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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.
