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1 Copyright © 2012 by ASME

Proceedings of ASME Turbo Expo 2012

GT2012

June 11-15, 2012, Copenhagen, Denmark

ASME2012-69143

OPTIMIZATION OF A TANDEM BLADE CONFIGURATION IN AN AXIAL

COMPRESSOR

Kaveh Ghorbanian

Sharif University of Technology

Dept. of Aerospace Engineering

11365-8639, Tehran, Iran

ghorbanian@sharif.ir

Mahdi Saeedipour

Sharif University of Technology

Dept. of Aerospace Engineering

11365-8639, Tehran, Iran

Saeedipour@ae.sharif.ir

Nazanin Rezaee Ghavamabadi

Sharif University of Technology

Dept. of Aerospace Engineering

11365-8639, Tehran, Iran

ghavam@ae.sharif.ir

ABSTRACT

A numerical investigation of possible tandem configuration

for transonic fan of high bypass ratio turbofan engines is

performed. A computational fluid dynamics code is developed

and validated for NASA rotor 67 at the near stall and near peak

efficiency operating points at 30%, 70%, and 90% spanwise

locations. Numerical simulation for different tandem

configurations at the near stall condition is performed. Mach

number contours plots at 10%, 70% and 90% spanwise

locations are presented for code validation. Contour plots of

pressure and velocity at midspan are provided for several

tandem redesigns. Further contour plots of Mach number,

pressure, temperature and static entropy at midspan are also

presented for the optimized configuration. Further, objective

functions are identified. Back propagation neural network is

trained to provide the NSGA-II algorithm with ample function

and data to initiate a multi-objective optimization. Finally,

through Pareto optimal front, the optimized solutions are

forwarded to the CFD solver for further analysis and

evaluation. A performance analysis is carried out for different

axial distance, percent pitch, and ratio of the front blade chord

length to the rear blade chord length. Results are compared with

those of a single airfoil and reveal that, at mid-span, tandem

blade rotor is from fluid mechanical point of view in advantage;

however, the performance of the rear rotor is strongly affected

by the performance of the front rotor. The results indicate that

the resulting high losses due to flow non-uniformities at the tip

region are of different nature and order and higher attention

should be paid to the design of the tandem configuration at the

tip region.

NOMENCLATURE

X:

Axial Distance between the center axis of the front

blade and the rear blade

D:

Amount of Rotation for the rear blade around the axis

of the fan

R:

The ratio of the chord of front blade to the chord of

the rear blade

NSGA:

Non-dominated sorting of genetic algorithm

BPNN:

Back Propagation Neural Network

TPR:

Total Pressure Ratio

TTR:

Total Temperature Ratio

TIE:

Total Isentropic Efficiency

TPE:

Total Polytropic Efficiency

INTRODUCTION

A possible solution to recent requirements that drives to

reduce the specific fuel consumption of turbofan engines is by

increasing the by-pass ratio of the fan which results in more

high-loaded fan blades. In addition, the number of fan stages

may be reduced and thus a higher thrust-to-weight may be

obtained. Consequently, the employment of high-loaded fan

rotor with transonic or low-supersonic velocities in the blade-

tip region in modern turbofan engines is of great interest due to

their benefits in terms of performance, compactness, reduced

weight, and cost.

The major limitation for this extra pressure rise is the

boundary layer separation on the blade suction surface and end

walls. One may recall that in general, while for subsonic inlet

Mach numbers of around 0.85, the flow experiences a relatively

large separated region that starts at the leading edge and

propagates down to 50% of the blade chord at midspan, the

separation pattern for low supersonic Mach numbers of around

1.15 is different and more complex which is shown in figure 1

[1]. Here, starting from the leading edge, the flow remains

attached to the blade surface to about 1/3 of the chord. Then, it

gets separated; however, it reattaches to the blade surface at

about 2/3 of the chord.

