High-fidelity simulations, e.g., large eddy simulation (LES), are often needed for accurately predicting pressure losses due to wake mixing and boundary layer development in turbomachinery applications. An unsteady adjoint of high-fidelity simulations is useful for design optimization in such aerodynamic applications. In this paper, we present unsteady adjoint solutions using a large eddy simulation model for an inlet guide vane from von Karman Institute (VKI) using aerothermal objectives. The unsteady adjoint method is effective in capturing the gradient for a short time interval aerothermal objective, whereas the method provides diverging gradients for long time-averaged thermal objectives. As the boundary layer on the suction side near the trailing edge of the vane is turbulent, it poses a challenge for the adjoint solver. The chaotic dynamics cause the adjoint solution to diverge exponentially from the trailing edge region when solved backward in time. This results in the corruption of the sensitivities obtained from the adjoint solutions. An energy analysis of the unsteady compressible Navier–Stokes adjoint equations indicates that adding artificial viscosity to the adjoint equations can dissipate the adjoint energy while potentially maintaining the accuracy of the adjoint sensitivities. Analyzing the growth term of the adjoint energy provides a metric for identifying the regions in the flow where the adjoint term is diverging. Results for the vane obtained from simulations performed on the Titan supercomputer are demonstrated.

References

References
1.
Gourdain
,
N.
,
Gicquel
,
L. Y.
, and
Collado
,
E.
,
2012
, “
Comparison of RANS and LES for Prediction of Wall Heat Transfer in a Highly Loaded Turbine Guide Vane
,”
J. Propul. Power
,
28
(
2
), pp.
423
433
.
2.
Jameson
,
A.
,
1995
, “
Optimum Aerodynamic Design Using CFD and Control Theory
,”
AIAA
Paper No. 95-1729.
3.
Lyu
,
Z.
, and
Martins
,
J. R.
,
2014
, “
Aerodynamic Design Optimization Studies of a Blended-Wing-Body Aircraft
,”
J. Aircr.
,
51
(
5
), pp.
1604
1617
.
4.
Economon
,
T. D.
,
Palacios
,
F.
, and
Alonso
,
J. J.
,
2013
, “
A Viscous Continuous Adjoint Approach for the Design of Rotating Engineering Applications
,”
AIAA
Paper No. 2013-2580.
5.
Wang
,
Q.
, and
Gao
,
J.-H.
,
2013
, “
The Drag-Adjoint Field of a Circular Cylinder Wake at Reynolds Numbers 20, 100 and 500
,”
J. Fluid Mech.
,
730
, pp.
145
161
.
6.
Blonigan
,
P.
,
Chen
,
R.
,
Wang
,
Q.
, and
Larsson
,
J.
,
2012
, “
Towards Adjoint Sensitivity Analysis of Statistics in Turbulent Flow Simulation
,”
Stanford Center of Turbulence Research Summer Program
, p.
229
.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.386.5323&rep=rep1&type=pdf
7.
Aceves
,
A.
,
Adachihara
,
H.
,
Jones
,
C.
,
Lerman
,
J. C.
,
McLaughlin
,
D. W.
,
Moloney
,
J. V.
, and
Newell
,
A. C.
,
1986
, “
Chaos and Coherent Structures in Partial Differential Equations
,”
Phys. D: Nonlinear Phenom.
,
18
(
1
), pp.
85
112
.
8.
Wilcox
,
D. C.
,
1998
,
Turbulence Modeling for CFD
, Vol.
2
,
DCW Industries
,
La Canada, CA
.
9.
Arts
,
T.
, and
de Rouvroit
,
M. L.
,
1992
, “
Aero-Thermal Performance of a Two-Dimensional Highly Loaded Transonic Turbine Nozzle Guide Vane: A Test Case for Inviscid and Viscous Flow Computations
,”
ASME J. Turbomach.
,
114
(
1
), pp.
147
154
.
10.
Lea
,
D. J.
,
Allen
,
M. R.
, and
Haine
,
T. W.
,
2000
, “
Sensitivity Analysis of the Climate of a Chaotic System
,”
Tellus A
,
52
(
5
), pp.
