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OCR for page 774
Experimental and CFD Analysis for Rotor-Stator
Interaction of a Waterjet Pump
H. H. Chun, W. G. Park, and J. G. Jun
(Pusan National University, Korea)
ABSTRACT
The numerical analysis of a waterjet propulsion
system was performed to provide a detail
understanding of complicated three-dimensional
viscous flow phenomena including the
interactions of intake duct, rotor, stator, and
contracted discharge nozzle. The incompressible
RAN S equations were solved on a moving, non-
orthogonal body-fitted multiblocked grid system.
To handle interface boundary between the rotor
and the stator, the sliding multiblock technique
using cubic spline and bilinear interpolation
method were applied. The numerical pressure
distribution on the intake duct surface was
compared with experiment and a good agreement
was obtained. The complex viscous flow feature
of the waterjet such as the secondary flow inside
of the intake duct, the recovery of axial flow by
the action of the stator, and tip vortex, etc. were
well understood by the figures of pressure
contours, velocity vectors, and streamlines.
INTRODUCTION
The waterjet propulsion system is widely used to
thrust high speed marine vessels because of its
capability of eliminating the cavitation problem,
which is done by increasing the static pressure at
impeller face above the vapor pressure through
diffusing the inlet duct. The waterjet propulsion
system has been also used for military tracked
vehicle to cross the river. The most of researches
on the waterjet propulsion has been carried out by
experiments and the computational work has been
rarely used. But, recently, the application of CFD
is continuously increased by virtue of the
advancement of numerical algorithms and
computer hardware. By the way, CFD works are
mostly devoted to the intake duct of waterjet
(Watson, 1998; Roberts, 1998), but, never
applied to the full system of waterjet propulsion
that includes all components of intake duct, rotor,
stator, and discharge nozzle.
The objective of present work is to apply the
OCR for page 775
Navier-Stokes equations to the full configuration
of the waterjet propulsion system. To the authors'
knowledge, the present work is the first attempt to
solve RAN S equations to analyze the flowfield of
the waterjet with interaction of the rotor and stator.
Especially, the case of a waterjet for the tracked
vehicle is a more obvious fact.
GOVERNING EQUATIONS
The three-dimensional unsteady incompressible
Navier-Stokes equations in a generalized
curvilinear coordinate system (:,t,q,() may be
written in a non-dimensional form as follows:
aq a (E-E ) a (F-F )
where q = tp, u, v, w]/J and E, F. G are the
convective flux terms, EV,Fv,Gv are viscous
flux terms. Equation (1) is solved by so called
iterative time marching method (Park and Sankar,
1993~. The solution procedure is briefly
reproduced here. First, let us consider the
momentum equation only. Since the momentum
equation is a parabolic type of partial differential
equation, it can be solved by the time marching
scheme as follows:
+ a (G-Gv )= 0
~ (qn+l _ qn )+ ~`En+l + ~O~Fn+l +'~`Gn+l
= &:,EVn+~ + 611 FVn+i + &`GVn+i (2)
The barred quantities denote the column vector
matrices consisted of momentum equations only.
The superscript denotes the physical time level.
The operators, 6`, Its and 6;represent spatial
differences in (-, A-, and (-direction, respectively.
If the Newton iteration method is applied to
efficiently solve the unsteady flow problem,
Equation (2) is rewritten as follows:
~ (qn+l~k+1 _ qn )+ o~En+l k+1 + ~ Fn+l'k+1
+~`Gn+l,k+1 = '~E n+l,k+1
+6l~Fv +l,k+1 +~`G n+l,k+1 (3)
where the second superscript, 'k+1', means the
iteration level. Following a local linearization of
E, F. G. Ev, Fv and Gv about the 'n+1' time
level and at the 'k' iteration level,
(FIT + BAA+ am B+ a; C)5q = t0R (4)
where CO is a relaxation factor. A, B and C are
the Jacobian matrices of the flux vectors
E-EV, F-FV, and G-GV, respectively.
Rn+17k is the residual vector defined as:
Rn+l,k _ q q
—(Gil E n+l,k + ~ F n+l,k + .~ G n+l,k )
+(°˘EV + 611 Fv + 6; G v ) (5)
Note that RHS of Equation (5) is the same form
of discretized momentum equations, Equation (3),
OCR for page 776
at 'k' iteration level. When R n+~'k goes to zero,
the momentum equations in their discretized form
are exactly satisfied at each physical time step.
Then, the solution is independent of Go, and any
approximations made in the construction of A, B.
and C. Next, let's consider the continuity equation.
