2.1 The governing equations and the wall measurement

This study solves the incompressible Navier–Stokes equations to compute the true solution, to take its statistics (e.g. time averages and correlations for linear stochastic estimation as well as evaluation of the agreement) and to advance prediction steps in the observer. We non-dimensionalize the system of the equations by the channel half-height, $h$ , the constant density, $\unicode[STIX]{x1D70C}$ , and the friction velocity, $u_{\unicode[STIX]{x1D70F}}\equiv \sqrt{\bar{\unicode[STIX]{x1D70F}}_{w}/\unicode[STIX]{x1D70C}}$ ( $\bar{\unicode[STIX]{x1D70F}}_{w}$ being the mean wall shear stress), and express it in three dimensions as:

with the continuity equation

where the friction Reynolds number is defined as $Re_{\unicode[STIX]{x1D70F}}\equiv u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}$ ( $\unicode[STIX]{x1D708}$ being the kinematic viscosity). The full nonlinear Navier–Stokes equations are solved for all these processes although fluctuating quantities relative to the time-averaged mean flow are used for post-processing and analyses in this study.

As described later in § 3.1, all the computation and the post-processing are performed in the wavenumber space, in which the discrete Fourier transform is defined as

Here, the discrete points in space and the wavenumbers are given by $x_{n_{x}}\equiv n_{x}L_{x}/N_{x}$ and $z_{n_{z}}\equiv n_{z}L_{z}/N_{z}$ ( $n_{x}\in [0,N_{x}-1]$ and $n_{z}\in [0,N_{z}-1]$ ) as well as $k_{x}\equiv 2\unicode[STIX]{x03C0}m_{x}/L_{x}$ and $k_{z}\equiv 2\unicode[STIX]{x03C0}m_{z}/L_{z}$ ( $m_{x}\in [-N_{x}/2,N_{x}/2-1]$ and $m_{z}\in [-N_{z}/2,N_{z}/2-1]$ ), where $L_{x}$ and $L_{z}$ are the domain lengths (i.e. the spatial periodicity) in the streamwise and spanwise directions, respectively. It should be remarked that $\tilde{u} (-k_{x},-k_{z})=\tilde{u} ^{\ast }(k_{x},k_{z})$ , where $\ast$ denotes complex conjugate. The other velocity components are similarly Fourier transformed only in the two periodic directions.

On both sides of the wall (i.e.  $y/h=\pm 1$ ), shear stresses in the streamwise and spanwise directions as well as pressure are recorded over the entire wavenumbers in time:

Unlike the series of studies by the Bewley & Henningson group (Bewley et al. Reference Bewley, Moin and Temam2001; Högberg et al. Reference Högberg, Bewley and Henningson2003; Bewley & Protas Reference Bewley and Protas2004; Chevalier et al. Reference Chevalier, Hœpffner, Bewley and Henningson2006; Colburn et al. Reference Colburn, Cessna and Bewley2011), we measure these physical quantities instead of the transverse derivative of the wall-normal vorticity and the second transverse derivative of the vertical velocity (the wall pressure is the same). To compare the results with their studies, we perform necessary conversions.

One such example is the conversion for the random measurement noise. Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006) imposed Gaussian white noise with zero mean on these three measured quantities across the entire wavenumber range. In appendix A, we attempt to examine equivalent measurement noise for evaluation of the filtering capability, but their normalizations associated with the measurement noise are unclear throughout the series of the papers (Hœpffner et al. Reference Hœpffner, Chevalier, Bewley and Henningson2005; Chevalier et al. Reference Chevalier, Hœpffner, Bewley and Henningson2006; Colburn et al. Reference Colburn, Cessna and Bewley2011). In reality, however, we can readily filter out uncorrelated noise by simply applying a sequential technique and separate core flow estimation problems from the measurement noise issue, as explained in appendix A. In the rest of the main part, we impose no measurement noise during sampling and also develop the estimation scheme without assuming measurement noise.

2.2 Linear stochastic estimation

We apply linear stochastic estimation (Adrian & Moin Reference Adrian and Moin1988) for the instantaneous estimation. As explained later, the correlation across different wavenumbers between the wall measurements and the velocity components in the flow is fairly small, while there is appreciable correlation across the opposite side of the wall measurements. Moreover, appendix B demonstrates that including time derivatives of the wall measurements improves the estimation only a little. Consequently, we express the estimated velocity (denoted by $\tilde{u} _{LSE}$ below) as linear summation of instantaneous measured quantities and minimize the least-squares error over time from the true solution (denoted by $\tilde{u} _{DNS}$ ). The cost function for the streamwise velocity in the interior domain then defines

where $\tilde{\unicode[STIX]{x1D749}}$ has six components including the streamwise and spanwise wall shear stresses as well as the wall pressure on both $y/h=\pm 1$ in the wavenumber space and $\boldsymbol{a}^{\dagger }$ consists of $a_{x}^{\ast }$ , $a_{z}^{\ast }$ and $a_{p}^{\ast }$ , which denote their coefficients (to be precise, their complex conjugate is taken for later convenience). We similarly define the cost functions for $v$ and $w$ , while we compute pressure via the observer. Although the acquired velocity field generally does not satisfy continuity, the Poisson equation in the observer rectifies it, as described later. Assuming the symmetry in the $y$ and $z$ directions as well as the homogeneity in the $x$ and $z$ directions for the turbulent statistics, we expect that the mean quantities vanish except for $u$ of $(k_{x},k_{z})=(0,0)$ , and (2.5) directly expresses the equation for the fluctuating components.

We determine the coefficients above by setting $\unicode[STIX]{x2202}J_{u}/a=0$ where $a$ is each of the six coefficients in $\boldsymbol{a}$ . Accordingly, the system of equations for each $(k_{x},y,k_{z})$ yields

where $\dagger$ expresses the complex conjugate transpose and the overbar the time average, and

Here, the subscripts $\pm h$ distinguish the measured quantities on the upper and the lower walls, and $\overline{\tilde{\unicode[STIX]{x1D749}}\tilde{u} _{DNS}^{\ast }}$ of a $6\times 1$ matrix in (2.6) is similarly defined. It is clear that the estimation of $\tilde{u} _{LSE}$ at a specific point is independent from those at other points in $y$ or other velocity components. Since the correlation matrix (2.7) is common to all these components, we can arrange the column vectors $\boldsymbol{a}$ as many as the number of the components to be estimated by replacing $\tilde{u} _{DNS}^{\ast }$ in (2.6) by a row vector of a velocity field, denoted by $\tilde{\boldsymbol{u}}_{DNS}^{\dagger }$ :

We should note that an estimate based on this method is unchanged by using different combinations of the measured quantities (e.g. in the studies of Bewley’s group) or by expressing the velocity field in different bases (e.g. orthogonal basis functions versus physical points), as long as they are uniquely given by linear combinations (i.e. with the same ranks). Suppose a different combination of the measured quantities can be converted by

where the inverse of the conversion matrix $\unicode[STIX]{x1D734}$ , which is $6\times 6$ , is assumed to exist. Likewise, the velocity vector including all three components across all the grid points is assumed to be decomposed as

where $\unicode[STIX]{x1D753}_{\boldsymbol{i}}$ denotes the $i$ th vector in different bases and $c_{i}$ its coefficient. When the bases constitute a complete set, the matrix $\unicode[STIX]{x1D731}$ becomes a square matrix of the full rank. If we adopt only a subset of it, the number of the columns in $\unicode[STIX]{x1D731}$ is fewer than the number of the components in $\tilde{\boldsymbol{u}}$ .

