Proceedings of GT2006
ASME Turbo Expo 2006: Power for Land, Sea and Air
May 8-11, 2006, Barcelona, Spain
GT2006-90503
BLADING AERODYNAMICS DESIGN OPTIMIZATION WITH MECHANICAL AND
AEROMECHANICAL CONSTRAINTS
H-D Li, L. He
School of Engineering
Durham University
Durham DH1 3LE
U. K.
ABSTRACT
A blading design optimization system has been developed
using an aeromechanical approach and harmonic perturbation
method. The developed system has the capability to optimize
aero-thermal performance with constraints of mechanical and
aeromechanical integrity at the same time. ‘Aerodynamic mode
shape’ is introduced to describe geometry deformation which
can effectively reduce the number of design parameters while
preserving surface smoothness. Compared to the existing
design optimization practices, the present system is simpler,
more accurate and effective. A redesign practice of the NASA
rotor-67 at the peak efficiency point shows that the aero
thermal efficiency can be improved by 0.4%, whilst the
maximum static stress has been increased by 33%.
Aeromechanical analysis of the optimized blade shows that the
aerodynamic damping of the least stable first flap mode is still
well above the critical value though the natural frequencies of
the first 5 modes have been reduced by 1~4%. The present
finding highlights the need for more concurrent integrations of
mechanics, aerodynamics and aeromechanics design
optimization.
NOMENCLATURE
F,G,H =
axial, tangential and radial flux vector
n
=
normal direction of mesh cell surface
NF
=
index of harmonics
u,v,w =
axial, tangential and radial flow velocities
S
=
source term
T
=
time
U
=
conservative variable vector
V
=
viscous terms
ρ
=
fluid density
τ
=
viscous stress components
ω
=
angular frequency
Y S Li, R. Wells
Siemens Industrial Turbomachinery Ltd.
Ruston House
PO Box 1 Waterside South
Lincoln LN5 7FD
U. K.
Subscripts
k
=
x, θ, r =
unsteady disturbance index
cylindrical polar co-ordinates
Superscripts
__
=
′
=
∼
=
time–averaged
unsteady perturbation
complex harmonic amplitude
INTRODUCTION
To consider aero thermal performance and mechanical
integrity at the same time in the design process is highly
demanded due to the fact that an ‘aerodynamic optimum’ may
not satisfy mechanical and aeromechanical criteria. Thus, an
aerodynamics only design optimization might lead to
considerable time delay and extra costs in the blade redesign
cycles. On the other hand, multi-stage effects on both
aerodynamics and aeromechanics have been identified as
significant in gas-turbines. The current and future designs all
follow the trend for higher performance and more compact
structures. Increased aero-thermal loading of each blade row
would naturally lead to intensified interactions between
adjacent rows with ever reducing intra-row gap spacing. The
impacts of blade rows interaction would not only influence
aero-thermal performance, but also blade mechanical integrity
through flow induced vibrations (flutter or forced response).
However, so far, most design optimization systems focus on
aerodynamics performance for either isolated blade row
[1][2][3] or steady flow multi-row environment using the
conventional mixing-plane model [4~7]. The difficulty of
integration of aero thermal performance design optimization
and aeromechanical design optimization in the multistage
environment lies on the fact that the performance prediction
1
Copyright © 2006 by ASME
and the aeromechanical analysis are very often on different
platforms using different software with different data structure.
Further more, aeromechanical analysis could be much more
time consuming than the aerodynamics simulation as in a
typical flutter analysis, several vibration modes at various flow
conditions have to be examined. A fully integrated design
optimization system involved with both components in the
multistage environment could be practically infeasible.
Though design optimization systems differ in various
aspects, geometry parametrization, flow solver and optimizer
are the most common elements. The challenge for the geometry
parametrization is to provide accurate and flexible
representations by using as few parameters as possible so that
the effort of optimization could be reduced. 3-D
turbomachinery blades can be described by B-spline
construction [5], Bezier-patches representation [8] or Tensorproduct Bezier surface (Hoschek and Lasser [9], Arnone et al.
