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

LES-INFORMED RESOLVENT-BASED ESTIMATION OF TURBULENTPIPE FLOW

Filipe R. Amaral and André V. G. Cavalieri
Department of Aerodynamics
Instituto Tecnológico de Aeronáutica
12228-900, São José dos Campos, SP, Brazil
filipefra@ita.br

ABSTRACT(2020) introduced a resolvent-based estimator for flow statistics, which was further generalized by Martini et al. (2020) for
time-domain estimates. For the latter case, in order to build thetransfer functions between the low-rank measurements and theflow state components, it is necessary to evaluate the resolvent
operator and feed the algorithm with the cross-spectral density (CSD) of nonlinear terms, treated as forcing. If the true
forcing CSD is used, optimal estimates of time-varying flow
quantities are obtained. Such estimates are not causal, as thefull time series of sensor data is required for estimation; extension to causal estimation, using only past sensor information,is proposed by (Martini et al., 2022). Amaral et al. (2021) successfully applied the methodology by Martini et al. (2020) to
direct numerical simulation (DNS) of turbulent channel flow,using wall shear-stress and pressure low-rank measurements.
Results show a close agreement between estimates and reference DNS fluctuations in the near wall region, and good agreement for large scale structures throughout the channel. A key
feature is the use of forcing statistics extracted from the DNS
database, which leads to an optimal estimator but requires expensive simulation and post-processing to obtain the forcing
CSD.
In the present paper we employ the aforementioned
resolvent-based methodology to estimate the space–time velocity fluctuation components of turbulent pipe flow at friction
Reynolds number Reτ ≈ 550 using wall shear-stress measurements. In addition to DNS, we also explore the capability ofwall-resolved large-eddy simulations (LES) to construct estimators. A first DNS database is used as the reference casefrom which we extract low-rank measurements of wall-shear
stresses. A second DNS and the other LES databases provide
the forcing (nonlinear) statistics to build the linear estimators.
Here we investigate the capability of cheaper LES databaseson the reconstruction of the space–time flow field, aiming toobtain a reliable and low-cost estimator that could be used forvarious high-Reynolds-number flows of practical interest.

A resolvent-based methodology is employed to obtain
non-causal spatio–temporal estimates of turbulent pipe flowfrom low-rank probe measurements of wall shear-stress fluctuations. DNS and LES pipe flow numerical simulations atfriction Reynolds number of 550 are used as databases. We
consider one of the DNS databases as the true spatio–temporal
flow field, from which the low-rank measurements are extracted. Such database is also employed to verify the accuracy of the linear estimators. The estimator needs a modelfor the nonlinear (or forcing) terms of the Navier-Stokes equations system, which are obtained from a DNS database andfrom a series of computationally cheaper LES databases withgrids coarser than the DNS. Comparisons between the reference DNS and the estimates indicate that sufficiently accurate
results can be achieved with cheaper LES containing up to
10% of the number of grid points of the DNS, with estimatesclosely matching the reference DNS results up to the bufferlayer and reasonable agreement up to the beginning of the log
layer.

INTRODUCTION
The estimation of space–time flow fluctuations fromnoisy, low-rank measurements is an interesting option for theunderstanding of the turbulence physics and for the designof flow control strategies. For wall-bounded turbulent flows,
wall quantities such as shear stress and/or pressure are usually employed as inputs for the estimation algorithms, as forpractical applications the measurement of such quantities iseasier to obtain than, e.g. the velocity components at a given
distance from the wall. Model-based methodologies can beused to build the estimator (Bewley & Protas, 2004; Hœpffner
et al., 2005; Chevalier et al., 2006; Colburn et al., 2011; Illingworth et al., 2018), although it is also possible to perform flow
estimations based solely on data (Encinar & Jiménez, 2019;
Sasaki et al., 2019; Guastoni et al., 2021). For both modeland data-driven methodologies the basic idea is to obtain relations between the measurements and the estimated flow state,
but the model-based methodologies have the additional advantage of providing insight on the underpinning physics.
Linearized models can be obtained using the resolvent
framework. In this case, the Navier-Stokes system is written
in the state–space form and the nonlinear terms are interpreted
as external forcing terms (McKeon & Sharma, 2010; Hwang& Cossu, 2010; Beneddine et al., 2016; Taira et al., 2017),
hence providing an input–output formulation. Towne et al.

