Numerical Study on Flow Induced Vibration of LOX Post in Liquid Rocket Engine Preburner

Chihiro Inoue 1 , Toshinori Watanabe 2 and Takehiro Himeno 2

1 School of Engineering The University of Tokyo

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan E-mail: inoue@aero.t.u-tokyo.ac.jp

2 Department of Aeronautics and Astronautics The University of Tokyo

ABSTRACT In a preburner of liquid rocket engines, some liquid-oxygen (LOX) posts, which introduced oxygen into combustion chamber, experienced severe flow-induced vibration due to unsteady cryogenic hydrogen flow. The mechanism of the vibration has not been fully understood because of the complexity of the flow field. In the present study, a new numerical method was developed to analyze the flow field of hydrogen, whose characteristic properties lie in its non-idealness and compressibility. The unsteady hydrogen flow inside the preburner unit was analyzed to investigate the details of hydrogen flow field as well as the mechanism of LOX post vibration. It was clarified that hydrogen flow inside the manifold and fluid dynamic forces on LOX posts were strongly affected by vortices shed from the junction at the upstream of the inlet. A baffle plate put inside the manifold was shown to reduce unsteady fluid forces on the LOX posts. INTRODUCTION In liquid rocket engines, a preburner is equipped at an upstream position of turbines to generate combustion gas for driving turbo pumps. A lot of liquid-oxygen (LOX) posts are arranged in the preburner to lead oxygen and hydrogen into the combustion chamber. A combustion chamber unit, schematically shown in Fig.1, is mainly consisted of a hydrogen manifold, an oxygen manifold, hundreds of LOX posts, and a combustion chamber. Figure 2 shows a cross-sectional view of hydrogen flow channel projected on the plane A-A’ in Fig.1. During the development of Space Shuttle Main Engine, some LOX posts were reported to experience harmful flow induced vibration (A.Kanmuri, 1983). Since the vibration was thought to be due to the unsteady hydrogen cross flow around the posts, some fluid dynamics studies have been carried out so far to clarify the mechanism of the vibration (S.S.Chen and J.A.Jendrzejczyk, 1984) (M.Williams, 1987). However, the detailed flow field in the preburner and the vibration mechanism have not been understood yet because of the complexity of the hydrogen flow field inside the preburner. The non-ideal characteristics of hydrogen itself inside the manifold also made it difficult to understand them. Recently, some LOX posts in a preburner of Japanese liquid rocket engine experienced a strong vibration, too. In the engine system, the hydrogen flow into the preburner was observed to have significant non-uniformity in the hydrogen manifold, though the details of the flow were still unclear.

To obtain full understanding of the flow and vibration phenomena, detailed flow simulations should be performed in which the effect of all the LOX posts are taken into account and the hydrogen is treated as compressible and non-ideal fluid.

In the present study, a new numerical method has been developed and proposed for simulating the complex flow field of compressible non-ideal hydrogen in the preburner. As a fundamental study, two-dimensional flow field, shown in Fig.2, was analyzed by the developed method to clarify the unsteady complicated flow field, and to investigate the mechanism of the flow induced vibration. The effect of a countermeasure of the strong vibration of LOX posts was also studied. NOMENCLATURE

D tube diameter [m] e internal energy [J/kg] f frequency of shed vortices [Hz] Hs area fraction of solid [ - ] Hv volume fraction of solid [ - ] J Jacobian [ - ] N�

normal vector of distance function [ - ] p pressure [Pa] q� heat flux [W/m2] R gas constant [m2/s2 � K] T temperature [K] Tc temperature at critical point [K] u� velocity vector [m/s] v� quasi-velocity vector [m/s] Vc volume at critical point [K] W velocity [m/s] � density [kg/m3] � first viscosity coefficient [Pa � s] � second viscosity coefficient [Pa � s] � distance function [m] � quasi-time [sec] � kinematic viscosity [m2/s]

, , � the generalized curved coordinates [m] � nabla operator [1/m] Subscripts m main tube b side tube

International Journal of Gas Turbine, Propulsion and Power SystemsOctober 2007, Volume 1, Number 1

Manuscript received on Jan. 9, 2007 Review completed on Sept. 25, 2007

Copyright © 2007 Gas Turbine Society of Japan

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NUMERICAL METHOD The difficulties in simulating hydrogen flow inside the preburner lie in compressible and non-ideal characteristics of hydrogen, and the complex configuration of the flow channel composed of hundreds of LOX posts. In the developed numerical method, what is called “Temperature-based CCUP (TCUP)” (T.Himeno et al., 2005) was used to analyze cryogenic hydrogen flow, and “Cut-cell” technique was incorporated to simulate the complex flow channel with the LOX posts. Basic Equations

The developed TCUP is based on CCUP(T.Yabe et al., 2001).While the combination of independent variables in the original CCUP was selected as ( , , )TQ u p�

� �, the combination was

changed to ( , , )TQ u T p � �

in TCUP. By solving temperature directly, heat transfer even in the incompressible flow field can be adequately simulated. The following Navier-Stokes equations were computed in the present simulation.

