3.1. Configuration and numerical set-up

Figure 1 depicts the computational domain. The inflow boundary, located 10 diameters upstream of the cylinder centre, prescribes a uniform velocity profile aligned with the square’s lateral sides at Reynolds number $Re=LU_{0}/\upsilon =100$ . Symmetry conditions are imposed on the lateral boundaries, positioned 7.5 diameters from the cylinder centre, while the outflow boundary is placed 30 diameters downstream. Table 1 summarises the physical and numerical parameters. The reference flow was obtained via DNS on a uniform fine mesh using a second-order semi-implicit temporal scheme. Periodic vortex shedding commenced at 90 s during the 400 s simulation. Pressure measurements for DA were acquired from probes located at the $1/4$ and $3/4$ circumferential positions between 300 s and 400 s to exclude initial transients. A coarse mesh was then constructed, yielding force coefficients with approximately 50 % error compared to the DNS. Three 100 s simulations were performed on this coarse mesh: (i) a standard unsteady laminar simulation; (ii) a DA simulation using the MAF strategy; and (iii) a DA simulation using the PAS strategy.

Figure 1.

Computational set-up and DA performance: (a) domain with pressure probes (grey indicates DA, blue indicates validation). Cost function convergence for (b) steepest descent and (c) conjugate gradient methods. (d) Normalised convergence history: MAF versus PAS strategies.

Figure 1. Long description

The figure contains multiple panels depicting different aspects of a computational setup and data assimilation performance. Panel A shows a sketch of the domain with pressure probes, where grey indicates data assimilation (DA) and blue indicates validation. The domain dimensions are labeled with L, 3L, 10L, 15L, and 40L. Panel B is a line graph showing the cost function convergence using the steepest descent method. The x-axis represents the iteration on a logarithmic scale from 10^0 to 10^4, and the y-axis represents the residual on a logarithmic scale from 10^-2 to 10^0. The graph shows a decreasing trend in residual with increasing iterations. Panel C is another line graph showing the cost function convergence using the conjugate gradient method. The x-axis represents the iteration from 0 to 10, and the y-axis represents the residual on a logarithmic scale from 10^-2 to 10^0. This graph also shows a decreasing trend in residual with increasing iterations. Panel D is a line graph comparing the normalized convergence history of two strategies, MAF and PAS. The x-axis represents t+ from 0 to 100, and the y-axis represents the residual on a logarithmic scale from 10^-6 to 10^0. The MAF strategy is represented by a red line, and the PAS strategy is represented by a blue line. The PAS strategy shows a more significant decrease in residual over time compared to the MAF strategy.

Table 1.

Numerical set-up for the simulations.

Table 1. Long description

A table summarizing physical and numerical parameters for simulations. The table has two columns: Model and Square cylinder. The Model column includes Diameter in meters, Reynolds number, Number of cells in reference mesh, Time step for reference mesh, Number of cells in coarse mesh, Time step for coarse mesh, and Observation sparsity. The Square cylinder column lists the corresponding values: 0.01 meters for Diameter, 100 for Reynolds number, 400000 cells for Number of cells in reference mesh, 0.01 for Time step for reference mesh, 23000 cells for Number of cells in coarse mesh, 0.1 for Time step for coarse mesh, and 1 for Observation sparsity.

Figures 1(b) and 1(c) illustrate the convergence trajectory during a single assimilation window for the MAF strategy (PAS exhibits similar behaviour). Both methods achieve comparable final residuals; however, the nonlinear conjugate gradient method demonstrates superior convergence rate and stability, while steepest descent exhibits slower asymptotic decay – consistent with theoretical expectations for first-order methods in non-convex optimisation. Figure 1(d) compares the normalised convergence histories for MAF and PAS. In the first cycle, significant deviation from the reference flow prevented the residual from reaching the specified convergence criterion, even with an increased iteration. In subsequent cycles, however, the residual decreased markedly at the cycle onset. This improvement stems from initialising the flow with the final state from the preceding cycle, and employing adjusted pressure observations. Crucially, the strategies diverged in their final residual levels: PAS achieved a lower residual after the third cycle, indicating minimal ongoing flow field impact on the assimilation, and suggesting system stabilisation. In contrast, the MAF strategy maintained a residual level nearly three times higher than PAS throughout the cycles.

