverse
of the the
6,
along
the
(see Fig. 3.2) are shown in Figs. velocity profiles are shown in Figs.
flow transferring
5, where
and
thins
boundary
1/3 traverse
cross-sectional Along
and
1 and
boundary
the
1/3 traverse.
fluid along
to the
vicinity
shape
is constant,
the I/3 traverse,
Of the the
5.11.a 5.12.a
5, the
along
layer
Y2
the duct
!/2 traverse.
the
boundary
thickening
is due
to natural boundary layer growth, but along the 1/2 traverse, the thickening due also to the common outward flow associated with the vortex pair. At
Stations
5 and
traverse
exhibit
a double
inflection
and
axial
5.5.c
and
5.5.d)
the
6, the
total
pressure behavior.
velocity
contours
and
velocity
The total (Figs.
profiles pressure
5.6.c
and
along contours 5.6.d)
the
is
1/_
(Figs. indicate
51 that
along
behavior high
traverses
adjacent
and
is observed.
This
is a result
momentum
sidewalls,
fluid
from
re-energizing
separation.
the
A break
of low momentum
creating
a flat
5.4.4
spot
6 (Fig.
ridge
than
Streamwise
vorticity The
second
in non-
vortex
These
flat
along
probably
the
layer
toward
flow
due
the
of duct
preventing
(y2 traverse)
region
5 (Fig.
to a
centerline,
is seen to be much
larger
5.6.c).
ducts
by a blunt
occurs
can
be
by the
of the transition of total pressure
generated
lateral
obstruction
vorticity
Oz +
duct loss.
different
(by pressure
in the
Oy +
vorticity
through
is much
wO ,
than axial
(4)
and by the
non-circular
OU
ducts. vorticity.
vorticity
equation:
BU
OU
+ a,N + a, o--; (3)
(2)
02 02 Oz2)(__--_) +__(.2 OyOz
02 + ( Oy2
The
skew-induced mean
-
layer. stresses
generation
straight
weaker steady
Reynolds
(1)
0-_) Ou
two
boundary
by the
Streamwise flows
vorticity
are represented
O 0-_ +-_x ( Oz
by
deflection
in a 2-D
created
vorticity.
in turbulent
stress-induced
mechanisms
in the exit plane significant regions
generation
is streamwise
stresses
pairs cause
circular
to as stress-induced
Generally,
very
region
midplane
peaking
a "ridge"
with spanwise vorticity. This mechanism is often streamwise vorticity. An example of this type is the
generated
mechanism
Reynolds
to the
boundary
This
at Station
first is vorticity
of a shear layer to as skew-induced
is referred
and
a double
convecting
vortlclty
mechanisms.
horseshoe
the
pair
flow
at the
field.
The presence of the vortex is undesirable, inasmuch as they
gradients) referred
layer
from
Y2 traverse,
vortex
core
occurs
velocity
5.6.d)
Streamwise
potential
fluid
in the
to the
of the
boundary
in the
transfer at Station
the
parallel
(5.2)
_ w2) + vv_,
(5)
(6)
(7)
where, fl'= The
LHS
OV
Oy
Oz'
fl_=
of equation
(5.2)
represents
The
term
on
convection. vorticity
OW
first
by vortex
of vorticity).
The
line
stretching
second
and
the
OU
OW
Oz
Ox'
the RHS
increase
terms
Ox
0_1
production
acceleration
on the
OU
in streamwise
represents
(streamwise
third
fl'-
OV
RHS
vorticity
by
of streamwise
causes
represent
amplification the
increase
in
vorticity due to lateral skewing (by transverse pressure gradients) of vorticity in the transverse directions. These are the terms associated with skew-induced vorticity. the
primary
gradient.
The
fourth shear
The
fifth
term stresses and
represents and sixth
is often terms
the
production
neglected on
the
RHS
of streamwise since
it contains
represent
the
vorticity
by
a streamwise production
of
52 streamwise
vorticity
by inhomogeneity
of the transverse
normal
stress
auisotropy
and by the secondary shear stress, respectively. These terms are responsible stressed-induced vorticity. The last term on the RHS represents the diffusion streamwise
vorticity
by viscous
forces.
The significance
of equation
(5.2)
transition planes
duct
configuration.
of symmetry,
to Quadrant
Since
to avoid
I (refer
can now be discussed the
confusion,
to Fig.
3.2)
of axial
of the
3,4,5
velocity
components
onto
a uniform
grid
by central
difference
approximations.
at Stations by dashed location prised _z
3,4,5
and 6 are
contour
lines.
where only
being
The
the peak
filled
circles
axial
in these
cm) and
then evaluating
fly and
the
approximate
vorticity
_z,
the
wall curvature
field is com-
axial
induces
5.3 and illustrated of the flow occurs
area
in Quadrant
pressure quires which
Fig.
gradient
5.13.a
creates
(Station primarily
there to be a thin layer was not resolved in the
tive
vorticity
will occur
the pressure gradient half of transition, the transverse global
pressure
negative
straight
is primarily wall curvature
of the
(see flow.
5), show
intensifies. tion
that This
(vortex
the negative
stretching)
Fig.
vortices
intensify.
tion. _zR/Ub
Between
5 and
1 the
no-slip
duct
condition
without
of wall curvature. sign which causes Also,
transverse re-
the
that
since
In the second a reversal of the
contracting
be expected
swirl
area
the
causes
reversal
a
of the
migrates vorticity surface
towards is caused
curvature
the
6, the
reversed
pressure
the
midplane
by streamwise along
the
gradient
magnitude
of the
peak
sidewalls.
concave GSrtler
assists
the developing vortex pair. 6), the vorticity is diffused
and
accelera-
duct
instabilities, that Taylor-
z=0
the
curtype
inward
In the transition by turbulent ac-
vorticity
drops
from
= 2.4 to 0.09.
Miau et al. [9] evaluated of two CR transition ducts aspect
concave
flow between 5.13.d, Station
Stations
5.1).
in Fig. 5.1. due to the
in the very near-wall region The generation of nega-
not generated by centrifugal vorticity in the same manner
In addition,
(negative y direction) duct extension (Fig.
The
transverse
cancel the vorticity generated in the first half however, as Figs. 5.13.b and 5.13.c (Stations 4
of the
and are the
a function changes
vorticity
strengthening
Although the vortices vature will accentuate
vorticity.
CR transition
It might
pressure gradient would effectively of transition. This is not the case, and
that
of positive vorticity present measurements.
in any
gradient
acceleration
3) shows
component
the
pressure gradients which were discussed in Section In the first half of transition, a global deceleration expansion.
vorticity
is represented
mark
1, the
com-
transverse
of axial
vorticity
plots
across
vorticity the
Contours
At Station
2, the
The
Negative
components
at Station
changes
will be restricted
interpolating
x 0.508
5.13.
occurs.
vorticity
Beginning
by
(0.508
in Fig.
vorticity
of transverse
zero.
shown
calculated
to the present
vorticity
discussion
duct.
at Stations
derivatives
6 was
transition
ponent the
and
sign
the following
relative
for of
ratios
(AR)
of two,
all the terms in equation of constant cross- sectional but
the
transition
lengths
(5.2) area. differed:
at the Both
exit plane ducts had
L/D=0.54
and
53 L/D=I.08.
The
vorticity than
results
at the exit
turbulence
and the
of their
plane
induced
present quadrant,
observed
in the equivalent the
ducts.
One
of one,
and
ratio
duct the
of three.
vorticity duct.
had
the
ducts followed
aspect had
feature
exhibit
area
by a contraction
considerably
vorticity suggest
that
geometry
and
the condition
of the flowfield More
Reynolds
distinguishing
feature
in the vicinity
of the vortex
as they pairs with are
the
The these
data
applicable
of the
vortices
deformation vortices
detailed common
present
the
ferences studied duct
pair.
sets
than
plane
the path
the
expansion the
that
negative
These
is very
results
sensitive
the vortex
is amplified
inlet
length
for the
respectively.
vorticity
between
at these
In Section
to the
core takes
or attenuated.
flows
were mentioned. Eaton
[31,32] results
away
from
the
afford
to determine
the
flow.
is taken
station
the
Based 70 cm
reported Mehta
studies in the
should
as occurs based
the
at Stations and
Bradshaw
In addition be noted.
where
5 and Eaton
the origin the geometry
length
the
of their length, other
boundary
region
is three-dimensional.
vortices
results
for the
6, respectively.
in a two-dimensional of the
transition
The
was 97 cm downstream most
to development Whereas
vortex
on their
development
present
inasmuch
comparisons
purposes,
as the location
on this,
in the
to make
conclusions
in-
the studies
for embedded
For comparative
by Pauley and
of 135 cm.
embedded layer
2). 30 and
structure
are relevant,
opportunity
if modelling duct
(Station
surface,
6. A
experimental
In particular,
turbulence
and
duct
length
detailed
and
flow
5 and
is the turbulence
mean flow
and
stations
at Stations
2.3, several
in the transition
generators
vortices
was measured
to the transition begins
boundary
In contrast, an area
is the transition
[30] and Pauley
is approximately
a development
tensor
measurements
first measurement from
stress
vortex
and Bradshaw present
with
duct.
