12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

TURBULENT MEAN FLOW ESTIMATION WITH STATE OBSERVERASSIMILATION OF VELOCITY MEASUREMENTS IN RANS MODELS

Olivier Marquet, Vincent Mons, Markus Zauner and Benjamin Leclaire
DAAA
ONERA, Université Paris Saclay
8 rue des Vertugadins, 92190 Meudon, France
olivier.marquet@onera.fr

INTRODUCTION

flow equations that is proportional to the difference between
reference data and numerical prediction. As such, the assimilation of data with state observer techniques is straightforward to
implement and virtually induces no supplementary cost compared to a baseline simulation. It has recently regained interest
in numerical (Zerfas et al., 2019) and experimental fluid mechanics (Saredi et al., 2021) due to its good performances instate estimation from limited data, in particular in the case offundamental turbulent flows (Di Leoni et al., 2020).

The full characterization of complex turbulent flows
is a challenging task in industrial applications. On the
one hand, scale-resolving computational fluid dynamics approaches, such as Direct Numerical Simulations (DNS) or
Large Eddy Simulations (LES), allow accurate estimation oftime-averaged and fluctuating quantities but they become very
expensive for flows with increasing Reynolds numbers. Therefore, small-scale structures are often not fully resolved but
modeled, which raises the issue of fidelity. On the other
hand, experimental techniques may provide accurate information about the full complexity of flows of interest. However, spatial and temporal resolution of measurements is often
sparse and/or limited to certain quantities, while boundary or
freestream conditions are often hard to characterize precisely.

In the present study, we are interested by improving theestimation of turbulent mean-flows as modeled with Reynolds
Averaged Navier-Stokes (RANS) equations thanks to the assimilation of sparse pointwise velocity. This objective was recently addressed by Zauner et al. (2022) for the mean-flow velocity around a square cylinder at the diameter-based Reynolds
number Re = 22000. Results of direct numerical simulations
exhibit broad-band high-frequency Kelvin-Helmholtz fluctuations in the shear layers that emerge at the leading-edge corners of the square cylinder, as well as low-frequency quasiperiodic vortex shedding in its wake. Simulations basedon the Unsteady Reynolds Averaged Navier-Stokes equations(URANS), such as those performed with the turbulent SpalartAllmaras model (Spalart & Allmaras, 1994), predict lowfrequency periodic-oscillation of the wake but fail to capture
the high-frequency fluctuations (Iaccarino et al., 2003). Recently, Zauner et al. (2022) have shown that the assimilation of
spanwise- and time-averaged pointwise velocity data (whichwere extracted from DNS) into an URANS model allows toimprove the low-frequency prediction but also to capture thehigh-frequency fluctuations of the shear layers. As for a wide
range of aerospace applications mean-flow estimates are often
sufficient, we here investigate the state-observer assimilationof pointwise velocities into a steady RANS model. Besides
the fact that solving such a steady model is more computationaly efficient than its unsteady counterpart, the state-observer
assimilation technique only requires time-independent and average pointwise velocity data. One of the difficulty raised by
the assimilation of pointwise velocity data is the appearanceof spurious flow oscillations around the measurement locations, as observed by Zauner et al. (2022) in the framework of
unsteady data-assimilation. Following Azouani et al. (2014);
Zerfas et al. (2019); Rebholz & Zerfas (2021), they used aninterpolant-based approach to reduce that detrimental effect on
the unsteady flow estimation. We thus here propose to investigate this interpolant-based approach in the present context
of assimilation of time-averaged pointwise velocities based on
RANS.

