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The Effects of Blade-Row Spacing on the Flow
Capacity of a Transonic Rotor
Randall M. Chriss t
NASA Lewis Research Center
Cleveland, Ohio
William W. Copenhaver
Steven E. Gorrell
Air Force Research Laboratory
Wright-Patterson Air Force Base, Ohio
ABSTRACT
Results from the ongoing Air Force Stage Matching
Investigation are presented. In the present work the effect of
upstream blade row wakes on the flow capacity of a
downstream stage (where the rotor sets the choking flow) is
investigated. An embedded stage was simulated by placing a
set of wake generators (similar to inlet guide vanes) in front
of a highly loaded single stage transonic core compressor.
The wake generator-to-rotor axial spacing was varied in
addition to the vane count. A complete parametric test matrix
was completed in order to determine which parameters were
important to the choking flow capacity.
The results show that for axial spacings above 50% of the
upstream axial blade chord, simple wake mixing alone fully
accounts for the upstream flow losses and a simple mass flow
rate correction based on the rotor face mass averaged total
pressure is sufficient. At spacings closer than this, other
effects or loss mechanisms may be present. If these loss
sources do exist, they are of unknown origin and magnitude
and so an embedded overflow condition for these spacings
cannot be ruled out. Research is ongoing that will attempt to
identify and clarify the relevant details associated with these
close blade row spacings.
INTRODUCTION
The turbomachinery research community has long sought to
establish the potential impact of unsteady flow phenomena
on compressor (and turbine) performance and reliability.
T
Unsteady flow phenomena has long been ignored in
turbomachinery design and its potential impact remains
largely unknown. In the present work the effect of upstream
blade row wakes on the choking flow capacity of a transonic
downstream stage is investigated.
The origins of this work date back to the late 1970s and early
1980s to at least two multistage experimental programs. The
first was an Allison five-stage compressor reported on by
Bettner and Alverson (1977) and the second was a Pratt &
Whitney closely coupled three-stage compressor (both
discussed by Wennerstrom (1989)). In both of these
experimental programs the second and sometimes third
stages overflowed severely (6% in one case). No explanation
could be found for such a huge flow discrepancy between the
design intent and actual flow rates. It was postulated that
some unknown unsteady phenomena, not accounted for in
the highly developed design systems in use in industry at that
time, could account for the differences. In addition, some
evidence from an early numerical study showed the existence
of wake effects on flow capacity (Scott and Hankey (1986)).
Since that time Mulac and Adamczyk (1992) have shown
that poor blockage assumptions can account for apparent
overflow conditions in multistage compressors.
This series of unexplained flow mismatches led to the
development of the Air Force Stage Matching Investigation
program.
The present work attempts to address the long-standing issue
regarding the overflow of embedded stages. A previous paper
Currently visiting researcher: Air Force Research Laboratory, WPAFB, Ohio
Presented at the International Gas Turbine & Aeroengine Congress & Exhibition
Indianapolis, Indiana — June 7–June 10, 1999
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axial spacing can be set to three values denoted as "close,"
"mid," and "far" as shown in Fig. 1. The spacings normalized
by the wake generator chord are given in Table 1.
by the authors (Gorrell, et al., (1997)) discussed efficiency
performance changes due to vane spacing and count. This
paper discussed preliminary experimental results, which
showed performance degradation with decreased wake
generator/rotor spacing. In addition, the paper showed that
the performance degradation was associated with the wake
generators at the lower vane counts. At the higher vane count
a wake generator/stator interaction could not be ruled out.
However, problems with the instrumentation layout
precluded definitive conclusions and further work is ongoing
to verify these results.
Wake Generator
Rotor Stator
EXPERIMENTAL APPROACH
In the design of the experiment, it was concluded that the
two-dimensional characteristics of upstream blade wakes
were the major source of unsteady disturbance in
downstream compressor stages. Although three-dimensional
disturbances are certainly present, it was felt that the twodimensional characteristics, together with reduced frequency,
were the simplest and most controllable and should therefore
be explored first. This therefore led to an experiment
subjecting a single stage compressor to upstream wakes of
differing two-dimensional characteristics.
Figure 1. Stage Matching Investigation rig cross-section.
Table 1 Wake Generator Axial Spacing (normalized by
local wake generator chord).