specifically,
AR=6
vorticity
For all ducts,
ducts 3.0,
of positive
for the
positive
had
feature
in the exit
if the initial
of the flowfield
of embedded
by Mehta
exhibited
vorticity
CR ratio
regions
vorticity
section.
vs. 1.5 and
[2,3]
of two
Results
The complete
vestigations
positive 1.08
will determine
Turbulence
transition
is
a length-to-diameter
small
plane
distinguishing
for the
plane
was constant.
exit
vorticity
a length-to-diameter
which
the duct
at the exit
and McCormick
of six and showed
in the
the
Another
L/D=0.54
the duct
through vorticity
of the transition.
through
and
rather results
in Quadrant
axial
exit
of negative
ducts
region
positive
Patrick
4),
Miau_s
is observed
in the
ratio
regions
the
through
area.
shorter
ducts:
large
between
negative
the exit
ducts.
of three
an aspect
and
corner
study,
of axial
3 and
between
in the
measurements
generation
(terms
axial vorticity
present
ratio
the
induced
of this investigation
duct
that
difference
vortieity
of their
from
an
cross-sectional
area equalled was
qm_drant
The results
which
axial of the
duct
for the AR=3
skew-
A notable
negative
5.13.c)
vorticity
other
A common
is that
5.5
Fig.
axial
the
Whereas
1 (upper calculated
5-7).
is that
sign.
showed
was primarily
(terms
study
plane is of opposite
analysis
layer,
results other
at dif-
investigators the
transition Also,
the
54 velocity deficit in the vicinity of the vortex cores is much larger in the transition duct than in the other studies. Mehta and Bradshaw placed their generators in the settling tential
chamber
so that
core flow velocity
and
Eaton,
test
section,
on the at the
were
made
Table
5.1.
(97
by the time
other
so that
somewhat
the velocity
hand,
cm).
Table
the vortices
5.1.
deficit
was initially
station in the
Velocity
deficit
percentage
at the
present, the
vortex
of the po-
the test section.
vortices
where
deficit
of the vortex
entered
their
data
Velocity
was a small
generated
a significant
downstream
deficit
Pauley
beginning
of the
but
had diminished
turbulence
measurements
core
region
is summarized
in
in the vicinity
cores.
x (cm) v/u,
Based
on
these
qualitatively Mehta and 5.5.1
results results
Pauley
& Eaton
Pauley
& Eaton
Pauley
& Eaton
[31]
in Fig. local
intensity
used
0.95
Mehta
& Bradshaw
[30]
90
0.95
Mehta
& Bradshaw
[30]
135
0.95
differences
alone,
similar
intensity
5.14.
Note
velocity
that
for reducing figures
contours
the
turbulence
the
and
present
Eaton
The
between
of the
the
results
than
will
to those
increase
Stations
intensity observed six
5.15-5.20.
intensity the
(u'/U)mat
5 and
Reynolds Negative
equations
(see
of
6 are
relative the
filled
circles
in Fig.
(peak
vorticity)
the
vortex
pair
thickness
evident.
local
Appendix
cores
,,_ 14% and
layer
6 is clearly
is defined Since
The
of the sidewall,
5 and
velocity.
vortex
in boundary
exhibits in the
here
bulk
data.
of the
intensity;
at Stations
response
hot-wire
location
measured
not
second-order
In the vicinity
6, respectively.
= 0, the turbulence the double inflection
and
all of the mark
5 and
in Figs.
that
of Pauley
u'/U
10% in places,
turbulence
Contours
results
component
of moderate
shown
it is expected
to the
contours
tongue
pair
0.75 0.90
an aid to interpretation.
the vortex
0.50
97 60
axial
subsequent
66
[31]
142
exceeds
C) were
[31]
[30]
turbulence
to the
0.40 0.50
& Bradshaw
be more Bradshaw.
shown
30 70
Mehta
Turbulence Axial
and
Present Present
5.14 as
extrudes
a
11% at Stations and
Along
distortion
by
the midplane
z
a double peak behavior which is a result of mean velocity profiles shown in Fig. 5.12.a. stress
components
contour
levels
at are
Stations
5 and
represented
by
6 are dashed
55 lines.
The
z/R
level
of symmetry
-- 0 is very
shown
in Fig.
between stress
5.20.
values
increase
in peak
axial
turbulence
axial
velocity
for the
are
higher
and _ Figs.
displaced
of the
intensity
(Fig.
5.14)
6 than
between
at 135 cm show u2
component
cm
show
the
plane
all the vortex
that
as the
a peak
in the
of peaks
near
in the
of symmetry.
The
vortex
present
region, outer
core
region
results
of the transverse
show
in the local
normal
of Mehta
along
the plane
results
of Pauley
also
show
of the
stress
a large
and They
on the
peak
also
about plane
to the
generators
component which
exhibits
two
peaks
were not observed
that
all the
normal
Anisotrop_y
(u 2 - v 2) is shown larger
core
at Station
nent
by as much the
& Eaton stresses
and
tical
normal
core
at Station
stress
of y/R
= 1.4,
Station
6, the are still
ticai
on
show
the
a
plane
of symmetry
and
components
=l:z/R
= 0.35,
vertical nearly
horizontal
stress
equal
are the
transverse
observed
vortex normal vorticity
midplane
at 97 cm suggest peak
axial
exceeds
similar
Anisotropy
of the
core stress
region.
5.22. equal.
the the
axial axial
Anisotro__py
components
in non-circular
axial
compolayer Pauley
of the the
axial
Near
is
vortex
boundary
behavior
in Fig.
than
transverse component
the
between
exceeds
less
behavior.
components.
to be nearly
stress
is everywhere
in the
of streamwise
report
core.
vertical
y--0
vicinity
transverse
(u 2 - w 2)) is shown
stresses
the
to a two-dimensional either
both
w 2 stress
the horizontal 5, the
component
than
vortex
the
data
In the
velocity
that
this double
At Station
transverse
about
and
to the
is present__directly
show
Eaton's
component.
is greater
of the
5, the
in the generation
the
& Bradshaw
vicinity
5.21.
This is in contrast
component Mehta
in the
in Fig.
and
exhibit component
horizontal
6, however,
axial
they
the
as 20%.
5. Pauley
stress
deficit
results located
will eventually
everywhere
velocity
symmetrically
bet._.ween the axial than
a large
6, the present
at Station
stresses
component
where
since
At Station
traced
at 97
(or lack of) in the vortex core re, on. It is likely that the vortices in the and Eaton study exhibited similar behavior to the present results at a the generators.
can be
the
Eaton
an additional located
peak
differences
and Bradshaw
of symmetry,
vortices.
symmetrically
These
The in peak
deficit Pauley
closer
pair.
sidewall. decrease
edge
behind
vortex
peak
roug____ half the peak magnitude the u 2 stress component differs
but edge
duct
an increase
The results The
sign
the Reynolds
5, and that
.to the
from
change
that
at the
location
of the
studies.
the
the
mldplane component
levels.
magnitude
increases.
near-wall
of symmetry
from
resulted
wall is approached
monotonically
show at Station
turbulence
the
components
is in contrast
which
6, the peak
stress
v 2 and w 2 are ne___r equal and are stress component u 2. Qualitatively,
significantly
about
U'_ stress
5.15-5.20
farther
stresses
of the high
5 and
component
at Station
and
magnitude
Stations
components of the axial
the _
distorted
in the vicinity
At both
stress
dit_cult-to-measure
1 and 2. In general,
are more
contour
for each
even
As expected,
Quadrants contours
pair
good,
normal and
the
In the
ver-
vortex vicinity
component.
At
component,
but
be..._tween the
ver-
(v 2 - w 2) is important ducts
(see Section
5.4.1).
56 Contours differences
of this quantity are plotted in Fig. 5.23. These plots show significant between Stations 5 and 6. It is well known that in two-dimensional
boundary
layers,
transverse
component
acts
normal
duct
flow,
both
Stations
layers
u-_ > u-'_ > u"_, where
surface
the above
component
6, the
at Station
Pauley
Eaton
4.39
and
of Ref. The
primary
stress
levels
where
which
transition
lower
anisotropy
to the
and
is pos-
intensified.
results
length
the
In the region
the auisotropy
has grown
similar
at
exceeds
anisotropy). where
region
wall boundary
reported
by
of 97 cm (see
Fig.
in Fig.
in the velocity mean
y and
this
stress
stress
data
mean
velocity
summit quantity
vorticity is of the
(see
same
Section
profile
plays
primary
as a result, shear
ridge
W_
is crossed.
an important
5.4.1).
