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A Kriging Approach for CFD/Wind Tunnel Data Comparison
J-C. Jouhaud�
, P. Sagaut�
, B. Labeyrie�
CERFACS�
Senior Researcher and�
PhD Student
Centre Europeen de Recherche et de Formation Avancee en Calcul Scientifique
42, Avenue Gaspard Coriolis, 31057 Toulouse Cedex, France
email: [email protected]
�
Professor, Laboratoire de Modelisation en Mecanique, University of Paris VI
4, place Jussieu 75252 Paris Cedex 05, France
email: [email protected]
Key-words: Kriging, Spatial Interpolators, Optimization, elsA Code, Error Estimation and Con-
trol, Validation of CFD Simulations
For submission to the Journal of Fluids Engineering
Revised version: 26/09/2005
1
Running head: A Kriging Approach for CFD/Wind Tunnel Data Comparison
Correspondence should be addressed to:
Dr. J-C. Jouhaud,
Centre Europeen de Recherche et de Formation Avancee en Calcul Scientifique, Team CFD
42, Avenue Gaspard Coriolis, 31057 Toulouse Cedex, France
Tel: (33) 561 19 30 51 fax: (33) 561 19 30 00 email: [email protected]
2
Abstract
A Kriging-based method for the parameterization of the response surface spanned by un-
certain parameters in computational fluid dynamics is proposed. A multiresolution approach
in the sampling space is used to improve the accuracy of the method. It is illustrated consider-
ing the problem of the computation of the corrections needed to recover equivalent free-flight
conditions from wind-tunnel experiments. Using the surface response approach, optimal cor-
rected values of the free-stream Mach number and the angle of attack for the compressible
turbulent flow around the RAE 2822 wing are computed. The use of the response surface
to gain an insight into the sensitivity of the results with respect to other parameter is also
assessed.
1 Introduction
Validation and assessment of data obtained using numerical simulation are recognized as crucial
steps in the development of reliable Computational Fluid Dynamics (CFD) tools for engineering
and academic research purposes. The importance of certification and validation strategies is now
so large that some international agreements on the validation process for numerical data have been
proposed [15, 16, 17, 18, 19, 20, 21, 22]. Some best practice rules and guidelines for numer-
ical models verification/validation/certification have been identified, which are now considered
as mandatory steps before a physical model or a numerical scheme can be considered as fully
assessed.
Despite a growing effort is devoted to the development of safe validation methodologies, both
practical and theoretical problems arise when designing the validation process in terms of com-
parison between numerical results and wind-tunnel data. The first problem is the availability of
sufficiently detailed reference databases, usually obtained through wind-tunnel experiments. The
second problem, which is the one mainly addressed in the present paper, deals with the uncertainty
3
in the definition of the validation cases.
The case of the turbulent statistically two-dimensional flow around a clean wing profile is retained
as an illustration of this problem in the present paper. In such a configuration, it is well known that
most wind tunnel experiments suffer some secondary effects because of the limited spatial extent
of the wind tunnel. The flow around the model is not fully identical to the flow around the same
profile in an unbounded space, and wind tunnel data must be corrected to mimic the corresponding
free-flight flow. In the present work, the compressible turbulent flow around the RAE 2822 wing
profile is selected. This flow is well known since it was selected as a validation case by the
sixteen partners of the EUROVAL European project [15] and by AGARD [16]. Significant outputs
of the EUROVAL dealing with this test case are corrected values of the Mach number and the
angle of attack to achieve equivalent free-flight conditions: it is recommended that numerical
simulations must be carried out using this corrected values [23], which are not equal to those of
the experimental configuration.
Therefore, the important question of the evaluation of these corrected values arise, since these
new values are the key of the validation process. Since they are not fully determined by the theory,
the corrections can be interpreted as uncertain values in the numerical simulations. The optimal
corrections can be defined as the ones which lead to the best overall agreement between a given set
of numerical simulations and the wind-tunnel data. A direct consequence is that optimal corrected
values are not strictly independent of the set of computations that will be assessed using them. A
crucial problem is therefore to develop a general strategy to compute this optimal corrections.
