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

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 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 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 ($y/h \approx 0.2$, where h is the half-channel height) 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 scale-dependent model ($\lambda$-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 around $15^\circ$ are predicted by the RWE and WBE, which generally recover the DNS results.


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. (2021Fu et al. ( , 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 (Hoepffner 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.

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 (-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.
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Mathematical description of the resolvent analysis
The incompressible Navier-Stokes equations are given by (2.1a) where   =   ℎ  is the friction Reynolds number,   is the friction velocity, ℎ is the half-channel height,  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  that contains the nonlinear interactions of velocity fluctuations while excluding the eddy-viscosity term is defined as where the superscript ′ denotes the fluctuation variable.Here, the eddy viscosity   is calculated from the semi-empirical expression by Cess (1958) and reported by Reynolds & Hussain (1972) as where the constants  = 0.426 and  = 25.4.By rearranging equation (2.1), the linearized Navier-Stokes equations hold, where  is the mean velocity, and   =   +  is the total viscosity.Note that equations (2.4) are equivalent to the incompressble Navier-Stokes equations (2.1).
From equations (2.2) and (2.4), 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  = − ′ • ∇ ′ .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 1 In the following sections, the tested prediction methods actually provide different approaches to model the re-defined forcing in (2.2).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  s are obtained by taking the Fourier transformation to equations (2.4) in the uniform spatial directions.For instance, in the fully developed turbulent channel flow, the Fourier transformation is taken in the streamwise () and spanwise () directions.The linearized equations (2.4) at  s = (  ,   ) can be therefore written in a discretized state-space form with  points in the wall-normal direction as where i = √ −1.From equation (2.6), the velocity û and pressure p are regarded as the response of the input forcing f  through the linear operator H  .Since we will focus on predicting the velocity field in this study, the linearized equation (2.7) can be reduced to the following form, û = R  • f  , (2.8) where R  is obtained from H  by deleting the rows corresponding to p at the left-hand side of equation (2.7) and columns corresponding to the constant 0 at the right-hand side.A typical application of equation (2.8) is to approximate the coherent structures of turbulence using the resolvent modes, which are obtained by taking the singular value decomposition to R  , i.e., û where ∞ =1 ψ,   ,  φ *

𝒌, 𝑗
= R  is the result of singular value decomposition of the resolvent operator R  , the response mode ψ,  and forcing mode φ,  are ordered by their singular value  ,  , the expansion coefficient  ,  = φ * ,  • f  is the projection of forcing on the −th resolvent forcing mode, and the superscript * denotes the Hermitian transpose.
Using equation (2.8), the CSD tensors can be calculated as, where ⟨•⟩ denotes the ensemble average.As in equation (2.10), the CSD tensor of the response is fully determined when the forcing CSD    , 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    , , after which the response  , can be directly derived from    , via equation (2.10).In the following, the typical methods that make use of the resolvent analysis to predict the turbulence field will be introduced.

The white-noise-based estimation
With the simplest assumption, the portion of forcing that excludes the eddy viscosity terms, as defined in equation (2.2), 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  can be expressed as, where   , as the energy of forcing, keeps constant at each node and in each direction.Substituting equation (2.11) into (2.10) and considering the resolvent modes in equation (2.9), it can be deduced that, (2.12) If we define ( ψ,  ψ * ,  ) as the CSD of the response at the -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  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.

The wall-distance-dependent and scale-dependent models
To improve the prediction accuracy of the white forcing, Gupta et al. (2021) propose the Walldistance-dependent Model (W-model) and Scale-dependent Model (-model) by modifying the profile of the forcing according to the profile of the eddy viscosity and the flow scales.The W-model and -model maintain the diagonal property of    , as in equation (2.11), 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., (2.13) where   denotes the diagonal of    , , i.e., the energy profile of forcing at .Based on the fact that the nonlinear interaction of turbulence is scale-dependent (Cho et al. 2018) where  = 2/( 2  + 2  ) 0.5 , and   () = 50/  + (2−50/  )tanh(6).The forcing energy profile is thereby calculated by   =     , .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 model cannot accurately predict the energy distribution of velocity fluctuation in the near-wall region, as will be further discussed in the following sections.

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  as the measured set of variables and  as the complete set of variables from all across the computational domain, where  = , the RBE could be briefly expressed as, (2.16) 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 (2.15) is actually the stationary point of the error matrix ê  , ê †  , , where the error ê  , is defined as the difference between the estimated forcing and the white-noiseassumed 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.,

,
(2.17) by which the stationary point of the error matrix ê  , ê †  , between the estimated forcing and the real one is obtained. , is the CSD tensor of the measurement noise at scale , 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, Wmodel, and -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   , 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.