In the
of magnitude
>
Along
the
and,
the primary
that
Section
(OU/Oy
flattens, zero
These
5.4.3).
of the velocity
order
cores.
rate-of-strain
(see
In the z-direction,
is another
to be positive
vortex
is nearly
sign as the
_
is observed of the
ridge
z directions
are depressed.
stress
5.18
vicinity
of the positive
axial
the
of streamwise
cores,
shown
role
vicinity
of
as the
primary
kinetic
energy.
stresses. The
These
normal
results
kinetic
are
energy
duction
shown
was
also
of kinetic
were
for
used
Stations
calculated.
energy
to calculate 5 and
is given
of this quantity
energy
are
associated
on
duct
upper
the
near-wall
region
is essentially 35%.
with and
high
lower
when these the
stations.
same,
but
production
walls
at
to the Between
the
The
production
derivatives,
of
the
pro-
OU
in Fig. 5.25.
less than
compared
5.24.
streamwise
- --uw
shown
is significantly
disproportionate between
are
turbulence
by:
P= Contours
the
6 in Fig.
Neglecting
OU
ergy
and
at a development
region
also changes
shear
production
shear
_
the
in both
in Fig. 5.19
vortex
For the
to the surface
exists
very
aforementioned
in this region
secondary
the
of positive
are a result
the
z=0
rate-of-strain
in the
pocket
qualitatively
stress
stress
with
midplane
shown
(negative
for a small
of negative
mean
The
shear
except
0) associated the
to the surface
pair
54).
to hold true in the near-wall tangent
for a vortex
u-_ is the
u_ is the component
of Ref.
component
region
component,
31).
everywhere regions
and
18.5
of the upper
5, a small
6 are
axial
region
the
acts normal
By Station
Fig.
is observed
pair at Station
The results
surface,
e.g.,
6. In the outer
5, however,
which
of the vortex
to the
(see,
inequality
5 and
at Station
itive.
tangential
to the
u-_ is the
peak
(5.3)
Generally,
rates.
A notable
Station
5.
that
Here,
observed
relatively Stations
kinetic
high
5 and
exception
increases
occurs in the
6, which
difference 6, the
of kinetic
production
at Station
small
energy
the
levels
peak by
seems
in kinetic production
approximately
en-
57
5.5.2
Turbulence
profiles
In this section, minor
axes
at Station stress
Reynolds
are presented 1.
In the
components
stress
profiles
and compared
following
profile
are relative
along
with
the duct
the
plots,
initial
the
to wall coordinates
by the bulk velocity
local
layer
boundary
thickness,
5._.
Normalized
of the
x, y, z laboratory
(b/R)
at Stations
1,5 and
Reb
in Fig. the
5.26.
are shown
Reb
1
0.31
0.31
0.29
0.29
5
-
0.21
0.58
0.18
6
-
-
0.65
0.25
sidewall
vortex
destabilizing flow away 5, the
axial
strong
attenuation
components levels. The between are
friction
normal
made
and
mean shear
the
lateral
in the
to scale in the
velocity
with duct
was
6. the
outer
region
components
show
region.
Near
In wall-bounded friction at
energy along
boundary
curvature and the developing the
boundary
wall
and
layer.
In
The
resulting
in the
5.12.a.
largest
common At
double Station
deviation
with
6, these
stress
but are still below the initial in the near-wall region decrease
velocity. are
6 along
kinetic
near
in Fig. the
shown
5 and
The
the wall at Station
shear
Stations
results
deficit
1 are
distributions
curvature.
shown
These
the
profiles.
of the
concave
of velocity
local
and
convergence
profiles
mid-planes
deduced.
that
a region stress
stress
initial
velocity
in the near-wall
5 and
at Stations
first to stabilizing convex second half of transition,
destabilizing
creates
at Station
shear
apparent
have increased to a certain extent, transverse normal stress components
Stations
known
were
wall
and
from
stabilizing
divergence
of the
ys-axis
measured
It is readily
experiences
the
behavior
390,000
measured
5.27
considerably
creates
flow
from
infection
5.28.
in Fig.
sidewalls is subjected convergence. In the pair
the
distributions
distributions
shown
deviate
lateral
addition,
stress
in Fig.
=
cm
stress are
axis
layer on the duct stabilizing lateral
6.
y2-axis
axis
semi-major
thickness
ys-axis
Normal
by the
5.2.
88,000
Reynolds
semi-major
profiles the
normal
is normalized
y2-axis
R = 10.214
-"
layer
Reynolds
is always directed along 3.2). All of the stresses
in Table
boundary
measured
designation
the wall coordinate
as summarized
Table
Station
Reference
and
and semi-
and not the
coordinates; that is, the v fluctuating velocity component the wall coordinate of interest, either y2 or Ys (see Fig. are normalized
semi-major distributions
layers,
the
Preston
1,5 and summarized
Reynolds tube
6 from
stresses
measurements which
in Table
the 5.3.
local
58 Table
5.8.
Normalized
drop
and
components velocity to
lag
Station
y2 =0
ys =0
1
4.06
4.06
5
2.70
4.89
6
3.23
4.38
behavior stress
vortex
pair
6.
normal
Several
stress,
5.3.
The
friction
in the near-wall
further
away
from
stresses
outweighs
transverse
the
to the
attenuation
First,
the
primary
of the
boundary
layer
axial
and
flow of the vortex
acts
to suppress
that
increase
as the in the
normal
stress
Stations
turbulence
(aU/ay2),
outward
appear
decrease
between
of the
stress friction
stresses
The
in the
is observed
rate-of-strain
by the common
the
it is presumed
sidewalls.
increase
contribute
shear
with
will eventually
duct
energy
and
normal
although
region
the
normal
6 correlates
velocity,
kinetic
is reduced
convergence
Reynolds 1 and
in turbulence
factors
wall region.
axial
Stations
in Table of the
components
a decrease
in the
between
shown
moves
transverse
rise
wall
development
all the
that
the
Reb = 390,000.
m/s
subsequent
near
the
velocity
x 100) at mid-planes,
Ub= 29.95 The
friction
(U,./U,
in the
hence
so
5 and near-
the wall shear
pair.
Second,
lateral
turbulence.
And
finally,
Fig. 5.12.a shows that between Stations 4 and 6, the near-wall flow to streamwise acceleration which suppresses turbulence generation.
is subjected The high
turbulence
are
levels
observed
of high primary
rates-of-strain
The history cantly
half
the flow along the boundary
destabilizing
divergence 6 are
shown
in Fig.
ited
by the
was
most
and
distributions shown
bevel
gear
serious
boundary
Wall
and
the
shear necessary and
Over
able to be measured, the net effect small. The trends show an increase decrease The
in the
combined
5, the
shear
is recognized further,
transverse effect
stress that
turbulence
normal
on the value
one
the
point
measurements
range
axis).
half
In the
kinetic
of the
of transition, curvature.
axis energy
at Stations
5 and
rotation.
surfaces
concave
convex
semi-minor
and
duct
is signifi-
at Stations profiles
are
6 were
lim-
This
limitation
at
Station
5 where
boundary
layer
that
was
of the above flow conditioning is relatively in the axial normal stress component and stress
components kinetic
the
is not
second
for probe
lower
turbulence
nearest
stress
axis)
to destabilizing
stabilizing duct
measurements
upper
is thinnest.
the
a result
deficit.
(semi-major
In the
experiences
layer
(semi-minor
is subjected
along
arrangement
along
layer
flow
proximity
velocity
z=O mld'pl_e
layer
divergence.
the
5.29
the
boundary
y=O mid-plane
measured
in Fig.
5.30.
of the with
the
lateral
decreases
stress
5 and
the
the
from
and
Normal
associated
of transition,
curvature the
mid-region
of the flow along
different
first
in the
wall the
is very
is observed
statistically along
between
energy
to decrease,
significant. y3 axis
were
Stations small.
1 and
6.
At Station although
To investigate made
a
at a bulk
it this
operat-
59 ing Reynolds
number
region
could
shown
in Fig. 5.31.
els and
the
or that
indicates
wall shear
speculate behavior
both
equilibrium
which
will be examined
more
closely
Considerations
The
calculations
by
performance
lating
predictions,
transition
must
also
duct
be included.
difficult
aspect
gradient
effects
Since
presence
of the
ity,
however,
relies
efforts
The first
group
prediction shown
for the
group
The present The
with
detailed
set should
be useful
simplest
turbulence
model
on the
to behave
concept like the
for incompressible
and
vt is the k is the
is the
dimensional important, are
not
flows inasmuch in agreement
where
flow is written
as:
only
as the with
viscosity,
even
diffusion
of vortic-
components.
into
two
groups.
parameters
and
et al. [33] has of embedded
length
models.
The
of particular
stress
(zero-order)
.
behavior.
the
stresses
Reynolds
Kronecker
(5.4)
shear stresses
flows.