In order to solve incompressible flow efficiently,
we need a relationship coupling changes in the
velocity field with changes in the pressure field
while satisfying the divergence-free constraint. In
the present study, the Marker-and-Cell(MAC)
approach (Viecelli, 1969) is used.
(I) [act ~ ) al ~ )
a ~ w (~ 11
where A(p/J)= (p/J)n+~ k+~ _ (p/J)n+~ k and ~ is a
relaxation factor. Again, when /\p goes to zero,
the continuity equation is exactly satisfied at each
time step. The spatial derivatives of convective
flux terms are differenced by using third order
accurate upwind QUICK scheme(Leonard, 1979)
to reduce unphysical oscillations for high
Reynolds number flows, and the spatial
derivatives of viscous terms and continuity
equation are differenced with central differencing.
The fourth order artificial damping term is added
to the continuity equation only. To capture the
turbulent flows, low Reynolds number k-e model
(Chien, 1982) is employed.
INITIAL AND BOUNDARY CONDITIONS
The governing equations are always solved in the
inertial frame. The use of inertial frame simplifies
the governing equations because centrifugal and
Coriolis forces do not appear explicitly. This
approach is suitable for rotating blade or
turbomachinery. Equation (1) requires initial
conditions to start the calculation as well as
boundary conditions at every time step. In present
work, it is assumed that the rotor is impulsively
started from rest. Thus, the rest condition is used
as initial condition and farfield boundary
condition. At outflow boundary, p=pOO is imposed
and the velocity is extrapolated from interior
nodes to account for the removal of vorticity from
the flow domain by convective processes. On the
body surface, the no slip condition is applied for
velocity components. The surface pressure is
determined by setting the zero normal pressure
gradient of pressure. To handle interface boundary
of block between the rotor and the stator, the
sliding multiblock technique using cubic spline
interpolation (2D interface) and bilinear interpola-
tion (3Dinterface) technique were applied. The
each grid of multiblocks is generated by the
elliptic grid generator.
RESULTS AND DISCUSSION
Before the flow analysis of waterjet propulsion
system, 3D cascade flows with rotor-stator
interaction are simulated to validate present code
and to test interpolation algorithm used for
interface boundary of rotor-stator. Figure 1 and 2
show H-grid system embedded by O-grid near the
blade of rotor and stator. The turbulent flows were
modeled by the Baldwin-Lomax model and Low
Reynolds number k-e model. The calculations
were performed at three rotating speeds of
Cx /{J=0.68, Cx /U=0.78, and Cx /U=0.96
OCR for page 777
where Cx is absolute inflow speed and U is
rotating speed of the rotor. Figure 3 shows time
averaged pressure coefficients of the rotor
compared with experiment (Dring, et al., 1982~.
This figure shows a good agreement with each
other. Figure 4 shows the formation of tip vortex.
1 1
., o
Figure 1: Rotor-stator grid system of
cascade
(a) Leading edge of rotor
(b) Trailing edge of rotor
Figure 2: Embedded O-grid system
|—Larnir~r ~ ~
~ BalahNin-Lorr~x I ')
— -- LowRe. k-e I $
it'
0 0.5 1
XIC
(a) Cx/U= 0.68
, ,
~ ~ . ~ ~
1 —
·-E~. ~
Larninar ·
~ Baldwin-Lornax ~
— - Low Re. k-e .;
_ ::-
WK
I_
0 0.5 1
X/C
(b) Cx/U= 0.78
1 t
o ~
fat
o
..
|—Laminar
~ Baldwin-Lomax
— - Low Re. k-e
If.
,::: - '
~ =,Nf
0.5 1
XIC
(c) Cx/U= 0.96
Figure 3: Time averaged Cp of a rotor at three
different rotational speeds.
OCR for page 778
-
- ~
- ~
- ~
63x25x35 and 55x21x35, respectively. The
number of grid points of tip clearance is 69x8x8.
The discharge nozzle has the grid points of
20x61x29.
