turbomachinery aerodynamic and aeromechanic design optimization using the adjoint method

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Durham E-Theses
Turbomachinery aerodynamic and aeromechanic design
optimization using the adjoint method
Wang, Dingxi
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2
TURBOMACHINERY AERODYNAMIC
AND AEROMECHANIC
DESIGN OPTIMIZATION
USING THE ADJOINT METHOD
Dingxi Wang
School of Engineering
University of Durham
The copyright of this thesis rests with the
author or the university to which it was
submitted. No quotation from it, or
information derived from it may i^e
published without the prior written
consent of the author or university, and
any information derived from it should be
acknowledged.
A thesis submitted to University of Durham
for the degree of Doctor of Philosophy
0 7 OCT 2008
July 2008
V'l/1 :/
Declaration
I declare that no material presented within this thesis has been submitted towards a
degree or qualification at this or any other university
Signiture:_
date: I ( o%\7^fS
Copyright© July 2008 by Dingxi Wang
The copyright of this thesis rests with the author. No quotation from it should be
published in any format, including electronic and the internet, without the author's
prior written consent. Any information derived from it should be acknowledged appro
priately.
Acknowledgements
I want to express my gratitude to my supervisor Prof. Li He for his offering me this
studentship, his patient guidance, his generosity in financial and spiritual support,
and his advice beyond academic issues. I would like to thank Prof. Roger Crouch
for acting as my second supervisor during the last several months of this study, being
my first year examiner and proofreading my thesis. Thanks go to Dr. Haidong Li
for his help during the early stage of this research work and his advice via loads of
emails during later stage of this research work. Dr. Xiuquan Huang deserves my
heart-felt gratitude for his enormous help in Durham. Thanks also go to Mrs. Ying
Dong for her friendship. Thank Mrs. Vera Menshikova and Mr. Orhan Aybay for
their harmonious accompany as officemates. Efforts of proofreading the thesis by Mr.
Jinguang Yang and Mr. Richard Williams are greatly appreciated. Siemens Industrial
Turbomachinery Ltd. Lincoln has been providing funding for the work. Discussions
with and suggestions from Dr. Yansheng Li, Dr. Tie Chen and Mr. Roger Wells at
Siemens are constructive.
Last, but not least, thanks go to my parents and my wife for their unconditional love.
Abstract
The thesis documents the investigation of the appUcation of the adjoint method to
turbomachinery blading design optimization, with emphasis on blading aerodynamic
design optimization in a multi-bladerow environment and concurrent blading aerody
namic and aeromechanic design optimization for a single bladerow.
Based on the nonlinear flow equations, a steady adjoint system has been developed
using the continuous adjoint approach. The capability of the conventional adjoint
system has been augmented by the introduction of an adjoint mixing-plane treatment.
This treatment is a counterpart of the flow mixing-plane treatment, enabhng the steady
adjoint equations to be solved in multi-bladerow computational domains. This allows
turbomachinery blades to be optimised to enhance their aerodynamic performance in
a multi-bladerow environment with matching between adjacent bladerows dealt with
in a timely manner.
The Nonlinear Harmonic Phase Solution method, a neat frequency domain method
catered specifically for turbomachinery aeromechanics prediction, has been chosen to
integrate with the adjoint method to calculate objective function sensitivities efficiently
for concurrent aeromechanic and aerodynamic design optimization for single row tur
bomachinery blades. The Nonlinear Harmonic Phase Solution method, unlike the
time-linearized methods, solves the unsteady flow equations at two or three carefully
selected phases of a period of unsteadiness. This approach not only can conveniently
turn a steady flow solver to one solving the unsteady flow equations efficiently, but also
provides a good basis on which the corresponding adjoint system can be formulated
and solved in a similar manner by extending a steady adjoint system.
In order to resolve the issue of having a good blading performance over a whole oper
ating range at a given operation speed, a multi-operating-point design optimization is
implemented by formulating an objective function of a weighted sum of performance
at more than one operating point.