The present paper aims at presenting a strategy for generating the response surface of numerical
tools, i.e. for describing the space of the solutions spanned by the numerical method, the turbu-
lence model and some configurational parameters (Mach number, angle of attack) based on the
Kriging approach [6]. The Kriging method is first used to estimate unknowns solutions which
have not been computed by interpolation in the uncertain parameter space, and, in a second step,
optimal corrections for wind-tunnel data are derived in an automatic way. The optimality is guar-
4
anteed, since the global extrema can be found. This systematic approach is to be compared with
the usual approach, in which corrections are found in a heuristic way.
The paper is structured as follows. Section 2 displays the governing equations, including the
turbulence models and the main features of the numerical method. The kriging method and the
present implementation with local refinement in the uncertain parameter space is presented in
section 3. The application to the selected case is presented in section 4. Conclusions are given in
section 5.
2 Governing Equations and Flow Solver
2.1 Physical model
The governing equations are the 3D Navier-Stokes equations which describe the conservation of
mass, momentum and energy of a viscous Newtonian fluid flow. Using Cartesian coordinates,
these equations can be expressed in a conservative form as follows:
������������� �� (1)
The state vector�
and the flux �� ���������� decomposed in an inviscid and a viscous Part, which
are expressed as follows:
� �� ������� ���"!$#&%���' �� � � ��� � )( � �+* ,�� � .- �/!��+*102# %
�3�� �� �4� 53� 536� � �7 # %
(2)
where � is the density, � the velocity, * the pressure and ! the total energy. For a Newtonian
fluid, the shear stress tensor 5 is given by:
58 :9 - � � � - � � 0 % 0;�=<�� � , (3)
with 9 the dynamic viscosity and < the second coefficient of viscosity. The Stokes assumption
5
reduces the Lame’s relation to � 9���� < �� . The heat flux 7 is given by Fourier’s law:
�7 ��� % ��� (4)
with � the temperature and � % the thermal conductivity coefficient. The dynamic viscosity 9 is
given by the Sutherland’s formulae:
9 �9�� ����� � ��� ������ ����� (5)
where 9�� is the dynamic viscosity at the reference temperature ��� and the constant ��� equals to��� �4�� K. With a constant Prandtl number, the heat conductivity can be written as � % 9������� with��� the specific heat at constant pressure and� � ��4"! � for air. For a caloric perfect gas, the state
equation is given by *8 ��$#%� where the gas constant # is equal to 287 (J/kgK) for air.
2.2 Numerical method
The present study is carried out using the elsA code developed at ONERA and CERFACS [1].
The elsA code solves the 3D compressible Navier-Stokes equations using a cell-centered finite-
volume method. Integrating the equation 1 over a domain & and applying Green’s divergence
theorem yield the following integral form:
')( � � ��*,+���-). ( � �/ *10 �� (6)
with �/ the outward normal of the boundary� & of the control volume & . The separated time/space
discretization process leads to the following delta form:
243 65�7 � ��3 �8 & 8 # -2 %5 0 (7)
where the residual # comes from the space discretization and depends on the conservative variable
field 5
. The Jacobian matrix2
comes from the implicit time discretization and3 5)7 �
5)7 � � 5 corresponds to the field correction also called the time increment.
6
In this paper, a standard multigrid [4] method combined with a local time stepping is used in order
to accelerate the convergence to steady solutions. The classical second order central scheme of
Jameson [3] is used for spatial discretization and a LU-SSOR implicit method [5] for solving the
time integration system (7).
2.3 Turbulence Modeling
Several turbulence models have been used for the present study, which are described below.
2.3.1 A One-Equation Turbulence Model: Spalart-Allmaras Model
Using the Spalart-Allmaras model [7], one has to solve the Reynolds-Averaged Navier-Stokes
equations and a transport equation for the eddy viscosity. Here, the eddy viscosity and the molec-
ular viscosity are respectively noted ��� and � . The Reynolds stresses are given by �������� ���� 0�� �where 0�� � �
�- .���.���� � .����.���� 0 and �� is the working variable which obeys to the transport equation:
� � ��� � ��� � � � ���� ��� �- � ��� � � 0 �0�� �� � ��� �� � � �� � �� �=� � �� - 9�� � �� 0 � ���
� �! � �� � ��� �" � � � � � � �� �* � (8)
where the dynamic eddy viscosity 9 � is obtained by the formulae:
9 � �� �� � � � # � � � $ �
$ � ��� �� �
# $ � ��9 (9)
Here, S is the magnitude of the vorticity,
�0 0 � ��" � * � � � � # � � � � � $� � $ � � �(10)
and * is the distance to the closest wall.