[10]). In either approach, at least 20 parameters are needed for
accurate description of each span wise section. Therefore more
than 100 parameters could be involved in the fully 3-D
geometry representation. In our current work, geometry
perturbations are represented by ‘aero mode shape’, which is
evolved from blade vibration mode shape but with zero
frequency. By using such kind aerodynamic mode shape, the
design parameters could be largely reduced to a few most
effective modes. However, there is always a tradeoff between
the flexibility of geometry representation and the number of
design parameters.
The flow solvers applied to numerical optimization are
closely linked to the selection of optimizer. In general,
optimization methods can be divided into gradient methods
such as Sequential Quadratic Programming (SQP) [5] and nongradient methods, e.g. Genetic Algorithm [8][10]. Non-gradient
methods have the ability of finding the global optimal while
gradient methods are relatively cheap. However, the number of
sensitivity equations is proportional to the number of design
parameters. Thus solving sensitivity equations is very time
consuming when the number of design parameters is large.
Implementation of the adjoint method [11] in sensitivity
analysis could avoid this problem by solving adjoint equations
which are independent of the number of design parameters. But
the derivation of adjoint equations and its corresponding
boundary conditions is never an easy job as it closely linked
with specific objective functions. In practice, the Response
Surface Method (RSM) and Design of Experiment (DOE) are
often coupled with either gradient methods or non-gradient
methods to reduce computational efforts.
As a compromise between fully integrated aerodynamics
aeromechanics design optimization and aerodynamics only
optimization, an aeromechanical approach for aerodynamics
design optimization with consideration of aeromechanical
constraints using harmonic perturbation analysis is pursued in
the current study. The harmonic methods have been developed
and successfully applied for flutter and forced response analysis
[12][13] and blade rows interaction problems [14]. Therefore
the same solver can be used for both aerodynamics
performance prediction and aeromechanical analysis.
Furthermore, the harmonic method can solve unsteady multirow interaction problems much faster than conventional time
marching methods. It has the potential to be applied to
multistage design optimization problems. Compared to
conventional gradient methods, the current method doesn’t
need to derive and solve separate gradient equations, but it is
more accurate than the finite difference approach for gradient
calculation as it separated perturbations from the mean flow.
The solution of harmonic perturbations can be utilized as
gradients of flow variables though it is not necessary to solve
the whole harmonic for steady state optimizations. Meanwhile,
‘aerodynamic mode shape’ is adopted to describe blade
geometry deformation, which simplified the parametrization
procedure and reduced the number of design parameters. An
additional advantage of using aerodynamic modes is that there
is no need for meshing individual design candidate in each
optimization loop, which could save some efforts in
interference with the meshing package. Requirement of
mechanical and aeromechanical integrity is considered as
constraints of the optimization problem.
AERODYNAMIC MODE
For the convenience of using the same methodology for
both aeromechanical analysis and aerodynamic optimization,
‘aerodynamic mode shapes’ are introduced to describe 3D
blade geometry deformation. The conventional 3-D design
features such as sweep and lean are effectively equivalent to
‘the first bending mode’, and the compound lean is equivalent
to ‘the second bending mode’ as shown on Fig. 1. Re-stagger of
blade sections in span wise direction could be simulated by ‘the
first torsion mode’. These aerodynamic modes can be defined
section by section or point by point as the result of FE model
analysis. For each mode, the modal amplitude is the design
parameter which will scale the geometry deformation of each
mesh point. As the mode shape of each individual point is
flexible, more complicated geometry deformation, e.g. change
of camber line and thickness can also be described point by
point and again the corresponding modal amplitude will scale
the local deformation.
The difference between the aerodynamic mode shape and
typical blade vibration mode shape is that these aerodynamic
modes have zero frequency and zero inter blade phase angle,
but with finite amplitude which represents the blade geometry
deformation. By using aerodynamic mode shape, conventional
sensitivity analysis of gradient methods could be replaced by
aerodynamic mode analysis, which is the same as the
aeromechanical analysis but with different mode shape. Instead
of getting unsteady flow field induced by blade vibration in the
aeromechanical analysis, aerodynamic mode analysis gives
perturbations of flow field due to the geometry deformation,
which is equivalent to the gradient of flow variables. This
aeromechanical approach not only avoids the complexity of
deriving and solving sensitivity equations, but also enables
aerodynamics performance design optimizations and
aeromechanical analysis to be carried out in a closely coupled
manner using the same solver.