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 (2.15-2.16), the predicted relative CSD tensor of velocity   near the wall can be estimated as, where  , should be the scalar energy of   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 (3.1), 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  R , the assumed reference layer used for estimating the relative CSDs in equation (3.1) is denoted as the quasi-reference layer  Q in this article.To obtain the relative CSD tensor used for the optimization, the value of  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)  2 (Baars et al. 2016) is introduced here, i.e., where û is the Fourier coefficient of  at scale , the overline denotes the complex conjugate, and  + 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  Q is set as the height beyond the inner peak where  2 ( ) = 0.3.When the height of  Q exceeds  R , it will be set as  R instead.With the above procedure, the value of  Q is adaptively determined.Details of the implementation of the

Modifications of the forcing profile with RBE Measured instantaneous velocity signal as input
Quasireference layer Wall

Reference layer
Modified forcing

Prediction region
Wall-normal direction 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  Q from the RBE, our goal is to minimize the relative error between the estimated CSD of the velocity  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  Q , which does not work well when  ⩾  Q .Thus, the modification range of the forcing is restricted below  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  ⩾  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  = (  ,   , ), the norm minimization problem is considered as follows, i.e., minimize 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   is obtained from the optimization, the forcing profile will be multiplied by a unified coefficient   to match the response energy at the reference layer for all the velocity components of , , and , respectively.
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 (  ,   , ) = (3.0/ℎ,30/ℎ, 1.3  /ℎ) corresponding to the large-scale motions (Smits et al. 2011) is selected, as in figure 2, where   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().The height of the quasi-reference layer  + Q is determined according to the LCS value defined in equation (3.2).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 (3.3) reaches the minimum under the constraints.The forcing and response profiles after modification are denoted in the orange color in figure 2() and (), respectively.The response energy profile after modification matches well with the DNS and RBE results.
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 -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 -model will also be included in the following validations.Note that the -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 -model will be applied in the time-resolved cases instead, where the flow is estimated at each spatio-temporal scale quantified by  = (  ,   , ).The modified eddy viscosity and forcing profiles in the -model will be calculated by equation (2.14) at each scale (  ,   ), keeping consistent with the original version.

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.To process the DNS data, the flow field is divided into blocks with a spatial domain of sizes   /ℎ = 4,   /ℎ = , and   /ℎ = 1 in each case.Given that the turbulent channel flow is statistically symmetric about the centerline  = ℎ, the flow data at  =  0 will be utilized together with that at  = 2ℎ−  0 when investigating the flow at  =  0 .The time periods of the blocks are set as 80Δ, 80Δ, and 120Δ for cases with   = 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   = 180, 550, and 950, respectively.

Case settings
Six cases are set to validate the prediction methods, as summarized in table 2.Besides the Reynolds number   , the height of the reference layer  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  + = 40 to the upper bound of the logarithmic region at /ℎ ≈ 0.2.The wall-normal direction  is discretized with 129, 201, and 257 Chebyshev polynomials in cases with   = 180, 550, and 950, respectively, with no-slip boundary conditions are applied at the walls.

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  ′ ,  ′ , and  ′ , which are important indexes to validate the DNS (e.g., Cheng et  prediction methods (e.g., Towne et al. (2020)).On the other hand, the RSS ⟨ ′  ′ ⟩ quantifies the correlation of  ′ and  ′ , 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   = 180, the only reference layer at  + = 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  + = 39, which is very close to  + = 40 used in the current case.As in figure 3(), 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   = 180 with a relative error of 13.2% at the inner peak.On the other hand, the predictions from the -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 -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.
With the increase of   , the RBE continues performing well when the reference layers are located at  + = 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  + = 200 with   = 950, the energy peak can be barely observed in the RBE-predicted streamwise RMS profile.Like the RBE, the energies predicted by the -model also tend to decrease when the reference layer lifts upward, which overestimates and underestimates the energy with  + R = 40 and  + R = 200, respectively.When  + R = 100, the -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 nearwall 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  ∈ ( 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.,  > 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  >  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 -model results become obviously larger than the DNS results at  >  R , which implies that the -model overestimates the forcing in the logarithmic region.The WBE underestimates the fluctuation energies at  ∈ ( 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  + 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 -model and WBE neither provide satisfying results with different Reynolds numbers and reference heights.
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   increases, the accuracy of prediction with the same  + 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 -model, the errors of which become minimum when  + 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.