3.2. Results and discussion

Figure 2 illustrates the temporal evolution of wake vorticity behind the square cylinder for the four simulations, using dimensionless time $t^{+}$ . Notably, the standard coarse simulation exhibits no vortex shedding, with a symmetric elongation of the wake throughout the simulation period. In contrast, both DA simulations develop wake asymmetry – a precursor to vortex shedding – by $t^{+}=10$ . By the end of the depicted period, vortex shedding is fully established in both DA cases, while the standard coarse simulation persists with symmetric wake elongation. It can further be noted from figure 2 that while DA cases employing the MAF strategy transition to vortex shedding, their recirculation zones remain elongated throughout the depicted period, resembling the standard coarse simulation. This elongation persists until $t^{+}\approx 40$ , after which the wake contracts, approaching the length more closely resembling that of the reference simulation. This delayed contraction can be interpreted in terms of the same dynamical mechanisms that operate in a Reynolds-number–driven bifurcation. The uncorrected coarse-grid solution behaves like an over-damped, steady attractor (analogous to the low- $Re$ steady state), whereas the MAF corrections provide a continuous, localised source of the modal content associated with vortex shedding. Because the injected disturbances must be amplified and convect downstream before they can reorganise the entire wake, the flow requires a finite time to leave the steady attractor and approach the shedding limit cycle; this explains why the wake remains elongated up to $t^{+}\approx 40$ before contracting toward the reference solution. Viewed in phase-space terms, the MAF-driven trajectory does not instantaneously reach the target attractor, but evolves along a low-dimensional path that closely follows the dominant manifold connecting the steady and periodic states (Noack et al. Reference Noack, Afanasiev, Morzyński, Tadmor and Thiele2003).

Figure 2.

Evolution of wake vorticity comparing the reference solution (Reference), standard coarse-grid simulation (Standard), and two DA strategies (MAF and PAS) at sequential time snapshots.

Further analysis of the cylinder side pressure signals reveals key differences. The PAS strategy partially mitigates a mean pressure offset evident in the MAF case. While neither DA simulation perfectly captures the oscillation amplitude, the PAS simulation accurately reconstructs the oscillation frequency. Fourier analysis of wake pressure data (figure 3) identifies a dominant peak at 0.15 Hz in the reference simulations, corresponding to Strouhal number $\sim0.15$ , consistent with that of Okajima (Reference Okajima1982) for a square cylinder at $Re=100$ . The coarse simulation without DA exhibits only a weak spectral peak near 0.14 Hz. Although the MAF strategy produces a peak at the correct frequency (0.15 Hz), its spectral energy level remains similar to the coarse simulation, indicating insufficient assimilation influence to overcome the coarse grid’s inherent limitations. In contrast, the PAS strategy yields a frequency spectrum nearly identical to the reference simulation.

Figure 3.

Time histories of pressure, streamwise velocity and their frequency spectra at probe points P1 and P2.

According to Menter & Egorov (Reference Menter and Egorov2010), the fundamental reason why standard coarse grid simulations fail to resolve the excitation and amplification of the instability lies in the inherent deficiencies caused by grid coarsening, which lead to excessive dissipation of the von Kármán vortices. In the assimilation process, the correction term forces the flow towards the observations, sustaining vortex-shedding instabilities by transferring energy from the model to the resolved flow. Figure 4 illustrates the temporal evolution of the correction fields for different strategies. A key distinction is the broader spatial influence of the PAS correction field compared to MAF. This is particularly marked in the initial phase, where PAS corrections operate globally across the domain, while MAF corrections remain strictly local. This behaviour originates in the mathematical foundation of PAS: the strategy leverages the intrinsic global propagation property of the elliptic adjoint equation. Adjustments to pressure implicitly facilitate information redistribution throughout the computational domain, mediated by the constitutive coupling between velocity and pressure in the incompressible NSEs. Consequently, the flow undergoes rapid transition from its initial steady state to the vortex-shedding state. In contrast, localised momentum corrections predominantly alter near-wall velocities while exerting minimal influence on flow regime transitions in wall-distal regions. This spatial limitation is directly evidenced in figure 3, where oscillations show delayed onset due to insufficient excitation. Another key observation in figure 4 is the attenuation of correction intensity as the predicted flow converges towards observations. This attenuation profile serves as a diagnostic for model capability: When the numerical model inherently reconstructs the flow field, corrections peak during initial assimilation – primarily resolving discrepancies with observations – and decay rapidly thereafter. Conversely, excessive dissipation from inadequate grid resolution limits predictive accuracy; here, corrections persist at significant magnitudes, providing sustained compensation for model-induced errors.

