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: jjouhaud@cerfacs.fr

�

Professor, Laboratoire de Modelisation en Mecanique, University of Paris VI

4, place Jussieu 75252 Paris Cedex 05, France

email: sagaut@lmm.jussieu.fr

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: jjouhaud@cerfacs.fr

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.

References

[1] L. Cambier and M. Gazaix, 2002, ”elsA: an efficient object-oriented solution to CFD com-

plexity,” 40th AIAA Aerospace Science Meeting & Exhibit, Reno, AIAA Paper, pp. 2002-

0108.

[2] D. C. Wilcox, 1994, ”Simulation of Transition with a Two-Equation Turbulence Model,”

AIAA Journal, Vol. 32, No 2, pp. 247-255.

[3] A. Jameson, W. Schmidt and E. Turkel, 1981, ”Numerical Solutions of the Euler Equations

by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes,” AIAA Paper, pp.

81-1259.

[4] A. Jameson, 1982, ”Steady State Solutions of the Euler Equations for Transonic Flow by a

Multigrid Method,” Advances in Scientific Comp., Academic Press, pp. 37-70.

[5] S. Yoon and A. Jameson, 1987, ”An LU-SSOR Scheme for the Euler and Navier-Stokes

Equations”, AIAA Papaer, pp. 87-0600.

26

[6] D. G. Krige, 1951, ”A Statistical Approach to some Basic Mine Valuations Problems on the

Witwatersrand,” J. of Chem., Metal. and Mining Soc. of South Africa, Vol. 52, pp. 119-139.

[7] P. R. Spalart anb S. R. Allmaras, 1994, ”A One-Equation Turbulence Model for Aerodynamic

Flows,” La Recherche Aerospatiale, Vol. 1, pp. 5-21.

[8] H. Deniau, 1996, ”Calcul d’ecoulements supersoniques pour la resolution des equations de

Navier-Stokes parabolisees: modelisation de la turbulence, traitement des poches superson-

iques,” PhD Thesis, Ecole Nationale Superieure de l’Aeronautique et de l’Espace.

[9] W. P. Jones and B. I. Launder, 1972, ”The Prediction of Laminarization with a 2-Equations

Model of Turbulence,” International Journal of Heat and Mass Transfer, Vol. 15, pp. 301-314.

[10] B. R. Smith, 1990, ”The k-kl Turbulence Model and Wall Layer Model for Compressible

Flows,” AIAA Paper, pp. 90-1483.

[11] M. A. Oliver and R. Webster, 1990, ”Kriging: a Method of Interpolation for Geographical

Information System,” INT. J. Geographical Information Systems, Vol. 4, No. 3, pp. 313-332.

[12] O. Dubrule, 1984, ”Comparing Splines and Kriging,” Computers and Geosciences, 10(2-

3):327-33.

[13] A. G. Journel and C. J. Huijbregts, 1978, ”Mining Geostatistics,” Academic Press, pp. 600.

[14] G. Christakos and D. T. Hristopulos, 1988, ”Spatiotemporal Environmental Health Mod-

elling,” Kluwer.

[15] W. Haase, F. Brandsma, F. Elsholz, M. Leschziner and D. Schwamborn editors, 1993, ”An

European Initiative on Validation of CFD-codes,” Notes on Numerical Fluid Mechanics,

Vieweg Verlag.

27

[16] P. H. Cook, M. A. McDonald and M. C. P. Firmin, ”Aerofoil RAE 2822 - Pressure Dis-

tributions and Boundary Layer and Wake Measurements,” AGARD AR 138, pp. A6-1 to

A6-77.

[17] T. J. Barber, 1998, ”Role of Code Validation and Certification in the Design Environment,”

AIAA Journal, Vol. 36, No 5, pp. 753-758.

[18] J. A. Benek, R. F. Lauer and E. M. Kraft, 1998, ”Validation Issues for Engine-Airframe

Integration,” AIAA Journal, Vol. 36, No 5, pp. 759-764.

[19] C. L. Rumsey, T. B. Gatski and A. Bertelrud, 1998, ”Prediction of High-Lift Flows Using

Turbulent Closure Models,” AIAA Journal, Vol. 36, No 5, pp. 765-774.

[20] P. J. Roache, 1998, ”Verification of Codes and Calculations,” AIAA Journal, Vol. 36, No 5,

pp. 697-702.

[21] A. Rizzi and J. Vos, 1998, ”Toward Establishing Credibility in Computational Fluid Dynam-

ics Simulations,” AIAA Journal, Vol. 36, No 5, pp. 668-675.

[22] U. B. Mehta, 1998, ”Credible Computational Fluid Dynamics Simulations,” AIAA Journal,

Vol. 36, No 5, pp. 665-667.

[23] T. Hellstrom, L. Davidson, 1994, ”Reynolds Stress Transport Modelling of Transonic Flow

Around the RAE2822 Airfoil,” AIAA Paper, pp. 94-0309.

[24] J. Sacks, S. Schiller and W. Welch, 1989, ”Designs for Computer Experiments,” Techno-

metrics, pp. 31:41-7.

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