3.1. Performance of the algorithm: multiple windows and spectral analysis

Figure 3.

Instantaneous streamwise-velocity fluctuations at $z = 0.49L_z$ , calculated relative to the true mean. (i) Adjoint-variational reconstruction; (ii) true fields; estimation errors. Panels ( $a$ ) and ( $b$ ) correspond to the initial ( $t=0$ ) and final ( $t=T$ ) times of the first window. The black boxes highlight the region where turbulence fluctuation is underestimated at $t=0$ , but better reconstructed at $t=T$ .

In this section, we focus on results from channel flow at $Re_\tau = 590$ . We start by considering fully resolved velocity observations outside $l_y^+ = 50$ . The unknown wall-attached layer includes the peak turbulent kinetic energy (TKE) and the majority of the contributions to the TKE budget, such as production, dissipation and turbulent transport. In terms of vorticity dynamics, no information about the flux of vorticity at the wall is available in the observations nor of the intense near-wall stretching and ejection events that lead to the extrema in the wall stresses (Eyink, Gupta & Zaki Reference Eyink, Gupta and Zaki2020; Wang et al. 2022a). In addition, beyond $y^+ = 50$ , the transport of turbulent enstrophy almost vanishes (Gorski, Wallace & Bernard Reference Gorski, Wallace and Bernard1994). Since an important dynamical region of the channel is absent from the observations, an accurate reconstruction of the turbulence is expected to be challenging.

A qualitative perspective on the adjoint-variational estimated state is provided in figure 3(i), for the first assimilation window $t \in [0, T]$ . At the initial time $t=0$ , the estimated streamwise fluctuations (panel $a$ -i) near the first observation plane at $y^+=50$ closely match the true state (panel $a$ -ii). At locations more distant away from the first observation plane, and closer to the wall, the estimation quality deteriorates, as highlighted within the black dashed boxes in panels ( $a$ -i) and ( $a$ -ii). The magnitude of the fluctuations is underestimated, due to the lack of sensitivity of observations to the initial state in the near-wall layer. The difference between the estimated and true field is quantified in panel ( $a$ -iii), which clearly shows the increase in errors away from the first measurement point. As the estimated state evolves forward using the Navier–Stokes equations, there is a noticeable improvement in the reconstructed turbulent kinetic energy and small-scale motions, which is highlighted within the dashed boxes in panels ( $b$ -i) and ( $b$ -ii). The estimation error is also reduced from $t = 0$ to $t = T$ , as shown in panel ( $b$ -iii). The duration of a single observation window, $T^+ = 31$ , may be insufficient for the estimated near-wall flow to reach a statistically stationary state. In addition, this time horizon may also be insufficient for all the scales near the wall to affect the outer flow and, hence, the observations. For these reasons, data assimilation across multiple windows has the potential to further improve the estimation accuracy.

Figure 4.

( $a$ ) Convergence history of the cost function when adjoint-variational approach is applied to six consecutive observation windows separately ( $T = 0.96$ , $T^+ = 31$ ). In all six panels, the cost function is normalised by the initial value of the first window, $\mathcal {J}_0$ . ( $b$ ) Volume-averaged error in the velocity versus time, for the Navier–Stokes evolution of the first guess (dashed), the adjoint-variational estimated state (solid) and the predicted flow starting from the final observation time $t = 6T$ (dotted).

The adjoint-variational data assimilation is repeated over six consecutive windows, which are sufficient to ensures that the estimated near-wall turbulence reaches a statistically stationary state. The convergence histories of the cost functions in these windows are shown in figure 4(a). All the cost functions are normalised by the value from the first guess in the first window, $\mathcal {J}_0$ . Within each assimilation window, after 100 forward-adjoint loops, the cost function always drops by more than two orders of magnitude. At the start of each subsequent window, however, the cost increases relative to the final iteration of the preceding one, which is expected because the new observations were not at all taken into account in the preceding assimilation. Nonetheless, the overall trend is a progressive decrease in the cost function, which, at the end of the sixth window, reaches $10^{-4}\mathcal {J}_0$ . Recall that the cost function is defined as the kinetic energy norm of the errors in reproducing the observations. As such, the cost reduction by the 4DVar algorithm translates to a reduction in the errors of reproducing the observed velocities to 1% of the initial values.