523
532
.
11.
Thuburn
,
J.
,
2005
, “
Climate Sensitivities Via a Fokker–Planck Adjoint Approach
,”
Q. J. R. Meteorol. Soc.
,
131
(
605
), pp.
73
92
.
12.
Ruelle
,
D.
,
2008
, “
Differentiation of SRB States for Hyperbolic Flows
,”
Ergodic Theory Dyn. Syst.
,
28
(
02
), pp.
613
631
.
13.
Ruelle
,
D.
,
2009
, “
A Review of Linear Response Theory for General Differentiable Dynamical Systems
,”
Nonlinearity
,
22
(
4
), p.
855
.
14.
Blonigan
,
P.
,
Gomez
,
S.
, and
Wang
,
Q.
,
2014
, “
Least Squares Shadowing for Sensitivity Analysis of Turbulent Fluid Flows
,” preprint arXiv:1401.4163.
15.
Garnier
,
E.
,
Adams
,
N.
, and
Sagaut
,
P.
,
2009
,
Large Eddy Simulation for Compressible Flows
,
Springer Science & Business Media
, Medford, MA.
16.
Moeng
,
C.-H.
, and
Wyngaard
,
J. C.
,
1989
, “
Evaluation of Turbulent Transport and Dissipation Closures in Second-Order Modeling
,”
J. Atmos. Sci.
,
46
(
14
), pp.
2311
2330
.
17.
Macdonald
,
C. B.
,
2003
, “
Constructing High-Order Runge-Kutta Methods With Embedded Strong-Stability-Preserving Pairs
,”
Ph.D. thesis
, Simon Fraser University, Burnaby, BC.http://people.math.sfu.ca/~cbm/mscthesis/cbm-mscthesis.pdf
18.
Roe
,
P. L.
,
1981
, “
Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes
,”
J. Comput. Phys.
,
43
(
2
), pp.
357
372
.
19.
Choi
,
H.
, and
Moin
,
P.
,
2012
, “
Grid-Point Requirements for Large Eddy Simulation: Chapman’s Estimates Revisited
,”
Phys. Fluids (1994-Present)
,
24
(
1
), p.
011702
.
20.
Bastien
,
F.
,
Lamblin
,
P.
,
Pascanu
,
R.
,
Bergstra
,
J.
,
Goodfellow
, I
. J.
,
Bergeron
,
A.
,
Bouchard
,
N.
, and
Bengio
,
Y.
,
2012
, “
Theano: New Features and Speed Improvements
,”
Deep Learning and Unsupervised Feature Learning
NIPS 2012
Workshop, p. 5590.https://arxiv.org/abs/1211.5590
21.
Bergstra
,
J.
,
Breuleux
,
O.
,
Bastien
,
F.
,
Lamblin
,
P.
,
Pascanu
,
R.
,
Desjardins
,
G.
,
Turian
,
J.
,
Warde-Farley
,
D.
, and
Bengio
,
Y.
,
2010
, “
Theano: A CPU and GPU Math Expression Compiler
,”
Python for Scientific Computing Conference
(
SciPy
).http://www-etud.iro.umontreal.ca/~wardefar/publications/theano_scipy2010.pdf
22.
Abarbanel
,
S.
, and
Gottlieb
,
D.
,
1981
, “
Optimal Time Splitting for Two-and Three-Dimensional Navier-Stokes Equations With Mixed Derivatives
,”
J. Comput. Phys.
,
41
(
1
), pp.
1
33
.
23.
Ford
,
W. F.
, and
Sidi
,
A.
,
1987
, “
An Algorithm for a Generalization of the Richardson Extrapolation Process
,”
SIAM J. Numer. Anal.
,
24
(
5
), pp.
1212
1232
.
24.
Celik
,
I.
, and
Zhang
,
W.-M.
,
1995
, “
Calculation of Numerical Uncertainty Using Richardson Extrapolation: Application to Some Simple Turbulent Flow Calculations
,”
ASME J. Fluids Eng.
,
117
(
3
), pp.
439
445
.
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