Data assimilation techniques (Hayase, 2015) are increasingly considered to overcome the above-mentioned respective
limitations, combining advantages of numerical and experimental approaches. It aims at merging experimental and numerical results in order to compensate for model and parameter uncertainties in numerical simulations on the one hand,
and to provide a full flow estimation from limited experimental data on the other hand. The outcome of data assimilation isthus an augmented flow prediction that combines the fidelityof the experimental/reference data and the comprehensive description that is offered by numerical simulation.
State-of-the-art data assimilation techniques include variational approaches, which rely on adjoint models to minimize
discrepancies between measurements and numerical flow estimation, and Kalman filter-based methodologies, which are derived from a Bayesian formulation of data assimilation. Whilethese two types of techniques may benefit from strong mathematical foundations, their application requires large computational costs, which may be of the order of one hundred baseline
simulations. The costs of variational approaches are mainly
driven by the number of optimization iterations to ensure convergence (Yegavian et al., 2015), while the computational requirements of Kalman filters are due to the need of propagating uncertainties in the flow state estimation, possibly throughMonte-Carlo techniques (Mons et al., 2016). In any case, theabove-mentioned computation cost may hinder the applicationof these data-assimilation techniques for complex flows.
In the present contribution, we study the use of an alternative and possibly over-looked data assimilation approach,namely nudging (Lakshmivarahan & Lewis, 2013), which may
also be referred to as state observer in the literature (Hayase,2015). It consists in adding a forcing term to the governing1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
hereinafter. The discretized non-linear equations, obtained using the FreeFEM open-source software (Hecht, 2012), are thensolved using a quasi Newton method. At the kth iteration ofthis algorithm, a linearized problem around the current solution (uk , pk , ν̃ k ) is solved with a direct LU solver, yielding
the correction (δ uk , δ pk , δ ν̃ k ). The solution is then updatedas (uk+1 , pk+1 , ν̃ k+1 ) + γ k (δ uk , δ pk , δ ν̃ k ), where the coefficient γ k is progressively increased with the number of iterations, unlike for the Newton method where it is kept fixedto γ = 1. This quasi-Newton method is crucial to avoid divergence of the algorithm, especially when considering thenudging approach described in the next section. At the inlet,
the boundary condition (u, v, ν̃) = (1, 0, 0) is imposed. At thecylinder surface, the no-slip boundary condition u = 0 is enforced in conjunction with ν̃ = 0. The conditions ∂∂ ν̃x = 0 and2(Re−1 + νt )S(u) · n − pn = 0 are used at the outlet, where ndenotes the normal vector. At the top and bottom boundaries,
symmetry conditions are imposed according to ( ∂∂ uy , v, ∂∂ ν̃y ) = 0.
Figure 1(b) displays isocontours of the streamwise meanflow velocity obtained by solving the RANS equations (1) with
the Spalart-Allmaras turbulent model (2). A symmetric recirculation region is also obtained in the wake of the square cylinder, but of much larger extent (up to x = 3.90) compared to thereference mean-flow (up to (up to x = 0.96, see figure 6-b).
The absolute value of the discrepancy between the two velocity fields is depicted in figure 1(c). The large error in the wakeis in agreement with the over-predicted extent of the recirculation region and can be explained by insufficient dissipation in
the Spalart-Allmaras turbulence model. The objective of thepresent study is to improve the estimation of the mean-flow
based on the SA-RANS model using state observer data assimilation and synthetic sparse measurements of the reference
mean-flow, as described in the next section.

The manuscript is organized in two parts. Section 1 describes the flow configuration, the numerical methods and theresults of standard RANS models. Section 2 is dedicated tothe state-observer assimilation method and its application to
the turbulent flow estimation around a square cylinder. The
pointwise- and interpolant-based approaches are first exposed,
the influences of the choice of the nudging coefficient and distance between measurements are then investigated, before concluding with a comparison between steady and unsteady data
assimilation.

1

TURBULENT FLOW PREDICTION WITH THE
STANDARD MODEL

We investigate the turbulent flow around a square cylinder at Reynolds number Re = U∞ D/ν = 22000, where ν is the
kinematic viscosity, U∞ is the upstream uniform velocity andD is the cylinder’s diameter. The origin of the cartesian coordinate system (x, y) is located at the center of the cylinder.
Hereinafter, all the spatial coordinates and flow variables are
made non-dimensional using D as length scale, U∞ as velocity
scale, and ρD2 as mass scale (per unit length), where ρ is thefluid density.
Three-dimensional Direct Numerical Simulation (DNS)of the flow, that is turbulent in the wake of the square cylinder, were performed by Dandois et al. (2018) with ONERA’s
FastS solver. The unsteady flow variables were spanwise- andtime-averaged to generate the two-dimensional mean-flow velocity ur , which is shown in figure 1(a). It is used not only
as a reference mean-flow but also to generate synthetic pointwise observations, as detailed later. The iso-contours of thestreamwise velocity, displayed in the figure, clearly indicate
the existence of a short symmetric recirculation region downstream the square cylinder, that extends up to x = 0.95. A
computationally efficient way to estimate this mean-flow is
to solve the following two-dimensional Reynolds-AveragedNavier-Stokes equations
 