Spacing
Close
Mid
Far
In the Stage Matching Investigation then, an embedded
second stage is simulated by placing a set of wake generators
(WGs), which are similar to inlet guide vanes, in front of a
heavily loaded single stage transonic core compressor.
x/c (mean)
0.12
0.26
0.56
x/c (hub)
0.10
0.26
0.60
x/c (tip)
0.14
0.26
0.52
Wake Generators
The axial spacing between the WGs and the rotor face can be
varied along with the WG vane count. Varying the axial
spacing varies the wake depth and width at the rotor face and
changing the WG count varies the reduced frequency in the
rotor passage. The three-blade-row compressor aerodynamic
performance was then mapped over a test matrix,
concentrating near the design corrected rotational speed.
Pratt & Whitney designed the wake generators with the intent
of producing wakes typical of modern, highly loaded low
aspect ratio front stage compressor stators. To simplify the
experiment, the wake generators were designed as
uncambered airfoils that do not turn the flow. Results from
measurements of stator wakes from rig tests were used as the
design target (see Creason and Baghdadi (1988)). These
wakes were produced in a stator that had no bow or sweep,
with a hub Mach number of 0.95 and a hub D-Factor of 0.55
and with a solidity of 1.64. For simplicity and in order to
isolate the effects of various wake parameters a twodimensional representation of the wake was desired.
In order to determine the two-blade-row performance (i.e.,
rotor/stator) from the three-blade-row experimental results,
the wake generators were calibrated in isolation to determine
their total pressure loss characteristics at each of the relevant
downstream locations corresponding to the location of the
rotor face. The two-blade-row performance was then
determined by "correcting" the three-blade-row performance
for the total pressure deficit at the rotor face based on the
calibration results. The two-blade-row flow capacity was
then compared for various spacings between the WGs and
rotor in order to assess changes in flow capacity.
The airfoils have a small leading edge radius with a relatively
blunt trailing edge radius as shown in Fig. 2. In the design
process, this produced the best combination of profile to base
drag in order to match the desired highly loaded stator wake.
Since the wake width and loss is a strong function of solidity,
the solidity of the wake generators was held constant from
hub to tip. This results in a tapered airfoil chord along the
radius as seen in the figure. The trailing edge is swept to
allow a constant non-dimensional mixing length from hub to
tip. With no diffusion in the flowpath the endwall losses are
STAGE MATCHING INVESTIGATION RIG
The experimental hardware was designed so that the wake
generator-to-rotor (where the wake generator models a stator)
2
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•
▪
expected to be low. There is no clearance and no fillet at
either the hub or tip.
1.9
1.85
D
1.7
w
ir:
.3
1.65
co
1,6
1.55
0 1.5
40 WG
"CLOSE .
SPACING
1.45
1.4
28
29
30
31
32
33
34
35
ACTUAL MASS FLOW, lbm/sec
(CORRECTED TO STANDARD DAY)
Figure 2. Wake generator meridional profile with cross
sections (flow from left to right).
Figure 3. Overall stage pressure ratio for the "clean"
inlet and 40 wake generator configurations.
The effects of periodic inlet disturbances in turbomachinery
are related to the reduced frequency of the disturbances. The
reduced frequency is defined as half the duration of a
disturbance in a blade passage divided by the time scale of
the disturbance itself.
A summary of the aerodynamic design parameters is given in
Table 2.
Table 2 SMI Aerodynamic Design Parameters
PARAMETER
Number of airfoils
Aspect Ratio — average
Inlet Hub/Tip Ratio
Flow/Annulus Area, lbm/sec/ft2
Flow/Frontal Area, lbm/sec/ft"
Flow Rate, lbm/sec
Tip Speed, Corrected (ft/sec)
Mm LE Hub
M rd LE Tip
PR, Rotor
PR, Stage
D factor, Hub
D factor, Tip
LE Tip Dia., in.
LE Hub Dia., in.
Reduced frequencies much less than one indicate that the
flow is quasi-steady. Reduced frequencies much greater than
one imply the unsteady effects dominate the flow. The wake
generator vane count can be set to 12, 24 or 40, and the
corresponding reduced frequencies are 1.87, 3.74 and 6.23
respectively and are typical values for in-service hardware.
Compressor Stage
The rotor and stator were designed at the Compressor Aero
Research Laboratory (CARL) by Law and Wennerstrom
(1989) and fabricated by Pratt & Whitney.
