AIAA 2002-4158
38th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit
7-10 July 2002, Indianapolis, Indiana
AIAA 2002-4158
Improvements of Inducer Inlet Backflow Characteristics Using 3-D Inverse
Design Method
Kosuke ASHIHARA*, Akira GOTO* , Kenjiro KAMIJO**
Hitoshi YAMADA***, Masaharu UCHIUMI****
ABSTRACT
The three-dimensional inverse design method was
applied to improve the inlet backflow characteristics of
highly loaded turbopump inducers for a liquid
hydrogen rocket engine. Flow mechanisms, both for
conventional and inverse design inducers, were
investigated
experimentally
by
flow
field
measurements and flow visualization, as well as
numerically by the application CFD.
The conventional inducer, which had been
designed for the H-IIA rocket LE-7A engine
turbopump, had a strong inlet backflow at the design
point. Optimizing the blade loading distribution
using the 3-D inverse design method and a CFD
analysis eliminated this inlet backflow.
Water model tests confirmed the elimination of
inlet backflow in the inverse design inducers.
However, it was confirmed that the suppressed inlet
backflow tended to make cavitation occur in the blade
passages and reduced suction performance.
Cavitation visualization and FFT analysis of unstable
phenomena were also performed in this study.
INTRODUCTION
Inducers are important components to achieve high
suction performance of turbopumps.
Especially for
a liquid rocket engine turbopump, a high-speed, highload and high-suction-performance inducer design is
essential to achieve a lightweight rocket design.
However, such an inducer is still a difficult component
to design and it often causes unstable turbopump
operation. It is a well-known fact that strong inlet
*
Ebara Research Co., Ltd. ashihara20004@erc.ebara.co.jp
** Tohoku University
*** National Aerospace Laboratory
**** National Space Development Agency of Japan
Copyright © 2002 The American Institute of Aeronautics and
Astronautics Inc. All rights reserved.
backflow and rotating cavitation in inducers may cause
mechanical failures in turbopumps and the entire
pumping system.
In conventional design practice, Inducer blades are
often designed with helical surfaces, and a
combination of a straight line and simple curves is
chosen for the camber lines of highly loaded inducers.
The suction performance of an inducer is a function of
the blade angle β at the leading edge. Usually, the
leading edge of the inducer is designed to have a knifeedge shape for high suction performance, and a
positive incident angle α is adopted to avoid
pressure surface cavitation(1).
According to the
design guideline(2), the ratio α/β is a characteristic
parameter of a high suction performance inducer and
the optimum ratio varies between 0.35 and 0.50.
However, unstable phenomena of inducers are affected
not only by the blade angle but also by several other
factors. For example, the inlet backflow is strongly
affected by the design inlet flow coefficient. So, it is
important to design an inducer by considering and
controlling several effects of inducer geometry on flow
fields.
Recently, computer technology and CFD
(Computational Fluid Dynamics) technology have
rapidly developed and complex internal flows within
turbomachinery can be predicted with reasonable
accuracy. The CFD-aided design has already become
daily practice for turbomachinery design.
It is
possible to make a qualitative evaluation of an
abnormal flow phenomenon such as inlet backflow.
Conversely, flow fields with cavitation are very
complex and difficult to predict numerically because
they include a phase change between liquid and gas.
A large number of flow models including cavitation
has been investigated (3)(4). However, these flow
models are limited to particular cases and numerical
calculations of cavitating flows in 3-D rotating
passages can still not be applied in practice.
Consequently, it is common to examine CFD results
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Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
for single-phase flows while paying attention to the
low-pressure region in the blade passage to improve
suction performance.
On the other hand, a new approach of
turbomachinery design using a three-dimensional (3D) inverse design method has been proposed (5). The
3-D blade shapes that satisfy specified blade loading
distributions (and the required 3-D pressure fields) can
be designed using the 3-D inverse design method. In
this new design approach, it is easy to control the
performance of turbomachinery while keeping
designated design specifications. Many successful
applications have already been published (6)(7)(8) both
for impellers and diffusers.
In this study, a new approach of 3-D inducer design
for controlling inlet backflow characteristic using the
3-D inverse design method and CFD was performed.
The highly loaded inducers with the same design
specification as the liquid hydrogen turbopump inducer
of the Japanese H-IIA rocket engine LE-7A are
designed with controlled blade loading distribution.
