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56th 3AF International Conference
on Applied Aerodynamics
28 — 30 March 2022, Toulouse – France
FP74-AERO2022-benali
Data assimilation for aerothermal mean flow reconstruction using aero-optical
observations: a synthetic investigation
M.Y. Ben Ali(1), O. L ´
eon(1), D. Donjat(1), H. B´
ezard(1), E. Laroche(1), V. Mons(2) and F. Champagnat(3)
(1)ONERA/DMPE, Universit´
e de Toulouse, France, mohamed yacine.ben ali@onera.fr
(2)ONERA/DAAA, Universit´
e Paris Saclay, France
(3)ONERA/DTIS, Universit´
e Paris Saclay, France
ABSTRACT
In this study, we investigate a Data Assimilation ap-
proach for the numerical reconstruction of turbulent aero-
thermal flows, and more particularly of their average den-
sity fields. The proposed approach relies on an ensem-
ble Bayesian inference method providing a derivative-
free optimization framework suitable for solving inverse
problems and quantifying model uncertainties. The orig-
inality of this work lies in the use of non-intrusive and
dense measurements such as those obtained by the optical
technique referred to as Background Oriented Schlieren
(BOS). Such a reconstruction problem is usually tackled
with a purely aero-optical modeling as in tomographic
BOS. The present approach introduces additional physi-
cal constraints by leveraging Reynolds Averaged Navier–
Stokes flow models. To validate the relevance of this ap-
proach, the developed framework is applied as a simple
calibration tool on numerical experiments involving a hot
jet impinging on a flat plate, providing approximate in-
ference for uncertain closure coefficients parametrizing a
k−ω(SST) turbulence model.
1. INTRODUCTION
Aero-thermal flows characterized by in-homogeneous
density fields are ubiquitous in aeronautical applications
where heat exchanges can play an important role in the
life-span of some mechanical components (cooled aero-
engine blades for example) or in the aerodynamic per-
formance of a structure (e.g. hot impinging jets for sur-
face anti-icing). Having an accurate modeling capabil-
ity of such aero-thermal flows would help manufactur-
ers to develop optimal cooling or heating devices. Usu-
ally, experimental wind tunnel tests and numerical sim-
ulations are deployed separately in the investigation of
such flows. On the one hand, numerical simulations may
provide full fields of flow state quantities such as tem-
perature, density or velocity. Among various techniques,
Reynolds Averaged Navier–Stokes (RANS) simulations
form a widespread and affordable approach. Yet, im-
portant limitations exist in RANS models that are usu-
ally detrimental for accurate simulations of aero-thermal
flows. In particular, the level of heat exchange at a wall
is generally unsatisfactorily predicted, which may require
the use of higher-fidelity but more expensive simulations
[6]. On the other hand, experimental techniques may
provide reliable but limited information about such aero-
thermal flows. For example, the Background Oriented
Schlieren (BOS) technique [4], which exploits the refrac-
tion of light in the traversed flow to provide light devi-
ation maps using a simple optical setup, can give some
quantitative insights into the flow density field. This den-
sity field may then be reconstructed from the observed
deviations based on a tomographic approach [13]. While
such a technique can be efficiently used to study free
flows, its application to bounded flows is more challeng-
ing because of the restrictions applying on the possible
points of view. Furthermore, tomographic reconstruc-
tions are performed relying on an aero-optical modeling
only and does not involve any prior or basic information
on the flow of interest. Finally, it should be emphasized
that this procedure can only provide estimates for the den-
sity field and not for other flow quantities such as the
temperature or the velocity. Based on estimation theory,
Data Assimilation (DA) [5] allows to overcome the re-
spective limitations of numerical simulations and experi-
ments, namely the lack of accuracy due to, among others,
modeling errors for the former and the scarcity of flow
information for the latter. In this study, we investigate an
1
ensemble-based Bayesian approach relying on filters de-
veloped for DA that supplies 2D maps of light-deviations
to a RANS solver. The goal here is twofold. On the one
hand, the present methodology aims at providing a full
flow estimate from light-deviation measurements. On the
other hand, it systemically enhances a baseline RANS
prediction, which could not be used alone to accurately
estimate the flow of interest.
