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American Institute of Aeronautics and Astronautics

1

Multi Objective Optimization of a Supersonic Axial Turbine Blade Row Shape

for a Rocket Engine Turbopump

Kaname Kawatsu1, Naoki Tani

2, Mitsuru Shimagaki

3, Masaharu Uchiumi

4 and Nobuhiro Yamanishi

5

Japan Aerospace Exploration Agency, Tsukuba, Ibaraki, 305-8505, Japan

and

Katsunori Mitsuhashi6 and Tsutomu Mizuno

7

IHI Corporation, Tonogaya, Mizuho-machi, Nishitama-gun, Tokyo, 190-1297, Japan

A rocket engine should be small and low weight, therefore, a turbopump for a rocket

engine must be smaller and have higher rotation speed than a conventional pump. However,

to achieve high thrust, the required power of the pump should be high enough to achieve

high specific impulse and thrust. To attain these requirements, a study of design

optimization with generic algorithm was applied to blade row shape of the supersonic axial

turbine. In this study, a multi-objective optimization was carried out to obtain a tradeoff

tendency between multi-objective functions, turbine performance and turbine structural

strength. In the present optimization, unsteady CFD was carried out in each optimization

population to estimate turbine efficiency more clearly since shock interaction between stator

and rotor is one of the most important points for supersonic turbine performance estimation.

The optimized result showed that there is a strong tradeoff between turbine efficiency and

diameter. The tradeoff information can be used to improve turbopump performance to

satisfy requirements as a component of a rocket engine.

Nomenclature

Corr = correlation function

D = turbine diameter

P = pressure

s = entropy

T = temperature

Y+ = normalized wall distance

= turbine total-static efficiency

Subscripts

0 = total value

ave = time average value

BL = baseline shape

in = inlet

max = maximum value

out = outlet

s = static value

1 Engineer, JAXA’s Engineering Digital Innovation Center, JAXA, AIAA Member.

2 Engineer, JAXA’s Engineering Digital Innovation Center, JAXA, AIAA Member.

3 Researcher, Engine System Research and Development Group, Space Transportation Propulsion Research and

Development Center, JAXA. 4 Associate Senior Engineer, Engine System Research and Development Group, Space Transportation Propulsion

Research and Development Center, JAXA. 5 Senior Engineer, JAXA’s Engineering Digital Innovation Center, JAXA, AIAA Member.

6 Engineer, Space Technology Group, IHI Corporation.

7 Professional Engineer, Space Technology Group, IHI Corporation.

47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit31 July - 03 August 2011, San Diego, California

AIAA 2011-5784

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

I. Introduction

N order to achieve high efficiency and high reliability with lower cost, an expander bleed cycle was chosen as an

engine cycle for the next generation booster stage rocket engine, called LE-X as is shown in Fig. 1 [1]

. In an

expander bleed cycle [2]

, the energy of the turbine driving gas is supplied by heat exchange around a main

combustion chamber. In order to achieve high engine performance, flow rate of a turbine driven gas must be small

since an expander bleed cycle is one of the open cycle liquid rocket engine. As a result, turbine expansion ratio must

be high to generate high work output compared to a closed cycle liquid rocket engine, like as staged combustion or

full expander cycle. In addition to the above feature, as a component of a rocket engine, weight should be small and

high shaft power output is required. In order to attain above points, high pressure ratio impulse turbine is usually

applied.

High pressure ratio of the turbine introduce that Mach number at both the nozzle exit and the rotor inlet becomes

supersonic [3]

. As a result, strong shock wave interaction can be generated between nozzle exit and rotor inlet, and

this interaction affects the turbine aerodynamic performance[4,5,6]

. However, there are few researches or data for a

supersonic low reaction turbine. Therefore, it is important to clarify a tendency of turbine blade row shape which

can achieve high performance. Especially, strong shock wave interaction can be generated between nozzle exit and

rotor inlet, and aerodynamic losses associated with the interaction should decrease the turbine performance.

Furthermore, the shock wave interaction between rotor and stator should depend on the phase between rotor and

stator. Therefore, unsteady CFD is required to estimate turbine efficiency more clearly since shock interaction

between stator and rotor is one of the important points for supersonic turbine performance estimation.

