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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
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),
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
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.
- - ~ - ~ - ~ 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
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
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
:` 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~.
(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
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
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
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
(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
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.
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.