A compressor map showing the overall stage pressure ratio
for the "clean" inlet and 40 wake generator configurations is
shown in Fig. 3.
ROTOR
33
0.961
0.750
40.00
17.502
34.46
1120
0.963
1.191
1.88
-0.545
0.530
19.000
14.250
STATOR
49
0.892
0.816
----0.82
0.69
-1.84
0.502
0.491
19.000
15.502
CALIBRATION PROCEDURE AND FLOW RATE
CORRECTION
The hardware was designed so that the wake generators
could be calibrated by simply moving the wake generators
forward in the casing and placing a traversing rake assembly
forward of the rotor. Three survey rakes fit in the traverse
ring, each at a different axial spacing coinciding with each of
the rig wake-generator-to-rotor axial spacings. The traverse
ring is computer controlled and its circumferential position is
continuously variable.
Measurements of the casing and hub endwall boundary layers
were obtained separately using a rake built for the purpose.
3
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These measurements were then combined with the previous
survey results and so constitute the complete set of total
pressure measurements across the blade passage at all three
axial locations. The survey results were then used to
determine the loss coefficient for each of the relevant
spacings.
in the overall mass averaged loss coefficient was established
as ±0.01.
APNASA SIMULATION
The experiment was simulated using Adamczyk's APNASA
flow solver using the average-passage formulation of the
Navier-Stokes equations to simulate the steady-state
performance effects of multistage turbomachinery blade row
interactions. The code is described in some detail in
Adamczyk, et al. (1990) and has an updated k-c turbulence
model described in Shabbir, et al. (1996). All three blade
rows were modeled in the simulation. Three configurations
were run to obtain three complete speed lines in addition to
running the WG in isolation. The three configurations were:
1) the clean inlet configuration (no wake generators present);
2) the 40 wake generator case at the close spacing and 3) the
40 wake generator case at the far spacing. The mesh size was
the same for these configurations and had 228 nodes axially
and 41 in the spanwise and pitchwise directions. The isolated
wake generator simulation had 137 nodes axially with 45 in
the spanwise direction and 61 in the pitchwise direction.
The total pressure loss (mass averaged) across the wake
generator was determined using the loss coefficient at each of
the three distances downstream of the WG trailing edge
which correspond to the three WG-to-rotor face spacings
used in the investigation. This mass averaged total pressure
deficit was then used to correct the three-blade-row results to
determine the embedded two-blade-row corrected mass flow.
This calibration procedure assumes that the rotor does not
strongly impact the WG loss generation, i.e., that the loss
measured at a given location downstream of the WG trailing
edge during calibration is identical to the WG loss which
exists when the rotor face is at the same location. Therefore,
this procedure ignores any loss other than that due to mixing.
STEADY STATE PERFORMANCE
INSTRUMENTATION AND UNCERTAINTY
The choking flow rate obtained from the simulations was
about 2% higher than that found in the experiment. This
"bias" difference between the CARL facility and APNASA
has been noted for other compressors and although of
unknown origin, is consistent and so must be systematic.
This bias difference could be associated with a systematic
error in the amount of untwist predicted in the stress analysis
system used in determining the "cold" rotor blade
manufacturing coordinates or there could exist a small bias in
the experimental measurement system. Another source for a
discrepancy could be a small error in the flow solver postprocessor.
Overall experimental performance results presented in this
study were obtained from an array of stage exit total pressure
and total temperature probes. Ten pressure and ten
temperature rakes (alternating and equally spaced
circumferentially) with eight spanwise measurements on each
rake made up the exit instrumentation. The rakes are located
2.1 stator axial chords downstream of the stator trailing edge.
Inlet conditions are determined from an array of 30 inlet
thermocouples for temperature and the inlet total pressure
was assumed equal to the settling chamber static pressure.
The exit rakes are positioned so that a pitchwise distribution
in pressure and temperature can be established for a
circumferentially periodic flow field. Placement is such that
one stator pitch distribution is measured.
WAKE CALIBRATION RESULTS
Due to considerable wake to wake variation, the calibration
results were obtained by surveying across four different vane
passages, which were then combined and averaged together.
