Energy and Power Engineering, 2013, 5, 51-55
doi:10.4236/epe.2013.54B010 Published Online July 2013 (http://www.scirp.org/journal/epe)
Simulation of Unsteady Water Film Flow on Pelton Bucket
Shen Na
School of Electrical Engineering, Guangzhou College of South China University of Technology, Guangzhou, China
Received January, 2013
ABSTRACT
In order to simulate the complicated unsteady flow in Pelton bucket, it is necessary to apply the animated cartoon ap-
proach. In this paper, a free jet and the inner surface of a bucket is described by boundary fitted grid (BFG) with
non-orthogonal curvilinear coordinates. The water flow is discretized in space and time for CFD (computational fluid
dynamics). The moving grids of water film are successfully projected onto the bucket’s inner surface by a projection
algorithm. The visualization result of the jet landing on bucket’s surface and the unsteady flow in the rotating buckets in
3D verifies the effectiveness of the proposed method.
Keywords: Water Film Flow; Simulation; Projection; Pelton Bucket
1. Introduction
Different from the reaction turbines such as Francis and
Kaplan turbines, the flow in Pelton turbine is essentially
unsteady in space and time. This is the reason why the
application of CFD (computational fluid dynamics) to the
Pelton turbines is much behind other turbines. With the
continuous improvement of computer technology and
physical model, the numerical simulation of inner com-
plicated flow in Pelton turbine also becomes possible. In
1994, Kubota and Nakanishi classified the unsteady flow
in Pelton turbines for the study of the scale effect [1]. In
1998, Kubota et al. first proposed a numerical method
called “BucFlAs” which solves the unsteady quasi-2D
flow in Pelton bucket for different space/time steps indi-
vidually [2]. Liu et al. described the comparative nu-
merical analysis of the flow in buckets at different
enlarging rates of jet in 2D [3]. In 2005, Zheng, Han, et
al. discussed the unsteady interference between free wa-
ter jet and rear surface of rotating buckets in Pelton tur-
bine in quasi-3D[4].
So far, there is no paper reporting the unsteady water
film flow in rotating Pelton bucket in 3D. In this paper,
the animated cartoon approach is applied to simulate the
unsteady flow in Pelton bucket. The cross section of free
jet and the free curved surface is discretized by BFG. The
flow particles may depart from the bucket surface be-
cause of error in simulation, it is necessary to project
those particles in each moving step. A projection algo-
rithm is proposed in this paper to solve the problem, and
the example verifies the reliability of this method.
Through the simulation, the visual results of free jet
landing on the buckets and the unsteady flow water film
on bucket surface are obtained.
2. Discretization of Pelton Bucket and Free
Jet
2.1. Discretization of Complicated Bucket
Surface
The boundary of Pelton bucket consists of splitter, cutout,
end-wall and brim, respectively as shown in Figure 1.
Since all the boundary is curved, the orthogonal grids
cannot describe the surface of Pelton bucket accurately. It
is necessary to apply BFG with non-orthogonal curvilinear
coordinates. The natural basis vector and their partial dif-
ferential can be calculated accurately by differential ge-
ometry[5]. Those parameters are used in the algorithm of
projection which will be mentioned later for CFD.
Outer-brim
Splitter Cutout
End-wall
Inner-brim
Cutout-
b
otto
m
Side-
b
rim
*Project: Important Science and Technology Specific Projects in Zhe-
j
iang Province (Grant No. 2008C11057); Teaching Reform Research
Project in Guangzhou College of SCUT (Grant No. JY110319) Figure 1. Complicated curved surface of Pelton bucket.

S. NA
52
2.2. Discretization of the Cross Section for Free
Jet
Since the free jet is the unsteady flow with time-varying
and the radius is gradually increased with the radius ex-
pansion rate. In order to simulate the unsteady flow in
Pelton turbine, it is necessary to discrete the cross section
of free jet into 641 BFG considering the jet radius expan-
sion rate as shown in Figure 2, where R0 is the given
contraction radius of jet, and Rref is the reference radius
of the Pelton turbine. Every jet node is defined by two
parameters jR and k
. Each jet cell has the attributes of
four surrounding corner nodes.
2.3. Discretization of Time and Space by iF
The concept of analyzing the unsteady flow with the nu-
merical method of animated cartoon approach is the ap-
plication of the space-time identity number iF[6]. It dis-
cretizes, with a short time interval, both the space and
time into a series of consecutive momentary frames. In
order to simulate the unsteady flow in the rotating Pelton
bucket, in this paper, the spatial distance and time-step of
the two adjacent buckets were evenly discretized into
NdivF = 40 under the definition of iF = 0, where the splitter
tip of relevant bucket first touches to the inner-most
streamline of the free jet as depicted in Figure 3.
Z
II III IV
V VI
-Rref R0
jR
k