The paper is organised as follows. Section 2 describes
the methodological ingredients. Section 3, first, intro-
duces the flow configuration, then discusses the results of
the DA process with synthetic experimental data. Section
4 concludes the study and opens up with perspectives.
2. DATA ASSIMILATION FRAME-
WORK
Given noisy BOS observations of an aero-thermal flow
in the form of mean optical deviation maps [13], our ob-
jective is to infer an approximate mean flow state rely-
ing on a RANS model that naturally involves physical
constraints on the flow quantities and that is considered
in this first work to be subject to parameters uncertainty
only. The present study thus aims at validating an ap-
proximate Bayesian model calibration method [10] that
will be extended to account for model-form uncertainties
[16] in future works.
2.1 A state augmented approach
The framework considered in the present study relies
on a state-space model [1], and more particularly on a
state augmentation approach recently proposed for in-
verse problems [8] and termed Ensemble Kalman Inver-
sion (EKI) [3]. This approach relies on an iterated ensem-
ble Kalman method [5] that blends observations and prior
knowledge for estimation of the true flow state. In this
sense, the present approach may be classified as a sequen-
tial DA technique were temporal dynamics is replaced by
an artificial one, providing an approximate Bayesian in-
ference method [16].
In this framework, the state-space model at iteration K
writes
XK=M
M
M(XK−1,ηm),(1a)
YK=H
H
HXK+ηo,(1b)
with the n-dimensional state XKand the m-dimensional
measurements YK. In a state-augmented framework, XK
contains flow physical variables (and particularly ρin our
case), augmented with the uncertain RANS controling
parameters. Eq. (1a) then models the artificial dynam-
ics with M
M
Membedding the non-linear RANS operator that
updates the state from iteration K−1 to iteration Kwith
some model uncertainty ηm. In the present numerical
study, the model dynamics is considered as perfect and
this system noise ηmis neglected. Eq. (1b) represents
the observation (or measurement) model in which H
H
His
a linear observation operator that projects the state XK
to the measurement space with some measurement noise
ηo. The linearity of the observation operator emerges
from the type of measurement here considered and de-
tailed in section 2.3. It is on this state-space model (1)
that an ensemble-based Kalman Filter is applied to pro-
vide flow state and parameters estimates. Further details
on the artificial dynamical model M
M
M, on the observation
model H
H
Hand on the ensemble Kalman method used are
provided in the following sections.
2.2 Aero-thermal RANS model
The aero-thermal RANS model constrains the estimated
solution with conservation equations for mass density ρ,
momentum ρUand total energy ρE. The RANS mod-
els are closed by relations for the Reynolds stress ten-
sor ρu′
iu′jfor the dynamic part, and for the turbulent heat
flux terms ρu′
iT′for the thermal part. These two modeled
terms entirely account for the unresolved physics and are
the ones embedding uncertain parameters subject to opti-
mization with the present approach.
Among the large variety of models available in the lit-
erature [15], we chose to use the k−ωSST (shear-stress
transport) [11] turbulence model since it was reported
in [6] to yield a very reasonable quantitative description
of the flow considered in this work and detailed in sec-
tion 3.1. Such a model is one of the eddy-viscosity mod-
els, which relies on Boussinesq’s hypothesis. As men-
tioned previously, the associated model-form uncertainty
is not accounted for in this work and we mainly con-
sider model-parameters uncertainty for calibration. Such
a calibration approach can be justified by the fact that
model parameters are usually empirically determined by
matching model results to some experimental or numer-
ical results obtained for canonical flow configurations,
and hence are not universal. We furthermore chose to
only consider a few number of parameters and a sensitiv-
ity analysis with the flow configuration described in sec-
tion 3.1 indicated the importance of at least four model
parameters, namely (β∗,σk2,σω2,β2). It can be noted
that these chosen parameters are mainly acting on the
outer part of the model (see [11]) that drives the free
shear-flow region of the considered impinging jet flow as
well as its turbulent/non-turbulent interfaces. Varying the
inner part of the model was observed to induce conver-
gence difficulties.