Tandem rotor configuration may be considered as a

candidate for achieving the envisioned higher overall blade

loading and hence increasing the thrust-to-weight ratio and

further to lower the flow separation tendency by shifting the

boundary layer growth to the rear rotor. While tandem

configuration are used either as flaps and slats on aircraft's

wing or as stators in gas turbine engines, to the best of the

author’s knowledge, it is not utilized in commercial gas turbine

rotors. A near term application of tandem blade rotor could be

either in the last stages of a high pressure compressor or in the

fan section.

One of the main investigations that is conducted to

determine the potential of tandem blade configuration for

improving the efficiency and the stable operating range of

2 Copyright © 2012 by ASME

compressor stages is done by Wennerstrom [2]. Brent and

Clemmons [3], Bammert et al [4], and Hasegawa et al [5] have

performed experimental investigation on tandem blade

compressor rotors. Saha and Roy [6],[7] conducted various

aerodynamic performance evaluations of a single and a tandem

cascade for a wide range of inlet angles. The objective was to

determine the high deflection capabilities of the tandem blade

configuration by comparing the results at off design with an

equivalent single one. In a recent attempt by J. Mc Glumphy et

al [8] a numerical investigation is performed on tandem rotor in

the rear stages of a core compressor.

Due to the complexity of the flow field structure and its

strong dependency to the tandem geometrical parameters, the

employment of optimization techniques is unavoidable. Multi-

objective optimization of an axial compressor blade, using

NSGA-II coupled with a second order polynomial-based

response function (RSA), to apply as an approximating part for

fitting data, obtained from a CFD solver is reported by Abdus

Samad and Kwang-Yang Kim [9]. A multidisciplinary

optimization method of radial compressor for micro gas turbine

applications is also introduced by Verstrade et al [10], in which

a similar framework is defined including CFD solver, neural

networks estimator and genetic algorithm as the optimizer

module. In a similar work done by X.D.Wang et al [11], using

improved NSGA-II rotor 37 is examined with objectives of

maximizing efficiencies at both design point and choked

working point.

The present research is motivated by performing both

systematic numerical evaluation and optimization of the tandem

configuration for transonic fan of high bypass ratio turbofan

engines. In this regard, the thermo-fluid behavior of a NASA

rotor 67 in a tandem configuration is investigated. The

designated NASA rotor 67 is a low aspect ratio transonic axial

fan rotor with 22 blades at a design speed of 16043 RPM,

figure 2, which is tested at NASA Lewis Research Center

[12],[13].

CFD CODE

A computational fluid dynamics code, based on a three-

dimensional time-marching Euler/Navier-Stokes analysis, is

developed. The program exploits a finite-volume, time-

marching numerical procedure in conjunction with a flexible,

coupled 2-D/3-D multiple grid block geometric representation.

An artificial dissipation mechanism with the second and fourth

order is added to the scheme to prevent oscillation in the

numerical solutions. The turbulence effects are captured by

using one of the common RANS models namely the k-ε

turbulence model which, in previous attempts, is successfully

employed for NASA rotor 67 fan. The flow is assumed to be

periodic in the pitchwise direction. The endwalls are modeled

as solid viscous surfaces with no-slip condition. Each

simulation undergoes a number of iterations that is specified by

the user prior to execution. Typically the RMS residual was less

than 10e-4 for a converged solution. Solutions that did not meet

these criteria within the number of user input iterations were

restarted and run until they converged.

Code validation is performed for NASA rotor 67 with 22

blades, a design pressure ratio of 1.63, and a tip speed of 1500

ft/sec. The numerical results are compared with available

experimental data described by Strazisar et al. [12]. The

validation is executed for two operating points – that is, near

stall and near peak efficiency.

MESH GENERATION

All computational domains in the current study are

structured grids that can be either single-block, or multi-block.

Breaking large domains into multiple smaller blocks allows for

parallel processing of solutions whereby multiple CPUs

perform computations on the individual blocks, then they

communicate with each other to arrive at a final solution. In

theory, a domain can be divided into any number of blocks,

provided enough CPUs are available. The disadvantage to

increasing the number of blocks is greater inter-block

communication time. The complex geometries such as the

tandem airfoil would be modeled as a multi-block mesh.