Reynolds and
viscosity are stress
(5.4)
/_ii is the
normal
simple
the
eddy
2
Equation
Reynolds
Reynolds
will predict
stress
Reynolds
stresses
OUj
energy. the
models
the ability
and
of
via pressure
in the region
Here,
viscosity
eddy
vorticity
mixing
algebraic
of Boussinesq.
kinetic
in the neighborhood
work of Liandr_t simple
calcu-
of turbulence the most
performance
flow
molecular
turbulent
turbulence
when
for
of course,
of the
flowfield
even
for this group.
.OUi
where
that
required,
can be divided overall
mean
5. Near-wall
standpoint,
Reynolds
the task of demonstrating
to predict
to
effects
even simple
of the
well with
data
based
assumed
is concerned
models
the
prediction
of the mean
reasonably
of
be neglected
of streamwise
with
the benefit
4, it is difficult
shown
modelling
The computational
features
have
cannot
configuration
primarily
region
5.7.
is the flowfleld
Accurate
levstress
of the Preston
Without
From a computational
modelling
present
is concerned
the primary
turbulence
model
pairs.
are
energy
in shear
at Station
predictions
process,
a larger
in the near-wall 3 and
et al. [7,8]
generation
on accurate
can be predicted
second
tensor
vortex
Stations
of turbulence
an inviscid
of flow separation.
that
vortices
the wall.
configuration
the initial
is essentially
the
Modelling
level desired.
of the present
pair.
near
effects
kinetic
by means
recovers
For accurate
The
on the information
Burley viscous
flows.
depends the vortex
measm_
and
for this case
decrease
stress
in Section
Modelling
overall
The
lead to the flow condition
Turbulence inviscid
is thicker profiles
the unchanged
stress
the shear
energy
stress.
at the intermediate
on the factors
layer
kinetic
confirm
either
measurements
and
of the shear
the flow is not in local
turbulence
6.6
results
that
the boundary
stress
behavior
to the increased
This
where
Shear
These
decreasing
is in contrast tubes.
of 88,000
be resolved.
For
stresses
(u--7_',
predicted the
delta
is applicable
present
function to three-
i #
by this
j)
are
equation
configuration,
6O equation transition
(5.4) may be adequate for prediction purposes through the end of the section, but will not be applicable if the flow is allowed to develop in the
rectangular in the
duct
generation
Since
gradients
flow solvers
normal
flow.
equation
secondary
reduces
are important
flows. with
to examine
For the primary (5.4)
stresses
are implemented
it is worthwhile
present
neglected,
Reynolds
of corner-generated
models,
for the
are
transverse
design-oriented
turbulence
viscosity
the
and diffusion
many
viscosity eddy
where
the
shear
algebraic behavior
stresses,
eddy of the
if streamwise
to:
OU
=
(5.5) OU
= Rearranging
these
equations
to solve
(5.6)
for the eddy
viscosity
yields:
(5.7)
= v,,, = _ l(OV)Oz where
the
eddy
y and
z subscripts
viscosity.
equations these
(5.7)
plots
were
The
are
and
for
in Figs.
These
Station
5.34 solid
computed
only
5.35.
and
symmetry that the and
strain
the
eddy
regions in the
shown about results
results,
rates-of-strain
5 is shown along
plotted 5.33.
To avoid
value
in terms
5.32
and
cross 5.35
for
where
and
plane. show
5.32.
in
Also
in
5.34),
(vt/v), 6 are
eddy
and
Although similar
viscosities
strain-rates
the
(5.7)
indicated
ratio
Station
the
the
trends
viscosity
eddy with
is
is greater viscosity respect
to
the midplane z=O, the magnitude of the differences is enough so should be considered only qualitative, In areas where the stress
in both viscosity
transverse
directions
components
where large deviations component strain rates
component
plotting.
are shown vanish
(5.8)
are
are
observed
nominally
of the
to be nearly
same
equal.
magnitude, Most
in their
embedded
of the
occur can be traced to either a large difference or to inadequate resolution of a large velocity
gradient, e.g., in the vicinity of the velocity ridge near the vortex core. This is in contrast to the results of Mehta and Bradshaw, who reported vt,y
of the
(denominators eddy
of a viscosity
in equations
in the
(anisotropy)
component
results
points
in Figs.
5.33
the
Equivalent
denominator
in Figs.
in Fig.
which
singular
lines
the
5% of its maximum
distributions
dependence
primary
at Station traverses
dashed
where
directional
of the
5 in Fig.
and
(between than
(5.8))
horizontal
calculated.
shown
admit
behavior
(5.8)
vortices
was
so ill-behaved
this flow. that the
so as to preclude
61 It was mentioned strain
often
terms
which
this
exhibit
more
account
is that
which terms
in Chapter
the
2 that
spectacular
behavior
for the extra
extra
boundary
strain
rates-of-strain
layer flows
than
rates.
cause
what
with extra
is predicted
It is argued
large
changes
that
rates-of-
by explicit
the
reason
in hlgher-order
for
terms
appear in the Reynolds stress transport equations. These higher-order are modelled in terms of the Reynolds stresses and require the specification
of empirical constants. Dimensionless turbulence structure parameters the cross-correlation coefficient are related to constants in turbulence
such as models.
Often
flows
such
when
applied
the constants
two-dlmensional to more
are determined
boundary
complex
flowfields.
computed
for the present
Ruw
= _"w/u_w
' and
R,,_ the
5.36,
and duct
5.37
R,w
sidewalls due
5, in the
region
value
and
is close
R,,w
to the
parameter
secondary in the
range
is the
shear-energy
of the
resultant
turbulence
of the
peak
is observed.
ratio
been
et al. [56]).
energy.
observed
For the
with
al.
but
to be constant flow, aly
with
R,_
a value
decreased
lower
walls, value.
shows
The
is also depressed of interest as the
surface
to twice
layer,
this
(see,
e.g.,
of 0.15
the
values
is defined
boundary
is
Station constant
has
parameter
to a wall
two shear-energy
At
a near
value
this parameter
Near
parameter
stations
parameter
normal
correlations
two-dimensional at both
',
shown
of 0.45.
layer.
and
dimensionless
In a two-dimensional
present
upper
the
This
in a plane
the
6, the
the
= _"_/v'w'
Another
parameter
stress
occur,
Station
of the flowfield, core.
stress
boundary
Near
is in agreement
vortex
cores,
of the
= _'_/ulv
6 are
value
intensities By
value. Rvw
shear
as
parameters
R_v
5 and
vortex
turbulence
correlation
shear
kinetic
also
the
of 0.25 for most
region
of the
0.55
at Stations
two-dimensional
convergence
two-dimensional
stress
in the
vicinity
structure
correlations
The primary the
of simple
performance
stress
evaluated
with
lateral
generally
shear
dimensionless
The shear
respectively.
in the
where
of these
= _-'_/v'w'
5.38,
to strong
in poor
flow.
be compared
of approximately
and
has
and
may
depressed
Rvw
on the results
resulting
Some
were
in Figs.
based
layers,
parameters
were
ratio the
parameter Bradshaw
computed:
(5.9)
= x/_'fi 2 + _"_2 /2k
(5.10) which
are
walls,
respectively.
in Fig.
applicable
5.39
and
away The
the
from
the
corner
al v shear-ratio
al= parameter
region
parameter is shown
on the
vertical
at Stations
in Fig.
5.40.
high turbulence levels are observed, the al_ parameter dimensional value and, like the shear stress correlation,
and 5 and
In the
horizontal 6 is shown
regions
where
is higher than the twois well below the initial
value in the region of the vortex pair. Along the horizontal walls, the alz parameter is elevated at Station 5, but returns to the initial value at Station 6. These results structure,
show
that
more
the
transition
so at Station
duct 5 than
produces at Station
a distortion 6. The
largest
of the
turbulence
distortion
occurs
62 in the
vicinity
of the
vortex
core
where lateral
region
convergence
suppresses
turbulence. Further ining
the
insight
data
into the
in terms
structure
of the
of the
invariants
turbulence
of the
can be gained
anisotropic
stress
by exam-
tensor.
This
analysis is based on the concept of physical realizability limits of turbulence which has been used extensively to study the return to isotropy of homogeneous turbulence
(see,
by Rotta
e.g.,
Lumley
[58], is defined
[57]).
The
tensor
where
must
I,II
satisfy
and
III
stress
tensor,
first
proposed
as:
u, i-
bij = This
anisotropic
the
are
}k6,
2k
Cayley-Hamilton
theorem:
_3 _ I_2
+ II),
tensor
invariants:
the
(5.11)
-
III
(5.12)
= 0
I = bii 1 = --bi, 2
II
III The first invariant,
I, is identically
ergy.
Invariant
defines
stress
tensor.
A positive
component ponents
III
that are
applying the as the strain to infinity,
is large,
large.
and
Limits
J
zero by definition indicates
a negative the
of the turbulence
of the ellipsoid
of III
on
b,i
= _bi._b._kbk_
the shape value
d
associated
that
value
indicates
anisotropic
stress
--_ u goes
to zero
and
bll
goes
in any one direction If all the turbulence
to -1/3.
5.41.
recast
The
shaded
area
realizable turbulence is also labeled. The
in terms
bulence
of the
is primarily
III
invariant
Fig.