� �u ut� � ��

� �� � � ��

� � (1)

� �u u ut

� ��� �� � �

��

� � � (2)

� � [ ]e u e u qt

� ��� � � �� � � � �

��� � � (3)

The stress tensor � consisted of pressure tensors pI and viscous force �T is described as follows: p � � �� I T (4)

( ) [( ) ( ) ]Tu u u� � � � � � � � �T I� � � (5) Eq.(1)-(3) are solved with TCUP scheme, the algorithm of

which is schematically illustrated in Fig.3. The change in the flow field during a computational time step �t is divided into an advection phase and a non-advection phase. The advection equation is solved by CIP(T.Yabe and E.Takei, 1988). The non-advection phase is divided into a diffusion phase and an acoustic phase again.

Hydrogen Property

The applicability of TCUP is not limited to ideal gas flow. It can treat any fluids whose state equations are known. In order to calculate density from temperature and pressure, the state equation is necessary. In the present study, Redlich-Kwong equation (O.Redrich and J.N.S.Kwong, 1949) (R-K eq.) was used, which is written as follows;

2

0.51 (1 )RT apb T b

� �� �

�� �

(6)

where the values of a and b are calculated from the condition at critical point as

( / ) 0Tcdp dV (7) 2 2( / ) 0Tcd p dV (8) Specific heat, coefficient of viscosity, coefficient of thermal

conductivity and sound speed are calculated based on Eq.(6)(Y.Wakamatu et al., 1987). The variation of the sound speed by temperature is shown in Fig.4 at the pressure of 30MPa. As shown in Fig.4, the calculated values based on Eq.(6) show good agreement with the experimental data(McCarty, R.D, 1975) for higher value of temperature than 100K, while the calculated values from the equation of state for ideal gas do not agree. It means that the cryogenic hydrogen should be computed as non-ideal fluid. Since hydrogen temperature and pressure are around 160K and 30MPa, respectively, in the present study, R-K eq. is judged to be applicable for computation.

Cut-cell Technique with Distance Function

In order to simulate the flow field inside the complex channel, Cut-cell technique was utilized in which the governing equations were discretized on the orthogonal grid system. The surface configuration of each object, such as the LOX post and the triangular baffle, were expressed as the iso-contour surface where the value of distance function was zero. The distance function at an arbitrary point in the domain was defined as the signed distance from surface of the nearest object to the point. The sign corresponds to fluid or solid portion as follows.

triangular baffle

strut

injector(LOX post)

flow

Fig. 2 Hydrogen flow channel inside preburner

Fig. 1 Schematic of combustion chamber

Fig.3 Algorithm of TCUP scheme

50 100 150 2000

500

1000

1500

experimentR-K equationideal gas

T[K]S

ound

spe

ed [m

/sec

]

Fig.4 Hydrogen sound speed at 30MPa

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0 outside objects(fluid)� � � 0 surface of objects� � 0 inside objects(solid)� � � Figure 2 shows the preburner configuration, that is, the iso-contour of the zero distance function, while Fig.5 shows the contours of the distance function defined inside the preburner. Both Fig.2 and Fig.5 are illustrated in an orthogonal cross grid. Compared with a body fitted grid, the complex objects are easily expressed by representing the surfaces implicitly as the contours of zero distance function. Though, of course, this method is not able to analyze flow inside surface boundary layers precisely because of lack of grid fineness near objects, the advantage of Cut-cell technique is to avoid the pre-process of mesh generation and to compute flow field with a small computational cost.

Since each object is represented implicitly in the computational domain, the viscous surface boundary condition was imposed by zero-velocity fix method (M.Watanabe et al., 2003) as follows. Normal vector of distance function is calculated as /N � � � �

� (9)

A variable v� is defined as /v u ��

� � (10)

v� is transferred into solid by the following advection equation by CIP.

