Energy and Power En gi neering, 2011, 3, 29-33
doi:10.4236/epe.2011.31005 Published Online February 2011 (http://www.SciRP.org/journal/epe)
Copyright © 2011 SciRes. EPE
Predication of 3-D Viscous Flowfield of a Centrifugal
Impeller
Limin Gao, Xudong Feng, Jian Xie
School of Power and Energy, Northwestern Polytechnical University, Xi’an, China
E-mail: gaolm@nwpu.edu.cn
Received September 10, 2010; revised November 5, 2010; accepted November 15, 2010
Abstract
A three-dimensional viscous code has been developed to solve Reynolds-averaged Navier-Stokes equations.
The governing equations in finite volume form are solved by two-step Runge-Kutta scheme with implicit
residual smoothing. The eddy viscous is obtained using the Baldwin-Lomax model. A prediction of the 3-D
turbulent flow and the performance in the “all-over controlled vortex distribution” centrifugal impeller with a
vaneless diffuser has been made for the compressor at design and off-design condition. The predicted effi-
ciency is a little higher than the experiment data. These results suggest that the present calculation code is
able to determine the flow development in the impeller and also the turbulence model in the centrifugal im-
peller should be improved.
Keywords: Centrifugal Impeller, Aerodynamic Performance, 3-D Viscous Flow Calculation, Design &
off-Design Conditions
1. Introduction
Centrifugal compressors are used widely in industry due
to their advantages of simple structure and high-pressure
ratio. However, their efficiency and stability are ad-
versely influenced by the present of impeller exit flow
non-uniformity. In recent years, as a result of improve-
ments in experimental techniques and numerical methods,
it has been possible to avoid the non-uniformity of exit
flow field in order to obtain improved efficiency and
stability of an impeller. The work of Eckardt’s [1] and
Krain’s [2] are most representative in all related experi-
mental research. Their studies indicated that both com-
plicated secondary flows and separated boundary layer
would cause the radial and circumferential non-uniform
flows at the outlet, and consequently the performance of
the centrifugal compressor decreases. Meanwhile, some
numerical codes have also been developed to study the
flowfield in centrifugal compressor by other researchers.
Most of them [2-4] have provided the detailed flowfield
to understand the complicated flow. However, CFD is
still a little immature and does not reach the engineering
implementation level. Further research is necessary to
develop a more accurate and faster, numerical predicting
code, which will provide the sophisticated tool to predict
the aerodynamic performance of a high-speed centrifugal
compressor for designing an impeller with higher pres-
sure-ratio and efficiency.
In the present work, a 3-D turbulent code has been
developed, and the prediction of 3-D viscous flowfield of
an AOCV centrifugal impeller has been carried out.
Therefore, it is hoped that the present study will provide
a useful predication tool to aid in the future experimental
work and the industrial design.
2. AOCV Centrifugal Impeller
The air compressor, which is applied in a large-scale
air-separation plant, is produced from SER Turboma-
chinery Research Center of the Xi’an JiaoTong Univer-
sity. Its impeller is a three-dimensional centrifugal im-
peller which is design using the All-Over Controlled
Vortex Distribution designing theory.
The impeller is shrouded and has 19 full blades with
an exit backswept of 50 degs. The inlet diameter is 0.222
m and the inlet blade height is 0.0606m. The exit diame-
ter is 0.340 m and the exit blade height is 0.0272 m. The
design speed of 16360 rpm gives a rotor exit tip speed of
290 m/s. The design mass flow rate is 2.8 m3/s at the
standard inlet conditions of 101, 325 N/m2 and 288.15 K.
The rotor tip Reynolds number (U2D2/ν) is 1.42 × 106.
The impeller is showed in Figure 1.
L. M. GAO ET AL.
30
Figure 1. AOCV Impeller.
3. Computational Method
3.1. Governing Equations
It is convenient to write the three–dimensional Reynolds
averaged Navier-Stokes equations in finite volume form
cast in the blade-relative frame using cylindrical coordi-
nates (r, θ, x):

IV I
UdVolHHndSG dVol
t
 
 

 
(1)
0
zz
z
IV
rr
rr
W
WW pe
W
UrWH rWWrpeHre
We
WW pe
EW
IW





 