The function �� is:
�� 8 &% � ���(' % ' ��� ' � )* # %. � ���! �- � ' � � 0 # � ���0 " � * � (11)
7
For large�, �� reaches a constant, so large values of
�can be truncated to 10 or so. The function
� � � is:
� � � � � exp- �%� ��� $ � 0 (12)
The constants are:
��� � ��4 � � ��� # ��� � ��4�� ��� # � �
� # " ��4�� �� � � )� � � � 7 � �� # � � :�4�� # � � � # � � � ! �
� � � � # � ��� �Turbulent heat transfer obeys a turbulent Prandtl number equal to 0.09.
2.3.2 Two-Equations Turbulence Models: � �� , ��� , ���Other turbulence models considered in the present study belong to the two-equations model fam-
ily. Following Deniau [8], all the classical two-equations models can be cast into the following
formulation:
.����. � �=� � � � � - 9 �������� 0 � � � � ��� ��% -�� � 0 � �.��"!. � ��� � � � � � - 9�� � ��$# 0 � � � � )%� � � � � � �
!%��! (13)
In Eq.13, the variable�
corresponds to quantities �� , � or � , respectively the isotropic modeled
dissipation rate, the specific dissipation rate and the caracteristic length, is the turbulence kinetic
energy and % -�� � 0 is a function depending only on�
and . These quantities permit to evaluate a
turbulence characteristic time scale � :
% Jones-Launder [9] � �� model: �: � &'% Wilcox [2] ��� model: �: � % Smith [10] � � model: �: () �
8
All these models are based on the Boussinesq assumption.�
represents the term of � production
and do not need additional closure hypothesis. � is a complementary term of dissipation which
appears in some models ( � �� and � � for example) and ! a term which plays a major role in
buffer region of boundary-layers. One obtains the following expressions:
� ��� ��� � �9 � - � � ��� � % � �
� � � 0 � �
� � ,9 � ��1� � � � �
In the equations 14 and 13, the coefficients � � � � � � � � � � � � � � � � ! are constants. � � and � � are
damping functions which depend on the turbulence models.
The Jones-Launder � �� model equations are given by the system (14):.��"�. � ��� � � � � - 9�� � �� � 0 � � � � � �� � �.�� &'. � �=� � � � �� � - 9�� � ���� 0 � �� � � ' ) � �&'� � � � ' � � � �
&' �� ��! (14)
with the following formulations:
9 � �� � � � � � �&' # � � exp � ��� � � 7� � � � �� � # # � � �� &'
� �� - ��� 01 - ��� 0! � � � ��
� - ��� 0 � � � - ��� 0 �� � � � �4�� exp
- �%# �� 0Concerning the coefficients, � � ��4���� , � ' ) � ��� , � ' � � , � � � �� and � ' � �� .The Wilcox � � model equations are given by the system (15):.��"�. � ��� - � � 0 � � -�- 9��������� 0 � 0 � ��� � ���
9
.�� . � ��� - � � � 0 � � -�- 9�� � ���� 0 � � 0 ��� � � ��� 1��� �
(15)
with the eddy viscosity 9 � � � and the closure coefficients � � � �� , � ) � �� , � � �4���� ,
� ��4��,! � , � ��4 � .In the case of the � � turbulence model proposed by Smith, the following transport equations are
considered: .��"�. � ��� - � � 0�� � -�- 9 � � �� � 0 � 0 � � � �� �� �� ) ( ���.�� (. � ��� - � ��� 0 � � -�- 9�� � ���� 0 � � 0� - � � ! � 0
� ) ��
� ) - � � - (� 0 � 0
� ����6� � � �� � (- � � � � 0 - (� 0 �
� � ���� � � - � � �� 0 (16)
where�
and � have been defined above and * is the distance to the wall. To complete system 16,
we add the following formulations:
9 � �9���� � # � � ) �� (� � ) � )� � - � �)���� 7 � ���� � 7 � �� �) 7 � ���� � 7 � � 0 � ��� # ��� exp- � � � - (� 0 � 0
� � � � � # � � � The constants are: " ��4�� � # � � � ( � �� � # � � �� # ! � � � .3 Response Surface Building using the Kriging Method