Closure for the RANS total energy equation is obtained
in the present work using the simple gradient diffusion
hypothesis to model the turbulent heat flux, such that
ρu′
iT′≃ − µt
Prt
∂T
∂xi
,(2)
2
where µtis the turbulent viscosity deduced from the k−ω
SST model for the Reynolds stress tensor and Prtis the
turbulent Prandtl number. This former parameter is the
only one driving the thermal part of the present model
and is thus added to the vector of uncertain parameters
that finally writes Ω= (β∗,σk2,σω2,β2,Prt). It may be
noted that finer thermal models could have been consid-
ered, but the present basic modeling is here somewhat
counter-balanced by the use of a Bayesian inference tech-
nique looking for optimal parameters.
Given a set of model parameters ΩKat iteration Kin
the framework defined in section 2.1 (with the other pa-
rameters of the RANS model left to their default values or
deduced from explicit relationships), solving the RANS
equations yields the n-dimensional flow state vector de-
fined as
χK= ((ρ1,U1,P
1,k1,ω1,T1),· · · ,(ρM,UM,P
M,kM,ωM,TM)),
(3)
where Pdenotes the mean pressure field and Mrefers
to the size of the computational grid. The over-bars and
the subscript ·Kare omitted on the flow state variables
for clarity. As mentioned in section 2.1, this state is aug-
mented to include the parameters ΩK, such that we define
XK= (χK,ΩK).(4)
It is highlighted that the state solution given by this aero-
thermal RANS model with its ”default” closure coeffi-
cients will be later-on referred to as the baseline state.
2.3 Observation model
The data YKas given in the observation model (1b) re-
sults from the application of the observation operator H
H
H
on the the flow state embedded in XK. This operation in-
volves performing a synthetic BOS observation [13] to
evaluate 2D visualization maps of ray deviation angles
in one projection plane (or camera). A multi-camera
setup will be considered in future works dealing with
non-axisymmetric flow configurations.
In a geometrical optics framework, the observed de-
viation angles result from the integral of the field of re-
fractive index gradient ∇nralong the path of light rays
traversing the in-homogeneous fluid flow medium. The
local refractive index is related to the fluid density by
the Gladstone–Dale relation nr−1=Gρ, where Gis a
constant that depends on the gas composition (air in the
present case) and the light wavelength. The three compo-
nents of light ray deviations in space can then be written
as
ε=G
nr0Zs∈ray
∇ρds(5)
where nr0is the refractive index of the medium surround-
ing the flow, which is assumed constant. As the ray path
depends itself on the density, the relationship is gener-
ally nonlinear. A common way to simplify the problem
is to use the paraxial approximation since the angles are
usually small [13], providing a linear framework for the
observation operator H
H
Hthat then amounts to a linearized
ray-tracing operator. Furthermore, the three components
εx,y,zof the field of light ray deviations then approxi-
mately correspond to deviation angles and are thus con-
sidered as such in the following sections.
Given the preceding definitions, the observation data
YKin (1b) can be defined as
YK= ((εx,εy,εz)1,·· · ,(εx,εy,εz)L)(6)
where Lis the number of light rays considered (or equiv-
alently the number of points considered in the 2D devia-
tion map). The dimension of YKis m=3Land the obser-
vation operator H
H
His of dimension m×nin matrix-form.
Finally, the experimental noise ηoin (1b) is assumed
to follow a Gaussian distribution N(0,R
R
R). The value of
the (diagonal) co-variance matrix R
R
Rwas determined by
a trial and error method to avoid convergence issues in-
duced by possible negative model parameter values. This
pragmatic approach is by no means optimal and proper
ways of defining it and of thresholding parameters will
be considered in future works.
2.4 Summary of the algorithm
Relying on the previously defined models and state-
augmented framework, an approximate solution to the in-
verse problem is obtained following the steps given by
Algorithm 1. Practical details on each step are provided
in the following paragraphs.