Mesh blocks can also follow different structure patterns.

Here the H-O-H multi-block mesh is used where an H-mesh is

served to model the inlet to the compressor blade row, a body-

centered O mesh is served to model the compressor blade itself

and finally another H-mesh is served to model the exit of the

blade row. It should be emphasized that for the single blade we

use one body-centered O mesh and for the tandem blades two

body-centered O meshes should be used. The H-mesh has

orthogonal grid lines with the i index corresponding to the axial

coordinate, the j index corresponding to the radial coordinate,

and the k index corresponding to the pitchwise coordinate. By

contrast, the body-centered O-mesh has an i index that follows

the curvature of the airfoil surface, and a k index that runs

normal to the surface. The H-O-H multi-block grid system and

the sample mesh for single and tandem blades of NASA Rotor

67 are illustrated in figure 3.

Finally a numerical technique intended to accelerate

solution convergence which is called multi-grid is incorporated

that is shown schematically in figure 4. The multi-grid process

involves carrying out early iterations on a fine mesh and later

iterations on progressively coarser virtual ones. The results are

then transferred back from the coarsest mesh to the original fine

mesh.

Further, the sensitivity of the results to the grid size is

examined until the grid independent solution is attained in order

to establish the minimum number of points for grid

independency of 3D solutions. This was ensured by taking a

baseline tandem mesh and incrementally increasing the number

of points on the airfoil surfaces in the axial direction to see if

the concerned parameters (e.g. total pressure ratio, total

temperature ratio, total isentropic efficiency and total polytropic

efficiency) changed for a given set of boundary conditions. The

number of airfoil surface points was increased until there was

no longer a significant variation. The same procedure was

repeated for the number of pitchwise points in the flow

passages, and later in 3-D for the number of radial points. The

number of points for a tandem mesh was controlled by the

single airfoil mesh generator, since the tandem meshes were

created by combining the individual forward and aft airfoil

meshes. The Number of Nodes for a single blade is stated in

table 1. As we can see the changes in the parameters that are

3 Copyright © 2012 by ASME

concerned are less than 1% for the nodes in cases 5 and 6, so as

the result of that we can say that the solution is independent of

the number of nodes that are used in the mesh.

OPTIMIZATION FRAMEWORK

The optimization method is based on coupling artificial

intelligence concept with non-dominated sorting of genetic

algorithm (NSGA-II) to optimize the tandem configuration

design in terms of maximizing efficiency and total pressure

ratio per stage. Based on CFD simulations, better performance

would occur when the chord of the front blade in comparison to

the rear blade is the same. The variables of this multi-objective

optimization task are the axial distance between the blades and

the amount of rotation of the rear blade around the axis of the

fan. The initial geometrical data and other values are shown in

table 2. The whole optimization framework is also illustrated in

figure 5. As evident, based on the geometry selection, the CFD

code builds the database and trains an Artificial Neural

Network (ANN) as the estimator module, using an Error back

propagation algorithm [14]. Considering the fact that a multi-

layer perceptron neural networks with at least two hidden layers

and sigmoid function threshold, can assist in estimation process

for any nonlinear systems, MLP with two hidden layers are

implemented in estimator module. To reach the optimum

architecture of NN, the number of neurons in first and second

layers should be selected as well as the neural networks

learning parameters. Thus the internal optimization process

would be beneficial when the number of population in database

is not numerous enough to reach the optimum NN architecture.

So the objectives of this multi-objective optimization are

validation and training errors to be minimized and its variables

are numbers of neurons in hidden layers (X1 and X2) and two

parameters of the training algorithm related to error back

propagation algorithm and threshold function (X3 and X4).

This process would be done by NSGA-II [15]. After completion

of the training phase, the optimized estimator is calibrated by

using optimum parameters to find the weight factors and used

to approximate date for the NSGA-II which is again

implemented as an optimizer module to reach the best point for

the main problem as is defined figure 6. Then; the best point is

selected from the Pareto optimal front based on the priority of

objectives in the design problem and reverted to the CFD

module, corroborating the accuracy and authenticity of the

optimization process.