3.1 What is Kriging ?
Spline interpolation [12] was originally developped for image processing. In GIS (Geographic
Information System), it is mainly used in visualization of spatial data, where the appearance of
interpolated surface is important. In geology and geomorphology, on the other hand, a different
10
interpolation method referred to as kriging is widely used. This method was developed by a
South African geologist G. Krige [6] in 1951. Since then, it has been extended to many fields
of application, including agriculture, human geography [11], epidemiology [14], biostatistics or
archeology.
Kriging predictors are called optimal since they are statistically unbiased (e.g. on the average, the
predicted value and the true value coincide) and they minimize prediction mean-squared error (see
below equation 23), a measure of uncertainly or variability in the predicted values.
3.2 Principle of the Method
Kriging uses the covariogram [13], a function of the distance and direction separating two loca-
tions in the uncertain parameter space, to quantify the spatial autocorrelation in the data. The
covariogram is then used to define the weights that determine the contribution of each data point
to the prediction of new values at the unsampled locations in the space spanned by uncertain pa-
rameters.
The main statistical assumption underlying Kriging, called here assumption A1, is that statistical
properties (such as mean, variance, covariance ...) do not depend on the exact spatial locations,
so the mean and variance of a variable at one location is equal to the mean and variance at an-
other location. Also, the correlation between any two locations depends only on the vector that
separates them, and not on their exact locations. When data cannot be assumed to satisfy this as-
sumption, detrending techniques are used. The assumption2 �
is very important since it provides
a way to obtain replication from a single set of correlated data and allows us to estimate important
parameters and make valid statistical inference.
3.3 Different Types of Kriging
Simple, ordinary and universal Kriging predicators are all linear predicators, meaning that pre-
diction at any location is obtained as a weighted average of neighbouring data. The difference
11
between these three models is in the assumption about the mean value of the variable under study:
simple Kriging requires a known mean value as input to the model, while ordinary Kriging as-
sumes a constant, but unknown mean and estimates the mean value as a constant in the searching
neighbourhood (assumption A2). Thus, these two approaches model a spatial surface as deviations
from a constant mean (assuming that the expectation of surface function is constant), where the
deviations are spatially correlated (application to steady problems). Universal Kriging models lo-
cal means as a sum of low-order polynomial functions of the spatial coordinates and then estimate
the coefficients in this model. This type of model is appropriate when there are heterogeneity in
the expectation of surface function (application to unsteady problems).
In the following, only ordinary Kriging will be considered. In fact, simple Kriging implies that
functions to be estimated have a known mean, which is not true in the present application. And,
universal kriging is not justified in the context of our applications, since we are only here interested
in steady flow simulations.
3.4 Mathematical Formulation
As seen above, the principle of Kriging method is to estimate, on a study region noted 0 , the
attribute value of a surface function � at a location � � -� ���40 where we do not know the true
value:
�� - � �/0 5���� �
� � - � �/0�� - � � 0 (17)
where �� - � � 0 is the estimator function and-� ���40 are the coordinates in the two-dimensional pa-
rameter space. The region 0 contains / data values � - � � 0 , � � � � / , and Kriging approach
consists in interpolating the value at a certain location by a weighted summation (� � (x) weight
functions) of the values surrounding sample points. The problem is how to determine the weight
function, which is the main issue in Kriging approach.
12
Assumption2 �
implies that the covariance � of the surface function � of two locations-� � � � �60
is given by a function of only the distance between these locations:
�8� � - � � 0 ��� - � � 02#1 � - 8 � ��� � � 8 0 (18)
where � is defined by the following formulation (assumption A2):
� - � � � � � 0 �!�� - � - � � 0�� 930 - � - � � 0�� 9302# (19)
Covariances are usually represented as a matrix called the covariance matrix:
� ������������
� � � - 8 � � � � �8 0 ����� � - 8 � � � � 5 8 0� - 8 � � � � �
8 0 � � ����� � - 8 � � � � 5 8 0...
.... . .