5.42.a.
is only
one
principle
that
two
principle
can
be
anisotropic contained
will always Following
on this plot
stress
be positive.
Pauley
tensor axial The
and Eaton,
level
of turbulence
invariants location
region
is bl 1
turbulence are are illustrated within
which
all
of the turbulence at the at Stations 1, 5 and 6 was
invariants.
component the
the
nature tensor
by
must go to zero as OU/Oz goes
on allowable These limits
represents
must lie. The Reynolds stress in the
largest
com-
defined
is 2k, which occurs when them turbulence is in the z direction, then u 2 = 2k and
is equal to 2/3. Following Lumley [571, the limits recast in terms of the tensor invariants II and III.
physically boundaries
Reynolds
tensor
The
en-
the
condition that the Reynolds stress in any direction rate in that direction goes to infinity. For example,
that can occur one-dimensional.
in Fig.
there
kinetic
with
At
Station
I, the
so it is expected at Station of each
1 are data
tur-
that
the
plotted
in
point
on the
63 invariant which
map
relative
has its origin
invariant
map.
location
This
in the
to positive
vector
layer
dimensional though, turbulence.
which
That
one component
high
of anisotropy.
level
tions
5 and
observed the
the
6 are
rates
layer
cluster
more
limit
indicating
5.7' Wall The
use
Since
flow
These
along 5.46.
two
suggests layer
which
tubes were
both
tion
velocities
used
pitot
terval
however,
The
and
are
plots
rates
are
strain
contraction dominant.
are
30 and
toward This
friction
on the
plots.
< 0.6, inagreement
downstream. obtained in Fig.
average
Bound-
At
Station
greater
than
implies
The
is satisfied
5 fric-
fines
individual
5, deviations 80.
in
at Stations 5.47.
of the
behavior that
shown
distribution
diameters
law-of-the-wall
in Fig. 5 show
Although
(n=1,2,5,6,7,8)
shown
outside
behavior
for s/s,.el good
5 and
measured
are shown at Station
relatively
Results
based
data
at 5.45.
velocity,
at Stations tube
behavior
are
analyzed in Fig.
at this station.
6, the law-of-the-wall 200.
was
local
Preston
y,, traverses
for y+ values
tend
flow
are shown the
exist
probes.
on these
apparent
At Station
with
6, the
coordinates,
non-dimensional
all profiles
between
the
hot-wire
superimposed
30 < y+ 4.125
Co
1.5463591
0.51544988
10.000000
Ca
0:0000000
0.00000000
0.0000000
Cz
0.0000000
0.00000000
0.0000000
C3 C4
0.0000000
0.00000000
0.0000000
0.0000000
0.00000000
0.0000000
C5
0.0000000
0.00000000
0.0000000
R = 10.214
cm
137 Table
A._.
Transition
Duct
Geometry
Variables
Data Station
x/R
air
b/R
1
-1.00
1.0000
1.0000
2.0000
1.0000
1.0000
2
1.00
1.0000
1.0000
2.0000
1.0000
1.0000
1.12
1.0003
0.9997
2.1047
1.0006
1,0192
1.24
1.0025
0.9978
2.2154
1.0047
1.0376
1.36
1.0078
0.9831
2.3331
1.0251
1.0555
1.48
1.0174
0.9746
2.4588
1.0439
1.0729
1.60
1.0316
0.9719
2.5934
1.0614
1.0897
1.72
1.0509
0.9548
2.7383
1.1006
1.1056
1.84
1.0752
0.9333
2.8947
1.1520
1.1198
3
4
5 6
AR
Ac,/Ac,.i
1.96
1.1041
0.9177
3.0645
1.2031
1.1319
2.08
1.1371
0.8784
3.2497
1.2945
1.1410
2.20
1.1734
0.8462
3.4528
1.3867
1.1464
2.32
1.2123
0.8117
3.6770
1.4935
1.1475
2.44
1.2527
0.7759
3.9261
1.6145
1.1441
2.56
1.2936
0.7396
4.2050
1.7491
1.1362
2.68
1.3341
0.7037
4.5199
1.8958
1.1241
2.80
1.3729
0.6693
4.879I
2.0512
1.1087
2.92
1.4093
0.6370
5.2932
2.2124
1.0911
3.04
1.4423
0.6078
5.7775
2.3730
1.0725
3.16
1.4712
0.5821
6.3524
2.5274
1.0544
3.28
1.4954
0.5606
7.0484
2.6675
1.0381
3.40
1.5147
0.5435
7.9106
2.7869
1.0248
3.52 3.64
1.5290 1.5385
0.5308 0.5224
9.0131 10.000
2.8806 2.9451
1.0152 1.0083
3.76
1.5439
0.5176
10.000
2.9828
1.0026
3.88
1.5460
0.5157
10.000
2.9979
1.0003
4.00
1.5464
0.5154
10.000
3.0000
1.0000
1.5464
0.5154
10.000
3.0000
1.0000
8.00 R -
_7
10.214
cm
APPENDIX HOT-WIRE An
all
new
present
integrated
study.
department output, the
signal.
ACQUISITION
data
from are
appendix
important
components
of the
was
ones
used
directly
performed
contains
of the more
system
previous
operating
operations
This
acquisition
differs than
SYSTEM
on the
acquired within
analog
on a digital
a description
operating
for the
the M.E.
anemometer
representation
of the
system
of
and
an
features.
Hardware
(TSI)
primary
IFA
converter
100 Intelligent and
Workstation B.1. The two
IFA
signal.
The
range
of the
a voltage
usable
span
The more
IFA
less than
signal
must
The
an RS-232 this
link
active
are
the
channels
generates
transducer must
number
controls
starting
a 16-bit
which
the
data voltage.
be stripped
to the
FIFO
first-in, output
to a DRQ3B
P_I_/_'_
(channel rate.
and DMA
high
(FIFO) are
sent
controller
INTENTJONALLY_!1
A/D
as possible,
The
a DMA
through through
in the
the
first),
the
the
A/D
binary
number
channel
converting
word
of
converter
representation
digitizer
data
(Direct
to
characters
before
data
The
of up
are controlled
the
word
or
ASCII
sample,
contain
rates
of one changing
contain
16-bit
conversions on rapidly
that
each
data
memory. card
voltage will have
data
to be sampled For
4-bits
16-bit
via
input
range
conversions
parameters
low 12-bits
voltage.
first-out
A/D
of collecting
The
address
the
the
the
output)
film sensors
of this
IFA 200 by sending
sampling The
(when
amplified.
Selectable
The
a DC offset and
and
a low-pass
(anemometer
and
as much
of a Model
contain
display wire
in Fig.
anemometers
input
simultaneous
link.
from
Most
and
consisting
of applying to the
158 digital
capacity
the
word.
to a hexadecimal
word-deep
the
unit
conditioners
To utilize
signals.
communication and
of the
this.
provide
gain
volts.
high-speed
output
computer
serial
Model
be offset
200 provides
signal
(A/D) Scientific
is illustrated
temperature
capable
and
Incorporated
VAXLab/VSII
of this equipment
The
is +5
much
Hz per channel
signals.
of the
digitizer
(DEC)
150 constant
mode)
Systems
IFA 200 analog-to-digital
is a two-channel
an amplifier
DC
range
anemometer
50,000
and
IFA200
transducer
study
two Model
is in the
a TSI
Corporation
conditioners.
filter
filter
are a Thermal
schematic
in this
housing
a high-pass
Analyzer,
A block
157 signal
high-pass
system
Equipment
100 used
cabinet
Model
filter,
Flow
a Digital computer.
158 slave
cable
system rather
reduction
of some
The
the
This
data
analog
B.1
hot-wire
in that,
overview
DATA
B
address the
binary
is loaded
into
a 64-
enter
FIFO
pass
that Memory
the
Access)
interface
computer.
PRECEDZNG
PAGE
BLANK
NOT
FILMED
140 The that
DRQ3B
allows
of up
Parallel
real-time
to 1.3 megaHz.
buffering,
that
ventional
single
can
the
time
The
I/O
DRQ3B
between
has
increase
DMA
but
module
of 16-bit
significantly
buffer
fills, it is dequeued, ing
DMA
collection
transfers.
requires
the
To improve
enqueued
before
automatically
any
result
B.2
Buffer-Swinging
in the
method
of data
of data
and
enqueued
to the
enqueued,
data
DRQ3B
is filling, the data the second buffer process again B.3
repeats
prior
begins
to any to the filling
second
when
can
the
buffer's
are
sent
the
next the will
the
to be
DRQ3B
intervention.
transfer
exceed
rates.
the
available
are forwarded
directly
of the
buffer-swinging
a continuous memory
the
last
the
and
are
buffer
buffer
fills,
second
to the hard transferred
to disk
stream buffers
the first
While
written directly data have been
Dur-
buffers
fills,
When
buffer.
buffer.
or more
or more
When
a buffer
must not exceed IFA200, or data
higher
transfer.
second
next
software
data
to con-
when
enqueueing
to capture
buffer.
data
as compared
quickly
Two
data
first
double
a buffer
by means
is used
to as
the
any
enabling
manner.
referred DMA,
two
the digitized
feature
rates
and
without
computer
in the first buffer are fills and first buffer's as the
receives
This
begins
automatically
this,
in the
transfer
buffers in the
allows
interface
at
rates
to enqueue
rates
memory
data
buffer
buffer
buffers,
following
DRQ3B transfer
call
Now,
sampling
disk
in the
single
buffer
between
acquisition.
operates
transfer
DRQ3B
next
To avoid
hard
data With
transferred.
at high
computer.
feature,
time between FIFO memory
the
the
time
data
71 megabyte
are
filling
is less down
Transferring to the
situation,
data
begins
The
memory
this
a unique
previous
buffer, no data are transferred. This time it takes to fill the 64-word-deep be lost.
digital
a software
dequeueing
is a high-speed
parallel
is the
buffer
disk. When to disk, the
the
first
buffer
data.