The novel idea in this technique, is implementing the

multi-objective algorithm twice; one for the main problem and

the other one for optimizing the parameters of ANN in training

process to reach the optimized estimator. In other words, apart

from the objectives, that is – the total pressure ratio and

isentropic efficiency, minimizing the error of approximating

model is applied to find the optimum weights and the number

of hidden layers of neural network. This internal optimization is

depicted in figure 7. The Pareto optimal front for the internal

multi-objective optimization is shown in figure 8 in which the

minimization of both validation error and training error is done

and the parameters of optimum neural network are chosen.

The Pareto optimal front for the main problem is shown in

figure 9 and the selected optimum point and final results are

provided in table 3. In order to verify the code, it is required to

produce the geometry based on the new results from the

optimization process and CFD analysis, corroborating the

optimum results. This procedure is done and the performance

results of optimum point shown in table 4 which shows the

accuracy of the framework.

CODE VALIDATION

As a step towards code validation, a detailed comparison is

made between the numerical results and experimental data for a

NASA rotor 67 stage at the near stall and near peak efficiency

operating points as outlined in table 5. Calculated Mach

number distributions at 30%, 70%, and 90% pitch are compared

with laser anemometer data reported in [12]. Also a comparison

for the rotor map between experimental data and numerical data

is depicted in figures 10 and 11. For discovering the amount of

differences in mass flow rate between different operating

points, percent of errors for each operating point is stated in

table 6. In this paper, results corresponding to both the near stall

condition and also near peak efficiency operating points are

reported. An eye inspection reveals that the numerical results

are in good agreement with the experimental data; these results

are shown in figure 13. Further, figure 14 illustrates the velocity

and pressure distribution along the rotor for the stage.

TANDEM CONFIGURATION STUDY

In order to generate a tandem configuration from a single

blade, the rotor 67 is simply split in two equal segments with

the same chord length at each span wise section and with the

same airfoil shape but with a half chord length. It means that,

we put the rear blade in a distance with respect to front blade in

which the distance between the central axes of the two blades in

the 50% of the blade span, is about 2 inches. It should be

mentioned that because of the 3D geometrical shape of the

tandem blades that has been considered here, by moving the

rear blade in horizontal and tangential directions, the distances

between blades change variously in different spans of the blade.

So the variables considered for the multi-objective optimization

task are the axial distance between the central axis of the front

blade and the rear blade that are passed through the central

gravity of the each airfoil as the axial distance and the amount

of rotation of the rear blade around the axis of the fan as the

pitch percent. In other words, as a starting point, we constrain

the tandem configuration to have not only the same total chord

length and blade height but also the coordinates of the leading

edge of the front rotor and the trailing edge of the rear rotor

should coalesce with the single rotor 67. It should be

emphasized that, for the reason of future multi-objective

optimization procedure and the required database, the authors

have decided to start the calculation with a tandem profiling as

everything except an ideal profile. Consequently, the thickness

and so the profile around the trailing edge of the front rotor and

the leading edge of the rear rotor are, except some rounding

effects, similar to the mid-chord of the single rotor. This

condition is hereafter referred to “tandem Blade-case 1”. A

numerical simulation for the abovementioned tandem

configuration is performed at the near stall condition. As a next

step, we start to shift tangentially and also horizontally the rear

4 Copyright © 2012 by ASME

rotor relative to the front rotor. 40 different positions are

considered with this method and after having investigated the

effects of changing the position of the rear blade in tangentially

and also horizontally directions, we start to change the ratio of

the chord of the front blade to the chord of rear blade and

investigate 10 diverse occasions ; which the characteristics of

10 different cases for various positions of the rear blade with

respect to front blade in tangential and horizontal directions and

4 different tandem cases with the different chord sizes for rear

and front blades are given in table 3.

It should be emphasized that, even in these cases, the

authors have preferred to start the numerical simulation, and

hence the database development, with the front- and rear blade

profiles. Important parameters for changing the position of rear

blade with respect to front blade are shown in figure 12.