To clarify the tendency of high efficiency supersonic turbine, multi-objective optimization was carried out in the

present study. As an optimization method, multi-objective generic algorism (MOGA) was applied since MOGA can

easily handle multi-objective optimization with large number of design variables and can search through a large

design space. In a gas turbine and airplane design, effectiveness of MOGA is widely demonstrated in applications

such as a compressor[7]

, a turbine[8]

and a cooling system[9]

. Furthermore, the feasibility of this optimization method

for rocket engine turbopump blade design was already demonstrated[10]

. Usually, engineering problems should be

tradeoff problems, such as weight and structural strength, multi-objective optimization is useful to clarify tradeoff

information. As objective functions, turbine efficiency and turbine diameter were chosen to clarify the tradeoff

between a parameter of turbine performance and a parameter of turbine structural strength. The shape optimization

is one of the best way to achieve the above requirements. However, shape optimization requires many number of

design parameters, and the present optimization uses unsteady CFD, therefore, optimization efficiency must be high.

In the present study, Kriging interpolation method was combined with MOGA[11]

.

Figure 1. LE-X Cycle Schematic and 3D Model

Oxidizer

Fuel

I


3

II. Computational Method

The optimization method for the present study is a real-coded multi-objective generic algorithm with constraint-

handling method by Oyama et al.[12]

. The parameters of MOGA are listed in Tab. 1. A Best-N selection with sharing

and Pareto-ranking method was applied. The selection method was SUS method, and crossover method was BLX-

0.5. One of the disadvantages of MOGA is a large computational cost, and sometimes it requires more than 1000

CFD runs. To reduce such a extremely high computational cost, Jeong[11]

combined the Kriging interpolation

method with MOGA. The Kriging method is used as a response surface, and MOGA optimization is done on that

response surface. This method is quite effective to reduce computational cost, and number of CFD runs can be

reduced from thousands to hundreds. Therefore, in the present optimization, MOGA with the Kriging interpolation

was used.

As a CFD solver, the CRUNCH CFD® [13]

was used. Presently, preconditioning method was applied, since the

velocity of flow field in the turbine is predicted to be supersonic but estimated lowest Mach number is around 0.3.

The standard k- model was applied as a turbulence model. These condition are also listed in Tab. 2. Figure 2 shows

the baseline blade row shape, which is consisted by two stators and two rotors. The grid number is about 1 million

and wall Y+ is about 200 on the premise of using wall function.

As objective functions, turbine efficiency and turbine diameter, which correlate with the turbine tip rotational

speed, were chosen to clarify the tradeoff between a parameter of turbine performance and turbine structural strength.

Turbine efficiency : Maximize

Turbine diameter : Minimize

Figure 3 shows control points of the design variables. The total design variables are 48. The shape of upstream of

1st stage nozzle throat is fixed to keep mass flow rate of turbine. For the shape deformation, grid morphing software

SCULPTORTM

1.8.7[14]

was used. The grid morphing technique has the following advantages. One is that

complicated grid re-generation method is not necessary, thus this method only needs initial grid generation and

definition of the control points. The other is a system generality, since shape optimization can be carried out only by

defining morphing control points.

Constraint functions are often considered in optimization problem. However, the present optimization is set to be

constraint free. Because, the mass flow rate is kept to be constant by fixing the throat area of 1st stage nozzle. And,

an unphysical result due to turbine blade shape changing is prevented by setting of the control points.

The flowchart of the present optimization is shown in Fig.4. In this procedure, the unsteady CFD simulation,

which requires high computational cost, was carried out with JAXA's supercomputer system, called “JSS”. The

turbine efficiency is evaluated with the simulation results for each population's shape. In the PC cluster, MOGA

optimizer sets next generation design parameters from the evaluation results, and design variables and objective

function values were transferred between JSS and PC cluster.