By similarly defining the cost function as given in (2.5), we can derive the coefficient matrix for the different set of the measured quantities $\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}$ as

If $\unicode[STIX]{x1D731}$ is a non-square matrix, $(\unicode[STIX]{x1D731}^{\dagger })^{-1}$ denotes a pseudo-inverse such that $\unicode[STIX]{x1D731}^{\dagger }(\unicode[STIX]{x1D731}^{\dagger })^{-1}=\boldsymbol{I}$ still holds. As mentioned above, the estimation of an individual component in $\boldsymbol{c}$ is independent from the other components, regardless of the number of bases to be truncated. Consequently, the estimation of the coefficients $\boldsymbol{c}$ based on the new measurement set $\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}$ becomes

If $\unicode[STIX]{x1D731}$ is a square matrix with the full rank, equation (2.12) recovers the estimation of $\tilde{\boldsymbol{u}}$ by multiplying above by $\unicode[STIX]{x1D731}$ from the left. This manifests that the estimate of the velocity field is the same regardless of the set of measured quantities (i.e.  $\tilde{\unicode[STIX]{x1D749}}$ or $\tilde{\boldsymbol{\unicode[STIX]{x1D702}}}$ ) and the set of the bases (i.e.  $\tilde{\boldsymbol{u}}$ or $\boldsymbol{c}$ ) as long as these sets are uniquely related by linear combinations. If $\unicode[STIX]{x1D731}$ is a non-square matrix, the pseudo-inverse does not recover an identity matrix, i.e. $\unicode[STIX]{x1D731}\unicode[STIX]{x1D731}^{-1}\neq \boldsymbol{I}$ ; therefore, the estimation differs from that based on the complete set.

By expressing the estimate as $\tilde{\boldsymbol{u}}\sim \boldsymbol{A}^{\dagger }\tilde{\unicode[STIX]{x1D749}}$ , it is clear from the number of the columns in $\boldsymbol{A}^{\dagger }$ that the reconstructed velocity field consists of as few as six independent vectors, to be precise, up to the rank of the correlation matrix (2.7) for each $(k_{x},k_{z})$ . This is generally true if the estimation is given by a linear combination of the measured quantities on the wall. For example, if $\unicode[STIX]{x1D753}$ represents a proper orthogonal decomposition (POD) mode (Lumley Reference Lumley1970) in the vertical direction, $\unicode[STIX]{x1D731}$ consists of a set of orthogonal bases. Since $\boldsymbol{A}^{\dagger }=\unicode[STIX]{x1D731}\boldsymbol{B}^{\dagger }\unicode[STIX]{x1D734}$ from (2.11), we can view that a linear combination of $\tilde{\unicode[STIX]{x1D749}}$ determines the weight of the orthogonal bases, which can be as many as the number of the dimensions in $\tilde{\boldsymbol{u}}$ , to form its estimate. However, since the dimensions of $\tilde{\unicode[STIX]{x1D749}}$ are as few as six, this limits the degrees of freedom. Although we may be able to prioritize these bases using the POD, it requires post-processing a correlation matrix across all the grid points in $y$ for each $(k_{x},k_{z})$ ; hence, we pursue a much less expensive approach in this study.

We can expand the dimensions of the correlation matrix, for example by including time derivatives of the measured quantities or nonlinear combinations of the wavenumbers. However, appendix B demonstrates that the inclusion of time derivatives, solving $12\times 12$ matrices, improves the estimation only a little. Lasagna et al. (Reference Lasagna, Fronges, Orazi and Iuso2015) actually reported limited improvement using multi-time-delay information for linear stochastic estimation. Moreover, our analysis in § 4.1 shows little correlation across different wavenumbers, which supports decoupling of the wavenumbers in the stochastic estimation. Thus, we focus on the linear approach with the aforementioned six measured quantities.

Likewise, if we apply linear stochastic estimation using all the grid points in the physical space on the wall, it drastically expands the degrees of freedom (i.e. the number of the grid points in $x$ and $z$ times the types of the measured quantities on the wall). In reality, however, this approach tends to merely ‘memorize’ flow patterns and would work poorly outside the sampling duration unless we calculate the coefficients in (2.6) by taking the number of samples much greater than the number of the grid points times the types of the measured quantities. This cautions us that some data assimilation techniques that collect large cross-correlation data must take a sufficient number of samples over time. We revisit this point later.

2.3 Feedback via the body-force terms

To sequentially feed the flow estimation into the dynamical system, the body force is imposed at every time step. In popular data assimilation techniques, such as the Kalman filter (Kalman Reference Kalman1960, in which we follow the notation in this paragraph), the forcing terms are optimized in time and space by solving the covariance equations in a linear time-invariant system. In this study, instead, a velocity field given by the linear stochastic estimation is forced simply with a constant weight in time (see the discussion in Suzuki Reference Suzuki2012). The idea behind this approach consists in the following simplifications: (i) in the Kalman filter, the covariance $\boldsymbol{P}$ tends to converge to a constant matrix which governs the forcing gain, $\boldsymbol{K}\equiv \boldsymbol{P}\unicode[STIX]{x1D643}^{\boldsymbol{T}}[\unicode[STIX]{x1D643}\boldsymbol{P}\unicode[STIX]{x1D643}^{\boldsymbol{T}}+\boldsymbol{R}]^{-1}$ , after a sufficient number of time steps. We ignore the transient time, which is much shorter than the sampling duration for the estimation, and simplify the time variation of $\boldsymbol{K}$ . (ii) The optimization of the feedback terms requires the inversion (to be precise, the transpose) of the transfer matrix from the velocity field to the wall measurement, called the observation matrix $\unicode[STIX]{x1D643}$ . Our approach replaces the inverse of the transfer matrix by the stochastic estimation from the wall measurement and the Kalman gain by a time-invariant function in $y$ (i.e.  $\boldsymbol{K}(\boldsymbol{y}-\unicode[STIX]{x1D643}\boldsymbol{x})\approx \unicode[STIX]{x1D716}f(y)(\unicode[STIX]{x1D643}^{-1}\boldsymbol{y}-\boldsymbol{x})$ , where $\boldsymbol{y}$ and $\boldsymbol{x}$ represent the wall measurement and the velocity field, respectively, $\unicode[STIX]{x1D716}$ and $f$ are defined below, and $\unicode[STIX]{x1D643}^{-1}\boldsymbol{y}$ is given by the stochastic estimation). We regard this system as a simple state observer, whose quantities on the left-hand side are expressed with hat marks (which are later defined as the observer quantities with the subscript ‘Obs’) and write the nonlinear system including the body-force terms as

where the velocity components above given by the linear stochastic estimation are computed via the inverse Fourier transform of (2.3):