RESULTS

Figure 13 shows the numerical contour plots of relative

Mach numbers at flows near peak efficiency and near stall at

30%, 70%, and 90% spanwise locations of a single NASA rotor

67 compared to experimental results [12]. As evident, the

results corresponding to the hub region and at the mid-span for

both conditions, near peak efficiency and near stall, are in better

agreement compared to the tip region. In the tip region and near

stall condition, the numerical results predict the sonic condition

relatively more downstream compared to experimental data.

The velocity and pressure distribution along the rotor for the

stage for a number of tandem configurations and the

geometrical arrangement for the single blade and tandem blade

in 14 cases that have been chosen from all 50 cases are also

given in figure 14. In figure 15 the blade loading charts for

these elected different tandem cases are given. It is evident that,

for the tandem blade-case 1 configuration, the performance of

both the front- and rear rotor is lowered. However, a shift of the

rear rotor in the clockwise direction will enhance the flow field

for the front rotor. Further, the results indicate that, for the

different tandem configurations, while the aerodynamic

performance of the front rotor may remain unaltered or to some

degrees improved, the contribution of the rear rotor compared

to the mid-to-trailing edge part of the single rotor is very weak

and at some point even destructing and negative. The

abovementioned observation is even truer for the tip region

indicating that the resulting high losses due to flow non-

uniformities at the tip region are of different nature and order.

Results indicate that by far a higher attention should be paid to

the design of the tandem configuration at the tip region, in

particular for transonic fan application. In addition, the

calculated optimum points are shown in table 4. The optimum 1

is the optimum point among the 50 data which are analyzed and

the optimum 2 is the optimum point from implementation of

framework. The contour plots for the Mach number, pressure,

temperature, and static entropy at 50% span of the optimum

point for the same chord configuration are shown in figure 16.

CONCLUSION

A systematic numerical investigation of possible tandem

configuration for transonic fan of high bypass ratio turbofan

engines is initiated. A computational fluid dynamics code,

solving the Navier-Stokes equations based on a flexible

multiple-block grid discretization scheme with a four-stage

Runge-Kutta time-marching finite volume solution technique

and using k-ε turbulence model, is developed. Code validation

is performed for NASA rotor 67 with 22 blades, a design

pressure ratio of 1.63, and a tip speed of 1500 ft/sec by

comparing the numerical results, at the near stall and near peak

efficiency operating points at 30%, 70%, and 90% spanwise

locations, with available experimental data from open literature.

Further, an optimization technique based on coupling

artificial intelligence concept with non-dominated sorting of

genetic algorithm (NSGA-II) is used to optimize the tandem

configuration design in terms of maximizing efficiency and

total pressure ratio per stage.

The present research is motivated by evaluating the tandem

configuration for achieving higher overall blade loading and at

the same time lowering the flow separation tendency by

shifting the boundary layer growth to the rear rotor. As a

starting point, the rotor 67 is simply split in two equal segments

with the same chord length at each spanwise section and we put

the rear blade in a distance with respect to front blade in which

the distance between the central axes of the two blades is about

2 inches. Further, the rear rotor is shifted tangentially and also

horizontally in the direction of the axes of Fan and by this

method we make 40 different cases and after having

investigated the effects of changing the position of the rear

blade in tangentially and also horizontally directions, we start

to change the ratio of the chord of the front blade to the chord

of rear blade and make 10 other cases. Numerical simulation

for the single blade and different tandem blades at the near stall

condition is performed. Contour plots for the pressure and

velocity are provided and 14 counters are shown in figure 14 as

the example.

It should be pointed out that to the best knowledge of the

authors, the adopted technique of coupled optimization

framework for analyzing tandem configuration is done for the

first time. Further, although the framework is applied for not

heavily populated database, nearly 50 analyzed models, the

results are promising and the developed coupled optimization

technique has the potential for further refinement and

improvement.

