# Nudging-based data assimilation of the turbulent flow around a square
cylinder

M. Zauner1  V. Mons1  O. Marquet1  B. Leclaire1  1DAAA, ONERA, Université
Paris Saclay, F-92190 Meudon, France

###### Abstract

We investigate the prediction of the turbulent flow around a canonical square
cylinder at $Re=22000$ solving the unsteady Reynolds-averaged Navier-Stokes
(URANS) equations. The limitations of URANS modelling are overcome through the
application of a data assimilation technique involving a feedback control
term, allowing to drive the URANS predictions towards reference data. Using
regularly-spaced temporally-resolved reference data that are extracted from
DNS, we assess the abilities of the data assimilation methodology in improving
the unsteady URANS predictions. While improving the low-frequency oscillations
of the wake flow, we also demonstrate the ability to recover the high-
frequency dynamics of Kelvin-Helmholtz instabilities. The influence of spatial
resolution of reference data is systematically investigated. The data
resolution which is required to improve the prediction of flow structures of
interest is related to their wavelength. Using a spacing of the order of one
cylinder length between data points, we already observe synchronisation of the
low-frequency vortex shedding that leads to a significant decrease in temporal
and spectral errors as computed by spectral proper orthogonal decomposition.
Furthermore, improved accuracy in terms of mean flow prediction of URANS is
achieved. Using spacing of the order of the wavelength of the Kelvin-Helmholtz
vortices, we even recover those shear layer structures which are absent in the
baseline simulation.

###### keywords:

##  1 Introduction

Fine characterization of turbulent flows is usually performed via experiments,
scale-resolving numerical approaches, or simulation methods relying on
turbulence models. Experiments reproduce the entire complexity of flow
physics, but the spatial and temporal resolutions of experimental data are
generally limited, which prevents from a full characterization of the flow of
interest. Direct numerical simulations (DNS) may provide such detailed results
with high spatio-temporal resolution for the full flow field. But those scale-
resolving simulations require massive computational power, that may even be
excessive when investigating high Reynolds number flows of interest for the
aeronautics, civil engineering or automotive industry. Lower-fidelity steady
or unsteady methods based on Reynolds-averaged Navier-Stokes (RANS) equations
are popular low-cost alternative approches. However, the accuracy of such
computations may largely be affected by the fidelity of the closure model for
the Reynolds stress tensor.

In order to overcome the respective limitations of the above mentioned
approaches, data assimilation (Lewis et al., 2006; Hayase, 2015b) appears as a
valuable tool. Data assimilation consists in merging experimental, or
reference data with computationally efficient numerical models. On one hand,
it allows to infer or compensate for uncertainties in input and/or model
parameters in simulations. On the other hand, it provides an augmented and
full flow estimation from the possibly sparse reference data. Popular data
assimilation techniques include adjoint-based (Le Dimet & Talagrand, 1986;
Papadakis & Mémin, 2008; Franceschini et al., 2020; Li et al., 2020) and
(ensemble) Kalman filter-based methodologies (Evensen, 2009; Suzuki, 2012;
Meldi & Poux, 2017; Da Silva & Colonius, 2020). Despite their significant
differences in terms of formulation and implementation, these two classes of
approaches may require a similar computation cost, which is typically of the
order of between $10$ and $100$ simulations (Mons et al., 2016).

As a cost-efficient alternative, we here investigate the application of the
nudging technique (Hoke & Anthes, 1976; Lakshmivarahan & Lewis, 2013), which
may also be referred to as state observer in the framework of control theory,
or more specifically measurement-integrated simulation (Hayase, 2015a) in the
context of Computational Fluid Dynamics (CFD). Nudging consists of adding a
feedback term to the flow governing equations that acts at specified
measurement locations and is proportional to the difference between reference
data and numerical prediction. Incompressible flows have generally been
considered in previous studies discussed below, and, depending on the
considered type of observation (namely velocity or pressure measurements), the
feedback term may act on the momentum equations (Imagawa & Hayase, 2010)
and/or on the pressure Poisson equation (Neeteson & Rival, 2020). The
intensity of this feedback term is tuned through a single scalar, which thus
allows to adjust to which extent the simulation is driven towards the
reference data. Nudging may be considered as a simplification of the Kalman
filter. Indeed, Kalman filtering also amounts to introduce a feedback term in
the governing equations that is proportional to the discrepancies between
observations and prediction, but this feedback term is there weighted through
a matrix gain instead of a single scalar. Nudging may also be categorized as a
special case of proportional–integral–derivative (PID) controller that only
includes the proportional contribution. The benefits of considering temporal
integral and derivative of the discrepancies between observations and
prediction have been recently investigated in (Neeteson & Rival, 2020; Saredi
et al., 2021). In contrast to previously mentioned data assimilation
approaches such as adjoint- and Kalman filter-based techniques, the
supplementary computational cost that is associated to nudging is essentially
negligible compared to that of a baseline simulation.

In the context of CFD, the application of nudging was first performed in a
purely numerical context by Hayase & Hayashi (1997); Imagawa & Hayase (2010)
considering the turbulent flow in a square pipe. Based on punctual
observations of the velocity field of a reference DNS, nudging was employed to
drive a second simulation towards the reference one. Nudging was able to
compensate for uncertainties in the initial condition (Imagawa & Hayase, 2010)
or numerical errors due to the use of a coarser grid in the second simulation
(Hayase & Hayashi, 1997), and thus successfully reproduced the reference
solution. Neeteson & Rival (2020) also considered nudging and more general PID
controllers to drive a coarse-grid simulation based on synthetic pressure data
of the flow past a square cylinder at $Re=100$. In Di Leoni et al. (2020),
nudging was applied to homogeneous turbulence, still relying on DNS. In
particular, nudging was used to recover a reference forced isotropic turbulent
flow, assuming that the forcing of the reference flow was unknown. Various
types of observations of the reference flow were considered among Eulerian
observations, as in Hayase & Hayashi (1997); Imagawa & Hayase (2010); Neeteson
& Rival (2020), Lagrangian data, or in spectral space. The effect of the
observation density on the ability of nudging in recovering unmeasured scales
was also investigated. While the reference data were generated with the same
numerical methods as those employed for data assimilation in the above
studies, nudging in conjunction with a turbulence model was investigated by
Buzzicotti & Di Leoni (2020) in the context of large eddy simulations of
isotropic turbulence. Filtered reference data were generated by DNS, and
nudging was performed assimilating the full reference data at all resolved
scales, and was thus only useful to correct model errors.

Aside from these purely numerical studies, Nisugi et al. (2004); Yamagata et
al. (2008) demonstrated the ability of nudging to deal with various types of
experimental data of the flow past a square cylinder at $Re=1200$, such as
wall pressure and particle image velocimetry (PIV) measurements, based on the
laminar Navier-Stokes equations. Suzuki et al. (2009a, b) performed the
assimilation of particle tracking velocimetry (PTV) measurements of flows past
airfoils at low Reynolds numbers. Their proposed methodology slightly differs
from that in the above studies in the fact that the measurements were
rearranged before the data assimilation procedure in order to be directly
comparable with a flow state estimated by DNS. To efficiently handle
experimental data of higher Reynolds number flows, Nakao et al. (2009)
considered relying on unsteady RANS (URANS) modelling to perform nudging.
Recently, steady nudging in conjunction with RANS was performed by Saredi et
al. (2021) based on three-dimensional PIV measurements of the mean flow around
a wall-mounted bluff obstacle. Nudging proved to be effective in correcting
deficiencies in the RANS prediction, such as the overestimation of the extent
of the recirculation region past the obstacle. However, while these studies
confirmed the feasibility of applying nudging to experimental data and
ambitious flow configurations, the use of only experimental data precluded a
quantitative assessment of the improvement in the estimation of the considered
high Reynolds number flows outside of the measurement locations.

In the present contribution, we aim to assess in detail the full potential of
the nudging approach for unsteady flow estimation in an aerodynamic context,
based on URANS modelling and synthetic velocity data. The unsteady turbulent
flow around a square cylinder at $Re=22000$ is chosen as flow configuration
(Lyn et al., 1995; Trias et al., 2015) because it exhibits several flow
phenomena occurring at various time and spatial scales that are of interest
for aeronautical application. They include low-frequency quasi-periodic vortex
shedding (VS) past the square cylinder, along with broad-band high-frequency
Kelvin-Helmholtz (KH) fluctuations in the shear layers at the top and bottom
sides of the cylinder. URANS simulations aim at capturing the fluctuation of
the statistically-averaged flow, and are most accurate when the fluctuation
frequency, such that of vortex shedding, is much lower than that of the
turbulence itself, which is modelled. For the present flow configuration,
Iaccarino et al. (2003) showed that, while being inevitably more expensive
than RANS simulations, URANS simulations significantly improve the prediction
of the time-independent mean flow. On the other hand, they do not necessarily
capture the small-scale high-frequency fluctuations associated to Kelvin-
Helmholtz vortices and, more generally, their accuracy remains determined by
the adequacy of the closure model for the Reynolds stress tensor (Bosch &
Rodi, 1998; Palkin et al., 2016), here the Spalart-Allmaras model (Spalart &
Allmaras, 1994). The objective of the paper is thus to assess the potential of
the nudging method in unsteady flow estimation, not only for the large-scale
low-frequency fluctuations that are solved by standard URANS simulations, but
also for the small-scale high-frequency fluctuations that are not captured by
these same simulations. Synthetic observations for the considered two-
dimensional URANS simulations will be generated through a spanwise average of
a three-dimensional DNS of the present flow configuration. Nudging will then
be employed to enhance the URANS predictions and reconstructing the full
reference flow from sparse velocity observations, whose spatial density will
be systematically varied. A crucial aspect of the present study will be to
correctly reproduce flow structures evolving coherently in space and time.
This will be assessed through the spectral proper orthogonal decomposition
(Towne et al., 2018).

The paper is organised as follows. Standard numerical simulations of the
turbulent flow around a square cylinder are described in §2. After introducing
the numerical methods used for DNS and URANS simulations in §2.1 and §2.2,
respectively, we describe methods to analyse dynamic as well as mean
properties of flow fields in §2.3. DNS and standard URANS results are
discussed in §2.4. The potential of nudged URANS simulations is then discussed
in §3. After a description of the nudging method and used data-sets of
measurements in §3.1, results of nudged simulations using a dense set of
measurements are compared to standard URANS and DNS data in §3.2. Then, the
performance of nudging with respect to low- and high-frequency flow phenomena
is discussed in sections §3.3 and §3.4, respectively. Implications for
experimental studies are discussed in §3.5. Finally, concluding remarks are
drawn in §4.

##  2 Standard URANS simulations compared to DNS

We investigate the turbulent flow developing around a square cylinder of
length $D$ facing an upstream uniform flow of velocity $U_{\infty}$. The
Reynolds number based on these reference length and velocity scales is
$Re=U_{\infty}D/\nu=2.2\cdot 10^{4}$. Three-dimensional direct numerical
simulation (DNS) presented in §2.1 captures the full range of flow scales,
unlike standard two-dimensional Unsteady Reynolds-Averaged Navier-Stokes
(URANS) equations introduced in §2.2 that capture only the low-frequency
dynamics of statistically averaged flow structures. A turbulence model then
accounts for the effect of incoherent structures so that the overall
computational costs are significantly lower for URANS computations compared to
DNS. In order to assess the quality of URANS predictions, instantaneous DNS
snapshots are averaged in the spanwise direction and provide reference data.
After describing respectively temporal and modal diagnosis tools in §2.3,
detailed comparison between DNS and URANS simulations is provided in §2.4.