 





 

 



 

z
e
 
2
2
0
0
2
2
Ir
GrW
Wp
rW
r
Tq


 

r

ˆ
with ˆˆ
x
x
qWi WiWi


rr
, the relative velocity; =
rotation speed;

= the stress tensor (containing both
the static pressure and the viscous stresses); and
22
porel
ICT r, the rothalpy. The system is closed
by an equatio
and a mixing
length turbulence model patterned after Baldwin-Lomax.
n state

.PE


of
10
r

2
5qq

The code solves the N-S equation in an integral con-
versation form using hexahedral control volumes formed
by a simple H-mesh. The integrals in the conservation
equations are replaced by discrete summation around the
faces of the computational cell, which is expressed as
follows:
 
,,
,,
,, ,,,,
ijk ijk
IVijk ijkijk
cell
UVol
t
HHArea SVolRU
 

(2)
R
in the above Equation (2) is called as the residual
on each 3-D grid element (i, j, k). Fluxes through cell
faces are calculated by linear interpolation of the density,
velocity, etc., between neighbor cell centers. Thus the
formal spatial accuracy is second order on the smoothly
varying meshes and global conversation is ensured. A
combined second and fourth derivative adaptive artificial
viscosity model with pressure gradient switching is
added to the discretized equations to eliminate spurious
“wiggles” and to control shock capturing. The basic al-
gorithm as described by Dawes (1988) [5] is similar to a
two-step Runge-Kutta method plus residual smoothing:
  
  

11
222
2
;
;
nn
ijk
n
ijk
n
t
URUU
VOL
t
URUU
VOL
UU
 
 
1
1
U
U
(3)
In order to improve the computational efficiency, the
discretized equations are solved using accelerating tech-
niques, such as the local-time stepping and multigrid
methods.
3.2. Boundary Condition
At inlet, total temperature and total pressure are fixed
and either flow angle or absolute swirl velocity held con-
stant depending on whether the relative flow is sonic or
supersonic. At outlet the hub static pressure is fixed. The
physical periodic condition upstream and downstream of
the impeller is imposed. No slip, no flux and adiabatic
wall conditions (or wall temperature) are applied on the
solid wall. Additionally, the derivative of the pressure
normal to the wall surface is set to zero.
3.3. Grid for Computation
There are 89987 calculation points with 29 circumferen-
tial points, 29 radial points and 107 streamwise points.
To capture the boundary layer, the grids are designed
clustering near the wall. The inlet for the 3-D flow cal-
culation was 0.1265 m upstream of the impeller inlet.
The exit radius of the vaneless diffuser for the calcula-
tion has a ratio of r3/r2 = 1.294 to the impeller outlet.
The computing grids are showed in Figure 2.
Copyright © 2011 SciRes. EPE
L. M. GAO ET AL.
Copyright © 2011 SciRes. EPE
31
ing requirements are satisfied: 1) the mass flow error
throughout the domain was deemed acceptable and flow
mass in inlet and outlet are in accordance, 2) both
RMSM (maximum computing error, which is expressed
as Equation (4)) and RMS (summation of computing
error, which is expressed as Equation (5)) through all
computing points decrease three or four orders; 3) the
flow variables (velocity or pressure) at the arbitrary
computational point is monitored to insure that the ve-
locity field was not changing.
4. Results of 3-D Prediction
The predicated meridional velocity vectors on three sur-
faces with three tangential positions are presented in
Figure 2. There is no backflow at the diffuser exit and
no separation in the shroud boundary layer all through
the compressor. Flow near the pressure side of the im-
peller passage is shown in Figure 3(a). A secondary
flow from the hub to shroud becomes evident in the sec-
ond half of the passage due to curvature of the passage
from the axial to radial direction. At mid-passage, Fig-
ure 3(b) shows the shroud boundary layer is thickening
in impeller and velocity vectors are nearly parallel to the
shroud and the hub. Near the suction side of the impeller,
centrifugal and curvature effects combine to give strong
secondary flows towards the shroud. This is especially
true in the second half of the passage, where flow angles
as high as 30 degs to the meridional direction is seen in
Figure 3(c). Clearly the meridional velocity near the
suction surface is larger than near the pressure surface.
(a)
Distribution of the static pressure on the mid-pitch
meridional surface through the impeller is presented on
the Figure 4. The value on the pressure contours is the
ratio of the local static pressure to the inlet total pressure.
From the Figure 4, it can be seen that the static pressure
is gradually increased and that the distribution of the
static pressure is uniform comparatively on the same
chord section.
(b)
Figure 2. Grid for 3-D flow computation, (a) Meridional
view; (b) Blade-to-blade plane near impeller leading edge. The measured and calculated total/total polytropic
impeller efficiency is presented on Figure 5. In the cal-
culation, the centrifugal impeller works on the design
and the off-design conditions at 100 percent design speed
respectively, totally eight work-conditions. According to
3.4. Convergence
Computing convergence is determined when the follow-
2
22
22
,,
zr
ijk
V
VV e
RMSM Maxtttt t