The authors would like to emphasize that the present paper

should be considered as the layout of groundwork for a

systematic numerical investigation of tandem rotor

configurations. Different airfoil profiles for the front and rear

rotors, axial distance, percent pitch, ratio of the front blade

chord length to the rear blade chord length, and other

geometrical considerations are under investigation. Finally,

further multi-objective optimization remains to be carried out

enabling a detailed and precise evaluation of tandem

configuration compared to a single airfoil.

ACKNOWLEDGMENTS

The present work has been supported in part by the office

of research at Sharif University of Technology.

5 Copyright © 2012 by ASME

REFERENCES

[1] Anderson, J., “Modern Compressible Flow with Historical

Perspectives,” McGraw Hill, Third Edition, 2003.

[2] Wennerstrom A.J., “Highly Loaded Axial Flow

Compressors: History and Current Developments,” Journal of

Turbomachinery, 112(4), pp. 567-578, 1990.

[3] Brent, J.A. and Clemmons, D.R., “Single-Stage

Experimental Evaluation of Tandem-Airfoil Rotor and Stator

Blading for Compressors,” NASA CR-134713, 1974.

[4] Bammert, K., Staude, R., Experimentelle Untersuchungen

an ebenen verzögernden Tandemgittern, VDI-Berichte Nr. 264,

pg. 81–89, Hannover, 1976.

[5] Hasegava, H., Matsuoka, A., and Suga, S., “Development

of Highly Loaded Fan with Tandem Cascade,” AIAA paper

2003-1065, 2003.

[6] Saha, A.K. and Roy, B., “Experimental Analysis of

Controlled Diffusion Compressor Cascades with Single and

Tandem Airfoils”, ASME paper number 95-CTP-41, 1995.

[7] Saha, A.K. and Roy, B., “Experimental Investigations on

Tandem Compressor Cascade Performance at Low Speeds”,

Experimental Thermal and Fluid Science, Vol. 14, pp. 263-276,

1997.

[8] McGlumphy, J., Ng, W., Wellborn, S., and Kempf, S., “3D

Numerical Investigation of Tandem Airfoils for a Core

Compressor Rotor,” ASME Paper, Vol. 132/031009-1, 2010.

[9] A.Samad and K.Y.Kim, “Multi-objective optimization of

an axial compressor blade” Journal of Mechanical Science and

Technology, 22 (2008).

[10] T.Verstraete, Z.Alisalihi,R.A Van den Braembussche,

“Multidisciplinary optimization of radial compressor for micro

gas turbine applications” ASME Journal of Turbomachinery,

Vol 132, 2010.

[11] X.D.Wang, C.Hirsch, Sh.Kang and C.Lacor, “Multi-

objective optimization of turbomachinery using improved

NSGA-II and approximation model” Computer Methods in

applied Mechanics and Engineering journal, accepted

manuscript, 2010.

[12] Strazisar, A.J., Wood, J.R., Hathaway, M.D., and Suder,

K.L., “Laser Anemometer Measurements in a Transonic Axial-

Flow Fan Rotor,” Technical Report NACA-TP-2879, 1989.

[13] Hathaway, M.D., “Unsteady Flows in a Single-Stage

Transonic Axial-Flow Fan Stator Row,” Technical

Memorandum-88929, 1986.

[14] Hagan, M.T., Demuth, H.D and Beale, M.H., “Neural

Network Design,” PWS publication, 1996.

[15] Deb, K., Agrawal, S., Pratab, A. and Meyarivan, T., “A

Fast Elitist Non-Dominated Sorting Genetic Algorithm for

Multi-Objective Optimization: NSGA-II”, In Proceeding of the

parallel Problem Solving from Natural VI Conference, 2000.

Table 1 – The incrementally changes in TPR, TTR, TIE and

TPE by increasing number of nodes in the mesh for a

single blade.