Algorithm 1 Iterative Bayesian inference procedure
Initialisation:K←0; generate initial ensemble {X(i)
0|
i∈{1···N}};
repeat
(a) Calculate {Xf,(i)
K+1}solving RANS equations,
(b) Perform observations H
H
HX f,(i)
K+1,
(c) Estimate {Xa,(i)
K+1}using Ensemble-based KF,
update K←K+1
until Exit convergence criterion.
Initialization A prior distribution of the model state is
required for initialization of the ensemble-based proce-
dure. In this study, different realizations of the state are
simply obtained by perturbing the set of model parame-
ters gathered in Ωfrom the baseline. A Latin hyper-cube
sampling method is selected to draw near-random values
for the parameters in a range of ±50% of their base-
line values, providing an ensemble of initial parameters
{Ω(i)
0|i∈ {1,··· ,N}} with Nthe number of ensemble
members set to 50 for the considered experiments. The
3
associated ensemble of initial flow states {χ(i)
0}is then
evaluated using the aero-thermal RANS model described
in section 2.2 to build the ensemble of initial augmented
states {X(i)
0}.
(a) RANS simulations are performed for every member
of the ensemble of parameter values estimated in the pre-
ceding inference iteration. These simulations are carried
out using the ONERA multi-physics platform CEDRE
[14]. The compressible RANS equations are discretized
by a finite volume method on a unstructured mesh. Con-
vergence of each RANS simulation are ensured with a
stopping criterion on the residual of conservative quanti-
ties that are required to drop by at least 1 order of magni-
tude.
(b) The 2D BOS synthetic observations are deviations
fields evaluated from numerically-obtained 3D density
flow fields (obtained from RANS simulations or LES re-
sults in the current study, see section 3). These optical
deviations are obtained by considering a synthetic cam-
era whose spatial location with respect to the flow field
and whose optical properties are first specified. A ded-
icated ray-tracing algorithm developed for tomographic
BOS [13] and coded to work on GPUs are used.
(c) An Ensemble Transform Kalman filter (ETKF) [2]
is used in the estimation of optimal values of turbulence
model parameters. Among others, this method is more
suitable for nonlinear models and high dimensional con-
figurations than classical Kalman filter [9]. The open-
source project PDAF (Parallel Data Assimilation Frame-
work [12]) is used to perform the ETKF analysis step.
Exit criteria This ensemble-based inference method-
ology is iterated to obtain converged estimates and no
stopping criterion are used in this validation work. Yet,
for practical applications with time and resources con-
straints, various criteria may be considered. For instance,
one may use the L2norm of the relative variations of the
parameters, the same norm for the observed quantity, or
also the relative reduction on Mean Root Mean Square
Error (MRMSE) on the state that will be discussed in the
following results.
3. RECONSTRUCTION OF SYNTHETIC
EXPERIMENTS
This section presents calibration results obtained with
synthetically generated data. These synthetic data pro-
vide known ground-truth to assess the validity of the
present inference method. The application case investi-
gated is a subsonic hot-jet flow impinging a flat plate.
3.1 Flow configuration
This impinging jet flow configuration is illustrated in
Fig. 1. A fully-developed turbulent air flow heated at
a total temperature TJof 130°C is issued from a round
pipe of diameter D=0.06m at x/D=0 with a centerline
mean axial velocity UJ=31.5ms−1. The bulk Reynolds
number is Reb≈60,000. The subsonic turbulent hot jet
impinges a flat plate located at x/D=3 and placed per-
pendicular to the jet axis. The simulation domain radi-
ally extends up to y/D=12 to capture the development
of the induced wall jet. This turbulent flow is statisti-
cally symmetric around the jet axis and 2D axisymmetric
RANS simulations were performed. Relying on a pre-
vious extensive experimental study [7], boundary condi-
tions for the jet mean axial velocity profile in the pipe exit
plane and for the entrained air flow mean velocity profile
at x/D=−3 are set using LDV measurements. Further-
more, the plate temperature distribution TPis defined us-
ing infrared temperature measurements obtained at ther-
mal equilibrium. The simulations are performed using
an unstructured mesh containing approximately 42,000
grid cells. Time integration is carried out using a first-
order implicit scheme and a first-order spatial scheme is
used to ensure stationarity of the solutions for the range
of parameters explored. Synthetic BOS observations are
performed with one camera oriented with its optical axis
parallel to the zaxis. The observation resolution is cho-
sen to provide dense deviation maps with a dimension of
(nx=500,ny=500), thus such that the size of observa-
tion data YKis m=3×2.5×105=7.5×105.