Figure 4.

Time evolution of correction fields for square cylinder flow: (a) $\partial _{i}F_{i}$ for MAF; (b) $\mathcal{S}$ for PAS.

4.1. Computational set-up

4.1.1. Generation of high-fidelity database

The high-fidelity reference database was obtained through wall-resolved LES of flow over an NACA0012 aerofoil at $Re=3.6\times 10^{5}$ and $12^{\circ}$ angle of attack. The wall-adapting local eddy-viscosity SGS model was employed, with validation against time-resolved particle image velocimetry measurements (Li et al. Reference Li, He, Ge and Liu2023a ). This enabled a fully quantitative benchmark for evaluating the DA’s reconstruction capabilities, with access to the true data at every point and resolved scale.

The computational domain employs a structured C-type grid as shown in figure 5. Periodic and freestream boundary conditions are imposed at the spanwise and outer boundaries, respectively. The domain dimensions – wake length 5 $c$ , outer radius $7.3c$ , and spanwise extent $0.3c$ – follow the validated configuration from Ribeiro & Wolf (Reference Ribeiro and Wolf2017) for $Re=4.1\times 10^{5}$ . The spanwise extent was chosen to ensure that two-point correlations of velocity fluctuations decayed to zero, following Jones et al. (Reference Jones, Sandberg and Sandham2008). The LES grid spacings follow the air flow criteria of Tamaki & Kawai (Reference Tamaki and Kawai2023): $\Delta \xi ^{+}=\Delta \xi\, u_{\tau }/{}\nu \lt 25$ (chordwise), $\Delta \eta ^{+}\approx 1$ (wall-normal, at wall), and $\Delta z^{+}\lt 13$ (spanwise), ensuring adequate resolution of near-wall streaks. The mesh also includes at least five grid points within the region $\eta ^{+}\lt 10$ . The time step $\Delta {t}_{\textit{LES}}^{+}=\Delta t_{\textit{LES}}\,U_{0}/c=2.2\times 10^{-5}$ maintained a maximum Courant–Friedrichs–Lewy (CFL) number less than 0.5. All simulations employed an in-house code (He et al. Reference He, Liu and Yavuzkurt2018) on a grid of $\sim45.7$ million points.

Figure 5.

Computational set-up. (a) Geometry schematic and cross-sectional view of the locally refined grid. (b) Instantaneous vortical structures ( $Q=10^{4}$ ) coloured by wall-normal distance. For clarity, grid lines are shown at reduced sampling rates (1 : 5 in $\xi$ , 1 : 4 in $\eta$ ).

Figure 6(a) shows chordwise and wall-normal grid spacings along the upper surface in viscous wall units. A sharp reduction in chordwise spacing occurs near the leading edge at $x/c=0.06$ , upstream of the laminar–turbulent transition ( $x/c\approx 0.08$ ). The criterion $\Delta \xi ^{+}\lt 25$ is satisfied in both transitional and turbulent regions. The wall-normal spacing satisfies $\Delta \eta ^{+}\approx 1$ across the entire upper surface. Figure 6(b) shows spanwise correlation coefficients at $x/c=0.4$ and $0.8$ , evaluated at the height of maximum turbulent kinetic energy (TKE). The correlation coefficient converges to zero at these locations, confirming that the $0.3c$ spanwise extent is sufficient to avoid spurious correlations. Velocity and Reynolds normal stress profiles are further examined at a mild pressure gradient location ( $x/c=0.42$ ) over the upper surface, where the turbulent boundary layer is fully developed and attached (figures 6 c,d). Comparison with DNS data for a flat-plate turbulent boundary layer at a matched friction Reynolds number ( $Re_{\tau }=252$ ; Schlatter & Örlü Reference Schlatter and Örlü2010) shows close alignment in both mean velocity and Reynolds normal stresses. These results suggest that the grid resolution is sufficient to resolve the energetic vortices in the boundary layer, as also depicted in figure 5(b), where the instantaneous vortical structures illustrate the transition and growth of the turbulent boundary layer along the upper surface.