###  2.1 Direct numerical simulations

Three-dimensional DNS of the turbulent flow around the square cylinder are
performed with the ONERA’s compressible solver FastS (Mary & Sagaut, 2002;
Dandois et al., 2018). The three-dimensional computational domain is formed by
a two-dimensional circle of radius $50D$ in the $\boldsymbol{x}=(x,y)$ plane,
centered around the square cylinder, which is extruded in the spanwise
homogeneous direction $z$ with length $L_{z}=4D$. The two-dimensional mesh
consists of approximately $2.6\cdot 10^{5}$ grids points and $N_{z}=960$
planes uniformly discretise the spanwise direction. Periodic boundary
conditions are applied in the spanwise direction. Homogeneous freestream
conditions are enforced at the outer boundaries, where only the horizontal
velocity component (parallel to upper and lower cylinder surfaces) is non-zero
and equal to $U_{\infty}$. The simulation is run close to incompressible flow
regime at $M=0.1$ enforcing no-slip conditions at the cylinder surface. In the
following, all quantities are nondimensionalised based on the reference length
$D$ and free stream parameters $U_{\infty}$, $\rho_{\infty}$. The DNS time
step is $\Delta t_{DNS}=3.3188\cdot 10^{-4}$, which corresponds to a maximal
CFL number of approximately $0.7$. More details about this DNS solver are
provided in Dandois et al. (2018); Mary & Sagaut (2002); Franceschini et al.
(2020).

The unsteady three-dimensional velocity fields $\boldsymbol{u}$ so obtained
can be decomposed as

$\displaystyle\boldsymbol{u}(\boldsymbol{x},z,t)=\bar{\boldsymbol{u}}_{r}(\boldsymbol{x},t)+\boldsymbol{u}^{{}^{\prime}}(\boldsymbol{x},z,t)$
(1)

where $\boldsymbol{\bar{u}}_{r}$ is the statistically-averaged two-dimensional
velocity field of interest while $\boldsymbol{u}^{{}^{\prime}}$ refers to the
associated three-dimensional deviation, not used in the following. Then, it is
assumed that $\boldsymbol{\bar{u}}_{r}$ can be extracted from DNS velocity
fields $\boldsymbol{u}$ by averaging instantaneous snapshots in the spanwise
direction according to

$\displaystyle\bar{\boldsymbol{u}}_{r}(\boldsymbol{x},t)=\frac{1}{L_{z}}\int_{0}^{L_{z}}\boldsymbol{u}(\boldsymbol{x},z,t)\,dz\sim\frac{1}{N_{z}}\sum_{k=1}^{N_{z}}\boldsymbol{u}(\boldsymbol{x},z=k\Delta
z,t)$ (2)

where $\Delta z\sim 0.004$ is the uniform discretization step in the spanwise
direction. This spatial average well approximates the statistical average if
the spanwise extent $L_{z}$ of the computational domain and the number of
planes $N_{z}$ are large enough. This two-dimensional field extracted from DNS
will be used as reference solution (hence the subscript $r$), not only to
assess the accuracy of the standard URANS simulation methods in §2.4, but also
to improve their predictive capability thanks to the nudging method in §3. To
complete the description of this reference data set, it is obtained by down-
sampling the DNS snapshots every $63$ time steps. The time interval $\Delta
t_{r}$ between snapshots in this reference data set is thus $\Delta
t_{r}=63\times\Delta t_{DNS}=0.0209$ convective time units.

###  2.2 Unsteady Reynolds-averaged Navier-Stokes simulations

(a) | (b)
---|---
|
Figure 1: (a) Sketch of the computational domain $\Omega$ for URANS
simulations, where red dots denote positions $(\mathbf{x}_{k})_{k=1\cdots M}$
of velocity measurements in the internal domain $\Omega_{I}$ (bounded by the
red frame). These so-called nudging points are uniformly separated in both
directions by a spacing-distance $\Delta s$. The domain that is external to
the measurement area is denoted as $\Omega_{E}$. (b) Typical triangular mesh
used for the standard and nudged URANS simulations around a square cylinder.

Rather than extracting the temporal variation of the spatially-averaged flow
from DNS simulations, one may rather estimate the latter through the two-
dimensional URANS equations which are supplemented with a turbulence model to
take into account the effect of the non-resolved component. The two-
dimensional resolved velocity ($\bar{\boldsymbol{u}}$) and pressure
($\bar{p}$) fields then satisfy the following (incompressible) flow equations

$\displaystyle\frac{\partial\bar{\boldsymbol{u}}}{\partial
t}+(\bar{\boldsymbol{u}}\cdot\boldsymbol{\nabla})\bar{\boldsymbol{u}}+\boldsymbol{\nabla}\bar{p}-\boldsymbol{\nabla}\cdot\left[2\left(Re^{-1}+\nu_{t}(\tilde{\nu})\right)\boldsymbol{\nabla}_{\mathrm{s}}\bar{\boldsymbol{u}}\right]$
$\displaystyle=$ $\displaystyle\boldsymbol{0}\;,$
$\displaystyle\boldsymbol{\nabla}\cdot\bar{\boldsymbol{u}}$ $\displaystyle=$
$\displaystyle 0,$ (3)

where $\boldsymbol{\nabla}_{\mathrm{s}}\bar{\boldsymbol{u}}$ refers to the
symmetric part of the velocity gradient and $\nu_{t}$ denotes the turbulent
eddy-viscosity. Here, the latter is determined using the Spalart-Allmaras
model (Spalart & Allmaras, 1994), where the following governing equation for
the intermediate eddy-viscosity-like variable $\tilde{\nu}$ is solved

$\displaystyle\frac{\partial\tilde{\nu}}{\partial
t}+(\bar{\boldsymbol{u}}\cdot\boldsymbol{\nabla})\tilde{\nu}$ $\displaystyle=$
$\displaystyle s(\tilde{\nu},\bar{\boldsymbol{u}}).$ (4)

The source term $s$ accounts for the production, dissipation and diffusion
terms for $\tilde{\nu}$. Details about the present implementation of the
Spalart-Allmaras model are provided in Franceschini et al. (2020).

The URANS equations (2.2)-(4) are discretised in space using the finite-
element method implemented in the software FreeFEM++ (Hecht, 2012). Second-
order polynomial elements $(P_{2})$ are employed for the velocity field, while
first-order elements $(P_{1})$ are considered for the pressure and eddy-
viscosity variables. For the relatively high Reynolds number flow of the
present contribution ($Re=22000$), streamline-upwind Petrov-Galerkin (SUPG)
(Brooks & Hughes, 1982) and grad-div (Olshanskii et al., 2009) stabilisations
are implemented. The rectangular computational domain is sketched in figure
1(a) with an inlet located at $x=-10$, an outlet at $x=15$ and top and bottom
planes at $y=10$ and $y=-10$, respectively. Remaining details of figure 1(a)
will be discussed later when introducing the nudging approach. The
unstructured mesh, shown in figure 1(b), consists of triangles with a total
number of $46228$ nodes. At the inlet, the boundary condition
$(\bar{u},\bar{v},\tilde{\nu})=(1,0,0)$ is imposed. At the cylinder surface,
the no-slip boundary condition $\bar{\boldsymbol{u}}=\boldsymbol{0}$ is
enforced in conjunction with $\tilde{\nu}=0$. The conditions
$\frac{\partial\tilde{\nu}}{\partial x}=0$ and
$2(Re^{-1}+\nu_{t})\boldsymbol{\nabla}_{\mathrm{s}}\bar{\boldsymbol{u}}\cdot\boldsymbol{n}-\bar{p}\boldsymbol{n}=\boldsymbol{0}$
are used at the outlet, where $\boldsymbol{n}$ denotes the normal vector. At
the top and bottom boundaries, symmetry conditions are imposed according to
$(\frac{\partial\bar{u}}{\partial
y},\bar{v},\frac{\partial\tilde{\nu}}{\partial y})=\boldsymbol{0}$.

Time integration is performed in a fully implicit way based on a second-order
accurate finite-difference approximation of the time derivative. A quasi-
Newton method is used to solve the resulting nonlinear problem at each
iteration. The time step of URANS simulation is denoted by $\Delta t$. Results
of the standard simulations presented in §2.4 are obtained with $\Delta
t=\Delta t_{r}$, i.e. matching the time interval of the reference data set.
For nudged URANS simulations in §3, the value of $\Delta t$ will be equal to
the time interval of data sets (not necessarily the reference data set)
inserted into the simulations through the feedback term. By varying the
sampling of these data sets, we will thus modify the time step $\Delta t$.

###  2.3 Temporal and spectral errors

We introduce in this paragraph various mathematical definitions for assessing
errors between the reference data set and the output of the standard or nudged
URANS simulations. Note that the latter have not yet been introduced, but we
will use the same notation for denoting the corresponding variables in both
simulations. In the following, both standard and nudged URANS solutions may be
referred to as the estimated flow, while the field $\bar{\boldsymbol{u}}_{r}$
which is extracted from DNS in (2) is referred to as the reference flow, as
mentioned above. Definitions of the instantaneous and time-averaged errors are
introduced in this section before considering the spectral error based on a
SPOD analysis of the data.

The instantaneous error field $e(\boldsymbol{x},t)$ between the estimated
velocity field $\bar{\boldsymbol{u}}$ and the reference one
$\bar{\boldsymbol{u}}_{r}$ is defined as

$\displaystyle
e(\boldsymbol{x},t)=\left({(\bar{u}_{r}-\bar{u})^{2}+(\bar{v}_{r}-\bar{v})^{2}}\right)^{\frac{1}{2}}.$
(5)

The instantaneous and time-averaged global errors are then respectively
defined as

$\displaystyle
E(t)=\left(\int_{\Omega}e(\boldsymbol{x},t)^{2}\,d\boldsymbol{x}\right)^{\frac{1}{2}},\quad\langle
E\rangle=\frac{1}{T_{e}}\int_{t_{i}}^{t_{i}+T_{e}}E(t)\,dt,$ (6)

where $\Omega$ denotes the entire computational domain, $t_{i}>0$ is an
arbitrary time that is chosen to avoid transient effects (occurring in
$[0,t_{i}]$) and $T_{e}$ refers to the considered time window for averaging
the instantaneous global error. Note that the time average is denoted
$\langle\cdot\rangle$, and thus $\langle\bar{\boldsymbol{u}}\rangle$ is the
time-averaged velocity field of the estimated flow. Its error compared to the
time-averaged reference velocity $\langle\bar{\boldsymbol{u}}_{r}\rangle$ is
then defined as

$\displaystyle
e_{\langle\bar{\boldsymbol{u}}\rangle}(\boldsymbol{x})=\left(\left(\langle\bar{u}_{r}\rangle-\langle\bar{u}\rangle\right)^{2}+\left(\langle\bar{v}_{r}\rangle-\langle\bar{v}\rangle\right)^{2}\right)^{\frac{1}{2}}.$
(7)