Beyond reproducing observations, it is crucial to evaluate the performance of the adjoint-variational approach in the near-wall layer where observations are not available. The estimation accuracy is quantified using the root-mean-square (r.m.s.) error between the estimated and true states,

(3.1) \begin{equation} \mathcal {E}_I(\boldsymbol {u}) = \langle \left \| \boldsymbol {u} - \boldsymbol {u}_r\right \|^2 \rangle _{\Omega _I}^{1/2}, \end{equation}

where $\langle \cdot \rangle$ denotes averaging and the subscript represents the averaging domain. The temporal evolution of the estimation error is plotted in figure 4( $b$ ). The dashed line reports the outcome of advancing the initial guess, which is obtained by linear stochastic estimation at $t=0$ , using the Navier–Stokes equations; the solid line is the outcome of performing the sliding-window adjoint reconstruction within the six time intervals. Starting with the LSE result, the estimation error monotonically increases, which is a manifestation of the chaotic nature of turbulence. In contrast, the error of the adjoint-estimated state progressively decreases, demonstrating the convergence to the true turbulent state. If the estimated state at $t = 6T$ is marched forward using the Navier–Stokes equations, the error starts to grow due to turbulence chaos (dotted line in figure 4 $b$ ). Despite the divergence of the predicted flow from the true state, the accuracy of the forecast remains comparable to that within the assimilation horizon for a short time.

Figure 5.

Comparison of the instantaneous horizontally averaged flow statistics between adjoint-variational estimation (black solid) and true field (red dashed) at (i) $t = T$ and (ii) $t = 6T$ . ( $a$ , $b$ , $c$ ) Root-mean-square fluctuations of streamwise velocity, spanwise velocity and pressure, evaluated by subtracting their true mean.

To determine whether the estimated state reaches statistical stationarity in the wall layer, we examine the wall-normal profiles of the r.m.s. fluctuations in figure 5 and the probability density function (p.d.f.) of the wall shear stresses and pressure in figure 6. At $t=T$ , the reconstructed fluctuations of velocity and pressure are all weaker than the true state (figure 5 a-i), which aligns with the qualitative comparison in figure 3. Additionally, the p.d.f.s of the estimated wall signals exhibit narrower tails than the true profiles, which indicates a reduced probability of the extreme events at $t=T$ . At the end of the sixth window, $t=6T$ , the reconstructed fluctuations and p.d.f.s closely match the true statistics. Combined with the diminishing estimation error in figure 4, these results validate that the adjoint-variational algorithm successfully converges to a statistically stationary turbulent state that shadows the true flow state in the outer layer and reproduces its observations.

The adjoint-variational reconstruction of streamwise fluctuation at $t=6T$ is visualised in figure 7( $a$ ), with a comparison against the true field and a linear stochastic estimation. Here, the LSE is performed using contemporaneous outer observations, at $t=6T$ . Inclusion of earlier observations into LSE could improve the estimation accuracy, but previous research has explored this strategy and concluded such enhancements are minimal (Encinar & Jiménez Reference Encinar and Jiménez2019). The two visualised horizontal planes are at the midpoint of the unknown layer ( $y^+ = 25$ ) and the location of peak turbulent kinetic energy ( $y^+ = 12$ ). At both planes, the adjoint estimation reproduces all the scales of the near-wall turbulence, while LSE only captures the large-scale motions. A quantitative assessment is provided in panel (b), which presents the wall-normal profiles of the horizontally averaged estimation errors. The largest error using LSE is 75 % of the maximum r.m.s. fluctuations, which implies that only 25 % of the near-wall turbulent kinetic energy can be estimated from outer motions when linear methods are adopted. The error of the adjoint-estimated field is only 17 % of the maximum r.m.s. fluctuations, which is 77 % lower than LSE and demonstrates the benefit of turbulence reconstruction using the full nonlinear Navier–Stokes equations.

Figure 6.

Comparison of the probability density function of wall signals between adjoint-variational estimation (black solid) and true field (red dashed) at (i) $t = T$ and (ii) $t = 6T$ . ( $a$ , $b$ , $c$ ) Fluctuations of wall shear stress $\tau _{xy}^{\prime }$ , $\tau _{yz}^{\prime }$ and pressure $p^{\prime }$ , calculated by subtracting the true mean and normalised by the true mean shear stress at the wall $\langle \tau _{\textrm {w}} \rangle$ .

Figure 7.

( $a$ ) Streamwise velocity fluctuations in the adjoint-variational estimated field, true state and linear stochastic estimation, at $t = 6T$ . The visualised fields are normalised by the true r.m.s. fluctuations at the corresponding wall-normal location. A version of this figure is featured in the recent review by Zaki (Reference Zaki2025, CC BY 4.0). ( $b$ ) Horizontally averaged estimation error of streamwise velocity, reconstructed using LSE (dashed) and adjoint approach (solid), normalised by the maximum value of r.m.s. streamwise fluctuations.

Figure 8.

One-dimensional Fourier spectrum of estimation error (blue solid) and the true streamwise velocity (red dashed), averaged within $t \in [5T, 6T]$ , evaluated at ( $a$ , $b$ ) $y^+ = \{25, 12\}$ . (i) Streamwise spectra averaged in the span; (ii) spanwise spectra averaged along the streamwise direction.