and $v_{LSE}$ and $w_{LSE}$ are similarly defined. Since only the fluctuating parts are estimated as $\tilde{u} _{LSE}$ , the time-averaged DNS velocity profile for $(k_{x},k_{z})=(0,0)$ is added above. Thus, these terms in (2.13) can be regarded as linear feedback, $\boldsymbol{K}\boldsymbol{y}\sim$ $\unicode[STIX]{x1D716}f\boldsymbol{A}^{\dagger }\tilde{\unicode[STIX]{x1D749}}$ , from the wall measurement (where $f_{x}$ , $f_{y}$ and $f_{z}$ are symbolically expressed as $f$ hereafter). The constant $\unicode[STIX]{x1D716}$ is the weight on the feedback, and $f$ provides the spatial distributions of the feedback, which are equivalent to the ‘patch function’ introduced in Suzuki, Ji & Yamamoto (Reference Suzuki, Ji and Yamamoto2009a ), Suzuki et al. (Reference Suzuki, Sanse, Mizushima and Yamamoto2009b ), Suzuki, Ji & Yamamoto (Reference Suzuki, Ji and Yamamoto2010). We normalize $f$ such that $\max [f]=1$ . Unlike those previous studies, it takes a non-zero value everywhere in this study because the linear stochastic estimation gives an estimate over the entire domain.

We have examined various ways of defining $f$ . For example, we have included it in the wavenumber space as $\tilde{f}(k_{x},y,k_{z})$ prior to the inverse Fourier transform by referring to the correlation in $y$ for each $(k_{x},k_{z})$ ; however, this hardly improves the estimation through the dynamical system compared with $f$ defined in the physical space. We have also tested various distributions of $f(y)$ , as briefly discussed later, but none of them are substantially better than a constant $f$ . As a result, we analyse only one spatially varying $f(y)$ explained in § 4.3 as well as $f\equiv 1$ with several constant $\unicode[STIX]{x1D716}$ .

3.1 Direct numerical simulation (DNS)

Incompressible wall-bounded channel flow is solved in three dimensions using DNS. This code has been validated in the past (a part of the verification can be found in figure 27(a) in appendix C), and the details of its numerics are described in Hasegawa & Kasagi (Reference Hasegawa and Kasagi2011). The main features and the points specific to this study are summarized below. Periodic boundary conditions are imposed in the streamwise and spanwise directions (see figure 1 for the coordinate system); therefore, the flow is considered to be perfectly homogeneous in these directions. The density and the viscosity are set to be constant, and the friction Reynolds number is set to be $Re_{\unicode[STIX]{x1D70F}}=100$ (normalized based on the channel half-height, $h$ ), corresponding to $u^{+}\equiv u_{centreline}/u_{\unicode[STIX]{x1D70F}}\approx 17.$ The flow is observed to be fully developed turbulence, yet with some low Reynolds number characteristics, which are discussed later in § 4.1.

The physical domain size is set to be $L_{x}=4\unicode[STIX]{x03C0}h$ , $L_{y}=2h$ and $L_{z}=4\unicode[STIX]{x03C0}h/3$ to make a fair comparison with Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006), and the $3/2$ law is used to remove aliasing errors: $L_{x}$ and $L_{z}$ are equally spaced with $N_{x}=N_{z}=64$ grid points, in which flow quantities are decomposed into $42\times 42$ Fourier modes in the streamwise and the spanwise directions, respectively. Unlike their study, however, the Chebyshev collocation method is used in the vertical direction, in which $L_{y}$ is discretized into $N_{y}=64$ spacings using $43$ polynomials. Compared with Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006) using $N_{y}=64$ uniform grid spacings with finite differencing, the resolution in $y$ is expected to be higher in this study. Thus, the velocity field is spatially differentiated with the spectral methods in all directions.

As opposed to solving the vertical velocity and the wall-normal vorticity, referred to as the $v{-}\unicode[STIX]{x1D714}_{y}$ formulation, in many other studies (Kim, Moin & Moser Reference Kim, Moin and Moser1987; Högberg et al. Reference Högberg, Bewley and Henningson2003; Bewley & Protas Reference Bewley and Protas2004; Chevalier et al. Reference Chevalier, Hœpffner, Bewley and Henningson2006), primitive variables, $u$ , $v$ , $w$ and $p$ , are solved in the current code instead. For pressure decoupling, a fractional step method (Kim & Moin Reference Kim and Moin1985) is used to march the aforementioned transformed velocity field in time: the second-order Adams–Bashforth scheme is adopted for the convection terms and the second-order Crank–Nicholson scheme for the diffusion terms. The time step is set to be $\unicode[STIX]{x0394}tu_{\unicode[STIX]{x1D70F}}/h=2.0\times 10^{-3}$ (i.e.  $\unicode[STIX]{x0394}t^{+}=0.2$ ). The solver is used both to solve the original Navier–Stokes equations as well as to run as an observer with the body force defined in § 2.3. The body-force term is inserted before solving the Poisson equation to impose the continuity every time step, as introduced in Suzuki et al. (Reference Suzuki, Ji and Yamamoto2009a , Reference Suzuki, Ji and Yamamoto2010).

The collocation method allows us to directly compute flow quantities on the wall. On the other hand, we must treat the no-slip boundary condition differently from staggered grid systems. Appendix C describes the details of the no-slip boundary treatment posterior to solving the Poisson equation.

3.2 Sampling statistics

After the flow becomes statistically stationary by running the DNS, we prepare two sets of unsteady flow fields (i.e. true solutions of the original Navier–Stokes equations) for the same conditions but with different initial conditions, plus another extra initial condition. During the first unsteady solution, referred to as the ‘sampling duration’, time-averaged quantities, necessary statistics and correlations are calculated using every time step; at the same time, the wall quantities are recorded also at every time step and the velocity components as well as pressure are recorded at every $100$ time steps over all points in the physical space. Likewise, the same unsteady quantities are recorded during the second solution set. To secure enough sampling in the wavenumber space, $50\,000$ time steps (i.e.  $100h/u_{\unicode[STIX]{x1D70F}}$ ) are computed for both sets of solutions, and their initial conditions are $t=200h/u_{\unicode[STIX]{x1D70F}}$ apart to make these three initial conditions statistically independent. No measurement noise is added to the wall quantities except for the analysis in appendix A.

Using the first solution set the coefficients $\boldsymbol{A}$ in (2.8) for the linear stochastic estimation are computed. Assuming that the two sets of solutions are statistically equivalent, these coefficients are commonly applied to the wall measurement for both solutions, and the discrepancies in the velocity field inside the domain from the true solution are compared between the two sets of the solutions. To analyse the observer, the flow is started with the extra initial condition but the second set of the wall measurements is used for the update steps; accordingly, it is monitored if the estimated flow field is sequentially assimilated to the true solution from an unknown initial condition.