To better understand the unsteady effect, a spectral analysis of the data is
performed which relies on the spectral proper orthogonal decomposition (SPOD)
described in Towne et al. (2018). Taking a temporal series of velocity
snapshots as inputs, the SPOD analysis provides a set of SPOD modes
$\boldsymbol{\Phi}=(\Phi_{u},\Phi_{v})^{T}$ that each oscillate at a single
non-dimensional frequency (Strouhal number) $St=fD/U_{\infty}$. To each
frequency corresponds a set of normalized SPOD modes
($\int_{\Omega}\boldsymbol{\Phi}^{*}\boldsymbol{\Phi}d\Omega=1$, where the
asterisk denotes the Hermitian transpose) ranked by their kinetic energy
$\lambda$, that are the eigenvalues of the cross-spectral density tensor which
is estimated from snapshots. In the following, we will focus on the dominant
SPOD modes, namely which are associated to the largest eigenvalue, for
frequencies of interest. Denoting hereinafter the SPOD modes of the reference
and URANS data set as $[\lambda_{r},\boldsymbol{\Phi}_{r}]$ and
$[\lambda,\boldsymbol{\Phi}]$ respectively, a spectral error field may be
defined as

$\displaystyle\Phi_{e}=\left(\left(\sqrt{\Phi_{u,r}^{*}\Phi_{u,r}}-\sqrt{\Phi_{u}^{*}\Phi_{u}}\right)^{2}+\left(\sqrt{\Phi_{v,r}^{*}\Phi_{v,r}}-\sqrt{\Phi_{v}^{*}\Phi_{v}}\right)^{2}\right)^{\frac{1}{2}}.$
(8)

In the following, the dynamical content of standard and nudged URANS
simulations will thus be assessed and compared to the reference solution
through their frequency content, the kinetic energy of the dominant SPOD
modes, and the shape of the latter through (8)

###  2.4 Comparisons between standard URANS and DNS

(a) | (b)
---|---
|
(c) | (d)
|
Figure 2: Results for (a-b) reference (DNS) and (c-d) estimated (standard
URANS) flows. (a,c): Instantaneous spanwise vorticity field $\bar{\omega}_{z}$
at $t=50$, where black iso-curves denote $\left<\overline{u}\right>=0$. (b,d):
Fourier spectrum of the streamwise velocity at the green monitor point in
(a,c) (full black line). For the sake of comparison, the DNS spectrum is
duplicated in (d) with the grey curve. Vertical dashed and dotted lines
indicate the low-frequency peak associated with large-scale vortex-shedding
($St_{VS}=0.137$ for DNS while $St_{VS}=0.126$ for URANS) and the high-
frequency bump linked to Kelvin-Helmholtz instabilities ($St_{KH}=4.384$ for
DNS), respectively.

A preliminary URANS simulation is performed so as to obtain a periodic
evolution of the aerodynamic coefficients. It allows to determine an instant
which minimises the discrepancy between the lift coefficients computed from
URANS and DNS simulations. The flow field at that instant is then used as
initial condition $t=0$ for the URANS simulation that is discussed below. By
doing so, the DNS and URANS solutions are initially phased in time and we can
thus properly compare the temporal evolution of the error between the two.

Figure 2 provides a first comparison of the flow extracted from DNS [figures
2(a-b)] and estimated by the URANS equations [figures 2(c-d)]. The
instantaneous vorticity contours are shown in figures 2(a) and (c) for $t=50$.
In both cases, we observe that large-scale clockwise (blue) and counter-
clockwise (red) vortices are shed in the wake of the square cylinder. Small-
scale structures are also visible in the reference flow field shown in figure
2(a). In the upper and lower shear layers, emerging from the leading edge
corners of the cylinder, they correspond to two-dimensional roll-up structures
associated with Kelvin-Helmholtz instabilities. In the wake, they interact
with the large-scale vortices of the von-Kármán vortex street. Such small-
scale structures are clearly absent in the estimated vorticity field that is
shown in figure 2(c). Large-scale and small-scales structures are associated
with different parts of the Fourier spectra displayed in figures 2(b,d), which
are obtained from the streamwise velocity $u$ monitored at a grid point
located in the upper shear layer, at $(x,y)=(0.25,0.75)$ [green dot in figures
2(a,c)]. The large-scale periodic vortex shedding is associated to several
peaks, one at the fundamental frequency $St_{VS}=U_{\infty}f_{VS}/D$
(indicated in both figures by the vertical dashed lines) and the others at
multiples of the fundamental frequency, i.e. $kSt_{VS}$ with $k\geq 2$. The
fundamental frequency is $St_{VS}=0.137$ for the DNS simulation and
$St_{VS}=0.126$ for the standard URANS simulation. The small-scale structures
that are observed in the shear layers in the reference snapshot (figure 2(a))
correspond to a high-frequency broadband bump which is centered around
$St_{KH}\approx 4.384$, as indicated by the vertical dotted line in figure
2(b). This high-frequency bump is absent in the spectrum of the URANS signal,
displayed with the black curve in figure 2(d). This confirms that the URANS
simulation does not capture the emission of Kelvin-Helmholtz vortices in the
shear layers.

Although the URANS simulation well approximates the periodic vortex-shedding
phenomenon, the difference between Strouhal numbers noticed above is
responsible for desynchronisation of the wake flow as illustrated in figure 3.
The instantaneous vorticity fields displayed in figures 3(a-c) and (d-f) for
DNS and URANS simulations, respectively, seem in phase after $2$ vortex-
shedding periods (a,d), but, after $4$ shedding cycles (b,e), large vortex
structures observed in URANS snapshots clearly lag behind corresponding
structures in DNS snapshots. After $19.5$ periods (c,f), the two solutions are
out-of-phase by a half-wavelength. This desynchronisation of the large-scale
structures results in large values of the instantaneous local error in the
wake of the cylinder, as illustrated by figures 3(g-i). The temporal evolution
of the instantaneous global error $E(t)$ in (6) is reported in figure 3(j)
from $t=0$ to $t=200$, which corresponds to a time window of $26$ vortex-
shedding periods. The error $E(t)$ increases during the first $20$ periods
($0\leq t\leq 150)$, in agreement with the previous observations, before
slightly decreasing. This effect is still related to the lack of
synchronisation between reference and estimated flows.

(a) | (b) | (c)
---|---|---
|  |
(d) | (e) | (f)
|  |
(g) | (h) | (i)
|  |

(j)
---

Figure 3:  Instantaneous vorticity fields of (a-c) the reference flow
$\boldsymbol{\bar{u}}_{r}$ (DNS) and (d-f) estimated flow
$\boldsymbol{\bar{u}}$ (standard URANS) at times (a,d) $t=15$, (b,e) $t=30$
and (c,f) $t=150$. (g-i) Instantaneous error fields $e(\boldsymbol{x},t)$
between the reference and estimated flow (definition in eq. 5) at the
corresponding instants. (j) Temporal evolution of the global error $E(t)$ (see
eq. 6) where the bottom axis reports the (nondimensional) time while the top
axis reports the number of low-frequency cycles determined as $t/\tau_{VS}=t\
St_{VS}$ with $St_{VS}=0.137$. The grey circles indicates time $t=15$, $30$
and $150$.

The statistical and dynamic discrepancies between estimated and reference
flows are further examined in the following, and we first consider the time
average of these flows. The streamwise velocity of the time-averaged reference
(DNS) and estimated (URANS) flows are depicted in figures 4(a) and 4(b),
respectively, while the associated discrepancy field
$e_{\langle\bar{\boldsymbol{u}}\rangle}(\boldsymbol{x})$ in (7) is displayed
in figure 4(c). Interestingly, the mean flows agree qualitatively well, and
the error field $e_{\langle\bar{\boldsymbol{u}}\rangle}(\boldsymbol{x})$
reaches overall significantly lower values than the discrepancy field
$e(\boldsymbol{x},t)$ for the instantaneous flow in figure 3(c). Largest
errors are observed in the shear layers that emerge from the cylinder upstream
corners. In the near wake, the recirculation region, which is delineated by
black curves, is slightly larger for the estimated flow ($L=0.55$) compared to
that of the reference flow ($L=0.45$). Substantial errors are also visible in
the far wake where streamwise velocity is overestimated by URANS.

(a) | (b) | (c)
---|---|---
|  |
Figure 4: (a-b) Streamwise component of the time-averaged (a) reference flow
$\langle\boldsymbol{\bar{u}}_{r}\rangle$ (DNS) and (b) estimated flow
$\langle\boldsymbol{\bar{u}}\rangle$ (standard URANS). (c) Associated
discrepancy field $e_{\langle\bar{\boldsymbol{u}}\rangle}$ as defined in (7).
In (a-b), black curves correspond to contours of zero streamwise time-averaged
velocity, which delineate the recirculation region.

The SPOD decomposition of the reference and estimated flows are both performed
by considering as input $8420$ snapshots that are sampled at $\Delta
t_{r}=0.021$ in a time interval of around $176$ convective times. Using the
Matlab implementation provided by Towne et al. (2018), this series is
typically divided in $3$ overlapping bins, each bin containing about $4200$
instantaneous snapshots, and overlapping by $50\%$ of their size. The time
interval of a bin is thus around $88$ convective time units.

(a) |  | (b)
---|---|---
|  |
(c) |  | (d)
|  |
Figure 5: SPOD analysis of the vortex-shedding phenomenon in the reference (DNS) and estimated (standard URANS) flows: streamwise velocity contours of the imaginary part of the dominant (a) DNS and (c) URANS SPOD modes for the fundamental frequency $St_{VS}$. Modes are phased so that each of them reaches its maximum value at the green monitor point. (b) Spectra showing the largest eigenvalue $\lambda$ as a function of the frequency $St$ in the low-frequency range $0\leq St\leq 1$ for DNS (black curve) and URANS (grey curve). The vertical lines indicate low-frequency peaks at $St_{VS}=0.137$ and its harmonics for DNS results. The modal discrepancy field $\Phi_{e}$ in (8) between the modes in (a) and (c) is reported in (d). (a) |  | (b)
---|---|---
|  |
Figure 6:  (a) Contours of streamwise velocity fluctuations are shown for the
real part of the dominant SPOD mode of DNS that is associated to Kelvin-
Helmholtz instabilities ($St_{KH}=4.384$, only captured with DNS). (b) Spectra
showing the largest eigenvalue $\lambda$ as a function of the frequency $St$
in the high-frequency range $1\leq St\leq 7$. Black and light grey curves
correspond to DNS and URANS results, respectively.

The SPOD analysis of the vortex-shedding phenomenon as predicted by DNS and
standard URANS is first illustrated in figure 5. Spectra of the most-energetic
SPOD modes for a low-frequency range ($0\leq St\leq 1$) as functions of
Strouhal number $St$ are reported in figure 5(b). DNS and URANS results are
denoted by black and light grey dots, respectively. As in the Fourier analysis
of figure 2, several peaks are observed at the fundamental frequency
$St_{VS}=0.137$ of the vortex-shedding phenomenon and at frequencies
corresponding to second and third harmonics at $St_{VS,2}=0.261$ and
$St_{VS,2}=0.398$, respectively. Frequencies of URANS modes (grey peaks) are
slightly lower, peaking at $St_{VS}=0.126$, $St_{VS,2}=0.261$, and
$St_{VS,2}=0.386$. Streamwise velocity contours of the imaginary part of the
dominant DNS and URANS SPOD modes for the fundamental frequency $St_{VS}$ are
displayed in figures 5(a) and 5(c), respectively. Both modes are phased so
that they reach their maximum amplitude at the monitor point indicated by the
green dot, which is the same as in figure 2. Their spatial structure is
typical of modes corresponding to the alternate shedding of vortices in the
wake, with a streamwise wavelength equal to $\sim 4$ cylinder lengths.
Furthermore, we show in figure 5(d) the so-called modal error field
$\Phi_{e}(\boldsymbol{x})$ which is defined in equation 8. Dominant errors are
observed in boundary and shear layers, as well as in the near-wake region,
which may be related to the slightly longer recirculation region predicted by
URANS.