The inlet backflow characteristics and suction
performance of the inducers are examined numerically
and experimentally, and compared with conventional
design inducers including the LE-7A liquid hydrogen
turbopump inducer.
NOMENCLATURE
B
number of blades
N
rotation speed, min-1
NPSH
net positive suction head, m
P
pressure, Pa
Q
flow rate, m3/min
V
absolute velocity, m/s
U
peripheral blade speed, m/s
W
relative velocity, m/s
m
meridional distance
r
radius, m
α
incidence angle, deg.
β
blade angle, deg.
φ = V1m U 1t
φ
flow coefficient,
U 12t
σ
cavitation number,
σ = NPSH
2g
ρ
density, kg/m3
ρU 22t
ψ
head coefficient,
ϕ = ( p 2 − p0 )
2
η
efficiency
ν
power coefficient
Subscripts
0
inlet
1
inducer leading edge
2
inducer trailing edge
m
meridional component
s
static value
t
inducer tip
θ
circumferential component
Superscripts
A*
normalized value
A
averaged value
+
pressure surface value
suction surface value
NUMERICAL METHOD
Inverse Design Method
The basic theory of the 3-D inverse design method
proposed by Zangeneh (1991)(5) is briefly described
here. In this theory, turbomachinery blades are
represented by sheets of vorticity, whose strength is
determined
by
a
specific
distribution
of
circumferencetially
averaged
swirl
velocity
rVθ defined as:
2π
B B
rVθ =
∫ rVθ dθ
2π 0
(1)
The rVθ is related to the blade loading through the
following expression in incompressible potential
flows:
+
−
Ps − Ps =
2π
∂rVθ
ρWmbl
B
∂m
(2)
Where, B is the number of blades, Wmbl is the
meridional component of relative velocity on the blade.
Blade loading or the pressure difference across the
+
−
blade, ( Ps − Ps ) , can be optimized by controlling
the distribution of ∂rVθ ∂m in Eq. (2).
The blade shape represented by blade wrap angle
or θ -coordinate of the blade camber is determined by
integrating the first order hyperbolic partial differential
equation, see Zangeneh (1991)(5) for details. This
equation is obtained as a result of the application of the
inviscid slip condition, which implies that the blade
shape must be aligned with the local velocity vector
induced by the vorticity. For this integration, a
distribution of wrap angle should be specified along a
quasi-orthogonal (e.g. along the trailing edge) as an
initial condition. This initial condition of the wrap
angle is called the “stacking condition” in the present
method.
The input data for this design method are : (i)
loading distribution (distribution of bound circulation
2πrVθ ), (ii) meridional geometry, (iii) velocity
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distribution at inlet, (iv) rotational speed, (v) blade
thickness distribution, (vi) number of blades, and
(vii) stacking condition. By controlling the loading
distribution, it is possible to design turbomachinery
blades having high efficiency, high suction
performance, super compact machine size, etc.
CFD Method
The CFD code used in the present study is a Dawes
code for incompressible steady flows (Walker and
Dawes, 1990) (9), which solves Reynolds averaged
Navier-Stokes equations using the Baldwin-Lomax
turbulence model and Chorin’s method of artificial
compressibility. It calculates one flow passage of
turbomachinery
assuming
periodic
boundary
conditions.
The computational grid used in this study is the Htype, and the grid was automatically generated with
non-uniform grids clustered near endwalls, blade
surfaces, and blade leading and trailing edges. The
number of grid points used in this study for the inducer
analysis was 201,761 per blade pitch. An Inducer has
a very small blade tip angle compared to other
turbomachinery blades, so the CFD analysis of an
inducer using the H-type grid may have grid distortion
problems. The grid number dependence study of
industrial inducer analysis using the Dawes code was
performed prior to the study(10). The results showed
that the grid number of 100,000 per blade pitch is
enough to evaluate the total head and minimum static
pressure of inducers within reasonable errors. The
inducer tip clearance was 0.5 mm and four grid cells
were placed in the gap region. The calculation time
for one case was approximately three hours using a
typical engineering workstation.