To further evaluate the estimation accuracy at different scales, we compute the streamwise spectrum of the adjoint-variational estimation error,

(3.2) \begin{equation} \mathcal {E}_{zt}(\hat {u}) = \langle | \hat {u} - \hat {u}_r |^2\rangle ^{1/2}_{zt}, \end{equation}

where $\hat {u}$ is the Fourier transform of $u$ in the horizontal directions. In the above expression, the error is averaged along the spanwise direction and within the last assimilation window $t\in [5T,6T]$ . Similarly, the spanwise spectrum $\mathcal {E}_{xt}(\hat {u})$ is evaluated by averaging the error along the streamwise direction. The resulting spectra at $y^+ = \{25, 12\}$ are plotted as blue solid lines in figures 8( $a$ ) and 8( $b$ ), respectively. Although the estimation errors are larger for longer wavelengths, this trend must be viewed relative to the high energy content in these scales, as shown by red dashed lines. Theoretically, if the estimated state is fully developed and independent of the truth, the estimation error would equal $\sqrt {2}$ times the true spectra. This argument follows from decomposing the mean-square error,

(3.3) \begin{equation} \mathcal {E}_{zt}^2(\hat {u}) = \langle |\hat {u}- \hat {u}_r|^2 \rangle _{zt} = \langle |\hat {u}|^2 \rangle _{zt} + \langle |\hat {u}_r|^2 \rangle _{zt} - 2 \langle |\hat {u}^* \hat {u}_r| \rangle _{zt} = 2\langle |\hat {u}_r|^2 \rangle _{zt}, \end{equation}

where the last equality used the assumption of a fully developed estimated state (as such $\langle |\hat {u}|^2 \rangle _{zt} = \langle |\hat {u}_r|^2 \rangle _{zt}$ ) which is independent of the true flow ( $\langle \hat {u}^* \hat {u}_r \rangle _{zt} = 0$ ). Conversely, if the estimated state is correlated with the truth, the error is lower than $\sqrt {2}$ times the true spectra. Our results in figure 8 show that the adjoint estimation error is an order of magnitude smaller than the true spectra at the energy-containing large scales, implying a strong positive correlation between the estimated and true large-scale motions. The error reduction is less significant for smaller scales that represent the dissipative eddies and at locations farther from observations. Nevertheless, the estimation error remains below the true spectrum across all the scales, which indicates that even the smallest Kolmogorov eddies in the reconstructed wall layer are correlated with the true flow. These results demonstrate the capability of our approach to reconstruct all the missing scales of near-wall turbulence, while reproducing velocity observations in the outer layer.

Figure 9.

Wall-normal distribution of the one-dimensional Fourier spectrum of estimation error, normalised by the spectrum of the true field. ( $a$ ) Streamwise velocity, ( $b$ ) spanwise velocity and ( $c$ ) pressure. (i,ii) Streamwise and spanwise spectra. The thickness of the unknown layer is $l_y^+ = 50$ . Panels ( $a$ -i) and ( $a$ -ii) featured in the recent review by Zaki (Reference Zaki2025, CC BY 4.0).

The wall-normal dependence of the error spectra is reported in further detail in figure 9 for the streamwise velocity, spanwise velocity and pressure. To ensure a consistent comparison across different flow quantities, the spectra of error are normalised by the true spectra of the corresponding variable. A key observation is that the large-scale motions, such as quasi-streamwise rolls, are accurately estimated throughout the entire unknown layer, while the accuracy of small scales deteriorates with the distance from the observed regions. These results are qualitatively consistent with the characteristics of the coherent structures in wall turbulence (Jiménez Reference Jiménez2018): longer eddies are also deeper in the wall-normal direction and the depth saturates until the eddies become attached to the wall. However, different from the forward perspective that focuses on the evolution of coherent structures, the adjoint approach reveals the turbulent dynamics from a dual, or inverse, perspective. To construct a Navier–Stokes solution that reproduces outer observations, the large scales across the entire wall layer and the small scales in the vicinity of outer observations must be accurately reconstructed. The remaining flow structures, especially the small scales far from the outer layer, are only partially observable from the outer flow.

We conclude this discussion by visualising the shear stresses and pressure at the wall (figure 10), which is the most distant location from observations in the observed outer layer. For all three components, the adjoint reconstruction (panels i) is almost identical to the true state (panels ii), despite slight mismatch in the small scales, as quantified by the spectra of the estimation error (panels iii). Recall that the thickness of the unknown layer, $l_y^+ = 50$ , exceeds the threshold that guarantees a perfect synchronisation. Although the estimation error cannot converge to machine zero, the reconstructed near-wall turbulence is sufficiently accurate to evaluate any flow quantity of interest, such as pressure gradients or vortical structures.

Figure 10.