Figure 6(b) shows the spectrum of the most dominant SPOD modes corresponding
to high-frequency fluctuations. A broadband peak centered around
$St_{KH}=4.384$ is clearly visible in the DNS spectrum (black) but not in the
URANS one (grey). In order to prevent modes from being corrupted by numerical
oscillations in regions of decreased grid resolution, we limit the domain that
is considered for SPOD targeting Kelvin-Helmholtz phenomena to $-1.5<x<1.5$
and $-1.5<y<1.5$. Furthermore, we consider $1440$ snapshots at a sampling rate
of $\Delta t_{r}=0.021$ in a time interval of around $30$ convective times
using $6$ overlapping bins, each containing about $33$ Kelvin-Helmholtz
cycles. In order to better visualise convective small-scale instabilities
originating from the front edge of the cylinder, a magnification of the upper-
side shear layer in figure 6(a) shows the real part of streamwise velocity
contours for the dominant SPOD mode from DNS results at $St_{KH}=4.384$.
Typical wavelengths of these structures are around $0.25$, i.e. a quarter of
the cylinder’s length. Similar spatial structures are also observed on the
lower side of the cylinder, but of much weaker magnitude. As expected when
considering the spatial symmetry of the time-averaged flow, a second SPOD mode
with spatial structures predominant on the lower-side shear layer is also
obtained but associated to a slightly lower kinetic energy.

The results of the SPOD analysis are summarized in table 1, which confirms a
rather good agreement between DNS and standard URANS in terms of the
prediction of low-frequency vortex-shedding structures. Interestingly, a small
discrepancy in the estimation of the fundamental frequency $St_{VS}$ (about
$10\%$) leads to large temporal errors due to desynchronisation effects. It
may thus be anticipated that adding sparse information to the low-frequency
content of the estimated flow may significantly reduce the corresponding
temporal error. In the next section, we will use the nudging approach to
improve the prediction of the large-scale structures oscillating at low
frequency, but also to recover the small-scales Kelvin-Helmholtz structures,
which are not captured with standard URANS simulations.

| $\Delta C_{L}$ | $\overline{C_{D}}$ | $St_{VS}$ | $\lambda_{VS}$ | $St_{VS,2}$ | $\lambda_{VS,2}$ | $St_{VS,3}$ | $\lambda_{VS,3}$ | $St_{KH}$ | $\lambda_{KH}$
---|---|---|---|---|---|---|---|---|---|---
DNS | $4.683$ | $2.091$ | $0.137$ | $14.847$ | $0.261$ | $0.359$ | $0.398$ | $0.261$ | $4.384$ | $0.000164$
standard URANS | $4.198$ | $2.017$ | $0.126$ | $15.082$ | $0.261$ | $0.442$ | $0.386$ | $0.440$ | $-$ | $-$
Table 1: Time-averaged and unsteady characteristics of the flow obtained with
DNS and standard URANS simulations. The first two columns report the maximal
lift variation and time-averaged drag coefficients, respectively. The
remaining columns report the dominant eigenvalue $\lambda$ for various
frequencies $St$ from the SPOD analysis, considering the fundamental harmnonic
for the von-Kármán vortex shedding ($VS$), its second (${VS,2}$) and third
(${VS,3}$) harmonics, and the dominant frequency for the broadband emission of
Kelvin-Helmholtz vortices ($KH$).

##  3 Nudging-based data-assimilation

Now that the shortcomings of the URANS prediction compared to the DNS have
been characterised, we will introduce a data assimilation technique with the
objective of alleviating them. The proposed nudging approach, which amounts to
incorporating pointwise measurements extracted from the reference DNS into the
URANS simulations, is introduced in §3.1 and is then applied in §3.2-§3.4
considering various scenarios in terms of temporal and spatial resolutions of
measurement data. First, results for a baseline configuration using dense
measurement data for nudging are presented in §3.2. As a further step, the
influence of reduced resolution of measurement data is assessed in §3.3 and
§3.4 with respect to low- and high-frequency phenomena, respectively.

###  3.1 Data assimilation methodology

####  3.1.1 Nudged equations and measurements

Nudging allows a dynamic adjustment of the estimated flow through the addition
of a feedback term in the URANS momentum equations in (2.2), which is
proportional to the discrepancies between flow measurements denoted
$\boldsymbol{m}$ and the estimated velocity field. The intensity of this
feedback term is adjusted through a parameter called $\alpha$. The estimated
velocity field, which is now denoted $\bar{\boldsymbol{u}}_{\alpha}$,
satisfies the so-called nudged equations

$\frac{\partial\bar{\boldsymbol{u}}_{\alpha}}{\partial
t}+(\bar{\boldsymbol{u}}_{\alpha}\cdot\boldsymbol{\nabla})\bar{\boldsymbol{u}}_{\alpha}+\boldsymbol{\nabla}\bar{p}_{\alpha}-\boldsymbol{\nabla}\cdot\left[2\left(Re^{-1}+\nu_{t}(\tilde{\nu}_{\alpha})\right)\boldsymbol{\nabla}_{\mathrm{s}}\bar{\boldsymbol{u}}_{\alpha}\right]=\alpha\,\mathcal{H}^{\dagger}[\mathcal{H}(\bar{\boldsymbol{u}}_{\alpha})-\boldsymbol{m}],$
(9)

which are supplemented by the divergence-free condition in (2.2) and the
Sparlart-Allmaras model (4) to determine the eddy-viscosity field
$\nu_{t}(\tilde{\nu}_{\alpha})$ in the above equation. The feedback term on
the right-hand-side involves the measurement operator $\mathcal{H}$, which
acts on the estimated velocity field $\bar{\boldsymbol{u}}_{\alpha}$ and
returns a vector $\mathcal{H}(\bar{\boldsymbol{u}}_{\alpha})$ in the
measurement space. Further details about the measurement operator and its
adjoint $\mathcal{H}^{\dagger}$ are provided below. Setting $\alpha=0$, one
recovers the standard URANS equations, while for $\alpha>0$, the feedback term
in the momentum equations drives the estimated flow
$\bar{\boldsymbol{u}}_{\alpha}$ towards measurements $\boldsymbol{m}$. The
rationale behind the choice of an appropriate value for $\alpha$ is detailed
in appendix A, where minor sensitivity of results is shown as soon as
$\alpha>1$. The value $\alpha=100$ is thus chosen in the following nudged
URANS simulations.

In the present study, the measurement operator $\mathcal{H}$ is defined so as
to extract the two components of a velocity field at selected spatial
locations, thus mimicking the Particle Image Velocimetry (PIV) approach but
without taking into account any measurement error that may occur in real
experiments. These $M$ ‘nudging points’, which are denoted by
$\boldsymbol{x}_{k}$ with ${k=1\cdots M}$, are sketched with red dots in
figure 1(a). They are chosen to be equidistantly distributed (in both
directions) by a spatial distance $\Delta s$ inside a nudging region
$\Omega_{I}$ which is delimited by the red frame in this figure. Note that the
complementary region is denoted as $\Omega_{E}$. In the following, the nudging
region is fixed to
$\Omega_{I}=\left[-1.5,3.5\right]\times\left[-1.5,1.5\right]$ and we will
investigate the effect of the spatial sampling $\Delta s$.

The measurements $\boldsymbol{m}$ are here extracted from the reference
velocity field $\bar{\boldsymbol{u}}_{r}$ from DNS results (see §2.1). More
precisely, $\boldsymbol{m}$ is defined as

$\displaystyle\boldsymbol{m}(t)=\mathcal{H}(\bar{\boldsymbol{u}}_{r}(t))=\left[\bar{\boldsymbol{u}}_{r}(\boldsymbol{x}_{1},t),\cdots,\bar{\boldsymbol{u}}_{r}(\boldsymbol{x}_{M},t)\right]^{T}=\left[\boldsymbol{m}_{1}(t),\cdots,\boldsymbol{m}_{M}(t)\right]^{T}$
(10)

where $\boldsymbol{m}_{k}$ is a vector that contains the two velocity
components at measurement point $\boldsymbol{x}_{k}$. The size of the data
vector $\boldsymbol{m}$ is therefore $2M$. At an instant $t$, the right-hand
side term in (9) is obtained from the discrepancy between the estimated flow
and this data vector, i.e.
$[\mathcal{H}(\bar{\boldsymbol{u}}_{\alpha})-\boldsymbol{m}(t)]=\left[\bar{\boldsymbol{u}}_{\alpha}(\boldsymbol{x}_{1})-\boldsymbol{m}_{1},\cdots,\bar{\boldsymbol{u}}_{\alpha}(\boldsymbol{x}_{M})-\boldsymbol{m}_{M}\right]^{T}$,
and the adjoint measurement operator $\mathcal{H}^{\dagger}$ then allows to
transform this discrepancy vector of size $2M$ into a field (in the modelled
flow space) of pointwise forcings at the same locations $\boldsymbol{x}_{k}$.

The temporal and spatial methods that are used to discretise the nudging
equation (9) are similar to those described in §2.2 for the standard URANS
simulations. Note that the feedback term is discretized as an implicit term so
as to improve the numerical stability of the algorithm, thus allowing to
choose the time step $\Delta t$ arbitrarily large. For all simulations
considered hereinafter, it is assumed that the measurement vector
$\boldsymbol{m}(t_{k})$ is available at every instant $t_{k}=k\Delta t$ of the
simulation. In other words, we do not make any difference between the time
step $\Delta t$ of the simulation and the time interval between measurements.
In the following, $\Delta t$ is therefore used to characterize the temporal
sampling of a data set that can be written as

$\displaystyle\left[\boldsymbol{m}(t_{1}),\cdots,\boldsymbol{m}(T_{m})\right]$
(11)

where $N_{m}$ is the number of measurement vectors and
$T_{m}=t_{N_{m}}=N_{m}\Delta t$ is the measurement time (or equivalently here
the time window of the simulations). In the next paragraph, it will be
detailed how the spatial sampling $\Delta s$, the temporal sampling $\Delta t$
and the measurement time $T_{m}$ are chosen to define several data sets that
are used to assess the nudging method in its ability to improve the fidelity
of the estimated flow at different spatial and temporal flow scales.