Handshaking Synchronization
of data
transfer
between
the
IFA
200 and
troller card is accomplished by an interlocked two-wire erly implemented, the handshake prevents data overrun from input
occurring. port
REQUEST t At the port plish
DMA this,
FUNCT
and
BUSY
time
the
transfers the
The
pin-to-pin
200
DMA
listed
The
are connected,
DMA
pins,
B.1.
0, STROBE
DMA
Interface
respectively,
on the
data
acquisition
from
the
IFA 200 to the
cable
had
to be modified
connections
Interface
in Table
OUT
via the
Cable
between and
the
equipment
the
ACK
pins
to the
IFA 200 output
DRQ3B
on
controller
card.
J2 output on the
with connectors
DRQ3B
the
DRQ3B CYCLE
port. TSI
con-
When propconditions
READY,
acquired,
for compatibility port
DRQ3B
handshake. and underrun
Cable,
was
J1 and
J2 input
and
the
j For data did
not
sup-
To accomthe
DRQ3B.
on the DMA
card
IFA are
141 transfer high.
to begin, When
the
strobe,
begins
QUEST
line
the
last
the
ACK
last
buffer
computer
low,
indicating low,
that
Once
the
digitizer
following
1. When
the
of the
has
IFA
200,
releases
to receive
been
activated
data
and
When sets
the the
data
CYCLE
are
3. When
IFA 200 receives not
the
of the
available,
DRQ3B
IFA
DRQ3B
REQUEST
200,
FIFO
places
the
it releases
it places starting B.4
IFA valid
the
high,
prepares the
the
DMA
the
the
value
the ACK value. the
ACK
line
case
is held
200 receives
BUSY
high
from
ACK
lines
and
that word.
the
asserts
the
is read
out
DRQ3B
line
CYCLE
line
(unless
a word
the
it
valid
REQUEST
it to go high low until that
data
data
CYCLE
to
DRQ3B,
DRQ3B
next
the
line
indicating
line of the
indicates
the
low,
to the
allowing
ACK
en-
in its 512-word-deep
ACK the
from
been
REQUEST
signal
indicating high
have
CYCLE
high
on
the
it to go high,
buffers
ACK
data
asserts
When
transfer:
memory).
cycle
REUntil
DRQ3B data.
line allowing
to transmit
STROBE
is full, in which
of FIFO and into for the next word. 4. When
and
receives
the
low from
line
CYCLE
available.
to receive
low from
and
BUSY
on the sampling the
are
computer,
for data
FIFO memory. The DRQ3B then asserts the IFA 200 that it has received the data 2.
data
ACK
occur
STROBE
the
asserts
DRQ3B
data.
of events
it reads
it turns
and
ready
the
0 pin on the
line, valid
by the
it is not
DRQ3B
receives
that
DRQ3B that
sequence
DRQ3B
OUT
memory
DRQ3B
on the
it is now ready
the the
FIFO
to the
indicating
is enqueued,
the FUNCT
this on the READY
its 64-word-deep
is enqueued
signal
indicating
asserts
IFA 200 senses
filling
buffer
queued,
the
of the
is ready DRQ3B,
REQUEST
low,
over.
Software Control
subroutine of the ing,
of the library.
DRQ3B This
experimental
plotting
main
study,
and
programs
accomplished
analysis.
written
in the
by sending
by means
B.5
Digital
rely
The hot-wire on an accurate
output variable
of the
The
through provides
real-time
data
subroutines
FORTRAN
ASCII
characters,
FORTRAN
WRITE
the
useful are
of DEC's
called
from
Control
an RS-232
serial
VAXlab
for all aspects
acquisition,
language. via
use
programs
signal
process-
user
supplied
of the
IFA200
is
communication
statement.
Sampling
signal and
package
including,
data
link,
velocity,
is accomplished
software
techniques evaluation
from
the
therefore
is made
here.
employed in the of the mean and
hot-wire the
If the
present variance
anemometer.
anemometer instantaneous
output voltage
The
study (see Appendix C) (E and e 2) of the analog usual
voltage,
assumption is an
is decomposed
ergodic
that
the
random
into a mean
and
142 D
fluctuating
component,
E = E @ e, the true
by a finite
average
of samples
N are taken:
of discrete
= rnmoo .,o
At
E. are
is the
T, the
the
time
number
are
required
the
waveform
increment
by
hot-wire
convergence
data
point
time
were
taken more
very
which
yields
and than
at
mean
it was adequate
For Since
high
and
observed
be estimated large
number
time
of 150,000 the
for all regions
above
total
not rates.
For set
samples was
transfer transition
time
time
statistical
and
duration quantities
necessary
were
(B.2)
sampling
sampling
single
values of the
(B.1)
= (E.(nAt)--_)2
a fixed only
sampling
mean-square that
-_
it was
duration
a total
and
can
a sufficiently
T is the
measurements, rate
of the
_
samples.
sampling
respectively,
The
that
samples,
is N = T/At.
digitizing
the
--_)2dt
between
average
= E.(.AO
digitized
of samples
measurements, seconds,
individual
for the
provided
E(t)dt
f,:°+r(E(0
--e_=limo¢_l where
samples,
temporal
to "capture" the
turbulence
at 10 kHz for each
data
monitored rate duct
and
15
point. for each
and
sampling
flowfield.
143 Table
B.I.
DMA
IFA Signal Name
Interface
Cable
200
Pin
Connections
DRQ3B
Port No.
Pin No.
Pin No.
Port No.
Signal Name
J1
02
17
J2
STROBE
READY GND
J1 J2
06 01
22 42
_
FUNCT OUT GND
BUSY GND
_
02 03
18 41
ACK GND
GND
04
40
GND
GND
07
39
GND
GND
11
38
GND
GND
12
37
GND
GND
13
36
GND
GND
15
35
GND
GND
17
34
GND
GND GND
19 2O
32 31
GND GND
GND
21
29
GND
GND
22
28
GND
GND
23
27
GND
GND
24
26
GND
CYCLE
REQUEST
DATA
OUT
00
39
01
DATA
IN 00
DATA
OUT
01
37
02
DATA
IN 01
DATA
OUT
02
35
03
DATA
IN 02
DATA
OUT
03
33
04
DATA
IN 03
DATA
OUT
04
31
05
DATA
IN 04
DATA
OUT
05
29
06
DATA
IN 05
DATA
OUT
06
27
07
DATA
IN 06
DATA
OUT
07
25
08
DATA
IN 07
DATA
OUT
08
26
09
DATA
IN 08
DATA
OUT
09
28
10
DATA
IN 09
DATA
OUT
10
30
11
DATA
IN 10
DATA
OUT
11
32
12
DATA
IN 11
DATA
OUT
12
34
13
DATA
IN 12
DATA
OUT
13
36
14
DATA
IN 13
DATA
OUT
14
38
15
DATA
IN 14
DATA
OUT
15
40
16
DATA
IN 15
Note:
Unlisted
32 pins
are not
connected.
32
0
144
%PA
100
Anemometer
Model Digital Display
158
AL Model 150 Constant Temperature Anemometer
Model 157 Signal Conditioner
Ch.l Ch.2
J
Hot-Wlre(a)
O U
Oh. i
e4
Ch. 2
+
N !
_
IFA
200
A/D
Converter DMA
Interfa¢
Cable
VAXLab/VSII VR260
Ip-
--
DRQ3B DMA Module
I/0
Display
w
m_
+
r
Processor
Hard Disk Drive
Fig.
CPU
BUS
i_.._
Main Memory
iii+_;+i.;+iiii Keyboard
B.I.
Schematic
e2
of hot-wire
data
acquisition
system.
APPENDIX HOT-WIRE C.1
TECHNIQUES
Introduction The
derivation
the Method mean
and
of these
of the mean
A technique
The
response
in this
equations,
for the
in situations
when
This,
wire technique flow
and
amounts used.