Because we decouple wavenumbers in the linear stochastic estimation, the number of samples, 50 000, is much greater than the degrees of freedom, i.e. six in this study. If we were to apply it based on the wall measurement at every grid point in the physical space, the degrees of freedom suddenly expands to $64\times 64\times 6=\text{24 576}$ , which is nearly the half of the number of samples. In such a case, linear stochastic estimation can find a linear combination that almost exactly reconstructs the sampled velocity fields up to the degrees of freedom but would not estimate the flow well beyond the sampling duration. Analogous problems occur when the degrees of freedom increase (e.g. by including coupling of the wavenumbers) or when the sampling is limited (e.g. by characterizing the flow with fewer reduced-order models or ensembles); thus, special attention being required for fair evaluation (cf.  Chevalier et al. Reference Chevalier, Hœpffner, Bewley and Henningson2006; Colburn et al. Reference Colburn, Cessna and Bewley2011).

4.1 Basic flow characteristics at $Re_{\unicode[STIX]{x1D70F}}=100$

We start with general observations of the flow to be estimated. For reference, we compare flow statistics with the database by Iwamoto, Suzuki & Kasagi (Reference Iwamoto, Suzuki and Kasagi2002), which has been extensively validated in the past. Figure 2 compares the time- and space-averaged streamwise velocity profile, i.e. the time average of $\overline{\tilde{u} }(k_{x}=0,y,k_{z}=0)$ , with those at higher Reynolds numbers. All cases collapse very well in the viscous sublayer ( $y^{+}\lesssim 10$ ). Our case, $Re_{\unicode[STIX]{x1D70F}}=100$ , indicates a low Reynolds number feature in the log layer, as reported by Moser, Kim & Mansour (Reference Moser, Kim and Mansour1999); namely, the velocity profile in this region overshoots those at high Reynolds numbers, as is expected.

Figure 2. Comparison of the time- and space-averaged streamwise velocity. Line patterns: ——, $Re_{\unicode[STIX]{x1D70F}}=100$ in the current study; - - -, $Re_{\unicode[STIX]{x1D70F}}=150$  (Iwamoto et al. Reference Iwamoto, Suzuki and Kasagi2002, and thereafter); – ⋅ – ⋅ –, $Re_{\unicode[STIX]{x1D70F}}=396$ ; $\cdots \cdots$ , $Re_{\unicode[STIX]{x1D70F}}=643$ . (a) Profiles near the wall. Thin grey lines indicate $u^{+}=y^{+}$ and $u^{+}=0.4^{-1}\log y^{+}+5$ . (b) Profiles across the channel.

Figure 3 similarly compares velocity fluctuations in our case with those at higher Reynolds numbers. Regardless of $Re_{\unicode[STIX]{x1D70F}}$ , the streamwise velocity fluctuation becomes the highest in the buffer layer, in which the linear and the log profiles intersect, and turbulent kinetic energy also forms a peak in this layer. In the physical scale, the peak of turbulent kinetic energy shifts away from the wall with decreasing Reynolds number. Its level near the centreline at $Re_{\unicode[STIX]{x1D70F}}=100$ is slightly higher than the other cases, possibly because the aspect ratio of our domain is $L_{x}/L_{z}=3$ as opposed to $L_{x}/L_{z}=2$ by Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002). It should be noted that we have confirmed the agreement with their database for the same condition in appendix C. In general, our simulation possesses fundamental turbulence characteristics although it indicates some low Reynolds number features.

Now, the correlations of the velocity components, $\tilde{u}$ , $\tilde{v}$ and $\tilde{w}$ , in the flow with the wall measurements, $\tilde{\unicode[STIX]{x1D70F}_{x}}$ , $\tilde{\unicode[STIX]{x1D70F}_{z}}$ and $\tilde{p}$ , are analysed in the wavenumber space. Through these results, the region reconstructible from the instantaneous wall information can be assessed. Four different wavenumber combinations, $(k_{x}^{\prime }h,k_{z}^{\prime }h)=(0.5,1.5)$ , $(0.5,6)$ , $(3,18)$ and $(5,30)$ , are analysed for the velocity in the flow; in particular, the second and third combinations are representative of low-/high-speed streaks and quasi-streamwise vortices, respectively, containing a large portion of turbulent kinetic energy and representing key features of near-wall turbulence (Smith & Metzler Reference Smith and Metzler1983; Kim et al. Reference Kim, Moin and Moser1987). We record the correlation coefficient for each wavenumber combination against each wall quantity over the entire wavenumber space during the sampling duration of $100h/u_{\unicode[STIX]{x1D70F}}$ .

Figure 3. Comparison of the fluctuating velocity components (database at higher Reynolds numbers is similarly taken from Iwamoto et al. Reference Iwamoto, Suzuki and Kasagi2002). Line patterns are the same as figure 2. (a) Root mean squares of three velocity components near the wall (a similar figure can be found in Moser et al. Reference Moser, Kim and Mansour1999). (b) Turbulent kinetic energy across the channel.

Contours for one of such coefficients, given by

are drawn at three heights in figures 4–6. The overbar above represents the time average over the aforementioned duration: every time step during $100h/u_{\unicode[STIX]{x1D70F}}$ is used for the evaluation in the wavenumber space. Here, the lower half-plane (i.e.  $k_{z}<0$ ) is omitted because it is complex conjugate of the upper half. Near the wall, a distinctive peak is formed at the same wavenumber as the velocity component. Aside from the peak, weak correlations are found in the vicinity of the same $k_{x}$ , indicating spanwise patterns of streamwise vorticity, but mostly less than $0.2$ . Although only a single combination (i.e.  $\tilde{u}$ versus $\tilde{\unicode[STIX]{x1D70F}_{x}}$ ) is shown here, other correlations across different wavenumbers are also investigated and confirmed to be always low except for the primary peak (i.e. the identical wavenumber). Accordingly, linear stochastic estimation is performed for each wavenumber combination in this study. Moreover, correlation of a velocity component in the flow with a product between two wall quantities should then be further reduced; hence, unlike unsteady laminar flows, the justification of nonlinear stochastic estimation is weak for such turbulent flows.

Figure 4. Correlation coefficients of $\tilde{u} (k_{x}^{\prime }h=0.5,y,k_{z}^{\prime }h=1.5)$ with $\tilde{\unicode[STIX]{x1D70F}}_{x}(k_{x},y/h=1,k_{z})$ . (a)  $y/h=0.942$ (peak correlation $0.978$ at the same wavenumber point indicated by broken lines). (b) $y/h=0.803$ ( $0.614$ ). (c) $y/h=0$ ( $0.129$ at a different wavenumber point).

Figure 5. Correlation coefficients of $\tilde{u} (k_{x}^{\prime }h=0.5,y,k_{z}^{\prime }h=6.0)$ with $\tilde{\unicode[STIX]{x1D70F}}_{x}(k_{x},y/h=1,k_{z})$ . (a)  $y/h=0.942$ (peak correlation $0.991$ at the same wavenumber point indicated by broken lines). (b) $y/h=0.803$ ( $0.902$ ). (c) $y/h=0$ ( $0.154$ at a different wavenumber point).