Table 1. Generic Algorithm Method and Parameters

Table 2. Computational Conditions

Fitness Parato Ranking with Shearing

Selection SUS

Crossover BLX-0.5

Alteration of Generetion Best-N

Mutation Rate 0.2

Generation No. 6

Population No. 16

Operating Fluid Hydrogen Hot Gas

Boundary Condition

Inlet : Static Pressure, Static Temperature, Velocity

Outlet : Static Pressure

Wall : Non slip, Adiabatic

Space Accuracy 2nd Order (TVD Scheime)

Turbulence Model Standerd k-

Time Accuracy 1st Order (Preconditioning with Dual Time Stepping)


4

Figure 2. Baseline Turbine Blade Row Shape and Grid Figure 3. Control Points of the Design Variables

Figure 4. Flowchart of the Present Optimization

Inlet BC

Outlet BC

Nozzle (1S)

1st Rotor (2R)

Stator (2S)

2nd Rotor (2R)

1S 1R 2S2R

FIX

Span Direction

1S 1R 2S 2R

Axial Direction

FIX

Rotational Direction

Unsteady CFD using

CRUNCH CFD

JSS(JAXA Supercomputer System)

原型変形後

Grid Deformation

using SCULPTOR

Grid Generation for

Baseline Shape

Evaluation of the

Objective Functions

MOGA Optimizer Finish

Check

Generation

Generation N < Nmax

Generation N = Nmax


5

III. Results and Discussion

A. CFD Simulation Results with Baseline Shape

In this section, CFD results with baseline shape is examined before discussion on optimized results. Especially,

characteristics of flow field in the turbine cascade and mechanism of aerodynamic losses are mentioned to enhance

understanding of tradeoff information from optimization results.

Figure 5 shows distribution of Mach number and entropy at mean diameter section of steady CFD simulation

result with baseline shape. The phase between rotor and stator is fixed in this steady CFD. High pressure ratio of the

turbine causes that Mach number at both nozzle exit and rotor inlet becomes supersonic. As a result, shock wave

interaction can be generated between nozzle exit and rotor inlet. Expansion waves are generated at the nozzle

trailing edge in the 1st stage. The expansion waves reflect at the stator wall and impinge to rotor inlet. At the same

time, the detached shock waves, which are generated at the leading edge of rotor blade, impinge to stator blade and

interact with the rotor blade suction surface boundary layer. On the other hand, the Mach number at the 2nd stage

interface between rotor and stator is almost subsonic. Thus, the rotor-stator interaction caused by shockwaves is not

observed. While, the flow speed, which is accelerated by expansion wave from leading edge of the stator, is dropped

to subsonic by the normal shock at the blade suction side of 2nd stage stator. These interaction and shock wave can

affect the turbine aerodynamic performance. According to the entropy distribution, distinguished increases of

entropy are observed at blade suction side of 1st rotor and 2nd stator. Respectively, these increase of entropy

indicate aerodynamic losses caused by interaction between detached shock wave and boundary layer at 1st rotor

blade suction side and normal shock wave at 2nd stator blade suction side.

Figure 5. Distribution of Mach Number and Entropy at Mean Diameter Section

of Steady CFD Simulation Result with Baseline Shape

Figure 6 shows vortex structure evaluated by using Q number[15]

, which is the second invariants of the velocity

tensor, colored by non-dimensionalized helicity, which indicates vortex curing direction. It can be observed that the

boundary layer separation caused by the interaction between shock wave and boundary layer appeared at the 1st

stage rotor blade suction side and 2nd stage stator blade suction side. Furthermore, corner vortexes at blade tip and

hub side of rotor blades can be observed. In the turbine flow, these shock interaction and corner vortex are stated as

factors of the aero dynamic loss.

The unsteady CFD results are shown in Fig. 7. The Mach number and entropy contours at mean diameter section

in several time steps with baseline shape are shown in this figure, these results indicate that the shock interaction

between rotor and stator change depending on rotor position. Especially, shock interaction between rotor and stator

of 1st stage depends on the phase between rotor and stator. Therefore, in the present optimization, unsteady CFD

was carried out in each optimization population to estimate turbine efficiency more clearly since shock interaction

between rotor and stator is one of the important points for supersonic turbine performance estimation.