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

No. of

Nodes

50232

56304

97104

12923

14361

14810

No. of

Elements

42120

47520

84800

115000

128000

132000

TPR

1.6394

1.6395

1.6355

1.631

1.6311

1.6313

TTR

1.1705

1.1705

1.1698

1.1689

1.169

1.1691

TIE

89.429

89.436

89.490

89.534

89.501

89.495

TPE

90.116

90.123

90.173

90.214

90.184

90.179

Table 2 – Geometrical Specification of Blade Configurations

Blade Category

X (in)

D (degree)

R

Single

---

---

---

Tandem– Case1

2

-8

1

Tandem– Case2

2

-5

1

Tandem– Case3

2

-3

1

Tandem– Case4

2

0

1

Tandem– Case5

2

3

1

Tandem– Case6

2

5

1

Tandem– Case7

2.15

-5

1

Tandem– Case8

2.15

-3

1

Tandem– Case9

1.85

-5

1

Tandem– Case10

1.85

-3

1

Tandem– Case11

2.7

-8

2

Tandem– Case12

2.7

-5

2

Tandem– Case13

2.7

-3

2

Tandem– Case14

2.7

0

2

Table 3 – Final results of CFD module at Single Blade

Design Point

TPR

TTR

TIE (%)

TPE (%)

Single

1.7405

1.1902

88.2175

89.0649

Tandem– Case1

1.6339

1.1902

74.0944

75.7101

Tandem– Case2

1.6517

1.2091

74.7682

76.3235

Tandem– Case3

1.6515

1.2089

74.5681

76.0964

Tandem– Case4

1.5277

1.1832

71.9707

73.4293

Tandem– Case5

1.6217

1.2039

72.4406

74.0188

Tandem– Case6

1.6253

1.2046

72.9518

74.5547

Tandem– Case7

1.5960

1.2016

71.4466

73.0671

Tandem– Case8

1.6105

1.2046

71.9448

73.5790

Tandem– Case9

1.5896

1.1993

73.9380

75.5416

Tandem– Case10

1.5413

1.1866

74.2586

75.7670

Tandem– Case11

1.7251

1.2137

81.4100

82.8367

Tandem– Case12

1.7364

1.2145

82.1335

83.5230

Tandem– Case13

1.7367

1.2143

82.1249

83.5020

Tandem– Case14

1.7259

1.2128

81.4513

82.8272

Table 4 – Comparison of framework and CFD Results for

optimum point

X

(in)

D

(deg)

R

TIE

(%)

TPR

Optimum point

(Framework)

2.5

-4.4

1

74.091

1.641

Optimum point

(CFD simulation)

2.5

-4.4

1

73.904

1.643

Table 5- Two operating Points Considered for Numerical

Code Evaluation

Operating Point

Mass Flow Rate (𝑚̇ )

Pressure Ratio

Near Peak Efficiency

34.573 kg/s

1.63

Near Stall

33.305 kg/s

1.69

6 Copyright © 2012 by ASME

Table 6- The Error between Numerical and Experimental

Mass flow rates for different operating points.

Experimental Data

Numerical Data

Error

(%)

Single Rotor

Single Rotor

Point

Mass Flow Rate

(kg/s)

Mass Flow Rate

(kg/s)

Near Stall

32.6876

29.532

9.65

Near Peak

Efficiency

34.6104

30.8209

10.94

Choke

34.96

31.8947

8.76

0.8<Inlet Mach<1.0 1.0<Inlet Mach<1.2

a b

Figure 1 – Flow Pattern around the airfoils with: a) Inlet

Mach number between 0.8 and 1.b) Inlet Mach number

between 1 and 1.2 [1].

R hub L.E.= 4 in

R hub T.E.=5 in

R shroud L.E.=9.6 in

R shroud T.E.=10 in Z1=3.6 in Z2=7.8 in

Figure 2 – NASA Rotor 67.

i

k

i

k

ik

ba

i

k

i

k

ik

ik

dc

Figure 3 –a) H-O-H multi-block grid system for Single Blade

of NASA Rotor 67, b) A sample mesh for Single Blade of

NASA Rotor 67, c) H-O-H multi-block grid system for

Tandem Blade of NASA Rotor 67, d) A sample mesh for

Tandem Blade of NASA Rotor 67.

Original

Mesh

First Coarse

Mesh (Virtual)

Next Coarse Mesh

(Virtual)

Figure 4 – Multi-Grid Scheme.

