Amano R.S., Sunden B. (Eds.) Computational Fluid Dynamics and Heat Transfer: Emerging Topics
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116 Computational Fluid Dynamics and Heat Transfer
The airfoil section design is a very important step for the aerodynamic design.
All the airfoils are designed by the section design and reasonable stackup.A three-
dimensional blade method was developed in this study as shown in Figure 3.57.
It can beseen that CFD codeforboth section analysis and three-dimensional blade
row analysis is very important. The development of the efficient and wide-range
application of CFD codes is very important. In this study, an effective numerical
method was developed for both two-dimensional and three-dimensional codes. In
this chapter, a brief introduction for the numerical method developed for two-
dimensional incompressible and compressible flows was provided.
3.10.1 Flow solver for section analysis
A time-marching algorithm was used in the present cascade flow computations
[2,20,22,23,29].The computationstarts witharough per turbation, which develops
under certain boundary conditions. In this approach, the governing equations are
replaced by a time-difference approximation with which steady or time-dependent
flowsof interestcanbe solved ateachtime level.The detailsofthe governingequa-
tions and the method of solving the equations are discussed in previous sections.
The discussion has been extended for many years for the optimum type of the
mesh that should be used for turbine or compressor bladeflow calculation [15,16].
The more orthogonal the grid, the smaller will be the numerical errors due to
the discretization. However, no one type of grid is ideal for blade-to-blade flow
calculation. In this study, the H-type mesh is used.Another problem for the blade-
to-blade mesh is the trailing edge mesh and the trailing edge Kutta condition [7].
The authors have noticed that the number of mesh points near the trailing edge
points strongly influence the loss calculation. Here, a realistic method is proposed;
that is, when the meshonthe trailing edgecircleis more than10points, an explicit
viscous Kutta condition [17] is applied on the training edge circle which allows
flow leaving the blade smoothly.
The mesh refinement study was conducted prior to the calculations [2,3]. It is
shownthatthemeshsizeof110×45with80pointsonthebladesurfaceintheblade-
to-blade section is sufficient. The H-mesh [2,3] was used in all the computations.
The computational mesh is shown in Figure 3.58. A typical convergence history
of the calculation is shown in Figure 3.59. It is demonstrated that the present code
has a good convergence. In this study, this code was used in the two-dimensional
airfoil section analysis and optimization.
3.10.2 Optimization
Turbine and compressor designsfacedthechallenge of high efficiency and reliable
blade design. Design techniques are typically based on the engineering experience
andmayinvolvemuchtrialanderrorbeforeanaccepteddesignisfound.Theability
of the three-dimensional codes and the abilitytoperform a three-dimensional flow
calculation are difficult to improve the design. The three-dimensional approach
is a way to help designers do some parameter study and compare the numerical
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CFD for industrial turbomachinery designs 117
Figure 3.58. Computational mesh.
Number of steps
Error x 1000
0 500 1000 1500 2000
10
0
10
−1
10
−2
10
−3
Figure 3.59. Typical convergence history.
study with testingin order to improvethedesign.The three-dimensional method is
used for parameterstudies, like lean, bow, and sweep effects.The two-dimensional
analysis developed in the study is used to optimize blade section airfoil.
The blade design often starts with two-dimensional airfoil section. Some of
the designparameters areobtained by through-flow optimizationand vibrationand
heat transfer analysis, for example, the solidity and numberofblades.And most of
the parameters are optimized through the sectional design. For example, leading-
and trailing-edge radii, stagger angle, and maximal thickness position. The final
three-dimensional analyses are used to make lean, bowed, and sweep modification
of the blades.
The optimization process used in this study was based on the constrained opti-
mization method [26]. If the objective function to be minimized is F(X) and
constraint functions is g
j
(X), the optimization problem can be formulated by the
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118 Computational Fluid Dynamics and Heat Transfer
objective and constraint functions as:
∇F(X) +
λ
j
∇g
j
(X) = 0 (78)
where λ
j
are the Lagrange multipliers and X is the vector of the objective. The
finite differencing methodcan beused to obtainthe gradients ofobjectiveand con-
straintfunctionswithrespecttodesignvariablesandconstraints. Manycommercial
optimization packages are available as an optimizer to design problems [26].
The objective function of this study is adiabatic efficiency. The optimization
objective is to obtain the highest efficiency under given constraints. Foe more
convenience, the total pressure loss is used as an objective variable to judge the
design.The definition of the total pressure loss in this study is:
ξ =
(p
∗
in
− p
∗
out
)
H
(79)
where H is the outlet dynamic head of the exit plane.
Two-dimensional section optimization
In this study, a compressor stator blade originally stacked up by using NACA 65
was redesigned and optimized. In this design, the thickness of the airfoil, chord,
stagger angle, and inlet and outlet flow parameters were fixed.The designed blade
was used to replace the old blade. The section design optimization was mainly
achieved through adjusting the section maximum thickness location. It was found
thatthe maximumthickness locationsinfluence thelosses andperformance. Inthis
study, five sections were selected as design section. All section designs used the
similar method and procedure. As an example, the results presented here are the
design information for 50% span section.
Figures 3.60 through 62 show the Mach number distributions of the design
bladeswithdifferentmaximumthicknesslocations.Figure3.63showstherelation-
ship between total pressure losses and location of the maximum thickness. It was
shown that the Mach number distributions changed with the maximum thickness.