(i) Adjoint-variational estimation of fluctuations of wall shear stress and pressure at $t = 6T$ when $l_y^+ = 50$ . (ii) The true fields. (iii) Spanwise spectra of the estimation error of wall quantities averaged in $t \in [5T,6T]$ (blue), compared with the spectra of the true field (red dashed). ( $a$ , $b$ , $c$ ) Two components of the wall shear stress and pressure, $\{\tau _{xy}^+, \tau _{yz}^+,p^+\}$ .

3.2. Influence of the unobserved layer thickness and of the Reynolds number

When the thickness of the unobserved layer is increased and the observations are more separated from the wall, we can anticipate that an accurate reconstruction of the near-wall turbulence becomes more difficult. From the perspective of the adjoint-variational optimisation algorithm, fewer observations imply that a greater number of Navier–Stokes solutions can replicate the available data with an acceptable level of accuracy. Accordingly, the landscape of the cost function may be expected to feature numerous local minima that can forestall the optimisation algorithm. Moreover, the first guess of the initial condition, which is constructed using linear stochastic estimation, deviates more substantially from the true state, which exacerbates the difficulty of convergence to the global minimum of the cost function. From the perspective of turbulence dynamics, an elevated $l_y$ results in fewer wall-attached eddies being captured in the observations (Marusic & Monty Reference Marusic and Monty2019), leaving more near-wall flow structures unobservable from outer measurements. To quantitatively determine the impact of $l_y$ on the estimation accuracy, we consider $l_y^+ = \{70, 90\}$ and perform adjoint-variational data assimilation for six consecutive windows as in § 3.1. The effect of Reynolds number will also be examined briefly.

We first analyse the adjoint-variational reconstruction with $l_y^+ = 90$ at $Re_{\tau } = 590$ . The evolution of the cost function, estimation error and flow statistics across the six assimilation windows are similar to figure 4 and thus are not repeated here. Within the last window $t \in [5T, 6T]$ , the spectra of estimation error are evaluated and visualised in figure 11( $a$ ) for the streamwise velocity. Within $y^+ \in [40, 90]$ , the dependence of estimation error on the wavelength is similar to the $l_y^+ = 50$ case in figure 9( $a$ ): longer wavelengths are reconstructed more accurately and the associated errors are less affected by the distance from the first observation plane. A particularly interesting region is $\{y^+ \lt 40, \lambda _x^+ \gtrsim 2000, \lambda _z^+ \gtrsim 200\}$ , where the error of near-wall large scales becomes almost independent of the wall-normal coordinate. However, the error of smaller scales within $y^+ \lt 40$ remains increasing with the distance from the observations until it saturates. These results are reminiscent of the bimodal behaviour of the energy spectra of wall turbulence and the associated two categories of coherent structures: outer large-scale motions and inner-layer streaks (Hutchins & Marusic Reference Hutchins and Marusic2007). The latter are predominately located at $y^+ \approx 15$ with characteristic wavelengths $\lambda _x^+ \approx 1000, \lambda _z^+ \approx 100$ , which correspond to the ‘inner peak’ of the energy spectra of streamwise velocity fluctuations. The instability of these streaks leads to the formation of quasi-streamwise vorticies and the regeneration cycle of near-wall turbulence. Based on the numerical experiments by Jiménez & Pinelli (Reference Jiménez and Pinelli1999), the near-wall cycle is self-sustained and almost independent from turbulent motions in the outer layer. Therefore, the inner-layer streaks are difficult to reconstruct using outer observations in $y^+ \gt 90$ . The other category of coherent structures, namely the outer large-scale motions, contribute to the ‘outer peak’ of the energy spectra. The sizes of these structures scale with outer units, $\lambda _x, \lambda _z \sim O(h)$ . Despite the residence of these structures in the outer layer, they have a significant influence on the inner-layer turbulence through the amplitude modulation mechanism, which facilitates an accurate estimation of near-wall large scales.

Figure 11.

( $a$ ) Wall-normal distribution of the one-dimensional Fourier spectra of estimation error of streamwise velocity, normalised by the spectra of the true state, $\mathcal {E}_{zt}(\hat {u}) / |\hat {u}_r|$ . ( $b$ ) Coherence spectra of streamwise velocity, $\mathcal {R}(\hat {u}(l_y),\hat {u}(y))$ . (i,ii) Streamwise and spanwise spectrum. Black dashed lines, $\mathcal {R} = 0.5$ ; black dash-dotted lines in panel ( $a$ ), $\lambda _z^+ = 2(l_y^+ - y^+)$ . The thickness of the unknown layer is $l_y^+ = 90$ .