DESIGN SPECIFICATION AND CFD
ANALYSIS
Tab.1
Design specification
INDUCER
Design
Blade Surface
Rotating speed
N min-1
Design flow rate Q m3/min
Inlet flow coefficient
φ1
Inlet tip diameter
D1t mm
Inlet hub diameter D1h mm
Exit tip diameter
D2t mm
Exit hub diameter D2h mm
Inlet tip blade angle β1t deg
Exit tip blade angle β2t deg
Blade number
Z
A
B
Conventional Conventional
design
design
Helical
42161
31.7
0.067
174
50
174
80
6.4
11.1
3
Helical
42161
31.7
0.08
161.8
50
168
88
7.5
11.5
3
C
D
3-D
3-D
Inverse Inverse
Design
Design
3-D
Ruled
42161
42161
31.7
31.7
0.08
0.08
161.8
161.8
50
50
168
168
88
88
3
3
Fig.1
Geometry of the studied inducers
Inducer Design Specifications
Four different inducers, designed for the liquid
hydrogen turbopump, were examined. Table 1 shows
the design specification of each inducer. Inducers A
and B are conventional helical-type inducers, while
Inducers C and D are 3-D Inverse design inducers.
Figure 1 shows the geometry of these four inducer
models.
The helical Inducer A had been designed by a
conventional method for the liquid hydrogen
turbopump of LE-7A engine. The meridional shroud
shape is an axial-flow one, and inlet and exit diameter
is 0.174m. The design flow coefficient of Inducer A
is 0.067. Inducer A is known to have an unstable
inlet backflow problem.
The other conventional helical Inducer B had been
designed to improve the back flow characteristic of
Inducer A. The meridional shroud shape is a mixedflow one, and inlet diameter is 0.1618m and exit
diameter is 0.168m. The design flow coefficient is
0.080 and the head coefficient is smaller than that of
Inducer A.
Inducer C, designed by the 3-D inverse method,
has a 3-D blade surface. The meridional shroud
shape is the same as that of Inducer B. The design
flow coefficient of Inducer C is 0.080 and the head
coefficient is the same as that of Inducer B. The
blade loading at the leading edge was set to zero to
achieve a no incidence condition. More than 50 cases
were studied to optimize the blade loading.
Inducer D was also designed by the 3-D inverse
design method, but its blade surface was modified to a
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ruled surface by linearly connecting the hub and the tip
to improve productivity and structural strength. The
design flow coefficient of Inducer D is 0.080 and the
head coefficient was increased from Inducer C. The
meridional shape along the shroud is the same as those
of Inducers B and C, while the hub radius was
increased from the inlet to achieve a high pressure rise
(see Fig.3). The leading edge blade loading was
maximized within the limit of backflow suppression so
that it has larger incidence to avoid pressure surface
cavitation.
The leading edges of Inducers A and B were
finished to knife-edge shapes and had sweep back
configurations, while the leading edges of Inducers C
and D had one-fourth elliptic shapes and linear
meridional configurations.
CFD analysis
CFD analyses by Dawes 3-D N-S code were
performed for the four inducers at the flow rates of
Q*=1.00(design point), Q*=0.94(minimum flow rate),
and Q*=1.06(maximum flow rate).
Figure 2 shows the total head coefficients of CFD
results. The predicted total head coefficient for each
inducer at design point Q*=1.00 was A:0.332, B:0.267,
C:0.266, and D:0.292. Inducer A has a very large
head coefficient and Inducer B has a smaller head
coefficient than Inducer A, because of its smaller exit
diameter. It is confirmed that Inducer C has almost
the same head coefficient as Inducer B, and Inducer D
has a larger head coefficient than Inducer B with the
same exit diameter.
a) Q*=0.94
Fig.2
CFD results – Head coefficient
Flow streamline predictions
Figure 3 shows the CFD analysis results of flow
streamlines at Q*=0.94, 1.00, and 1.06. Here, the
streamlines were calculated based on circumferentially
averaged meridional velocities. Closed streamlines
can be predicted if the inducers have an inlet backflow.
At the design flow rate of Q*=1.00, a large backflow
region is observed upstream of Inducer A. In the case
of Inducer B, which has a larger design flow
coefficient than Inducer A, the backflow region
became smaller than that of Inducer A, but it still can
be observed. In the case of Inducer C, designed by
the 3-D inverse design method, with zero bladeloading at the leading edge (i.e., no incidence
condition), backflow is not observed. In the case of
Inducer D, although it was designed for a higher head
coefficient and a larger incidence angle than Inducer C,
b) Q*=1.00
CFD results – Flow streamline
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Fig.3
c) Q*=1.06
backflow was also avoided at the design flow rate.