Y0
Figure 2. Discretization of cross section for free jet with
butterfly BFG.
Proceeding bucket
Relevant bucket
Relative path of splitter tip
Inner-most representative
stream line of free jet
Space-time discrete
origin i
F
=0
Free jet
Figure 3. Discretization of time and space by frame.
So, the elapsed space-time through each discrete frame
is
 tF=2/(  ·NB·NdivF) (1)
where Ω is the angular speed of the buckets, NB the
number of buckets, and NdivF the number of discrete
frames in a bucket pitch.
3. Projection of Arbitrary Position on
Curved Surface
After the free jet contacts the bucket, free jet will move
freely along curved surface. Therefore, the moving grids
are different from the boundary fitted grids of the bucket.
The moving grids flow along the curved surface, and
they may depart from the bucket surface due to the si-
mulation error, it is necessary to project moving grids
onto the bucket surface in each moving step.
So far, the curved surface of a bucket was discretized
into the nodes of BFG without the attribute of surface. In
order to find the arbitrary position of the curved surface,
however, the curved surface of the respective local panel
shall be defined as shown in Figure 4. The reliability of
the BFG was clarified by the differential geometry to
find the arbitrary position, in order to predict the un-
steady flow on the rotating bucket for CFD.
3.1. Projection of Spatial Point onto Curved
Panel
Let’s assume that a spatial point Ppre (=XRpre
’ieR’i) was
given near to the node-C of BFG in the Cartesian frame
as described in Figure 5. In order to find the position of
the point Ppre on the curved surface, it is necessary to
define the local curved surface of a panel by using the
nearest node-C.
First, Ppre was transformed from the Cartesian frame to
a natural frame at the nearest node-C as follows:
Ppre = Ppre
j gCj
= Ppre
1gC1 + Ppre 2gC2 + Ppre 3gC3 (2)
s
t
g
1
g
2
g
3
Figure 4. Local curved panel at each node.

S. NA 53
where gC1, gC2 and gC3 are the basis vectors in natural
frame.
The natural component ΔPpre
j was computed as a dis-
placement from PC to Ppre in the local natural frame as
shown in Figure 6:
Ppre
j ≈ (Ppre j/Xpre ’i)(XRpre ’i – XRC ’i)
= RC
j
’i (XRpre
’i – XRC ’i) (3)
where RC
j
’i is the matrix of transform tensor from natural
frame to Cartesian frame calculating by using the cofac-
tor C’i
j of RC
’i
j and the determinant det(RC
’i
j) of RC
’i
j as:
RC
j
’i = C’i j/det(RC
’i
j) = (-1)’i+jM’i j/det(RC
’i
j) (4)
where, M’i
j is the minor removed the row-j and the col-
umn-’i from RC
’i
j which is the matrix of transform tensor
from Cartesian frame to natural frame[5].
In (2), the first two terms in the right hand side showed
the projection of Ppre on the tangential flat panel
(gC1xgC2), and the last term was the depth from Ppre to the
projected point on the tangential flat panel as follows:
Ptan = Ppre
1gC1 + Ppre 2gC2 (5)
Pdep = Ppre
3gC3 (6)
The height ΔPhigh
3 from the P tan on the tangential flat
panel to the curved panel of bucket was predicted by the
differential geometry as shown in Figure 7 as:
Phigh = Phigh
3gC3 (7)
Phigh
3 = (1/2){LC(Ppre 1)2
+ (MC1+ MC2)Ppre
1 Ppre 2+ NC(Ppre 2)2}(8)
C
N
NW
WP
p
re
X
R
Y
R
Z
R
Figure 5. Ppre surrounded by four nodes including node-C.
X
R
Y
R
Z
R
C
N
NW
W
P
C
P
pre
P
pre
g
C1
g
C2
P
pre1
g
C1
P
pre2
g
C2
Figure 6. Ppre defined by natural frame at node-C.
where, LC, MC1, MC2 and NC can be computed by differ-
ential geometry[5].
In conclusion, the position ΔPpost on the curved panel
was determined as a point projected from ΔPpre onto the
tangential flat panel based on the node-C in the natural
frame as follows:
Ppost = Ptan + Phigh
= Ppre
1gC1 + Ppre 2gC2 + Phigh 3gC3 (9)
3.2. Verification of Prediction
In order to prove the reliability of this algorithm of pro-
jection, we make a comparison for the moving grids be-
tween projection and without projection. Figure 8 shows
the result of comparison at iF = 5. Figure 8(a) clearly
shows that some moving grids are departing from the
boundary surface of the bucket without using the projec-
tion. The water film flows along the boundary surface of
buckets, it is just close to the surface but not departing
from the surface. All the moving grids are close to the
curved surface of buckets after using the algorithm of
projection as shown in Figure 8(b). Therefore, this pro-
jection algorithm is necessary and reliable for the un-
steady simulation of flow particles in Pelton bucket.
4. Simulation Result
A Pelton turbine having the geometrical specific speed
B/Dref = 0.35 with 18 buckets was selected as the nu-
merical model for the unsteady flow investigation. Under
the optimum unit speed nDH = 40 rpm, the ratio of splitter
g
C1
g
C2
C
N
NW
W
P
tan
P
high

