# A resolvent-based prediction framework for incompressible turbulent channel
flow with limited measurements

Anjia Ying1  Tian Liang1  Zhigang Li1  Lin Fu1,2,3,4<EMAIL_ADDRESS>1
Department of Mechanical and Aerospace Engineering, The Hong Kong University
of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2 Department of
Mathematics, The Hong Kong University of Science and Technology, Clear Water
Bay, Kowloon, Hong Kong 3 HKUST Shenzhen-Hong Kong Collaborative Innovation
Research Institute, Futian, Shenzhen, PR China 4 Center for Ocean Research in
Hong Kong and Macau (CORE), The Hong Kong University of Science and
Technology, Clear Water Bay, Kowloon, Hong Kong

###### Abstract

A new resolvent-based method is developed to predict the space-time properties
of the flow field. To overcome the deterioration of the prediction accuracy
with the increasing distance between the measurements and predictions in the
Resolvent-Based Estimation (RBE), the newly proposed method utilizes the RBE
to estimate the relative energy distribution near the wall rather than the
absolute energy directly estimated from the measurements. Using this extra
information from RBE, the new method modifies the energy distribution of the
spatially uniform and uncorrelated forcing that drives the flow system by
minimizing the norm of the cross-spectral density (CSD) tensor of the error
matrix in the near-wall region in comparison with the RBE-estimated one, and
therefore it is named as the Resolvent-informed White-noise-based Estimation
(RWE) method. For validation, three time-resolved direct numerical simulation
(DNS) datasets with the friction Reynolds numbers $Re_{\tau}=180$, 550, and
950 are generated, with various locations of measurements ranging from the
near-wall region ($y^{+}=40$) to the upper bound of the logarithmic region
($y/h\approx 0.2$) for the predictions. Besides the RWE, three existing
methods, i.e., the RBE, the $\lambda$-model, and the White-noise-Based
Estimation (WBE), are also included for the validation. The performance of the
RBE and $\lambda$-model in predicting the energy spectra shows a strong
dependence on the measurement locations. The newly proposed RWE shows a low
sensitivity on $Re_{\tau}$ and the measurement locations, which may range from
the near-wall region to the upper bound of the logarithmic region, and has a
high accuracy in predicting the energy spectra. The RWE also performs well in
predicting the space-time properties in terms of the correlation magnitude and
the convection velocity. We further utilize the new method to reconstruct the
instantaneous large-scale structures with measurements from the logarithmic
region. Both the RWE and RBE perform well in estimating the instantaneous
large-scale structure, and the RWE has smaller errors in the estimations near
the wall. The structural inclination angles (SIAs) around $15^{\circ}$ are
predicted by the RWE and WBE, which generally recover the DNS results.

###### keywords:

##  1 Introduction

Credible predictions of turbulent flows have long been an essential concern
for numerical and experimental studies. However, in many cases, only part of
the flow information could be obtained due to the inaccuracy of the numerical
models or the limitations of the experimental measurement techniques. For
instance, in the numerical studies, the popular wall-modelled large-eddy
simulation (WMLES) resolves only the large-scale flow motions beyond the local
grid-scale and models the effect of the near-wall small-scale motions with
RANS-like methods to balance the accuracy and cost for turbulence simulation,
see, e.g., Larsson et al. (2016) and Fu et al. (2021, 2022). However, the
velocity fluctuation in the near-wall region, which is an important turbulence
property, is missing or inaccurate in WMLES (Bae et al., 2018). As for the
experimental studies, the measuring points are usually sparsely placed in
space due to technical limitations, and important flow information might be
lost. To derive the missing flow information in the unresolved or unmeasured
regions, many researchers attempt to estimate the turbulence statistics from
limited sets of available flow data, the methodologies of which could be
generally categorized into the data-driven approaches (Townsend, 1976; Marusic
et al., 2010; Baars et al., 2016; Guastoni et al., 2021) and the physics-based
approaches (McKeon & Sharma, 2010; Hwang & Cossu, 2010).

The data-driven approaches predict the flow statistics based on the
fundamental research on the statistical properties of wall-bounded turbulence,
where the widely recognized attached eddy model (AEM) (Townsend, 1976) and
inner-outer interaction model (IOIM) (Marusic et al., 2010; Baars et al.,
2016) are proposed and extensively validated. The AEM considers the turbulence
properties in the logarithmic region to be characterized by a collection of
self-similar energy-containing eddies whose roots are attached to the wall.
From the basic concept of attached eddies, many kinds of statistics of
turbulence could be derived, such as the logarithmic profile of the variance
of the streamwise velocity fluctuations. From AEM, the logarithmic laws could
be further extended to describe the wall-normal distributions of higher-order
even moments (Meneveau & Marusic, 2013; de Silva et al., 2015; Yang et al.,
2016). On the other hand, the IOIM states that the near-wall turbulence is
influenced by the large-scale motions (LSM) and very-large-scale motions
(VLSM) via the superposition and modulation effects, which also paves the way
to predict the near-wall turbulence using the velocity data in the logarithmic
region. Recently, the consistency between the AEM and the IOIM has been
demonstrated by Cheng & Fu (2022), which enables isolating the attached eddies
at a given single scale and further predicting their separated superposition
effect in the near-wall region. With the isolated attached eddies at a given
length scale, the turbulence properties can be further clarified (Cheng et
al., 2022; Cheng & Fu, 2023). However, in practice, the IOIM needs the given
data at the near-wall region to calculate the transfer kernel for predicting
the footprint of the attached eddies in the near-wall region, which could not
be directly applied for the predictions when the second-order flow statistics
are not available. Meanwhile, the AEM only considers the attached eddies that
are dominant in the logarithmic region, which is not the case in the inner and
buffer layers where the smaller detached eddies are significant. Besides the
above models describing the turbulence, the convolutional neural network (CNN)
is also used for the data-driven prediction of the turbulence properties
(Güemes et al., 2021; Guastoni et al., 2021). For instance, using the model
parameters trained from the existing datasets, Guastoni et al. (2021)
reconstruct the flow field with the shear stress measurements at the wall.

In addition to the data-driven approaches, the physics-based ones predict the
turbulence field based on the Navier-Stokes equations, which govern the flow
dynamics. In general, the physics-based approaches rearrange the Navier-Stokes
equations to the form of the linearized relationship between the nonlinear
forcing (input) and the response of velocity, pressure, and temperature
(output) (McKeon & Sharma, 2010; Hwang & Cossu, 2010). Specifically, when the
linearized relationship is defined in the frequency domain, the operator that
builds the linear relationship is named the resolvent operator (McKeon &
Sharma, 2010). Taking Fourier transformation to the linearized Navier-Stokes
equations in all the uniform spatial and temporal directions, the linearized
relationship between the nonlinear forcing and the response at each scale is
extracted, where the nonlinear forcing involves the convolutions from all the
other scales (McKeon, 2017). So far, the relationship between the forcing and
the response is equivalent to the original form of the Navier-Stokes equations
without any assumptions. The response could be fully recovered as long as the
nonlinear forcing is completely known (Morra et al., 2021). However, the
completed knowledge of the nonlinear forcing is unavailable as long as the
flow data are not totally known. Despite this point, important turbulent
properties can be extracted from the resolvent operator itself when the mean
velocity profile is known. For instance, the forcing and response modes
ordered by their gains could be obtained after taking singular value
decomposition to the resolvent operator. When the gain of the leading mode
dominates those of the sequential modes, which is referred to as the low-rank
behaviour (Pickering et al., 2021), it can effectively construct a low-
dimensional description of the turbulence (Sharma & McKeon, 2013; Moarref et
al., 2013; McKeon, 2017). Assuming the low-rank behaviour, Beneddine et al.
(2016) predict the streamwise velocity spectra by fitting the amplitude of the
leading response mode, which minimizes the square of the error between the
predicted profile and the measurements at different positions. However, the
energy of the leading resolvent mode is not always predominant over the
following modes (Morra et al., 2021), which means that only taking the leading
mode inevitably sacrifices much information.

Rather than considering only the leading mode, another group of physics-based
approaches implicitly take advantage of the low-rank behaviour by treating the
unknown nonlinear forcing as white in space (Hwang & Cossu, 2010; Morra et
al., 2019; Madhusudanan et al., 2019). Since the white-noise assumptions imply
that the energies of nonlinear forcing modes are equal to each other among all
the modes (Towne et al., 2018), the energies of the response modes are
proportional to the gains accordingly. By modeling a part of the forcing with
the eddy viscosity terms (Cess, 1958; Reynolds & Hussain, 1972) in the
linearized Navier-Stokes equations and assuming that the remaining forcing is
white noise, the accuracy of prediction on the fluctuation statistics is much
improved (Hwang & Cossu, 2010; Morra et al., 2019). Later, Madhusudanan et al.
(2019) estimate the superposition effect of the large-scale structures on the
near wall region using the eddy-viscosity model with the remaining forcing
assumed as white noise. Gupta et al. (2021) further improve the work of
Madhusudanan et al. (2019) by modifying the forcing profile according to the
wall-normal distributions of the eddy viscosity and flow scales.

Besides the above approaches that assume a predefined forcing profile, the
Kalman filter-based approaches (Hœpffner et al., 2005; Chevalier et al., 2006)
and resolvent-based approaches (Towne et al., 2020; Martini et al., 2020) are
also proposed. Illingworth et al. (2018) estimate the large-scale structures
in wall turbulence with the H2-optimal approaches. Illingworth et al. (2018)
also demonstrate significant improvement in the predictions when the eddy-
viscosity term is introduced in the linearized Navier-Stokes equations. Towne
et al. (2020) predict the two-point space-time statistics in the near-wall
region of turbulent channel flow by estimating the minimum forcing that could
fully reproduce the measurements, namely the RBE. Since the approach only
estimates the minimum forcing that recovers the observations, it
underestimates the forcing that generates large responses at other positions
but has minor effects on the measurements (Karban et al., 2022). In terms of
practical applications, the measurements are not guaranteed to be located in
the vicinity of the prediction region, which means that the direct
applications of RBE are limited. Later, Martini et al. (2020) generalizes the
RBE approach by taking the forcing color and sensor noise into account when
constructing the transfer function for estimation. An optimal linear estimator
of the flow states can be obtained by taking the real forcing CSD tensors as
input. In practice, the forcing statistics can be estimated from additional
sensors or approximated from reasonable models. The improved RBE (Martini et
al., 2020) recovers the original one (Towne et al., 2020) when the forcing is
assumed to be white in space and the sensor noise is neglected. Amaral et al.
(2021) predict the turbulent channel flow with wall shear stress and pressure
using the improved RBE, showing that the flow information can be better
predicted when the actual forcing spectra are already known. In practice, a
method that can avoid rather predefined or already-known forcing statistics is
indeed needed to fill the gap between theory and real applications.

With the current physics-based methods, the predictions based on the white-
noise assumptions (Morra et al., 2019; Madhusudanan et al., 2019; Gupta et
al., 2021) could represent the general properties of the turbulence, but the
accuracy is limited. On the other hand, without known forcing statistics, the
RBE (Towne et al., 2020) performs well when the prediction layer is located
near the measurement layer but becomes invalid when the prediction location
moves far away from the measurement location. In cases where the measurements
are not close to the prediction region, both the methods based on white-noise
assumptions and the RBE are not expected to provide accurate predictions on
the flow information. However, these two kinds of methods could compensate for
each other in a sense. The new method proposed in this study builds the
skeleton of forcing based on the white noise to maintain the predicted
response energy even when the measurement is far away from the prediction,
while refines the forcing profile with near-wall relative energy profile from
the RBE results. Through the above procedure, the advantages of these two
kinds of methods are combined in the new approach.

The remainder of this article is organized as follows. In section 2, existing
prediction methods are reviewed and discussed. In section 3, the prediction
method is derived and illustrated. In section 4, the newly proposed method is
validated by comparing the prediction with the results from the DNS data and
existing prediction methods. Discussions and concluding remarks are presented
in section 5.

##  2 The existing methods

In this section, the mathematical description of the resolvent analysis is
introduced, followed by a brief review of the existing methods to predict the
flow field, including the WBE (Morra et al., 2019), the Wall-distance-
dependent Model (W-model) and the Scale-dependent Model ($\lambda$-model)
(Gupta et al., 2021), as well as the RBE (Towne et al., 2020; Martini et al.,
2020). Basic ideas of the to-be-reviewed methods will form the cornerstone of
our newly proposed method derived in section 3.

###  2.1 Mathematical description of the resolvent analysis

The incompressible Navier-Stokes equations are given by &ut+u⋅u =
-p+1Reτ⋅(u+uT),
⋅u = 0, where $Re_{\tau}=\frac{u_{\tau}h}{\nu}$ is the friction Reynolds
number, $u_{\tau}$ is the friction velocity, $h$ is the half-channel height,
$\nu$ is the kinematic viscosity, and the superscript T denotes transpose.
Following the previous studies (Illingworth et al., 2018; Morra et al., 2019;
Towne et al., 2020), the forcing $\boldsymbol{f}$ that contains the nonlinear
interactions of velocity fluctuations while excluding the eddy-viscosity term
is defined as

$\boldsymbol{f}=-\boldsymbol{u}^{\prime}\cdot\bnabla\boldsymbol{u}^{\prime}-\frac{1}{Re_{\tau}}\bnabla\cdot[\frac{\nu_{t}}{\nu}(\bnabla\boldsymbol{u}^{\prime}+{\bnabla\boldsymbol{u}^{\prime}}^{\rm
T})],$ (1)

where the superscript ′ denotes the fluctuation variable. Here, the eddy
viscosity $\nu_{t}$ is calculated from the semi-empirical expression by Cess
(1958) and reported by Reynolds & Hussain (1972) as

$\displaystyle\nu_{t}=\frac{\nu}{2}\left\\{1+\frac{\kappa^{2}Re_{\tau}^{2}}{9}(2y-y^{2})^{2}(3-4y+2y^{2})^{2}[1-{\rm
exp}(\frac{-Re_{\tau}y}{A})]^{2}\right\\}^{\frac{1}{2}}-\frac{\nu}{2},$ (2)

where the constants $\kappa=0.426$ and $A=25.4$. By rearranging equation
(2.1), the linearized Navier-Stokes equations hold,
&u′t+¯u⋅u′+u′⋅¯u+p′-1Reτ⋅[νTν(u′+u′T)] = f,
⋅u′= 0, where $\overline{\boldsymbol{u}}$ is the mean velocity, and
$\nu_{T}=\nu_{t}+\nu$ is the total viscosity. Note that equations (2) are
equivalent to the incompressble Navier-Stokes equations (2.1).