Figure 6. Correlation coefficients of $\tilde{u} (k_{x}^{\prime }h=3.0,y,k_{z}^{\prime }h=18.0)$ with $\tilde{\unicode[STIX]{x1D70F}}_{x}(k_{x},y/h=1,k_{z})$ . (a)  $y/h=0.942$ (peak correlation $0.776$ at the same wavenumber point indicated by broken lines). (b) $y/h=0.803$ ( $0.120$ ). (c) $y/h=0$ ( $0.101$ at a different wavenumber point).

By focusing on the same wavenumber between $\tilde{\unicode[STIX]{x1D70F}_{x}}$ on the wall and the velocity components, the correlation coefficients are plotted across the channel in figure 7. At low wavenumbers, i.e.  $(k_{x}h,k_{z}h)=(0.5,1.5)$ and $(0.5,6.0)$ , high correlation is retained beyond the viscous sublayer ( $y/h\lesssim 0.8$ ), but all three correlations decay toward the centreline. At higher wavenumbers, the correlations sharply drop beyond the viscous sublayer. At all wavenumbers, the streamwise velocity correlation is appreciably higher than the other two components, indicating that the streamwise shear-stress measurement is effective to estimate the streamwise velocity component.

Figure 8 similarly plots the correlations of $\tilde{\unicode[STIX]{x1D70F}_{z}}$ and $\tilde{p}$ at $(k_{x}h,k_{z}h)=(0.5,6.0)$ . In contrast to the correlation with $\tilde{\unicode[STIX]{x1D70F}_{x}}$ above, these two wall quantities, $\tilde{\unicode[STIX]{x1D70F}_{z}}$ and $\tilde{p}$ , are primarily responsible for $\tilde{w}$ and $\tilde{v}$ (these trends are more or less similar at other wavenumbers, not shown). The correlation with $\tilde{p}$ is lower than those with $\tilde{\unicode[STIX]{x1D70F}_{x}}$ and $\tilde{\unicode[STIX]{x1D70F}_{z}}$ ; namely, the pressure signal is less effective than the shear stresses. We can confirm this by applying linear stochastic estimation with individual components (the results are not shown). Regardless of the combinations, all the profiles decay as low as $0.2$ at the centreline. These results imply inherent difficulty with instantaneous estimation near the centreline based on the wall measurements.

Figure 7. Correlation coefficients of $\tilde{\unicode[STIX]{x1D70F}_{x}}(y/h=1)$ with three velocity components at the same wavenumber. Line patterns: ——, with $\tilde{u}$ ; - - -, with $\tilde{v}$ ; – ⋅ – ⋅ –, with $\tilde{w}$ . (a)  $(k_{x}h,k_{z}h)=(0.5,1.5)$ . (b) $(0.5,6.0)$ . (c) $(3.0,18.0)$ . (d) $(5.0,30.0)$ .

At low wavenumbers, we find discernible correlation recovery on the opposite side of the wall (i.e.  $y/h=-1$ ) in figures 7(a–b) and 8. We attribute this phenomenon to large-scale motions across the channel represented by lower-order Chebyshev polynomials. This motivates us to process wall quantities on both sides simultaneously in the stochastic estimation. By expanding the degrees of freedom only from three to six, we can appreciably improve the accuracy of estimation (comparison not shown).

Figure 8. Correlation coefficients of the other wall quantities with three velocity components at $(k_{x}h,k_{z}h)=(0.5,6.0)$ . Line patterns are the same as figure 7. (a) $\tilde{\unicode[STIX]{x1D70F}_{z}}(y/h=1)$ . (b) $\tilde{p}(y/h=1)$ .

4.2 Instantaneous estimation: linear stochastic estimation

Next, we discuss the results of instantaneous velocity estimate using the linear stochastic estimation explained in § 2.2. To make the comparison of our results with those by Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006) easier, we define the correlation coefficient in the same manner:

where $u_{LSE}^{\prime }$ and $u_{DNS}^{\prime }$ denote the streamwise velocity fluctuation based on the linear stochastic estimation and that of the true solution from the original DNS, respectively. Here, the overbar represents time average over the $100h/u_{\unicode[STIX]{x1D70F}}$ period but takes a sample every $100$ time steps for physical quantities, a total being $500$ samples. We similarly define the correlation coefficients for the other velocity components, $v^{\prime }$ and $w^{\prime }$ , as well as the Reynolds stresses, $u^{\prime }v^{\prime }$ , $v^{\prime }w^{\prime }$ and $w^{\prime }u^{\prime }$ .

Figure 9(a) shows the correlation coefficients of the three velocity components near the wall. Here, we plot the estimates for the two sets mentioned in § 3.2: (i) the first set during which statistics for the linear stochastic estimation are taken (thin grey lines) and (ii) the second set sufficiently after the sampling duration (thick black lines). In $y^{+}\leqslant 20$ , the correlations between the two sets are nearly the same. These profiles also agree very well with those of the extended Kalman filter by Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006) in figure 6 of their paper. To be precise, our streamwise velocity correlation tends to be slightly lower than theirs, while the vertical and spanwise velocity correlations are somewhat better.

Figure 9. Correlation coefficients of velocity fluctuations. Line patterns: ——, $\text{Cor}(u_{LSE}^{\prime },u_{DNS}^{\prime })$ ; – – –, $\text{Cor}(v_{LSE}^{\prime },v_{DNS}^{\prime })$ ; – ⋅ – ⋅ –, $\text{Cor}(w_{LSE}^{\prime },w_{DNS}^{\prime })$ . Thick black lines indicate the estimates sufficiently after the sampling duration, and thin grey lines those during the sampling duration. (a) Profiles near the wall. (b) Profiles across the channel.

Figure 9(b) reveals that all the correlations drop toward the centreline. In particular, the correlations after the sampling duration are clearly lower than those during the sampling duration beyond the viscous sublayer ( $|y/h|\lesssim 0.8$ ). This implies that the statistics between the two periods, although $100h/u_{\unicode[STIX]{x1D70F}}$ apart, are not exactly equivalent, and the linear stochastic estimation ‘memorizes’ a part of flow patterns rather than takes statistical correlation. Interestingly, the streamwise velocity generally gives the highest correlation except near the centreline for the result outside the sampling duration. The estimation of spanwise velocity fluctuations is least successful over the entire channel.

We also plot other quantities evaluated by Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006). The correlation profiles of all the Reynolds stresses in figure 10(a) are again very close to their results (see figure 8 in Chevalier et al. Reference Chevalier, Hœpffner, Bewley and Henningson2006). Not surprisingly, $\text{Cor}(v^{\prime }w_{LSE}^{\prime },v^{\prime }w_{DNS}^{\prime })$ yields the lowest correlation because the individual estimates of $v^{\prime }$ and $w^{\prime }$ are worse than $u^{\prime }$ .

Figure 10. Profiles of other quantities near the wall. Distinction between the thick and thin lines is the same as figure 9. (a) Reynolds stresses. Line patterns: ——, $\text{Cor}(u^{\prime }v_{LSE}^{\prime },u^{\prime }v_{DNS}^{\prime })$ ; – – –, $\text{Cor}(v^{\prime }w_{LSE}^{\prime },v^{\prime }w_{DNS}^{\prime })$ ; – ⋅ – ⋅ –, $\text{Cor}(w^{\prime }u_{LSE}^{\prime },w^{\prime }u_{DNS}^{\prime })$ . (b) Errors defined in (4.3). Line patterns: ——, $\text{Err}(u_{LSE}^{\prime },u_{DNS}^{\prime })$ ; – – –, $\text{Err}(v_{LSE}^{\prime },v_{DNS}^{\prime })$ ; – ⋅ – ⋅ –, $\text{Err}(w_{LSE}^{\prime },w_{DNS}^{\prime })$ .