Keywords: design optimization, aerodynamics, aeromechanics, adjoint method, gra
dient, objective function, blade shape
Contents
1 Introduction 1
1.1 Background and Motivation 1
1.2 Turbomachinery Blading Design Optimization 2
1.3 Essential Elements for Blading Design Optimization 4
1.4 Key Methodologies 7
1.5 Overview of the Thesis 8
2 Literature Review 10
2.1 Blading Aerodynamics Calculation 10
2.1.1 Flow Mixing-plane Treatment 11
2.2 Blading Aeromechanics Calculation 13
2.2.1 Unsteady Flow Solution Methods for Aeromechanics 15
2.2.1.1 Time Domain Methods 15
2.2.1.2 Frequency Domain Methods 16
2.3 Methods for Turbomachinery Blading Design Optimization 20
2.4 Development and Application of the Adjoint Method 23
2.4.1 External Steady Flow Application 26
2.4.2 External Unsteady Flow Application 27
2.4.3 Internal Steady Flow Application 27
2.4.4 Internal Unsteady Flow Application 28
2.5 Summary 29
3 Steady Nonlinear Flow Model and Solution Methods 31
3.1 Reynolds Averaged Navier-Stokes Elquations 31
3.2 Turbulence Models 33
3.2.1 Baldwin Lomax Model 34
3.2.2 Spalart Allmaras Model 35
3.3 Boundary Conditions 36
3.4 Flow Mixing-plane Treatment 38
3.5 Discretization of the RANS Equations 42
3.5.1 Spatial Discretization 42
3.5.2 Temporal Discretization 45
3.6 Summary 46
4 Development of Steady Flow Adjoint System 47
4.1 Adjoint Formulation Principle 47
4.2 Steady Adjoint Equations 50
4.3 Boundary Conditions 54
4.4 Adjoint Mixing-plane Treatment 58
4.5 Solution of the Adjoint Equations 63
4.6 Summary 65
5 Blading Aerodynamic Design and Optimization System 66
5.1 Gradient Calculation 66
5.2 Objective Function 67
5.3 Shape Parameterization 69
5.4 Single-Operating-Point Design Optimization Process 73
5.5 Multi-Operating-Point Design Optimization 74
5.6 Summary 77
6 Verification and Application of Aerodynamic Design Optimization
System 78
6.1 Verification of Gradient Calculation 78
6.1.1 Verifying the Adjoint Solver 78
6.1.2 Verifying the Adjoint Mixing-plane Treatment 83
6.2 Application of Aerodynamic Design Optimization System 87
6.2.1 Redesign of NASA Rotor 67 at a Single Operating Point . . .. 88
6.2.2 Redesign of DLR Compressor Stage at a Single Operating Point 93
6.2.3 Redesign of ATC Compressor 99
u
6.3 Summary 110
7 Nonlinear Harmonic Phase Solution Method for Unsteady Flows 112
7.1 Governing Equations 112
7.2 Boundary Conditions 114
7.3 Validation of the Unsteady Flow Solver 116
7.3.1 Validation against a semi-analytic method 116
7.3.2 Validation against 3D linear cascade experimental data 118
7.4 Summary 122
8 Development of Unsteady Flow Adjoint System 126
8.1 Unsteady Adjoint Systems 126
8.1.1 Adjoint Formulation Principle 127
8.1.2 Unsteady Adjoint Equations 129
8.1.3 Boundary Conditions 131
8.2 Worksum and its Gradient 135
8.3 Verifying the Unsteady Adjoint Solver 136
8.4 Summary 138
9 Concurrent Aeromechanic and Aerodynamic Design Optimization 139
9.1 Introduction 139
9.2 Objective Function 142
9.3 Structural Modehng and FE-CFD Interpolation 142
9.4 Design Optimization Process 144
9.5 Multi-Operating-Point Design Optimization 146
9.6 Design Optimization Case Studies 148
9.6.1 Redesign of 2D Rotor Blade Configuration 148
9.6.2 Redesign of DLR Rotor at a Single Operating Point 151
9.6.3 Redesign of DLR Rotor at Multiple Operating Points 157
9.7 Summary 162
10 Conclusions and Recommendations for Future Work 164
10.1 Conclusions 164
10.2 Recommendations for Future Work 167
ni
Bibliography 169
A Adjoint Navier-Stokes Equations 180
B Adjoint Viscous Solid Wall Boundary Condition 185
C Objective Function Gradients 190
D Nonlinear Harmonic Phase Solution Method versus Time-linearized
Linear Harmonic Method 193
E Time-Domain Unsteady Adjoint Equations 197
IV