...� - 8 � 5 � � �8 0 � - 8 � 5 � � �
8 0������ � � � - �/0
������������
(20)
Similarly, for plain explanation, covariance vector � is introduced:
� - � �/0 ������������
� - 8 � � � � �8 0� - 8 � � � � �8 0
...� - 8 � � � � 5 8 0
� ����������
(21)
In order to optimize the estimator function, one has to choose the weight functions� � (x) (i=1,. . . ,n)
which minimize the variance of estimator functions represented here by the mean square error
( � 0 ! ):
���� -
� � 0 � 0�! �� - � � 0 :� (22)
where � 0 ! is defined by the following formula:
� 0 ! - �� - � � 0�0 !�� �� - � � 0 � � - � 0 � # �
!�� �� - � � 02# � � ! � � - � � 02# � � � ! � � - � � 0��� - � � 02#13
5�� � �
5�� � ��� � - � � 0 � � - � � 0 � - 8 � ��� � � 8 0 � � � � � 5�
��� ��� � - � � 0 � - 8 � � � � � 8 0 �
%-� � 0 � � -
� � 0 � � � � � � % - � � 0 � - � � 0 (23)
The result is:
� -� �/0 � � �
� - � �/0 (24)
In ordinary Kriging, the sum of weight functions� � - � � 0 is equal to 1 at any location in 0 :5�
��� �
� � - � � 0 � (25)
To enforce this constrain, a Lagrange multiplier < - � � 0 is introduced. The original problem (22)
with constrain (25)is then rewritten as:
���� -
� � 0 � 0�! �� - � � 0 � � % - � � 0 , < - � � 0� �� (26)
where , is the identity operator, leading to
� 7 - � � 0� � � �7 � 7 - � � 0 (27)
where
� 7 - � � 0 ����������������
��-� �/0
��-� � 0...
� 5 - � � 0< - � � 0
� ��������������
(28)
� 7 ����������������
� � � - 8 � � � � �8 0 ����� � - 8 � � � � 5 8 0 �
� - 8 � � � � �8 0 � � ����� � - 8 � � � � 5 8 0 �
......
. . ....
�� - 8 � 5 � � �
8 0 � - 8 � 5 � � �8 0������ � � � - �/0 �� � ����� � �
� ��������������
(29)
14
and
� 7 - � 0 ����������������
� - 8 � � � �8 0� - 8 � � � �8 0
...� - 8 � � � 5 8 0�
� ��������������
(30)
In ordinary Kriging, covariogram function is arbitrarily chosen from typical theoretical functions,
or estimated from the observed data. Here, to estimate the surface function, the following model
covariogram is used:
� -�� 0 � � exp� � �� � � � (31)
This surrogate expression for the covariogram is commonly used in cases in which the exact cor-
relations are not known [24], since it yields the definition of a non-oscillatory interpolation proce-
dure. The linear system (27) (inversion of matrix C for each � � ) is solved using the mathematical
library of LAPACK (Linear Algebra PACKage).
3.5 Kriging Computational Suite
In this work, the Kriging method has been implemented in a Kriging Computational Suite which
is coupled with elsA solver. This suite (see fig. 1) is divided in four stages :
1. Definition of the following data:
(a) Range of variation of the uncertain parameters.
(b) Sampling in the selected subspace for uncertain parameters.
2. CFD Computations with elsA:
15
(a) Realization of the simulations for each sampling point in the uncertain parameter
space.
3. Data Processing:
(a) Computations of the values taken at sampling points by the function to be interpolated.
(b) Creation of data files for Kriging method.
4. Kriging Method:
(a) Reconstruction of the surface function.
(b) Computation of the mean square error of kriging method.
(c) Visualization of the surface function.
(d) Determination of the zone to be refined in the uncertain parameter space and return to
the first stage.
Within this multiresolution framework in the uncertain parameter space, the Kriging surface re-
sponse is built using all data contained at all levels. It is worth noting that the present imple-
mentation is non-intrusive, since it does not require any modification of the basic CFD tool: the
coupling between the Kriging tool and the CFD solver is performed via data file transfers. Another
important feature is the capability of using a multiresolution approach in the uncertain parameter
space to minimize the error in the response surface interpolation. In the present work, a local grid
refinement in the uncertain parameter space is used.