In addition, the normalized error is defined as

for the streamwise velocity fluctuation and the other velocity components are similarly defined. The results of these errors in figure 10(b) are consistent with figure 9(a) in the sense that the agreement is similar to Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006). To be precise, our estimate of $u^{\prime }$ deviates slightly more for greater $y^{+}$ , while our other two components give better estimates than theirs. Because we impose no measurement noise, the errors almost vanish near the wall except in the very small $v^{\prime }$ region. The differences due to the sampling duration are again relatively small, and we analyse the data only after the sampling duration (i.e. the second set) in the rest of the discussion.

These results consistently suggest that if we can remove the measurement noise, the linear stochastic estimation can essentially estimate velocity fields as good as the extended Kalman filter. Both approaches take the same information from the wall, but the major difference is that the former one estimates the velocity field from instantaneous information while the latter one optimizes the forcing distribution and accounts for the dynamics in the course of time. General expectation is the observer solving the Navier–Stokes equations compensates the information propagating from the wall. However, the fact that the agreement away from the wall is comparable between the two approaches may imply that the extended Kalman filter only suppresses the measurement noise and hardly contributes to improvement of the estimation beyond the viscous sublayer.

Figure 11. Profiles of turbulent quantities across the channel. Line patterns: ——, original Navier–Stokes; – – –, linear stochastic estimation. (a) Turbulent kinetic energy ( $\cdots \cdots$ , the squared difference). (b) Reynolds shear stress ( $\cdots \cdots$ , total shear stress).

We elucidate the reason for poor estimates away from the wall. Figure 11(a) compares the turbulent kinetic energy profile of the linear stochastic estimation with that of the true solution together with their squared difference. We notice that the turbulence intensity in the linear stochastic estimation diminishes beyond the buffer layer. When the correlation between the wall quantity and the velocity in the flow drops, as seen in § 4.1, the magnitude of the coefficients in (2.6) decreases. Namely, if the estimation is less certain, the linear stochastic estimation produces smaller values to minimize errors, resulting in over-suppression of turbulent motion, especially in the log layer. Likewise, figure 11(b) reveals that the contribution of the $\overline{u^{\prime }v^{\prime }}$ shear-stress component is much lower in the linear stochastic estimation near the centreline. On the other hand, the mean streamwise velocity profile is invariant since we determine the linear coefficients $\boldsymbol{a}^{\dagger }$ in (2.5) relative to the mean flow; namely, the shear stress by the molecular viscosity remains the same between the true solution and the linear stochastic estimation.

In this sense, the correlation coefficient is somewhat misleading since $\int \int |u_{LSE}^{\prime }|^{2}\,\text{d}x\text{d}z$ in the denominator of (4.2) diminishes toward the centreline, elevating its value. It is rather intuitive if we plot the covariance normalized by the auto-correlation as

Figure 12 now reveals that the estimation is mostly unsuccessful away from the wall and indicates the limitation of instantaneous measurement based on the wall quantities. Since both correlation coefficients and normalized errors in Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006) essentially show the same profiles as our profiles, their estimation must have the common problem. The key for further improvement is to infer motions away from the wall, especially beyond the peaks of turbulent kinetic energy.

Figure 12. Normalized covariance given by (4.4). Line patterns: ——, $\widehat{\text{cov}}(u_{LSE}^{\prime },u_{DNS}^{\prime })$ ; - - -, $\widehat{\text{cov}}(v_{LSE}^{\prime },v_{DNS}^{\prime })$ ; – ⋅ – ⋅ –, $\widehat{\text{cov}}(w_{LSE}^{\prime },w_{DNS}^{\prime })$ .

We subsequently visualize the discrepancy in the wavenumber space. To measure the success of the estimation, we similarly define the normalized covariance between the linear stochastic estimation and the true solution for each $(k_{x},k_{z})$ :

where the time average again takes every time step for wavenumber components. Figure 13 draws its contours for $\tilde{u}$ on selected constant $k_{z}$ planes corresponding to figure 7. Regardless of the wavenumbers, the normalized covariance reaches approximately unity near the wall: its thickness decreases with increasing $k_{z}$ . In the vicinity of the centreline (i.e.  $y=0$ ), the covariance is high only at low wavenumbers. We also display the contours for $\tilde{v}$ and $\tilde{w}$ on $k_{z}h=6.0$ in figure 14, showing the same trends. Thus, the stochastic estimation can reconstruct only very large motions, and its capability rapidly deteriorates from the buffer layer with increasing wavenumber.

Figure 13. Contours of normalized covariance for $\tilde{u}$ given by (4.5). The domain extends over the entire $k_{x}-y$ space. (a) $k_{z}h=1.5$ . (b) $k_{z}h=6.0$ . (c) $k_{z}h=18.0$ . (d) $k_{z}h=30.0$ .

Figure 14. Contours of normalized covariance at $k_{z}h=6.0$ . (a) for $\tilde{v}$ . (b) for $\tilde{w}$ .

In contrast to the turbulence estimation based on the wall measurement, Baars, Hutchins & Marusic (Reference Baars, Hutchins and Marusic2016) reported success in estimation of near-wall turbulent statistics from large-scale motions in the log layer and beyond. By extracting coherent motions from pointwise measurement in the log layer, they conversely infer streamwise velocity fluctuations beneath it using linear stochastic estimation even at higher Reynolds numbers (i.e. $Re_{\unicode[STIX]{x1D70F}}>O(10^{3})$ ). On the other hand, their study also reveals the limitation of the instantaneous estimation in the buffer layer, in which incoherent motions dominate the peak of the energy spectrum. This is consistent with our observation below that it is difficult to infer instantaneous motions in the log layer from those in the viscous sublayer.

4.3 State observer combined with the linear stochastic estimation

Finally, an observer is established by imposing the body-force term in the Navier–Stokes equations, as explained in § 2.3. In this approach, a part of the estimation is given instantaneously by the linear stochastic estimation, while the rest is advanced from the previous time step using the DNS. Therefore, it is expected that the over-suppression of turbulent kinetic energy found in the previous section can be relaxed by propagating the information toward the centreline from the wall. Through this process, the pressure field is simultaneously produced, which is also evaluated below.

Figure 15. Time histories of the squared velocity difference given by (4.6). Line patterns: $\cdot \,\cdot \,\cdot \,\cdot \,\cdot$ , $\unicode[STIX]{x1D716}=1$ ; $\cdots \cdots$ , $\unicode[STIX]{x1D716}=2$ ; $-\cdot \cdot -\cdot \cdot -$ , $\unicode[STIX]{x1D716}=5$ ; – ⋅ – ⋅ –, $\unicode[STIX]{x1D716}=10$ ; – – –, $\unicode[STIX]{x1D716}=20$ ;— —, $\unicode[STIX]{x1D716}=50$ ; ——, linear stochastic estimation. (a) Linear scale over the entire sampling duration. (b) Log scale in the initial part.