Although the estimation accuracy in figure 11 is consistent with the physical property of coherent structures in wall turbulence, the present results should not be simply interpreted as a confirmation of previous findings. The coherent structures are generally extracted using isosurfaces of two-point correlations that only quantify the linear dependence across different locations, while the adjoint-variational approach uses information from the full Navier–Stokes equations. In fact, it is illustrative to compare the spectra of estimation error with the coherence spectrum,

(3.4) \begin{equation} \mathcal {R}\left (\hat {q}_1(\boldsymbol {k},l_y), \hat {q}_2(\boldsymbol {k},y) \right ) = \frac {|\langle \hat {q}_1(\boldsymbol {k},l_y) \hat {q}_2^*(\boldsymbol {k},y)\rangle | }{ \langle |\hat {q}_1(\boldsymbol {k},l_y)|^2\rangle ^{1/2} \langle |\hat {q}_2(\boldsymbol {k},y)|^2\rangle ^{1/2} }, \end{equation}

which quantifies the correlation between the observation $\hat {q}_1(\boldsymbol {k},l_y)$ and the flow variable within the unknown layer $\hat {q}_2(\boldsymbol {k},y)$ in Fourier space. To ensure convergence of the coherence spectrum, the average in (3.4) is performed over 400 advective time units. The coherence spectra of the streamwise velocity fluctuation are plotted in figure 11( $b$ ) and the contour lines of $\mathcal {R} = 0.5$ are reproduced in figure 11( $a$ ) (black dashed lines) for a more direct comparison. A key observation is that the coherence tends to underestimate the observability of near-wall turbulence from outer measurements, especially for the near-wall large scales. For example, at $\lambda _z^+ = 300$ and $y^+ \leq 20$ , the coherence is below $0.2$ , suggesting a weak correlation between these structures and the outer observations, while the adjoint-variational estimation error is only $50\, \%$ of the true spectrum. At an arbitrary location $y$ within the unknown layer, the accurately reconstructed spanwise waves can be better demarcated by $\lambda _z \geq 2(l_y - y)$ (dash-dotted line in panel $a$ -ii). This relation is derived by assuming the structures with wavelength $\lambda _z$ centred at $y = l_y$ have a wall-normal depth proportional to $0.5 \lambda _z$ .

Figure 12.

Filtered streamwise velocity fluctuation of ( $a$ ) adjoint-variational estimation and ( $b$ ) the true state, visualised by isosurfaces of $u^{\prime } = -0.12$ within $y^+ \in [0,90]$ . Blue isosurfaces, the filtered wavelengths satisfy $\mathcal {R}(\hat {u}(\lambda _x,l_y),\hat {u}(\lambda _x,y)) \geq 0.5$ and $\mathcal {R}(\hat {u}(\lambda _z,l_y),\hat {u}(\lambda _z,y)) \geq 0.5$ . Green isosurfaces, same filter for $\lambda _x$ as blue, but $\lambda _z^+ \geq 2(l_y^+ - y^+)$ .

The predictions of near-wall structures from outer measurements using the adjoint approach and based on properties of the coherence spectra are further differentiated in figure 12. The blue isosurfaces represent the streamwise fluctuations extracted using a coherence-based filter, where the wavelengths $(\lambda _x,\lambda _z)$ of the filtered field satisfy $\mathcal {R} (\hat {u}(\lambda _x,l_y),\hat {u}(\lambda _x,y) ) \geq 0.5$ and $\mathcal {R} (\hat {u}(\lambda _z,l_y),\hat {u}(\lambda _z,y) ) \geq 0.5$ . The same streamwise filter is adopted for the green isosurfaces, but the spanwise filter is changed to $\lambda _z \geq 2(l_y - y) + 2\triangle z$ . In the vicinity of observed outer layer, both filters capture a wide range of scales in the adjoint-variational estimation (panel $a$ ) that match the true state (panel $b$ ). Deeper towards the wall, the coherence-based filter fails to identify the majority of the large-scale structures that are accurately reconstructed by the adjoint approach. Another advantage of adjoint-variational data assimilation is the quantification of causality among different flow structures, or events, in turbulence. Only the structures that influence the observed signals can be accurately reconstructed. In other words, the flow structures in the unknown layer can be classified as ‘observation-attached’ and ‘observation-detached’ eddies, although the term ‘attached’ takes on a more general meaning since it entails an influence on the observations within the observation horizon and through the dynamics of the Navier–Stokes equations. The ‘observation-attached’ eddies in the present case consist of the small-scale motions near the observed region and the large scales that penetrate across the wall layer. These structures align with the domain of dependence of observations, which is determined by marching the adjoint equations backwards in time (Wang et al. Reference Wang, Wang and Zaki2022b ). The ‘observation-detached’ eddies are primarily near-wall structures, ranging from self-sustained streaks to Kolmogorov eddies that are not encoded in the outer observations.

Figure 13.

