ABSTRACT
In this work, the transient process of surge has been investigated numerically in a gas turbine engine. A one-dimensional
stage-by-stage mathematical model has been developed which can describe the system behavior during aerodynamic
instabilities. It is demonstrated that, these instabilities can be stabilized by the use of active control strategies, such as air
bleeding and air injection. Both steady and unsteady active control systems were considered. In the steady case, mass is
removed at a fixed rate from the diffuser, or mass is injected at a fixed rate into the first stage of the compressor. In
unsteady control, the rate of bleeding or injection is linked with the amplitude and the frequency of the upstream pressure
disturbances. Results show that both steady and unsteady strategies eliminate surge disturbances and suppress the
instabilities. Therefore, they extend the stable operating range of compressor. It is also shown that smaller amount of
compressed air needs to be removed in the unsteady control case. Also, a variable area diffuser is shown to be able of
suppressing surge instabilities. Active control of instabilities, caused by sinusoidal perturbations of inlet total pressure, was
also investigated, which showed to destabilize the stable operating condition the design point.
Key words: Surge Avoidance, Gas Turbine, Active Control
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1- PhD Student (Corresponding Author): khaleghi@aut.ac.ir
2- Assistant Prof.
3- Assistant Prof.
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NOMENCLATURE
A = Area
Am = Amplitude of Surge Disturbance
Cf = Friction Factor
DH = Hydraulic Diameter
DA = Diffuser Area
E = Internal Energy Per Unit mass
Fx = Axial Force
f = Frequency of Surge Disturbance
fdis =
Frequency of Inlet Total Pressure
Perturbation
K1-K5 = Constants
mB = Mass Flow Rate of Bleed or Injected air
p = Static Pressure
Q = Heat Production
t = Time
U = Axial Velocity
UBX = Axial Velocity of Bleed or injected air
W = Work
5 =
Density
Introduction
Turbo machines are used in a wide variety of
engineering applications for power generation and
propulsion. There are two major fluid dynamic
instabilities in compression systems, known as
rotating stall and surge. Surge is a large amplitude
oscillation of the total annulus averaged flow through
the compressor; whereas in rotating stall, one can
finds from one to several cells of severely stalled flow
rotating around the circumference, although the
annulus averaged mass flow remains constant in time
once the pattern is fully developed. Therefore,
rotating stall is the two-dimensional or threedimensional
disturbance localized to the compressor
and characterized by regions of reduced or reversed
flow that rotate around the annulus of the compressor
[1-4]. To avoid these dangers, compressors have been
designed to operate away from the peak operating
point.
A compression system mathematical model was
developed using lumped-volume techniques which
make certain assumptions about compressibility
within the system.
The lumped volume model uses an isentropic
relationship to relate the time-dependant change in
density to a time-dependant change in total pressure,
and uses a steady-state form of the energy equation
[5]. A stage-by-stage mathematical model was
presented by Davis [6] which removed assumptions
inherent in lumped-volume models. A one
dimensional model developed by Garrard and Davis
[7-10] was found to predict the flow oscillations of
surge cycles due to perturbations of fuel flow rate. A
one dimensional model was developed to predict
surge disturbance propagation and engine response
during surge and surge recovery, due to perturbation
of total pressure and temperature, and exit nozzle area
[11]. Moore and Greitzer [12] developed a 2-D model
for rotating stall and surge. Their analysis was
extended to the compressible flow regime by
Bonnaure [13] and Hendricks [14]. It also was further
modified to include actuation by Feulner [15], who
also converted the model to a form compatible with
control theory. Paduano used controllable inlet guide
vanes for elimination of rotating stall [16,17]. Pinsley
[18] studied centrifugal surge control using throttle
valves as actuators. The effect of bleeding on the
control of instabilities was studied by Eveker [19],
Yeung [20] and Murray [21]. The reported amounted
of bleeding by Yeung to achieve operating
enhancements ranges from 1 to 10 percent based on
the mean flow. Niazi and Stein [22] developed a
three-dimensional viscous flow solver and studied the
fluid dynamic phenomena that lead to the onset of
instabilities in centrifugal and axial compressors and
the effect of bleeding on the control of instabilities.