The
individual
use
required
are
being
has
the
velocity data
and
error minimization here
slant-wire
probes
in order
of the
hot-wires
velocity
relation
Before Method
continuing,
A that
present
skew
angles,
interference
effects
from
associated
cannot ever,
be the
with
angular by suitable
probe
configurations, with
Me_hod
Method
with
the the
probe
sweep
of this
design along A are
this
with
illustrated
A will be discussed
body
as the
can
be rotated
interference
effects
can be minimized probe
an estimate in Fig. response
itself.
of their
C.1.
The
equations
and
is required.
regions impact are
with
B, for zerolittle
region
or no of rota-
probe
supports
be expected; the
Two
The
associated
the
can
and
by an effective
Method
from
hot-wire
variables.
360 ° with
data
least-
normal
is a significant
no valid
hot-wire
using
procedure
with
be
to the
The
flow
a restriction
there
region,
region
of the
note
can
compensated
and
is governed
calibration
B. Whereas
supports,
,4 where
Within
to
is usually
to the
yaw large
of sensitivity
flow variables.
yaw flow angles
and
is necessary.
use of custom-made
a simple
Method
probe
Method
neglected.
shifted
and only
mean
when probes
positions
sensitivity
it is important
is not
for the
pitch
available
This
of
a hot-
the
prohibitive
range
field
axis
is not based
recalibration
the
sequential
to maximize to pitch
frequent
of rotational
to solve
on the
so that
to-moderate tion
relies
to deduce
technique
on extensive
commercially
components.
number
criteria
proposed
cooling
stress
probe
[61] proposed
is used
becomes
here
stress
to the
Their
limits
is proposed
Reynolds
Kool
relies
and that
probe
at a large
technique
rather
accumulated slant-wire
and
slant-wire
but
and
B)
development
[46].
which
normal
for of the
(Method
of the
[45] and
flow
conditions.
for calibration
advantage
a single
mean
rotatable
form
technique
is nominally
similar
equations
by A1-Beirutty
De Grande
law relation,
time
mean
response
flow
field under
cooling
for by obtaining squares
of the
primary
response The working
For details
to references
hot-wire
a single
of data technique
rotatable
in which
The
Their
is referred
is not unique.
on an empirical
developed
derivation.
in itself,
turbulence
calibration.
equations
measurement the
hot-wire
in this appendix.
without
reader
A single
is applicable rotation.
chapter
the
Method
and turbulence
are presented
turbulence
are presented
use
C
angular
possible
howposition
slant-wire
of interference of this
developed.
restriction
for on
146
C.2
Generalized
Empirical
The
starting
for the
equations
point
for both
Method
Cooling derivation
A and
is given
Relation mean
and
turbulence
B is the empirical
cooling
response
law relation.
on whether a linearized or non-linearized For a non-linearized system, the relation
by: (ENL)
where
of the
Method
The form of this equation is dependent hot-wire anemometer system is used.
Law
E is the instantaneous
2
E20 -- BNLUe
bridge
voltage
(C.l.a)
n
and [7, is the instantaneous
velocity; E0_ and B are the intercept and slope of the wire the calibration stream is normal to the wire. Conversely, output
is linearized,
the cooling
relation EL
is given
calibration when the
cooling
curve when anemometer
by:
= BLUe
(C.l.b)
These equations can be presented in a general form applicable to both linear]zed and
non-linearized
signals
as: (E 2 - A) 2/''
where,
for a linearized
Eo) 2,
following
oped
for the
extracted C.3
The
mean
Method
velocity
A, this
to Positions
to Positions there is the from
one
discuss transverse
mean case,
vector into
and
can
how
the
probe
flow angle
and
from
-
BNL,
turbulence which
the
can be evaluated
four
angular
rn -- n
response
equations
linearized
system
2, and
supports situation in the
x-z
by
in magnitude
positions,
such
mean
bridge
corresponding
be accomplished
1 and
this
B
are
devel-
equations
are
Development
3 and 4. Without risk that the wire
of the
later
the
Equation
C.2,
m = 2
case.
is rotated
in Fig.
B = B2L,
A = E_,
non-linearized
Flow
a hot-wire shown
analysis,
as a special
Mean
A = O,
system: E2NL,
E 2 =
axis
-
for a non-linearized
In the
(C.l.c)
system:
E 2 = (EL and
= B21mUe2
by
rotating
rotating
will occur,
wire
rendering
7, defined
direction
the
if beforehand, as shown
when
1,2,3
and
is recorded. wire
about
mean velocity into a position
can be avoided, plane,
voltage
an inclined
a normal
knowing the will be rotated
and
as Positions
about the
4 For
the
same
yaxis
direction, however, where interference
data
invalid. an estimate
in Fig.
We
will
of the
C.2, is known.
147 The reader
should
a restriction For
inherent
Method
by rotating to-moderate present
this
in mind in the
B, the
with
that
this is strictly
response
four
wire
a single inclined skewness levels,
itself
direction direction
for
keep
equations positions
wire about the problem
Method
B,
and,
section,
mean
flow response
will be presented. Except both of these methods. When position
the shown
Combining
where
therefore,
no
that U.
time-averaged in Fig.
equations
C.2,
form the
not
in Fig.
C.2
can
be achieved
the
flow
following
of equation four
of the
in the
A and
discussion
(C.l.a)
flow x-
B is in the expressions In the remainder of
Method
Method
B
is applicable
is applied
response
mean
is nominally
Method (C.l.a).
for both
the
following
knowledge
equations
at each
to
wire
result:
(.E,_- ._)_/,,, = BW"(U2), (E_2_ ._)2/,,, = B_/,,,(U2)_
(C.2.a)
(E_ _ ._)2/,,,= B2/,,,(U_)_ (E, 2_ _02)2/m= B2/",(U.2),
(c.2.c)
(C.2.a)
(,_12
equations
noted,
and
the x-axis. Unlike Method A, for zeroof probe support interference does not
difference between Method A and cooling velocity term in equation
the
limitation
themselves. shown
is required apriori, other than and that V and W are less than
The primary the effective
a physical
&: (C.2.b)
-- F__2o)2/m/(E22
and
equations
-- Eo2)2/m
=
(C.2.b)
(C.2.d)
(C.2.c)
& (C.2.d)
yields: (C.3.a)
-(Ue2)l/(Ue2)2
(z__- _,_, )_/" /( E,_- ._)_/" = (uJ)_/(uJ),, Rearranging
where,
equations
for convenience,
By eliminating (C.4.a)
(C.3.a)
and
(C.4.b),
the
and
(C.3.b)
gives:
(uJ)_-s,_(u,2)2= 0
(C.4.a)
(U._)3 - _s4(U,2)4 = 0
(C.4.b)
following
definitions
are employed:
s,2 = (E,2- _0_)_/m/(E2 _- _o_)_/_
(C.5.a)
_34= (E3_ - _,_,)_/"/(E,2- _,)_/"_
(c.s.b)
the dependence we have
placed
on the wire the
calibration
restriction
that
slope the
same
(B)
in equations
hot-wire
must
148 be
used
in Positions
require, same
however, as the
intercept
one used
values
In order ponents, mean
the
used
equations cooling
components
in terms
the
(C.5.a)
(C.4.a)
(U, V, W),
and and
(C.4.b)
the
E34,
expressed
do
not
2 be
the
calibration
terms
velocity
(Ei_2, Es4)
of the must
be
(e 2) voltages.
(C.4.a)
and
by substituting
(C.4.b)
E = E + e into
in a binomial
com-
in terms
mean-square
in equations
expression
We
appropriate
for the mean be
voltage
and
4. 1 and
(C.5.b):
must
(E)
voltages
resulting
3 and
in Positions
and
(U,)
mean
E12 and
in Positions
is used
4, so long as the
velocity
of measurable
E02)2/', expanding at second order:
that
3 and
of measurable terms,
likewise,
in equations
to solve
voltage
expressed
hot-wire
in Positions
are
in terms
•The
2, and,
the
effective
velocity
expressed
1 and that
series,
e/_
and
can
be
(E 2 -
truncating
Ce2 + _2__o2
(E'-F_,g)2/'=(.E2-F_,g)2/'[l+(41m)__,2__g
]
(C.6.a)
where: C=2/m+(4/m)( Time-averaging
this
expression
-
(C.6.b)
E_ _ _02 )
1)(_2
yields: Ce 2
(E=- gg)_/-_= ($_ - &_)_/=[1+ _ _ Eo _] Substituting
equation
(C.6.c)
into
equations
(C.5.a)
and
(C.6.c)
(C.5.b)
gives:
r_12 = r_i/_2
(C.7.a)
(c.7.b) where: el
el
2
:_, = (&__ &_)_/,-[_+ _ - _1 C2
c2 2
:_ = (_ - ,&_)_/,-[_ + E_- _1
(C.7.c)
v
I
03e32
_:_= (_ - &_)_/,,,[_+ ._ - E_] 04e42
S4 = ('E'_ -
The (C.4.a)
exact and
expression
(C.4.b)
-Eo2)2/"[ 1+
for the effective
can be expressed
cooling
in terms
_42 _ _021
velocity of the
mean
term,
Ue 2, in equations
velocity
components
149 (U, V, shown
W)
and
in Fig.