At the minimum flow rate of Q*=0.94, the backflow
regions of Inducers A and B expanded from Q*=1.00,
but backflow was not observed in the cases of Inducers
C and D. At the maximum flow rate of Q*=1.06, the
backflow regions of Inducers A and B shrank from
Q*=1.00, but they can still be observed. However,
backflow was not observed in the cases of Inducers C
and D at all flow rates.
From these results, it is expected that the controlled
leading edge loading design using the 3-D inverse
design method (Inducers C and D) is effective to
improve the inlet backflow characteristic of the
inducer from the conventional design inducer
(Inducers A and B).
Inlet velocity predictions
Figure 4 shows CFD results for the inlet velocity
distribution at 5mm upstream from the leading edge at
the hub. The figure shows the radial distribution
from hub to tip of the tangential velocity and the axial
velocity at flow rates of Q*=0.94, 1.00, and 1.06.
The tangential velocity CT and the axial velocity CZ
are normalized using the peripheral velocity at the
trailing edge tip. A velocity distribution change can
be predicted if the inducer has an inlet backflow. In
the case of Inducer A, all flow rate results show that
regions with large tangential and negative axial
velocities exist at the tip side. In the case of Inducer
B, similar velocity changes as Inducer A are observed
at the flow rates of Q*=1.00 and 0.94, but these are
much smaller than those of Inducer A. In the cases of
Inducers C and D, velocity change is not observed at
all flow rates.
Experimental facility
The test models of Inducers A, B, C, and D were
made by cutting a block of stainless steel using a 5axis numerically controlled milling machine, and water
model tests were performed according to ISO
standards. The model inducers were scaled down by
40% from the design scale and tested at a rotational
speed of 3,000rpm and 3,600rpm. Similar to the
actual rocket engine turbopump layout, a guide vane
was arranged downstream from the inducer. The
scaled down model of the actual guide vane was made
by Rapid Prototyping using Selective Laser Sintering
(SLS).
EXPERIMENTAL RESULTS AND
DISCUSSION
a) Q*=0.94
Fig.5
b) Q*=1.00
Fig.4
c) Q*=1.06
CFD results – Inlet velocity
Experimental set-up
Figure 5 is a schematic diagram of the
experimental set-up. The overall performance test
and the flow velocity measurements were performed at
a rotational speed of 3,000 rpm and the suction
performance test and the FFT analysis test were
performed at a rotational speed of 3,600 rpm. The
wall static pressure difference between the suction pipe
and the discharge pipe (downstream from the guide
vane) was measured in the Q-H performance tests and
the suction performance tests.
The inlet flow
velocities 3 mm upstream from the leading edge of the
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hub were measured using a 3-hole Pitot tube. The
model inducers were scaled down by 40% from the
design scale so that this Pitot tube location corresponds
to 5 mm upstream from the leading edge of the hub in
the design scale. The FFT analysis of wall static
pressure oscillation under cavitating conditions was
performed using a pressure sensor located 28.5mm
upstream from the leading edge at the hub.
observed at all flow rates. In the case of Inducer D, a
small backflow region is observed at the minimum
flow rate of Q*=0.94.
Overall performance results
Figure 6 shows the experimental results of headflow and efficiency-flow characteristics. All test data
were showed in non-dimensional coefficients.
Inducer D showed almost the same head coefficient as
Inducer A in the operating range between Q*=0.94 and
1.06, and it is confirmed that Inducer D achieved a
very high loading with a small exit diameter compared
to other inducers. The efficiency of Inducer C was
lower than those of other inducers due to the zero
incidence design at the leading edge. Note here that
the guide vane best matched the original Inducer A.
a) Q*=0.94
b) Q*=1.00
Fig.7
Fig.6
Experimental results – Overall performance
Inlet velocity measurement results
Figure 7 shows the results of the inlet velocity
measurements.
The figure shows the radial
distributions from hub to tip of tangential and axial
velocities at flow rates of Q*=0.94, 1.00, and 1.06.