Figure 5- The Main optimization framework.

Figure 6-The optimization problems formulation.

Data Base

Optimizer Module

TrainingNSGA-II

Approximation

After training

Figure 7-The internal optimization diagram.

7 Copyright © 2012 by ASME

Figure 8- Pareto optimal front for the internal problem.

Figure 9- Pareto optimal front for the Main problem.

Rotor Total Pressure Ratio Rotor Adiabatic Efficiency

Mass Flow Rate/Mass Flow Rate at Choke

(Choke flow rate=34.96Kg/s)

a

b

Near Peak

Efficiency

Near Stall

Near Peak

Efficiency

Near Stall

0.82

0.86

0.90

0.94

1.3

1.4

1.5

1.6

1.7

1.8

0.90 0.92 0.94 0.96 0.98 1.00

0.90 0.92 0.94 0.96 0.98 1.00

Figure 10- Rotor map from Experimental Data [12]: a) Total

Adiabatic Efficiency. b) Total Pressure Ratio.

Figure 11- Rotor map from Numerical Data: a)Total

Isentropic Efficiency. b)Total Pressure Ratio.

X

D

Figure 12 – Parameters used for changing the position of

the rear blade with respect to front blade in the NASA Rotor

67.

8 Copyright © 2012 by ASME

Near Peak Efficiency

(Experimental Results)

Near Peak Efficiency

(Numerical Results)

Near Stall

(Experimental Results)

Near Stall

(Numerical Results)

a)

a)

b)

b)

c)

c)

Figure 13 - Experimental [12] and Numerical Contour Plots of Relative Mach Numbers at Flows Near Peak Efficiency and Near

Stall at three Spanwise Locations of a Single NASA Rotor 67:

a) 90%, b) 70%, c) 30%

9 Copyright © 2012 by ASME

Pressure Contour

Velocity Contour

3D Geometry of the Rotor

A

B

C

10 Copyright © 2012 by ASME

Pressure Contour

Velocity Contour

3D Geometry of the Rotor

D

E

F

11 Copyright © 2012 by ASME

Pressure Contour

Velocity Contour

3D Geometry of the Rotor

G

H

I

12 Copyright © 2012 by ASME

Pressure Contour

Velocity Contour

3D Geometry of the Rotor

J

K

L

13 Copyright © 2012 by ASME

Pressure Contour

Velocity Contour

3D Geometry of the Rotor

M

N

O

Figure 14 –The 3D Geometry, the Pressure and Velocity Contours for 15 situations that have been mentioned in Table 2.

A)Single Blade-B) Tandem Blade : Case1-C) Tandem Blade : Case2-D) Tandem Blade : Case3-E) Tandem Blade : Case4-F)

Tandem Blade : Case5-G) Tandem Blade : Case6-H) Tandem Blade : Case7-I) Tandem Blade : Case8-J) Tandem Blade : Case9-

K) Tandem Blade : Case10-L) Tandem Blade : Case11-M) Tandem Blade : Case12-N) Tandem Blade : Case13-O) Tandem Blade :

Case14.

14 Copyright © 2012 by ASME

A

B

C

D

E

F

15 Copyright © 2012 by ASME

G

H

I

J

K

L

16 Copyright © 2012 by ASME

M

N

Figure 15 –The Blade Loading Charts for 14 situations that have been mentioned in Table 2.

A) Tandem Blade : Case1-B) Tandem Blade : Case2-C) Tandem Blade : Case3-D) Tandem Blade : Case4-E) Tandem Blade :

Case5-F) Tandem Blade : Case6-G) Tandem Blade : Case7-H) Tandem Blade : Case8-I) Tandem Blade : Case9-G) Tandem Blade

: Case10-K) Tandem Blade : Case11-L) Tandem Blade : Case12-M) Tandem Blade : Case13-N) Tandem Blade : Case14.

a

b

c

d

Figure 16 – Contour plots for: a) Pressure, b) Mach number, c) Temperature, and d) Static Entropy at 50% spanwise location of

the optimum point for the same chord configuration.