FLOW SOLVER
The flow solver adopted here is based on the linear
harmonic method [14][15]. In this method, an unsteady flow
variable can be decomposed into a time-averaged part and an
unsteady perturbation, e.g.,
U = U + U′
2
(1)
Copyright © 2006 by ASME
NF ~
U ′( x , θ, r, t ) = ∑ ( U k e
k =1
iω k t
~
+U − k e − iω k t )
(2)
~
~
where U k and U − k are a pair of complex conjugates and NF is
the number of harmonics for the disturbance with given
frequency ωk. Substituting the above expression for the
conservative variables into the integral form of the unsteady
Navier-Stokes equation and time-averaging them, the resultant
time-averaged equation and harmonic equations are given as
∫∫ [Fn x + (G − Uv mg )n θ + Hn r ] ⋅ dA
δA
= ∫∫∫ Si ⋅ dV + ∫∫ [V x n x + V θ n θ + V r n r ] ⋅ dA
δV
~
V
=
∫∫∫
δ
V
(3)
δA
− iω ⋅U
∫∫∫
δ
k
⋅ dV +
~
(Si ) k ⋅ dV +
~
∫∫δ [F n
k x
~ ~
~
+ (Gk − U k v mg )nθ + H k n r ] ⋅ dA
A
~
∫∫δ [V n
x x
~
~
+ Vθ nθ + Vr n r ]k ⋅ dA
static stress analysis. Therefore it is a very fast process. At the
mean time, mode analysis is carried out so that all the vibration
mode shapes and their corresponding natural frequencies can be
found. If only the reduced frequency should be controlled for
flutter free design, these frequencies can be constrained. At the
second stage, when a temporal optimized design has been
found, flutter and/or forced response analysis will be conducted
to find out the exact aerodynamic damping value and forced
response level. The reason for carrying out detailed damping
evaluation is that the reduce frequency is not sufficient to rule
out flutter risk. This approach has the advantage that the
number of calls to forced response/flutter analysis is minimal as
it is relatively more time consuming than the aerodynamics
analysis.
(4)
A
(k = 1,2,⋅, , , N F )
These governing equations are discretized in space using
the cell-centred finite volume scheme, together with the
blended 2nd and 4th order artificial dissipation to damp
numerical oscillations. The time-averaged equation as well as
the harmonic equations are solved in a domain consisting of
one passage in each row by using the 4-stage Runge-Kutta time
marching scheme, accelerated by a time-consistent multi-grid
technique (He, 2000[16]) or conventional multi-grid with local
time stepping.
On solid blade/endwall surfaces, the log-law is applied to
determine the surface shear stress and the tangential velocity is
left to slip. Both 1-D and quasi-3D non-reflective boundary
conditions are available to be applied to the harmonic solution
at both the inlet and the exit.
Now the linear and nonlinear harmonic perturbations of
both aerodynamic modes and aeromechanical modes can be
solved using the same solver with different options.
MECHANICAL AND AEROMECHANICAL INTEGRITY
A typical blade design procedure starts with aerodynamics
and performance design with mechanical design constraints.
Aeromechanics analysis as well as mechanical analysis must be
conducted after aerodynamics design to examine if it is
mechanically sound. The main drawback of such a procedure is
that a good aerodynamics design may not satisfy mechanical
and aeromechanical requirement. Therefore aerodynamics
redesign needs to be taken to satisfy requirements for
mechanical integrity. To reduce the time and cost in
aerodynamics redesign and to minimize the number of design
iterations, mechanical
and
aeromechanical
integrity
requirement can be considered as constraints in the design
optimization procedure. Here, mechanical and aeromechanical
constraints are considered in two stages during the optimization
process. First of all, the static stress level of each design
scenario is evaluated by using FE analysis software ANSYS
and the maximum stress level has been constrained according
to the Goodman Diagram. By using moving grid technique,
there is no need of re-meshing each design scenario for the
OPTIMIZATION SYSTEM
Fig. 2 shows the diagram of the integrated optimization
system. It starts with a baseline design and a few identified aero
modes which are sensitive to the objective functions. The
aerodynamic perturbations of each mode can be found by
harmonic analysis of each individual mode shape. At the same
time, a number of design scenarios are constructed by
superposition of all the aero modes with a multiplier of the
mode amplitude. Based on linear assumption of perturbations
of each individual mode (which is the same for any sensitivity
analysis), the objective function of each reconstructed design
scenario can be evaluated by superimposing perturbations with
the same multiplier used for design scenario construction. In
order to check mechanical validity, FE analysis is carried out
for each design scenario to find out its maximum stress level as
well as vibration characteristics. A direct search is performed
here to find out the best of the constructed designs within the
constrained design space. The direct search is well suitable for
cases with a few design parameters. Also the direct search has
the capability to jump over local optimums where conventional
gradient based optimizers could be trapped in.