####  3.1.2 Measurement data-sets

Data-set group | $St$ | $\tau$ | $\lambda_{x}$ | $\Delta t$ | $N_{m}$ | $T_{m}=N_{m}\Delta t$ | $\tau/\Delta t$ | $T_{m}/\tau$
---|---|---|---|---|---|---|---|---
VS (§3.3) | $0.137$ | $7.335$ | $4.00$ | $0.209$ | $908$ | $189.85$ | $35$ | $26$
KH (§3.4) | $4.384$ | $0.228$ | $0.25$ | $0.021$ | $1441$ | $30.13$ | $11$ | $132$
VS+KH (§3.2) | $0.137$ | $7.335$ | $4.00$ | $0.021$ | $8421$ | $176.1$ | $349$ | $24$
| $4.384$ | $0.228$ | $0.25$ |  | $11$ | $772$
Table 2:  Description of three groups of measurement data-sets which are named from the targeted flow phenomenon by the nudging approach: the von-Kármán Vortex Street (VS), the Kelvin-Helmholtz vortices (KH) and both phenomena (VS+KH). Their frequency $St$ and period $\tau=1/St$ as captured by DNS are recalled in the second and third columns, respectively. These data-set groups are characterized by the time interval between consecutive measurements $\Delta t$ (fifth column), the number of measurement snapshots $N_{m}$ (sixth column) and the total measurement time $T_{m}$ (seventh column). The last two columns report the number of snapshots per cycle (of the target flow phenomenon) and the total number of cycles. $\Delta s$ | $0.03125$ | $0.0625$ | $0.125$ | $0.25$ | $0.5$ | $0.75$ | $1$ | $1.25$ | $1.5$ | $1.66$ | $2$
---|---|---|---|---|---|---|---|---|---|---|---
Data-set group | (a) | (b) | (c) | (d) | (e) | (f) | (g) | (h) | (i) | (j) | (k)
VS (§3.3) | $128$ | $64$ | $32$ | $16$ | $8$ | $5.33$ | $4$ | $3.2$ | $2.66$ | $2.41$ | $2$
KH (§3.4) | $8$ | $4$ | $2$ | $1$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
VS+KH (§3.2) | $(128;8)$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
Table 3: Test matrix indicating, for the three data-set groups VS, KH and
VS+KH in table 2, the number of measurement points $N_{s}=\lambda_{x}/\Delta
s$ that sample the wavelength $\lambda_{x}$ of the corresponding spatial
structures (indicated in the fourth colmun of table 2) when varying the
spatial sampling $\Delta s$. For instance, the data-set VS(a) corresponds to
$(\lambda_{x}=4.00,\Delta s=0.03125)$, thus $N_{s}=128$.

Three groups of data-sets are defined in table 2 to specifically investigate
the ability of nudging to improve the prediction of low-frequency vortex
shedding [group VS], to reproduce high-frequency Kelvin-Helmholtz vortices
[group KH], and to capture both phenomena [group VS+KH]. The frequency $St$,
period $\tau=1/St$ and typical wavelength $\lambda_{x}$ of the structures that
are associated to each flow phenomenon are first recalled in the first four
columns of this table. The time interval $\Delta t$ between consecutive
measurements and the total time $T_{m}$ of the simulations are then chosen to
properly capture one of the two phenomena (VS or KH) or both of them (VS+KH).
The ratios $\tau/\Delta t$ and $T_{m}/\tau$, which are indicated in the last
two columns, correspond to the number of measurement snapshots per cycle and
to the total number of cycles, respectively. They confirm that the temporal
resolution and time window of the data assimilation experiments are
sufficiently large to capture the physics of interest.

It may be noted that data-sets of group VS+KH combine the temporal resolution
of group KH ($\Delta t=0.021$) and the time window of group VS
($T_{m}=211.5$), which makes them sufficiently well resolved for describing
the high-frequency shedding of Kelvin-Helmholtz vortices ($12$ snapshots per
cycle), while also capturing a significant number ($29$) of the low-frequency
von-Kármán vortex shedding cycles. Data-sets in this group (VS+KH) are
therefore characterized by a large number of measurement snapshots ($N_{m}\sim
10^{4}$), which makes them less convenient for parametric studies. They will
mainly be used in §3.2 to demonstrate the ability of the nudging method to
reconstruct both phenomena based on measurements that are dense in both time
and space.

The influence of the spatial resolution $\Delta s$ on the performances of the
data assimilation procedure is mainly investigated using the data-set groups
VS and KH in §3.3 and §3.4, respectively. Table 3 summarizes the values of
spatial sampling $\Delta s$ that are tested for each data-set group. More
precisely, it indicates the number of measurement points
$N_{s}=\lambda_{x}/\Delta_{s}$ that sample the characteristic wavelength
$\lambda_{x}$ of the corresponding flow structure. The data-sets of group VS
cover a wide range of spatial resolutions, from the highest resolution with
$128$ points per wavelength when $\Delta s=0.03125$, to the lowest spatial
resolution with $4$ points per wavelength for $\Delta s=1$. The spatial
resolution covered by the data-sets of group KH is more restricted, from $8$
points to only $1$ point per wavelength of Kelvin-Helmholtz vortices.

###  3.2 Results with the data-set of highest spatio-temporal resolution
(data-set VS+KH(a))

We first investigate results of the nudging method using the data set
VS+KH(a), which corresponds to the highest temporal and spatial resolutions
investigated here, as detailed in tables 2 and 3. Results from the nudged
URANS simulation are first examined and compared with standard URANS through
the instantaneous vorticity field at $t=50$, similarly as in figure 2 which
reported DNS and standard URANS results. Small-scale structures, which are
missing in the URANS prediction without nudging (figure 7(a)), are clearly
recovered with nudging (figure 7(c)), and the thus estimated flow looks very
similar to the reference one (figure 2(a)), especially in the nudging region.
More downstream ($x>3.5$), we may already note that these small scale
structures are missing, as further discussed in the following.

Figures 7(b) and (d) display the Fourier spectrum of the streamwise velocity
at $(x,y)=(0.25,0.75)$ for the estimated flow without and with nudging,
respectively. As mentioned in §2.4, the high-frequency content ($1<St<10$)
corresponding to Kelvin-Helmholtz fluctuations in the reference flow (grey
curve in figure 7(b)) is not captured by the estimated flow without nudging
(black curve in figure 7(b)). On the other hand, the Fourier spectrum
corresponding to the URANS prediction with nudging (black curve in figure
7(d)) is perfectly superimposed to the reference spectrum, for all
frequencies. In particular, the low-frequency peak corresponding to vortex-
shedding (dashed vertical line) is reached at $St_{VS}=0.137$ with nudging,
instead of $St_{VS}=0.126$ without nudging, which is in prefect agreement with
the reference value.

(a) |  | (b)
---|---|---
|  |
(c) |  | (d)
|  |
Figure 7:  Results for (a-b) standard and (c-d) nudged URANS with the data-set
VS+KH(a). (a,c): Instantaneous spanwise vorticity field $\bar{\omega}_{z}$ at
$t=50$, where black iso-curves denote $\left<\overline{u}\right>=0$. The
extent of the nudging region is delineated by black lines in (c). (b,d):
Fourier spectrum of the streamwise velocity at the green monitor point in
(a,c) (full black line). For the sake of comparison, the DNS spectrum is also
reported (full grey line, well-hidden in (d)). Vertical dashed and dotted
lines indicate the low-frequency peak associated with large-scale vortex-
shedding and the high-frequency bump linked to Kelvin-Helmholtz instabilities,
respectively.

This improvement in the prediction of the fundamental frequency $St_{VS}$ is
associated with a better synchronisation of the estimated flow with the
reference one. This effect, which could have already been identified by
comparing figure 7(c) with 2(a), is clearly visible in figure 8 which displays
three instantaneous snapshots of the reference and nudged flows in figures
8(a-c) and 8(d-f), respectively. The large scale vortices are now perfectly
synchronised, in contrast with standard URANS and figure 3). The small-scale
structures seem also satisfactorily captured in the nudging region, but not
further downstream. This defect, which is also identifiable through the
examination of the spatial error distribution in figures 8(g-i), may be due to
imperfections of the turbulence model, leading to excessive dissipation of the
von-Kármán vortices. Nevertheless, the induced discrepancies remain
significantly smaller than the ones for standard URANS in figure 3(g-i). The
large improvement in the estimated flow through the nudging approach is
further confirmed by considering the temporal evolution of the global error
$E(t)$ (defined in eq. 6) as illustrated in figure 8(j). This error is
strongly reduced for the nudged simulation (black curve) compared with the
standard one (grey curve). In the former case, $E(t)$ decreases during an
initial transient and eventually maintains a fairly constant low level. This
is in contrast with the standard URANS results, for which $E(t)$ significantly
increases and then oscillates due to insufficient synchronisation with respect
to the reference solution, as discussed in §2.4.

(a) | (b) | (c)
---|---|---
|  |
(d) | (e) | (f)
|  |
(g) | (h) | (i)
|  |

(j)
---

Figure 8:  Instantaneous vorticity fields of (a-c) the reference flow $\boldsymbol{\bar{u}}_{r}$ and (d-f) the estimated flow $\boldsymbol{\bar{u}}$ with nudging relying on the data-set VS+KH(a) at times (a,d) $t=15$, (b,e) $t=30$ and (c,f) $t=150$. (g-i) Instantaneous error fields $e(\boldsymbol{x},t)$ between the reference and nudged flows (see eq. 5). (i) Temporal evolution of the global error $E(t)$ (see eq. 6) between the nudged and reference flow (black curve) and between the URANS and reference flows (grey curve). The bottom axis reports the (nondimensional) time while the top axis reports the number of low-frequency cycles determined as $t/\tau_{VS}=t\ St_{VS}$ with $St_{VS}=0.137$. The grey circles indicates time $t=15$, $30$ and $150$ (a) |
---|---
(b) |
Figure 9:  Temporal evolution of (a) external error $E_{E}$ and (b) internal
error $E_{I}$ (see (12)) for standard URANS (grey lines) and nudged URANS with
the data-set VS+KH(a) (black lines).

In order to get a better insight in the extrapolation capabilities of nudging,
the evaluation of $E(t)$ is decomposed according to

$\displaystyle
E(t)^{2}=E_{I}(t)^{2}+E_{E}(t)^{2}=\int_{\Omega_{I}}e(\boldsymbol{x},t)^{2}\,d\boldsymbol{x}+\int_{\Omega_{E}}e(\boldsymbol{x},t)^{2}\,d\boldsymbol{x}.$
(12)

This decomposition allows to distinguish between an internal error $E_{I}(t)$
defined over the nudging domain $\Omega_{I}$ (see figure 1), which is directly
influenced by the measurements, and an external error $E_{E}(t)$ defined over
the complementary domain $\Omega_{E}=\Omega\setminus\Omega_{I}$. Both error
components $E_{E}$ and $E_{I}$ are shown as functions of time in figures 9 (a)
and (b), respectively.

Initially, the internal and external errors in the estimated flow are of the
same order of magnitude. The internal error $E_{I}$ for nudged URANS (black
curve) achieves a large reduction by three orders of magnitude in about $5$
convective times, which indicates an excellent agreement with the reference
flow in $\Omega_{I}$. More time is necessary to decrease the external error
$E_{E}$ as the corrected flow in the nudging region has to be convected in the
complementary region $\Omega_{E}$. The external error remains however
significantly larger than the internal one and thus dominates the total error
$E$. The reduction in the external error compared to standard URANS results
(grey curve) is still noticeable (by one order of magnitude), which confirms
the significant benefits of nudging even outside of the nudging region
$\Omega_{I}$.

So far, the potential of nudging the URANS equations for unsteady data
assimilation using a measurement data-set with high spatio-temporal resolution
has been demonstrated. In the following, we will study the performance of the
nudging procedure with respect to the spatial and temporal resolutions of the
measurement data as characterised by $\Delta s$ and $\Delta t$, respectively.