The next section of this study contains the model
description. In the third section, the results of using
steady and unsteady control for uniform inlet flow are
presented. Air bleeding, air injection and variable
area diffuser are used as control systems. Although
one-dimensional models are not able to simulate
rotating stall, they are shown to properly enable
simulation of surge instability and study of active
control. In the fourth section, the results of using
steady and unsteady control for inlet flow with total
pressure perturbation are presented.
Modeling
Figure 1 shows the engine geometry consisting of
compressor, a diffuser, a combustion chamber, a two
stage turbine and an exhaust duct with a convergent
nozzle. Dimensions are given in mm. The compressor
geometry and characteristics are taken from Rolls-
Royce C-141, which its geometry and experimental
data were available in house.
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99 Mech. & Aerospace Eng. J. Vol. 2, No. 2, Nov. 2006
Fig. 2 Compressor characteristic performance of the
third stage.
Fig. 1Geometry of the engine.
Governing Equations
The governing equations are the unsteady, onedimensional
equations of continuity, momentum and
energy. These equations for an inviscid flow are
expressed in the conservative form (Equations 1 to 4).
It must be noted that these 1-D equations are used for
simulation of compressor and turbine flows and not
for the combustion chamber. The combustion
chamber is considered as a zero dimensional
component which energy release is modeled by an
increase in total temperature as:
(1)
(2)
(3)
(4)
Component Characteristics
To provide stage force (FX) and shaft work (Wshaft)
input to the momentum and energy equations, a set of
quasi-steady stage characteristics must be available
for closure. The stage characteristics provide the
pressure and temperature variation across each stage
as a function of normalized corrected mass flow rate.
The compressor has four stages and an inlet guide
vane (IGV) with different pressure and temperature
characteristics. During transition to surge, the steady
stage forces derived from the steady characteristics
are modified for dynamic behavior via a first-order
time lag equation. Appropriate time constants must be
used for each stage to provide the correct transient
behavior. The pressure and temperature
characteristics of the third stage are given in Fig.2.
Characteristics of other stages are also modified for
dynamic behavior.
Burner is considered as a zero dimensional
component. The energy release from the combustion
chamber is considered by an increase in total
temperature. The air-fuel ratio and combustion
chamber loss of a typical engine are used in the
model. The ratio of exit to inlet total pressure of
combustion chamber is 0.96. Turbine stages
characteristics, with specified power rating, were
obtained in order to take the matching condition into
account.
Numerical Scheme and Boundary Conditions
The method of characteristics is used as the
Numerical scheme to solve the governing equations.
A variable time step is used to satisfy the Courant
condition. For more details of MOC one may refer to
reference [23].
Specified total pressure and temperature during
normal forward flow is the inlet boundary conditions.
The exit boundary condition is the specification of
S ,
x
N
t
M =
+
,
=
e
M U
2 ,
+
= +
Ue pU
U p
U
N
( )
( )
.
2 1
2
ln
2
2
+
+
+
=
Adx
dm
e p
dx
U p dA
A
U
Adx
W
Adx
Q t
Adx
dm U
D
U U
C
Adx
FX
dx
dA
A
U
Adx
dm
dx
U d A
S
B
B
shaft
B Bx
H
f
B
& &
&
&
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Mech. & Aerospace Eng. J. Vol. 2, No. 2, Nov. 2006 100
exit Mach number or static pressure. During
reverse flow the inlet is converted to an exit boundary
with the specification of the ambient static pressure.
Therefore, both the inlet an exit boundaries function
as exit boundaries during a surge cycle. The boundary
conditions are properly applied to the combustor to
take the zero dimensional modeling into account. The
initial values are determined for the boundary
conditions at some specified compressor operating
points.
Control Strategies
In this study, two types of active control systems are
considered: steady and unsteady. In steady control,
including steady air bleeding and air injection, a fixed
fraction of mass flow rate is removed from or injected
to the compression system. In unsteady case, the mass
flow rate of removed or injected air is linked to the
pressure fluctuation upstream of compressor during
instabilities. Figure 3 illustrates the schematic of the
unsteady control system which is used in the present
study.
Fig. 3 Schematic of the unsteady control system.
The Validation of Results
To obtain the stable operating conditions, the
equations are solved by a time marching technique.