P
pre

P
post
Figure 7. Height from tangential flat panel to curved sur-
face.
(a) Before projection at iF = 5 (b) After projection at iF = 5
F -
jection at iF = 5.
igure 8. Comparison of water sheet before and after pro

S. NA
54
tip speed = 1.4. The expansion rate of jet radius was as-
sumed as k =
Rj 0.1%. Consider the cartoon frames be-
tween two adjacent buckets NdivF = 40, discretization
number along jR direction NdivR0 = 40, and discretization
number along k
direction Ndivθ0 = 20.
4.1. Visualization of Landing Jet on Inner
The anding on bucket surface real-
Surface of Bucket
block diagram of jet l
ized by CFD was shown in Figure 9.
By using the algorithm in the above block diagram, the
jet nodes landing on bucket surface were simulated nu-
merically in 3D as shown in Figure 10 for different
frames.
At iF = 0 where is the origin of the discretization in
space and time, outer margin of the free jet just touch the
splitter tip of the bucket. Since the shape of the cutout is
complicated, there is a little water entering the bucket.
Free jet entering the bucket is increasing with iF in-
creased, until the whole jet enters the bucket. At iF = 40,
the whole bucket has already entered the rotating bucket.
After iF = 40, the following bucket starts touching and
intercepting the free jet. From iF = 90, it is no longer the
whole jet entering the relevant bucket. The jet entering
the bucket is less and less with the continuous increasing
of iF. From iF = 128, there is no more jet entering the
bucket. The visualization of numerical results meets the
flow phenomena in Pelton bucket; and they are very
useful to study the mysterious flow in Pelton bucket.
Prepare position of jet
nodes at contraction
position
Get jet nodes at X = X
st
cross section with
considering k
Rj
Find landing jet nodes on
bucket surface
Find OMB
alon
g
cutout
Retransform landing
jet nodes from
stationary to runner
Calculate relative
velocity W at each
landing jet node
Find trailing edge
surface of flow-off jet
including IMB at each
frame
End landing jet on
bucket surface ()
Prepare discharge dQ of
each jet cell at contraction
position
Start landing jet
i
F
= 0 i
F
= 20
i
F
= 40 i
F
= 60
i
F
= 80
i
F
= 100
i
F
= 120
i
F
= 127
Figure 10. Landing jet on surface at different frames.
4.2. Visualization of Water Film on Rotating
Bucket
According to the numerical setting up above, the simula-
tion result of the water film flowing on the rotating
bucket is obtained. As shown in Figure 11, the free jet is
increasing with iF increased. The water film in the rotat-
ing bucket flows towards to the out-brim of the bucket
orderly. This simulation result can supply the powerful
technical support for the optimization design of the Pel
ton bucket.
5. Conclusions
The complicated unsteady flow is simulated by the ani-
mated cartoon approach, and we can draw out followin
conclusions:
1) The curved surface of a bucket is described by BFG
with non-orthogonal curvilinear coordinates and the
cross section of the free jet is discrete with butterfly BFG
for CFD.
2) The moving grids are successfully projected onto
the curved bucket’s surface by a projection algorithm
pro
3) The visualization result of the jet landing on buck-
kets
on bucket
surface () start
Transform bucket from
runner frame to stationary
frame
-
g
posed in the paper.
Figure 9. Block diagram for finding landing jet on surface.
et’s surface and the unsteady flow in the rotating buc
in 3D verifies the effectiveness of the proposed

S. NA
55
technical support for the optimization design of the Pel-
ton bucket.
REFERENCES
[1] T. Kubota and Y. Nakanishi, “Classification of Flow in
Pelton Turbines for Study of Scale Effect,”17th
IAHR-Symposium, Beijing, Vol. G-6, 1994, pp. 865-876. iF = 1 iF = 2
[2] T. Kubota, J. Xia, H. Takeuchi, et al., “Numerical Analy-
sis of Free Water Sheet Flow on Pelton Buckets,” Pro-
ceedings of 19th IAHR Symposium, Singapore, 1998, pp.
316-329.
[3] J. Liu, C. X. Wei and F. Q. Han, “Effect of Enlarged Free
Jet on Energy Conversion in Pelton Turbine,” Journal of
Hydrodynamics, Series B, Vol. 18, No. 2, 2006, pp.
211-218.
iF = 3 iF = 4 [4] A. L. Zheng, F. Q. Han, Y. X. Xiao, et al., “Unsteady
Interference between Jet and Rear Surface of Rotating
Buckets in Pelton Turbine,” 8th Asian International Fluid
Machinery Conference, Yichang, China, 2005, pp.
302-310.
[5] N. Shen and T. Kubota, “Curved Surface of Pelton Buck-
et Based on Differential Geometry,”APPEEC2012,
Shanghai, Vol. 3, 2012.
I
F = 5 iF = 6
Figure 11. Unsteady water film at different fram
[6] F. Q. Han, N. Shen, L. X. Li, et al., “Unsteady
Separation of Jet Branch by Cutout of Rotating
Pelton Bucket,” SCIENCE CHINE Technological
Science, Vol. 54, No. 2, 2011, pp. 302-310.
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es.
method. The simulation result can supply the powerful