0.0 1.0

s/smax (-)

1.0 2.5

Absolute Mach # (-)flood contour line

0.0 1.6

Absolute Mach # (-)

Normal Shock WaveInteraction between Detached

Shock Wave and Boundary layer


6

Figure 6. Iso-surface of Q Number ( Q=2x10

5) Countered with Non-dimensionalized Helicity

of Steady CFD Simulation Result with Baseline Shape

Figure 7. Mach Number and Normalized Entropy Contours at Mean Diameter Section

of Unsteady CFD Simulation Result with Baseline Shape

1S1R

2S2R

1S1R

2S 2R

Blade Tip Corner Vortex

Hub Side Corner Vortex

Interaction between Shock Wave and Boundary Layer

Normal Shock Wave

Non-dim. Helicity (-)

-1.0 1.0

ClockwiseCounter Clockwise

0.0

5/8 pitch

7/8 pitch

1/8 pitch

3/8 pitch

0/8 pitch

4/8 pitch

8/8 pitch

1.0

s/smax (-)

0.0 1.6

Absolute Mach # (-)


7

B. Optimized Results

The object functions for the present optimization are shown in Fig. 8. Each value is normalized by the baseline

shape result. As the generation proceeds, results become to have a gathering near the pareto line, which is described

with red broken line in this figure. The maximum improvement of turbine efficiency is 1.7% increase, and turbine

diameter is 5% reduction. According to this figure, it seems to be that there is a strong tradeoff between turbine

efficiency and turbine diameter.

Figure 9 shows distribution of pressure and temperature at each stage inlet and outlet. In this figure, baseline

shape, "BL" and two typical shape of the optimization results are compared. The one of the two has higher

efficiency and nominal diameter, "HE", the other one has nominal efficiency and smaller diameter, "SD".

Comparing with "BL", optimized results "HE" and "SD" has a tendency to increase the road allocation and decrease

the reaction degree at 2nd stage.

To examine the unsteady characteristic of each shape, time series behavior of turbine efficiency of each shape

are shown in Fig. 10. In this figure, the turbine efficiency is normalized with time averaged value of turbine

efficiency with baseline shape. As can be seen in this figure, amplitude and frequency of turbine efficiency

fluctuation of all shape are much the same.

Figure 8. Plots of Objective Functions

Small Diameter & Nominal Efficiency (SD)

High Efficiency & Nominal Diameter (HE)

Base Line (BL)

Pareto Line

D/D

BL

(η-ηBL)/ηBL

Total Value Static Value

P0/P

0in, P

s/P

0in

(-)

T0/T

0in

, T

s/T

0in

(-)

0.00

0.20

0.40

0.60

0.80

1.00

Inlet 1S Exit 1R Exit 2S Exit Outlet

Ps/

P0

in, P

0/P

0in

(-)

BL

HE

SD

0.00

0.20

0.40

0.60

0.80

1.00

Inlet 1S Exit 1R Exit 2S Exit Outlet

Ts/

T0

in, T

0/T

0in

(-)

BL

HE

SD

(a) Pressure (b) Temperature


8

Figure 9. Distribution of Physical Value at Each Stage

Figure 10. Time Series Behavior of Turbine Efficiency

Figure 11. Comparison of Mach Number Distribution at Mean Diameter Section

0.98

0.99

1.00

1.01

1.02

1.03

1.04

6 7 8 9 10

BL

HE

SD

Time

η/η

ave

,BL

1pitch

1pitch

1/8 pitch

3/8 pitch

5/8 pitch

7/8 pitch

0.0 1.6

Absolute Mach # (-)

BL HE SD

1.0 2.5

Absolute Mach # (-)

contour lineflood


9

Figure 11 shows comparison of Mach number distribution at mean diameter section of each shape. It can be

observed as a mutual characteristic of optimized shapes, "HE" and "SD", that the normal shock at 2nd stage rotor is

not observed and the velocity of flow field between 2nd stage rotor and stator is supersonic, contrary to baseline

shape. Thus, the interaction between 2nd stage rotor and stator can be observed in the case of optimized results "HE"

and "SD".

In the case of baseline shape, the unsteady characteristic of the turbine efficiency depends on interaction between

stator wake and rotor blade in the 2nd stage. But, the fluctuation of the turbine efficiency caused by shock

interaction between rotor and stator at 2nd stage in the case of optimization results.

In order to clarify the relation between design parameters and objective functions, the correlation function (Corr)

was used as a evaluation parameter. The Corr shows a tendency of similarity between two data arrays, in this case

the data arrays are design parameter and objective function. If the absolute value of the Corr is large, correlation

between the selected two arrays is strong. And its sign indicates tendency of the correlation. Figure 12 shows Corr

between design parameters and objective functions of pareto-ranked population. According to this results, strong

correlation with the tradeoff tendency are observed about several design parameters. The first point is the shroud

diameter. Increase of shroud diameter cause decrease of the turbine efficiency, since the turbine angular momentum

is proportional to the blade diameter with same blade road. The second point is the inlet blade angle of 2nd stage

stator blade. The leading edge at hub side has strong correlation with objective functions. The angle affect to work

of the blade, is shown in Fig. 13. Increase of this angle causes higher turbine efficiency with higher blade load.