−2 0 2
x/D
0
1
2
3
4
y/D
Pipe
Entrained flow
Cover and Teflon
Flat plate
Observed region
Figure 1: Computation domain for the hot-jet flow im-
pinging on a flat plate.
3.2 Numerical twin experiments
The validity of the present framework for the estimation
of RANS model parameters using BOS observations was
first evaluated using synthetic data from two numerical
twin experiments performed with the RANS model it-
self. This is an idealized setup where the model used
4
Table 1: Mean values and standard deviations before
(”prior”) and after (”estimate”) estimation by the itera-
tive Bayesian inference method along with the pseudo-
true values of the three parameters chosen for the two
twin experiments.
(a) Twin experiment RANS1.
β∗σω2β2
Statistic Mean Std. Mean Std. Mean Std.
Prior 0.09 2.6e−02 0.856 2.5e−01 0.0828 2.4e−02
Estimate 0.1076 2.7e−05 0.658 3e−04 0.1029 3.4e−5
Pseudo-true 0.1076 - 0.663 - 0.103 -
(b) Twin experiment RANS2.
β∗σω2β2
Statistic Mean Std. Mean Std. Mean Std.
Prior 0.09 2.6e−02 0.856 2.5e−01 0.0828 2.4e−02
Estimate 0.0455 1.1e−04 0.694 1.4e−02 0.0887 5e−04
Pseudo-true 0.0459 - 0.772 - 0.0942 -
can perfectly match the experimental observations, thus
for which no model-form uncertainty exists. In these nu-
merical experiments, some values of the RANS model
parameters were arbitrarily chosen and were used to gen-
erate synthetic data considered as experimental observa-
tions. These values are later-on referred to as the pseudo-
true values. It is emphasized that these cases of study do
not provide realistic solutions and should only be consid-
ered as artificial experiments. For this twin experiments
study, only the three parameters (β∗,σω2,β2)are consid-
ered to simplify the analysis and to restrict the problem
to a purely aerodynamic one.
In the first twin experiment, named RANS1, the pa-
rameters were chosen to be close to their baseline val-
ues, within a ±25% range. This provides a first simple
test-case with pseudo-true values within the prior dis-
tribution. Baseline and pseudo-true values are given in
Table 1a. The second experiment, named RANS2, ex-
plores the case were the state to recover is more different,
with parameters further away from the baseline as shown
in Table 1b. Notably, a maximum relative difference of
about 50% is set on β∗, that is almost out of the prior
distribution. The pseudo-true density fields obtained for
the twin-experiments RANS1and RANS2are given in
Fig. 2(a) and Fig. 3(a) respectively, illustrating the qual-
itative difference between the two cases, in comparison
with the initial baseline state (Fig. 2(b) and reproduced
in Fig. 3(b)). Finally, the initial data-model discrepancies
in terms of root mean square error (RMSE) between the
pseudo-true state and the initial baseline state for the two
twin-experiments are displayed in Fig. 2(c) and Fig. 3(c).
This (normalized) RMSE between two states, φ, is de-
fined as
RMSE(φref,φk) = p(φref −φk)2
max(φref)−min(φref ).(7)
As shown in these figures, the initial discrepancies reach
maximum values of about 10% of the maximum density
variations in the pseudo-true state for RANS1and up to
40% for RANS2.
These discrepancy maps provide indications on the
differences of density fields between the baseline (prior
sample mean) and the pseudo-true states, but state obser-
vations are actually performed on BOS deviation fields.