0v N v�

�� �

�

� � �� (11)

Where� is quasi-time and Eq.(11) is iterated several times. Then v� is transformed into u� inside solid as u v �

� �� (12) As the result of this procedure, velocity on a solid surface is fixed to be zero approximately when it is discretized in the computational domain. Zero-velocity fix method is applied only inside objects. In order to simulate turbulent phenomena, k �� type turbulence model was introduced. When a distance from a body surface is needed, the value of distance function can be used.

As shown in Fig.6, the volume fraction of objects in a cell and the area fraction of objects on the cell boundary are defined at every cell to represent the flow field inside preburner. They are calculated from the distance function. Hs at arbitrary cross section can be calculated by Eq.(13): � � 0.5, 0.5, / ( / )Hs min max n n n �� �� � � � �� �� �

(13)

At cell boundary, the area fraction is � �0.5Hs i � and � �0.5Hs i � at

0.5i � and 0.5i � , respectively. The volume fraction Hv can be calculated as summation of the area fraction within

0.5i � � � 0.5i � as Eq.(14):

0.5

0.5

1 ( )i

Si

Hv i H dJ J

�

�

� � (14)

where 1/ J is the volume of the cell with i . Hs and Hv are introduced to calculate the flux into a cell from

its boundary. For example, the net flow area at 0.5i � and the net flow volume at i are as follows.

[0.5 ( 0.5)]SH iJ �

� � (15)

1 [0.5 ( )]VH iJ

� (16)

If ( 0.5) 0.5iHs � , the cell boundary at 0.5i � is filled with a solid, and the flux does not get across there. It was confirmed that a single cylinder could be represented by at least 4 grids in diameter within 0.5% error of the original volume. To incorporate the Cut-cell technique into TCUP, the zero- velocity fix method was set in prior to the diffusion phase and area/volume fractions were used to compute flux in the acoustic phase. The numerical scheme was validated by comparisons with fluid dynamic forces on a 2D cylinder and pressure drop through a tube bank (C.Inoue et al., 2005). RESULTS AND DISCUSSIONS

The hydrogen flow field inside the preburner was computed by the developed numerical method. At first, the flow channel without junction at the upstream of the inlet is adopted as a basic case with uniform inlet flow (Fig.2). However, in the practical case, the upstream flow channel of the preburner has a junction as shown in Fig.11. One of the two flow paths of the junction comes from cooling pipes of main combustion chamber (MCC), while the other one is from those of nozzle skirt area (NSA). It was reported that some LOX posts suffered severe vibration by hydrogen flow when strong non-uniformity occurred in the manifold. Since the non-uniformity is attributed to the junction, the flow field in the channel with that was further analyzed. The diameter of the LOX post is about10mm and that of inlet pipe is about 150mm. Uniform Inlet Flow Case The flow channel is shown in Fig.2. The structure is composed of a triangular baffle, 15 struts and 216 LOX posts. Inlet velocity was set to be 100m/sec. Inlet temperature and pressure were assumed to be 166K and 27.9MPa, respectively. Outlet condition was that the hydrogen flew out perpendicularly from the plane of Fig.2 in the LOX posts region. The numerical region of 500mm×400mm area was represented with 250×200 stencils. Reynolds number defined at the preburner inlet was 75.3 10� based on the inlet diameter. Typical computed flow field is shown in Fig.7. Hydrogen flows into the preburner area and is divided by the triangle baffle. Though the flow is unsteady, it is almost symmetric in upper and lower sides of the manifold. The hydrogen passes through the area between the struts and flows into the LOX posts region. Velocity at the point ‘A’ in Fig. 7 and horizontal fluid force on the LOX post, which was reported to experience considerable fluctuation in a flow channel with junction (Fig.11), are shown in

��� ���

���

Fig. 5 Contour lines of distance function in preburner

Fig. 6 Calculation of volume and area fraction

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Fig.8. In this case, the horizontal force is much larger than the vertical one. The point ‘A’ is located at just upstream the LOX post. The fluid dynamic force was obtained through integration of pressure on the post surface. The positive value of horizontal force means the fluid force acts in the same direction as the inlet flow. Figure 8 shows velocity at point ‘A’ fluctuates related to vortices shed from the triangle baffle as shown in Fig.9. The vortices are observed to block a flow path between the struts periodically and induce velocity and pressure fluctuations at the point ‘A’. The velocity and force in Fig.8 was analyzed by FFT, and the obtained spectrum is shown in Fig. 10. The frequency of the shed vortices from the triangle baffle is 600Hz as shown in Fig.10(a). There is another peak at about 100Hz, which is thought to be induced by the strut at the upper side of point ‘A’. Figure 10(b) shows that the fluid force on the post also has the same frequency as the velocity at point ‘A’ corresponding to the vortices shed from the triangle baffle.

