Increasing aerothermal and aero-elastic performance requirements and constraints are closely linked in modern blading designs. There is thus a need for more concurrent interaction between the disciplines at earlier stages of a design process. Presented in this paper are the development, validation, and demonstration of the adjoint approach to concurrent blading aerodynamic and aero-elastic design optimizations. A nonlinear harmonic phase solution method is adopted to solve the unsteady Reynolds-averaged Navier–Stokes equations. The flow field response in terms of both the mean aerothermal performance and aero-elastic stability to a geometrical perturbation can be obtained by three “steadylike” flow solutions at three distinctive temporal phases. This unsteady flow solution method is computationally very efficient and provides a convenient and consistent base for formulating the corresponding adjoint equations. The adjoint system for the unsteady flow solver is solved effectively by a relatively simple extension of the method and techniques previously developed for a steady flow adjoint solver. As a result, the sensitivities of both the steady (time-mean) flow loss and the aerodynamic damping/forcing to detailed blade geometry changes can be very efficiently obtained by solving equivalently three steadylike adjoint equations. Several case studies are presented to illustrate the validity and effectiveness of this new concurrent approach.

1.
Vahdati
,
M.
,
Sayma
,
A. I.
,
Marshall
,
J. G.
, and
Imregun
,
M.
, 2001, “
Mechanisms and Prediction Methods for Fan Blade Stall Flutter
,”
J. Propul. Power
0748-4658,
17
(
5
), pp.
1100
1108
.
2.
He
,
L.
, and
Ning
,
W.
, 1998, “
An Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines
,”
AIAA J.
0001-1452,
36
(
11
), pp.
2005
2012
.
3.
Hall
,
K. C.
,
Thomas
,
J. P.
, and
Clark
,
W. S.
, 2002, “
Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique
,”
AIAA J.
0001-1452,
40
(
5
), pp.
879
86
.
4.
Jameson
,
A.
, 1988, “
Aerodynamic Design Via Control Theory
,”
J. Sci. Comput.
0885-7474,
3
(
3
), pp.
233
260
.
5.
Giles
,
M. B.
, and
Pierce
,
N. A.
, 2000, “
An Introduction to the Adjoint Approach to Design
,”
Flow Turbul. Combust.
,
65
(
3/4
), pp.
393
415
.
6.
Mohammadi
,
B.
, and
Pironneau
,
O.
, 2004, “
Shape Optimization in Fluid Mechanics
,”
Annu. Rev. Fluid Mech.
0066-4189,
36
, pp.
255
279
.
7.
Wang
,
D. X.
, and
He
,
L.
2008, “
Adjoint Aerodynamic Design Optimization for Blades in Multi-Stage Turbomachines: Part I—Methodology and Verification
,” ASME Paper No. GT2008-50208.
8.
Wang
,
D. X.
,
He
,
L.
,
Li
,
Y. S.
,
Wells
,
R. G.
, and
Chen
,
T.
, 2008, “
Adjoint Aerodynamic Design Optimization for Blades in Multi-Stage Turbomachines: Part II—Validation and Application
,” ASME Paper No. GT2008-50209.
9.
Nadarajah
,
S. K.
, and
Jameson
,
A.
, 2002, “
Optimal Control of Unsteady Flows Using a Time Accurate Method
,” AIAA Paper No. 02-5436.
10.
Nadarajah
,
S. K.
, and
Jameson
,
A.
, 2006, “
Optimum Shape Design for Unsteady Three Dimensional Viscous Flows Using a Non-Linear Frequency Domain Method
,” AIAA Paper No. 06-3455.
11.
Thomas
,
J. P.
,
Hall
,
K. C.
, and
Dowell
,
E. H.
, 2003, “
A Discrete Adjoint Approach for Modeling Unsteady Aerodynamic Design Sensitivities
,” AIAA Paper No. 03-0041.
12.
Duta
,
M. C.
,
Giles
,
M. B.
, and
Campobasso
,
M. S.
, 2002, “
The Harmonic Adjoint Approach to Unsteady Turbomachinery Design
,”
Int. J. Numer. Methods Fluids
0271-2091,
40
, pp.
323
332
.
13.
He
,
L.
, 2008, “
Harmonic Solution of Unsteady Flow Around Blades With Separation
,”
AIAA J.
0001-1452,
46
(
6
), pp.
1299
1307
.
14.
Moffat
,
S.
, and
He
,
L.
, 2003, “
Blade Forced Response Prediction for Industrial Gas Turbines, Part I: Methodologies
,” ASME Paper No. GT2003-38640.
15.
Ning
,
W.
,
Moffat
,
S.
,
Li
,
Y. S.
,
Wells
,
R. G.
, and
He
,
L.
, 2003, “
Blade Forced Response Prediction for Industrial Gas Turbines, Part 2: Verification and Application
,” ASME Paper No. GT2003-38642.
16.
He
,
L.
, 2003, “
Unsteady Flow and Aeroelasticity
,”
Handbook of Turbomachinery
,
2nd ed.
,
E.
Logan
and
R.
Roy
, eds.,
Dekker
,
New York
, Chap. 5.
17.
Yang
,
H.
, and
He
,
L.
, 2004, “
Experimental Investigation of Linear Compressor Cascade With 3-D Blade Oscillation
,”
J. Propul. Power
0748-4658,
20
(
1
), pp.
180
188
.
18.
Wang
,
D. X.
, 2008, “
Turbomachinery Aerodynamic and Aeromechanic Design Optimization Using the Adjoint Method
,” Ph.D. thesis, Durham University, Durham, UK.
19.
Dunker
,
R.
,
Rechter
,
H.
,
Starken
,
H.
, and
Weyer
,
H.
, 1984, “
Redesign and Performance Analysis of a Transonic Axial Compressor Stator and Equivalent Plane Cascades With Subsonic Controlled Diffusion Blades
,”
ASME J. Eng. Gas Turbines Power
0742-4795,
106
, pp.
279
287
.
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