Finally, the inlet blade angle of 1st stage rotor blade also has strong correlation with objective function. The angle

affects to strength of expansion waves as is shown in Fig. 14. These expansion waves impinge to the neighboring

rotor blade and affect to onset of detached shock wave at blade leading edge as is shown in this figure.

Figure 12. Corr between Design Parameters and Objective Functions of Pareto-ranked Population

Design Parameters (Span Direction) Design Parameters (Axial Direction)

Co

rr

Co

rr

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

HU

R_1S

_R

HU

R_1R

_R

HU

R_2S

_R

HU

R_2R

_R

SH

R_1S

_R

SH

R_1R

_R

SH

R_2S

_R

SH

R_2R

_R

Turbine Efficiency

Turbine Diameter-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

HU

R_

1S

_Z

HU

R_

1R

_Z

HU

R_

2S

_Z

HU

R_

2R

_Z

SH

R_

1S

_Z

SH

R_1

R_1

_Z

SH

R_1

R_2

_Z

SH

R_1

R_3

_Z

SH

R_

1R

_4

_Z

SH

R_

2S

_Z

SH

R_

2R

_1

_Z

SH

R_

2R

_2

_Z

SH

R_

2R

_3

_Z

SH

R_

2R

_4

_Z

Turbine Efficiency

Turbine Diameter

Shroud Diameter

Design Parameters (Rotational Direction)

Co

rr

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

HU

R_

1S

_T

H

HU

R_

1R

_1

_T

H

HU

R_

1R

_2

_T

H

HU

R_

1R

_3

_T

H

HU

R_

1R

_4

_T

H

HU

R_

2S

_1

_T

H

HU

R_

2S

_2

_T

H

HU

R_

2S

_3

_T

H

HU

R_

2S

_4

_T

H

HU

R_

2R

_1

_T

H

HU

R_

2R

_2

_T

H

HU

R_

2R

_3

_T

H

HU

R_

2R

_4

_T

H

SH

R_

1S

_T

H

SH

R_

1R

_1

_T

H

SH

R_

1R

_2

_T

H

SH

R_

1R

_3

_T

H

SH

R_

1R

_4

_T

H

SH

R_

2S

_1

_T

H

SH

R_

2S

_2

_T

H

SH

R_

2S

_3

_T

H

SH

R_

2S

_4

_T

H

SH

R_

2R

_1

_T

H

SH

R_

2R

_2

_T

H

SH

R_

2R

_3

_T

H

SH

R_

2R

_4

_T

H

Turbine Efficiency

Turbine Diameter

2nd Stator Camber

Angle Hub Side

1st Rotor Camber

Angle Shroud Side


10

Figure 13. Tradeoff between Objective Functions with Inlet Blade Angle of 2nd Stator

Figure 14. Tradeoff between Objective Functions with Inlet Blade Angle of 1st Rotor

IV. Conclusion

In this study, a multi-objective optimization using unsteady CFD simulation result was carried out to obtain a

tradeoff tendency between multi objects, turbine performance and turbine structural strength, for a supersonic axial

turbine blade row shape. The unsteady CFD simulation results indicate that the shock interaction between rotor and

stator change depending on rotor position. Especially, shock interaction between rotor and stator of 1st stage

depends on the phase between rotor and stator. And, according to the present optimization results, the following

points can be pointed out.

The strong tradeoff tendency can be seen between turbine efficiency and diameter from the distribution of

pareto-ranked populations.

As a mutual trend of these pareto-ranked populations, increase of 2nd stage load comparing baseline shape

is indicated in the static pressure distribution in the turbine row.

Hub Side

Higher Efficiency

Smaller Diameter

0.0 1.6

Relative Mach# (-)

HE SD

HE

SD

Inlet Blade Angle

Shroud SideHigher Efficiency

Smaller Diameter

HE

SD

Inlet Blade Angle 0.0 1.6

Relative Mach# (-)

HE SD


11

Amplitude and frequency of turbine efficiency fluctuation of all shape are much the same, but the

mechanism of that are different between baseline shape and optimized shapes. This difference is

depending on the normal shock on suction surface of 2nd stage stator.