From equations (1) and (2), the inclusion of the eddy viscosity model into the
linearized Navier-Stokes equations changes both the definition of forcing and
the linearized relationship between the forcing and response. In McKeon &
Sharma (2010), the total forcing is defined as
$\boldsymbol{f}=-\boldsymbol{u}^{\prime}\cdot\bnabla\boldsymbol{u}^{\prime}$.
The forcing re-defined in this study is equivalent to the remaining portion of
the forcing in McKeon & Sharma (2010) after excluding the eddy-viscosity term
$\frac{1}{Re_{\tau}}\bnabla\cdot[\frac{\nu_{T}}{\nu}(\bnabla\boldsymbol{u}^{\prime}+{\bnabla\boldsymbol{u}^{\prime}}^{\rm
T})]$. In the following sections, the tested prediction methods actually
provide different approaches to model the re-defined forcing in (1). The total
forcing, on the other hand, is equal to the summation of the portion that is
modeled by the eddy viscosity terms and the remaining portion that is modeled
by the prediction methods.

The linearized Navier-Stokes equations in each spatial scale
$\boldsymbol{k}_{\rm s}$ are obtained by taking the Fourier transformation to
equations (2) in the uniform spatial directions. For instance, in the fully
developed turbulent channel flow, the Fourier transformation is taken in the
streamwise $(x)$ and spanwise $(z)$ directions. The linearized equations (2)
at $\boldsymbol{k}_{\rm s}=\left(k_{x},k_{z}\right)$ can be therefore written
in a discretized state-space form with $N$ points in the wall-normal direction
as &M qks(t)t = A_k_s q_k_s (t) + Bf_k_s (t),
m_k_s (t) = Cq_k_s (t) + n_k_s (t), where $\boldsymbol{q}_{\boldsymbol{k}_{\rm
s}}(t)=\left[\boldsymbol{u}^{\rm T}_{\boldsymbol{k}_{\rm
s}}(t),p_{\boldsymbol{k}_{\rm s}}(t)\right]^{\rm T}$,
$\boldsymbol{m}_{\boldsymbol{k}_{\rm s}}(t)\in\mathbb{C}^{N_{\rm m}}$ is the
system observation, $\boldsymbol{n}_{\boldsymbol{k}_{\rm
s}}(t)\in\mathbb{C}^{N_{\rm m}}$ is the measurement noise, and $N_{\rm m}$ is
the number of observations. The expressions of the operators $\mathcal{M}$,
$\mathcal{A}_{\boldsymbol{k}_{\rm s}}$, $\mathcal{B}$, and $\mathcal{C}$ are
provided in Appendix A. Further, taking the Fourier transformation to equation
(2.1$a$) in the temporal direction, the following linear relationship at each
spatio-temporal scale $\boldsymbol{k}=\left(\boldsymbol{k}_{\rm
s},\omega\right)$ holds,

$\hat{\boldsymbol{q}}_{\boldsymbol{k}}=\mathcal{H}_{\boldsymbol{k}}\cdot\hat{\boldsymbol{f}}_{\boldsymbol{k}}\\\
,$ (3)

where $\hat{\boldsymbol{u}}_{\boldsymbol{k}}$, $\hat{p}_{\boldsymbol{k}}$ and
$\hat{\boldsymbol{f}}_{\boldsymbol{k}}$ are the Fourier coefficients of
velocity, pressure and forcing at scale $\boldsymbol{k}$, respectively, and
the resolvent operator $\mathcal{H}_{\boldsymbol{k}}$ is expressed as,

$\mathcal{H}_{\boldsymbol{k}}=\left(-{\rm
i}\omega\mathcal{M}-\mathcal{A}_{\boldsymbol{k}_{\rm s}}\right)^{-1},$ (4)

where ${\rm i}=\sqrt{-1}$. From equation (3), the velocity
$\hat{\boldsymbol{u}}_{\boldsymbol{k}}$ and pressure
$\hat{p}_{\boldsymbol{k}}$ are regarded as the response of the input forcing
$\hat{\boldsymbol{f}}_{\boldsymbol{k}}$ through the linear operator
$\mathcal{H}_{\boldsymbol{k}}$. Since we will focus on predicting the velocity
field in this study, the linearized equation (4) can be reduced to the
following form,

$\hat{\boldsymbol{u}}_{\boldsymbol{k}}=\mathcal{R}_{\boldsymbol{k}}\cdot\hat{\boldsymbol{f}}_{\boldsymbol{k}},$
(5)

where $\mathcal{R}_{\boldsymbol{k}}$ is obtained from
$\mathcal{H}_{\boldsymbol{k}}$ by deleting the rows corresponding to
$\hat{p}_{\boldsymbol{k}}$ at the left-hand side of equation (4) and columns
corresponding to the constant 0 at the right-hand side. A typical application
of equation (5) is to approximate the coherent structures of turbulence using
the resolvent modes, which are obtained by taking the singular value
decomposition to $\mathcal{R}_{\boldsymbol{k}}$, i.e.,

$\hat{\boldsymbol{u}}_{\boldsymbol{k}}\\\
=\sum_{j=1}^{\infty}\left(\boldsymbol{\hat{\psi}}_{\boldsymbol{k},j}\sigma_{\boldsymbol{k},j}\boldsymbol{\hat{\varphi}}_{\boldsymbol{k},j}^{\ast}\right)\cdot\hat{\boldsymbol{f}}_{\boldsymbol{k}}=\sum_{j=1}^{\infty}\sigma_{\boldsymbol{k},j}\boldsymbol{\hat{\psi}}_{\boldsymbol{k},j}\left(\boldsymbol{\hat{\varphi}}_{\boldsymbol{k},j}^{\ast}\cdot\hat{\boldsymbol{f}}_{\boldsymbol{k}}\right)=\sum_{j=1}^{\infty}\sigma_{\boldsymbol{k},j}\boldsymbol{\hat{\psi}}_{\boldsymbol{k},j}\boldsymbol{\beta}_{\boldsymbol{k},j},$
(6)

where
$\sum_{j=1}^{\infty}\left(\boldsymbol{\hat{\psi}}_{\boldsymbol{k},j}\sigma_{\boldsymbol{k},j}\boldsymbol{\hat{\varphi}}_{\boldsymbol{k},j}^{\ast}\right)=\mathcal{R}_{\boldsymbol{k}}$
is the result of singular value decomposition of the resolvent operator
$\mathcal{R}_{\boldsymbol{k}}$, the response mode
$\boldsymbol{\hat{\psi}}_{\boldsymbol{k},j}$ and forcing mode
$\boldsymbol{\hat{\varphi}}_{\boldsymbol{k},j}$ are ordered by their singular
value $\sigma_{\boldsymbol{k},j}$, the expansion coefficient
$\boldsymbol{\beta}_{\boldsymbol{k},j}=\left(\boldsymbol{\hat{\varphi}}_{\boldsymbol{k},j}^{\ast}\cdot\hat{\boldsymbol{f}}_{\boldsymbol{k}}\right)$
is the projection of forcing on the $j{\rm-th}$ resolvent forcing mode, and
the superscript ∗ denotes the Hermitian transpose.

Using equation (5), the CSD tensors can be calculated as,

$S_{\boldsymbol{uu},\boldsymbol{k}}=\left\langle\hat{\boldsymbol{u}}_{\boldsymbol{k}}\hat{\boldsymbol{u}}_{\boldsymbol{k}}^{\ast}\right\rangle=\mathcal{R}_{\boldsymbol{k}}\cdot\left\langle\hat{\boldsymbol{f}}_{\boldsymbol{k}}\hat{\boldsymbol{f}}_{\boldsymbol{k}}^{\ast}\right\rangle\cdot\mathcal{R}_{\boldsymbol{k}}^{\ast}=\mathcal{R}_{\boldsymbol{k}}S_{\boldsymbol{ff},\boldsymbol{k}}\mathcal{R}_{\boldsymbol{k}}^{\ast},$
(7)

where $\left\langle\cdot\right\rangle$ denotes the ensemble average. As in
equation (7), the CSD tensor of the response is fully determined when the
forcing CSD $S_{\boldsymbol{ff},\boldsymbol{k}}$ is known. Thus, in many
existing methods (e.g., Morra et al. (2019), Towne et al. (2020), Gupta et al.
(2021)), the estimating or modeling object is actually
$S_{\boldsymbol{ff},\boldsymbol{k}}$, after which the response
$S_{\boldsymbol{uu},\boldsymbol{k}}$ can be directly derived from
$S_{\boldsymbol{ff},\boldsymbol{k}}$ via equation (7). In the following, the
typical methods that make use of the resolvent analysis to predict the
turbulence field will be introduced.

###  2.2 The white-noise-based estimation

With the simplest assumption, the portion of forcing that excludes the eddy
viscosity terms, as defined in equation (1), can be modeled to be white as in
some researches (Hwang & Cossu, 2010; Madhusudanan et al., 2019; Morra et al.,
2019), which means it is spatially uniform and uncorrelated. Note that, in
this approach, the total forcing as defined by McKeon & Sharma (2010) is the
summation of the eddy-viscosity portion and the white-noise-assumed portion.
With the white-noise assumption of the WBE approach, the CSD tensor of forcing
at each scale $\boldsymbol{k}$ can be expressed as,

$S_{\boldsymbol{ff},\boldsymbol{k},{\rm{WBE}}}=E_{\boldsymbol{k}}\cdot\mathcal{I},$
(8)

where $E_{\boldsymbol{k}}$, as the energy of forcing, keeps constant at each
node and in each direction. Substituting equation (8) into (7) and considering
the resolvent modes in equation (6), it can be deduced that,

$S_{\boldsymbol{uu},\boldsymbol{k},{\rm{WBE}}}=E_{\boldsymbol{k}}(\mathcal{R}_{\boldsymbol{k}}\mathcal{R}_{\boldsymbol{k}}^{\ast})=E_{\boldsymbol{k}}\sum_{j=1}^{\infty}\sigma_{\boldsymbol{k},j}^{2}(\boldsymbol{\hat{\psi}}_{\boldsymbol{k},j}\boldsymbol{\hat{\psi}}_{\boldsymbol{k},j}^{\ast}).$
(9)

If we define
$(\boldsymbol{\hat{\psi}}_{\boldsymbol{k},j}\boldsymbol{\hat{\psi}}_{\boldsymbol{k},j}^{\ast})$
as the CSD of the response at the $j$-th mode, the resultant response CSD can
be interpreted as the linear summation of the CSDs of all the resolvent modes
weighted by the gains $\sigma_{\boldsymbol{k},j}^{2}$. The white forcing, as
the simplest form, can be utilized to estimate the coherent structures (Hwang
& Cossu, 2010; Madhusudanan et al., 2019; Gupta et al., 2021) and the spectra
of turbulence (Morra et al., 2019). However, the resultant accuracy of
prediction with the initial white forcing is far from engineering usage (Towne
et al., 2020) even if the forcing is partially modeled by the eddy-viscosity
term.

###  2.3 The wall-distance-dependent and scale-dependent models

To improve the prediction accuracy of the white forcing, Gupta et al. (2021)
propose the Wall-distance-dependent Model (W-model) and Scale-dependent Model
($\lambda$-model) by modifying the profile of the forcing according to the
profile of the eddy viscosity and the flow scales. The W-model and
$\lambda$-model maintain the diagonal property of
$S_{\boldsymbol{ff},\boldsymbol{k}}$ as in equation (8), while modifying the
vertical energy distribution. Extended from the work of Jovanović & Bamieh
(2005) in laminar flow, Gupta et al. (2021) propose the W-model by assuming
that the vertical profile of forcing energy is proportional to that of the
eddy viscosity, i.e.,

$W_{\boldsymbol{k}}=E_{\boldsymbol{k}}\nu_{t},$ (10)

where $W_{\boldsymbol{k}}$ denotes the diagonal of
$S_{\boldsymbol{ff},\boldsymbol{k}}$, i.e., the energy profile of forcing at
$\boldsymbol{k}$. Based on the fact that the nonlinear interaction of
turbulence is scale-dependent (Cho et al., 2018), Gupta et al. (2021) further
propose the $\lambda$-model with the modified eddy viscosity, i.e.,

$\nu_{t,{\boldsymbol{k}}}=\frac{\lambda}{\lambda+\lambda_{m}}\nu_{t},$ (11)

where $\lambda=2\pi/(k_{x}^{2}+k_{z}^{2})^{0.5}$, and
$\lambda_{m}(y)=50/{Re_{\tau}}+(2-50/{Re_{\tau}}){\rm{tanh}}(6y)$. The forcing
energy profile is thereby calculated by
$W_{\boldsymbol{k}}=E_{\boldsymbol{k}}\nu_{t,{\boldsymbol{k}}}$. The
strategies of modifying the forcing profiles by Gupta et al. (2021)
efficiently improve the prediction of the vertical spatial correlation of
streamwise velocity fluctuations in turbulent channel flow. However, the
W-model and $\lambda$-model cannot accurately predict the energy distribution
of velocity fluctuation in the near-wall region, as will be further discussed
in the following sections.