The resulting spanwise loss profiles are shown in Figs. 4a
and 4b. These profiles indicate a nearly constant loss across
the span for both the 24 and 40 WG cases. This tends to
verify the design intent of a two-dimensional wake flow
although the results for the far case indicate some evidence of
secondary flow effects near the hub and case.
This research effort involves a parametric study of the
influence of small changes in blade row spacing and solidity
on the choking mass flow rate. To determine the uncertainty
of the mass flow measurement, a repeatability study was
conducted early in the program. The uncertainty was
established through a series of repeat tests (with the 24 WG
configuration) over five different testing days spanning a
month, with common rig configurations and corrected
speeds. This series of tests included rig disassembly and
reinstallation. The resulting uncertainty in the mass flow
measurement was ±0.075 lb m/sec (Gorrell, et al. (1997)).
Uncertainty in the loss coefficient based on uncertainty in the
pressure measurement was established as ±0.001. However,
due to considerable wake-to-wake variability, the uncertainty
4
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•
•
The wake decay results shown in Fig. 5 indicate a similar
mixing trend to that found in the literature. In addition, the
values fall within the design target band from the results of
Creason and Baghdadi (1988).
40 WAKE GENERATOR CASE
1
0.9
0.8
- --0--- CLOSE SPACING
MID SPACING
- --0
FAR SPACING
0.7
4
ul
CALIBRATION WAKE DECAY COMPARISON
0.6
•
0.9
0
❑40
WAKE GENERATOR CASE
- -011-- 24 WAKE GENERATOR CASE
-
- STAUTER DATA CURVE FIT
----- RAJ & LARSEMINARAYANA DATA
BROOKFIELD MODEL (NO SWIRL)
-
z 0.5
O
•
Si 0.4
4.
0.7
0.6
0.3
" 0.5
0.2
P4'
r4 0.4
E.
0.1
0
w
0
0.25
0.5
0.75
0.3
4 o
a
1
SPANWISE MASS AVERAGED LOSS COEFFICIENT, CO
2
••••••...
4 0.1
C.)
N
Figure 4a. Spanwise mass-averaged loss profiles for the
40 wake generator case (0) = total pressure loss/inlet
dynamic pressure).
0
-0- CLOSE SPACING
MID SPACING
-0
FAR SPACING
z 0.7
'11 0.6
0
z 0.5
0
.
4
0.3
0.2
0.1
0
1
At the close spacing the agreement between the experimental
calibration results and the APNASA results for the wake
generator in isolation is reasonable. However, the simulation
under predicts the level of loss incurred at the far spacing.
The loss results shown for the APNASA three-blade-row
case does include blade row interaction effects. These results
show increased loss due to the presence of the rotor even for
a very low backpressure.
1-1
•
0.75
The overall loss coefficients (computed from the calibration
results) as a function of axial spacing are shown in Fig. 6.
The results obtained from the APNASA flow simulations are
also presented. Several features can be noted from this figure.
First the design loss intent (based on a 2-D design) closely
matches the result without the endwall losses included. As
expected, the 2-D design does not capture the endwall loss
effects. In addition, the endwall losses are a significant
portion of the overall loss.
0.9
0.8
0.5
Figure 5. Calibration wake decay comparisons to the
literature.
24 WAKE GENERATOR CASE
1
0.25
NORMALIZED DISTANCE FROM TE, K/C
0
0.25
0.5
0.75
1
SPANWISE MASS AVERAGED LOSS COEFFICIENT, 0)
Notable in the APNASA results is the wake loss mixing rate
seen in the slope of the line for the wake generator in
isolation. The APNASA loss curve for the three-blade-row
simulation has a comparable slope. It is clear from the results
that in the presence of the rotor an additional loss is incurred
Figure 4b. Spanwise mass-averaged loss profiles for the
24 wake generator case (co = total pressure loss/inlet
dynamic pressure).
5
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apparent underflow results. The embedded flow results for
the two wake generator vane counts are similar except at the
closest spacing.
(an increase in the loss coefficient of about 35%) which is
similar for both the close and far cases.
0.2
The flow rate correction is based on calibration results and so
does not include any blade row interaction or shock/wake
interaction losses. These losses, if they exist at all, would
tend to increase the computed corrected flow rate, i.e., the
embedded flow rate. In addition to this, the upstream wakes
may effect the blockage development within the rotor and
this would directly effect its choking flow capacity. Small
blockage changes within a rotor are difficult to sort out even
with results from numerical simulations.