An aeromechanical analysis is then called for flutter and
forced response analysis to check if the found optimal design
satisfies aeromechanical constraints. If the found optimal
design doesn’t meet aeromechanical criteria, the system will
step back to the search subroutine to find out the next optimal
which also satisfies all the aerodynamics and mechanical
constraints. This local loop is continuously performed until the
sound aeromechanical design is found. The design loop is
finally closed by updating the baseline design with the found
optimal which is still not the maximum optimal, otherwise the
loop will stop and output the maximum optimal.
DESIGN PRACTISE
The optimization case considered here is to redesign
NASA rotor-67, which is a low aspect ratio transonic axial flow
fan rotor. The rotor design speed is 16043RPM and it has 22
blades. The predicted design point pressure ratio is 1.72 at a
mass flow rate of 34.07 kg/s. The Reynolds number based on
the mid-span chord length and the inlet free stream velocity is
1.0E6.
The objective function of this design optimization problem
is chosen to be the efficiency at design mass flow rate point.
Two 3-D design features, sweep and compound lean, have been
explored simultaneously with the following constraints:
Geometry constraints:
3
Copyright © 2006 by ASME
The maximal displacement of the geometry in each
direction should be constrained. For example, the axial
displacement of the blade is limited by the spacing between
adjacent rows. Here, we assume the following constraints.
∆x < 50% tip chord; ∆θ < 50% pitch
Performance constraints:
To achieve the design target, design mass flow rate and
design pressure ratio should be kept as close as possible to the
original values.
| δMassFlowRate |
< 0.5%
Design MassFlowRate
| δ Pr essureRatio |
< 0.5%
Design PressureRatio
Mechanical integrity:
The maximal stress level for mechanical integrity can be
determined by looking at the Goodman diagram with the
material property. Here we assume titanium blade and the stress
is limited as,
Maximum static stress < 800 M Pa.
Aeromechanical constraint:
Flutter safe aerodynamic damping > 0.5% (Log dec)
Fig. 3 shows the mesh distribution in a meridional plane
and a span wise section respectively. 110×35×29 meshes in
axial direction, pitch wise direction and radial direction are
used in the current study. The Mach contours at 90% span
section in comparison with the experimental data are plotted on
Fig. 4, which shows satisfactory numerical accuracy of the
present prediction.
Geometry change and its corresponding effects
The geometry change between NASA rotor 67 and the
optimized design is shown on Fig. 5, where both 3D shape
change and 2D section changes are plotted. It can be seen that
the optimized design is the combination of forward sweep and
backward compound lean (the suction surface leans toward the
pressure surface). The effects of compound lean have been
argued by many researchers [17][18]. Conventional theory
based on subsonic turbine design practice suggests that the
forward compound lean reduces loading on both root and tip
sections and increases loading on the mid-span section.
Therefore, high loss generated near end walls are driven
towards the middle passage and mixed with main flow, which
will reduce the overall loss generation. However, more recent
studies suggest that it could be case dependent [19]. In the
present design practice, the effect of compound lean is not
significant as it can be seen by the loading distribution shown
on Fig. 6. The surface pressure distribution on 25%, 50% and
85% span sections are plotted on Fig. 6. First of all, the overall
loading hasn’t been changed between two designs. Secondly, it
is observed that at the mid span section and the near hub
section where the passage shock was moved forward and the
strength has been reduced. This phenomenon is largely due to
the forward sweep effect and it can also be identified by
looking at the pressure contours distribution on both the
pressure surface and the suction surface as shown on Fig. 7.
The suction surface signature of the passage shock wave was
broken at about 80-85% span (Fig 7b) and the shock has been
moved forward to the leading edge, while on the pressure
surface it is weakened.