###  2.4 The resolvent-based estimation

The RBE method (Towne et al., 2020) estimates the minimum L2-norm forcing that
can fully recover the measured signal. Denoting $\boldsymbol{m}$ as the
measured set of variables and $\boldsymbol{u}$ as the complete set of
variables from all across the computational domain, where
$\boldsymbol{m}=\mathscr{C}\boldsymbol{u}$, the RBE could be briefly expressed
as,

$S_{\boldsymbol{ff},\boldsymbol{k},{\rm{RBE}}}=T_{\boldsymbol{ff},\boldsymbol{k}}\cdot
S_{\boldsymbol{mm},\boldsymbol{k}}\cdot
T_{\boldsymbol{ff},\boldsymbol{k}}^{\ast},$ (12)

where the estimation operator
$T_{\boldsymbol{ff},\boldsymbol{k}}=(\mathscr{C}\mathcal{R}_{\boldsymbol{k}})^{\dagger}$,
$S_{\boldsymbol{mm},\boldsymbol{k}}=\mathscr{C}S_{\boldsymbol{uu},\boldsymbol{k}}\mathscr{C}^{\rm
T}$ is the CSD tensor of the measured variables at scale $\boldsymbol{k}$,
$S_{\boldsymbol{ff},\boldsymbol{k},{\rm{RBE}}}$ is the estimated CSD tensor of
forcing, the expression of the observation matrix $\mathscr{C}$ can be found
in equation (30) in Appendix A, and the superscript † denotes the pseudo-
inverse. With the estimated forcing, the CSD tensor of the complete set of
variable $\boldsymbol{u}$ is calculated by

$S_{\boldsymbol{uu},\boldsymbol{k},{\rm
RBE}}=\mathcal{R}_{\boldsymbol{k}}S_{\boldsymbol{ff},\boldsymbol{k},{\rm{RBE}}}{\mathcal{R}_{\boldsymbol{k}}}^{\ast}.$
(13)

The RBE has been validated to be efficient for predicting the field where the
turbulence is highly correlated with the measured reference points (Towne et
al., 2020; Yang et al., 2020). On the other hand, since the RBE method only
estimates the “observed” forcing, its prediction deteriorates with the
decrease of correlation between the signals of measurements and the predicted
locations.

Later, Martini et al. (2020) demonstrate that the estimated forcing in the
original version of RBE in equation (12) is actually the stationary point of
the error matrix
$\overline{\boldsymbol{\hat{e}}_{f,\boldsymbol{k}}\boldsymbol{\hat{e}}_{f,\boldsymbol{k}}^{\dagger}}$,
where the error $\boldsymbol{\hat{e}}_{f,\boldsymbol{k}}$ is defined as the
difference between the estimated forcing and the white-noise-assumed forcing
here. Since the real forcing is not white, the estimation of the original RBE
is not optimal to minimize the relative error between the estimated forcing
and the real one. Martini et al. (2020) then propose the improved RBE that
incorporates the effect of the forcing color and the measurement noise, i.e.,

$T_{\boldsymbol{ff},\boldsymbol{k},{\rm{opt}}}=S_{\boldsymbol{ff},\boldsymbol{k}}(\mathscr{C}\mathcal{R}_{\boldsymbol{k}})^{\dagger}\left[(\mathscr{C}\mathcal{R}_{\boldsymbol{k}})S_{\boldsymbol{ff},\boldsymbol{k}}(\mathscr{C}\mathcal{R}_{\boldsymbol{k}})^{\dagger}+S_{nn,\boldsymbol{k}}\right]^{-1},$
(14)

by which the stationary point of the error matrix
$\overline{\boldsymbol{\hat{e}}_{f,\boldsymbol{k}}\boldsymbol{\hat{e}}_{f,\boldsymbol{k}}^{\dagger}}$
between the estimated forcing and the real one is obtained.
$S_{nn,\boldsymbol{k}}$ is the CSD tensor of the measurement noise at scale
$\boldsymbol{k}$, which is set as zero since the DNS data at a given reference
layer can be directly provided as measurements without introducing additional
errors in this study. When the real forcing color from additional sensors is
used to inform the transfer function, the optimized RBE performs better than
the original RBE in estimating the turbulent channel flow given the same
amount of measurements. However, the optimized RBE needs the knowledge of
forcing statistics obtained from additional sensors or forcing models, which
is not considered in this study. Thus, the RBE method mentioned in the
following refers to its original version by Towne et al. (2020), which will be
used to develop our newly proposed methods and provide comparison results.

The above existing methods can be categorized into two groups. First, the WBE,
W-model, and $\lambda$-model assume that the forcing is uncorrelated in space,
then explicitly model the forcing energy with predefined profiles. These
approaches aim to describe the forcing statistics throughout the flow field.
On the other hand, the information of the measurements is only utilized to
determine the overall forcing energy $E_{\boldsymbol{k}}$, while the estimated
relative energy distribution of forcing is independent of the measurements.
The advantage of these methods with predefined profiles is that the predicted
response can generally reflect the energy distribution of the response to some
extent, no matter how far the measurement layers are located from the
prediction region. However, the performance of the prediction is highly
dependent on the specific form of the predefined profiles, which implies that
the accuracy of this group of methods could not be as good as expected, even
if the measurements and predictions are close to each other, as can be seen in
section 4. Second, the RBE infers the forcing statistics from the measurements
without imposing assumptions on the form of the forcing profile. The RBE has
been validated to be efficient for predicting the field where the turbulence
is highly correlated with the measured reference points (Towne et al., 2020;
Yang et al., 2020). However, since it only estimates the “observed” forcing,
its prediction accuracy deteriorates with the increasing distance between the
measurements and the predicted locations.

##  3 Derivations of the Resolvent-informed White-noise-based Estimation
method

From the above discussions of the existing methods, the group of methods
assuming predefined profiles and the RBE compensate with each other in a
sense. Specifically, the group of methods assuming predefined profiles
performs better in estimating the general energy distribution of the forcing
and response when there is a long distance between the measurements and the
prediction region. On the other hand, the RBE is efficient in providing
reasonable estimations when the measurements are located near the prediction.
According to Holford et al. (2023), the energy spectra of turbulence can be
well recovered using the spatially uncorrelated forcing with optimal profiles.
In this study, following the strategy of modifying the spatially uncorrelated
forcing profile, we aim to propose an adaptive method to adjust the spatially
uniform and uncorrelated forcing profile based on reliable inference informed
by the RBE.

As already discussed, when the measured reference layer is not close to the
prediction region, the RBE cannot be directly applied for prediction due to
the deterioration of accuracy. However, the RBE can, instead, be utilized to
estimate the relative energy profile of the response with respect to an
assumed reference layer that is closer to the wall than the actual reference
layer. Based on the basic RBE formula (12-13), the predicted relative CSD
tensor of velocity $u_{i}$ near the wall can be estimated as,

$\hat{S}_{u_{i}u_{i},\boldsymbol{k},{\rm
RBE}}=\frac{S_{u_{i}u_{i},\boldsymbol{k},{\rm
RBE}}}{S_{mm,\boldsymbol{k}}}=\mathcal{R}_{\boldsymbol{k}}(\mathscr{C}\mathcal{R}_{\boldsymbol{k}})^{\dagger}\cdot(\mathscr{C}\mathcal{R}_{\boldsymbol{k}})^{\dagger\ast}{\mathcal{R}_{\boldsymbol{k}}}^{\ast},$
(15)

where $S_{mm,\boldsymbol{k}}$ should be the scalar energy of $u_{i}$ at a
single reference height so that it can be eliminated from the denominator by
the numerator. Note that there is no requirement on the specific value of the
reference height in equation (15), a pretty high accuracy can be obtained for
predicting the relative CSD tensor. This desirable property of the RBE in
estimating the relative response statistics provides a standard for the
modification of the initially assumed white forcing profile. To be
distinguished from the actual reference layer, which is denoted as $y_{\rm
R}$, the assumed reference layer used for estimating the relative CSDs in
equation (15) is denoted as the quasi-reference layer $y_{\rm Q}$ in this
article. To obtain the relative CSD tensor used for the optimization, the
value of $y_{\rm Q}$ should be determined first. According to Towne et al.
(2020), the accuracy of RBE decreases as the wavenumbers and frequency
increase. As the wavenumbers and frequency are closely related to the flow
scale, we choose to determine the quasi-reference layer according to the flow
scale in the wall-normal direction. The purpose of this step is to let the RBE
provide a reasonable estimation of the widest possible wall-normal extent. To
quantify the wall-normal scale, the linear coherence spectrum (LCS)
$\gamma^{2}$ (Baars et al., 2016) is introduced here, i.e.,

$\gamma^{2}(\boldsymbol{k})=\frac{\left|\left\langle\hat{u}(y_{\rm
Q}^{+})\overline{\hat{u}(y_{\rm
P}^{+})}\right\rangle\right|^{2}}{\left\langle\left|\hat{u}(y_{\rm
Q}^{+})\right|^{2}\right\rangle\left\langle\left|\hat{u}(y_{\rm
P}^{+})\right|^{2}\right\rangle},$ (16)

where $\hat{u}$ is the Fourier coefficient of $u$ at scale $\boldsymbol{k}$,
the overline denotes the complex conjugate, and $y_{\rm P}^{+}=15$ corresponds
to the height of the near-wall inner peak (Smits et al., 2011). Since the LCS
at a given distance increases as the flow scale enlarges, the LCS could be an
effective index to quantify the flow scale. Based on the value of LCS
preliminarily estimated from the WBE, the quasi-reference layer $y_{\rm Q}$ is
set as the height beyond the inner peak where
$\gamma^{2}(\boldsymbol{k})=0.3$. When the height of $y_{\rm Q}$ exceeds
$y_{\rm R}$, it will be set as $y_{\rm R}$ instead. With the above procedure,
the value of $y_{\rm Q}$ is adaptively determined. Details of the
implementation of the LCS calculation and discussions about the impact of the
threshold LCS on the prediction accuracy are provided in Appendix C.

With the information of the turbulence statistics below $y_{\rm Q}$ from the
RBE, our goal is to minimize the relative error between the estimated CSD of
the velocity $u$ from the modified forcing and the RBE-estimated one. As will
be revealed in section 4.3, the RBE just provides the reliable prediction of
the relative energy profile below the quasi-reference layer $y_{\rm Q}$, which
does not work well when $y\geq y_{\rm Q}$. Thus, the modification range of the
forcing is restricted below $y_{\rm Q}$. For the region beyond the quasi-
reference layer, there is no reliable information to further improve the
forcing profile there. Thus, a conservative strategy is adopted by setting the
forcing to be unity at each node for $y\geq y_{\rm Q}$, which corresponds to
the spatially uniform forcing, as also assumed by the WBE reviewed in section
2.2. Derivations of the explicit relationship between the relative energy
profiles of forcing and response are provided in Appendix D. For each scale
$\boldsymbol{k}=(k_{x},k_{z},\omega)$, the norm minimization problem is
considered as follows, i.e.,

$\displaystyle\mathop{{\rm{minimize}}}\limits_{W_{\boldsymbol{k}}}$
$\displaystyle\left\|\frac{S_{uu,{\boldsymbol{k}}}(W_{\boldsymbol{k}})|_{y<y_{\rm
Q}}}{S_{uu,\boldsymbol{k},y_{\rm
Q}}(W_{\boldsymbol{k}})}-\hat{S}_{uu,\boldsymbol{k},{\rm RBE}}|_{y<y_{\rm
Q}}\right\|,$ (17) $\displaystyle~{}{\rm{subject~{}to~{}}}$
$\displaystyle~{}0\leq W_{\boldsymbol{k}}(y)\leq 1,~{}~{}\frac{{\rm
d}W_{\boldsymbol{k}}(y)}{{\rm d}y}\geq 0,~{}~{}$ $\displaystyle\forall
y\in[0,y_{\rm Q}),$ $\displaystyle W_{\boldsymbol{k}}(y)=1,~{}~{}$
$\displaystyle\forall y\in[y_{\rm Q},2h],$

where the norm $\|\bcdot\|$ is defined as
$\left\|\bcdot\right\|=\int_{0}^{y_{\rm Q}}(\bcdot)^{2}{\rm d}y$. This
constrained optimization problem is solved with the interior point method
(Momoh et al., 1999). By minimizing the norm of the error matrix, the energy
spectrum of the modified response is optimized with respect to the RBE-
estimated one. After the relative profile of $W_{\boldsymbol{k}}$ is obtained
from the optimization, the forcing profile will be multiplied by a unified
coefficient $E_{\boldsymbol{k}}$ to match the response energy at the reference
layer for all the velocity components of $u$, $v$, and $w$, respectively.

Figure 1: Schematic sketch of the RWE.

The above construction process of the newly proposed RWE method is sketched in
figure 1. To illustrate the actual modification process of the forcing and its
effects on the response energy profile, the scale with
$(k_{x},k_{z},\omega)=(3.0/h,30/h,1.3u_{c}/h)$ corresponding to the large-
scale motions (Smits et al., 2011) is selected, as in figure 2, where $u_{c}$
is the mean velocity at the half-channel height $h$. The forcing energy is
uniformly distributed in the vertical direction before the modifications, with
the corresponding response energy profile much larger than the DNS results in
the near-wall region, as figure 2$(b)$. The height of the quasi-reference
layer $y_{\rm Q}^{+}$ is determined according to the LCS value defined in
equation (16). During the modification procedure, the forcing energy is
reduced in the near-wall region to let the predicted relative energy profile
approach that by the RBE till the norm of the error matrix in equation
(LABEL:eq:minmimize_norm) reaches the minimum under the constraints. The
forcing and response profiles after modification are denoted in the orange
color in figure 2$(a)$ and $(b)$, respectively. The response energy profile
after modification matches well with the DNS and RBE results.

Figure 2: Sketch of the modification process at
$(k_{x},k_{z},\omega)=(3.0/h,30/h,1.3u_{c}/h)$. ($a$) Energy profile of
forcing; ($b$) Energy profile of the streamwise velocity.