In this case with uniform inlet flow, the velocity and the force fluctuations around LOX posts behind the triangle baffle are mainly affected by shed vortices from the baffle.

Flow Field with Junction without Baffle Plate

Figure 11 shows the flow channel with junction. Hydrogen from cooling pipes of main combustion chamber (MCC) flows perpendicularly into the channel from cooling pipes of nozzle skirt area (NSA). Inlet velocity ratio was set to be 0.7, where the velocity was 58.8m/sec from cooling pipes of MCC and 41.2m/sec from those of NSA. Temperature and pressure at both inlets were assumed to have same values of 166K and 27.9MPa, respectively. Outlet condition was that the hydrogen flew out perpendicularly from the plane of Fig.11 in the LOX posts region.

Typical result of flow velocity field is shown in Fig.12. Hydrogen flows into the preburner region and is divided into upper and lower side by the triangle baffle. The notable flow features in the flow field are vortices shed from the junction and strong circumferential non-uniformity inside the manifold. Since these phenomena were not observed in the flow field with uniform inlet flow, they are thought to arise from the junction.

Figure 13 shows time variation of velocity at the point ‘A’ in Fig.12, which is located at just upstream the LOX post that suffered strong vibration. As shown in Fig.13, the velocity periodically fluctuates with large amplitude. Figure 14 shows the results of FFT analysis of the velocity shown in Fig.13. A remarkable spectrum peak exists around the frequency of 200Hz. Hibara et al. revealed that vortices were shed from a T-junction periodically in a certain condition. Strouhal number Sr of the vortices was represented as follows (H.Hibara et al., 2004);

� � 0.352 0.21.12 Reb

m

fDSrW

� ��

(17)

Dimensionless parameters are defined as the ratio of diameters /m bD D� � , the ratio of velocity /m bW W� � and Re /m mW D �� . Hibara et al. conducted experiments under the

(a) Frequency spectrum of velocity

(b) Frequency spectrum of horizontal force

Fig. 10 Frequency spectrum of velocity and force on the LOX post

Fig. 9 Instantaneous velocity vector diagram

behind the triangle baffle

Fig. 8 Velocity at point’A’ and horizontal force

on the LOX post suffered strong vibration

Fig. 7 Typical velocity vector diagram of uniform inlet flow case

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condition where 3 4Re 5.0 10 ~ 5.0 10 � � , 2.04 ~ 5.20� and

0.33 ~ 2.00� . Though the condition of the present analysis is not fully correspondent to those of the experiment, the frequency of vortex shed from the junction can be calculated as 139Hz if Eq.(17) is assumed. Since the peak frequency of velocity at point ‘A’ is observed around 200Hz, similar to the above value, a possible source of the fluctuation is attributed to the vortices from the junction.

Figure 15 shows fluid dynamic force on three LOX posts, which are the post suffered strong vibration, the upper side neighbor and the lower side one. Figure 15(a) shows horizontal force, while Fig.15(b) shows vertical force. As shown in Fig.15(a), the horizontal force on each LOX post shows same temporal tendency with each other. Figure 15(b) shows that only the fluid force on the post with strong vibration has large amplitude with negative sign, that is, the fluid force in the downward direction. Figure 16 shows the frequency spectrum of horizontal and vertical force on the post with strong vibration. The peak frequency in both

cases is seen to be about 200Hz, which is correspondent to that of the velocity at point ‘A’. Hence, the cause of the fluid force fluctuation on LOX posts may also be attributable to the vortices from the junction and non-uniformity inside the manifold.