The overall loss generation inside the blade passage is then
calculated to show the contribution of different parts. The
spanwise distribution of the axial and pitchwise mass averaged
loss and the axial distribution of the spanwise and pitchwise
mass averaged loss are plotted on Fig. 8. Compared to the
original design, loss generation from the leading edge to the
mid chord part is almost the same. But the loss generation from
the mid chord afterward is consistently lower than the original
design. In span wise direction, loss generation in the mid-span
region didn’t change very much which confirms that the
compound lean effects is small. This might be due to the fact
that though the backward compound lean is applied, the
resultant geometry is still nearly straight. However, loss
generation above 70% span height is consistently lower in the
optimized design. This is due to the weakening of passage
shock as well as forward sweep effects on the tip leakage
vortex, which can be identified by the iso-surfaces of the loss
distribution near the tip region shown on Fig. 9. Compared to
NASA 67, the loss core of the optimized design, which is
located on the suction side about 70% downstream from the
leading edge, is smaller and the overall high loss region is
confined in a shorter area.
Performance evaluation& mechanical integrity
Table 1 lists the comparison of performance parameters
between NASA rotor 67 and the optimized design produced by
the current design optimization system. A 0.4% increase of
efficiency has been achieved at the design condition while
satisfying all constraints. To verify the off design performance
of the optimized design, a performance characteristic
calculation has been carried out. Fig. 10 shows the comparison
of predicted fan characteristic curve with the experiment data of
NASA rotor 67. The predicted performance agrees with the
experiment data very well, which confirms the performance
prediction capability of the current numerical method.
Compared to NASA rotor 67, the optimized design has higher
efficiency when reducing back pressure towards the choke
condition and the choke mass flow rate has been increased.
When the back pressure increased towards stall, the efficiency
and pressure ratio are almost the same as the NASA rotor 67
but the stall margin has been improved. This is mainly due to
the forward sweep effect which was observed previously by
Denton and Xu [18].
The mechanical and aeromechanical integrity have been
examined along the design optimization procedure. Comparison
of the static stress distribution is plotted on Fig. 11, where the
maximum stress has been shifted from the hub leading edge to
the mid chord. Also the high stress area is concentrated in the
centre of the optimized blade which is corresponding to the
compound lean. The maximum static stress level has been
increased by ~33% though it is still within the safety threshold.
This indicates that further improvement of aerodynamics
performance has been prevented by mechanical constraints.
Flutter stability analysis of the NASA rotor 67 shows that
the first flap mode at 2 nodal diameters in a forward traveling
wave mode is least stable. Thus the aerodynamic damping of
the optimized design of this mode has been examined and it is
slightly higher than the original design. It is interesting to
notice that though the natural frequency of the first five modes
of the optimized design are consistently lower than NASA67 by
1%~4%, the aerodynamic damping values are not necessarily
lower than the original design. A forced response analysis
should be carried out with both damping and forcing
calculation at the engine crossing point flow condition. There is
no practical data available for this task to be conducted though
4
Copyright © 2006 by ASME
the system is ready for performing such a task. Nevertheless,
the current finding still shows the necessity for more concurrent
integration of aerodynamics and aeromechanics design
optimization.
CONCLUSIONS
A 3-D design optimization method for aerodynamics
performance optimization with consideration of mechanical and
aeromechanical integrity has been developed. The developed
system uses an aeromechanical approach with ‘aerodynamic
mode’ describing geometry deformation and the harmonic
perturbation method has been adopted as the gradient solver.
The redesign practice of NASA rotor-67 shows that the present
optimization approach can achieve a 0.4% increase in peak
efficiency while all the aerodynamics and aeromechanical
constraints are well satisfied. The loss reduction in the current
design practice seems caused by the weakening of passage
shock and the reducing of loss core near the tip of the suction
surface. Flutter analysis shows the minimal damping value of
the first flap mode is not decreased though the vibration
frequency is reduced. Nevertheless, the maximum blade static
stress has been increased by 33%. The present case study and
results highlight the need for more concurrent mechanic,
aerodynamic and aeromechanic design optimizations.