Figure 6.

Computation grid statistics. (a) Grid spacings along the aerofoil upper surface in the viscous wall units. (b) Spanwise correlation $R_{u'u'}$ . (c) Mean chordwise velocity. (d) Reynolds normal stress in viscous wall units at $x/c=0.42$ . Circles denote flat-plate DNS at $Re_{\tau }=252$ .

4.1.2. The DA set-up

Conventional state estimation typically relies on DNS with matching grids and discretisation to eliminate numerical uncertainties (Zaki Reference Zaki2025). In contrast, our DA framework adopts a deliberately coarser mesh within the same domain (table 2), which is insufficient for standalone LES, and introduces discrepancies between the reference and assimilation models. Thus the DA process is more inclined towards a weak-constraint approach (Li et al. Reference Li, He and Liu2023b ) to account for computational uncertainties, such as subgrid model errors and discretisation errors, making it more applicable to realistic flow conditions. Rather than correcting the model itself, we focus on mitigating its adverse effects on resolved flow structures (e.g. excessive dissipation), ensuring accurate state estimation at the resolved scales.

Table 2.

Mesh set-up for the simulations.

Table 2. Long description

A table comparing simulation parameters and control volumes. The table has four rows and six columns. The columns are labeled Simulation, Resolution, max(Δξ+), Δξ+, max(Δη+), and Control volume (106). The rows are labeled LES, CLES, and DA-CLES. Row 1: Simulation, LES; Resolution, Fine; max(Δξ+), 20; Δξ+, 12; max(Δη+), 1; Control volume (10⁶), 45.7. Row 2: Simulation, CLES; Resolution, Coarse; max(Δξ+), 60; Δξ+, 60; max(Δη+), 5; Control volume (10⁶), 1.7. Row 3: Simulation, DA-CLES; Resolution, Coarse; max(Δξ+), 60; Δξ+, 60; max(Δη+), 5; Control volume (10⁶), 1.7.

Table 3 details the DA configurations examined in this study. All cases share boundary conditions identical to those of the reference simulation, with adjoint conditions defined in § 2.2. Second-order discretisation is applied consistently to the primary-adjoint system. The DA time step is set to $\Delta {t}_{c}^{+}=\Delta t_{c}\,U_{0}/c=2.2\times 10^{-4}$ , ensuring a maximum CFL number less than 0.5. Synthetic observations are generated from the high-fidelity database. The primary velocity field is initialised from coarse-grid free evolution. Observations are assimilated every 25 steps.

Table 3.

DA cases set-up.

Table 3. Long description

The table presents various data assimilation (DA) configurations examined in a study. It consists of five rows and five columns. The columns are labeled Sections, Simulation, Mesh, Observation (x/c times z/c), and Temporal sparsity (Δt0+). The rows detail different DA cases with specific configurations. Row 1: Sections 4.2-4.3, Simulation L00, Mesh not specified, Observation [0, 1] times [-0.15, 0.15], Temporal sparsity 0.0055. Row 2: Sections 4.2-4.3, Simulation Nudging, Mesh not specified, Observation [0, 1] times [-0.15, 0.15], Temporal sparsity 0.0055. Row 3: Sections 4.2-4.3, Simulation LSE, Mesh not specified, Observation [0, 1] times [-0.15, 0.15], Temporal sparsity 0.0055. Row 4: Sections 4.2-4.3, Simulation CLES, Mesh Coarse, Observation not specified, Temporal sparsity infinity. Row 5: Sections 4.4, Simulation L30-L70, Mesh not specified, Observation [0.3, 1] times [-0.15, 0.15], Temporal sparsity 0.0055. The table compares different simulations, mesh types, observation ranges, and temporal sparsity values used in the study.