CALIBRATION & APNASA RESULTS
(FOR 40 WARE GENERATOR CASE)
0.19
0.18
0ga
N
0.17
wE:
>z
W 0.16
H
NU
E8
0.15
0.14
0.13
0.12
40 WAKE GENERATOR CASE
1
DESIGN GOAL
N
0.11
0. 1
a
EXP WITHOUT ENDWALL BL
11111111!
0.2
3 to
o
!III!
0.3
0.4
0.5
4.3
0 6
IA
DISTANCE FROM WARE GENERATOR TRAILING EDGE
(NORMALIZED BY MEAN WAKE GENERATOR CHORD)
"C>
0
gOg
Zr4
ZHZ 0.98
OWN
000
Cu=
OW
0
Figure 6. Overall loss from calibration and APNASA
results for the 40 wake generator case (co = total pressure
loss/inlet dynamic pressure).
2?..04
HM3
O PE,
CORRECTED FOR ROTOR FACE TOTAL PRESSURE DEFICIT (EXP BASED ON CAL RESULTS) -
0.97
N D
N0
Overall, the results from the wake generator calibration
verify that the wake generators create wakes similar to the
design intent, i.e., they produce wakes similar to those found
behind highly loaded first stage stator vanes. In this way the
general usefulness of the results may be enhanced.
g
O
0.96
z
0.95
0.2
0.3
0.4
0.5
0 6
WAKE GENERATOR TO ROTOR FACE AXIAL SPACING
(NORMALIZED BY MEAN WAKE GENERATOR CHORD)
FLOW CAPACITY RESULTS
Figure 7. Choking flow rate as a function blade row
spacing for the 40 wake generator configuration at the
design rotational speed.
The choking flow capacity (which for this stage is set in the
rotor) results, at the design corrected rotational speed, for the
40 and 24 wake generator cases are shown in Figs. 7 and 8,
respectively. The flow capacity changes for the 12 wake
generator count case are omitted here since they were small
and below the experimental flow rate measurement
uncertainty.
Clearly, at some wake generator-to-rotor axial spacing the
wake mixing will be complete and any effects from blade
row interaction or rotor blockage phenomena will diminish to
zero. For this stage, this point is apparently reached (within
the uncertainty) near the "far" spacing for both wake
generator vane counts. At spacings closer than this other
mechanisms, such as those just mentioned, are at work.
These effects, having been detected far down the choke line,
would only be expected to increase, as the stage is
backpressured.
Both the three-blade-row choking flow rate as well as the
two-blade-row (i.e., the three-blade-row corrected) choking
flow rate, hereafter referred to as "embedded" choking flow
rate, are shown in the figures. As seen in the figures, the
actual three-blade-row flow rate drop is significant for both
the 40 and 24 wake generator count cases. These results,
when corrected for mass averaged total pressure loss at the
rotor face based on the calibration results, indicate no
overflowing of the embedded stage as was originally
expected when the experiment was designed. Instead, an
6
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a simple one-dimensional mass flow rate correction begins to
break down as the blade rows interact.
24 WARE GENERATOR CASE
I
I
If loss sources are present at spacings closer than 50% they
are of unknown origin and magnitude and so an embedded
overflow condition for these spacings cannot be ruled out.
Research is ongoing that will attempt to identify and clarify
the relevant aerothermodynamic flow details associated with
these close blade row spacings.
N
EXP
Ba 0.99
0 0
W 0H
H W 4
0
CORRECTED FOR ROTOR FACE TOTAL PRESSURE DEFICIT
(BASED ON CAL RESULTS)
fl
zz
•
r.4 0 . 9 8
ot4
0 0
O
7-
N
C.) 64
zH CC
4
ACKNOWLEDGEMENTS
EXP
W E.0.97
oio
N 0
O N
H
The authors would like to thank Mr. Ron Berger, Dr. Herb
Law, Mr. Robert Wirrig, Dr. Greg Bloch, and Mr. Terry
Norris for acquiring the experimental data. Thanks also go
out to Mr. Mark Celestina and Mr. Rick Mulac for help in
implementing and operating the NASA Lewis APNASA
flow solver. Gratitude is also extended to Dr. Tony Strazisar,
Dr. John Adamczyk, Prof. Ted Okiishi, Prof. Nick Cumpsty,
Prof. Ed Greitzer, Prof. Frank Marble and Dr. Choon Tan for
many useful and informative suggestions.