In the next section, validations of the RWE will be conducted in terms of the
prediction capability of the near-wall statistics as well as the instantaneous
flow field with the DNS data and several representative existing prediction
methods. The WBE (Morra et al., 2019) and RBE (Towne et al., 2020) will be
included in the following validations, since they provide the initial forcing
profile and the reference for optimization for the newly proposed RWE method,
respectively. Besides, the $\lambda$-model (Gupta et al., 2021) considers the
effects of flow scale on the estimated forcing profile, which is found to
perform better than the W-model and WBE (named as B-model by Gupta et al.
(2021)) for estimating the large-scale motions in the near-wall region. Thus,
the $\lambda$-model will also be included in the following validations. Note
that the $\lambda$-model is originally used in cases where temporal
information of velocity is unknown. The forcing is thus assumed to be white in
time in those cases (Gupta et al., 2021), the response CSD tensor of which can
be obtained via the algebraic Lyapunov equation. Meanwhile, in this study, the
$\lambda$-model will be applied in the time-resolved cases instead, where the
flow is estimated at each spatio-temporal scale quantified by
$\boldsymbol{k}=(k_{x},k_{z},\omega)$. The modified eddy viscosity and forcing
profiles in the $\lambda$-model will be calculated by equation (11) at each
scale $(k_{x},k_{z})$, keeping consistent with the original version.

##  4 Results

In this section, the DNS data with three friction Reynolds numbers equal to
180, 550, and 950 are used to provide reference measurements at corresponding
locations and validate the tested methods in predicting the flow properties of
turbulent channel flows.

###  4.1 Descriptions of the DNS database

The code used to compute the extensively validated DNS database for channel
flows (Del Alamo & Jiménez, 2003; Hoyas & Jiménez, 2008) is utilized to
generate the time-resolved channel flow data with $Re_{\tau}$ = 180, 550, and
950. Details of the DNS setups are listed in table 1. To provide time-resolved
results, the sampling time intervals $\Delta t^{+}=\Delta
t\cdot\frac{u_{\tau}^{2}}{\nu}$ are set as 2.13, 4.81, and 5.09 for cases with
$Re_{\tau}=180$, 550 and 950, respectively. The normalized total simulation
time $(u_{\tau}\cdot T)/h$ is larger than 5.0 in each case to obtain
statistically convergent results. To assess the DNS dataset generated in this
study, comparisons of the mean and root-mean-squared velocity profiles with
the open source DNS database (Del Alamo & Jiménez, 2003; Hoyas & Jiménez,
2008) are provided in Appendix B.

To process the DNS data, the flow field is divided into blocks with a spatial
domain of sizes ${L_{x}}/h=4\pi$, ${L_{z}}/h=\pi$, and ${L_{y}}/h=1$ in each
case. Given that the turbulent channel flow is statistically symmetric about
the centerline $y=h$, the flow data at $y=y_{0}$ will be utilized together
with that at $y=2h-y_{0}$ when investigating the flow at $y=y_{0}$. The time
periods of the blocks are set as $80\Delta t$, $80\Delta t$, and $120\Delta t$
for cases with $Re_{\tau}=180$, 550 and 950, respectively, with $75\%$ overlap
in the temporal direction. The rectangular window function is used when
conducting spectral analyses. With the above setups for data processing, the
numbers of blocks are 1128, 752, and 360 for cases with $Re_{\tau}=180$, 550,
and 950, respectively.

$Re_{\tau}$ | $Nx$ | $Nz$ | $Ny$ | ${L_{x}}/h$ | ${L_{z}}/h$ | ${L_{y}}/h$ | $(u_{\tau}\cdot T)/h$
---|---|---|---|---|---|---|---
180 | 768 | 512 | 97 | $12{\pi}$ | $4{\pi}$ | 2 | 11.45
550 | 1536 | 1536 | 257 | $8{\pi}$ | $4{\pi}$ | 2 | 8.80
950 | 3072 | 2304 | 385 | $8{\pi}$ | $3{\pi}$ | 2 | 5.39
Table 1: Parameters of the incompressible channel DNS setups.

###  4.2 Case settings

Six cases are set to validate the prediction methods, as summarized in table
2. Besides the Reynolds number $Re_{\tau}$, the height of the reference layer
$y_{\rm R}$ where the measurements are obtained is also treated as an
independent variable to test the sensitivities of the methods to the location
of measurements, which ranges from the near-wall region at $y^{+}=40$ to the
upper bound of the logarithmic region at $y/h\approx 0.2$. The wall-normal
direction $y$ is discretized with 129, 201, and 257 Chebyshev polynomials in
cases with $Re_{\tau}=180$, 550, and 950, respectively, with no-slip boundary
conditions are applied at the walls.

Case | 180-40 | 550-40 | 550-100 | 950-40 | 950-100 | 950-200
---|---|---|---|---|---|---
$Re_{\tau}$ | 180 | 550 | 550 | 950 | 950 | 950
$y_{\rm R}^{+}$ | 40 | 40 | 100 | 40 | 100 | 200
$y_{\rm R}/h$ | 0.22 | 0.073 | 0.18 | 0.042 | 0.11 | 0.21
Table 2: Case settings for validation.

###  4.3 Reynolds stress profiles

In this section, the root-mean-squared (RMS) velocities and the Reynolds shear
stress (RSS) are investigated to study the ensemble effect of fluctuations
with all the spatio-temporal scales along the height. Specifically, the RMS
velocities quantify the fluctuation energies of $u^{\prime}$, $v^{\prime}$,
and $w^{\prime}$, which are important indexes to validate the DNS (e.g., Cheng
et al. (2019)) or prediction methods (e.g., Towne et al. (2020)). On the other
hand, the RSS $\left\langle u^{\prime}v^{\prime}\right\rangle$ quantifies the
correlation of $u^{\prime}$ and $v^{\prime}$, which is closely related to the
skin friction of wall-bounded turbulence (Fukagata et al., 2002) and has been
investigated via the resolvent analysis for flow control (Luhar et al., 2015;
Nakashima et al., 2017).

When $Re_{\tau}=180$, the only reference layer at $y^{+}=40$ is used for the
predictions. According to the evaluations of RBE by Yang et al. (2020), the
best performance of RBE in terms of estimating the RMS profile of the
streamwise velocity is achieved when $y^{+}=39$, which is very close to
$y^{+}=40$ used in the current case. As in figure 3$(a)$, the RMS profiles
predicted by RBE are pretty consistent with the DNS results in all three
directions, with only $2.0\%$ relative error at the inner peak of the
streamwise RMS velocity profile. The RWE also performs well when
$Re_{\tau}=180$ with a relative error of $13.2\%$ at the inner peak. On the
other hand, the predictions from the $\lambda$-model and WBE both deviate
quite a lot from the DNS results, with maximum errors of $25.7\%$ and
$109.7\%$ at the inner peaks of the streamwise RMS velocity. Note that this
case is considered to be the least challenging one with the smallest Reynolds
number and the closest reference layer to the wall, the large prediction
errors in the $\lambda$-model and WBE results indicate that the predefined
forcing profiles cannot properly model the forcing effects if not adjusted by
additional flow information, especially in the near-wall region.

Figure 3: Comparisons of the RMS profiles for cases with $Re_{\tau}=180$ ($a,d,g,j$), 550 ($b,e,h,k$) and 950 ($c,f,i,l$). The solid lines denote the DNS results. The dashed lines denote the predictions from the RWE ($a,b,c$), $\lambda$-model ($d,e,f$), RBE ($g,h,i$), and WBE ($j,k,l$), where the lines with square, lower triangles and upper triangles denote the RMSs of the streamwise velocity, spanwise velocity, and vertical velocity, respectively. The vertical dotted lines denote the reference layers corresponding to the dashed lines with the same colors. Case | 180-40 | 550-40 | 550-100 | 950-40 | 950-100 | 950-200
---|---|---|---|---|---|---
RWE | 13.2 | 11.82 | 3.3 | 16.6 | 10.1 | 8.9
RBE | 2.0 | 5.9 | -28.0 | 7.7 | -20.6 | -47.4
$\lambda$-model | 25.7 | 24.5 | -8.5 | 17.5 | -8.0 | -24.0
WBE | 109.7 | 122.1 | 113.4 | 103.6 | 115.2 | 115.7
Table 3: Summary of the relative percentage errors of the predicted streamwise
RMS velocity at the inner peak.

With the increase of $Re_{\tau}$, the RBE continues performing well when the
reference layers are located at $y^{+}=40$, with the maximum error of $7.7\%$
at the inner peak in Case 950-40. However, as the reference layer moves away
from the wall, the RBE-predicted energy decreases rapidly, which is considered
to be attributed to the decrease of coherence between the signals at the
reference layer and the near-wall region. Especially, when the reference layer
is located at $y^{+}=200$ with $Re_{\tau}=950$, the energy peak can be barely
observed in the RBE-predicted streamwise RMS profile. Like the RBE, the
energies predicted by the $\lambda$-model also tend to decrease when the
reference layer lifts upward, which overestimates and underestimates the
energy with $y_{\rm R}^{+}=40$ and $y_{\rm R}^{+}=200$, respectively. When
$y_{\rm R}^{+}=100$, the $\lambda$-model-predicted results match fairly well
with the DNS results. The WBE overestimates the energies of the fluctuations
of all the velocity components in all the tested cases, which indicates that
the WBE cannot be directly used to estimate the energy magnitude in the near-
wall region. Despite this, the capacity of WBE to estimate the distributions
of the relative energy spectra should be further investigated in the following
sections. Compared with the above methods, the RWE performs steadily well in
all the cases with various Reynolds numbers and choices of the reference
layers, with the maximum error equal to $16.6\%$ at the inner peak of
streamwise RMS velocity in Case 950-40.

Besides the RMS velocity profiles below the reference layer, those in the
higher region in each case are also depicted in figure 3 with translucent
curves. From the RWE results, the predicted profiles match well with the DNS
results in the wall-normal range of $y\in(y_{\rm R},0.2h)$ for all the RMS
velocities with different Reynolds numbers and reference layers. Moreover, for
the same Reynolds number, the profiles predicted by the RWE nearly overlap
with each other for each Reynolds number, which again demonstrates the
insensitivity of the RWE results to the choice of reference layer. For the
region that is higher than the upper bound of the logarithmic region, i.e.,
$y>0.2h$, the RWE results become larger than those from the DNS, which should
be attributed to the mismatch of the RWE-modeled forcing profile in the outer
layer with the real forcing in the DNS results. On the other hand, the other
three considered methods cannot provide satisfying results at $y>y_{\rm R}$.
For the RBE, the magnitudes of the RMS velocities shrink rapidly when the
prediction location lifts up from the reference layer. On the contrary, the
$\lambda$-model results become obviously larger than the DNS results at
$y>y_{\rm R}$, which implies that the $\lambda$-model overestimates the
forcing in the logarithmic region. The WBE underestimates the fluctuation
energies at $y\in(y_{\rm R},0.2h)$, which in turn overestimates the magnitudes
in the outer layer.

The RSS profiles are depicted in figure 4. The RWE-predicted RSS profiles,
which are obtained from the measurements of the streamwise velocity at the
reference layers, are fairly consistent with the DNS results in all the cases.
The RBE results that are depicted with upper triangles match well with the DNS
results when $y_{\rm R}^{+}=40$. However, when the reference layer increases
to 100 and 200, the magnitudes of the RBE-predicted RSS profiles become
obviously underestimated. In the meantime, the $\lambda$-model and WBE neither
provide satisfying results with different Reynolds numbers and reference
heights.

Figure 4: Comparisons of the Reynolds shear stress profiles for cases with
$Re_{\tau}=180$ ($a$), 550 ($b$) and 950 ($c$). The solid lines and dashed
lines denote the DNS results and predictions, respectively. The lines with
square, upper triangles, lower triangles, and diamonds denote RWE, RBE,
$\lambda$-model, and WBE results, respectively, where only the RWE results are
depicted with opaque colors. The vertical dotted lines denote the reference
layers corresponding to the dashed lines with the same colors.

From the above discussions, the impacts of the friction Reynolds numbers and
measuring locations on the prediction accuracy of the tested methods can be
concluded as follows. When the $Re_{\tau}$ increases, the accuracy of
prediction with the same $y_{\rm R}^{+}$ by the RBE and RWE does not show an
obvious difference. Meanwhile, the variation of the measuring locations
significantly influences the prediction accuracy of all the methods except for
the newly proposed RWE. Specifically, the increasing height of the reference
layer reduces the predicted fluctuation energy at a given height for the RBE
and $\lambda$-model, the errors of which become minimum when $y_{\rm
R}^{+}=40$ and 100, respectively. Since the measurements are usually not
guaranteed to be fixed at an ideal location in the WMLES or experiments, the
insensitivity of the choices of reference layers implies a large potential of
RWE to practical engineering applications.

###  4.4 One-dimensional energy spectra of the fluctuation velocities

Besides the RMS profiles, the energy spectra, which describe the energy
distributions among different scales, provide a more comprehensive picture to
display the predicted flow properties. In this section, the distributions of
one-dimensional energy spectra at different heights as functions of $k_{x}$,
$k_{z}$, or $\omega$ will be studied. For brevity, only the results of
streamwise velocity $u$, as the most interesting physical quantity, in the
cases where the reference layers are the farthest away from the wall for the
corresponding Reynolds numbers, i.e., Cases 180-40, 550-100, and 950-200, are
chosen for the validations in this section.

Figure 5 depicts the premultiplied one-dimensional energy spectra of the
streamwise velocity $u$ with the reference layer located at $y^{+}=40$, where
the location of the reference layer is denoted with the white dashed line in
each case. The black dashed line denoting the contour of $0.5S_{{\rm
DNS,max}}$ is marked in each case to highlight the energy-concentrating
region, where $S_{{\rm DNS,max}}$ is the maximum premultiplied spectral energy
of the DNS result in the same case. The values of the premultiplied energy are
normalized by $S_{{\rm DNS,max}}$. When the reference layer is located at
$y^{+}=40$ with $Re_{\tau}=180$, the RBE works well in predicting the energy
distributions at different heights and flow scales, with the maximum error
lower than $0.25S_{{\rm DNS,max}}$ according to figure 6 that depicts the
relative error. This result is consistent with the previous work that also
uses the RBE to predict the energy spectra with the reference layer at
$y^{+}=37$ (Towne et al., 2020). The energies predicted by the RWE also match
fairly well with the DNS results, whose maximum error is about $0.46S_{{\rm
DNS,max}}$. On the other hand, the relative errors that are larger than
$1.0S_{{\rm DNS,max}}$ are observed in the results of the $\lambda$-model and
WBE. For the $\lambda$-model, the maximum error occurs at the heights around
the inner peak $y^{+}=15$ and the scales that are slightly smaller than the
energy-concentrating scales predicted by the DNS, as denoted by the black
dashed lines. This indicates that the deviations between the $\lambda$-model
and DNS results mainly concentrate at small scales near the inner peak. On the
other hand, the WBE overestimates the energy within a wide wall-normal range
from $y^{+}=3$ to 20, which indicates that the WBE results cannot be directly
applied to the predictions of the magnitude of the energy spectra.