One-dimensional Fourier spectra of the estimation error of ( $a$ ) spanwise velocity and ( $b$ ) pressure, evaluated at different wall-normal locations and normalised by the spectrum of the respective true field. (i,ii) Streamwise and spanwise spectra.

The spectra of estimation error of the spanwise velocity, shown in figure 13( $a$ ), are analogous to the streamwise velocity in figure 11( $a$ ). The quasi-streamwise rolls are reconstructed with the highest accuracy. The discrepancy between the adjoint estimation error and the coherence spectra ( $\mathcal {R}(\hat {w},\hat {w})=0.5$ , black dashed lines) is more evident here than in figure 11( $a$ ). The coherent structures identified using $\mathcal {R}(\hat {w},\hat {w})=0.5$ never touch the wall, while the adjoint approach successfully reconstructs the large-scale $w^{\prime }$ structures that are attached to the wall. The comparison between error spectra of pressure and the coherence spectra $\mathcal {R}(\hat {p},\hat {p})=0.5$ is provided in figure 13( $b$ ). As reported in previous studies (Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2014), the coherence-based pressure structures are moderately elongated along the spanwise direction. An accurate reconstruction of these structures can be difficult since the outer pressure data are not available for the adjoint approach. Due to the non-local property of pressure, the outer pressure field, especially near the edge of observed region $y = l_y$ , cannot be fully determined from the velocity measurements. Nevertheless, the adjoint approach effectively captures the dominant coherent structures of pressure in the near-wall layer.

The adjoint reconstructions of the wall shear stresses and pressure are visualised in figures 14(i), and compared with the true fields in panels (ii). Overall, the estimation is less accurate than in the $l_y^+ = 50$ case reported in figure 10. Here, most of the streaks and local extrema of stresses are captured, but their locations and magnitudes are not precisely reproduced. The estimation errors in Fourier space are quantified in figure 14(iii). The errors in small scales are approximately $\sqrt {2}$ times the true spectra of the corresponding wall signal, which indicates that the reconstructed dissipative eddies are nearly independent from the true state. For longer waves, the errors are lower than the true spectra, which is consistent with the qualitative description in panels (i) and (ii). These structures can also be extracted using the same low-pass filter as in figure 12 for the green isosurfaces. At the wall, the filtered wavelengths, $\lambda _x^+ \geq 3000$ and $\lambda _z^+ \geq 180$ , approximately coincide with the intersections between the spectra of error and the true state in figure 14(iii). The filtered structures of the streamwise wall shear stress and pressure are visualised in figure 15. The locations and shapes of the estimated large scales are similar to the true filtered fields, which demonstrates that accurate prediction of the near-wall large scales is essential to reproduce the outer large-scale observations.

Figure 14.

(i) Adjoint-variational estimation of fluctuations of wall shear stress and pressure at $t = 6T$ when $l_y^+ = 90$ . (ii) The true fields. (iii) Spanwise spectra of the estimation error of wall quantities averaged in $t \in [5T,6T]$ (blue), compared with the spectra of the true field (red dashed). ( $a$ , $b$ , $c$ ) Two components of the wall shear stress and pressure, $\{\tau _{xy}^+, \tau _{yz}^+,p^+\}$ .

Figure 15.

Large-scale structures of the ( $a$ ) streamwise wall shear stress and ( $b$ ) wall pressure, visualised using (i) adjoint-variational estimation and (ii) the true state. Large scales are extracted by applying a low-pass filter ( $\lambda _x^+ \geq 3000, \lambda _z^+ \geq 180$ ) to the full field.

Figure 16.

Influence of thickness of the unknown layer and Reynolds number on the estimation accuracy, quantified using the correlation coefficient between the estimated and true state. The correlations are averaged along the horizontal directions and in time, $t \in [5T,6T]$ . ( $a$ , $b$ , $c$ ) Fluctuations of streamwise velocity, spanwise velocity and pressure: $l_y^+ = \{50,70,90\}$ (blue, green, black); $Re_{\tau }=\{392,590\}$ (dashed, solid).

Figure 16 illustrates the impact of the thickness of the unknown layer $l_y$ and of the Reynolds number on the estimation accuracy. The figure reports the correlation coefficient between true and estimated states,

(3.5) \begin{equation} \mathcal {C}_{xzt}(q) = \frac {\langle q q_r\rangle _{xzt}}{\langle q^2\rangle _{xzt}^{1/2} \langle q_r^2\rangle _{xzt}^{1/2}}. \end{equation}

The average in (3.5) is performed along the horizontal directions and over the last assimilation window $t \in [5T,6T]$ . Within the observed region ( $y \gt l_y$ ), the correlations of the streamwise and spanwise fluctuations are equal to unity since the adjoint approach accurately reproduces the velocity observations, and the correlation of pressure remains higher than $0.95$ despite the lack of pressure data. Outside the observed region, the correlation remains close to unity when $l_y^+ = 50$ . As the unknown layer is expanded ( $l_y^+ = 70,90$ ), the correlation decays with distance from the first observation plane towards the wall. Note that when the profiles of these correlations are plotted versus $y^+$ , they appear approximately independent of $Re_{\tau }$ . The trend indicates that the decay in the correlations is primarily due to the inaccurate estimation of the inner-layer streaks and smaller scales. These structures dominate the kinetic energy of the near-wall turbulence and their sizes scale with viscous units.