Method
the
A and
[Method
Reynolds
C.2.
stresses
as follows
Method
(u-7_,
(recalling
i,j
that
= 1,2,3)
different
at the four expressions
wire
positions
are required
for
B):
A I (see
Appendix
D for derivation) w
__
V2
(U,2), =Ko,iU2{K,,,[I
+
_= [(v)' + -O-g] _-{]+
(C.8.a)
+ K_,,sN[(-ff) + _1 +
v2 (-U-j2)2 =KoaU2{KI,2[1
+ _-g] + [(_)
+ gzah_g[(W)
. ('Ue2)3
2 + U2
U)+
21
=Ko,sU
+ _--_]
_-_
{_(KI,a
+ h2ngKs,s)[1
+ -_1
_ + 1_(K,,a + h2BNKa,a)[("_ + [( V )2 + _-'_l -
(K,,3
_-_1}
2 W - hBNK3,z)[("-_-)
W)2
w2 + -U-'_]
(C.S.c)
+ _-'_1}
m
1/2
(U¢_),
=Ko,4U2{_(K1,4
+ h2BNK3,4)[1
+ -_]
"_ + 1_( K .,, + h2BNKa,4)t(W) + [(v)= + _1 + (K,,4 "
]Method
B[ (see
(Ue2)I
.
=
-- hBNI{.3,I)[(-'O-
Appendix
=I'(o,IU2{[1
W
) Jr
A, reference
_
_- _--_]-_ KI,I[(
+K3,,h2BN[(--_
IV)2
2 + _1 w2
(C.S.d)
_-_
_---_']}
[47] for derivation)
U)2 + _-_] v_
w2 +_--_1+2K2,,[(
V
(C.9.a) _'_ )+_-_]}
150
132
t,2 + K1,2[(V)2+ -U-_] (U+2)2=go,2U2{[1+ -_1
(c.9.b)
2 w 2 + _-_]_ + K3,2hBN[(_-) 2K2,2[(u) + _]} v2 m
m
u2 + (U,2)3 =g0,3U 2{[1 + _-_]
V 2 + _._ 132]
K3,3h_N[(-_)
+ K1,_[(w)2+ ,1,2] u2 + 2K2,3[(W) (U,2)4
u2 re" h 2 r,'V32 2 {[1 + _-'_] + ,,3,4 BNtt _':
=Ko,,tU
(C.9.c)
+ _-_] }
v2 + _"_] (C.9.d)
W)2
+ K1,4[(-_where
K0, K1, K2
tions the
of the tangential
cient.
The
equations
and//'3
wire
cooling functional
(C.8.a)
stituted
into
equations,
(the
inclination
second
angle
through
(C.8.d)
I(ov + Klv(V/U)
and
hnN
variables (C.4.b),
Method
wire
cooling is the
are
are
through
func-
(h_¢)
cooling
in Appendix
following
Method
position) coefficient
binormal
given
(C.9.a) the
A and
+ _']}
denotes
normal
or equations and
to both
_-)
the
(k);
for these
(C.4.a)
2K2,,[(W
subscript
(a),
coefficient forms
equations
applicable
_
+ _--_]-
and
coeffi-
D. When
(C.9.d)
non-linear
are
sub-
algebraic
B, result:
+ K2v(V/U) 2 + K3,,(W/U) 2 + Cv = o
(C.10.a)
Ko_ + K,_(W/U) + K_(W/U) 2+ K_(V/U) _+ C_ = 0
(C.10.b)
where u2
_
v 2
w 2
(C.11.a)
c_ = Ko_(_--_) + g,,,(_-_) + K2,,(_-_) + K3,,(_--_) m
.
U2 .
m
.
U_
c= = I;0w(_--_) + x;,=(_-_) + g2=(_-_) and
where,
for convenience,
the
w
W 2
following
l) 2
(C.ll.b)
+ g3w(_-_)
definitions
have
been
employed:
_ethod AJ Kot, = Ko,l Kl,1
-
Ko,2 K1,2 E12
Klo =2K0,1K2,1 + 2K0,2K2,2E12 h'2v =K0,_ - K0,2E12 K3_ =hBN(Iio,lI;3,1
-- Ko,2K3,2E12)
(C.12.a)
151 and, I
r
p"
2
Kow=(_)[Ko,3(_,_ + hBNN3,3) 2
- Ko,_(KI,, + hBNK3,41:34] (C.12.b) + Ko,4(K1,4 K_w
--
hBNK3,4_34 2
]
=Ko,,,
I':3,,,=Koa - Ko,, _.34 [Method
B]
=Ko,a - Ko,2_12 Klv =2Ko,1K_,l + 2KoaK2,2E]2 K2. = Ko,1KI,a - Ko,_K1,2E]2 2 ," K3,, =hBN(I_o,IK3,1 -- Ko,2K3,2Ea2) KOt)
(C.13.a)
and, --/(0,3
_w _w
terms
which
Cv and
would
to mean
result
which
nents
(V/U)
(C.14.a) first-order
+ KI,,,(W/U)
(C. 14.a)
tions
can and
and
and (W/U).
accurate
are may
and
fluctuating
the
terms
without
introducing
represent
two coupled for the
be obtained
solutions directly
(v/U)o = -KodK_. (tWU)0 = -X;ow/K_.,
mean
velocity and
Cw
serious
Under
these
(C.14.a)
non-linear
for (V/U) as:
(W/U).
Cv and
2 = 0
relative
mean
relative
2 = 0
2 + K3w(V/U)
second-order
neglected,
and
errors
velocity.
correction
2 + K3,,(W/U)
simultaneously
for the
components
cooling
ratios (V/U) reduce to:
+ K2w(W/U)
account
velocity
effective
(< 10%),
+ K_,,(V/U)
_34)
(C.10.b)
can be neglected
velocity equations
When
]x_0,4K3,4
for the
field
(C.10.b)
(C.14.b)
be solved
(C.14.b)
the
expression
intensity and
Ko,, + K1,,(V/U)
Equations
(C.10.a)
neglecting
errors in the computed mean conditions, the mean response
Ko,,
-/(0,4K1,4_34
-- h 2BN( Ko,3 K3,3 -
in the
(C.10.a)
(C.13.a)
= Ko,3 K1,3
from
In a low turbulence
Y]34 q- 2Ko,4K2,4_34
Cw in equations
components
in equations
Ko,4
--21(0,3I(2,3
_w The
--
(C.14.b) algebraic velocity
terms (I¥/U)
equacompo-
in equations which
are
152 which
are appropriate
For the more non-linear
case,
algebraic
"_-i/U2,
equations
at least
which, become
(C.14.a)
and
unknowns of the
solved
for the
values
the
mean
(U)
at
of the
each
effective
can
until
ratios
positions
_, to
the coefficient difficulty, ratios
ma-
equations (V/U)
and
the
1 is obtained,
Reynolds stress tensor. The calculated mean velocity components by simulta-
the
The
Reynolds
improved
stress
and
(W/U)
by
averaging
its
calculated
in Fig.
C.2.
When
after
1 (equation equation
some
tensor.
are known,
shown
following
mean This
velocity iteration
is achieved.
(V/U)
at Position
(C.2.a),
at Position
then this
2, v2/V
are required normal-wire
component
(C.10.b).
convergence
velocity
velocity
and
to update
be determined
four
cooling
equation
(C.10.a) used
is repeated
velocity
into
equations
are subsequently
Once
arbitrarily,
two
initial estimates are made using equations (C.15.a) and (C.15.b). results are then used to solve the turbulence response equations
solving
proce.dure
ui/U
and/or
flows.
represent
W/U,
slant
mean
skewed
(C.10.b) equations
to overcome
(to be developed shortly) for the complete stresses can then be used to correct the neously
(V/U,
rotation In order
in slightly and
independent
If this is done
are first
(W/U)
(C.10.a)
five more
requires
ill-conditioned.
(C.14.b)
(W/U) after The calculated
in seven
positions.
and
equations
Therefore,
in turn,
five different
may
(V/U)
however,
_-6/U 2, _'C/U2).
for closure, trix
for evaluating
general
(C.8.a) for the
the axial
value the
or (C.9.a)) axial
with
expression
velocity
mean
the
wire
for
the
is substituted with
the
wire
rearrangement:
[- t odAI u, ={(g2 _ _,o_)2/m [1+ C,F,_I(_,[- g2,)]I(Bo)_/'}'/_ {K,,,B + _] + V + 2,r_., 1 [(_.)
[Method
v_._ _ ]
_ [(-_--)_ + g_,,h _N
+ w'_
(c.16)
_"-_11_1/2 + _.,
B1
u, ={(&_ - _7)_/-'[a+ c,7,_/(_,_,- t,2,)]/(B