Although the WBE overestimates the magnitude of the energy, it is still worth
investigating the relative spectral energy predicted by the WBE. To depict the
relative spectral energy distributions in the WBE results, we set the upper
bound of the color bar as the maximum spectral energy predicted by the WBE
rather than $S_{\rm DNS,max}$. The relative distributions of the energies
predicted by WBE are roughly consistent with those from the DNS, while
deviations could also be found, e.g., the inner peak for the streamwise
velocity from WBE is located at about $y^{+}\approx 10$, which should be at
$y^{+}\approx 15$ according to the DNS results.

Among the above-discussed methods, the RBE provides the most accurate results,
followed by the RWE. Although the RWE result does not totally recover the RBE
results, it provides a better result over the WBE and $\lambda$-model.

Figure 5: Premultiplied energy spectra of streamwise velocity as a function of
the wall-normal distance $y$ and the streamwise wavenumber $k_{x}$
($a,d,g,j,m$), spanwise wavenumber $k_{z}$ ($b,e,h,k,n$), and frequency
$\omega$ ($c,f,i,l,o$) in Case 180-40, from the results of DNS ($a,b,c$), RWE
($d,e,f$), RBE ($g,h,i$), $\lambda$-model ($j,k,l$), and WBE ($m,n,o$). The
black dashed lines denote the contour of $0.5S_{{\rm DNS,max}}$, and the white
dash-doted lines denote the height of the reference layer. The values shown in
the figures are normalized by $S_{{\rm DNS,max}}$. Figure 6: Relative error of
the premultiplied energy spectra of streamwise velocity as a function of the
wall-normal distance $y$ and the streamwise wavenumber $k_{x}$ ($a,d,g,j,m$),
spanwise wavenumber $k_{z}$ ($b,e,h,k,n$), and frequency $\omega$
($c,f,i,l,o$) in Case 180-40, from the results of DNS ($a,b,c$), RWE
($d,e,f$), RBE ($g,h,i$), $\lambda$-model ($j,k,l$), and WBE ($m,n,o$). The
values shown in the figures are normalized by $S_{{\rm DNS,max}}$. The black
solid and dashed lines denote the contours of $0.75S_{{\rm DNS,max}}$ and
$0.25S_{{\rm DNS,max}}$, respectively.

With the increase of the reference height $y_{\rm R}$, as in figures 7 and 9,
the prediction accuracy of RBE deteriorates, obviously. The RBE-predicted
energy-concentrating regions denoted by the black dashed lines shrink towards
the vicinity around $y^{+}=15$ in Case 550-100, which finally disappear in
Case 950-200, indicating that the near-wall energy is seriously underestimated
by RBE when $y_{\rm R}^{+}=200$. On the other hand, as the reference layer
lifts up, the magnitude of the $\lambda$-model-predicted energy decreases
gradually. In Case 550-100, the $\lambda$-model results match well with the
DNS results, with the relative error lower than $0.35S_{{\rm DNS,max}}$, as in
figure 8. When $y_{\rm R}^{+}$ increases to 200, the energy magnitude
predicted by the $\lambda$-model continues decreasing and consequently becomes
lower than the DNS results by more than $0.75S_{{\rm DNS,max}}$ in the small
scales at around $y^{+}=15$, as in figure 10. The WBE significantly
overestimates the magnitude of energy, while the relative energy distribution
roughly reflects the patterns of the DNS results. Compared to the WBE, the
$\lambda$-model has a higher accuracy, which highlights the importance of
including the scale effects when determining the forcing profile. Meanwhile,
the RWE keeps the highest accuracy in the streamwise energy distributions with
$k_{x}^{+}$, $k_{z}^{+}$, and $\omega^{+}$, with the maximum error of
$0.33S_{{\rm DNS,max}}$. The good performance of the RWE supports the
capability of the adaptive modification process as derived in section 3.

Figure 7: Same as figure 5, but in Case 550-100. Figure 8: Same as figure 6,
but in Case 550-100. Figure 9: Same as figure 5, but in Case 950-200. Figure
10: Same as figure 6, but in Case 950-200.

In order to quantitatively analyze the patterns of the near-wall predictions
from different methods as the reference layer moves away from the wall, the
relative errors at the inner peak ($y^{+}=15$) and along different scales
quantified by $k_{x}^{+}$, $k_{z}^{+}$, and $\omega^{+}$ are shown in figure
11. The values are normalized by $S_{{\rm DNS,max}}$, as consistent with those
in figures 6, 8, and 10. The green curves denoting the relative errors of the
WBE results deviate too much from the reference zero value, and thus they
cannot be fully depicted in the figures. The predicted energies from the RBE
and $\lambda$-model show a clear trend to decrease compared to the DNS results
as the reference layer lifts up. For instance, in Case 180-40, the spectral
energy predicted by the $\lambda$-model is larger than the DNS results at
almost all the scales, whose maximum error is more than $1.37S_{{\rm
DNS,max}}$ with a positive value. When the reference layer lifts up to
$y^{+}=200$, the maximum relative error of the $\lambda$-model predictions
becomes $0.7S_{{\rm DNS,max}}$ with a negative value instead. With the
increasing height of the reference layer, the RBE and $\lambda$-model achieve
their best performance with $y_{\rm R}^{+}=40$ and 100, respectively. On the
other hand, the relative error of the RWE results is pretty small in the cases
tested. Meanwhile, with the increase of flow scales, the spectral energy
predicted by the RWE tends to increase compared to the DNS results, which can
also be observed in the predictions from the other tested methods.

Figure 11: Relative error of the predicted energy at $y^{+}=15$ along
$k_{x}^{+}$ ($a,d,g$), $k_{z}^{+}$ ($b,e,h$), and $\omega^{+}$ ($c,f,i$) in
Cases 180-40 ($a,b,c$), 550-100 ($d,e,f$), and 950-200 ($g,h,i$). The values
of the curves are normalized by $S_{{\rm DNS,max}}$.

###  4.5 Two-dimensional spectra of the fluctuation velocities

In this section, the two-dimensional auto- and cross-spectra of the
fluctuation velocities in the near-wall region ($y^{+}=10$) are analyzed to
investigate the energy distributions and correlations at different spatial
scales. For brevity, the results for the auto-spectra of the streamwise
velocity and the cross-spectra of the streamwise and wall-normal velocities
are presented for the analyses.

First, the results of streamwise velocity in Cases 180-40, 550-100, and
950-200 at the near-wall plane at $y^{+}=10$ are presented in figure 12. In
order to show the relative energy spectrum predicted by the WBE, the separate
color bar with its upper bound set as the maximum spectral energy predicted by
the WBE is used. Meanwhile, the relative errors of the premultiplied spectra
are depicted in figure 13, with the black solid and dashed lines denoting the
contours of $0.75S_{{\rm DNS,max}}$ and $0.25S_{{\rm DNS,max}}$ of the DNS
results, respectively. When $Re_{\tau}=180$ with $y_{\rm R}^{+}=40$, it is
observed that the predictions of RWE and RBE are relatively consistent with
the DNS results. On the other hand, the $\lambda$-model and WBE obviously
overestimate the spectral energy. This observation is consistent with that in
the one-dimensional spectra as discussed in section 4.4. With the increase of
the reference height $y_{\rm R}^{+}$, the predicted energies from the RBE and
$\lambda$-model decrease correspondingly. When $y_{\rm R}^{+}=200$, only a
small amount of the RBE-predicted energy is observed in the large-scale region
with $\lambda_{x}/h\geq 2$, which indicates that the RBE becomes invalid when
the reference layer in this case. In section 4.4, we have observed that the
$\lambda$-model-predicted one-dimensional energy distributions match well with
the DNS results with a wide range of $k_{x}^{+}$ and $k_{z}^{+}$ values.
However, when considering the energy distributions as binary functions of
$\lambda_{x}$ and $\lambda_{z}$, the $\lambda$-model results are not
consistent with the DNS. From figure 13, the energy predicted by the
$\lambda$-model is obviously lower than the DNS result, in the energetic
region denoted by the black dashed lines. In the RWE results, the energy-
concentrating regions that are surrounded by the black dashed lines in figure
12 match well with the DNS results. From figure 13, the largest relative error
in the RWE results occurs in Case 180-40, which is equal to $0.48S_{\rm
DNS,max}$. In Cases 550-100 and 950-200, the relative error is always lower
than $0.4S_{\rm DNS,max}$. As consistent with the one-dimensional results, the
WBE results are overestimated by more than 4 times in magnitude for all the
cases. Whereas, the WBE results fairly reflect the energy distributions from
the DNS.

Figure 12: Premultiplied energy spectra as a function of the streamwise
wavelength $\lambda_{x}$ and spanwise wavelength $\lambda_{z}$ from the
results of DNS ($a,b,c$), RWE ($d,e,f$), W-model ($g,h,i$), RBE ($j,k,l$), and
WBE ($m,n,o$), at $y^{+}=10$ in Case 180-40 ($a,d,g,j,m$), Case 550-100
($b,e,h,k,n$), and Case 950-200 ($c,f,i,l,o$). The values shown in the figures
are normalized by $S_{{\rm DNS,max}}$. Figure 13: Relative errors of the
predicted premultiplied energy spectra normalized by the maximum energy
spectra density from DNS as a function of the streamwise wavelength
$\lambda_{x}$ and spanwise wavelength $\lambda_{z}$ from the results of RWE
($a,b,c$), W-model ($d,e,f$), RBE ($g,h,i$), and WBE ($j,k,l$), at $y^{+}=10$
in Case 180-40 ($a,d,g,j$), Case 550-100 ($b,e,h,k$), and Case 950-200
($c,f,i,l$). The values shown in the figures are normalized by $S_{{\rm
DNS,max}}$.

Then, the cross-spectra between $u^{\prime}$ and $v^{\prime}$ at $y^{+}=10$
are investigated, as shown in figure 14 with the relative errors presented in
figure 15. The values in figures 14 and 15 are normalized by the peak absolute
value of the DNS results, as denoted by $\left|S_{{\rm DNS}}\right|_{\rm max}$
here. The region with large values of negative correlations between the
streamwise and wall-normal velocities are observed in each case from the DNS
results, which is slightly offset towards smaller flow scales than the auto-
spectra as in figure 12. In Case 180-40, the RBE provides the most accurate
result of the cross-spectra. The RWE result is larger in magnitude than the
DNS results for about $0.3\left|S_{{\rm DNS}}\right|_{\rm max}$ in the energy-
concentrating region in Case 180-40. Meanwhile, the relative errors of the RWE
results are much smaller than those in the $\lambda$-model and WBE results. As
the reference layer lifts up to $y^{+}=100$ and 200, both the RBE and
$\lambda$-model obviously underestimate the correlations between $u^{\prime}$
and $v^{\prime}$. Meanwhile, the RWE keeps a relatively low prediction error
in the estimation of the cross-spectra in Cases 550-100 and 950-200.

Figure 14: The same as figure 12, but for the premultiplied cross-spectra
between $u^{\prime}$ and $v^{\prime}$ at $y^{+}=10$. The values shown in the
figures are normalized by $\left|S_{{\rm DNS}}\right|_{\rm max}$. Figure 15:
The same as figure 13, but for the relative errors of the premultiplied cross-
spectra between $u^{\prime}$ and $v^{\prime}$ at $y^{+}=10$. The values shown
in the figures are normalized by $\left|S_{{\rm DNS}}\right|_{\rm max}$.

###  4.6 Space-time properties of the fluctuation velocities near the wall

The space-time correlation is an important property of turbulence (He et al.,
2017). According to Taylor’s frozen hypothesis (Taylor, 1938), the spatial
distributions of the turbulent motions are carried past a fixed point by the
convection velocity without significant changes. However, Taylor’s frozen
hypothesis cannot accurately describe the complex space-time properties of
turbulence (He & Zhang, 2006; He et al., 2017). In this section, we will test
the capability of the newly proposed RWE method in estimating the space-time
properties of turbulence in the near-wall region.

The space-time correlation $R\left(r,\tau\right)$ is defined as the
correlation of the velocity fluctuation at $(x,t)$ and $(x+r,t+\tau)$, which
can be calculated by,

$R\left(r,\tau\right)=\frac{\left\langle
u^{\prime}(x,t)u^{\prime}(x+r,t+\tau)\right\rangle}{\sqrt{\left\langle
u^{\prime}(x,t)^{2}\right\rangle\left\langle
u^{\prime}(x+r,t+\tau)^{2}\right\rangle}}=\frac{\sum_{\boldsymbol{k}}S_{uu,\boldsymbol{k}}e^{{\rm
i}(k_{x}r-\omega\tau)}}{\sum_{\boldsymbol{k}}S_{uu,\boldsymbol{k}}},$ (18)

for stationary stochastic processes in the $x$ and temporal direction such as
the fully developed turbulent channel flow. With equation (18), the space-time
correlation can be estimated using the predicted spectra
$S_{uu,\boldsymbol{k}}$. The space-time correlations of the streamwise
velocity obtained from different methods are depicted in figure 16, where the
slopes of the black and red dashed lines respectively denote the local mean
streamwise velocity $\left\langle u\right\rangle$ and the convection velocity
$U_{c}$, as defined by

$U_{c}=\mathop{\rm{arg~{}max}}\limits_{U}\int_{-\infty}^{\infty}R\left(Ut,t\right)dt,$
(19)

following Choi & Moin (1990). From the DNS results in figure 16($a$-$c$), the
magnitude of the space-time correlation at a given spatio-temporal point
increases with the increasing $Re_{\tau}$, which is due to inner-flow scaling
(Kunkel & Marusic, 2006) for the flows in the near-wall region. In all three
cases, the red dashed line that denotes the convection velocity $U_{c}$ does
not overlap with the black one that denotes the local mean streamwise velocity
$\left\langle u\right\rangle$, where $U_{c}>\left\langle u\right\rangle$. The
deviations of $U_{c}$ and $\left\langle u\right\rangle$ in the near-wall
region are consistent with previous observations (Kim & Hussain, 1993; Del
Alamo & Jiménez, 2003). Since the convection velocity $U_{c}$ quantifies the
propagation of the coherent structures (Kim & Hussain, 1993; He et al., 2017),
the near-wall flow structures are propagated with a larger velocity than the
local streamwise velocity.