 


 



 
 



(4)
2
22
22
,,
,, 1,,
**
IM JM KMzr
ijk ijk
V
VV e
tttt t
RMS IM JM KM







 


 



 
 



(5)
L. M. GAO ET AL.
32
(a)
(b)
Figure 4. static pressure contour on the mid-pitch surface.
Figure 5. performance map of the AOCV centrifugal im-
peller.
cient
the experiment, the y-axis is the impeller polytropic effi-

**
22
**
22
lg
1
lg
PP
TT
 , and the x-axis is defined as
the mass rate coefficient

10 2
22
10
in
QQ Dnb
polytropic efficiency is 90.125%
. The
measured maximum,
correspondingly the mass rate coefficient is 0.546; the
predicted maximum efficiency is 91.533% and the mass
coefficient is 0.6960. At the same time, the measured
minimum polytropic efficiency is 87.125%, correspond-
ingly the mass rate coefficient is 0.678, the predicted
minimum efficiency is 84.209% and the mass coefficient
is 0.833. From the Figure 5, the predicted efficiency is a
little higher than that measured in experiment. The 3-D
flow calculation with the time-marching method gives
(c)
Figure 3. Meridional views of velocity vectors.
Copyright © 2011 SciRes. EPE
L. M. GAO ET AL.33
the broader work condition than the measurement.
5. Conclusion
A three-dimensional viscous code has been developed to
solve Reynolds-averaged Navier-Stokes equations in
finite volume form with time-marching method and the
eddy viscous is obtained using the Baldwin-Lomax
model. A prediction of the three-dimensional turbulent
flow and the performance in the AOCV centrifugal im-
peller with a vaneless diffuser has been made for the
compressor design and off-design condition. The pre-
dicted efficiency is a little higher than that tested in ex-
periment. The presented results suggest that the present
calculation code is able to determine the flow develop-
ment in the impeller and the turbulence model in the
centrifugal impeller should be improved.
erences
entrifugal Compressor I
] W. N. Dawes, “A Numerical Analysis of the Three-Di-
cous Flow in a Transonic Compressor Ro-
rison with Experiment,” ASME Journal of
6. Ref
[1] D. Eckardt, “Detailed Flow Investigations within a High
Speed Cmpeller,” Transaction of
the ASME, Journal of Fluids Engineering, Vol. 98, No.
3390, 1976, pp. 390-402. doi:10.1115/1.3448334
[2] H. Krain and W. Hoffman, “Verification of an Impeller
Design by Laser Measurements and 3D-Viscous Flow
Calculations,” ASME Paper, No. 89-GT-159, 1989.
[3] C. Hah and H. Krain, “Second Flows and Vertex Motion
in a High-Efficiency Backswept Impeller at Design and
off-Design Condition,” ASME Journal of Turbomachin-
ery, Vol. 112, No. 1, 1, 1990, pp. 7-13. doi:10.1115/1.29
27425
[4] R. Kunz and B. Laskshminarayana, “Navier-Stokes In-
vestigation of a Transonic Centrifugal Compressor Stage
Using an Algebraic Reynolds Stress Model,” AIAA Pa-
per, 1992, pp. 92-3311.
[5
mensional Vis
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Turbomachienry, Vol. 109, No. 1, 1987, pp. 83-109. doi:
10.1115/1.3262074
Copyright © 2011 SciRes. EPE