The tangential velocity CT and the axial velocity CZ
are normalized using the peripheral velocity at the
trailing edge tip. In the case of Inducer A, it is
confirmed that regions with large tangential and
negative axial velocities exist at the tip side at all flow
rates. This negative axial velocity region indicates
the onset of inlet backflow. In the case of Inducer B,
the backflow region is not observed at the design flow
rate of Q*=1.00 and the maximum flow rate of
Q*=1.06, but a medium-sized backflow region is
observed at the minimum flow rate of Q*=0.94. In
the case of Inducer C, the backflow region is not
c) Q*=1.06
Experimental results – Inlet velocity
A comparison of CFD results (See Fig.4) and
experimental results confirms that the velocity
distribution in the backflow region of Inducer A did
not show such a good agreement in terms of absolute
value, but the qualitative tendency of backflow region
size showed a good agreement. In the case of Inducer
C, both CFD results and experimental results showed
uniform velocity distributions with no inlet backflow.
However, in the case of Inducers B and D, the
qualitative tendency of backflow region did not show
such a good agreement between CFD results and
experimental results. At the design flow rate of
Q*=1.00, CFD results overestimated the backflow
region of Inducer B, and at the minimum flow rate of
Q*=0.94, CFD results underestimated the backflow
region of Inducer D.
From these results, it is confirmed that the
controlled leading edge loading design using the 3-D
inverse design method (Inducers C and D) is effective
to improve an inlet backflow characteristic of the
inducer from that of the conventionally designed
inducer (Inducers A and B). In particular, the zero
leading edge loading design (Inducer C) can suppress
inlet backflow over the whole range of operating flow
rates of the inducer.
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=0.04 and showed high suction performances at all
flow rates. However, In the case of Inducers C and D,
the head coefficients started decreasing at a relatively
high cavitation number and showed lower suction
performances than Inducers A and B. The suction
performance of Inducers C and D depended on the
flow rate and they rapidly deteriorated at the maximum
flow rate of Q*=1.06.
a) Q*=0.94
b) Q*=1.00
Fig. 9-a Cavitation visualization at Q*=0.94
Fig.8
c) Q*=1.06
Experimental results – Suction performance
Tab.2
Cavitation number at 10% head
INDUCER
A
B
C
D
FLOW RATE Q*
0.94
1.00
1.06
0.0253
0.0199
0.0194
0.0296
0.0294
0.0305
0.0448
0.0455
0.0581
0.0416
0.0541
0.0762
Suction performance results
Figure 8 shows the suction performance test results.
The suction performance tests were performed at flow
rates of Q*=0.94, 1.00, and 1.06, and these graphs
present the head coefficient change for various inlet
pressure conditions (represented by cavitation number).
Table 2 shows the suction performances of the
inducers, which were evaluated by calculating the
cavitation number at 10% head-drop points. In the
case of Inducers A and B, the head coefficients were
kept high under a very low cavitation number of σ
Fig. 9-b Cavitation visualization at Q*=1.00
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cavitation mainly occurred at the front suction surfaces
of the leading edge region and did not spread into the
blade passages at all flow rates. However, in the
cases of Inducers C and D, which are designed using
the 3-D inverse design method and for which backflow
characteristics were improved, it was confirmed that
cavitation occurred in the blade passages and spread
toward the trailing edges of the inducers. This
tendency increased at larger flow rates.
Note here
that tip vortex cavitation, typically observed in
conventional inducers, was not clearly observed for the
inverse design Inducers C and D.
Fig. 9-c Cavitation visualization at Q*=1.06
Cavitation visualization results
Figure 9 shows the cavitation visualization results.
The tests were performed at flow rates of Q*=0.94,
1.00, and 1.06, and the inlet conditions were chosen at
a cavitation number of σ=0.04. In the cases of
Inducers A and B, which are designed by a
conventional method and tend to have inlet backflow,
a) Q*=0.94
FFT analysis results of inlet pressure
Figure 10 shows the FFT analysis results for inlet
pressure at flow rates of Q*=0.94, 1.00, and 1.06, and
a cavitation number of σ =0.04. Inducer A has
broadband noise and a strong peak at 73.7Hz at the
design flow rate of Q*1.00. The broadband noise
was due to the large inlet backflow, and the strong
peak at 73.7Hz was caused by rotating cavitation. A
high-speed camera also confirmed that there was
rotating cavitation. Rotating cavitation was strong
enough to cause a vibration problem in the present
experimental facility. At all flow rates, rotating
cavitation was observed in the case of Inducer A. In
the case of Inducer B, only rotation frequency (60Hz)
b) Q*=1.00
Fig.10
Experimental results - FFT analysis results
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c) Q*=1.06
and low noise were observed, and there was no
evidence of rotating cavitation at all flow rates. In
the cases of Inducers C and D, the results showed
extremely low-level noise and no evidence of rotating
cavitation at all flow rates.