To validate the results, the predicted overall
characteristic of the compressor (for stable
conditions) is compared with the experimental data in
Fig.4. Close agreement between the model steady
state results and experimental data, especially near the
surge point, is obtained. Table 1 shows the
comparison between the surge point obtained from
the model and the experimental surge point. As
shown in figure 4 the experimental curve is
sufficiently close to the theoretical curve near the
surge point. Such results can be attributed to the fact
that one-dimensional modeling is quite close to the
nature of surge.
Tab. 1 Experimental and computational surge
point of compressor.
Fig. 4 Compressor overall characteristics.
Uniform Inlet Flow
In steady control, a fixed fraction of the mass flow
rate is removed through a valve which can be placed
at the diffuser or at the interstage of compression
system, or a fixed fraction of the mass flow rate is
injected into the first stage of the compressor. Figure
5 shows the static pressure at the compressor face
versus time. Steady bleeding from diffuser, equal to
4.3% of mean mass flow rate was applied to the
unstable operating condition at point B (shown in
Fig.6). As shown, this amount of bleeding can
remove surge disturbance.
Fig. 5 Inlet static pressure fluctuation.
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To study the effect of bleeding, steady bleeding from
diffuser and also interstage (second stage) was
considered. Bleeding equal to 3 % and 4.3 % of mean
flow rate was applied to the unstable condition at
point B. For 70 % and 100 % of bleeding which is
equal to 3 % and 4.3 % of mean flow rate
respectively, the stable controlled operating points are
shown as points C and D in Fig.6. In Fig.6, the
horizontal axis is the mass flow rate after bleed valve
and the vertical axis is the static pressure ratio of the
compressor. For 70 % of bleeding (3 % of mean flow
rate), the computed mass flow rate is 15.75 kg/s and
the corresponding overall static pressure ratio is 2.37.
For 100 % of bleeding (4.3 % of mean flow rate), the
mass flow rate is 15.65 kg/s and the static pressure
ratio is 2.35. This reduction of static pressure ratio is
due to removing more compressed air in 100 %
bleeding.
To investigate the effect of interstage bleeding,
the same amount of bleeding (4.3 % of mean flow
rate) was applied to the unstable condition at point B.
Mass is removed from the interstage (second stage)
and the new operating point is shown as point E
which has the mass flow rate of 15.6 kg/s and static
pressure ratio of 2.38. As illustrated, interstage
bleeding results in higher pressure ratio.
3.3 % of the mean flow rate was injected into the
first stage, during the unstable condition at point B, to
study the effect of injection on the performance of the
compressor. The new stable operating condition is
point F in Fig.6. The mass flow rate of point F is
15.85 and the corresponding pressure ratio is 2.5.
Fig. 6 Compressor characteristic performance for
steady control
The steady bleeding is inefficient and must be turned
off during design operation. In unsteady control, the
removed mass flow rate is linked to the pressure
fluctuation upstream of the compressor. Although
such strategy is not possible in one-dimensional
analysis, using periodic functions for bleeding mass
flow rate is shown to improve the stable operating
range. The amount of mass, which is removed from
diffuser, is linked to the amplitude and frequency of
surge disturbance. The amplitude and frequency of
pressure fluctuation during surge is found to be 12
Kpa and 80 Hz from Fig.5.
Three forms of periodic functions are used for
bleeding control. In the following equations, K1 , K2 ,
K3 , are chosen to be 1, 0.9, 0.5 and I/4
respectively:
(5)
(6)
(7)
In the above equations, "P1", "t", "Am", "f" and ""
are respectively ambient pressure, time, amplitude,
frequency of the fluctuations and the phase lag. The
constants K1, K2, K3 are chosen to ensure the bleed
rate is less than 3 % of mean flow rate. The parameter
m& B is averaged mass flow rate and is set to be 2.3 %
of the mean mass flow rate. Figure 7 is given for
better understanding the trend of control function.
Results are shown in figure 8. Point G, H and I are
new stable operating points corresponding to equation
5-7 respectively. For point G the computed mass flow
rate is 15.6 kg/s and the static pressure ratio is 2.4.