METHODS
Resolvent-based estimator
We begin writing the linearized Navier-Stokes (LNS)
equations in the discretized state-space form in time domain,
i.e.

M

1

dqq(t)= A q (t) + B f ,dty (t) = C q (t) + n (t).

(1a)
(1b)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
In this equation, q = [uux u r u θ p ]T , where u x , u r and u θ indicate streamwise, radial and azimuthal velocity fluctuations,
respectively, and p indicates the pressure fluctuation. Moreover, t denotes time, A is the linearized Navier-Stokes operator, B is the actuation matrix that restricts the forcing terms toappear only in the momentum equation, y is the system observation (measurements), C is the observation matrix that selects
Ns sensor readings from the state vector (in the present paper,wall-shear stresses in the axial and azimuthal directions), andn is the measurement noise. M is a diagonal matrix whose
entries are set to one and zero for the momentum and continuity equations, respectively. Dependency on longitudinal andazimuthal wavenumbers, α and m, respectively, as well as onwall-normal variable y, were dropped to simplify notations.
M − A ) is written in
For pipe flow, the LNS operator L = (−iωMcylindrical coordinates (Luhar et al., 2014) and linearization isaround the mean turbulent profile, considered as known.
Equation 1 can be written in the frequency domain as
C (−iωM
M − A)−1 B] f̂f (ω) + n̂n(ω),
ŷy(ω) = [CPeriodic boundary conditions were assumed in the streamwise
and azimuthal directions. For all simulations the pipe lengthis Lz = 10R, where R is the pipe radius. Table 1 shows theparameters for all cases, including the number of streamwise(Nz ) and azimuthal wavenumbers (Nθ ), the number of meshpoints in the radial direction (Nr ), the mesh discretization in
the streamwise (∆z+ ), azimuthal ((R∆θ )+ ) and radial (∆y+ ) directions and the mesh points ratio with respect to the DNS case
(N/NDNS ). Plus symbols denote inner (wall and/or viscous)
units. All simulations contain 2981 snapshots and the time
steps based on outer units is ∆t = 0.2 for all simulations. Cases
starting with D denote DNS, whereas letter L indicate LES,
carried out using (Smagorinsky, 1963) subgrid scale model,
with a Smagorinsky constant set as Cs = 0.05. As estimations
lose accuracy for large wavenumbers, only the lowest 16 and
32 streamwise and azimuthal wavenumbers were used to construct the estimators. Welch’s method (Welch, 1967) was employed to evaluate the forcing and state components statistics,
with blocks containing N f f t = 256 time steps and 75% overlap. A Hann window was applied to each block to minimize
spectral leakage.
The simulations were validated with reference DNS results by El Khoury et al. (2013) (not shown here). Cases D1,
D2 and L1 show good agreement with the reference simulations, regarding mean flow profile, axial, azimuthal and radial velocity fluctuations. The coarser grid cases (L3, L6 andL8) progressively deteriorate the agreement, with the L8 caseshowing strong mismatch with all quantities.
Wall shear-stress measurements in the axial and azimuthal
directions were extracted from the D1 database, which is taken
as ground truth. The other databases, i.e. D2, L1, L3, L6 andL8, were employed to construct the estimator transfer funcP f f ). It is important
tions by extracting the forcing statistics (Pto remark that the estimators employed in this study have no
information from the D1 case. The LES databases, which havecoarser grids, significantly reduce the estimates computationalcost, since the number of grid points to run the simulations are
small, but in turn lead to suboptimal estimators. In this paper we employ P f f obtained in simulations that have different
grids than the measurements database (D1) and after the evaluation of the transfer functions results are interpolated to a gridequivalent to that of the measurement database.