16
Stage 4 : Kriging Method
Stage 3 : Data Processing
Refinement
Stage 1 : Definition of the parameters
Stage 2 : CFD Computations with elsA
Figure 1: Kriging Computational Suite.
4 Application to CFD/Wind Tunnel Comparison for RAE2822 Pro-
file
4.1 Definition of the test case
The proposed methodology for identification of optimal wind-tunnel data corrections is illustrated
here considering the two-dimensionnal, steady, turbulent, compressible flow around the RAE2822
wing profile. This case has been extensively used for validation of Navier-Stokes codes applied to
transonic airfoil flow sinc it was retained as a international test case by AGARD [16] and within
the EUROVAL Project [15]. A large number of simulations with different flow parameters and
turbulence modelshave been carried out. These simulations are related to numerous experimental
work available in the literature.
In the work the case investigated experimentaly by Cook et al. [16] is retained. These case is
referred as Case 9 in the EUROVAL project and corresponds to the following experimental pa-
rameters:
��� ��4"!)� �4� � � � � � #�� � � � � � ' (32)
where ��� , � and #�� are the free-stream Mach number, the angle of attack and the chord-based
17
Reynolds number, respectively. The transition of turbulence flow takes place at �� �
� �4�� �on both upper and lower side of the airfoil. In fact, transition trips were used in the experiments,
and transition location is prescribed in the simulations.
In the EUROVAL [15] validation project, empirically-derived wind-tunnel corrections are applied
to the above flow parameters (32) and the values referred to as mandatory flow parameters for
Case 9 are the following:
��� ��4"!)� �4� � � "! � � #�� � � � � � ' (33)
These corrected values were obtained searching for the values which yield the best overall agree-
ment between experimental values of the drag and the lift and those computed by ten research
groups using different numerical methods, turbulence models and computational grids. The tar-
geted values of the aerodynamic forces are:
� ��4�� � � � � ( ��4 � � � (34)
Following the work of EUROVAL, we propose here to use Kriging method in order to find the
optimal optimized corrected values of ��� � � and � for which four classical turbulence models
(Spalart-Allmaras, Jones-Launder, Wilcox and Smith) will be in best agreement with experimental
results (34). The sensitivity with respect to the location were transition to turbulence is prescribed
( ��
values) will also be investigated.
4.2 Reference Simulations
Before searching for the optimal corrected values, some reference simulations have been carried
out using the four selected turbulence models described above for uncorrected values of the free-
stream Mach number and the angle of attack. All the reference simulations have been carried out
on the same computational grid. The mesh is a C-mesh made of 2 blocks defined 177 x 65 points
18
(see Fig. 2), which corresponds to a fine space discretization. It extends 10 chords lenghts in front,
upper and lower side, as well as downstream of the trailing edge. There are 258 cells located on
the airfoil’s surface.
Figure 3 displays the Mach number contours computed using Spalart-Allmaras model: this flow
is characterized by a supersonic zone with a shock on the upper surface. Pressure coefficient
distributions on the airfoil are also plotted for all turbulence models in Figure 3. It can be noticed
that computations are in good agreement with experimental data with only few discrepancies, in
particular concerning the shock location. The corresponding values of drag and lift coefficients
and relative errors are given in Table 1.
Results � � ( 3 � 3 � (Spalart-Allmaras’s model 0.01629 0.7510 3.03 � 6.26 �
Wilcox’s model 0.01720 0.7833 2.38 � 2.45 �
Smith’s model 0.01537 0.8371 8.51 � 4.25 �
Jones-Launder’s model 0.01828 0.7582 8.81 � 5.58 �
Experimental data 0.0168 0.803
Table 1: Drag and lift coefficients and relative errors in reference simulations (without corrected
values of the Mach number and the angle of attack).
4.3 Kriging Interpolation, Cost Function Definition and Optimal Corrections
In order to compute optimal corrected values of control parameters, it is necessary to define a cost
function to be minimized. It is chosen here to use a multiobjective cost function, which combines
the relative errors computed on both lift and drag coefficients using the four selected turbulence
models. The computed corrections will be expected to be robust, meaning that they should lead
to an improvement of computational data/wind-tunnel data for a wide class of numerical models.