ACTUAL (3 BLADE ROW)
aH
4
0.96
0.95
■
0.2
I
0.3
■
I
0.4
0.5
0 6
WAKE GENERATOR TO ROTOR FACE AXIAL SPACING
(NORMALIZED BY MEAN WAKE GENERATOR CHORD)
Figure 8. Choking flow rate as a function blade row
spacing for the 24 wake generator configuration at the
design rotational speed.
REFERENCES
Celestina, M.L., Beach, T.A. and Barnett,
Adamczyk,
M., 1990, "Simulation of Three-Dimensional Viscous Flow
Within a Multistage Turbine," ASME Journal of
Turbomachinery, Vol. 112, pp. 370-376.
CONCLUSIONS
The present work was an attempt to address the longstanding question regarding the overflow of embedded
transonic compressor stages. The experiment was designed to
isolate and investigate the effects of upstream twodimensional wakes on downstream stage flow capacity.
Simplifications were made in the design of the experiment
(such as using uncambered wake generators) and some
important flow features such as freestream turbulence level
were not representative of a real embedded compressor stage.
The experiment, designed to investigate very large flow
capacity changes (as seen in the prior studies mentioned), did
not include many of the real effects present in embedded
transonic compressor stages.
Bettner, J.L., and Alverson, R.F., 1977, "Turbine Engine
High Flow Compressor," AFAPL-TR-77-23, WrightPatterson, Air Force Base, Ohio.
Brookfield, J.M., Waitz, I.A. and Sell, J., 1996, "Wake
Decay: Effect of Freestream Swirl," ASME Paper no. 96-GT495.
Creason, T. and Baghdadi, S., 1988, "Design and Test of a
Low Aspect Ratio Fan Stage, " AIAA Paper no. 88-2816.
Cumpsty, N.A., 1997, Private communication.
Given these limitations, the results show that for axial
spacings beyond 50% of the upstream blade chord, simple
wake mixing alone fully accounts for the flow losses and a
simple correction based on the rotor face mass averaged total
pressure deficit is sufficient. At spacings closer than this,
other effects or loss mechanisms may exist. These may
include: 1) Shock/wake or other two-dimensional blade row
interaction loss sources not accounted for in the calibration
procedure; 2) Increased blockage development in the rotor
due to the upstream wakes or secondary flows; and 3) Threedimensional or other effects not accounted for in the
experiment. As was noted by Cumpsty (1997), the concept of
Gorrell, S.E., Copenhaver, W.W. and Chriss, R.M., 1997,
"Effects of Upstream Wakes on the Performance of a
Transonic Compressor Stage." Presented at the thirteenth
International Symposium on Air Breathing Engines,
Chattanooga, Tenn., (ISABE 97-7070).
Law, C.H. and Wennerstrom, A.J., 1989, "Two Axial
Compressor Designs For a Stage Matching Investigation,"
AF-WAL-TR-89-2005.
7
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Mulac, R.A. and Adamczyk, J.J., 1992, "The Numerical
Simulation of a High-Speed Axial Flow Compressor,"
ASME J. of Turbomachinery, Vol. 114, No. 3.
Raj, R. and Lakshminarayana, B., 1973, "Characteristics of
the Wake Behind a Cascade of Airfoils," Journal of Fluid
Mechanics, Vol. 61, pp. 707-730.
Scott, J.N., and Hankey, W.L., 1986, "Navier-Stokes
Solutions of Unsteady Flow in a Compressor Rotor," ASME
Journal of Turbomachinery, Vol. 108, No. 2, pp. 206-215.
Shabbir, A., Zhu, J. and Celestina, M.L., 1996, "Assessment
of Three Turbulence Models in a Compressor Rotor," ASME
Paper 96-GT-198.
Stauter, R.C., Dring, R.P. and Carta, F.O., 1991, "Temporally
and Spatially Resolved Flow in a Two-Stage Axial
Compressor: Part 1 — Experiment," ASME J.
Turbomachinery, Vol. 113, No. 2, pp.219-226.
Wennerstrom, A.J., 1989, "Low Aspect Ratio Axial
Compressors: Why and What it Means," ASME J. of
Turbomachinery, Vol. 111, No. 4.
8
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