(2)

where ω denotes frequency and hats are used for Fouriertransformed quantities. Following (Martini et al., 2020), it isT q ) bepossible to obtain the optimal linear transfer function (T̂
tween the system observation (ŷy) and the estimated flow statecomponents (q̃qˆ ), i.e.
T q ŷy,q̃qˆ = T̂(3)

T q is the transfer function and dependency on frequency
where T̂
ω was dropped to simplify notations. Martini et al. (2020)
T q that is based on the minimization
derived a expression for T̂
of the error between the true ( f̂f ) and estimated ( ˆf̃f ) forcing
terms and is given by−1

H P f f H ∗ + P nn )T q = R B P f f H ∗ (H
T̂

,

(4)

M − A )−1 is the resolvent operator, H = C R B
where R = (−iωMC ) and acis the resolvent operator including the observation (C
∗
∗
B
P
n
n
P
f
f
⟨
⟩
tuation (B ) matrices, nn = n̂ n̂ and f f = ⟨ f̂ f̂ ⟩ are CSDs
of sensor noise and forcing, respectively. The asterisk (∗) indicates a Hermitian transpose. More details on the estimator
derivation can be found in Martini et al. (2020).
To build the transfer function (Eq. 4), it is necessary to
P f f ). When true forcing
specify a priori the forcing CSD (Pstatistics are known, Eq. 4 provides the optimal linear estimator. Other models for the forcing CSD provide sub-optimal
estimators.
The snapshots are reconstructed according to the procedures described in Amaral et al. (2021) and are briefly addressed below. First, it is necessary to take the inverse Fourier
T q , Eq. 4, in order to retransform of the transfer function T̂turn to time domain and obtain T q . Hence, the time domaintransfer function T q must be convolved with the measurements/observations y (t) to evaluate the state estimate in time
domain q̃q. Finally, double inverse Fourier transforms in the azimuthal and longitudinal directions are taken in order to returnfrom wavenumber domain to physical space.

RESULTS
Figures 1, 2 and 3 show sample snapshots of the streamwise velocity fluctuations from the D1 database, filtered to retain only the lower axial and azimuthal wavenumbers, and corresponding estimates obtained using D2, L1, L3, L6 and L8
forcing statistics. Results are shown at radial distances fromthe pipe wall of y+ = (1 − r+ ) ≈ 15 and 100 and 200. Plus superscripts denote inner (viscous) scaling units, r is the radial
coordinate and y is the distance from the pipe wall. In the figures we use a pseudo-spanwise coordinate z = rθ (λz = rλθ )for enabling comparisons with structures found in planar wallbounded flows.
When considering the buffer layer, at a wall-normal distance of y+ ≈ 15, the resemblance between DNS results andthe estimates is remarkable, even when the L8 forcing is employed. This is somehow expected, since (Amaral et al., 2021)
showed that assuming the forcing statistics as spatial white
noise to build the estimator also provides accurate estimatesfor distances very close to the wall. Moving further fromthe wall, the estimates are not as accurate, especially for thecoarser LES (L8) estimator, although most of the large-scale

Numerical simulations
To generate the databases, we employed numerical simulations conducted with the Openpipeflow code (Willis, 2017).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 1: DNS and LES numerical simulations parameters.

Case

Reτ

Nr

Nz

Nθ

∆z+

(R∆θ )+∆y+

N/NDNS

D1

550.3

128

528

528

10.4

6.5

0.07-6.3

1.000

D2

550.3

128

528

528

10.4

6.5

0.07-6.3

1.000

L1

548.5

96

384

384

14.2

8.9

0.4-14.7

0.397

L3

568.6

96

192

192

28.6

18.0

0.4-15.2

0.099

L6

551.8

64

64

64

86.1

54.1

0.7-22.9

0.007

L8

509.6

32

32

32

159.3

100.1

0.7-22.9

0.002

Figure 1: Comparison between streamwise velocity component instantaneous snapshot of filtered DNS (D1) and resolvent-based estimates using wall measurements of shear stress and considering the D2, L1, L3, L6, and L8) forcing statistics at
y+ ≈ 15. Fluctuations shown in outer units.

Figure 2: Comparison between streamwise velocity component instantaneous snapshot of filtered DNS and resolvent-based estimates at y+ ≈ 100. See comments in the caption of figure 1.

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

Figure 3: Comparison between streamwise velocity component instantaneous snapshot of filtered DNS and resolvent-based estimates at y+ ≈ 200. See comments in the caption of figure 1.
CONCLUSIONS

structures present in the DNS snapshots are still recognizable
in all but the L8 estimator. Regarding the radial and azimuthal
velocity components, not shown here, similar results as thoseof the streamwise velocity component were observed.