When the errors in the forcing term are uncorrelated in space, and its magnitude is uniform, we can apply the criterion in Suzuki et al. (Reference Suzuki, Ji and Yamamoto2009a ) for the weight of the forcing, $\unicode[STIX]{x1D716}$ , in (2.13). In such a case, $\unicode[STIX]{x1D716}$ must overcome the maximum amplification rate of the system (in the current flow, the system is linearly stable; hence, it is difficult to directly determine the threshold). However, the errors are partially correlated in space, and the accuracy of the linear stochastic estimation is significantly deteriorated outside the viscous sublayer. Thus, several constant weights are examined below. In figure 15, time histories of the squared velocity difference given below are plotted for various  $\unicode[STIX]{x1D716}$ :

Here, the subscript $\text{Obs}$ denotes a quantity from the observer, which is equivalent to that with a hat mark in (2.13). Since we impose the initial condition inconsistent with the wall information for the rest of the time, the discrepancy starts with the maximum value and soon becomes stationary for all the cases. This shows a sequential data assimilation feature, but the observer can suppress the discrepancy by approximately one third at best. The least weight yields the slowest initial convergence, and the slope becomes steeper with increasing $\unicode[STIX]{x1D716}$ . We should note that the least weight here, $\unicode[STIX]{x1D716}=1$ , is still strong in general control applications, and the magnitude of $\unicode[STIX]{x1D716}=O(10)$ is actually necessary to reduce the errors to the level equivalent to the linear stochastic estimation. This approximately corresponds to the frequency of streamwise-vortex passages above the wall. To remove the initial transient states, we take statistics from $t=10h/u_{\unicode[STIX]{x1D70F}}$ in the following discussion.

Figure 16. Normalized covariance given by the observer with different weights. Line patterns: $\cdot \,\cdot \,\cdot \,\cdot \,\cdot$ , $\unicode[STIX]{x1D716}=1$ ; $\cdots \cdots$ , $\unicode[STIX]{x1D716}=2$ ; $-\cdot \cdot -\cdot \cdot -$ , $\unicode[STIX]{x1D716}=5$ ; – ⋅ – ⋅ –, $\unicode[STIX]{x1D716}=10$ ; – – –, $\unicode[STIX]{x1D716}=20$ ;— —, $\unicode[STIX]{x1D716}=50$ ; ——, linear stochastic estimation. (a) $\widehat{\text{cov}}(u_{Obs}^{\prime },u_{DNS}^{\prime })$ . (b) $\widehat{\text{cov}}(v_{Obs}^{\prime },v_{DNS}^{\prime })$ . (c) $\widehat{\text{cov}}(w_{Obs}^{\prime },w_{DNS}^{\prime })$ . (d) $\widehat{\text{cov}}(p_{Obs}^{\prime },p_{DNS}^{\prime })$ . No linear stochastic estimation available for pressure.

Subsequently, we compare the normalized velocity covariances given in (4.4) for various $\unicode[STIX]{x1D716}$ in figure 16(a–c). To achieve good agreement near the wall, we find that very strong forcing is necessary, especially for $v$ and $w$ . The magnitude of the velocity fluctuations in the viscous sublayer is much smaller than that near the peak of turbulent kinetic energy, as shown in figure 11. In the observer, strong motions in the buffer layer, which are inaccurately estimated from the wall information, evolve over time and dominate over weak fluctuations near the wall, which are accurately imposed by the linear stochastic estimation. Thus, we must force the right hand side strongly enough to overcome it and to retain accurate motions near the wall.

Away from the wall, on the other hand, smaller weights can improve estimate. For the streamwise velocity in figure 16(a), $\unicode[STIX]{x1D716}=1$ gives the highest covariance in $|y/h|\lesssim 0.4$ (this will be clear in figure 17), and the covariance degrades with increasing weight in this region. Likewise, $\unicode[STIX]{x1D716}=5$ provides the highest covariance for the spanwise velocity over a wide range of $y$ in figure 16(c), while the linear stochastic estimation predicts best across the entire channel for the vertical velocity in figure 16(b). If we compare the correlation coefficients instead (not shown), the best weight changes, but the trend of better estimation with a smaller weight away from the wall is more or less similar. By relaxing the forcing, we can improve the over-suppression of turbulent kinetic energy in the log layer, and this benefit seems to surpass the influence of strong motions inaccurately estimated in the buffer layer.

Regarding the estimated pressure field, the normalized covariance monotonically improves with increasing $\unicode[STIX]{x1D716}$ , as shown in figure 16(d). Because of the nature of the wall boundary condition, i.e.  $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}y\approx 0$ , pressure exerted by the motions in the buffer layer penetrates the viscous sublayer. As a result, the pressure estimation on the wall is less successful compared with that of velocity fluctuations. On the other hand, such large-scale motions partly influence pressure near the centreline, leading to better estimates than those of velocity fluctuations.

Figure 17. The weight of the forcing $\unicode[STIX]{x1D716}$ providing the best covariance at each $y$ (denoted by ○) and a quartic function best fitting to them (denoted by a solid line), which is used as the spatially varying $\unicode[STIX]{x1D716}f(y)$ ( $\text{max}[\unicode[STIX]{x1D716}]\equiv 100$ ). (a) For streamwise velocity. (b) For spanwise velocity.

Referring to figure 16, we eventually show that the normalized covariance can be improved by varying $\unicode[STIX]{x1D716}$ as a function of $y$ . As a simple example, figure 17 plots the weight $\unicode[STIX]{x1D716}$ that provides the best normalized covariance in figure 16 at each $y$ (points at which the linear stochastic estimation gives the best covariance are eliminated). We then fit a quartic function to those points assuming symmetry and introduce such spatially varying $\unicode[STIX]{x1D716}f(y)$ ( ${\leqslant}100$ is imposed) into the observer for the streamwise and spanwise velocity components. For the vertical velocity estimation, the linear stochastic estimation always yields the best covariance; hence, we impose constant $\unicode[STIX]{x1D716}=100$ across the channel for simplicity (a plot for $v^{\prime }$ is omitted in figure 17).

Figure 18. Normalized covariance of the observer with the spatially varying $\unicode[STIX]{x1D716}f(y)$ given in figure 17 (denoted by a thicker black line). For reference, the envelope of the best estimate in figure 16 is plotted by dots together with the linear stochastic estimation by a thin grey line for (a–c). (a) $\widehat{\text{cov}}(u_{Obs}^{\prime },u_{DNS}^{\prime })$ . (b) $\widehat{\text{cov}}(v_{Obs}^{\prime },v_{DNS}^{\prime })$ . (c) $\widehat{\text{cov}}(w_{Obs}^{\prime },w_{DNS}^{\prime })$ . (d) $\widehat{\text{cov}}(p_{Obs}^{\prime },p_{DNS}^{\prime })$ . No linear stochastic estimation available, but the $\unicode[STIX]{x1D716}=50$ case from figure 16(d) is plotted by a grey dashed line for reference.