1
(u · ∇)u + ∇p − ∇ · 2+ νt (ν̃) S(u) = 0
Re∇ · u = 0,2

To improve the estimation of the turbulent mean flow
based on the RANS equations, a state observer technique is investigated for the assimilation of pointwise velocity measurements of the reference mean flow ur .
The two components of the velocity fields are extracted asa set of M discrete points (xi )i=1···M . As shown in figure 2(a),they are here equally spaced in a measurement domain denotedΩm that will be specified later. The distance between thosemeasurements is denoted d. The data vector m, which gathers
these pointwise velocity measurements is formally defined as(1)

where u = (u, v) and p denote the mean velocity vector andpressure, respectively. To obtain the above equations, theBoussinesq assumption has been used to express the Reynolds
stress tensor as a function of S(u), denoting the strain-rate
tensor of the mean velocity. The (non-dimensional) eddyviscosity field νt is thus introduced in the momentum equations. In the present study, it depends on the variable ν̃ satisfying the following Spalart-Allmaras equation(u · ∇)ν̃ = s(ν̃, u),

TURBULENT FLOW ESTIMATION WITH
STATE OBSERVER ASSIMILATION OF
POINTWISE VELOCITY

m = H (ur ),(3)

where H is the so-called measurement operator. Unlike otherdata-assimilation method relying on optimization techniques,
the velocity data vector is directly injected into the momentum
equations (1), yielding(2)

where s refers to the source, diffusion and dissipation terms(Spalart & Allmaras, 1994).
For this convection-dominated flow, the Streamline Upwind Petrov Galerkin (SUPG) finite-element method (Brooks& Hughes, 1982) is used to discretize the standard RANS
equations (1) and (2). The rectangular computational domain
is defined in −10 ≤ x ≤ 15 and −10 ≤ y ≤ 10. An anisotropic
mesh adaptation based on the reference mean velocity andpressure (Fabre et al., 2018) is performed to obtain the meshmade of triangles for a total of ∼ 5 · 104 vertices that is used

 


1
(uγ · ∇)uγ + ∇pγ − ∇ · 2+ νt (ν̃) S(uγ )Re= γ I [m − H (uγ )],(4)

where uγ and pγ denote the reconstructed velocity and pressure fields. The right-hand-side term is a source term proportional to the error between the measurement of the velocity

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

(b)

(c)

Figure 1: Streamwise velocity of (a) the spanwise- and time-averaged flow obtained from Direct Numerical Simulations(DNS) and (b) the mean-flow computed as solution of the standard Reynolds Averaged Navier-Stokes (RANS) equations
using the turbulent Sparlat-Allamaras model. (c) Magnitude of the discrepancies between the DNS and RANS meanvelocity fields.
(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 2: (a) Punctual measurements. (b,c) Linear interpolation of the measurements in (a) on two symmetric triangularmeshes. Note that a layer is added to the mesh shown (c) but the values corresponding to the external nodes are set tozero, allowing a smoother interpolation at the boundaries of the measurement domain. Second, third and fourth rows
correspond to the forcing I [m − H (uγ )] in (4), the streamwise velocity of the reconstructed solution, and corresponding
error velocity field with respect to the reference (DNS) solution, respectively, using the punctual measurements shown in(a) (first column), or the linear interpolation shown in (b,c) (second and third columns).

field estimated with the RANS model and the data vector. The
coefficient γ allows to vary the magnitude of this source term,
while the operator I is introduced to map the pointwise error
to the space of velocity estimated by the RANS model. Various choices for this operator are investigated in the present
study.
The first one is the dual measurement operator, i.e. I = H † .
In that case, the forcing in the momentum equations is localized around the measurement points, as shown in figure2(d). Such forcing may induce spurious structures in the re-constructed flow, as noticed by Zauner et al. (2022), wheninvestigating assimilation of sparse unsteady velocity in unsteady RANS models. In the present study, these spurious oscillations are also observed in the reconstructed mean velocity
displayed in figure 2(g).
Interpolant-based nudging as proposed by Azouani et al.
(2014) is a simple way to get a smoother forcing and reconstructed flow. In that case, the operator I can be defined as
the product of two operators, i.e. I = M L , where L is aninterpolation of the measurement points in the measurement312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