Discussion
From these experimental results, it is confirmed
that the design with no loading (or controlled loading)
at the leading edge using the 3-D inverse design
method is effective to suppress the inlet backflow of
the inducer. However, there is a trade-off relationship
between the inlet backflow characteristic and the
suction performance.
These results were obtained in the water model
tests, but the actual rocket inducer operates in liquid
hydrogen. It is well known that the thermodynamic
effect of liquid hydrogen suppresses cavitation
occurrence and improves suction performance
significantly. However, inlet backflow phenomena of
an inducer may reduce the thermodynamic effect and
the suction performance in reality.
In the case of Inducer C or D with suppressed inlet
backflow, it was confirmed that cavitation appeared in
the blade passages and reduced the suction
performance in water model tests. However, if the
thermodynamic effect of liquid hydrogen is to be
improved due to the suppression of inlet backflow, the
suction performance may be improved for Inducers C
and D under the actual environment of liquid hydrogen.
The effects of inlet backflow on the thermodynamic
effect need to be clarified in a future study.
CONCLUDING REMARKS
The 3-D inverse design method was applied for
designing highly loaded turbopump inducers for rocket
engine application.
The conventional inducer, which had been
designed for the LE-7A rocket engine turbopump, had
a strong inlet backflow at the design point. This inlet
backflow was eliminated by optimizing the blade
loading distribution using the 3-D inverse design
method and a CFD analysis.
Water model tests confirmed the elimination of
inlet backflow in the inverse design inducers.
Pressure oscillation was maintained at a very low level
and no evidence of rotating cavitation was observed.
However, the suppressed backflow tended to make
cavitation occur in the blade passages and reduced
suction performance. The value of inlet backflow
suppression design needs to be evaluated including its
effects on the thermodynamic effects of liquid
hydrogen.
REFERENCES
(1) Brennen, C., “Hydrodynamics of pumps”, 1994,
Concepts ETI, Inc., pp. 132-138.
(2) Jakobsen, J. K., 1971, “Liquid rocket engine
turbopump inducers”, NASA SP-8052, pp.17-24.
(3) He, W., 1997, “The Improvement of the
Cavitation Performance of a Cooling Water
Circulation Pump “, FEDSM97-3381.
(4) Singhal, A. K., Vaidya, N., Leonard, A. D., 1997,
“Multi-Dimensional Simulation of Cavitating
Flows Using a PDF Model for Phase Change”,
FEDSM97-3272.
(5) Zangeneh, M., 1991, “A Compressible Three
Dimensional Blade Design Method for Radial
and Mixed Flow Turbo-machinery Blades,” Int.
J. Numerical Methods in Fluids, Vol. 13, pp.
599-624.
(6) Zangeneh, M., Goto, A., and Harada, H., 1998,
“On the Design Criteria for Suppression of
Secondary Flows in Centrifugal and Mixed-Flow
Impellers,” ASME Journal of Turbomachinery,
Vol.120, pp723-735.
(7) Goto. A., and Zangeneh, M., 1998, “Hydrodynamic Design of Pump Diffuser Using Inverse
Design Method and CFD”, ASME FEDSM984854.
(8) Ashihara, K., Goto, A., 1999, ”Improvements of
Pump Suction Performance Using 3D Inverse
Design Method,” ASME FEDSM99-6846.
(9) Walker, P. J. and Dawes, W. N., 1990, “The
Extension and Application of Three-Dimensional
Time-Marching Analysis to Incompressible
Turbomachinery Flows,” ASME Journal of
Turbomachinery, Vol. 112, pp. 385-390.
(10) Ashihara, K., Goto, A., 2002, “Effects of Blade
Loading on Pump Inducer Performance and Flow
Fields,” ASME FEDSM2002-31201
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