Figure 5: Configuration of waterjet propulsor
Figure 4: Tip vortex formation
After the code validation, the present iterative
time marching procedure has been applied to the
flow within the waterjet propulsion system which
consists of four rotors and nine stators as shown
in Figure 5, which was previously experimented
with a 1/5 scale model (shun, 2001) in a towing
tank of Pusan National University, Korea,
followed by the ITTC 1996 standard test method
for waterj et. The computation has been performed
at 4.3x105 of Reynolds number based on the
resultant velocity and chord length at r=0.7R. The
mean inflow velocity through the intake duct is
1.2 m/s and the rotor rotates at 4000 rpm. The
grid was elliptically generated as shown in Figure
6. The number of grid points of intake duct is
60x30x60. The grids of rotor and stator have
(a) Grid of intake duct
(b) Grid of impeller and nozzle
Figure 6: Grid system of the waterjet
Figure 7 shows the velocity vectors and pressure
contours along the symmetry plane and shows
OCR for page 779
non-uniform velocity and pressure distribution at
exit of intake duct, that is, at the face plane of
impeller rotor. This non-uniformity affects
significantly the performance of waterj et. Figure 9
shows the cross-sectional pressure contours at
designated downstream locations of Figure 8. The
streamlines given by Figure 9 show the secondary
flow of vortex that is formed by the change of
cross-section shape from rectangle to circle and,
consequently, the variation of cross-sectional
pressure distribution along the streamwise
direction.
(a) Velocity vectors
(b) Pressure contours
Figure 7: Velocity and pressure contours along
the symmetry plane
3 4
~ ,
Figure 8: Locations of designated cross-section
Level p
18 0.68
17 0.36
16 0.04
15 .0.28
14 -1.24
13 -1 47
10 1.89
9 -2.21
6 245
4 -2.85
3 -3.17
2 -3.49
1 -3.81
Level p
18 0.68
17 0.36
16 0.04
15 ·0.28
14 -1.24
13 -1.47
12 -1.57
11 -1.69
10 -1.89
9 -2.21
8 -2.34
7 -2.41
6 -2.45
5 -2.s3
4 -2.85
3 -3.17
2 -3.49
1 -3.81
.
(a) at cross-section 1-1
(b) at cross-section 2-2
Figure 9: Pressure contours at several cross-
section
OCR for page 780
Level p
18 0.68
17 0.36
16 0.04
1 5 4~.28
14 -1.24
1 3 -1.47
1 2 -1.57
11 1.69
10 -1 .89
9 -2.21
8 -2.34
7 -2.41
6 -2.45
5 -2.53
4 -2.85
3 -3.17
2 -3.49
1 -3.81
p
18 0.68
17 0.36
16 0.04
5 41.28
14 -1.24
13 -1.47
12 -1.57
1 1 -1.69
10 -1.89
9 -2.21
8 -2.34
7 -2.41
6 -2.45
5 -2.53
4 -2.85
3 a3.17
2 -3.49
1 -3.81
(c) at cross-section 3-3
interfaces may become one way to check the
appropriateness of interpolation algorithm on the
block interfaces.
(d) at cross-section 4-4
Figure 9: continued.
Figure l l (a) shows the surface pressure along the
ramp side of the duct, compared with experiment
measured at the position of Figure 1 labs. A fairly
good agreement between the present calculation
and experiment is obtained. To illustrate the
smoothness of flow properties across the block
interface boundary of sliding multiblock
technique, the velocity vectors in the body-fixed
frame, streamlines in the inertial frame, and
pressure contours across the block interface
boundary are drawn in Figure 12. This
smoothness of flow properties across block
(a) Side view
(b) Bottom view
(c) Top view
Figure 10: Streamlines within the intake duct
OCR for page 781
:`
f
_.- oB 16 2.4 SZ ~ ~
Surface distance from start of intake ramp ~
(a) Surface pressure distribution along the ramp
~ \ 1
E
(b) Locations of pressure tabs
Figure 11: Surface pressure, compared with
experiment (shun, 2001 3
(a) Velocity vectors in the body-fixed frame
Figure 12: Velocity vectors, streamlines, and
pressure contours across the block
interface boundary.
(b) Streamlines in the inertial frame
(c) Pressure contours
Figure 12: continued.
Figure 13 and 14 show the limiting streamlines
and pressure contours on the suction and pressure
side of the rotor, respectively. Figure 15 and 16
show those of the stator. Figure 17 shows time
averaged surface pressure distribution on the rotor
and stator surface, where the pressure coefficient,
Cp, is defined as Cp = (p _ pOO )/(pv2 / 2~.
OCR for page 782
(a) Limiting streamlines
(a) Limiting streamlines
(b) Pressure contours
Figure 13: On the suction side of the rotor
(b) Pressure contours
Figure 15: On the suction side of the stator
(b) Pressure contours
Figure 14: On the pressure side of the rotor
(b) Pressure contours
Figure 16: On the pressure side of the stator
OCR for page 783
Here, PTE denotes the static pressure at the
trailing edge and PT denotes the total pressure
at the stagnation point. The chordwise pressure of
the rotor is plotted at r/R=0.4, 0.7, O.9S as shown
in Figure 17(a) and pressure of the stator is
plotted at r/R=0.6, 0.8, 0.95 as shown in Figure
17(b). In Figure 17(a), the tendency of lowering
pressure toward trailing edge is due to the
decrease of area of the through-flow as shown in
Figure ll~b). The increase of pressure toward
trailing edge of the stator as shown in Figure
17(b) is come from the fact that the pressure near
the trailing edge is affected by high pressure in
the bottleneck of the discharge nozzle.