Strong correlations with the tradeoff tendency are observed about several design parameters.

First, about the radial direction design parameters, the shroud diameter has strong correlation with

objective functions. Second, the inlet blade angle of 2nd stage stator at hub side, the angle affects to work

of the blade. Increase of this angle cause higher turbine efficiency with higher blade work. Finally, the

inlet blade angle of 1st stage rotor also has strong correlation with objective functions, the angle affects to

strength of expansion waves. These expansion waves impinge to neighbor rotor blade and affect to the

formulation of the detached shock wave at the blade leading edge.

Acknowledgments

Present work was supported by IHI Corporation Space Technology Group, JAXA Space Transportation

Propulsion Research and Development Center. All CFD simulations presented in this paper are carried out on the

JAXA Supercomputer System (JSS). The optimization method, which is used in this work, had been developed by

Dr. Shimoyama, et. al. (Tohoku Univ.) at Fujii and Oyama Laboratory in JAXA/ISAS. The authors greatly

appreciate their contributions and supports.

References

1Kurosu, A., et al., “LE-X –Japanese Next Liquid Booster Engine–,”, 44th AIAA/ASME/SAE/ASEE Joint Propulsion

Conference and Exhibit, AIAA 2008-4665, July 2008.

2G. P. Sutton, “Turbopumps, a Historical Perspective,”, 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and

Exhibit, AIAA 2006-5033, July 2006.

3C. D. Colclough, “Design of Turbine Blades Suitable for Supersonic Relative Inlet Velocities and the Investigation of Their

Performance in Cascades: Part 1 Theory and Design,”, Journal Mechanical Engineering Science, Vol.8 No.1, 1966.

4Denton, J. D., "Loss Mechanisms in Turbomachines,", Journal of Turbomachinery, Transaction of the ASME, Vol.115,

pp621-656, 1993.

5Tanaka, M., et al., “Investigation of the Two-dimensional Performance of Supersonic Impulse Turbine Blade Cascades (2nd

Report, Flow and Losses in Blade Passage and Some Design Criteria),”, Bulletin of JSME, Vol. 27, No.255, 1984.

6T. P. Moffitt, “Design and Experimental investigation of a Single-Stage Turbine with a Rotor Entering Relative

Mach Number of 2,”, NACA RM E58F20a, 1958. 7Oyama, A., et al., “Transonic Axial-Flow Blade Optimization Using Evolutionary Algorithms and a Three-Dimentional

Navier-Stokes Solver,”, AIAA Journal of Propulsion and Power, Vol.20, No.4, pp.612-619, 2004.

8Griffin, L.W., et al., “Design and Analysis of Turbines for Space Applications,”, Proceedings of 33rd AIAA Fluid Dynamics

Conference and Exhibit, AIAA 2003-3933, June 2003.

9Schumacher, T., et al., “Automatic Optimization of Aircraft Ventilation System Components with CFD,”, Proceeding of

IACC2007 on Disc [CD-ROM], Paris, France, 2007.

10Tani, N., et al., “Feasibility Study of Multi objective Shape Optimization for Rocket Engine Turbopump Blade Design,”, 44th

AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, AIAA 2008-4659, July 2008.

11Oyama, A., et al., “Aerodynamic Multiobjective Optimization Using Design Exploration of a Flapping Airfoil Using a

Navier-Stokes Solver,”, Journal of Aerospace Computing, Information, and Communication, Vol. 6, No. 3, pp. 256-270, 2009.

12Jeong, S., Minemura, et al., “Optimization of Combustion Chamber for Diesel Engine Using Kriging Model,”, Journal of

Fluid Science and Technology, Vol. 1, pp. 138-146, 2006.

13http://www.craft-tech.com/

14 http://gosculptor.com/

15Jeong, J., et al., “On the identification of a vortex,”, Journal of Fluid Mechanics, No. 285, pp. 69-94, 1995.

Download - [American Institute of Aeronautics and Astronautics 47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit - San Diego, California ()] 47th AIAA/ASME/SAE/ASEE Joint Propulsion

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