In the current paper nondeterministic computational fluid dynamics (CFD) computations of three-dimensional (3D), developing, and statistically steady turbulent flow through an asymmetric diffuser with moderate adverse pressure gradient are presented. The inflow condition is assumed to be uncertain. The inlet streamwise velocity is supposed to be a stochastic process and described by the Karhunen–Loève (KL) expansion. In addition, the inlet turbulence intensity and turbulent length scale are assumed to be uncertain. The nonintrusive polynomial chaos (NIPC) expansion is used to propagate the inflow uncertainties in the flow field. The developed code is verified using a Monte Carlo (MC) simulation with 1000 Latin Hypercube samples on a planar asymmetric diffuser. A very good agreement is observed between the results of MC and polynomial chaos expansion methods. The verified uncertainty quantification method is then applied to stochastic developing turbulent flow through a 3D asymmetric diffuser. It was observed that the eigenvalues of covariance kernel rapidly decay due to the large correlation lengths and thus a few terms in the truncated KL expansion are used to describe the stochastic inlet velocity. For the KL expansion, the mean and the standard deviation are set to those measured experimentally. The uncertain inlet condition has a significant influence on the numerical results of velocity and turbulence fields specially in the developing region before the shear layers meet. It is concluded that one of the reasons for discrepancies between experimental and deterministic CFD results is the uncertainty in inflow condition. A sensitivity analysis is also performed using the Sobol’ indices and contribution of each uncertain parameter on outputs variance is presented.

References

References
1.
Fishman
,
G.
,
2013
,
Monte Carlo: Concepts, Algorithms, and Applications
,
Springer Science & Business Media
,
New York
.
2.
Ghanem
,
R. G.
, and
Spanos
,
P. D.
,
2003
,
Stochastic Finite Elements: A Spectral Approach
,
Courier Corporation
,
Mineola, New York
.
3.
Wiener
,
N.
,
1938
, “
The Homogeneous Chaos
,”
Am. J. Math.
,
60
(
4
), pp.
897
936
.
4.
Ghanem
,
R.
, and
Kruger
,
R. M.
,
1996
, “
Numerical Solution of Spectral Stochastic Finite Element Systems
,”
Comput. Methods Appl. Mech. Eng.
,
129
(
3
), pp.
289
303
.
5.
Ghanem
,
R.
,
1999
, “
Ingredients for a General Purpose Stochastic Finite Elements Implementation
,”
Comput. Methods Appl. Mech. Eng.
,
168
(
1–4
), pp.
19
34
.
6.
Ghanem
,
R.
,
1999
, “
Stochastic Finite Elements With Multiple Random Non-Gaussian Properties
,”
J. Eng. Mech.
,
125
(
1
), pp.
26
40
.
7.
Ghanem
,
R.
, and
Red-Horse
,
J.
,
1999
, “
Propagation of Probabilistic Uncertainty in Complex Physical Systems Using a Stochastic Finite Element Approach
,”
Physica D
,
133
(
1–4
), pp.
137
144
.
8.
Debusschere
,
B. J.
,
Najm
,
H. N.
,
Pebay
,
P. P.
,
Knio
,
O. M.
,
Ghanem
,
R. G.
, and
Maitre
,
O. P. L.
,
2004
, “
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
,”
SIAM J. Sci. Comput.
,
26
(
2
), pp.
698
719
.
9.
Reagana
,
M. T.
,
Najm
,
H. N.
,
Ghanem
,
R. G.
, and
Knio
,
O. M.
,
2003
, “
Uncertainty Quantification in Reacting-Flow Simulations Through Non-Intrusive Spectral Projection
,”
Combust. Flame
,
132
(
3
), pp.
545
555
.
10.
Mathelin
,
L.
,
Hussaini
,
M. Y.
, and
Zang
,
T. A.
,
2005
, “
Stochastic Approaches to Uncertainty Quantification in CFD Simulations
,”
Numer. Algorithms
,
38
(
1–3
), pp.
209
236
.
11.
Hosder
,
S.
,
Walters
,
R. W.
, and
Perez
,
R.
,
2006
, “
A Non-Intrusive Polynomial Chaos Method for Uncertainty Propagation in CFD Simulations
,”
AIAA
Paper No. 2006-891.
12.
Hosder
,
S.
,
Walters
,
R. W.
, and
Balch
,
M.
,
2008
, “
Efficient Uncertainty Quantification Applied to the Aeroelastic Analysis of a Transonic Wing
,” AIAA Paper No. 2008-729.