The highest attainable prediction accuracy of the wall stresses and pressure, given fully resolved outer observations, is summarised in figure 17. When $l_y^+ \lesssim 30$ , we can achieve a perfect synchronisation of near-wall turbulence by continuously injecting observations into the Navier–Stokes solution (Wang & Zaki Reference Wang and Zaki2022). Therefore, the correlation coefficient between the true and estimated quantities is precisely equal to unity. Beyond thirty viscous units, synchronisation is no longer feasible. However, the wall stresses and pressure can still be accurately reconstructed for $l_y^+ \leq 50$ using the adjoint-variational approach. As the observations are farther separated from the wall, the correlation coefficient monotonically decreases, reaching $ (\mathcal {C}(\tau _{xy}^{\prime }), \mathcal {C}(\tau _{yz}^{\prime }), \mathcal {C}(p^{\prime }) ) = (0.42, 0.26, 0.49)$ at $l_y^+ = 90$ . Unlike the linear state estimation methods, the adjoint reconstruction of wall stresses and pressure are not only correlated with the outer flow, but also dynamically and causally consistent with the Navier–Stokes evolution reproducing the outer observations.

3.3. Effect of measurement resolution and filtering

The discussion thus far has assumed access to fully resolved velocity data in the outer layer, which is difficult to acquire in practice. It is also desirable to investigate whether the estimation accuracy achieved in the previous sections is robust against limited measurement resolution in the outer layer. Intuitively, if the data resolution is in the limit of excessively coarse data resolution, the estimation quality of the wall layer will be significantly compromised. In this section, we consider what can be regarded as moderately coarse velocity data in the outer layer, and focus on the estimation accuracy of wall stresses and pressure at $Re_{\tau } = 590$ .

Figure 17.

Estimation accuracy of wall signals versus thickness of the unknown layer, when fully resolved outer observations are adopted. The accuracy is quantified using the correlation coefficient between the true wall signals and their estimations. Symbols ( $\circ , +, \times$ ) correspond to the two components of wall shear stress and pressure ( $\tau _{xy},\tau _{yz},p$ ). A version of this figure featured in the recent review by Zaki (Reference Zaki2025, CC BY 4.0).

The outer velocity fields that furnish the observations are first filtered and then sub-sampled. Specifically, the fully resolved fields obtained from DNS are box-filtered along the horizontal directions,

(3.6) \begin{equation} \tilde {\boldsymbol u}(\boldsymbol x,t) = \int G(\boldsymbol r) \boldsymbol u(\boldsymbol x - \boldsymbol r,t) {\rm d} \boldsymbol r. \end{equation}

The filter function $G(\boldsymbol {r})$ is defined as

(3.7) \begin{equation} G(\boldsymbol r) = G(r_x,r_z) = \frac {1}{\triangle _{fx}\triangle _{fz}} H\left (\frac 12 \triangle _{fx} - |r_x| \right ) H\left (\frac 12 \triangle _{fz} - |r_z|\right ), \end{equation}

where $H(\cdot )$ is the Heaviside step function and $(\triangle _{fx},\triangle _{fz})$ are filter widths in the streamwise and spanwise directions. The filtered velocity is then sub-sampled on every eighth grid point in each dimension, including space and time. As a result, the sparse observations are separated by $\triangle x_m^+ = 77$ , $\triangle y_m^+ \in [12,52]$ and $\triangle z_m^+ = 39$ , and their temporal resolution is $\delta t_m^+ = 0.51$ , which are comparable to typical resolutions of experimental measurements of wall turbulence (Lu et al. Reference Lu, Xiang, Zaki and Katz2024). Although the Kolmogorov eddies are not resolved by the observations, such a set-up has been demonstrated sufficient to accurately reconstruct the turbulence in between the observation sites, when observations are available throughout the channel volume (Wang & Zaki Reference Wang and Zaki2021), while here, in contrast, we only consider outer observations. The filter width is selected slightly smaller than the observation spacing, $\triangle _{fx}^+ = 58$ and $\triangle _{fz}^+ = 29$ , which correspond to the size of six DNS grid cells in $x$ and $z$ , respectively, $\triangle _f / \triangle _{DNS} = 6$ . The observation operator $\mathcal {M}$ can therefore be expressed as $\mathcal {M}(\boldsymbol {u}(\boldsymbol {x})) = \tilde {\boldsymbol {u}}(\boldsymbol {x})|_{\boldsymbol {x} = \boldsymbol {x}_m}$ . To separate the influence of filtering and sub-sampling, we also consider a case where the observations are not filtered before sub-sampling, denoted as $\triangle _f / \triangle _{DNS} = 0$ .