It is then instructive to examine the associated discrepan-
cies in measured light deviations, as shown in Fig. 4 for
RANS1. Only the RMSE for two components of the de-
viation field are shown (εxand εz) since the third one (εy)
is one order of magnitude smaller. First of all, these maps
emphasize that dense observations are obtained on the en-
tire jet flow. Second, since εxprovides information about
differences in density gradients along the xaxis, the main
errors are naturally concentrated in the impingement re-
gion and in the wall jet. Discrepancies on εzhowever
mainly reflect differences in the jet shear-layers. Finally,
the normalized error amplitudes are of the same order of
magnitude (here about 8% on εx) as the relative discrep-
ancies on the density field in Fig. 2(c).
Calibrations were then performed on these two test-
cases. Fig. 5 shows the convergence of the spatial average
of RMSE, denoted MRMSE, in log-scale on the observed
deviations for the two experiments. It is highlighted that
these global relative discrepancies on the sample means
significantly decrease to less than 0.1% of the baseline
maximum deviation difference within 4 iterations.
Convergence of the samples and of the sample-mean
values of the model parameters is furthermore shown in
Fig. 6. The final estimates are given in Tables 1a and 1b
that list the means and standard deviations of the param-
eters before and after inference, together with the corre-
sponding pseudo-true values. From these results, it can
be emphasized that mean estimates perfectly match the
pseudo-true values of the experiment RANS1(less than
1% error on all the parameters) while a satisfactory match
is obtained for the more distant case RANS2. Indeed,
a maximum 11% error on σω2is obtained, but interest-
ingly this parameter is also identified as less sensitive as
the estimate is associated with a larger sample standard-
deviation (Std.) in Table 1b. The observed convergence
on the parameters estimates is associated with span re-
duction of the ensemble members in Fig. 6, with stan-
dard deviations reduced by 1 to 3 orders of magnitude
depending on the parameter. Regardless the experiment,
β∗, the coefficient involved in turbulence dissipation, was
retrieved with at least 2 significant digits as shown in Ta-
ble 1a. Compared to the other two considered parame-
ters, reduced standard deviations are obtained, thus point-
ing out its major role in the dynamic part of the present
RANS model.
It may be highlighted here that running the iterative
inference method on the experiment RANS2required an
5
0 2
x/D
0
1
2
3
4
y/D
(a)Pseudo −true
0 2
x/D
0
1
2
3
4
y/D
(b)Baseline
0 2
x/D
0
1
2
3
4
y/D
(c)RMSE
0.000
0.000
0.000
0.020
0.020
0.020
0.020
0.040
0.060
0.080
0.81
0.87
0.94
1.00
ρ/ρin f
0.81
0.87
0.94
1.00
ρ/ρin f
0.000
0.105
Figure 2: Density distributions of (a) the pseudo-true state for the twin experiment RANS1and of (b) the baseline solution
(also corresponding to the prior samples mean). (c) Map of RMSE(ρp-t ,ρbsl)between the two previous fields.
0 2
x/D
0
1
2
3
4
y/D
(a)Pseudo −true
0 2
x/D
0
1
2
3
4
y/D
(b)Baseline
0 2
x/D
0
1
2
3
4
y/D
(c)RMSE
0.000
0.000
0.060
0.060
0.060
0.120
0.120
0.120
0.180
0.240
0.300
0.360
0.81
0.87
0.94
1.00
ρ/ρin f
0.81
0.87
0.94
1.00
ρ/ρin f
0.000
0.386
Figure 3: Density distributions of (a) the pseudo-true state for the twin experiment RANS2and of (b) the baseline solution
(also corresponding to the prior samples mean). (c) Map of RMSE(ρp-t ,ρbsl)between the two previous fields.
200 300
nx
300
400
500
ny
(a) RMSE on εx
200 300
nx
(b) RMSE on εz
0.000
0.078
0.000
0.078
Figure 4: 2D maps of RMSE of observed deviations be-
tween the pseudo-true state and the baseline state for
RANS1.
empirical reduction in the variance of R
R
R, as already em-
phasized in section 2.3. Such an approach yielded real-
istic estimates but it also brought less confidence in the
experimental data and resulted in a apparent slow conver-
gence on the ensemble members. It nevertheless provided
more stability to the process and a narrower spread of the
MRMSE on BOS measurements as shown in Fig. 5.