19
x
z
-5 0 5 10-10
-5
0
5
10
x
z
0 0.5 1
-0.5
0
0.5
Figure 2: View of the computational grid.
x
z
0 0.5 1-0.5
0
0.5
1
x
-C
p
0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
SmithWilcoxSpalart-AllmarasJones-LaunderExperiment
Figure 3: Flow around the RAE2822 wing, Mach number contours for Spalart-Allmaras and -Cp
stations.
The mathematical expression of the cost function used in this study is
����� � � �����
� �� � � � � �� � � � ������
�����
� ( � � � ( � � �� ( � � ������
(35)
where � � � � and � ( � � � are experimental values of the drag and lift coefficients, respectively, and
20
� � ��- � - � �� 0 ��� - ��� 0;��� - � � 0 ��� - 0 2 0�0 (36)
� ( � ��- � ( - � �� 0;��� ( - ��� 0;��� ( - � � 0;��� ( - 0 2 0�0 (37)
where � -���� * � � 0 (resp. � ( -���� * � � 0 ) refers to he value of the drag coefficient (resp. the lift
coefficient) computed using the turbulence model named��� * � � .
The construction of the surface response of the cost function defined above in the two-dimensional
space spanned by the free-stream Mach number and the angle of attack is first addressed. These
two parameters are considered as uncertain parameters. All other parameters, both numerical
and physical ones, such as prescribed location of transition to turbulence, numerical viscosity
parameters, ... are kept unchanged with respect to the reference simulations.
For the sake of physical consistency, it is assumed that corrections to be imposed to wind-tunnel
parameters must correspond to small relative variations in order to prevent any bifurcation of the
solution (e.g.: relaminarization, shock disappearance, ...). The allowed range of variation of these
two parameters are
� ����� � � � �� ��� � �4"! � ��� � � � �4"! � (38)
The grid in the uncertain parameter space spanned by these values is shown in Fig.4. The basic
uniform grid corresponds to the black circles, while the local refined grid used to improve he
accuracy of the response surface corresponds to white triangle. The local refinement region was
defined to improve the accuracy around the global minimum of the surface response computed on
the first grid level.
The surface response of the cost function computed using the coarse grid resolution in the- � � � � 0
plane is shown in Fig.5 while the one computed using the locally refined grid is presented in Fig.6.
The global minimum of the cost function is found at � � ��4"!)� ! and � � � � using the coarse
21
resolution based response surface, while it is found at � � �4"!)� ! and � � � � using the
locally refined grid. These values are not identical to the ones proposed in the EUROVAL Project,
but are close to them.
α
Mac
h
2.6 2.8 3 3.2 3.4
0.725
0.73
0.735
0.74
First IterationSecond Iteration
Figure 4: Location of sampling points in the- � � � � 0 plane for building of the response surface
via Kriging interpolation. Black circles: first grid level; white triangle: second grid level.
α2.6
2.8
3
3.2
Mach0.725
0.73
0.735
0.74
Error
0.2400.2170.1950.1760.1590.1440.1290.1170.1050.0950.0860.0770.0700.0630.0570.0510.0460.0420.0380.034
α
Mac
h
2.6 2.8 3 3.20.725
0.73
0.735
0.74Error
0.2400.2170.1960.1770.1600.1450.1310.1180.1070.0960.0870.0790.0710.0640.0580.0520.0470.0430.0390.035
Figure 5: Surface function of the cost function in the- � � � � 0 plane - coarse resolution sampling
22
α2.6
2.8
3
3.2
Mach0.725
0.73
0.735
0.74
Error
0.2400.2170.1960.1770.1600.1450.1310.1190.1070.0970.0880.0790.0720.0650.0590.0530.0480.0430.0390.035
α
Mac
h
2.6 2.8 3 3.20.725
0.73
0.735
0.74Error
0.2400.2170.1960.1770.1600.1450.1310.1190.1070.0970.0880.0790.0720.0650.0590.0530.0480.0430.0390.035
Figure 6: Surface function of the cost function in the- � � � � 0 plane - locally refined resolution
The response surface build using the Kriging estimator can also be used to gain insight into the
sensitivity of the solution with respect to some computational parameter. To illustrate this point,
the sensitivity of the results and the dependency of the optimal corrections for the free-stream
Mach number with respect to another partially arbitrary parameter, namely the location where the
transition to turbulence is imposed in the computation, ��, is investigated. The uncertainty domain
considered here is defined as
�4"! � � � � � � �4"! �4� � �� � � �� � �4 � ! (39)
As in the previous case, a two-level grid resolution is used to build the response surface. The
response surfaces built using the coarse grid resolution and the locally refined fine grid are shown
in Figs. 7 and 8, respectively. The optimal values found on the two surfaces are the same: � � �4"!)� ! and �
� � "! � � � �
, revealing that the previous optimal value of the free-stream Mach
number can be kept unchanged.