^* Initialisation is acquired from coarse-grid-based free evolution.

For benchmarking, steady RANS simulations with the shear stress transport (SST) model and pure LES without DA (denoted as ‘CLES’) are performed using the same mesh. Note that resolving instability wave activation (e.g. Tollmien–Schlichting waves) and turbulence boundary-layer growth in the CLES case would require higher mesh resolution than used here (Mao et al. Reference Mao, Zaki, Sherwin and Blackburn2017). Nevertheless, it provides a relevant baseline for comparison. To assess the effectiveness of the present DA framework, we also reconstruct the velocity field from wall-pressure measurements using two established methods. The first method is a continuous-based ‘nudging’ approach (Clark Di Leoni et al. Reference Clark Di Leoni, Mazzino and Biferale2020; Zauner et al. Reference Zauner, Mons, Marquet and Leclaire2022), which incorporates a feedback gain in the pressure Poisson equation. The same observations are embedded in this method, and simulations are conducted with the same computational set-up, described by the equation

(4.1) \begin{align}\partial _{i}\partial _{i}\tilde{p}=\partial _{i}\left[-\rho\partial _{t}\tilde{u}_{i}-\rho \tilde{u}_{\kern-1pt j}\partial _{\kern-1pt j}\tilde{u}_{i}+\rho\partial _{\kern-1pt j}\left[\big(\nu +\nu _{\textit{sgs}}\big)\partial _{\kern-1pt j}\tilde{u}_{i}\right]\right]+K\left(\boldsymbol{Y}-\mathbb{H}(\boldsymbol{q})\right).\end{align}

The non-dimensional feedback gain $K$ is set to $4\times 10^{6}$ , the largest value that can be used without causing divergence for this mesh and implementation. The second method uses linear stochastic estimation (LSE) with a proper orthogonal decomposition (POD)-based variant (Discetti et al. Reference Discetti, Raiola and Ianiro2018; Encinar & Jiménez Reference Encinar and Jiménez2019). A detailed description of this method and its verification on the training set is provided in Appendix E. In all DA cases, the assimilation window spans $\Delta T^{+}=\Delta T\,U_{0}/c=8.25$ , during which 7500 instantaneous fields are stored. The analysis is based on the last 5000 snapshots. Additionally, by systematically varying sensor coverage, we examine how spatial localisation of pressure data governs the fidelity of vorticity flux modulation.

4.1.3. On the physical interpretation of the correction and regularisation

A central challenge in DA for turbulent flows is the selection of the regularisation coefficient $\zeta$ , which simultaneously influences the optimisation conditioning, the physical accuracy of the reconstructed flow, and the suppression of measurement noise through its modulation of small-scale fluctuations (see Appendix D for details). To appreciate the physical significance of $\zeta$ , it is instructive to consider the multi-scale nature of turbulence. In a turbulent flow, energy is injected at large, energy-containing scales, and transferred through the inertial range to progressively smaller scales until it is dissipated. In LES, the computational grid can only resolve scales larger than the filter width $\varDelta$ , so that motions at scales smaller than $\varDelta$ – the SGS motions – are unresolved. These unresolved motions represent physically meaningful turbulent fluctuations that participate in the energy cascade and contribute to the Reynolds stress tensor through the SGS stress. In the present DA framework, the regularisation coefficient $\zeta$ controls the effective scale separation between the resolved, assimilated turbulent motions and those that are filtered out.