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   ,   , or  will be studied.For brevity, only the results of streamwise velocity , 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  with the reference layer located at  + = 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 0.5 DNS,max is marked in each case to highlight the energy-concentrating region, where  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  DNS,max .When the reference layer is located at  + = 40 with   = 180, the RBE works well in predicting the energy distributions at different heights and flow scales, with the maximum error lower than 0.25 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  + = 37 (Towne et al. 2020).The energies predicted by the RWE also match fairly well with the DNS results, whose maximum error is about 0.46 DNS,max .On the other hand, the relative errors that are larger than 1.0 DNS,max are observed in the results of the -model and WBE.For the -model, the maximum error occurs at the heights around the inner peak  + = 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 -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  + = 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  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  + ≈ 10, which should be at  + ≈ 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 -model.
With the increase of the reference height  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  + = 15 in Case 550-100, which finally disappear in Case 950-200, indicating that the near-wall energy is seriously underestimated by RBE when  + R = 200.On the other hand, as the reference layer lifts up, the magnitude of the -model-predicted energy decreases gradually.In Case 550-100, the -model results match well with the DNS results, with the relative error lower than 0.35 DNS,max , as in figure 8.When  + R increases to 200, the energy magnitude predicted by the -model continues decreasing and consequently becomes lower than the DNS results by more than 0.75 DNS,max in the small scales at around  + = 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 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  +  ,  +  , and  + , with the maximum error of 0.33 DNS,max .The good performance of the RWE supports the capability of the adaptive modification process as derived in section 3.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 ( + = 15) and along different scales quantified by  +  ,  +  , and  + are shown in figure 11.The values are normalized by  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 -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 -model is larger than the DNS results at almost all the scales, whose maximum error is more than 1.37 DNS,max with a positive value.When the reference layer lifts up to  + = 200, the maximum relative error of the -model predictions becomes 0.7 DNS,max with a negative value instead.With the increasing height of the reference layer, the RBE and -model achieve their best performance with  + 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.

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 ( + = 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 nearwall plane at  + = 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  in section 4.4.With the increase of the reference height  + R , the predicted energies from the RBE and -model decrease correspondingly.When  + R = 200, only a small amount of the RBE-predicted energy is observed in the large-scale region with   /ℎ ⩾ 2, which indicates that the RBE becomes invalid when the reference layer in this case.In section 4.4, we have observed that the -model-predicted one-dimensional energy distributions match well with the DNS results with a wide range of  +  and  +  values.However, when considering the energy distributions as binary functions of   and   , the -model results are not consistent with the DNS.From figure 13, the energy predicted by the -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 0.48 DNS,max .In Cases 550-100 and 950-200, the relative error is always lower than 0.4 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.Then, the cross-spectra between  ′ and  ′ at  + = 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 | DNS | 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 | DNS | 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 -model and WBE results.As the reference layer lifts up to  + = 100 and 200, both the RBE and -model obviously underestimate the correlations between  ′ and  ′ .Meanwhile, the RWE keeps a relatively low prediction error in the estimation of the cross-spectra in Cases 550-100 and 950-200.

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  (, ) is defined as the correlation of the velocity fluctuation  at (, ) and ( + ,  + ), which can be calculated by, for stationary stochastic processes in the  and temporal direction such as the fully developed turbulent channel flow.With equation (4.1), the space-time correlation can be estimated using the predicted spectra  , .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 ⟨⟩ and the convection velocity   , as defined by following Choi & Moin (1990).From the DNS results in figure 16(-), the magnitude of the space-time correlation at a given spatio-temporal point increases with the increasing   , 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   does not overlap with the black one that denotes the local mean streamwise velocity ⟨⟩, where   > ⟨⟩.The deviations of   and ⟨⟩ in the near-wall region are consistent with previous observations (Kim & Hussain 1993;Del Alamo & Jiménez 2003).Since the convection velocity   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 -model and WBE deviate even larger from the DNS results.For the 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 -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  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  + 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   /ℎ ⩾ 2 are estimated by the RBE at  + = 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 -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 spacetime correlation distributions predicted by the WBE also appear to increase in magnitude as the reference layer lifts up.
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   as a function of  + in all the 6 cases with   = 180 − 950 and the reference layer, ranging from  + = 40 to /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  + 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 -model underestimate the convection velocity when  + 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 -model and WBE increase correspondingly, just like the phenomenon observed in the RBE results.A nonphysical increase of   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 -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  + R = 40 and 100, the relative errors between the RWE and the DNS results are less than 15%.When  + R = 200, the RWEpredicted 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.