13.
Bettis
,
B. R.
, and
Hosder
,
S.
,
2010
, “
Quantification of Uncertainty in Aerodynamic Heating of a Reentry Vehicle Due to Uncertain Wall and Freestream Conditions
,”
AIAA
2010-4642.
14.
Raisee
,
M.
,
Kumar
,
D.
, and
Lacor
,
C.
,
2015
, “
A Non-Intrusive Model Reduction Approach for Polynomial Chaos Expansion Using Proper Orthogonal Decomposition
,”
Int. J. Numer. Methods Eng.
,
103
(
4
), pp.
293
312
.
15.
Najm
,
H. N.
,
2009
, “
Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics
,”
Annu. Rev. Fluid Mech.
,
41
(
1
), pp.
35
52
.
16.
Loeven
,
G. J. A.
, and
Bijl
,
H.
,
2010
, “
The Application of the Probabilistic Collocation Method to a Transonic Axial Flow Compressor
,”
AIAA
Paper No. 2010-2923.
17.
Petrone
,
G.
,
De Nicola
,
C.
,
Quagliarella
,
D.
,
Witteveen
,
J.
, and
Iaccarino
,
G.
,
2011
, “
Wind Turbine Performance Analysis Under Uncertainty
,”
AIAA
Paper No. 2011-544.
18.
Liu
,
Z.
,
Wang
,
X.
, and
Kang
,
S.
,
2014
, “
Stochastic Performance Evaluation of Horizontal Axis Wind Turbine Blades Using Non-Deterministic CFD Simulations
,”
Energy
,
73
, pp.
126
136
.
19.
Pečnik
,
R.
,
Witteveen
,
J. A.
, and
Iaccarino
,
G.
,
2011
, “
Uncertainty Quantification for Laminar-Turbulent Transition Prediction in RANS Turbomachinery Applications
,”
AIAA
Paper No. 2011-660.
20.
Gopinathrao
,
N. P.
,
Mabilat
,
C.
, and
Alizadeh
,
S.
,
2009
, “
Non-Deterministic Thermo-Fluid Analysis of a Compressor Rotor-Stator Cavity
,”
AIAA
Paper No. 2009-2278.
21.
Han
,
X.
,
Sagaut
,
P.
, and
Lucor
,
D.
,
2012
, “
On Sensitivity of RANS Simulations to Uncertain Turbulent Inflow Conditions
,”
Comput. Fluids
,
61
, pp.
2
5
.
22.
Congedo
,
P. M.
,
Duprat
,
C.
,
Balarac
,
G.
, and
Corre
,
C.
,
2013
, “
Numerical Prediction of Turbulent Flows Using Reynolds-Averaged Naiver–Stokes and Large-Eddy Simulation With Uncertain Inflow Conditions
,”
Int. J. Numer. Methods Fluids
,
72
(
3
), pp.
341
358
.
23.
Ko
,
J.
,
Lucor
,
D.
, and
Sagaut
,
P.
,
2008
, “
Sensitivity of Two-Dimensional Spatially Developing Mixing Layers With Respect to Uncertain Inflow Conditions
,”
Phys. Fluids
,
20
(
7
), p.
077102
.
24.
Abdelaziz
,
O.
, and
Radermacher
,
R.
,
2010
, “
Modeling Heat Exchangers Under Consideration of Manufacturing Tolerances and Uncertain Flow Distribution
,”
Int. J. Refrig.
,
33
(
4
), pp.
815
828
.
25.
Han
,
X.
,
Sagaut
,
P.
,
Lucor
,
D.
, and
Afgan
,
I.
,
2012
, “
Stochastic Response of the Laminar Flow Past a Flat Plate Under Uncertain Inflow Conditions
,”
Int. J. Comput. Fluid Dyn.
,
26
(
2
), pp.
101
117
.
26.
García-Sánchez
,
C.
,
Philips
,
D.
, and
Gorlé
,
C.
,
2014
, “
Quantifying Inflow Uncertainties for CFD Simulations of the Flow in Downtown Oklahoma City
,”
Build. Environ.
,
78
, pp.
118
129
.
27.
Cervantes
,
M.
, and
Engstrom
,
T.
,
2008
, “
Pulsating Turbulent Flow in a Straight Asymmetric Diffuser
,”
J. Hydraul. Res.
,
46
(
1
), pp.
112
128
.
28.
Salehi
,
S.