(b)

Figure 3: Evolution of the global velocity error Eg in (6) as a function of (a) the nudging coefficient γ (for d = 0.5) and (b)the measurement distance d (for γ = 10). Circle and square symbols respectively denote extrapolation and interpolation
errors.
(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Effect of the nudging coefficient γ. (a-c) Streamwise velocity of the reconstructed field using equally-spaced

measurements (d = 0.5) and (d-f) error velocity field to the reference (DNS) solution for (a,d) γ = 0.1, (b,e) γ = 1 and(c,f) γ = 10.
space and M performs a mapping from the measurement tothe solution space, as detailed in the following. In the present
study, these operators are defined after discretization of thegoverning equations with the finite element method describedin the previous section. The measurement domain Ωm is discretized using a Delaunay triangulation of the set of measurement points, as shown by the grid displayed in figure 2(b). This
grid is symmetric with respect to the y = 0 axis so as to obtaina momentum forcing that satisfies the reflectional flow symmetry (see figure 2-e). Although we do not elaborate more onthat point in the following, we still remark that satisfying thisforcing symmetry is crucial to obtain accurate flow reconstruction, especially in case of very sparse data measurements. The
linear interpolation of the data measurement is then obtainedusing the linear Lagrange finite element function defined on
this mesh. For the streamwise reference velocity, it is definedasto the operator M , the forcing in the momentum equation is
finally obtained by an interpolation of the velocity error fromthe measurement mesh to the solution mesh. More details canbe found in appendix B in Zauner et al. (2022). The interestof the interpolant-based approach compared to the pointwisebased approach is clearly revealed when examining the interpolated function and forcing displayed in figures 2(b) and (e),respectively, that are now distributed in the measurement domain. This prevents from the appearance of spurious oscillations in the reconstructed solution, as shown in 2(h). A minor
drawback of this approach is that, for errors of same magnitude, those located at the boundaries of the measurement domain tend to be larger than those located inside the measurement domain. This effect is simply due to the shape of the finite element functions φui (x) corresponding to boundary nodes
i, that vanish non-smoothly out of the measurement domain.
Since the forcing function is discontinuous at the boundariesof the measurement domain, the iso-contours of the reconstructed velocity exhibit abrupt spatial variations around theboundaries of the measurement domain. This is particularly
visible in figure 2(e) at the top and bottom boundaries of themeasurement domain. To suppress that effect, we propose toextend the boundaries of the measurement mesh by one layerin every direction, as shown in figure 2(c). The additional
nodes located at the boundaries of this new mesh are ghost
nodes where the interpolated solution vanishes, as shown inM

L ur (x) = ∑ miu φui (x),

(5)

i=1

where φui (x) are the Lagrange P1 elements defined on the measurement grid. Using the definition of Lagrange elements, we
obtain that the interpolated function at measurement points isjequal to the measurement data, i.e. L ur (x j ) = mu . Thanks

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
up to x = 1, which is still slightly larger compared to the
DNS result of 0.96. It may be worth noting that d = 0.125
remains a measurement resolution that is coarse comparedto the flow structures close to the cylinder. In other words,
measurements alone, even in this case, can clearly not fully
characterize the flow, and interpolant-based nudging based on
RANS here enables more accurate evaluation of high-gradientregions. Within the boundary layers, however, we still observe
increased levels of velocity errors.
For separated flow configurations, unsteady RANS model
can significantly improve the accuracy of the mean-flow estimation, as shown by Iaccarino et al. (2003). Indeed, thecomponent of the Reynolds stress tensor induced by the lowfrequency fluctuations strongly contributes to reduce the sizeof the mean flow recirculation. Recently, Zauner et al. (2022)
performed the assimilation of unsteady pointwise velocity with
the URANS model. Figure 6(b) show the streamwise velocityof the mean flow obtained with the unsteady data-assimilationof pointwise uniform measurements, distributed in the domain−1.5 ≤ x ≤ 3 and −1.5 ≤ y ≤ 1.5 with a spacing of d = 1.5.
The results obtained using the present steady data-assimilation
approach are shown in Figure 6(a). Despite a significant improvement of the mean-flow estimation obtained with RANS
model, we should note that the nudged URANS simulations
provide better results and refer to Zauner et al. (2022) for further discussion.