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 Since all the tested methods can directly predict the CSD tensor of the velocity from the measurements, the transfer kernel   can be estimated from the auto-spectra coefficient  ,    (  ,   , ) and the cross-spectra coefficient  ,  (  ,   , ) from the estimated CSD tensor.In Gupta et al. (2021), the large-scale fluctuations are estimated with the linear model as in equation ( 4.3).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  + = 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  + = 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 -model and WBE.Specifically, the relative errors in the RWE results are smaller than those in the RBE results, with a maximum error of 0.13 DNS,max in the premultiplied energy spectra.On the other hand, the WBE and -model overestimate and underestimate the fluctuation energy at  + = 100, 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 0.26 DNS,max appears at  + = 10 for the large-scale motions with (  /ℎ,   /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 (  /ℎ,   /ℎ) = (2.5, 0.6) with the positive value equal to 0.17 DNS,max .The relative errors in the RBE predictions reach the extreme points at (  /ℎ,   /ℎ) = (6.2,1.6) and (2.5, 0.4).At the extreme points, the errors from RBE are equal to −0.34 DNS,max and 0.32 DNS,max , both of which are larger than those from RWE at the same points.The -model-predicted energy is lower than the DNS for the scales with   /ℎ ⩾ 6, with the relative error larger than 0.5 DNS,max , which indicates that the fluctuation energy predicted by the -model is much overestimated in the near-wall region.As for the WBE, the relative error is larger than 0.3 DNS,max when   /ℎ ⩾ 1.8 and 0.4 ⩽   /ℎ ⩽ 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 -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 -model is lower than that from the WBE.However, when applied in time-resolved cases as in this study, the energies predicted by the -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 -model.The deviations between the results generated by the forcing with different temporal properties indicate that the -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.
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,  = arctan [(/(Δ * )] , (4.5) where Δ * = arg max Δ   (Δ), and is the correlation function between the wall shear stress and the streamwise velocity fluctuation with a spatial delay of Δ, and  , is the cross-spectra between the wall shear  , respectively.When the reference layer moves to  + = 200, as in figure 21(), the RBEpredicted SIA is accurate when  + ⩾ 175.However, when the horizontal layers approach the wall, the SIA predicted by the RBE decreases, which becomes around 12 • when  + = 100.On the other hand, the RWE has the highest accuracy, with the root-mean-squared error equal to 0.81 • .Meanwhile, the WBE-predicted results are also close to the DNS results, whose root-mean-squared error is 1.14 • .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.

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   = 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, -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  + = 40 at the near-wall region to /ℎ ≈ 0.2 at the upper bound of the logarithmic region are set in cases with   = 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 -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  + = 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 -model also decreases when the reference layer lifts.The -model reaches its best performance with the reference layer at  + = 100, and overestimates and underestimates the near-wall energy when  + = 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  + = 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  + = 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  + = 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  + = 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 -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 • .
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  Q ⩽  R when determining the quasi-reference layer, the current RWE will degenerate to WBE when  R → 0. To effectively make use of the wall measurements, the way to determine  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 with where the expressions of R  and (,  + ) can be found in equations (2.7-2.8) and Appendix A, respectively.Since the LCS calculated in equation (C 2) is determined separately in each scale , 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  + Q at the wavelength (  ,   ) = (2/  , 2/  ) with  = 10  /  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.
In this study, we use the threshold LCS value (denoted as LCS 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 LCS t on the estimation accuracy, the RWEs with five values of LCS t equal to 0.1, 0.2, 0.3, 0.4, and 0.5 are set to estimate the streamwise RMS .
(C 3) The estimation errors are summarized in figure 24.A universal law between the relative error and LCS t clearly exists in all the tested cases when LCS t ⩾ 0.2.Although the tested cases with LCS t = 0.2 appear to have the minimum errors, we notice that there are sharp increases of the relative error when 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 LCS t is too low, in which case   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 LCS t to be 0.3 to avoid the possible sharp increase of estimation error at LCS t = 0.2.Thus, the threshold value of LCS t = 0.3 is adopted as a default setting for all the cases with RWE in this study.
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 where R ,,  denotes the sub-matrix of the resolvent operator R  by selecting its rows for   and columns for the forcing in   direction, • denotes the element-wise matrix multiplication, and  Q is the node number between  Q and the wall.

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   = 180, 550, and 950.Details of the DNS setups are listed in table 1.To provide time-resolved results, the sampling time intervals Δ + = Δ •  2   are set as 2.13, 4.81, and 5.09 for cases with   = 180, 550 and 950, respectively.The normalized total simulation time (  • )/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 rootmean-squared velocity profiles with the open source DNS database (Del Alamo & Jiménez 2003; Hoyas & Jiménez 2008) are provided in Appendix B.

Figure 4 :
Figure 4: Comparisons of the Reynolds shear stress profiles for cases with   = 180 (), 550 () and 950 ().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, -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.

Figure 14 :
Figure 14: The same as figure 12, but for the premultiplied cross-spectra between  ′ and  ′ at  + = 10.The values shown in the figures are normalized by | DNS | max .

Figure 15 :
Figure 15: The same as figure 13, but for the relative errors of the premultiplied cross-spectra between  ′ and  ′ at  + = 10.The values shown in the figures are normalized by | DNS | max .

Figure 20 :
Figure 20: Same as figure 19, but with the prediction layer at  + = 10.

Table 3 :
Summary of the relative percentage errors of the predicted streamwise RMS velocity at the inner peak.