In the predicted results from the four considered methods, quite diverse
patterns are observed. In Case 180-40, the RWE and RBE results match well with
the DNS results in terms of the correlation magnitude. However, the convection
velocities predicted by the RWE and RBE are slightly lower than the DNS
results, as the slopes of the predicted red dashed lines are smaller than
those of the DNS. On the other hand, the space-time correlations predicted by
the $\lambda$-model and WBE deviate even larger from the DNS results. For the
$\lambda$-model, the regions where the space-time correlations are larger than
0.2 concentrate at the vicinity of the red dashed line corresponding to the
convection velocity, which conflicts with the turbulence nature (He et al.,
2017). The space-time correlation predicted by the WBE is smaller than that
from the DNS at a given spatio-temporal point. Moreover, the convection
velocities predicted by the $\lambda$-model and WBE appear to be smaller than
the local mean streamwise velocity, with relative errors larger than those of
the RWE and RBE results in magnitude.

When the reference height $y_{\rm R}$ increases from 40 to 200, the space-time
correlations predicted by RWE tend to become larger compared to the DNS
results, which indicates that the large-scale motions are overestimated as the
reference layer lifts up. The RBE-predicted results show a much larger
overestimation of the space-time correlations, with the dark red region where
the correlation magnitude is larger than 0.6 occupying the depicted area when
$y_{\rm R}^{+}=200$. Meanwhile, the convection velocity predicted by the RBE
is obviously larger than the DNS results in Case 950-200. As discussed in
figure 12, only the turbulent motions with large scales with
$\lambda_{x}/h\geq 2$ are estimated by the RBE at $y^{+}=10$ in Case 950-200,
which means that the space-time correlations predicted by the RBE are
dominated by the large-scale flow motions with high spatio-temporal coherence.
The $\lambda$-model-predicted correlations are large in magnitude in the
region close to the red dashed line that denotes the convection velocity,
which totally deviates from the patterns of the DNS results. The space-time
correlation distributions predicted by the WBE also appear to increase in
magnitude as the reference layer lifts up.

Figure 16: Space-time correlation of the streamwise velocity from the results
of DNS ($a,b,c$), RWE ($d,e,f$), $\lambda$-model($g,h,i$), RBE ($j,k,l$), and
WBE ($m,n,o$) at $y^{+}=10$ in Cases 180-40 ($a,d,g,j,m$), 550-100
($b,e,h,k,n$), and 950-200 ($c,f,i,l,o$). The slope of the black dashed lines
denotes the mean streamwise velocity at the prediction layers, while the red
dot-dashed lines denote the convection velocity calculated by equation (19).

From the above discussions on the space-time properties, it is noticed that
the predicted near-wall convection velocities are quite different among the
results from different methods. To further investigate the variations of the
predicted convection velocity with different Reynolds numbers and reference
heights, figure 17 shows convection velocity $U_{c}$ as a function of $y^{+}$
in all the 6 cases with $Re_{\tau}=180-950$ and the reference layer, ranging
from $y^{+}=40$ to $y/h=0.2$. The relative errors of the predicted convection
velocities for different methods along the height are shown in figure 18.
Distinct characteristics are observed in the convection velocities predicted
by different methods. For RBE, the convection velocity matches well with the
DNS results when $y_{\rm R}^{+}=40$ with all the three considered friction
Reynolds numbers that are equal to 180, 550, and 950. In Case 950-40, the RBE-
predicted convection velocity is lower than the DNS results near the wall by
about $14\%$, which is still the most accurate compared to the results from
other prediction methods. With the increase of the reference height, the near-
wall convection velocity by RBE increases and thus deviates from the DNS
results, which is attributed to the underestimated near-wall small-scale flows
with smaller convection velocities compared to those of the large-scale
motions. On the other hand, the WBE and $\lambda$-model underestimate the
convection velocity when $y_{\rm R}^{+}=40$, which should be induced by the
overestimated small-scale flows near the wall. With the increase of reference
height, the predicted convection velocities by $\lambda$-model and WBE
increase correspondingly, just like the phenomenon observed in the RBE
results. A nonphysical increase of $U_{c}$ when approaching the wall in the
WBE results is observed at the near-wall region in Cases 550-100 and 950-200,
making the WBE-predicted near-wall convection velocity the largest among all
the methods. The $\lambda$-model provides the most accurate predictions of the
convection velocity in Cases 550-100 and 950-200. In all the tested cases, the
RWE steadily provides fairly well results in predicting the convection
velocity. When $y_{\rm R}^{+}=40$ and 100, the relative errors between the RWE
and the DNS results are less than $15\%$. When $y_{\rm R}^{+}=200$, the RWE-
predicted convection velocity shows an enlarged deviation with respect to the
DNS results with the maximum error equal to $16.4\%$ near the wall, which is
still smaller than that in the WBE and RBE results.

Figure 17: Convection velocity as a function of $y^{+}$ for Cases 180-40
($a$), 550-40 ($b$), 550-100 ($c$), 950-40 ($d$), 950-100 ($e$) and 950-200
($f$). Figure 18: Relative error of the convection velocity as a function of
$y^{+}$ for Cases 180-40 ($a$), 550-40 ($b$), 550-100 ($c$), 950-40 ($d$),
950-100 ($e$) and 950-200 ($f$).

###  4.7 Instantaneous large-scale flow structure of the streamwise velocity
near the wall

According to the inner-outer interaction model (Marusic et al., 2010; Cheng &
Fu, 2022), the footprint of the large-scale motions on the lower region can be
calculated as

$u_{L}^{+}\left(x^{+},y^{+},z^{+}\right)=F_{x}^{-1}\left\\{H_{L}\left(\lambda_{x}^{+},y^{+}\right)F_{x}\left[u_{o}^{+}\left(x^{+},y_{o}^{+},z^{+}\right)\right]\right\\},$
(20)

where $u_{o}^{+}$ is the streamwise velocity fluctuation at $y_{o}^{+}$ in the
logarithmic region, and $F_{x}$ and $F_{x}^{-1}$ denote FFT and inverse FFT in
the streamwise direction, respectively. Gupta et al. (2021) extend the
transfer kernel $H_{L}$ to quantify the correlation between $u^{+}(y^{+})$ and
$u_{o}^{+}(y_{o}^{+})$ at a given two-dimensional length scale of
$(\lambda_{x}^{+},\lambda_{z}^{+})$ rather than just the streamwise scale
$\lambda_{x}^{+}$. The re-defined transfer kernel $H_{L}$ can be calculated as

$\displaystyle H_{L}\left(\lambda_{x}^{+},\lambda_{z}^{+},y^{+}\right)$
$\displaystyle=\frac{\left\langle\hat{u}\left(\lambda_{x}^{+},\lambda_{z}^{+},y^{+},z^{+}\right)\overline{\hat{u}}\left(\lambda_{x}^{+},\lambda_{z}^{+},y_{0}^{+},z^{+}\right)\right\rangle}{\left\langle\hat{u}\left(\lambda_{x}^{+},\lambda_{z}^{+},y_{0}^{+},z^{+}\right)\hat{u}\left(\lambda_{x}^{+},\lambda_{z}^{+},y_{0}^{+},z^{+}\right)\right\rangle}=\frac{S_{uu,yy_{o}}(k_{x},k_{z})}{S_{uu,y_{o}y_{o}}(k_{x},k_{z})}$
(21)
$\displaystyle=\frac{\sum_{\omega}S_{uu,yy_{o}}(k_{x},k_{z},\omega)}{\sum_{\omega}S_{uu,y_{o}y_{o}}(k_{x},k_{z},\omega)}.$

Since all the tested methods can directly predict the CSD tensor of the
velocity from the measurements, the transfer kernel $H_{L}$ can be estimated
from the auto-spectra coefficient $S_{uu,y_{o}y_{o}}(k_{x},k_{z},\omega)$ and
the cross-spectra coefficient $S_{uu,yy_{o}}(k_{x},k_{z},\omega)$ from the
estimated CSD tensor. In Gupta et al. (2021), the large-scale fluctuations are
estimated with the linear model as in equation (20). Following Gupta et al.
(2021), we will also conduct the estimation of large-scale structures using
the transfer function estimated from the tested methods. Case 950-200 where
the reference layer is the farthest away from the wall among all the tested
cases will be used to investigate the estimated footprint at two prediction
heights of $y^{+}=100$ and 10 that correspond to the logarithmic region and
the near-wall region, respectively.

Figure 19 shows the estimated large-scale structures of the streamwise
velocity fluctuation at $y^{+}=100$. From the instantaneous flow field shown
in the left column, the RWE and RBE predictions well reflect the
characteristics of the instantaneous flow structures. The good performance of
RWE and RBE can also be seen in the spectral space, where the relative errors
are smaller compared to those of the $\lambda$-model and WBE. Specifically,
the relative errors in the RWE results are smaller than those in the RBE
results, with a maximum error of $0.13S_{{\rm DNS,max}}$ in the premultiplied
energy spectra. On the other hand, the WBE and $\lambda$-model overestimate
and underestimate the fluctuation energy at $y^{+}=100$, respectively.

Figure 19: Comparisons of the predicted footprint of the large scale motions
of $y_{\rm R}^{+}=200$ to the prediction layer of $y^{+}=100$ in Case 950-200
from DNS ($a,b,c$), RWE ($d,e,f$), W-model ($g,h,i$), RBE ($j,k,l$), and WBE
($m,n,o$). Panels ($a,d,g,j,m$) are the instantaneous flow field,
($b,e,h,k,n$) are the energy spectra, and ($c,f,i,l,o$) are the relative error
normalized by the maximum spectral energy in the DNS results. The values of
fluctuation velocity shown in ($a,d,g,j,m$) are normalized by the mean
velocity at $y=h$. The value of spectral energy and relative error shown in
($b,e,h,k,n$) and ($c,f,i,l,o$) are normalized by $S_{{\rm DNS,max}}$,
respectively.

As the prediction layer approaches the wall, the magnitude of relative errors
between the predictions from all the tested methods becomes larger, as in
figure 20. For the RWE methods, the largest magnitude of error equal to
$0.26S_{{\rm DNS,max}}$ appears at $y^{+}=10$ for the large-scale motions with
$(\lambda_{x}/h,\lambda_{z}/h)=(6.2,1.6)$, where the predicted energy is lower
than the DNS result. Meanwhile, another extreme point for error from RWE is
found at $(\lambda_{x}/h,\lambda_{z}/h)=(2.5,0.6)$ with the positive value
equal to $0.17S_{{\rm DNS,max}}$. The relative errors in the RBE predictions
reach the extreme points at $(\lambda_{x}/h,\lambda_{z}/h)=(6.2,1.6)$ and
$(2.5,0.4)$. At the extreme points, the errors from RBE are equal to
$-0.34S_{{\rm DNS,max}}$ and $0.32S_{{\rm DNS,max}}$, both of which are larger
than those from RWE at the same points. The $\lambda$-model-predicted energy
is lower than the DNS for the scales with $\lambda_{x}/h\geq 6$, with the
relative error larger than $0.5S_{{\rm DNS,max}}$, which indicates that the
fluctuation energy predicted by the $\lambda$-model is much overestimated in
the near-wall region. As for the WBE, the relative error is larger than
$0.3S_{{\rm DNS,max}}$ when $\lambda_{x}/h\geq 1.8$ and
$0.4\leq\lambda_{z}/h\leq 0.8$. Considering the above comparisons, the RWE
performs better than the other tested methods, with the smallest relative
error in the estimated energy spectra at both the logarithmic region and the
near-wall region.

In Gupta et al. (2021), the $\lambda$-model and WBE (namely B-model) are also
used to predict the large-scale motions in the near-wall region, with the
assumption that the input forcing is white in time, which leads to the same
conclusion that the energy predicted by the $\lambda$-model is lower than that
from the WBE. However, when applied in time-resolved cases as in this study,
the energies predicted by the $\lambda$-model and WBE both tend to decrease
compared to those using the white-in-time forcing in Gupta et al. (2021),
which causes the obvious underestimation of energy by the $\lambda$-model. The
deviations between the results generated by the forcing with different
temporal properties indicate that the $\lambda$-model could be improved by
considering the effect of temporal scales when applied in the time-resolved
cases. On the other hand, the validity of the RWE-kind method for estimating
large-scale motions in cases where only the spatial data are available should
be further tested.

Figure 20: Same as figure 19, but with the prediction layer at $y^{+}=10$.