###  3.3 Nudging the low-frequency large-scale structures (data-set group VS)

We now analyse the performance of the nudging approach to specifically improve
the accuracy of low-frequency phenomena ($St<1$) using data-sets of lower
resolution. More specifically, we now consider data-set group VS in table 2
which is characterized by the temporal sampling $\Delta t=0.209$, this value
being $10$ times larger than for the data-set group VS+KH used in §3.2.
Moreover, for most of the data-sets in group VS, the spatial sampling $\Delta
s$ is also decreased compared to previous results. As summarised in table 3,
the investigated values of $\Delta s$ amount to $2\leq N_{s}\leq 128$, where
$N_{s}=\lambda_{x}/\Delta s$ is the number of nudging points per
characteristic wavelength $\lambda_{x}\sim 4$ of the large-scale vortices.

(a) |  | (b)
---|---|---
|  |
(c) |  | (d)
|  |
(e) |  | (f)
|  |
(g) |  | (h)
|  |
Figure 10:  Results for nudged URANS simulations in data-set group VS with
spatial spacings (a-b) $\Delta s=1.66$, (c-d) $\Delta s=1.5$, (e-f) $\Delta
s=0.5$, and (g-h) $\Delta s=0.125$. (a,c,e,g): Instantaneous spanwise
vorticity field $\bar{\omega}_{z}$ at $t=50$, where black iso-curves denote
$\left<\overline{u}\right>=0$, while dots refer to nudging points. (b,d,f,h):
Fourier spectrum of the streamwise velocity at the green monitor point in
(a,c,e,g) (full black line), the reference spectrum to recover is also
reported (full grey curve). Vertical dashed lines indicate the low-frequency
peak associated with large-scale vortex-shedding.

Figure 10 shows the vorticity fields at $t=50$ and the Fourier spectra of the
streamwise velocity at point $(x,y)=(0.25,0.75)$, similarly as in figures 2
and 7, which are obtained for nudged URANS simulations with decreasing values
of the spatial sampling $\Delta s$. In the Fourier spectra, results
corresponding to nudged URANS are displayed with black curves, while the grey
curves correspond to results of the reference flow when sampled at $\Delta
t=0.209$, i.e. the temporal sampling of the present data-set. This spectrum is
therefore different from the reference spectrum in figures 2 and 7 that
corresponds to a temporal sampling ten times smaller, ($\Delta t=0.021$).
Their comparison clearly shows the filtering effect when down-sampling the
reference flow.

Let us first examine results obtained with the largest spatial samplings and
displayed in figure 10(a-b) for $\Delta s=1.66$ and in figure 10(c-d) for
$\Delta s=1.5$. For $\Delta s=1.66$, the nudged simulation is not synchronized
with the measurement data. The largest amplitude peak is obtained for a
frequency that is slightly smaller than in the reference simulation (see
figure 10(b)) and is the same as in the standard URANS results
($St_{VS}=0.126$ versus $St_{VS}=0.137$ in the reference simulation). Note
that, in the displayed vorticity field, we clearly observe undesired small-
scale wake-like structures around some nudging points. By decreasing the
spatial sampling to $\Delta s=1.5$, the vortex-shedding frequency of the
nudged simulation already locks to the reference frequency (figure 10(d)).
Furthermore, this appears to be true not only for the fundamental frequency
$St_{VS}$, but also for its second and third harmonics. For the present set-
up, the smallest number of measurements points that is necessary to lock the
low-frequencies that are associated to large-scale vortex shedding is thus
around only $N_{s}\sim 2.5$ points per characteristic wavelength. However,
despite this frequency locking, it may be noticed that the estimated flow is
still not perfectly in phase with the reference one for $\Delta s=1.5$, as
further discussed in the following. Moreover, the largest amplitude peak is
under-estimated, as well as the power spectral densities at higher
frequencies. A better flow estimation is obtained by further decreasing the
spatial sampling to $\Delta s=0.5$ (figure 10(e-f)). The large-scale vortices
are now well synchronized with the reference flow (figure 2(a)) and the
dominant peak in the spectrum is almost perfectly recovered. Moreover, the
spurious small-scale structures around nudging points that were observed for
larger spatial samplings are not visible anymore. Decreasing the spacing
between nudging points to $\Delta s=0.125$ adds more small-scale features in
the nudged URANS simulation, as illustrated by figure 10(g), which gets even
closer to the reference snapshot of figure 2(a). In addition, the
corresponding spectrum in figure 10(h) agrees perfectly well with the down-
sampled reference one.

(a)
---
(b)
Figure 11:  (a) Dominant Strouhal number and (b) phase angle with respect to
reference results as a function of the spacing between nudging points $\Delta
s$, or equivalently the number $N_{s}$ of nudging points per characteristic
wavelength, for nudged URANS simulations of data-set group VS. Results are
obtained from the Fourier analysis of the signal at the same monitor point as
in figure 10. Filled symbols correspond to the case where the estimated
shedding frequency is $St_{VS}=0.137$, as in DNS results, while open symbols
correspond to $St_{VS}=0.126$, as in standard URANS.

To better assess the frequency locking and phasing effects when varying the
number of nudging points, we display in figures 11(a) and (b) the frequency of
the largest amplitude peak $St_{VS}$ and the phase shift $\phi$, respectively,
as a function of $\Delta s$ (and $N_{s}$). As noticed above, the fundamental
frequency of the nudged simulations is equal to that in standard URANS results
($St_{VS}=0.126$) for low spatial resolution ($N_{s}<2.5$) and locks to the
frequency of the reference flow ($St_{VS}=0.137$) for higher spatial spatial
resolution ($N_{s}>2.5$). This frequency locking does not guarantee a perfect
flow synchronization. Indeed, for $2.5\leq Ns\leq 5.75$, we observe a
significant phase shift between the nudged and reference temporal evolutions
of the streamwise velocity at the monitor point in figure 10, which vanishes
only for higher spatial resolution ($N_{s}\geq 8$).

(a)
---
(b)
Figure 12: Time-averaged errors as a function of the spacing between nudging
points $\Delta s$, or equivalently the number $N_{s}$ of nudging points per
characteristic wavelength, for nudged URANS simulations of data-set group VS.
(a) Total error $\langle E\rangle$ in (6), which may be decomposed as the sum
of (b) internal $\langle{E_{I}}\rangle$ (black symbols) and external errors
$\langle{E_{E}}\rangle$ (grey symbols). The reported values are normalised by
the total time-averaged error for the standard URANS solution. Filled symbols
correspond to the case where the estimated shedding frequency is
$St_{VS}=0.137$, as in DNS results, while open symbols correspond to
$St_{VS}=0.126$, as in standard URANS.

The effect of the spatial resolution of measurement points on the time-
averaged global error $\langle{E}\rangle$ in (7) is now examined through
figure 12(a). In this figure, $\langle{E}\rangle$ is normalised by its value
for standard URANS results. For the smallest number of nudging points
$N_{s}=2\,(\Delta s=2)$, the normalized error is slightly smaller than one,
indicating a very weak improvement compared to the standard simulation. When
increasing to $N_{s}=2.66$, i.e the smallest number of nudging points leading
to a frequency locking, the normalized error decreases to $0.3$, which
corresponds to an improvement of $70\%$ compared to the standard URANS
results. Interestingly, for $2.66\leq N_{s}\leq 5.75$, the evolution of the
error follows that of the phase shift shown in figure 11(b). In this range of
spatial resolution, the decrease of the error is thus strongly related to the
decrease of the phase shift noticed before. Finally, for $N_{s}\geq 8$, the
time-averaged error reaches a constant value that is close to $0.05$ times the
standard URANS error.

To assess the extrapolation capability of the nudged simulations,
$\langle{E}\rangle$ is decomposed into internal and external components,
$\langle{E}_{I}\rangle$ and $\langle{E}_{E}\rangle$, corresponding to the
error inside and outside the nudging region, respectively, similarly as in
§3.2. The time-averaged internal and external errors, normalised by the total
error of the standard simulation, are displayed as a function of $N_{s}$ in
figure 12(b) using black and grey dots, respectively. We clearly note that the
external error is the dominant component, whatever the number of nudging
points, as for the data-set VS+KH(a) discussed in §3.2. For a small number of
nudging points ($N_{s}<5.75$), these two errors are of similar order of
magnitude, and they decrease in a similar way when increasing the number of
nudging points. This is related to the improved flow synchronisation
(frequency locking and phase-shift decrease) which has been previously noticed
for that range of spatial resolution and occurs everywhere in the flow. Once
the number of nudging points is large enough to perfectly synchronise the flow
($N_{s}>5.75$), the internal error strongly decreases unlike the external
error that reaches a constant value around $5\%$ of the total error obtained
by standard URANS simulation. This constant value is at the origin of the
plateau reached by the total error in figure 12(a). This irreducible error may
be attributed to (turbulence) model errors which limit the extrapolation
capability of the nudging approach. Its interpolation capability, i.e. the
ability to estimate the flow in between nudging points, is nevertheless very
good as the internal error decreases to $10^{-4}$.

(a)
---
(b)
Figure 13: (a) Frequency and (b) kinetic energy of the dominant SPOD modes
corresponding to vortex-shedding as a function of the spatial sampling $\Delta
s$ for nudged URANS simulations of data-set group VS. The dashed and dash-
dotted horizontal lines in (b) correspond to energy of the SPOD mode for the
URANS and reference data set, respectively. The spatial structure of few modes
are displayed in figure 14. Filled symbols correspond to the case where the
estimated shedding frequency is $St_{VS}=0.137$, as in DNS results, while open
symbols correspond to $St_{VS}=0.126$, as in standard URANS.

Having carefully characterised in time the flow estimations for the data-set
group VS, we now analyse these results in the frequency space through SPOD, as
performed in previous sections. For the present data-set, SPOD is computed
based on overlapping bins that cover about $63$ time units corresponding to
approximately $9$ vortex-shedding cycles. The frequency $St_{VS}$ and
eigenvalue $\lambda_{VS}$ (kinetic energy) corresponding to the largest
amplitude peak (see figure 5(b)) are shown in figure 13(a) and (b)
respectively, as a function of the spacing between measurements $\Delta s$.
For $\Delta s>1.5$, the frequency of the nudged mode is equal to that of the
URANS mode ($St_{VS}=0.126$), while for $\Delta s\leq 1.5$, it locks to the
frequency of the reference mode ($St_{VS}=0.137$), in agreement with previous
observations deduced from a pointwise temporal signal (figure 11). Figure
13(a) thus confirms the global aspect of frequency locking or $\Delta s\leq
1.5$. Examining now the estimated kinetic energy $\lambda_{VS}$ of dominant
SPOD mode at $St_{VS}$ in figure 13(b), it is almost equal to that of the
reference mode (dash-dotted line) once the frequency is locked to the
reference frequency. Using $N_{s}=2$ points per wavelength ($\Delta s=2$ in
the figure), the kinetic energy is still close to that of the URANS mode
(dashed line). But a slight increase to $N_{s}=2.66$ points ($\Delta s=1.5$)
yields an almost perfect agreement with the reference value. Further
increasing the number of nudging point (decreasing their spacing $\Delta s$)
does not significantly improve the estimation of $\lambda_{VS}$. The spatial
structure of the corresponding mode is shown in figures 14(a,c,e,g) for
several spatial spacings $\Delta s$. In addition, the corresponding spectral
error fields $\Phi_{e}$ in (8) are reported in figures 14(b,d,f,h). For the
SPOD mode associated to the unlocked frequency (figure 14(a-b)), the largest
errors are obtained close to some of the nudging points downstream of the
cylinder. They are associated to errors in the cross-stream velocity (not
displayed here) since the streamwise velocity is zero on the symmetry axis
$y=0$. These specifics defects vanish once the frequency is locked to the
reference frequency (figures 14(c-d)). Largest errors are now obtained in
between nudging points, in a (red) pocket immediately downstream of the
cylinder ($x<1.5$) and in two symmetric (yellow) pockets around $x=2$. Both
the shape and the intensity of the errors are actually similar to those for
standard URANS (figure 5(d)). Increasing the number of nudging points to
$\Delta s=0.5$ (figures 14(e-f)) allows for decreasing these errors in the
near wake. One may directly identify for the mode in figure 14(e) a change in
the structures around $2\leq x\leq 4$ compared to previous cases, these
structures being now closer to the ones in the reference mode (figure 5(a)).
However, modal errors are significantly reduced only for spacings $\Delta
s\leq 0.25$, as for the case $\Delta s=0.125$, which is illustrated in figures
14(g-h). The residual errors in the far wake ($x>8$) are barely modified when
further increasing the number of nudging points. Accurately recovering the
detailed shape of the dominant SPOD mode associated to vortex shedding thus
seems to require more spatially dense measurements ($N_{s}\geq 16$ nudging
points per characteristic wavelength) compared to the estimation of its
frequency and kinetic energy/magnitude ($N_{s}\sim 2.5$). However, it may be
emphasised that the estimation of the shape of such a mode, which is already
acceptable with standard URANS (see figure 5), may be considered as only
secondary to the correct prediction of associated frequency and kinetic
energy. This is supported by the low values for the time-averaged global
errors $\left\langle E\right\rangle$ that are already reached for $N_{s}\geq
2.66$, i.e. after frequency locking and phasing are achieved, and also with a
correct estimation of $\lambda_{VS}$, as discussed previously.