0 2 4 68 10 12 14 16 18
Iterations
10−5
10−3
10−1
MRMSE
Sample mean (RANS 1)
Sample mean (RANS 2)
Sample (RANS 1)
Sample (RANS 2)
Figure 5: Convergence of MRMSE(Yp-t ,YK)during the
assimilation procedure for the two twin experiments. Yp-t
refers to the pseudo-true deviations.
Given the relative accuracy of the estimated model pa-
rameters, the resulting density flow field for the experi-
ment RANS2is now examined, this test-case being less
favorable than RANS1. Comparing Fig 3 and Fig. 7, the
resulting mean estimate of density field has been drasti-
cally improved, with maximum RMSE values of about
0.5%.
6
02468 10 12 14 16 18
Iterations
0.5
1.0
1.5
Normalized Value
(a)β∗
02468 10 12 14 16 18
Iterations
0.5
1.0
1.5
Normalized Value
(b)σω2
02468 10 12 14 16 18
Iterations
0.5
1.0
1.5
Normalized Value
(c)β2
Sample mean (RANS 1) Sample mean (RANS 2) Exp (RANS 1) Exp (RANS 2) Baseline
Figure 6: Convergence of the three model parameters during the assimilation procedure for the two twin experiments, for
the sample and for the sample-mean. The values of the parameters are normalized by the baseline values.
0 2
x/D
0
1
2
3
4
y/D
(a)Pseudo −true
0 2
x/D
0
1
2
3
4
y/D
(b)Estimate
0 2
x/D
0
1
2
3
4
y/D
(c)RMSE
0.000
0.0000.000
0.000
0.000
0.002
0.002
0.002
0.004
0.81
0.87
0.94
1.00
ρ/ρin f
0.81
0.87
0.94
1.00
ρ/ρin f
0.000
0.386
Figure 7: Density distributions of (a) the pseudo-true state for the twin experiment RANS2and of (b) the posterior solution
obtained by the ensemble-based method (sample-mean at iteration 18). (c) Map of RMSE(ρp-t ,ρK=18)between the two
previous fields.
These first results provide support for the relevance of
the present ensemble-based inference method of RANS
models and approximate reconstruction of density fields
from 2D maps of BOS measurements only. However, as
previously emphasized, the idealized synthetic data here
considered were obtained relying on the RANS model
used for inference and challenges induced by model-form
inadequacy should be expected with more realistic flow
measurements.
3.3 Application to synthetic LES data
To investigate the case where more realistic flow observa-
tions are available, the present calibration approach was
applied on synthetic data obtained by Large Eddy Simu-
lation (LES) on the same impinging jet configuration [6].
In this study, two sets of model parameters are consid-
ered: a first set of 3 ”aerodynamic” parameters as in the
previous twin experiments and referred to as ”aero only”
in the following figures; and the full set of 5 model pa-
rameters Ωdefined in section 2.2 to better account for
thermal effects, referred to as the ”full set” later-on.
Applying the present calibration method yielded the
convergence history on the MRMSE of the BOS observa-
tions presented in Fig. 8. For both sets of parameters, the
relative global errors decreased below 1% and reached a
plateau after 7 −8 iterations. While no significant differ-
ences appear on this global result (where a slightly better
reduction was achieved with the full-set), the estimated
values of the ensemble members actually converge to dif-
ferent values for the two considered sets of parameters.