In this paper we employed resolvent-based estimators to
obtain the space–time flow field of turbulent pipe flow fromwall-shear stress measurements. We compared the performance of the estimator when modelling the forcing terms with
DNS and LES databases. Satisfactory results were obtainedwith the forcing statistics from LES, especially up to the buffer
layer. The accuracy level of a LES estimator containing approximately 10% of the grid points of the DNS database isvery close to what is obtained with the DNS estimator. LESinformed resolvent-based estimation is thus a viable approach
for accurate estimates of turbulent flow at high Re.

Figure 4 displays normalized correlations (Corr, left+
frame), r.m.s errors (Err, middle frame) and variance (⟨q′ q′ ⟩ ,right frame) for the streamwise velocity component. Such metrics are defined in Amaral et al. (2021) and, for brevity, we
refer to this paper for further details on the metrics definitions.
Accurate estimates correspond to low normalised error Err,close to 0, and high correlation Corr, close to 1.
All estimators are accurate up to y+ ≈ 10, showing correlation and r.m.s. errors of approximately 0.95 and 0.35, respectively. Moving farther from the wall, only estimators D2, L1
and L3 remain accurate, especially regarding the correlation
and r.m.s. metrics. It is interesting that estimator L3, which
has a grid with less than 10% of the points used for the DNSbased estimator, could attain such accuracy. This indicates thatthe large scales of interested are well calculated in the LES, as
expected, and their statistics may be used to build an accurate
estimator at a fraction of the computational cost of the DNSbased estimator considered in Amaral et al. (2021). L6 and L8
estimators, on the other hand, provided results that are comparable to cases including eddy-viscosity model on the linear
operator and white-noise forcing statistics modelling, strategies that our group previously employed to obtain estimatesof channel flow after low-tank measurements (Amaral et al.,2021). Overall, the quantitative metrics in figure 4 confirm thequalitative results shown in figures 1-3. Similar results, notshown here, were obtained for the azimuthal and radial velocity components.

Acknowledgements
F.R.A. received funding from the São Paulo Research
Foundation (FAPESP/Brazil), grant #2019/02203-2. A.V.G.C.
was supported by the National Council for Scientific and Technological Development (CNPq/Brazil), grant #313225/20206. F.R.A. and A.V.G.C. were also funded by FAPESP/Brazil,
grant #2019/27655-3.

REFERENCES
Amaral, Filipe Ramos, Cavalieri, André Valdetaro Gomes,
Martini, Eduardo, Jordan, Peter & Towne, Aaron 2021
Resolvent-based estimation of turbulent channel flow using
wall measurements. Journal of Fluid Mechanics 927, A17.
Beneddine, Samir, Sipp, Denis, Arnault, Anthony, Dandois,
Julien & Lesshafft, Lutz 2016 Conditions for validity ofmean flow stability analysis. Journal of Fluid Mechanics
798, 485–504.
Bewley, Thomas R. & Protas, Bartosz 2004 Skin friction andpressure: the “footprints” of turbulence. Physica D: Nonlinear Phenomena 196 (1-2), 28–44.
Chevalier, Mattias, Hœpffner, Jérôme, Bewley, Thomas R. &Henningson, Dan S. 2006 State estimation in wall-boundedflow systems. Part 2. Turbulent flows. Journal of Fluid Mechanics 552 (1), 167.
Colburn, C. H., Cessna, J. B. & Bewley, T. R. 2011 State estimation in wall-bounded flow systems. Part 3. The ensemble
Kalman filter. Journal of Fluid Mechanics 682, 289–303.
El Khoury, George K., Schlatter, Philipp, Noorani, Azad, Fischer, Paul F., Brethouwer, Geert & Johansson, Arne V.
2013 Direct Numerical Simulation of Turbulent Pipe Flow

Normalized r.m.s. error as a function of the wavenumbers
for the studied estimators are shown in figure 5 for wall distances (planes) of y+ ≈ 15 and 100. The large structures, which
are characterised by small α and m, are accurately estimatedfor the D2, L1 and L3 cases, with virtually zero r.m.s. error atboth planes. The estimates for smaller structures (large α andm), on the other hand, display higher r.m.s. error, especiallyfor the y+ ≈ 100 plane. For the coarser L6 and L8 estimators,
even for the y+ ≈ 15 plane, the accuracy of smaller structure
estimates is quite low.