Point H has mass flow rate of 15.7 kg/s and static
pressure ratio of 2.37. Point I has the minimum mass
flow rate equal to 15.5 kg/s with the static pressure
ratio of 2.37. The minimum mass flow rate obtained
is related to the equation 7 and the maximum pressure
ratio is related to equation 5. This behavior may be
attributed to the effect of different shapes of the
equations shown in figure 7. The similar shapes to the
nature of surge may lead to higher pressure ratios or
lower mass flow rates. As shown, smaller amount of
compressed air need to be removed in unsteady
control, and also it leads to operating point with
= + [ (t f + )]
P
m m m K Am B B B sin 2 . .
1
1 & & &
( )
( )
+
+ +
= +
t f
t f
P
m m m K Am B B B cos 2 . .
sin 2 . .
2
1
2 & & &
( )
( )
( )
+
+ +
+ +
= +
t f
t f
t f
P
m m m K Am B B B
cos 2 . .
cos 2 . .
sin 2 . .
3
2
1
3 & & &
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Archive of SID
Mech. & Aerospace Eng. J. Vol. 2, No. 2, Nov. 2006 102
higher pressure ratio.
Fig. 7 Schematic shape of three control functions.
Fig. 8 Compressor characteristic performance
for unsteady control
Variable Area Diffuser
To study the effect of variable area diffuser, which is
located before the burner, the area of the diffuser is
changed with time. The area variation is linked to the
amplitude and frequency of surge disturbance. The
following periodic control law was chosen in this
study:
(8)
DAd is the design area of the diffuser. Constants K4
and K5 are chosen to be 0.2 and 0.1 respectively so
that the area variation does not exceed 2.5% of the
design area.
This control law was applied to the unstable
operating condition at point B shown in figure 6.
Figure 9 shows the inlet mass flow rate versus time.
As illustrated, using an appropriate form of diffuser
area variation eliminates the surge instability and
leads to stable controlled condition.
The area of diffuser decreases periodically by
using the above control law. As a result, compressor
back pressure decreases periodically. Reduction of
back pressure has the same effect of bleeding.
Therefore, variable area diffuser is capable of
eliminating compressor instabilities.
Fig. 9 Inlet mass flow rate (variable area
diffuser).
Intel Total Pressure Perturbation
Instabilities Due to Inlet Total Pressure Perturbation
As mentioned in the previous sections, Transient
interaction of shock waves and boundary layer at the
entry may lead to sinusoidal variations of pressure
that can affect compressor instabilities. AS suggested
by Tesch and Steenken [24], the following form for
modeling of the perturbation was considered:
(9)
In the above equation, (PT) SS is the steady state inlet
total pressure, "t" is time and "fdis" is the perturbation
frequency. Instead of amplitude, a parameter named
amplitude-percent was used.
To study the engine response, we first applied the
above sinusoidal perturbation with the amplitudepercent
of 5 and frequency of 10 Hz to the stable
condition at surge point. Since no surge disturbance
was observed, we increased the frequency at constant
amplitude-percent. The first surge disturbance, with
( + )
=
1 5
4
1 / P sin 2f K.t . .
DA K Am
DA DA d
d
( ) ( ) ( )
Amplitude _ Percent Sifn(2 t ).
P P P
dis
T T SS T SS
= +
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Fig. 12 Compressor overall characteristics
the frequency of 25 Hz, was captured when the
frequency of perturbation was increased to 30 Hz.
The inlet mass flow rate versus time is given in
Fig.10.
Fig. 10 Inlet mass flow rate (pressure
perturbation)
Fig.11 Minimum frequency of the perturbations
that leads to surge
The same methodology was applied to the stable
operating condition at surge point for other
magnitudes of amplitude-percent. Figure 11 shows
the minimum frequency of the perturbations that
leads to surge for various amplitude-percent
magnitudes. It also indicates the surge frequency.
Steady Control Results
To study the effect of steady bleeding, 4 % of the
mean mass flow rate was extracted from the diffuser.
This amount of bleeding was applied to the stable
operating condition at point A which has been
destabilized with inlet perturbation of total pressure
of amplitude percent of 5 and frequency of 30 Hz. As
shown in Fig.11 this perturbation is capable of
destabilizing the stable condition at point A which is
closed to the surge point (Fig.12). Fig.13 shows the
inlet mass flow rate versus time for this case. As
illustrated, this amount of bleeding can remove surge
disturbance completely.
Fig. 13 Inlet mass flow rate (Steady bleeding).
Table. 2 Minimum amount of required steady
bleeding for various inlet perturbations.