The connection between the regularisation weight and turbulence scales can be interpreted through the Hessian eigenstructure. The linearised NSEs operator (2.24) embeds two fundamental turbulence time scales at each wavenumber $\boldsymbol{k}$ : a convective time scale $\tau _{conv}(\boldsymbol{k})$ , governing the advection of turbulent structures by the mean flow, and a diffusive time scale $\tau _{\textit{diff}}(\boldsymbol{k})$ , governing their dissipation by the effective viscosity. The Hessian eigenvalue at wavenumber $\boldsymbol{k}$ is determined by the competition between these two time scales (2.25): at low wavenumbers, where $\tau _{\textit{con}v}\lt \tau _{\textit{diff}}$ , advection dominates and the eigenvalues remain comparatively large, so the inverse problem is better conditioned; at high wavenumbers, where $\tau _{\textit{con}v}\gt \tau _{\textit{diff}}$ , diffusion becomes increasingly important and the eigenvalues decay, rendering the inverse problem increasingly ill-conditioned. The regularisation balance (2.28) connects $\zeta$ to these turbulence time scales: under Taylor’s frozen turbulence hypothesis, the balance reduces to $\zeta\,\|\boldsymbol{k}_{c}\|^{2}=\xi _{u}\,{\tau }_{\textit{diff}}^{2}(\boldsymbol{k}_{c})$ , suggesting that the regularisation coefficient may be related to the diffusive lifetime of turbulent structures at the cut-off wavenumber $\boldsymbol{k}_{c}$ . The cut-off may therefore be interpreted as lying near the wavenumber at which the Hessian changes from being predominantly advection-dominated to increasingly diffusion-dominated. In other words, $\zeta $ is not an arbitrary numerical parameter; it encodes the time scale information of the turbulent dissipation process at the assimilation cut-off.

Although this analysis yields a theoretical relationship between $\zeta$ and $\boldsymbol{k}_{c}$ based on Hessian conditioning arguments (2.30), the resulting estimate should be viewed as a guideline for establishing an initial value rather than a definitive prescription. The optimal regularisation weight depends critically on problem-specific factors including the resolution of the observational data, the spatial discretisation of the control variable, and the particular flow physics being assimilated. The difficulty lies in the dual role of regularisation: it (i) stabilises the ill-conditioned inverse problem by improving the Hessian conditioning, and (ii) imposes an implicit spectral filter that determines the assimilated turbulent scales. Consequently, even with theoretical guidance from (2.30), an a posteriori verification procedure is essential to ensure that the chosen $\zeta$ yields both numerical stability and physical accuracy.

To investigate this trade-off, we examine the DA performance across nine regularisation weights spanning the range $[10^{-15}, 10^{-9}]$ , corresponding to wavenumbers $kc$ in the range $[10^1, 10^3]$ . This sweep enables evaluation of convergence characteristics, spectral fidelity and spatial reconstruction quality as functions of the regularisation strength. Figure 7(a) presents the convergence histories for three regularisation coefficients: $\zeta _{{k_{1}}}=10^{-15}$ , $\zeta _{{k_{2}}}=7\times 10^{-12}$ and $\zeta _{{k_{3}}}=5\times 10^{-11}$ , corresponding to $k_{1}c = 200$ , $k_{2}c = 85$ and $k_{3}c = 65$ at the first assimilation cycle, respectively. The monotonic acceleration of convergence with increasing $\zeta$ is apparent: the largest regularisation weight ( $\zeta _{{k_{3}}}$ ) yields the steepest residual reduction, dropping below 10⁻² within five iterations, while smaller weights require more iterations to reach comparable residual levels. This aligns with theoretical expectations: Tikhonov–Sobolev regularisation modifies the eigenvalue distribution by adding a term proportional to $\zeta\,\|k\|^{2}$ to each eigenvalue, lifting the smallest eigenvalues and thereby reducing the condition number of the Hessian matrix. In the high-wavenumber regime, the regularised eigenvalues scale as $\lambda _{\textit{reg}}(k)\sim \zeta\,\|k\|^{2}$ , flattening spectral decay and preventing round-off errors in modes associated with small eigenvalues. Larger $\zeta$ values improve the Hessian conditioning by bounding the smallest eigenvalue from below, which translates directly to faster convergence rates in the gradient-based optimisation algorithm.

Figure 7.

Effect of regularisation scale on convergence and spectral fidelity: (a) convergence history of the residual; (b) minimum residual attained as a function of $\zeta$ ; (c) angular-averaged PSD of the pressure field at $y^{+}=5$ ; and (d) corresponding PSD of the converged scalar source $\mathcal{S}$ at $y^{+}=5$ .