From the instantaneous flow field for large-scale motions as discussed above,
the structures are observed to delay backward as the prediction layer
approaches the wall. The spatial delay of the flow structure is quantified by
the SIA (Marusic & Heuer, 2007), which is defined as,

$\theta={\rm{arctan}}\left[(y/({\Delta}x^{\ast})\right],$ (22)

where ${\Delta}x^{\ast}=\mathop{\rm{arg~{}max}}\limits_{{\Delta}x}R_{\tau
u}\left(\Delta x\right)$, and

$R_{\tau u}\left(\Delta x\right)=\frac{\left<\tau\left(t\right)u\left(t+\Delta
x/U\right)\right>}{\sqrt{\left<\tau^{2}\right>\left<u^{2}\right>}}=\frac{\sum_{\boldsymbol{k}}S_{\tau
u,\boldsymbol{k}}e^{{\rm i}k_{x}\Delta x}}{\sigma_{\tau}\sigma_{u}}$ (23)

is the correlation function between the wall shear stress and the streamwise
velocity fluctuation with a spatial delay of $\Delta x$, and $S_{\tau
u,\boldsymbol{k}}$ is the cross-spectra between the wall shear and the
velocity fluctuations at ${\boldsymbol{k}}$. To further investigate the
ability of the tested methods in predicting the spatial distribution of the
flow structures, the SIAs from the predictions are analyzed in the following.
Since the logarithmic region does not exist clearly for the channel flow with
$Re_{\tau}=180$, only the Cases 550-100, 950-100, and 950-200 will be
discussed.

Figure 21 depicts the SIAs in the tested cases based on the reference
positions located inside the logarithmic region from $y^{+}=100$ to $y/h=0.2$
(Jiménez, 2018). According to the DNS results, the inclination angles keep
almost constant around $15^{\circ}$ at reference positions with different
heights, which is consistent with the results in Marusic & Heuer (2007). In
Case 550-100, all the tested methods fairly predict the inclination angle of
around $15^{\circ}$ except for the $\lambda$-model. In fact, the
$\lambda$-model-predicted SIAs are always around $10^{\circ}$ in all the three
cases shown in figure 21. In Case 950-100, the RBE-predicted SIA is larger
than the DNS result by about $3^{\circ}$ when $y^{+}=100$. As the height lifts
up, the SIA predicted by RBE reaches the maximum value of about $19^{\circ}$
when $y_{\rm R}^{+}=125$, then gradually decreases and finally overlaps with
the DNS results when $y_{\rm R}^{+}=200$. The RWE and WBE perform well in Case
950-100, with the root-mean-squared error compared to the DNS result equal to
$0.96^{\circ}$ and $1.55^{\circ}$, respectively. When the reference layer
moves to $y^{+}=200$, as in figure 21$(c)$, the RBE-predicted SIA is accurate
when $y^{+}\geq 175$. However, when the horizontal layers approach the wall,
the SIA predicted by the RBE decreases, which becomes around $12^{\circ}$ when
$y^{+}=100$. On the other hand, the RWE has the highest accuracy, with the
root-mean-squared error equal to $0.81^{\circ}$. Meanwhile, the WBE-predicted
results are also close to the DNS results, whose root-mean-squared error is
$1.14^{\circ}$.

Based on the above analyses of the flow structures predicted by the tested
methods, the RWE has the best performance in estimating both spectral energy
distributions and the structural inclination angles of the large-scale motions
of the turbulent channel flow. Meanwhile, the WBE also performs well in
predicting the SIAs, indicating that the relative distribution of the coherent
structures can be fairly estimated by the WBE which just utilizes the white-
noise-assumed forcing.

Figure 21: Structure inclination angles of the predicted fluctuation velocity
in ($a$) Case 550-100, ($b$) Case 950-100 and ($c$) Case 950-200.

##  5 Conclusions

In this study, a new method for predicting the turbulent channel flow is
proposed, which is denoted as RWE. The RWE modifies the spatially uniform and
uncorrelated forcing profile based on the relative value of response energy in
the near-wall region estimated by the RBE (Towne et al., 2020). Comprehensive
validations are made to investigate the ability of RWE to predict the spectra
of the incompressible turbulent channel flow. Time-resolved DNS datasets with
$Re_{\tau}=180$, 550, and 950 are used to provide limited layers of reference
measurements for the predictions. The DNS results are further used as a
standard to evaluate the accuracy of different prediction methods, including
the RWE, $\lambda$-model (Gupta et al., 2021), RBE, and the method using the
WBE. To test the sensitivity of the prediction methods on the locations of
measurements and the friction Reynolds numbers, several reference layers
ranging from $y^{+}=40$ at the near-wall region to $y/h\approx 0.2$ at the
upper bound of the logarithmic region are set in cases with $Re_{\tau}=180$,
550 and 950, resulting in a total number of 6 cases for the validations.

The root-mean-squared (RMS) profiles of the velocity fluctuations are first
tested for all 6 cases. The RMS profiles predicted by the RWE are pretty
consistent with the DNS results. Specifically, the relative deviations of the
streamwise RMS peaks between the RWE and DNS results are less than $11\%$ in
all the tested cases, indicating that this newly proposed method is robust in
the friction Reynolds numbers and the measurement locations. On the other
hand, the performances of the RBE and $\lambda$-model show a dependence on the
specific measurement locations. The prediction accuracy of RBE is close to
that of RWE when the reference layer is located at $y^{+}=40$ that is in the
vicinity of the wall. With the increase of the reference height, the accuracy
of RBE near the wall deteriorates rapidly. The near-wall energy predicted by
the $\lambda$-model also decreases when the reference layer lifts. The
$\lambda$-model reaches its best performance with the reference layer at
$y^{+}=100$, and overestimates and underestimates the near-wall energy when
$y^{+}=40$ and 200, respectively. In all the tested cases, the WBE largely
overestimates the near-wall energy, indicating that the WBE is not suitable to
be directly used for the estimations of near-wall energy magnitude.

The prediction capability of the considered methods is further investigated by
examining the energy spectra. The RBE continuously shows a good prediction
capacity at specific reference layers at $y^{+}=40$. Meanwhile, it is observed
that the relative spectra predicted by the WBE can fairly reflect the energy
distribution of the DNS results. Compared with the other tested methods, the
RWE better predicts the energy distribution of velocity fluctuations at
different scales along the heights ranging from the wall to the reference
layer. On the other hand, compared to the DNS results, the predicted energy of
all the considered methods tends to decrease as the flow scale decreases,
while the RWE keeps the minimum relative error.

The space-time properties of the predicted flow field are also investigated,
including the magnitude of the space-time correlation and the convection
velocity. In Case 950-200, the magnitude of space-time correlations is
overestimated by all the methods, while the RWE and WBE provide the closest
results compared to the DNS. On the other hand, the RBE obviously
overestimates the correlation magnitude, especially at the near-wall region
$y^{+}=10$ due to the underestimation of the small-scale motions. The
convection velocity of the coherent structures is also studied. The RBE and
RWE results match well with the DNS when the reference layer is located at
$y^{+}=40$ that is close to the wall. However, the predicted convection
velocity in the near-wall region by the RBE and RWE increases with the
increasing height of the reference layer, which reaches the maximum error when
$y^{+}=200$. In Case 950-200, the RWE-predicted convection velocity is larger
than the DNS result by $16.4\%$ at the near-wall region, which is smaller than
the WBE and RBE.

The instantaneous flow field near the wall can be predicted based on the
estimated CSD sensor and the reference velocity signal in the logarithmic
region. The large-scale structures are well predicted by the RWE and RBE. On
the other hand, the $\lambda$-model and WBE underestimate and overestimate the
energies of the near-wall large-scale motions, respectively. The SIAs from the
tested methods are researched to further investigate the characteristics of
the predicted flow structures. The WBE and RWE perform well in predicting the
SIAs in Cases 550-100 and 950-200 when compared with the DNS results, i.e., at
around $15^{\circ}$.

From the above discussions, the strategy to modify the forcing profile
informed by the RBE shows a strong potential in predicting the turbulent
statistics of incompressible channel flows. Meanwhile, there are some
discussions on the application scope of the current RWE method and its future
extensions. (1) In this study, we focus on estimating the near-wall statistics
based on the measurements in the higher region, especially for the cases where
the reference layer is located in the logarithmic region. A related problem
could be the estimation of the flow patterns in the logarithmic region based
on the wall measurements, which is explored by Amaral et al. (2021) using the
RBE and Guastoni et al. (2021) using the CNN. Due to the restrictions of
$y_{\rm Q}\leq y_{\rm R}$ when determining the quasi-reference layer, the
current RWE will degenerate to WBE when $y_{\rm R}\rightarrow 0$. To
effectively make use of the wall measurements, the way to determine $y_{\rm
Q}$ in the current RWE should be modified, which can be explored in future
studies. (2) The current RWE approach only uses one type of observation at one
reference layer. It is clear that the prediction accuracy could be further
improved with different measurements at multiple reference layers with
suitable modeling approaches. Such a topic is to be explored in the future
work. (3) The method used in this study relies on the availability of time-
resolved measurement data. The RWE could be more practical if it can predict
the flow with the measurement data that is sparse in time, which might be
realized using the assumptions of the stochastic forcing as used by (Hwang &
Cossu, 2010; Gupta et al., 2021). (4) Possible extensions of the RWE to other
types of turbulent flows, such as the pipe flows, boundary layers, and jet
flows, could be explored in the future. For wall-bounded turbulence, the basic
steps of the RWE can be directly applied. According to Monty et al. (2009) and
Lee & Sung (2013), the differences between the channel/pipe flows and boundary
layers are obvious in the very-large-scale motions, which are energetic in the
outer region. In the inner region where the RWE is applied, the differences in
flow structures are not as obvious as those in the outer region. Thus, the RWE
is believed to be valid in estimating the wall-bounded turbulence, including
the turbulent pipe flows and boundary layers. For jets, the approach of RWE
developed in this study could not be directly applied, since the flow pattern
is different from that of the wall-bounded turbulence. In Schmidt et al.
(2018), the inclusion of the eddy viscosity has been demonstrated to largely
improve the agreement with the resolvent modes and the SPOD modes, which
implies the potential of linear estimation of jets in the future. However, to
our knowledge, the applicability of the resolvent-based estimation for jets
from limited measurements is not fully investigated. Future studies are needed
to explore the applications of the RWE-like methods in estimating the jets.
(5) The validity of the RWE in predicting the compressible or stratified
turbulence has not been tested, which should be comprehensively investigated
in the future.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

The data that support the findings of this study are available on request from
the corresponding author, LF.

## Acknowledgments

Lin Fu acknowledges the fund from the Research Grants Council (RGC) of the
Government of Hong Kong Special Administrative Region (HKSAR) with RGC/ECS
Project (No. 26200222) and RGC/GRF Project (No. 16201023), the fund from CORE
as a joint research center for ocean research between QNLM and HKUST, and the
fund from the Project of Hetao Shenzhen-Hong Kong Science and Technology
Innovation Cooperation Zone (No. HZQB-KCZYB-2020083).

##  Appendix A Linearized Navier-Stokes operator

Recall the discretized state-space form of the linearized Navier-Stokes
equations as &M qks(t)t = A_k_s q_k_s (t) + Bf_k_s (t),
m_k_s (t) = Cq_k_s (t) + n_k_s (t). The operators
$\mathcal{M}\in\mathbb{C}^{4N\times 4N}$, $\mathcal{A}_{\boldsymbol{k}_{\rm
s}}\in\mathbb{C}^{4N\times 4N}$, $\mathcal{B}\in\mathbb{C}^{4N\times 3N}$ in
equation (A$a$) are expressed as

$\displaystyle\mathcal{M}=\left[\begin{array}[]{cccc}\mathcal{I}_{N}&0&0&0\\\
0&\mathcal{I}_{N}&0&0\\\ 0&0&\mathcal{I}_{N}&0\\\ 0&0&0&0\\\
\end{array}\right],{~{}~{}}\mathcal{A}_{\boldsymbol{k}_{\rm
s}}=\left[\begin{array}[]{cccc}\mathcal{L}_{\boldsymbol{k}_{\rm
s}}&-\bnabla^{\rm T}\\\ \bnabla&0\\\
\end{array}\right],{~{}~{}}\mathcal{B}=\left[\begin{array}[]{cccc}\mathcal{I}_{N}&0&0\\\
0&\mathcal{I}_{N}&0\\\ 0&0&\mathcal{I}_{N}\\\ 0&0&0\\\ \end{array}\right].$
(24)

The spatial linearized Navier-Stokes operator
$\mathcal{L}_{\boldsymbol{k}_{\rm s}}\in\mathbb{C}^{3N\times 3N}$ in equation
(24) for the momentum equations is calculated as,

$\mathcal{L}_{\boldsymbol{k}_{\rm s}}=L_{0}+{\rm
i}{k_{x}}{L_{x}}+{L_{y}}\mathscr{D}+{\rm
i}{k_{z}}{L_{z}}+{L_{2}}(-k_{x}^{2}+\mathscr{D}^{2}-k_{z}^{2}),$ (25)

with

$\displaystyle
L_{x}=\left[\begin{array}[]{ccc}\overline{\mathscr{u}}&-\mathscr{v}_{T}^{\prime}&0\\\
0&\overline{\mathscr{u}}&0\\\ 0&0&\overline{\mathscr{u}}\\\
\end{array}\right],{~{}~{}}L_{y}=\left[\begin{array}[]{ccc}-\mathscr{v}_{T}^{\prime}&0&0\\\
0&-2\mathscr{v}_{T}^{\prime}&0\\\ 0&0&-\mathscr{v}_{T}^{\prime}\\\
\end{array}\right],{~{}~{}}L_{z}=\left[\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
-\mathscr{v}_{T}^{\prime}&0&0\\\ \end{array}\right],$ (26) $\displaystyle
L_{0}=\left[\begin{array}[]{ccc}0&\mathscr{u}^{\prime}&0\\\ 0&0&0\\\ 0&0&0\\\
\end{array}\right],{~{}~{}}L_{2}=\left[\begin{array}[]{ccc}-\mathscr{v}_{T}&0&0\\\
0&-\mathscr{v}_{T}&0\\\ 0&0&-\mathscr{v}_{T}\\\ \end{array}\right],$

where $\overline{\mathscr{u}}={\rm
diag}\left[\overline{\boldsymbol{u}}\right]$, $\overline{\boldsymbol{u}}$ is
the mean streamwise velocity profile, $\mathscr{v}_{T}=\frac{1}{Re_{\tau}}{\rm
diag}\left[\frac{1}{\nu}\nu_{T}\right]$,
$\mathscr{v}_{T}^{\prime}=\frac{1}{Re_{\tau}}{\rm
diag}\left[\frac{1}{\nu}\nu_{T}^{\prime}\right]$, $\mathscr{D}$ and ′ denote
${\p}/{\p y}$. The Hamiltonian operator $\bnabla\in\mathbb{C}^{N\times 3N}$ in
equation (4) is calculated as,