,
2013
, “
Computation of Developing Turbulent Flow Through a Straight Asymmetric Diffuser With Moderate Adverse Pressure Gradient
,” Master's thesis, University of Tehran, Tehran, Iran.
29.
Durst
,
F.
,
Fischer
,
M.
,
Jovanović
,
J.
, and
Kikura
,
H.
,
1998
, “
Methods to Setup and Investigate Low Reynolds Number, Fully Developed Turbulent Plane Channel Flows
,”
ASME J. Fluids Eng.
,
120
(
3
), pp.
496
503
.
30.
Kaltenback
,
H.
,
Fatica
,
M.
,
Mittal
,
R.
,
Lund
,
T.
, and
Moin
,
P.
,
1999
, “
Study of the Flow in a Planar Asymmetric Diffuser Using Large Eddy Simulations
,”
J. Fluid Mech.
,
390
, pp.
151
185
.
31.
Sbrizzai
,
F.
,
Verzicco
,
R.
, and
Soldati
,
A.
,
2009
, “
Turbulent Flow and Dispersion of Inertial Particles in a Confined Jet Issued by a Long Cylindrical Pipe
,”
Flow, Turbul. Combust.
,
82
(
1
), pp.
1
23
.
32.
Wang
,
L.
,
2008
, “
Karhunen–Loève Expansions and Their Applications
,” Ph.D. thesis, London School of Economics and Political Science, UK.
33.
Launder
,
B.
, and
Sharma
,
B.
,
1974
, “
Application of the Energy-Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc
,”
Lett. Heat Mass Transfer
,
1
(
2
), pp.
131
137
.
34.
Weller
,
H. G.
,
Tabor
,
G.
,
Jasak
,
H.
, and
Fureby
,
C.
,
1998
, “
A Tensorial Approach to Computational Continuum Mechanics Using Object-Oriented Techniques
,”
Comput. Phys.
,
12
(
6
), pp.
620
631
.
35.
Xiu
,
D.
, and
Karniadakis
,
G.
,
2002
, “
The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations
,”
SIAM J. Sci. Comput.
,
24
(
2
), pp.
619
644
.
36.
Witteveen
,
J. A.
, and
Bijl
,
H.
,
2006
, “
Modeling Arbitrary Uncertainties Using Gram–Schmidt Polynomial Chaos
,” AIAA Paper No. 2006-896.
37.
Sudret
,
B.
,
2008
, “
Global Sensitivity Analysis Using Polynomial Chaos Expansions
,”
Reliab. Eng. Syst. Saf.
,
93
(
7
), pp.
964
979
.
38.
Hosder
,
S.
,
Walters
,
R. W.
, and
Balch
,
M.
,
2007
, “
Efficient Sampling for Non-Intrusive Polynomial Chaos Applications With Multiple Input Uncertain Variables
,”
AIAA
Paper No. 2007-1939.
39.
Sobol’
,
I.
,
1990
, “
On Sensitivity Estimation for Nonlinear Mathematical Models
,”
Mat. Model.
,
2
(
1
), pp.
112
118
.
40.
Obi
,
S.
,
Aoki
,
K.
, and
Masuda
,
S.
,
1993
, “
Experimental and Computational Study of Turbulent Separating Flow in an Asymmetric Plane Diffuser
,”
Ninth Symposium on Turbulent Shear Flows
, Kyoto, Japan, Aug. 16–18, pp.
305.1
305.4
.
41.
Buice
,
C.
, and
Eaton
,
J.
,
2000
, “
Experimental Investigation of Flow Through an Asymmetric Plane Diffuser
,”
ASME J. Fluids Eng.
,
122
(
2
), pp.
433
435
.
42.
Iaccarino
,
G.
,
2001
, “
Prediction of a Turbulent Separated Flow Using Commercial CFD Codes
,”
ASME J. Fluids Eng.
,
123
(
4
), pp.
819
828
.
43.
El-Behery
,
S. M.
, and
Hamed
,
M. H.
,
2011
, “
A Comparative Study of Turbulence Models Performance for Separating Flow in a Planar Asymmetric Diffuser
,”
Comput. Fluids
,
44
(
1
), pp.
248
257
.
44.
Durbin
,
P. A.
,
1995
, “
Separated Flow Computations With the k–ε–v2 Model
,”
AIAA J.
,
33
(
4
), pp.
659
664
.
45.
Loh
,
W. L.
,
1996
, “
On Latin Hypercube Sampling
,”
Ann. Stat.
,
24
(
5
), pp.
2058
2080
.
This content is only available via PDF.
You do not currently have access to this content.