To obtain the minimum mass flow rate
needed to be removed for stabilizing the instabilities,
we reduce the steady removed mass. Bleeding equal
to 2.2 % of mean mass flow rate was found to be
optimum. The same methodology was used in the
case of perturbations of higher amplitude percent. The
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Mech. & Aerospace Eng. J. Vol. 2, No. 2, Nov. 2006 104
frequency of perturbations was chosen from Fig.11 to
ensure that the inlet perturbations can destabilize the
compressor. Table.2 shows the minimum amount of
steady bleeding needed for various amplitude percent
magnitudes. As the amplitude percent increases,
higher amount of mass flow rate is needed to be
removed in order to stabilize the instabilities.
Unsteady Control Results
The frequency of surge disturbance is chosen from
Fig.11 and the pressure fluctuation during surge is
also obtained from computations for each case. As
discussed, unsteady one dimensional bleeding with
periodic forms, can reduce the amount of bleeding.
Therefore, it is more efficient. A sinusoidal form
(equation 10) is used to consider periodic bleeding.
(10)
In equation 10, P1, t, Am, and f are the ambient
pressure, time, the amplitude and the frequency of the
fluctuations, respectively. "" is the phase lag and is
chosen to be /4. mB is the averaged mass flow rate
which is needed to be removed. The constant K4 is
chosen to ensure that the bleed rate does not exceed
20 % of the averaged removed mass flow rate. The
above control law was applied to the same operating
conditions in steady part. Tab.3 shows the minimum
averaged mass flow rate needed to be removed in
order to stabilize the instabilities. As shown, smaller
amount of compressed air need to be removed in
unsteady control, as compared to steady case.
CONCLUSION
A one-dimensional unsteady computer code has been
developed which enables simulation of surge
disturbance propagation through entire jet engines.
The effect of active control on the instabilities was
studied and the following observations and lower
conclusions were obtained:
1- steady control can eliminate surge disturbance. If
the amount of bleeding air increases, the new stable
operating point has lower mass flow rate and pressure
ratio.
2- using air injection, as the control system, the new
operating point has higher pressure ratio and also
higher mass flow rate, as compared to air bleeding.
3- if the bleeding air is injected into the first stage of
the compressor, the required amount of bleeding
reduces.
4- Interstage bleeding leads to a new stable operating
point with higher pressure ratios compared to
bleeding from the diffuser, so it is more efficient.
5- smaller amount of compressed air need to be
removed in unsteady control. This leads to a new
operating point with higher pressure ratio.
6- compressor back pressure reduces as the diffuser
area decreases. Therefore, variable area diffuser can
be used to stabilize compressor instabilities.
7- inlet perturbations can destabilize the stable
operating condition at design point.
Acknowledgment
This research was supported by Amirkabir University
of Technology, Iran, Tehran, which in greately
appreciated.
References
1. Eveker, K.M., Gysling, D. L., Nett, C. N., and
Sharma, O.P., "Integrated Control of Rotating Stall
and Surge in High-Speed Multistage Compression
Systems", ASME J. Turbomach., Vol. 120, , pp.440-
445, 1998.
2. Greitzer, E.M., " Surge and Rotating Stall in Axial
Flow Compressors–Part 1, Theoretical Compression
System Model", ASME J. Eng. for Power, Vol. 98,
No.2, pp.190-198, 1976.
3. Greitzer, E.M., " Review- Axial Compressor Stall
Phenomena, " ASME J. Fluids Eng., Vol. 102,
pp.134-151, 1980.
4. Greitzer, E.M., " The Stability of Pumping
Systems- The 1980 Freeman Scholar Lecture",
ASME J. Fluid Eng., Vol.103, pp. 193-242, 1981.
5. Davis, M.W. Jr., “A Stage-by-Stage Dual-Spool
Compression System Modeling Technique”, ASME
Gas Turbine Conf., ASME Paper 82-GT-189,
London, 1982.
6. Davis, M.W. Jr., “Stage-by-Stage Poststall
Compression System Modeling Technique”, J.
Propulsion, Vol. 7, No. 6, 1991.
7. Hale, A.A., and M.W. Davis, Jr., “Dynamic
Turbine Engine Compressor Code, DYNTECC –
Theory and Capabilities”, AIAA Paper -92-3190,
1992.