figure 2(c). Thus, they are not associated to new measurement
data. They are rather added to obtain finite-element functions
at the boundary nodes that smoothly vanish to zero. The newforcing function and reconstructed solution, shown in figure2(f) and (i), respectively, are thus smoother around the boundaries of the measurement domain. When comparing the error
distributions shown in figure 2(j,k,l) for the three approaches
described above, we may conclude that the interpolant-based
approaches provide a significant improvement of the velocity
reconstruction inside the measurement domain by suppressing spurious oscillations related to the pointwise nature of themeasurements. The one-layer extension of the measurement
grid allows to further reduce the errors not only around theboundaries of the measurement domain but also further downstream. In the rest of the paper, we will focus on results obtained with that last approach.
We now examine the effect of the parameter γ on the accuracy of the flow reconstruction when using measurements thatare equally spaced by the distance d = 0.5 in the domain Ωmdefined as −1 ≤ x ≤ 4 and −1.5 ≤ y ≤ 1.5. To that aim, we
introduce the ratio of global velocity errors defined as
Z
Eg =
Ωt

e(uγ )(x)2 1 Z122/e(u)(x)2 ,

(6)

Ωt

q
where e(uγ )(x) = (uγ − ur )2 + (vγ − vr )2 is the local errorof the reconstructed velocity field uγ = (uγ , vγ ). The denominator corresponds to the error of the RANS solution. The
spatial domain Ωt can be the measurement domain Ωm wheninvestigating the interpolation error or defined as −3 ≤ x ≤ 10
and −5 ≤ y ≤ 5, when investigating the extrapolation error.
Both errors are plotted in figure 3(a) as a function of the parameter γ, using circles for the interpolation error and squares
for the extrapolation error. In both cases, a notable decrease of
the error is observed for γ > 0.01 with a saturation observed forγ ∼ 10. For large values of γ, the extrapolation error becomesslightly dominant, as observed in Zauner et al. (2022). The reconstructed velocity u(γ) and the distribution of the local errore(uγ ) are displayed in figure 4 for (a,d) γ = 0.1, (b,e) γ = 1 and(c,f) γ = 10. When increasing the value of γ, the size of therecirculation region clearly decreases and gets closer to that of
the reference mean-flow. For γ = 10, the local error is below
0.1 everywhere in the wake of the cylinder. The largest value(around 0.4) is obtained around the shear layers existing at the
top and bottom of the square cylinder. In these regions, the velocity gradients in the reference mean flow are strong and thelinear-interpolation of measurements spaced by half a cylinder
diameter (d = 0.5) is insufficient to improve the mean-flow estimation.
Looking at figure 3(b) showing the global interpolation andextrapolation errors as functions of d, decreasing the distance
between measurements below d = 0.5 has only minor effecton the reconstructed flow velocity in the far wake. Therefore, we will focus on the near wake region of the squarecylinder. Figure 5 displays (a,d,g) the interpolated measurements, (b,e,h) the reconstructed velocity and (c,f,i) the error
field for measurement distances (a-c) d = 0.5, (d-f) d = 0.25,
and (g-i) d = 0.125. The decrease of the error is significant
for d = 0.125 and velocity gradients in the shear layers are
now more accurately reconstructed. Local errors outside of
the boundary layer drop well below 0.3, which is particularly
evident in the near-wake region. The recirculation region delineated by the black curve in (h) approximates time-averaged
DNS data reasonably well. The recirculation length extends

CONCLUSION
The present investigation of steady data assimilation suggests great potential using an interpolant-based nudging approach. Using sufficiently well-resolved velocity measurements, it can reduce errors near the cylinder by more than one
order of magnitude, even if it is less accurate than the moreexpensive unsteady nudging approach. The comparison ofthis highly-efficient data-assimilation method with variational
data-assimilation approaches deserves further investigations,especially for turbulent flow configurations that do not exhibit low-frequency fluctuations, as the backward-facing step
(Franceschini et al., 2020) or the periodic hill (Volpiani et al.,2021).