New simulations using the corrected values computed using the kriging-based response surface,
��� �4"!)� ! and � � � � , have been carried out to check that these new parameters yield an
23
Mach0.725 0.73 0.735 0.74
xt
0.03
0.04
Error
0.05800.05640.05480.05330.05180.05040.04900.04760.04630.04500.04380.04260.04140.04020.03910.03800.03700.03600.03500.0340
Mach
xt
0.725 0.73 0.735 0.74
0.025
0.03
0.035
0.04Error
0.05500.05360.05230.05100.04970.04850.04720.04610.04490.04380.04270.04160.04060.03960.03860.03760.03670.03580.03490.0340
Figure 7: Surface function of the cost function in the- � � � � � 0 plane - coarse resolution sampling
Mach0.725 0.73 0.735 0.74
xt
0.03
0.04
Error
0.05800.05630.05470.05310.05150.05000.04850.04710.04570.04440.04310.04180.04060.03940.03830.03720.03610.03500.03400.0330
Mach
xt
0.725 0.73 0.735 0.74
0.025
0.03
0.035
0.04Error
0.05800.05630.05470.05310.05150.05000.04850.04710.04570.04440.04310.04180.04060.03940.03830.03720.03610.03500.03400.0330
Figure 8: Surface function of the cost function in the- � � � � � 0 plane - locally refined resolution
effective improvement in the prediction of drag and lift. All other parameters are kept unchanged
with respect to the reference simulations. The computed values and the associated relative errors
are displayed in Table 2. By comparison with Table 1 it is observed that all values but one (lift co-
efficient predicted using the Smith’s model) are improved using the corrected parameters, showing
24
the efficiency of the method.
Results � � ( 3 � 3 � (Spalart-Allmaras’s model 0.01642 0.7568 2.26 � 5.75 �
Wilcox’s model 0.01717 0.8019 2.20 � 1.36 �
Smith’s model 0.01601 0.8461 4.70 � 5.37 �
Jones-Launder’s model 0.01749 0.7815 4.11 � 2.68 �
Table 2: Drag and lift coefficients and relative errors in reference simulations with corrected values
of the Mach number and the angle of attack.
5 Conclusion
A Kriging-based method for the parameterization of the surface response spanned by uncertain
parameters in CFD calculations is proposed. It was shown using the case of the flow around a
two-dimensional wing that this method is efficient. The most interesting features of the proposed
method is that it is non-intrusive, i.e. it does not involve any modification of the basic CFD
tool, and that it was coupled with a multiresolution approach in the uncertain parameter space to
increase the accuracy.
It was shown that such a tool makes it possible to compute optimal corrections for wind-tunnel
parameters to recover free-flight conditions. Here, the optimality is associated to the fact that the
proposed corrections lead to best overall agreement for a aggregated cost function which includes
both drag and lift but also a relevant set of turbulence models. The new corrected values are
observed to yield a significant improvement in the prediction of both drag and lift in almost all
cases.
The use of the Kriging-based response surface to evaluate the sensitivity of the solution was also
25
illustrated, considering the location where the transition to turbulence is prescribed as an uncertain
parameter.
The present surrogate modeling approach is fully general, in the sense that it does not rely on any
assumptions about the nature of the uncertain parameters and the features of the computational
model.
A last comment is that the Kriging approach can easily be applied to discontinuous function or
function with strong gradients in the uncertain parameter space using the local approach imple-
mentation, i.e. by limiting the interpolation to closest sampling points. Since it does not rely on
any explicit decomposition on a polynomial basis, the implementation of multiresolution-based
Kriging computational suites is easy.
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