This is made evident in Fig. 9A and Fig. 9B. Particularly,
the inclusion of the parameters related to the RANS ther-
mal model significantly modified the convergence values
of the parameters σω2and β2. Interestingly, regardless
of the chosen set of parameters, the ensemble values of
β∗did not diverge much from the baseline. Furthermore,
examining the ensemble spread, it can be observed that
it also led to some increase of the variance of σω2and
β2, especially for the later parameter. This suggests that
the parameters related to the thermal model might offer
a higher degree of freedom to the estimated state. A
7
0 4 8 12 16 20 24 28
Iterations
10−3
10−2
10−1
MRMSE
Sample mean (Full-set)
Sample mean (Aero-only)
Sample (Full-set)
Sample (Aero-only)
Figure 8: Convergence of MRMSE(Yp-t ,YK)during the
assimilation procedure of the LES data for the two sets of
chosen parameters.
finer investigation of the uncertainty within the thermal-
model-part is yet to be addressed in a future work. Re-
garding the two parameters Prtand σk2, reductions from
their initial baseline values are observed, suggesting an
enhancement of temperature diffusion and turbulent ki-
netic energy diffusion compared to the baseline.
The previously observed convergence on the model pa-
rameters and on the global observation error can also
be confirmed by comparing prior and posterior density
fields, displayed in Fig. 10 and Fig. 11 respectively, along
with the true-state (LES field) and the associated RMSE
maps. Significant improvements in terms of relative er-
rors are obtained. With the prior baseline state, max-
imum relative errors of about 35% were found in the
jet shear layer in Fig. 10(c), while maximum errors be-
tween 5−10% are obtained with the posterior solution in
Fig. 11(c). More particularly, the wall jet is almost per-
fectly reproduced and most of the discrepancies are found
in the initial jet shear layer.
Density profiles for prior and estimate samples are fur-
thermore presented in Fig. 12 and Fig. 13, along with
the LES results. As observed in Fig. 12, the baseline
and the prior samples clearly deviate from the LES re-
sults in terms of jet shear layer thickness. The esti-
mated density profiles are however in better agreement
compared to the baseline, even if differences in terms of
profiles shape are observed. In the wall jet region dis-
played in Fig. 13, the density profiles of the estimate
sample mean are also in a better agreement compared
to the baseline results. The remaining local discrepan-
cies observed between the estimate samples mean and
the LES data will be investigated in a future work. It
is suggested that such discrepancies could be the result
of inadequacies on model-parametrization, the proposed
aero-thermal RANS model being very rigid. We particu-
larly emphasize that only global model parameters were
considered in the assimilation process while the studied
flow shows at least two different regions (free jet and wall
jet), where local model adaptations may be more relevant,
and where the isotropic Boussinesq’s assumption may not
be entirely valid. Stated differently, the LES mean den-
sity field might not reside in the space spanned by the
prior ensemble since the present approach can only find
approximate solutions within this span [8].
4. CONCLUSION
In this study, a DA technique based on an ensemble
method was numerically investigated for mean density
reconstruction of aero-thermal flows. In the process, 2D
maps of light deviation angles obtained by synthetic BOS
were considered as input observations and a RANS flow
model was chosen to provide physically-plausible flow
constraints. To assess the relevance of such an approach,
this approximate Bayesian inference method was used as
a simple model calibration tool. To validate the devel-
oped framework, a k−ωSST RANS model was con-
sidered on the application case of a hot jet impinging
on a flat plate. Only a few number of coefficients in
this RANS model were selected as uncertain parameters.
First, two numerical twin experiments were conducted
using the same RANS model, demonstrating the capabil-
ity of the approach to retrieve accurate density fields and
pseudo-true values of the considered parameters when
model-form uncertainty is negligible. Second, results ob-
tained with more realistic observation data issued from
a LES on the same impinging jet configuration provided
relatively satisfactory estimates with potential for signif-
icant improvements. These last synthetic results high-
lighted the importance of model-form inadequacy since
the present approach can only find approximate solutions
in the span of the generated prior ensemble. Despite this
limitation, this iterative calibration approach is still found
to be valuable when 2D BOS observations are experimen-
tally available since it allows to reduce data mismatch af-
ter just a few number of RANS iterations compared to
a much more expensive LES. In future works, this effi-
cient approximate Bayesian estimation technique will be
applied to real experimental results. Furthermore, high-
dimensional parametrization will be considered to rely on
more complex turbulence models (e.g. second order tur-
bulence models such as Reynolds Stress Models). Fur-
thermore, the inclusion of model-form uncertainties will
also be investigated to provide a diagnostic tool for tur-
bulence closure models.
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