REFERENCES
Azouani, A., Olson, E. & Titi, E. S 2014 Continuous data assimilation using general interpolant observables. Journal of
Nonlinear Science 24 (2), 277–304.
Brooks, A. N. & Hughes, T. J. R. 1982 Streamline
upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 32, 199–259.
Dandois, J., Mary, I. & Brion, V. 2018 Large-eddy simulation of laminar transonic buffet. Journal of Fluid Mechanics
850, 156–178.
Di Leoni, P. C., Mazzino, A. & Biferale, L. 2020 Synchronization to big data: Nudging the navier-stokes equations
for data assimilation of turbulent flows. Physical Review X
10 (1), 011023.
Fabre, David, Citro, Vincenzo, Ferreira Sabino, D, Bonnefis,
Paul, Sierra, Javier, Giannetti, Flavio & Pigou, Maxime
2018 A practical review on linear and nonlinear global approaches to flow instabilities. Applied Mechanics Reviews
70 (6).

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 5: Reconstructed flow velocity around the upper side of the square cylinder for (a-c) d = 0.5, (d-f) d = 0.25 and(g-i) d = 0.125. Streamwise velocity of (a,d,g) the linearly interpolated measurement and (b,e,h) the reconstructed flow.
(c,f,i) Error velocity fields. The coefficient is fixed to γ = 10. Grey lines in (a,d,g) show the measurement grid while thethick black curves in (b,e,h) delimit the recirculation regions.
(a)

(b)

Figure 6: Comparison of mean-flow estimations obtained with (a,c) RANS and (b,d) URANS models. Streamwise velocityof the solution obtained (a,b) with standard models and (c,d) with assimilation of pointwise measurements spaced by the
distance d = 1.5 in the domain −1.5 ≤ x ≤ 3 and −1.5 ≤ y ≤ 1.5.

Franceschini, Lucas, Sipp, Denis & Marquet, Olivier 2020
Mean-flow data assimilation based on minimal correction of
turbulence models: Application to turbulent high reynolds
number backward-facing step. Phys. Rev. Fluids 5, 094603.
Hayase, T. 2015 Numerical simulation of real-world flows.
Fluid Dynamics Research 47 (5), 051201.
Hecht, F. 2012 New development in FreeFem++. Journal of
Numerical Mathematics 20, 251–265.
Iaccarino, G., Ooi, A., Durbin, P. A. & Behnia, M. 2003
Reynolds averaged simulation of unsteady separated flow.
International Journal of Heat and Fluid Flow 24 (2), 147–
156.
Lakshmivarahan, S. & Lewis, J. M. 2013 Nudging methods:
A critical overview. Data Assimilation for Atmospheric,
Oceanic and Hydrologic Applications (Vol. II) pp. 27–57.
Mons, V., Chassaing, J.-C., Gomez, T. & Sagaut, P. 2016 Reconstruction of unsteady viscous flows using data assimilation schemes. Journal of Computational Physics 316, 255–
280.
Rebholz, L. G. & Zerfas, C. 2021 Simple and efficient continuous data assimilation of evolution equations via algebraic
nudging. Numerical Methods for Partial Differential Equations 37 (3), 2588–2612.
Saredi, E., Ramesh, N. T., Sciacchitano, A. & Scarano, F. 2021

State observer data assimilation for rans with time-averaged
3d-piv data. Computers & Fluids 218, 104827.
Spalart, P. R. & Allmaras, S. R. 1994 A one-equation
turbulence model for aerodynamic flows. La RechercheAérospatiale 1, 5–21.
Volpiani, Pedro Stefanin, Meyer, Morten, Franceschini, Lucas,
Dandois, Julien, Renac, Florent, Martin, Emeric, Marquet,
Olivier & Sipp, Denis 2021 Machine learning-augmented
turbulence modeling for rans simulations of massively separated flows. Phys. Rev. Fluids 6, 064607.
Yegavian, R., Leclaire, B., Champagnat, F. & Marquet, O.
2015 Performance assessment of piv super-resolution withadjoint-based data assimilation. In 11th International symposium on particle image velocimetry, Santa Barbara, USA,pp. 14–16.
Zauner, M., Mons, V., Marquet, O. & Leclaire, B. 2022
Nudging-based data assimilation of the turbulent flow
around a square cylinder. Journal of Fluid Mechanics 937.
Zerfas, C., Rebholz, L. G., Schneier, M. & Iliescu, T. 2019
Continuous data assimilation reduced order models of fluid
flow. Computer Methods in Applied Mechanics and Engineering 357, 112596.

6