Figure 18.

(i) Adjoint estimation of wall signals at $t = 6T$ using filtered and sub-sampled outer observations beyond $l_y^+ = 50$ . (ii) The true fields. (iii) Spectra of the estimation error when observations are fully resolved outer data (black), sub-sampled data, $\triangle _f / \triangle _{DNS} = 0$ (blue), or filtered and sub-sampled data $\triangle _f / \triangle _{DNS} = 6$ (green). Spectra of the true fields (red). ( $a$ , $b$ , $c$ ) Two components of the wall shear stress and pressure, $\{\tau _{xy}^+, \tau _{yz}^+,p^+\}$ .

The data assimilation procedures and the sliding-window strategy are almost the same as in §§ 3.1 and 3.2, with only two differences. First, due to the limited data resolution in the outer layer, the adjoint-variational algorithm is now applied to reconstruct the full initial state, not just the wall layer. The other difference is the first guess of the initial state. Linear stochastic estimation is still adopted to approximate the initial velocity field in the wall layer, but only the coarse observations are fed into LSE, not the fully resolved data. Within the observed outer layer, the initial velocity is estimated between observation locations using spline interpolation of the sparse velocity data.

Given filtered and sub-sampled data with $l_y^+ = 50$ , the adjoint estimation of the wall stresses and pressure at $t = 6T$ are visualised in figure 18(i). Compared with the true fields in panel (ii), only a few differences in the small scales can be identified, and the estimation quality is comparable to the results using fully resolved observations in figure 10. Due to the small differences in these qualitative comparisons, it is necessary to perform a more detailed analysis of the influence of data resolution by examining the errors in spectral space, as shown in panels (iii). The reconstruction is slightly more accurate when the outer observations are filtered before the sub-sampling (from blue to green lines), because the filtering operation incorporates more information about the flow field. Regardless of the filter width, the error of adjoint estimation using under-resolved outer observations (blue, green lines) are higher than the results using fully resolved data (black lines) across all the scales. This trend can be explained from the assimilation perspective: as the outer observations become coarser, they can be traced back to a larger set of turbulent states, therefore increasing the fundamental difficulty of searching for the true turbulent state. Nevertheless, the estimation error using limited outer observations remains an order of magnitude smaller than the true spectra at large scales, which aligns with the high estimation quality in panels (i). The reconstructed smaller scales are still correlated with the true state, as evidenced by the estimation error being lower than $\sqrt {2}$ times the true spectra. In summary, the estimation accuracy of wall signals is compromised but not significantly affected by the decreased resolution of the data.

Figure 19.

Same as figure 18, but the thickness of the unknown layer is $l_y^+ = 90$ .

The robustness of the estimation accuracy against measurement resolution is further verified at $l_y^+ = 90$ in figure 19. In panels (i), the energy-containing large-scale structures reconstructed using under-resolved data closely resemble the true state in panels (ii), although discrepancies are evident at other scales. The estimation error in Fourier space is largely unaffected by the measurement resolution or filtering, as illustrated in panels (iii), with only a slight increase in error at large scales when using coarser observations. These results indicate that as observations move further from the wall, the decoded wall signals remain predominantly sensitive to the large scales in the outer layer, which are well captured by the observations despite the coarser data resolution.

Figure 20.

Estimation accuracy of wall signals versus thickness of the unknown layer. Estimation using fully resolved outer data (black); estimation using filtered and sub-sampled data (green). The accuracy is quantified by the correlation coefficient between the true wall signals and their estimations. Symbols ( $\circ , +, \times$ ) correspond to the two components of wall shear stress and pressure ( $\tau _{xy},\tau _{yz},p$ ).

Combining the estimation accuracy using filtered and sub-sampled data with figure 17, we summarise the influence of measurement resolution and amount of observations on the highest attainable accuracy of wall signals in figure 20. The estimated wall stresses and pressure remain accurate up to $l_y^+ = 50$ , with a correlation coefficient exceeding 0.95. This high level of accuracy is maintained for the herein considered sub-sampled / under-resolved outer observations, whose density is 1/4096 of the requirement for DNS. Beyond $l_y^+ = 50$ , the estimation accuracy diminishes with increasing thickness of the unknown layer, yet remains robust against moderate coarse-graining of the outer observations. Taken altogether, these results underscore the potential for an accurate estimation of wall signals even with reduced measurement resolutions in the outer layer, highlighting the effectiveness of the adjoint-variational approach that strongly enforces the full Navier–Stokes equations.