(a) | (b)
---|---
|
(c) | (d)
|
(e) | (f)
|
(g) | (h)
|
Figure 14:  SPOD analysis of the vortex-shedding phenomenon for nudged URANS
simulations in data-set group VS with spatial spacings (a-b) $\Delta s=1.66$,
(c-d) $\Delta s=1.5$, (e-f) $\Delta s=0.5$, and (g-h) $\Delta s=0.125$.
(a,c,e,g): Streamwise velocity contours of the imaginary part of the dominant
SPOD mode for the fundamental frequency $St_{VS}$. While the frequency of the
mode in (a) is $St_{VS}=0.126$, remaining modes oscillate at $St=0.137$. Modes
are phased so that each of them reaches its maximum value at the green monitor
point. (b,d,f,h): Corresponding modal discrepancy field $\Phi_{e}$ in (8).

Finally in this section, the quality of the mean flow as estimated by the
nudged URANS simulations for data-set group VS is discussed. The streamwise
mean velocity field is depicted in figures 15(a,c,e,g) for several values of
$\Delta s$ and can be compared to the reference and URANS mean flows displayed
in figures 4(a) and (b), respectively. The corresponding error fields
$e_{\left\langle\bar{\boldsymbol{u}}\right\rangle}$ in (7) are reported in
figures 15(b,d,f,h). For the lowest spatial resolution ($\Delta s=1.66$),
significant improvement is already observed in the far wake outside of the
nudging region, which is remarkable as the fundamental frequency $St_{VS}$ is
not locked to the reference one in this case. Once it is locked for $\Delta
s=1.5$ (figures 15(c-d)), the error increases in the near wake, similarly as
the spectral error of the SPOD mode in figure 14(d) for the same spacing. This
mean-flow error can be decreased by increasing the number of nudging points.
Significant improvements may already be identified for $\Delta s=0.5$ (figures
15(e-f)). For the largest spatial resolution ($\Delta s=0.125$) displayed in
figure 15(g-h), the mean-flow error drops to below $3\%$ in the near and far
wake. Only the error in the shear layers cannot be further decreased when
increasing the number of nudging points.

(a) | (b)
---|---
|
(c) | (d)
|
(e) | (f)
|
(g) | (h)
|
Figure 15:  Mean-flow results for nudged URANS simulations in data-set group
VS with spatial spacing (a,b) $\Delta s=1.66$, (c,d) $\Delta s=1.5$, (e,f)
$\Delta s=0.5$ and (g,h) $\Delta s=0.125$. (a,c,e,g): Streamwise component of
the mean flow. Black curves correspond to the isocontours of zero streamwise
velocity. (b,d,f,h): Mean-flow error field
$e_{\left\langle\bar{\boldsymbol{u}}\right\rangle}$ in (7). The extent of the
nudging region is delineated by black lines.

A more quantitative description of the mean-flow error is provided in figure
16 which displays the spatially-integrated mean-flow error
$\int_{\Omega}e_{\langle\bar{\boldsymbol{u}}\rangle}(\boldsymbol{x})dS$, where
$\Omega$ is the domain of integration. Figure 16(a) illustrated this global
error, divided its the value for standard URANS results, as a function of
$\Delta s$ for two domains having the same streamwise extent ($-1.5\leq x\leq
12$) but different cross-stream ones. Squares correspond to the mean-flow
error when integrated in a domain of size $-4\leq y\leq 4$, which is identical
to the $y$-extension used for the figures. In this case, the global mean-flow
error seems only slightly decreased $(by\sim 30\%)$ even for the smallest
value of $\Delta s$. However, when restricting the domain $\Omega$ to the
extension of the nudging region in the cross-stream direction ($-1.5\leq y\leq
1.5$), the global mean-flow error (circles) now exhibits a significant
decrease for $\Delta s\leq 1$, while it remains fairly constant for $\Delta
s>1$. This evolution is better understood by examining the internal (black
circles) and external (grey circles) components of this error, which are
displayed in figure 16(b). For $\Delta s>1$, the decrease of the external
error is actually balanced by the increase of the internal error. These two
opposite effects were already visible in figure 15(a-d). For $\Delta s\leq 1$,
both internal and external mean-flow errors decrease almost monotonically. The
external contribution reaches a constant value for $\Delta s\leq 0.25$ while
the internal one still decreases, which is reminiscent of the evolution of the
time-averaged spatially-integrated errors in figure 12. To conclude, the
nudging approach using the low-frequency data-set also allows for an
improvement in the mean-flow prediction, not only inside but also outside of
the nudging region. This extrapolation capability is mainly significant for a
domain of limited extension in the cross-stream direction, which seems related
to the extension of the nudging region and appears to be directly downstream
of the latter, according to the present results.

(a)
---
(b)
Figure 16:  (a) Spatially-integrated mean-flow error
$\int_{\Omega}e_{\langle\bar{\boldsymbol{u}}\rangle}(\boldsymbol{x})dS$ of the
nudged URANS simulations of data-set group VS as a function of the spatial
sampling $\Delta s$. This integrated error is evaluated over a spatial domain
$\Omega$ that extends in the streamwise direction between $-1.5\leq x\leq
12.5$ and is restricted in the cross-stream direction to (squares) $-4\leq
y\leq 4$ or (circles) $-1.5\leq y\leq 1.5$. The reported values are normalised
by the corresponding error of the standard URANS solution. (b) The integrated
error in the restricted domain $-1.5\leq y\leq 1.5$ is decomposed as the sum
of internal (black) and external (grey) errors to the nudging region.

###  3.4 Nudging the high-frequency small-scale structures (data-set group KH)

As detailed in §2.4, DNS results capture the shedding of Kelvin-Helmholtz (KH)
vortices in the shear layers of the cylinder. As illustrated by figure 6, a
broadband peak is observed around frequency $St=4.384$ in the spectrum
obtained by the SPOD analysis, and the dominant SPOD mode is characterized by
spatially oscillating structures of wavelength $\lambda_{x}\sim 0.25$ located
in the upper shear layer. This high-frequency small-scale structures are in
general not captured by standard URANS simulations (Iaccarino et al., 2003;
Palkin et al., 2016) and could also not be obtained in the present study,
whatever the mesh and temporal resolutions for the URANS calculations.
Therefore, we now investigate the potential of the nudging approach to recover
this unsteady phenomenon by using the data-sets of group KH (see tables 2 and
3). While data-sets in this group generally correspond to higher temporal and
spatial samplings compared to previously discussed data-sets (group VS in
§3.3), it may be noticed that these resolutions are actually moderate compared
to the time and spatial scales that are associated to the KH vortices. The
time sampling $\Delta t$ indeed corresponds to around $12$ snapshots per KH
cycle, while the investigated values of $\Delta s$ amount to a number of
nudging points per wavelength of the KH vortices verifying $1\leq N_{s}\leq
8$. Results are here directly analysed based on a SPOD analysis that is
adapted to KH phenomena as performed in §2.4.

(a) | (b)
---|---
|
(c) | (d)
|
(e) | (f)
|
(g) | (h)
|
Figure 17:  SPOD analysis of Kelvin-Helmholtz phenomena as estimated by nudged
URANS simulations for the data-set group KH with (a,b) $N_{s}=1$ ($\Delta
s=0.25$), (c,d) $N_{s}=2$ ($\Delta s=0.125$), (e,f) $N_{s}=4$ ($\Delta
s=0.0625$) and (g,h) $N_{s}=8$ ($\Delta s=0.03125$) nudging points per
wavelength of the KH instabilities. (a,c,e,g) Real part of the streamwise
component of the SPOD mode for $St_{KH}=4.384$ and (b,d,f,h) SPOD spectrum for
nudged URANS (black), the DNS results (grey) are also reported for the sake of
comparison.

The real part of of the streamwise component of the dominant SPOD mode at
frequency $St_{KH}=4.384$ obtained with nudging is shown in the left-hand-side
plots of figure 17 for the different values of the spatial sampling $\Delta s$
in data-set group KH. SPOD spectra are also displayed in the right-hand-side
plots.

The SPOD mode in figure 17(a), already captures fluctuations that are
reminiscent of Kelvin-Helmholtz structures visible in the reference SPOD mode
in figure 6(a). This is remarkable, as the associated spatial sampling $\Delta
s=0.25$ corresponds to $N_{s}=1$, i.e. one nudging point per wavelength of
coherent vortex structures. However, the spectrum in figure 17(b) indicates
significantly under-estimated amplitudes of all dominant modes around
$St_{KH}=4.384$. In particular, the amplitude $\lambda_{KH}$ of the mode in
figure 17(a) corresponds to only $\sim 4\%$ of the reference value. The
nudging approach seems to excite modes that are associated with KH
instabilities, but their energy amplification is not large enough to reach
sufficient amplitude, probably because of the eddy-viscosity (obtained here
with the Spalart-Allmaras model) that dissipates too quickly the energy at
that frequency. Increasing the spatial resolution of nudging points leads to
increasingly pronounced spectral peaks at $St_{KH}=4.384$, as more energy is
introduced to promote KH structures. This can be confirmed in figures 17(c,d)
for $\Delta s=0.125\,(N_{s}=2)$. Eventually, the spatial pattern obtained for
smaller spatial samplings (see figures 17(e) and (g)) is very similar to the
reference SPOD mode (figure 17(a)). The good performance of nudging is also
evident in the SPOD spectra in figures 17(f) and (h). $N_{s}=8$ nudging points
per KH wavelength (figures 17(g-h)) thus appear sufficient to satisfactorily
predict the kinetic energy of the modes around $St_{KH}=4.384$. In particular,
the estimated amplitude $\lambda_{KH}$ corresponds to $\sim 80\%$ of the
reference value.