However, fast convergence alone does not guarantee optimal solutions. Figure 7(b) reveals a non-monotonic relationship between regularisation strength and the minimum attainable residual. As $\zeta$ increases from $10^{-15}$ to approximately $10^{-11}$ , the residual decreases, stabilising the inversion against numerical noise. However, beyond this point, further increases in $\zeta$ lead to a rise in the residual, indicating that excessive regularisation over-penalises the control variable and degrades the fit to the data. Theoretically, the unconstrained least squares solution should yield the smallest possible residual; any regularisation necessarily increases this residual by constraining the solution space. The observed non-monotonic behaviour – where moderate regularisation actually reduces the residual compared to the near-unconstrained case – demonstrates that numerical conditioning concerns can dominate theoretical optimality in practical implementations. This finding underscores the importance of regularisation not merely as a mathematical convenience but as a numerically essential component that enables stable convergence to physically meaningful solutions.

The spectral analysis in figures 7(c) and 7(d) offers critical insight into how regularisation affects the multi-scale structure of the solution. Figure 7(c) displays the angular-averaged power spectral density (PSD) of the pressure fluctuations at $y^{+}=5$ , comparing the LES reference with the DA reconstructions obtained using the three representative regularisation weights. For $\zeta _{k_1}$ , the spectrum matches the LES reference at low and medium wavenumbers, but deviates at high wavenumbers, indicating that insufficient regularisation fails to suppress spurious energy accumulation in the poorly constrained high-wavenumber range. This energy pile-up is a direct consequence of the ill-posed inverse problem: insufficient regularisation permits amplification of components associated with small singular values, or equivalently weakly observable directions of the Hessian, which manifests as oscillatory artefacts rather than physically meaningful small-scale content (Benning & Burger Reference Benning and Burger2018). In turbulence estimation from limited observations, only flow events that materially influence the measurements can be robustly recovered, whereas scales that are weakly linked to the observations remain ambiguous (Zaki Reference Zaki2025). Because the high-wavenumber modes correspond precisely to the diffusion-dominated regime of the Hessian where eigenvalues are small, these modes are the least constrained by the observational data. In the absence of sufficient regularisation, the optimiser exploits these under-determined degrees of freedom to reduce the cost functional, populating them with energy that minimises the residual in a least squares sense, but bears no physical relation to the actual dissipation range motions. The excess high-wavenumber energy is therefore best understood as a numerical artefact of the ill-conditioned inversion – an amplification of weakly constrained small-scale components – rather than a genuine recovery of fine-scale turbulence. As the regularisation increases to $\zeta _{{k_{2}}}$ , the spectrum aligns closely with the LES reference across all resolved wavenumbers, demonstrating optimal regularisation that balances data fidelity and model constraints. However, further increasing $\zeta$ to $\zeta _{{k_{3}}}$ causes the spectrum to deviate, with enhanced low-wavenumber energy and a steep drop-off at high wavenumbers. This behaviour can be interpreted as an implicit spatial filtering operation: strong regularisation effectively imposes a wavenumber cut-off beyond which fluctuations are heavily damped, analogous to truncating the turbulent energy cascade at an artificially low scale.

Figure 7(d) further elucidates this scale-dependent filtering by examining the spectrum of the scalar source field at the same wall-normal location. Insufficient regularisation ( $\zeta _{{k_{1}}}$ ) results in an unphysical accumulation of high-wavenumber energy, manifested as elevated spectral levels at $kc\gt 50$ . This behaviour reflects spectral contamination induced by the ill-conditioned inverse problem: weakly constrained small-scale degrees of freedom absorb residual misfit and produce a spurious pile-up of energy at high wavenumbers, resulting in a non-physical spectrum (Benning & Burger Reference Benning and Burger2018). Moderate regularisation ( $\zeta _{{k_{2}}}$ ) suppresses this spurious high-wavenumber energy while preserving the physically meaningful intermediate-scale content. Excessive regularisation ( $\zeta _{{k_{3}}}$ ) over-filters the solution, removing not only spurious noise, but also physically relevant intermediate-scale structures. From a dimensional standpoint, the Tikhonov–Sobolev regularisation adds a term proportional to $\zeta\, \| k\| ^{2}$ to each eigenvalue of the Hessian, defining a cut-off wavenumber $k_{c}$ and a corresponding assimilation length scale $\ell_{c}=2\pi /\| k_{c}\|$ . Corrections at wavelengths shorter than $\mathcal{O}(\ell_{c})$ are strongly penalised, so that $\ell_{c}$ may be interpreted as the smallest turbulent scale that the assimilation retains. Rather than a mere numerical smoother, the coefficient $\zeta$ therefore acts as a scale-selection parameter. The corresponding cut-off length scale is governed by two factors: the representational capacity of the LES, and the informational content of the wall-pressure observations (Kim Reference Kim1989). The cut-off may therefore be interpreted as marking the spectral boundary beyond which model fidelity and observational information become insufficient to distinguish physically meaningful corrections from spurious ones, and its proximity to the grid spacing $\varDelta$ reflects the convergence of these two limits near the upper end of the resolved range.