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

Figure 4: Flow state comparison metrics for the streamwise velocity fluctuation component. Dash-dotted grey curve in theleft frame denote DNS (D1) results. Frames, from left to right: correlation, normalized r.m.s. and variance. The DNS
variance refers solely to wavenumbers retained for estimation.

Figure 5: Normalized r.m.s. error as a function of (α,m) state comparison metrics for D2, L1, L3, L6 and L8 estimators.

at Moderately High Reynolds Numbers. Flow, Turbulenceand Combustion 91 (3), 475–495.
Encinar, Miguel P. & Jiménez, Javier 2019 Logarithmic-layer
turbulence: A view from the wall. Physical Review Fluids
4 (11), 114603.
Guastoni, Luca, Güemes, Alejandro, Ianiro, Andrea, Discetti, Stefano, Schlatter, Philipp, Azizpour, Hossein & Vinuesa, Ricardo 2021 Convolutional-network models to predict wall-bounded turbulence from wall quantities. Journalof Fluid Mechanics 928, A27.
Hœpffner, Jérôme, Chevalier, Mattias, Bewley, Thomas R. &Hennington, Dan S. 2005 State estimation in wall-boundedflow systems. Part 1. Perturbed laminar flows. Journal of
Fluid Mechanics 534, 263–294.
Hwang, Yongyun & Cossu, Carlo 2010 Linear non-normal energy amplification of harmonic and stochastic forcing in the
turbulent channel flow. Journal of Fluid Mechanics 664,
51–73.
Illingworth, Simon J., Monty, Jason P. & Marusic, Ivan 2018
Estimating large-scale structures in wall turbulence using
linear models. Journal of Fluid Mechanics 842, 146–162.
Luhar, Mitul, Sharma, Ati S. & McKeon, Beverley J. 2014
Opposition control within the resolvent analysis framework.
Journal of Fluid Mechanics 749, 597–626.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022with the primitive equations: I. The basic experiment.
Monthly Weather Review 91 (3), 99–164.
Taira, Kunihiko, Brunton, Steven L., Dawson, Scott T. M.,
Rowley, Clarence W., Colonius, Tim, McKeon, Beverley J.,
Schmidt, Oliver T., Gordeyev, Stanislav, Theofilis, Vassilios& Ukeiley, Lawrence S. 2017 Modal analysis of fluid flows:
An overview. AIAA Journal 55 (12), 4013–4041.
Towne, Aaron, Lozando-Durán, Adrián & Yang, Xiang 2020
Resolvent-based estimation of space–time flow statistics.
Journal of Fluid Mechanics 883, A17.
Welch, Peter D. 1967 The use of Fast Fourier Transform forthe estimation of power spectra: A method based on time
averaging over short, modified periodograms. IEEE Transactions on Audio and Electroacoustics 15 (2), 70–73.
Willis, Ashley P. 2017 The Openpipeflow Navier–Stokes
solver. SoftwareX 6, 124–127.

Martini, Eduardo, Jordan, Peter, Cavalieri, André V. G.,
Towne, Aaron & Lesshafft, Lutz 2020 Resolvent-based optimal estimation of transitional and turbulent flows. Journalof Fluid Mechanics 900, A2.
Martini, Eduardo, Jung, Junoh, Cavalieri, André V. G., Jordan,Peter & Towne, Aaron 2022 Resolvent-based tools for optimal estimation and control via the Wiener-Hopf formalism.
Journal of Fluid Mechanics (accepted) 937, A19.
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. Journal of Fluid Mechanics
658, 336–382.
Sasaki, Kenzo, Vinuesa, Ricardo, Cavalieri, André V. G.,
Schlatter, Philipp & Henningson, Dan S. 2019 Transfer
functions for flow predictions in wall-bounded turbulence.
Journal of Fluid Mechanics 864, 708–745.
Smagorinsky, Joseph 1963 General circulation experiments

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