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Copyright © 2006 by ASME
Table 1 Comparison between NASA rotor 67 and the optimized design
Mass Flow
Pressure Ratio
Efficiency
Static Stress
NASA 67
34.07 kg/s
1.7246
92.01%
493 MPa
Optimized Design
34.05kg/s
1.7244
92.41%
654 MPa
Change
0.06%
0.01%
0.40%
32.6%
Table 2 Flutter analysis
Frequency (Hz)
Aero-damping, ND=2
(Log_dec)
NASA67 Optimized
NASA67
Optimized
design
design
599.181
592.170
1.21%
1.41%
1303.717 1299.516
1910.860 1818.677
2705.744 2659.112
3093.419 2973.350
Mode No.
1
2
3
4
5
Sweep
Lean
Compound Lean
Tip
Hub
Axial direction
Circumferential direction
Fig. 1 Description of the aerodynamic mode
LOOP
Start
N
Nonlinear steady flow
Base Flow Cal.
Loop
N=1,N_DESP
Y
Max Optimal?
Return
Y
Linear perturbation solutions
Reconstruction from
perturbed design parameters
Gradient Cal.
Scenarios
Reconstruction
AeroMechanical
Valid?
FE mode analysis (ANSYS)
Stress & Freq.
Assess
N
Aero-damping Cal.
Forced Resp. Cal.
Optimal
Search
Aeromechanics analysis
Update Baseline
Design
Fig. 2 Flow chart of the optimization system
6
Copyright © 2006 by ASME
Fig. 3 Mesh distribution in a span wise section and the meridional plane
1.35
1.42
93
0.
1.
35
0.81
Fig. 4 Mach contours comparison at 90% span section (left: Calc; right: Exp)
Optimized blade
NASA rotor 67
Fig. 5a 3D blade shape comparison between NASA67 and the optimized design
7
Copyright © 2006 by ASME
85% Span
NASA67
Optimized Design
50% Span
PS
SS
25% Span
Fig. 5b Blade profiles at different radial sections and an axial view at mid chord
Cp
NASA67
Optimized Design
1.4
NASA67
Optimized Design
Cp
Cp
1.5
NASA67
Optimized Design
1.4
1.4
1.3
1.3
1.2
1.2
1.2
1.1
1.1
1.1
1.3
1
1
1
0.9
0.9
0.9
X/C
X/C
0
0.25
0.5
(a) 25% span
0.75
1
0.8
0
0.25
0.5
0.75
(b) 50% span
1
0.8
X/C
0
0.25
0.5
0.75
1
(c) 85% span
Fig. 6 Surface pressure distribution on different span wise sections
a) Pressure surface pressure contours
b) Suction surface pressure contours
Fig. 7 Comparison of static pressure on pressure and suction surfaces
8
Copyright © 2006 by ASME
R/Span
Loss
1
0.04
0.9
0.8
0.035
Optimized design
NASA 67
0.7
0.03
0.6
0.025
Optimized design
NASA 67
0.5
0.02
0.4
0.015
0.3
0.01
0.2
0.005
0.1
L.E.
Loss
0
0.025
0.05
0.075
0.1
0
T.E.
1
X/C
2
(a) Spanwise distribution of axial and pitchwise averaged loss
(b) Axial distribution of pitchwise and spanwise averaged loss
Fig. 8 Mass averaged loss generation in the passage
Trailing
edge
Leading
edge
Optimized
Design
Loss Core
(0.23-0.5)
Trailing
edge
Leading
edge
NASA67
Fig. 9 Iso-surfaces of passage loss distribution
9
Copyright © 2006 by ASME
1.9
93
NASA 67
Optimized design
Experiment
92
1.85
92
1.8
91
91
89
1.7
1.65
88
1.6
87
1.55
86
1.5
Efficiency
Pressure Ratio
1.75
90
Efficiency
NASA67
NASA67
Lean+Sweep
Lean+Sweep
93
90
89
88
87
1.45
85
29
30
31
32
33
34
1.4
29
35
Flow rate
86
8530
29
31
30
32
31
3332
Flow rate Flow rate
3433
3534
(a) Efficiency-Mass flow curve
(b) Pressure-Mass flow curve
Fig. 10 Fan performance characteristic curves
Original Design
Optimized design
Max Stress
Fig. 11 Blade static stress contours
10
Copyright © 2006 by ASME