$\bnabla=\left[\begin{array}[]{ccc}{\rm
i}{k_{x}}\mathcal{I}_{N}&\mathscr{D}&{\rm i}{k_{z}}\mathcal{I}_{N}\\\
\end{array}\right].$ (27)

The observation matrix $\mathcal{C}$ in equation (A$b$) is determined by the
location of the measured variable. In this study, the state matrix is
assembled as

$\boldsymbol{q}_{\boldsymbol{k}_{\rm
s}}(t)=\left[\boldsymbol{u}^{\ast}_{\boldsymbol{k}_{\rm
s}}(t),p^{\ast}_{\boldsymbol{k}_{\rm
s}}(t)\right]^{\ast}=\left[u^{\ast}_{\boldsymbol{k}_{\rm
s}}(t),v^{\ast}_{\boldsymbol{k}_{\rm s}}(t),w^{\ast}_{\boldsymbol{k}_{\rm
s}}(t),p^{\ast}_{\boldsymbol{k}_{\rm s}}(t)\right]^{\ast}.$ (28)

Denoting $\left(u_{1},u_{2},u_{3}\right)=\left(u,v,w\right)$, when the
measured variable is the velocity $u_{i}~{}(i=1,2,3)$ at the $n$-th node, then
the observation matrix $\mathcal{C}\in\mathbb{C}^{1\times 4N}$ should be
expressed as

$\mathcal{C}=\left[\mathcal{O}_{1\times\left[{(i-1)N+n-1}\right]}~{}~{}1~{}~{}\mathcal{O}_{1\times\left[{4N-(i-1)N-n}\right]}\right].$
(29)

Note that, in other cases, such as when the state matrix is assembled in
another approach or the measurements are located at multiple wall-normal
locations, the expression of the measurement operator should be redetermined
in an analogous manner. For instance, in equation (12), the resolvent operator
$\mathcal{R}_{\boldsymbol{k}}$ is utilized, which excludes the rows
corresponding to the pressure state. In this case, when the measured variable
is the velocity $u_{i}~{}(i=1,2,3)$ at the $n$-th node, the observation matrix
as denoted by $\mathscr{C}$ should be calculated by

$\mathscr{C}=\left[\mathcal{O}_{1\times\left[{(i-1)N+n-1}\right]}~{}~{}1~{}~{}\mathcal{O}_{1\times\left[{3N-(i-1)N-n}\right]}\right].$
(30)

##  Appendix B Assessment of the DNS dataset generated in this study

In figure 22, the DNS dataset generated in this study is compared with the
open source DNS database (Del Alamo & Jiménez, 2003; Hoyas & Jiménez, 2008)
for the mean streamwise velocity and the RMS velocities. Very small
differences are found between the two datasets, which demonstrates that the
DNS data generated in this study are reliable.

Figure 22: Comparisons of the mean and RMS profiles between the open-source
DNS results (Del Alamo & Jiménez, 2003; Hoyas & Jiménez, 2008) and those
generated in this study.

##  Appendix C Discussions on the determination of quasi-reference layer

The LCS ($\gamma^{2}$) used to determine the quasi-reference layer $y_{\rm Q}$
is estimated using the WBE in each spatio-temporal scale. This quantity is
obtained by firstly estimating the CSD tensor with the white-noise-assumed
forcing through equation (9), i.e.,

$S_{\boldsymbol{uu},\boldsymbol{k}}=\mathcal{R}_{\boldsymbol{k}}S_{\boldsymbol{ff},\boldsymbol{k}}\mathcal{R}_{\boldsymbol{k}}^{\ast}=E_{\boldsymbol{k}}\cdot\mathcal{R}_{\boldsymbol{k}}\mathcal{R}_{\boldsymbol{k}}^{\ast}.$
(31)

From the definition of $\gamma^{2}$ in equation (16), and considering equation
(31), $\gamma^{2}$ is calculated by,

$\displaystyle\gamma^{2}(\boldsymbol{k})$
$\displaystyle=\frac{\left|\left\langle\hat{u}(y_{\rm
Q}^{+})\overline{\hat{u}(y_{\rm
P}^{+})}\right\rangle\right|^{2}}{\left\langle\left|\hat{u}(y_{\rm
Q}^{+})\right|^{2}\right\rangle\left\langle\left|\hat{u}(y_{\rm
P}^{+})\right|^{2}\right\rangle}$ (32)
$\displaystyle=\frac{S_{\boldsymbol{uu},\boldsymbol{k}}^{2}\left(y_{\rm
Q}^{+},y_{\rm P}^{+}\right)}{S_{\boldsymbol{uu},\boldsymbol{k}}\left(y_{\rm
Q}^{+},y_{\rm Q}^{+}\right)S_{\boldsymbol{uu},\boldsymbol{k}}\left(y_{\rm
P}^{+},y_{\rm P}^{+}\right)}$ $\displaystyle=\frac{\left|\mathscr{C}(u,y_{\rm
Q}^{+})\mathcal{R}_{\boldsymbol{k}}\mathcal{R}_{\boldsymbol{k}}^{\ast}\mathscr{C}^{\rm
T}(u,y_{\rm P}^{+})\right|^{2}}{\left[{\mathscr{C}(u,y_{\rm
Q}^{+})\mathcal{R}_{\boldsymbol{k}}\mathcal{R}_{\boldsymbol{k}}^{\ast}\mathscr{C}^{\rm
T}(u,y_{\rm Q}^{+})}\right]\cdot\left[{\mathscr{C}(u,y_{\rm
P}^{+})\mathcal{R}_{\boldsymbol{k}}\mathcal{R}_{\boldsymbol{k}}^{\ast}\mathscr{C}^{\rm
T}(u,y_{\rm P}^{+})}\right]},$

where the expressions of $\mathcal{R}_{\boldsymbol{k}}$ and
$\mathscr{C}(u,y^{+})$ can be found in equations (4-5) and Appendix A,
respectively.

Since the LCS calculated in equation (32) is determined separately in each
scale $\boldsymbol{k}$, the height of the quasi-reference layer calculated
from the LCS is also determined scale-by-scale. When the threshold LCS is 0.3
as adopted in this study, the value of $y_{\rm Q}^{+}$ at the wavelength
$(\lambda_{x},\lambda_{z})=(2\pi/k_{x},2\pi/k_{z})$ with
$\omega=10u_{\tau}/k_{x}$ in Cases 180-40, 550-100, and 950-200 are depicted
in figure 23. It can be observed that the height of the quasi-reference layer
tends to increase when the flow scale is larger.

Figure 23: Distributions of the quasi-reference layer $y_{\rm Q}^{+}$ at
different wavelengths with $\omega=10u_{\tau}/k_{x}$ in Cases 180-40, 550-100,
and 950-200.

In this study, we use the threshold LCS value (denoted as ${\rm LCS}_{\rm t}$)
to determine the height of the quasi-reference layer, which has an important
impact on the estimation result of the RWE. To efficiently modify the forcing
profile, an ideal choice of the quasi-reference layer should be as high as
possible under the premise of keeping the RBE valid for estimating the near-
wall statistics. To test the impact of ${\rm LCS}_{\rm t}$ on the estimation
accuracy, the RWEs with five values of ${\rm LCS}_{\rm t}$ equal to 0.1, 0.2,
0.3, 0.4, and 0.5 are set to estimate the streamwise RMS profiles in Cases
950-40, 950-100, and 950-200. To quantify the relative estimation error in the
tested cases, the integrated relative error of the streamwise RMS profile is
defined as

$\epsilon=\left[\frac{\int_{0}^{y_{\rm R}}\left(u_{\rm RMS,RWE}-u_{\rm
RMS,DNS}\right)^{2}{\rm d}y}{\int_{0}^{y_{\rm R}}\left(u_{\rm
RMS,DNS}\right)^{2}{\rm d}y}\right]^{0.5}.$ (33)

The estimation errors are summarized in figure 24. A universal law between the
relative error and ${\rm LCS_{t}}$ clearly exists in all the tested cases when
${\rm LCS_{t}}\geq 0.2$. Although the tested cases with ${\rm LCS_{t}}=0.2$
appear to have the minimum errors, we notice that there are sharp increases of
the relative error when ${\rm LCS_{t}}$ decreases from 0.2 to 0.1 in Cases
950-100 and 950-200. Such increases in the estimation error are due to the
underestimation of the fluctuation energy in the near-wall region when ${\rm
LCS_{t}}$ is too low, in which case $y_{Q}$ is too high so that the RBE
becomes invalid to predict the near-wall statistics. To ensure the estimation
accuracy in higher friction Reynolds numbers beyond 950, it is safer to
conservatively set the value of ${\rm LCS_{t}}$ to be 0.3 to avoid the
possible sharp increase of estimation error at ${\rm LCS_{t}}=0.2$. Thus, the
threshold value of ${\rm LCS_{t}}=0.3$ is adopted as a default setting for all
the cases with RWE in this study.

Figure 24: Impact of the threshold LCS on the estimation error of the
streamwise RMS profile.

##  Appendix D Relationship between the relative energy profiles of forcing
and response

In this study, the relative forcing energy is modified between the wall and
the quasi-reference layer $y_{\rm Q}$, which is set to be unity upon $y_{\rm
Q}$ then. The range of the relative forcing energy to be modified is the same
as the range of the target response estimated from RBE, both of which are
below the quasi-reference layer. In the optimization problem of this study,
the relationship between the forcing and response energy below the quasi-
reference layer is expressed by

$\hat{\mathcal{S}}_{\boldsymbol{k}}=\frac{{\mathscr{R}_{\boldsymbol{k},1}\cdot\hat{\mathcal{W}}_{\boldsymbol{k}}}+{\mathscr{R}_{\boldsymbol{k},2}\cdot\hat{\mathcal{U}}}}{{\mathscr{R}_{\boldsymbol{k},{N_{Q}},1}\cdot\hat{\mathcal{W}}_{\boldsymbol{k}}}+{\mathscr{R}_{\boldsymbol{k},{N_{Q}},2}\cdot\hat{\mathcal{U}}}},$
(34)

where $\hat{\mathcal{S}}_{\boldsymbol{k}}=\frac{{\rm
diag}\left[S_{{u_{i}u_{i}},\boldsymbol{k}}\right]|_{y<y_{\rm Q}}}{{\rm
diag}\left[S_{{u_{i}u_{i}},\boldsymbol{k}}\right]|_{y_{\rm
Q}}}\in\mathbb{C}^{N_{\rm Q}-1}$ is the relative energy profile below the
quasi-reference layer. The relative forcing energy profile
$W_{\boldsymbol{k}}(y)$ is split in equation (34) as
$W_{\boldsymbol{k}}=[\hat{\mathcal{W}}_{\boldsymbol{k}}^{\rm
T},\hat{\mathcal{U}}^{\rm T}]^{\rm T}$, where
$\hat{\mathcal{W}}_{\boldsymbol{k}}\in\mathbb{C}^{N_{\rm Q}-1}$ is the
relative forcing energy below $y_{\rm Q}$ that is to be determined,
$\hat{\mathcal{U}}=[1,...,1]_{N-N_{\rm Q}+1}$ is that beyond $y_{\rm Q}$. The
linear operators in equation (34) are expressed by
$\mathscr{R}_{\boldsymbol{k},1}=\mathcal{C}_{1}\mathscr{R}_{\boldsymbol{k}}\mathcal{C}_{1}^{\rm
T}$,
$\mathscr{R}_{\boldsymbol{k},2}=\mathcal{C}_{1}\mathscr{R}_{\boldsymbol{k}}\mathcal{C}_{2}^{\rm
T}$,
$\mathscr{R}_{\boldsymbol{k},{N_{Q}},1}=\mathcal{C}_{N_{Q}}\mathscr{R}_{\boldsymbol{k}}\mathcal{C}_{1}^{\rm
T}$,
$\mathscr{R}_{\boldsymbol{k},{N_{Q}},2}=\mathcal{C}_{N_{Q}}\mathscr{R}_{\boldsymbol{k}}\mathcal{C}_{2}^{\rm
T}$, and

$\displaystyle\mathcal{C}_{1}=\left[\begin{array}[]{cc}\mathcal{I}_{(N_{Q}-1),(N_{Q}-1)}&\mathcal{O}_{(N_{Q}-1),(N-N_{Q}+1)}\\\
\end{array}\right]\in\mathbb{C}^{(N_{Q}-1)\times N},$ (35)
$\displaystyle\mathcal{C}_{2}=\left[\begin{array}[]{cc}\mathcal{O}_{(N-N_{Q}+1),(N_{Q}-1)}&\mathcal{I}_{(N-N_{Q}+1),(N-N_{Q}+1)}\\\
\end{array}\right]\in\mathbb{C}^{(N-N_{Q}+1)\times N},$
$\displaystyle\mathcal{C}_{N_{Q}}=\left[\begin{array}[]{ccc}\mathcal{O}_{1,(N_{Q}-1)}&1&\mathcal{O}_{1,(N-N_{Q})}\\\
\end{array}\right]\in\mathbb{C}^{1\times N},$
$\displaystyle\mathscr{R}_{\boldsymbol{k}}=\sum_{j=1}^{3}\mathcal{R}_{\boldsymbol{k},i,j}\circ\overline{\mathcal{R}_{\boldsymbol{k},i,j}}\in\mathbb{C}^{N\times
N},$

where $\mathcal{R}_{\boldsymbol{k},i,j}$ denotes the sub-matrix of the
resolvent operator $\mathcal{R}_{\boldsymbol{k}}$ by selecting its rows for
$u_{i}$ and columns for the forcing in $x_{j}$ direction, $\circ$ denotes the
element-wise matrix multiplication, and $N_{\rm Q}$ is the node number between
$y_{\rm Q}$ and the wall.

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