The change in the Mach number distributions will influence the boundary-layer
developmentandinfluencethelossesofthesection.Itwasfoundthatthereisanopti-
mumlocationofthemaximumthickness.Studieshavealsoshownthatthislocation
changeswithflowconditionsandstaggeroftheairfoil.Afteroptimizationstudy,the
airfoilsectionwithhighestefficiencywasselected.Theairfoilwasstackedupusing
a smooth leading edge curve method. Before the airfoil was selected as the final
blade, the three-dimensional analysis was performed to investigate the benefits of
thethree-dimensionalbladefeatures.Three-dimensional analysis showed thatafter
sectionoptimizationtheairfoilperformancewasimproved,asshowninFigure3.64.
However, the end wall region did not show significant improvement. Some three-
dimensional features were used to reduce the losses and increase the efficiency.
Three-dimensional blading
Three-dimensional CFD analysis codes were used to guide the three-dimensional
final modification of the two-dimensional section stackup. Some parameters were
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CFD for industrial turbomachinery designs 119
0.65
0.75
0.75
0.3
0.65
0.15
0.05
0.35
0.65
0.85
0.45
0.35
0.55
0.05
0.55
0.25
0.55
0.55
0.45
Figure 3.60. Mach number distributions for maximum thickness at 15% chord.
0.65
0.55
0.35
0.55
0.65
0.45
0.75
0.35
0.25
0.65
0.45
0.15
0.25
0.05
0.55
0.25
0.05
0.35
0.65
0.55
0.15
0.25
0.45
Figure 3.61. Mach number distributions at maximum thickness at 25% chord.
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120 Computational Fluid Dynamics and Heat Transfer
Figure 3.62. Mach number distributions at maximum thickness at 45% chord.
0.35
0.4
0.45
0.5
0.55
15 25 35 45
Maximum thickness position (%)
Total pressure loss (%)
Figure 3.63. Total pressure lossesVs maximum thickness location.
selected to do the study based on design experience. In this redesign, the three-
dimensional feature-bowed blade was used to reduce the secondary losses. In
the three-dimensional blading, the optimizer will not be used in this study,
because optimization with optimizer like the three-dimensional code is very time
consuming. Although the mesh sizes, turbulence models and mixing between the
blade rows strongly influence the optimization results, the process propose here
provide a way to drive a better design. In the current study, three-dimensional
optimization was obtained through parameter study.
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CFD for industrial turbomachinery designs 121
0
10
20
30
40
50
60
70
80
90
100
0 3 6 9 12 15
Total pressure loss (%)
Radial height (%)
After optimization
After redesign sections
NACA 65 sections
Figure 3.64. Total pressure losses along with blade span.
(a) (b) (c)
Figure 3.65. Staticpressuredistributiononthesuctionsurface.(a)OriginalNACA
section blade. (b) Before 3D optimization. (c)After 3Doptimization.
In this study, only bow feature is applied. The bow location and degree of the
bowwereselectedas studyparameters. Itwasfoundthat the15degreebowlocated
at 30% of the spancan eliminate separation and get better performance.The three-
dimensional analysis showed that after two-dimensional section optimized design,
the flow in most of the spanwise locations remained attached and well behaved
although it had a small region of end wall separation on the section side, as shown
in Figures 3.65 through 68. After the bow was introduced the flow around the
airfoil on both tip and root end wall was attached. The total pressure losses were
reduced,asshowninFigure3.64.Thisstudyshowedthatthetotalsectionefficiency
increased by about 3% as compared with the originalblade. Both two-dimensional
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122 Computational Fluid Dynamics and Heat Transfer
(a) (b) (c)
Figure 3.66. Axialvelocitycontour nearbladesuction surface.(a)OriginalNACA
section blade. (b) Before 3D optimization. (c)After 3Doptimization.
(a) (b)
Figure 3.67. Velocity vector near the root section near the trailing edge suction
surface. (a) Before 3D optimization. (b)After 3D optimization.
section design and three-dimensional optimization increased the efficiency with
similar percentage. It has been shown that both two-dimensional optimization
and three-dimensional study are important. However, three-dimensional blading
depends more on the experience of the designer.
It is important to point out that, based on the design experience, the three-
dimensionalanalysisresultsnormally arenotaccepted asfinalshape.The selection
of the final configuration was accepted through small adjustments based on expe-
rience to obtain high performance and stability. After three-dimensional analysis,
somesmalladjustmentsweremadebasedonthe possiblemanufacturinguncer tain-
ties and applications. For example, the compressor stator design in this study was
recambered to increase exit angle by about 1.5 degrees to increase the compressor
surge margin.
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CFD for industrial turbomachinery designs 123
(a) (b)
Figure 3.68. Velocity vector near the tip section near the trailing edge suction
surface. (a) Before 3D optimization. (b)After 3D optimization.
The flow field around turbine and compressor blades exhibits very complex
flow features. The flow field involves various types of loss phenomena, and hence
high-level flow physics is required to produce reliable flow predications.The blade
design process is a very time consuming process and the optimization process is
very complicated. The method developed in this study for the airfoil section and
bladedesign and optimizationis one successful processand very easy foradapting
in the aerodynamic design. In this study, a two-dimensional code was used for
sectiondesigninsteadofthree-dimensionalcode, whichprovidesaneconomic way
for design and optimization. The three-dimensional blades were optimized after
stacking up the two-dimensional sections. The lean, bowed, and swept features of
the three-dimensional blade were created during three-dimensional optimization.
It was shown that both two- and three-dimensional design and analysis were the
backbones of the blade aerodynamic designs.The results show that both the airfoil
shape optimization andthe three-dimensional optimization can beused to improve
the performance of compressor and turbine blade.
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