Figure 17 shows time series of hydrogen velocity vector diagram inside the manifold. The vortex shed from the junction and the circumferential non-uniformity can be clearly observed. Figure 17(a)(b) and (c) show the shed vortex, indicated as a circle, is transported downstream through upper flow channel of the triangle baffle. In the later period, however, the downstream movement of the vortex is obstructed by the clockwise circumferential flow

Fig. 11 Hydrogen flow channel with junction

Fig. 12 Typical velocity vector diagram with junction

Fig. 13 Velocity fluctuations at point ’A’

Fig. 14 Frequency spectrum of velocity fluctuations at point ’A’

(a) Horizontal force on LOX posts

(b) Vertical force on LOX posts

Fig. 15 Unsteady fluid dynamic force on LOX posts

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comes from lower side of the triangle baffle. As shown in Fig.17(d) and (e), when the vortex is stagnated at the downstream of the triangle baffle, the hydrogen flow branch through the upper side of the triangle baffle is forced to bend downward toward the LOX post that suffered strong vibration. The bended flow is considered to cause the downward force on the LOX post with strong vibration as shown in Fig.15. In the period of Fig.17(f), on the other hand, the bended flow goes downstream smoothly, and the fluid dynamic force on the LOX post thereby decreases. It is concluded, hence, that the effects of the shed vortices from the junction and non-uniformity of the flow field are combined to cause strong fluctuations near the LOX post at the frequency of the shed vortices.

Flow Field with both Junction and Baffle Plate One of methods for alleviating LOX post vibration was to put a

baffle plate inside the manifold at the counter side of the preburner inlet as shown in Fig.18. To verify the effect of the plate, the flow field inside the channel with both the junction and the plate was computed. Owing to the proposed numerical method, the plate could easily be put in the computational region. The boundary conditions were same as those in the case without the baffle plate. The baffle plate was modeled as a non-permeable solid plate.

Figure 19 shows typical velocity vector diagram in the preburner equipped with the baffle plate. Though shed vortices from the junction are still observed in the present case, the strong non-uniformity inside the manifold is clearly prevented due to the plate. The upper and lower side flows inside the manifold are almost symmetric.

Velocity at the point ‘A’ in Fig.12 is compared with that of the case without the baffle in Fig.20. The velocity fluctuations in the case with baffle are observed to be smaller. Figure 21 shows frequency spectrum of the velocity in both cases with and without the baffle plate. The peak around 200Hz due to the shed vortices is clearly seen to decrease because of the effect of the baffle.

Fig. 16 Frequency spectrum of force on the LOX post

(a) (b)

(c) (d)

(e) (f)

Fig. 17 Time series of velocity field with junction without baffle

Fig. 18 Hydrogen flow channel with baffle plate

Fig. 19 Typical velocity vector diagram with baffle plate

Fig. 20 Comparison of velocity fluctuations at point ’A’

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Figure 22 shows fluid dynamic force on the post with strong

vibration. For comparison, the results of the case without the baffle plate are also plotted in the figure. As shown in Fig.22(a)(b), the horizontal and vertical forces on the post are substantially reduced by the baffle. Figure 23 shows FFT result of horizontal and vertical forces on the post in the flow channel with and without the baffle. Comparison between the two results show the horizontal and vertical fluid forces caused by shed vortices, the frequency of which is around 200Hz, are dramatically reduced due to the baffle plate.

Time series of the flow velocity fields are shown in Fig.24. Obviously, the circumferential non-uniformity inside the manifold disappears because of the baffle plate effect. Vortices shed from the junction are observed at upper side of the triangle baffle, and they are smoothly transported downstream inside the manifold. This is because the non-uniform flow, that makes the shed vortices stagnate, is prevented through the effect of the baffle plate. Due to the baffle, non-uniformity disappears and the flow passing through the upper side of the triangle baffle is not bended strongly toward the LOX post. Hence, the periodical fluid dynamic force on the LOX post is thought to decrease.

DISCUSSION ON LOX POST VIBRATION

In the flow channel with the junction and without the baffle plate, the dominant phenomena for the LOX post vibration are observed to be the shed vortices from the junction and the strong circumferential non-uniformity. If a LOX post is assumed to be a cantilever, the 1st bending natural frequency is in the order of 100Hz, which is well consistent with the frequency of vortices shed from the junction, that is around 200Hz. The vortex shedding, therefore, can be an exciting source of the LOX post vibration. If there is no baffle plate at the downstream region in the manifold, the shed vortices from the junction strongly affect the unsteady fluid dynamic characteristics of the LOX post combined with the effect of the clockwise non-uniform flow inside the manifold. The baffle plate prevents the non-uniformity, and hence, the effect of the shed vortices is reduced. The unsteady fluid dynamic force on the LOX posts is thereby alleviated.