In figure 18, we compare the results for the set of such $\unicode[STIX]{x1D716}f(y)$ with the envelopes of the best estimate in figure 16 as well as the linear stochastic estimation. Since a result of the linear stochastic estimation is unavailable for the pressure estimation, we overlay the $\unicode[STIX]{x1D716}=50$ case for reference in figure 18(d). The spatially varying weight predicts the streamwise and spanwise velocity components better than the other two profiles except very close to the wall. In particular, the improvement for $\widehat{\text{cov}}(u_{Obs}^{\prime },u_{DNS}^{\prime })$ is appreciable over a wide range of $y$ . In contrast, the covariance for the vertical velocity approximately remains the same since we only impose strong uniform forcing across the channel. Likewise, the degree of success in the pressure estimation is similar to the strongest uniform forcing of $\unicode[STIX]{x1D716}=50$ . Although we have not fully optimized the weight, these results demonstrate that we can improve the estimate in a certain degree by tuning the forcing function. On the other hand, the improvement of estimation for the two velocity components hardly helps the improvement of the remaining components. The comparison of the correlation coefficients in figure 19 shows consistent results.

Figure 19. Comparison of the correlation coefficients. Line patterns: ——, $\text{Cor}(u_{Obs}^{\prime },u_{DNS}^{\prime })$ ; – – –, $\text{Cor}(v_{Obs}^{\prime },v_{DNS}^{\prime })$ ; – ⋅ – ⋅ –, $\text{Cor}(w_{Obs}^{\prime },w_{DNS}^{\prime })$ . Thick black lines denote the results of the spatially varying $\unicode[STIX]{x1D716}f(y)$ and thin grey lines those of the linear stochastic estimation.

It is evident from figure 20(a) that the spatially varying weight helps relax the over-suppression of turbulent kinetic energy away from the wall compared with the linear stochastic estimation. However, the errors themselves somewhat increase near the centreline, indicating that the elevated turbulence intensity does not accurately fill large-scale motions. Figure 20(b) clarifies that the shear stress also slightly increases beyond the buffer layer. Unlike the linear stochastic estimation, the sum of the molecular and the eddy viscosities must comprise the total shear stress, which dictates the mean velocity profile. As a result, $\overline{u}(y)$ in the observer slightly deviates from the true solution. Nonetheless, the spatially varying weight provides the best estimate in the most regions for the streamwise and spanwise velocity components. Although we may speculate improvement by further relaxation of the forcing or by further enhancement of turbulent kinetic energy near the centreline, we found no other effective profile of $\unicode[STIX]{x1D716}f(y)$ among many other trials.

Figure 20. Turbulent quantities generated by the spatially varying weight. Line patterns: ——, original Navier–Stokes; – – – (black), spatially varying $\unicode[STIX]{x1D716}f(y)$ ; – – – (thin grey), linear stochastic estimation. (a) Turbulent kinetic energy ( $\cdots \cdots$ , the squared difference). (b) Reynolds shear stress ( $\cdots \cdots$ , total shear stress).

Figure 21. Comparison of instantaneous flow fields in a three-dimensional view. Iso-surfaces of the second invariant of the velocity gradient tensor $\unicode[STIX]{x1D64C}_{+}$ are displayed to visualize vortex cores. Contours of the streamwise velocity are also drawn. (a) True solution on the top ( $\unicode[STIX]{x1D64C}_{+}=-0.02$ ). (b) Estimate using the spatially varying weight on the bottom ( $\unicode[STIX]{x1D64C}_{+}=-0.01$ ).

Figure 22. Comparison of instantaneous flow fields in a cross-sectional view at $x/h=3.0$ . Streamwise velocity contours are drawn with velocity vectors on perpendicular planes. (a) True solution. (b) Estimate using the spatially varying weight.

In figure 21, an instantaneous flow field estimated using the spatially varying weight is compared with the true solution at one of the best instants in a three-dimensional view. Their cross-sections are also compared in figure 22. Although large-scale streaks creeping near the wall are estimated relatively well, those detached from the wall as well as quasi-streamwise vortices away from the wall are mostly smeared, as expected from the results above. It is rather surprising from figure 22 that such distinctive structures are still difficult to be reconstructed based on the wall measurement with the full resolution. We should remember, however, that any linear combinations of the flow representation cannot improve the estimation, and the weak correlation across different wavenumbers would make nonlinear estimation ineffective.

In appendix D, we apply the ensemble Kalman filter (Evensen Reference Evensen1994) analogous to the work by Colburn et al. (Reference Colburn, Cessna and Bewley2011) to the current channel geometry. Although there are some differences in the flow solver architecture, we examine a few possible scenarios to replicate their work. Counter to their conclusion, the correlations near the centreline are less than $0.5$ for all the cases (our results are displayed based on the normalized covariance, but this statement still holds). This is rather consistent with our analyses above. Thus, our results imply that the observer solving the dynamical system improves the accuracy in the viscous sublayer in a limited extent relative to the instantaneous estimation, i.e. the linear stochastic estimation. Although Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006) reported comparable results using the extended Kalman filter in $y^{+}\leqslant 20$ , it is fair from the discussions in appendices to ascribe the major contribution of the estimation in this region to the instantaneous estimation rather than the observer solving the dynamical system or the optimized feedback.

We observe that a series of our analyses reveal generic viewpoints in turbulent flow estimation based on wall measurement. The results of the following two analyses indicate that the linear part of the stochastic estimation, which is decoupled in the wavenumber space, dictates instantaneous estimation: one is poor correlation across different wavenumbers between the quantities on the wall and the velocity components in the flow, as shown in § 4.1; the other is limited improvement in the estimation by including time derivatives of the measured quantities and by using the ensemble Kalman filter, performed in appendices B and D, respectively. In addition, the analysis in § 2.2 manifests that the results of the estimation should be independent of the flow representation. Therefore, we cannot expect much improvement by incorporating nonlinearity or temporal variation in the estimation scheme or by changing it from the wavenumber space to the physical space except that they may help ‘memorize’ flow patterns during the sampling duration. The linear stochastic estimation can accurately reconstruct velocity components in the viscous sublayer, but inaccurately estimated motions in the buffer layer inevitably disrupt the estimation toward the centreline. Even by introducing an observer that solves the full Navier–Stokes equations, we can improve the estimation in a limited extent beyond the viscous sublayer. All these results may imply inherent difficulty of turbulent flow estimation even using sequential data assimilation techniques. Such techniques can remove unbiased uncorrelated measurement noise, but even a simple sequential observer has the same function, as demonstrated in appendix A. In fact, the accuracy of the estimation by using the extended Kalman filter (Chevalier et al. Reference Chevalier, Hœpffner, Bewley and Henningson2006) is only as good as the linear stochastic estimation. There is a study reporting nearly perfect estimation of a similar turbulent channel flow based on analogous wall measurements using the ensemble Kalman filter (Colburn et al. Reference Colburn, Cessna and Bewley2011). However, which function of their method truly enables the success is not understood.