8. Garrard, G.D. “ATEC: The Aerodynamic Turbine
Engine Code For the Analysis of Transient and
Dynamic Gas Turbine Engine System Operation-Part
1, Model Development”, ASME pp. 96-GT-193,
1996.
9. Garrard, G.D “ATEC: The Aerodynamic Turbine
Engine Code For the Analysis of Transient and
Dynamic Gas Turbine Engine System Operations—
Part 2: Numerical Simulations”, ASME pp. 96-GT-
194, 1996.
= + [ (t f + )]
P
m m m K Am B B B sin 2 . .
1
4 & & &
www.SID.ir
Archive of SID
105 Mech. & Aerospace Eng. J. Vol. 2, No. 2, Nov. 2006
10. Garrard, G.D., Davis, M.W., Jr., Wehofer, S., and
Cole, G. “A One-dimensional, Time Dependent
Inlet,Engine Numerical Simulation for Aircraft
Propulsion Systems”, ASME Paper # 97-GT-333,
1997.
11. Khaleghi, H., Boroomand, M. and Tousi, A. M.,
"A Stage by Stage Simulation and Performance
Prediction of Gas Turbine", The 43rd AIAA
Aerospace Exhibit and Meeting, Nevada, 2005
12. Moore, F.K. and Greitzer, E.M., "A Theory of
Post-Stall Transients in Axial Compression Systems-
Part I- Development of Equations and Part IIApplication,"
ASME J. Eng. for Gas Turbine and
Power, Vol. 108, pp. 68-97, 1986.
13. Bonnaure, L.P., "Modeling High Speed
Multistage Compressor Stability, ", M.Sc. Dep’t. of
Aeronautics and Astronautics, Thesis, 1991.
14. Hendricks, G.J., Bonnaure, L.P., Longley, J.P.,
Greitzer, E.M. and Epstein, A.H., "Analysis of
Rotating Stall Onset in High Speed Axial Flow
Compressors", AIAA Paper No. 93-2233, 1993.
15. Feulner, M.R., Hendricks, G.J., and Paduano,
J.D., "Modeling for Control of Rotating Stall in High
Speed Multistage Axial Compressors", ASME Paper
No. 94-GT-200, 1994.
16. Paduano, J., Valvani, L., Epstein, A., Greitzer, E.,
and Guenette, G., "Modeling for Control of Rotating
Stall", Automatica, Vol. 30, No. 9, pp. 1357-1373,
1994
17. Paduano, J., Epstein, A., Valvani, L., Longley, J.,
Greitzer, E., and Guenette, G., "Active Control of
Rotating Stall in a Low-Speed Axial Compressor", J.
Turbomach., Vol. 115, pp. 48-56, 1993.
18. Pinsley, J.E., Guenette, G.R., Epstein, A.H. and
Greitzer, E.M., "Active Stabilization of Centrifugal
Compressor Surge", J. Turbomach., Vol. 113, pp.
723-732, 1991.
19. Eveker, K.M. and Nett, C.N., "Model
Development for Active Surge Rotating Stall
Avoidance Aircraft Gas Turbine Engines", The 1991
American Control Conf., 1991.
20. Yeung, S. and Murray, R.M., "Reduction of Bleed
Valve Requirements for Control of Rotating Stall
Using Continues Air Injection", The 1997 IEEE Int.
Conf. on Control Application, Hartford, CT, pp.683-
690,1997.
21. Wang, Y. and Murray, R.M., "Effects of the
Shape of Compressor Characteristic on Actuator
Requirements for Rotating Stall, Control", The 1998
American Control Conf., 1998.
22. Niazi, S., Stein, A. and Sankar, L.N., "Numerical
Studies of Stall and Surge Alleviation in a High-
Speed Transonic Fan Rotor," AIAA Paper 2000-
0226, 2000.
23. Manning, J.R., “Computerized Method of
Characteristics Calculation for Unsteady Pneumatic
Line Flows”, J. Basic Eng., pp. 231-240, 1968.
24. Tesch W.A., Steenken W.G., “Dynamic Blade
Row Compression Model for Stability Analysis”,
AIAA paper No. 76-203, AIAA 14th Aerospace
Science Meeting, Washington D.C., pp. 26-28, 1976.
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