Q
0.3 1 1 1 1 1 __
0.15
I ~13~ r/R-0 7 1 it_
_
O _
-
-0.3 _
-0.45
~-
1 1 1 1
-0.15 0 0.15 0.3 0.45 0.6 0.75 0.9 1.05
XIC
(a) Of the rotor
12
10
8
6
4 _
~ ~ 1
0 O rlR=0.6 |
~ ~ r/R=0.8 1
o
-2
1 1 1 1 1 1 1
-0.15 0 0.15 0.3 0.45 0.6 0.75 0.9 1.05
(b) Of the stator
Figure 17: Time averaged surface pressure
distribution of the rotor and the stato23
0 10 20 30 40 50 60 70 80 90
Azimuthal angle(deg.)
(a) at r/R=0.36
3~
-
O 2
O
ID
s"
E -1
. _
A -2
-3
i,, 3
. _
0 2
-
E -1
as -2
..............................................................................................................................
a a vx
i 0 0 Vi
"~ ~ 0 0 v.
A_'
0 10 20 30 40 50 60 70 80 90
Azimuthal angle(deg.)
(b) at r/R=0.66
0 10 20 30 40 50 60 70 80 90
~ _ .. . . . . .
(c) at r/R=0.95
Figure 18: Velocity components before the face of
rotor impeller
OCR for page 784
4
3
-
0 2
o o
E -1
._
0 -2
z
-3 —
4
3
:-
0 2
-
t15 1
o
(,, O
E -1
0 -2
.
1 ..
0 10 20 30 40 50 60 70 80 90
Azimuthal angle(deg.)
(a) at r/R=0.62
l : ~=~
,,
0 10 20 30 40 50 60 70 80 90
Azimuthal angle(deg.)
(b) at r/R=0.80
4
. ~
j 3 1 1
.... .~
l
-3
.............................................. ~
A ~
0 10 20 30 40 50 60 70 80 90
. . ... . . . . , . ~
(c) at r/R=0.96
Figure 19: Velocity components after the trailing
edge of rotor impeller
4
o 3
:-
0 2
.o o
E -1
._
z -2
u, 3
:-
0 2
tts 1
o
u' o
E -1
. _
z -2
0 4 8 12 16 20 24 28 32 36 40
Azimuthal angle(deg.)
(b) at r/R=0.76
tn
~, 3
-
._
0 2
1
O
E -1
._
0 -2
-3
~~ ~
,
0 4 8 12 16 20 24 28 32 36 40
Azimuthal angle(deg.)
(a) at r/R=0.56
;~.~ N
~ .....
. ...
.. O O vx
O O v~
. v,
0 4 8 12 16 20 24 28 32 36 40
. .. . . , . ~
(c) at r/R=0.95
Figure 20: Velocity components after the trailing
edge of stator
OCR for page 785
This high pressure in the bottleneck can be
identified by Figure 21-23. Figure 18-20 show the
axial, redial, and tangential velocity component of
cross-sectional plane before and after the rotor,
and after the stator. These figures indicate that
nearly uniform inflow toward the rotor impeller
(Figure 18) attains to high tangential component
by the rotation of rotor (Figure 19) and, then,
recovers nearly no-tangential component flow
(Figure 20~. The negative radial component of
Figure 20 is due to the contraction of cross-
sectional area near the trailing edge of stator. The
surface pressure distribution on the rotor, stator,
and hub is shown in Figure 21. Figure 22 shows
the velocity vectors and pressure contours in the
symmetry plane after the trailing edge of stator,
i.e., in the discharge nozzle. In Figure 22(b), the
pressure continuously decreasing makes the flow
accelerate strongly to discharge from nozzle exit,
having high momentum (i.e., high thrust). Figure
23 shows the pressure distribution in the stator
and discharge nozzle along r/R=0.5 plane. Figure
24 and 25 show the streamlines past the rotor and
the stator, respectively.