We have shown that applying nudging with sufficiently well-resolved
measurement data allows to reproduce KH vortices, which are no captured by the
standard URANS equations. It may be emphasised that nudging improves URANS
simulations in a ’global’ sense and is not simply ’over-ruling’ local values
in shear layers. This is particularly evident for the case with large spacing
between nudging points (see figure 17(a)), where local maxima and minima in
SPOD modes associated with KH instabilities do not coincide with nudging
positions.

###  3.5 Modelling and experimental considerations

As discussed in §3.3 and §3.4, nudging here appears to be able to
significantly improve URANS results even when considering moderately dense
data with respect to the phenomenon of interest. For both large-scale vortex
shedding and Kelvin-Helmholtz instabilities, with a time interval between
measurements that is below one tenth of the considered characteristic period,
less than $N_{s}=3$ nudging points per characteristic wavelength is already
enough to correct the frequency content of URANS. The kinetic energy that is
associated to the (SPOD) modes that oscillate at the frequencies of interest
is also satisfactorily recovered with $N_{s}=3$ and $N_{s}=8$ for large-scale
vortex shedding and Kelvin-Helmholtz phenomena, respectively. The detailed
shape of these modes may be further adjusted when increasing the measurement
spatial density. If one is only interested in mean quantities, a spatial
sampling of $\Delta s=0.25$, or equivalently $N_{s}=4$ if the cylinder size is
here considered as the characteristic length, is sufficient to significantly
improve the estimation of the mean flow compared to standard URANS for the
present flow configuration.

It may be worth emphasising that the present results have been obtained in
conjunction with a specific RANS turbulence model, namely the Spalart-Allmaras
one. Through the consideration of other models, one could possibly further
decrease the requirements in terms of temporal and spatial samplings of the
measurements, in particular concerning the estimation of the Kelvin-Helmholtz
instabilities. Indeed, the study of Palkin et al. (2016) confirmed for a
similar flow configuration as the present one, namely the flow past a circular
cylinder at $Re=1.4\times 10^{5}$, that URANS calculations based on Reynolds
stress models may include such high-frequency phenomena, contrary to eddy-
viscosity models, thanks to a lower effective viscosity, among others. It
could thus be imagined in this case that less nudging points could be required
in order to amplify and satisfactorily sustain Kelvin-Helmholtz phenomena. In
other words, it could be expected that the correct estimation of the kinetic
energy that is associated to these fluctuations requires less than eight
points per characteristic wavelength when relying on a Reynolds stress model.

Aside from the choice of the turbulence model in the URANS framework, more
generally, it seems worth investigating the use of nudging to enhance the
estimation of turbulent flows in conjunction with other modelling approaches
such as hybdrid and multiscale techniques (Sagaut et al., 2013; Chaouat,
2017). Compared to URANS, methods such as Partially Averaged Navier–Stokes
(PANS) (Girimaji, 2006) or Partially Integrated Transport Modeling (PITM)
(Chaouat & Schiestel, 2005), among others, may enable to relax the assumption
of scale separation between coherent/mean and turbulent phenomena. In
addition, such approaches generally provide clearer and explicit definitions
of the resolved and modeled scales, and thus may disambiguate which aspects,
i.e. scales, in reference/experimental data the estimated flow is supposed to
recover.

In the perspective of using the current nudging approach in an actual
experimental context, it is interesting to note that the above mentioned
temporal and spatial resolutions may be considered as quite realistic in an
experimental point of view, i.e. if one were to consider performing a PIV
experiment of the present flow. Similar to the current nudging procedure, PIV
measurements are also typically characterised by regular spatial sampling of
measurements within a rectangular-shaped measurement domain. Current state-of-
the-art in high frame rate PIV includes cameras with a typical sensor size of
roughly $2000$ pixels. Using standard seeding conditions and current
processing algorithms, an interrogation window would consist of 16 pixels with
75% overlap (Raffel et al., 2018). The final vector spacing would be $\Delta
s_{PIV}=0.03$, which roughly corresponds to the lowest spatial sampling
investigated here. Considering a water-flow experiment and a cylinder size of
e.g. $20\ mm$ (leading to a flow velocity of roughly $1.1\ m/s$), the
frequency of shear-layer instabilities would be roughly equal to $230\ Hz$.
The present requirement of acquisition to be performed at a roughly ten times
higher frequency (thus $2.3\ kHz$) is thus also reachable, as a high frame
rate PIV system can nowadays operate with full laser energy and camera
resolution at frequencies of $2-3\ kHz$ minimum.

Note that this remains of course a rapid and preliminary verification. Turning
to an actual experiment would raise a number of additional challenges (local
spatial averaging due to the PIV interrogation windows, measurement noise,
etc.), which will be addressed in future steps. However, it is already quite
promising that the amount of data necessary for the nudging method to yield
excellent results is at first order compatible with current experimental
capabilities.

##  4 Conclusion

Data assimilation for the prediction of the turbulent flow past a square
cylinder has been performed through nudging and in conjunction with URANS
modelling. Through a careful examination of mean and dynamical flow features
in both physical and spectral spaces, this study has confirmed the potential
of nudging in the state estimation of a complex flow that exhibits a wide
range of spatio-temporal scales. Despite its conceptual simplicity, nudging
was indeed able to significantly enhance the URANS estimation based on sparse
velocity observations from DNS. With very few measurements per characteristic
wavelength, nudging allowed to improve the characterisation of large-scale
vortex shedding. Even more interestingly, nudging was able to overcome the
significant modelling limitations of URANS in the prediction of high-frequency
Kelvin-Helmholtz instabilities. The latter were successfully recovered through
nudging, while they were absent from the present baseline URANS results, based
on a still moderate number of measurements per characteristic wavelength. SPOD
and mean-flow analyses confirmed that nudging allowed a global improvement in
the flow prediction, and in particular outside of the region that is directly
driven by the measurements. As nudging virtually induces no supplementary
computational cost, and that URANS may be considered as a relatively
affordable simulation method, the use of nudging in conjunction with URANS
thus appears as a particularly cost-efficient data assimilation approach for
the high-fidelity unsteady characterisation of complex turbulent flows.

Future studies could be dedicated to the consideration of other high-Reynolds
number turbulent flows. In particular, it could be of interest to investigate
configurations which do not necessarily exhibit low-frequency quasi-periodic
phenomena in addition to broadband fluctuations at significantly higher
frequencies, as in the present case, and which could therefore be more
challenging for URANS. This could motivate the consideration of more advanced
modelling approaches in conjunction with nudging. Perhaps more importantly,
future work could deal with the consideration of actual experimental data. As
discussed in this study, this may be encouraged by the fact that the
identified requirements of nudging in terms of spatio-temporal resolution of
the measurements seem compatible with the possibilities that are offered by
optical techniques.

## Acknowledgements

This work was partly supported by the HOMER project from the European Union’s
Horizon 2020 research and innovation program under grant agreement No. 769237.

## Declaration of interests

The authors report no conflict of interests.

##  Appendix A Influence of nudging parameter $\alpha$

$\alpha$ | $0$ | $10^{-5}$ | $10^{-4}$ | $0.001$ | $0.01$ | $0.1$ | $1$ | $1000$
---|---|---|---|---|---|---|---|---
$\left\langle E\right\rangle$ | $30$ | $29.6$ | $26$ | $4.8$ | $1.86$ | $1.58$ | $1.52$ | $1.52$
$\left\langle E_{I}\right\rangle$ | $7.8$ | $7.8$ | $7.0$ | $1.24$ | $0.090$ | $0.032$ | $0.026$ | $0.024$
$\left\langle E_{E}\right\rangle$ | $22$ | $21.6$ | $19.0$ | $3.4$ | $1.78$ | $1.54$ | $1.5$ | $1.48$
$\left\langle||f_{x}||^{2}\right\rangle$ | 1 | $1.00$ | $0.94$ | $0.20$ | $9.1\cdot 10^{-3}$ | $5.1\cdot 10^{-4}$ | $9.9\cdot 10^{-6}$ | $1.1\cdot 10^{-11}$
$\left\langle||f_{y}||^{2}\right\rangle$ | 1 | $1.00$ | $0.94$ | $0.15$ | $5.9\cdot 10^{-3}$ | $4.1\cdot 10^{-4}$ | $8.7\cdot 10^{-6}$ | $1.0\cdot 10^{-11}$
$\alpha^{2}\left\langle||f_{x}||^{2}\right\rangle$ | $0$ | $1.7\cdot 10^{-8}$ | $1.6\cdot 10^{-6}$ | $3.5\cdot 10^{-5}$ | $1.5\cdot 10^{-4}$ | $8.6\cdot 10^{-4}$ | $1.7\cdot 10^{-3}$ | $1.9\cdot 10^{-3}$
$\alpha^{2}\left\langle||f_{y}||^{2}\right\rangle$ | $0$ | $2.9\cdot 10^{-8}$ | $2.7\cdot 10^{-6}$ | $4.3\cdot 10^{-5}$ | $1.7\cdot 10^{-4}$ | $1.2\cdot 10^{-3}$ | $2.6\cdot 10^{-3}$ | $2.9\cdot 10^{-3}$
Table 4: Sensitivity of time-averaged total, internal, external errors
(respectively $\langle{E}\rangle$, $\langle{E}_{I}\rangle$, and
$\langle{E}_{E}\rangle$), and forcing magnitudes ($\langle||f_{x}||\rangle$
and $\langle||f_{y}||\rangle$) to the nudging coefficient $\alpha$ for the
data-set of group VS with $\Delta s=0.125$ (see tables 2-3). In the fourth and
fifth rows, the values of $\langle||f_{x}||\rangle$ and
$\langle||f_{y}||\rangle$ are normalized by their values for standard URANS.

In this appendix, the sensitivity of the reconstruction results with respect
to the parameter $\alpha$ in the nudged URANS equations (9) is investigated.
Time-averaged global errors $\langle{E}\rangle$, $\langle{E_{I}}\rangle$ and
$\langle{E_{E}}\rangle$ (see (6) and (12)) are summarized in rows 2 to 4 of
table 4 for values of the nudging coefficient that range from $\alpha=0$
(corresponding to standard URANS) to $\alpha=1000$. The considered measurement
data-set belongs to group VS with $\Delta s=0.125$. Fifth and sixth rows in
table 4 report the time average of the norm of the forcing vector

$\boldsymbol{f}=\mathcal{H}(\bar{\boldsymbol{u}}_{\alpha})-\boldsymbol{m},$
(13)

which thus allows to quantify the discrepancies between estimated and
reference solution at the measurement locations, distinguishing between the
streamwise and crossflow components.

Table 4 confirms that the magnitude of the components of $\boldsymbol{f}$
(fifth and sixth rows) is reduced by more than three orders of magnitude for
$\alpha\geq 0.1$ compared to standard URANS and, as expected, converges
towards zero as $\alpha$ is further increased. On the other hand, the
effective amplitude of the forcing that is introduced in the nudged URANS
equations (9), which may be evaluated as
$\alpha\langle||\boldsymbol{f}||\rangle$, converges towards constant values,
as the errors with respect to the reference data seem to decrease
proportionally to increasing values of the nudging coefficient (seventh and
eighth rows in table 4). As this effective amplitude but also the global
errors in the second to fourth rows of table 4 show minor sensitivity with
respect to $\alpha$ for $\alpha\geq 1$, it was decided to maintain
$\alpha=100$ for all runs in the present contribution.

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