Fig. 21 Frequency spectrum of velocity at point ’A’

(a)Comparison of frequency spectrum of horizontal force

(b) Comparison of frequency spectrum of vertical force Fig. 23 Frequency spectrum of force on the LOX post

(a) Comparison of horizontal force

(b) Comparison of vertical force

Fig. 22 Comparison of fluid dynamic force on LOX posts

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CONCLUSIONS

Hydrogen flow in the channel of the preburner was numerically analyzed by the developed numerical method, which includes effects of compressible and non-ideal characteristics of hydrogen, and the complex configuration of the flow channel with many LOX posts. The principal concern was focused on the fluid dynamic mechanism of LOX post vibration. The effects of junction in the upstream of the preburner inlet and the baffle plate in the preburner flow channel were studied based on the flow simulation.

In the flow channel without the junction, the vortices shed from the triangle baffle were dominant for the flow fluctuations. From the upstream junction, periodical vortex shedding was observed. The shed vortices induced strong fluctuation in the flow field, and the fluctuation eventually resulted in the large unsteady fluid dynamic force on the LOX posts. The junction also caused large circumferential non-uniformity in the flow field of the manifold. Both effects of the non-uniformity and of the vortices were combined together to cause strong flow fluctuations near the LOX post, which suffered strong vibration. When the baffle plate was put at the end of the manifold, the large non-uniformity was prevented, and the force on the LOX post drastically decreased.

For the prediction of flow field inside real configuration of preburner and for more precise estimation of fluid dynamic forces on LOX posts, three-dimensional simulation will be required.

ACKNOWLEDGEMENT

The present work was partly supported by the 21st Century COE Program, “Mechanical Systems Innovation, The University of Tokyo”, by the Ministry of Education, Culture, Sports, Science and Technology. The authors would like to express their sincere gratitude for the support.

References A.Kanmuri, Y.Wakamatu and K.Toki, 1983, “Technical Problems during Development of Rocket Engines with High Pressure Liquid Oxygen and Hydrogen”, NAL TM-523.(in Japanese) C.Inoue, T.Watanabe and T.Himeno, 2005, “Numerical Study on Flow Induced Vibration in a Rocket Engine Preburner by Hydrogen Flow”, Asian Joint Conference on Propulsion and Power 2005 proceeding AJCPP-A2043 pp.1-6 H.Hibara, T.Muramatsu, N.Hirata and K.Sudo, 2004, “Flows in T-Junction Piping System (1st Report, Flow Characteristics and Vortex Street Formed by Branched Pipe Flow”, JSME Ser.B, vol.70, No.693, pp.1192-1200(in Japanese). McCarty, R.D, 1975, “Hydrogen Technology Survey - Thermophysical Properties”, NASA SP-3089. M.Sussman, P.Smereka, and S.Osher, 1994, “A Level-Set Approach for Computing Solutions to Incompressible Two-Phase Flow”, Journal of Computational Physics, vol. 114, pp.146-159. M.Watanabe, R.Kikinis and C.F.Westin, 2003, “Blood Field Simulation and Vessel Segmentation in Level Set Frame Work for MR Angiography”, JSME Ser.B, vol.69, No.687, pp.2408-2415(in Japanese). M.Williams, 1987, “Navier-Stokes calculation of the flow around simplified LOX posts arrays”, AIAA 87-0488. O.Redrich and J.N.S.Kwong, 1949, “On the Thermodynamics of Solutions.�”, Chemical Reviews, Vol. 44, pp233-244. S.S.Chen and J.A.Jendrzejczyk, 1984, “Flow Induced Vibration of the SSME LOX posts”, NASA CR-175608 T.Himeno, T.Watanabe and A.Konno, 2005, “Numerical Analysis for Propellant Management in Rocket Tanks”, AIAA Journal of Propulsion and Power, vol.21, No.1, pp76-86. T.Yabe and E.Takei, 1988, “A New Higher-Order Godunov Method for General Hyperbolic Equations”, Journal of Physical Society of Japan, Vol. 57, No. 8, pp.2598-2601. T.Yabe, F.Xiao, and T.Utsumi, 2001, “The Constrained Interpolation Profile Method�for Multiphase Analysis”, Journal of Computational Physics, vol. 169, pp.556-593. Y.Wakamatu, T. Karita and A.Kanmuri, 1987, “Fast Computation of Physical Properties of Propellants and Combustion Gases in Liquid Oxygen and Liquid Hydrogen Rocket Engines”, NAL TM-564 (in Japanese). http://www.pratt-whitney.com/news_pk_images_turbopumps.asp

(a) (b)

(c) (d)

(e) (f)

Fig. 24 Time series of velocity field with baffle plate

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