(a) Velocity vectors
Levi p
· 18 10.00
17 8.75
16 7.50
'' 15 6.25
14 5.00
2 3.7S
9 0.00
1~1 8 -1.25
~ 7 -2.50
@~ 6 .375
5 -5.00
4 -6.25
3 -7.50
· 2 -8.75
· 1 -10.00
(b) Pressure contours
Figure 22: Velocity and pressure in the symmetry
plane of discharge nozzle
Figure 21: Surface pressure distribution on the
rotor, stator, and hub surface
Figure 23: Pressure contours of stator and nozzle
along r/R=0.5 plane
OCR for page 786
(a) Front oblique view
(a) Side view
(b) Side view
(b) Top view
(c) Near the rotor blade
Figure 24: Streamlines past the rotor
(c) Rear oblique view
Figure 25: Streamlines past the stator
OCR for page 787
2001. (in Korean)
Dring, R.P., Joslyn, H.D., Hardin, L.W., and
Wagner, J.H.,"Turbine Rotor-Stator Interaction,"
Journal of Engineering for Power, Vol.4, 1982.
Leonard, B.P.,"A Stable and Accurate Convective
Modeling Procedure Based on Quadratic
Upstream Interpolation," Computer Methods in
Applied Mechanics and Engineering Vol.19
, ,
1979.
CONCLUSION
The numerical analysis of a waterjet propulsion
system was performed to provide a detail
understanding of complicated three-dimensional
viscous flow phenomena including rotor and
stator interaction and contracted discharge nozzle.
The incompressible RAN S equations were solved
on a moving, non-orthogonal body-f~tted
multiblocked grid system. To handle interface
boundary between the rotor and the stator, the
sliding multiblock technique using cubic spline
and bilinear interpolation method were applied.
To validate the present code, 3D cascade flows
with rotor-stator interaction were simulated and,
then, compared with experiments. Good
agreement with experiment has been obtained. It
has been also shown that the pressure and velocity
distribution at the face of rotor impeller are not
uniform because of curved intake duct. This
inflow after passing the rotor attains to high
tangential component by the rotation of rotor and,
then, recovers nearly non-tangential component
flow again by the action of stator. The streamlines
show the complicated flow feature of the waterjet.
REFERENCES
Chien, K.Y.,"Prediction
Boundary-Layer Flows with a Low-Reynolds
Number Turbulence Model," AIAA Journal,
Vol.20, 1982.
Chun, H.H., "A Study on the Resistance and
Propulsion of a Tracked Vehicle," Agency of
Defense Development, Rept.No. UD00003 1 CD,
of Channel and
Park, W.G. and Sankar, L.N.,"An Iterative Time
Marching Procedure for Unsteady Viscous
Flows," ASME-FED, Vol.20, 1993.
Roberts, J.L.,"The Influence of Hul1
Boundary Layers on Waterjet Intake
Performance," Ph.D. Thesis, University of
Tasmania, Australia, 1998.
Viecelli, J.A.,"A Method for Including Arbitrary
External Boundaries in the MAC Incompressible
Fluid Computing Technique," Journal of
Computational Physics, Vol.4, 1969.
Watson, S.J.P.,"The Use of CFD in Sensitivity
Studies of Inlet Design," Proceedings of
International Conference on Wateriet Propulsion
The Royal Institute of Naval Architects, Paper No.
8, 1998.
OCR for page 788
DISCUSSION
M. Abdel-Maksoud
Potsdam Model Basin, Germany
Thank you for the interesting presentation.
Did you investigate the effect of the applied
boundary condition at the outlet of the waterjet
on the results?
AUTHORS' REPLY
The velocity was obtained by extrapolating from
interior values and scaling to match the mass
conservation. The pressure was obtained by the
Neumann boundary condition. We also have
tried another simulation which had large
reservoir region, outside of nozzle exit and, then,
imposing free stream velocity profile and
hydrostatic pressure. In this another simulation,
we can not find any significant difference with
the previous simulation.
DISCUSSION
I.-Y. Koh
Naval Surface Warfare Center, Carderock, USA
How do you measure the rotor forces (torque and
thrust) in a hull?
AUTHORS' REPLY
I am sorry that I told that I measured the thrust
when I presented my paper. In fact, the thrust
was not measured directly. As you know, the
thrust of the rotor measured directly is not
meaningful. Therefore, the gross thrust of a
waterjet is indirectly measured by the
momentum change between the inlet and the
outlet by measuring the flux. However, the
torque was measured in the same way as the self-
propulsion test of a conventional ship using a
dynamometer. The detailed experimental
procedure and all the measured data together
their analysis will be published soon.
Representative terms from entire chapter:
intake duct