Spatial fields in figure 8 confirm these findings. With insufficient regularisation, the scalar source field shows streak-like patterns contaminated by fine-scale, noise-like oscillations. These spurious features manifest as irregular, high-frequency spatial variations superimposed on the physically meaningful large-scale structures, and bear no resemblance to the organised near-wall turbulence structures (streaks, quasi-streamwise vortices) expected from the physics of wall-bounded turbulence. The corresponding pressure field similarly exhibits unphysical fine-scale fluctuations that are inconsistent with the known spatial coherence of pressure in turbulent boundary layers. At the opposite extreme, overly strong regularisation produces an excessively smooth scalar source field that lacks the fine-scale detail present in the reference. Near-wall streaks appear diffuse and blurred, with significantly reduced spatial gradients, indicating that the regularisation has removed the dynamically important small-scale vortical motions that sustain the near-wall turbulence cycle through the self-sustaining process of streak instability and vortex regeneration (Jiménez Reference Jiménez2018). This over-smoothing directly impacts the reconstructed pressure field, removing small-scale coherent structures characteristic of wall turbulence.

Figure 8.

Effect of regularisation scale: (a) scalar source $\mathcal{S}$ distribution; (b) pressure field distribution at $y^{+}=5$ .

This analysis of convergence behaviour, spectral content and spatial structure reveals that Tikhonov–Sobolev regularisation functions as a scale-aware control mechanism rooted in turbulence physics. The regularisation coefficient $\zeta$ sets an effective cut-off wavenumber $k_{c}$ (equivalently, a length scale $\ell_{c}=2\pi /k_{c}$ ), which governs the balance between numerical stability and physical fidelity. This scale-selective filtering has important implications for practical DA in turbulent flows, and the choice of $\zeta$ should therefore be calibrated for each problem so that the assimilated field retains the essential multi-scale structure of the turbulence while avoiding spurious small-scale corrections.

4.2. Algorithm performance: statistics of the benchmark case

Figures 9, 10, 11 and 12 provide a spatially resolved assessment of reconstruction quality by showing instantaneous fluctuation fields together with the corresponding pointwise absolute error maps. Specifically, for each quantity $q$ (pressure fluctuation $p\textrm{'}$ or spanwise velocity fluctuation $w\textrm{'}$ ), we report the instantaneous error field

(4.2) \begin{align}\epsilon (\boldsymbol{x},t)=\left| q(\boldsymbol{x},t)-q_{\textit{ref}}(\boldsymbol{x},t)\right| ,\end{align}

where $q_{\textit{ref}}$ is the reference LES field at the same instant. Snapshots are shown at three representative phases: initialisation ( $t^{+}=0$ ), transition ( $t^{+}=1.35$ ), and the statistically stationary state ( $t^{+}=2.7$ ). The $x{-}y$ plane at $z/c=0$ , and the near-wall $x{-}z$ plane at $y^{+}=5$ , are used to highlight, respectively, the separation/reattachment dynamics and the near-wall multi-scale content.

Figure 9.

Instantaneous pressure fluctuations and error distributions in the $x{-}y$ plane at $z/c=0$ .

Figure 10.

Instantaneous pressure fluctuations and error distributions in the $x{-}z$ plane at $y^{+}=5$ .

Figure 11.

Instantaneous spanwise velocity fluctuations and error distributions in the $x{-}y$ plane at $z/c=0$ .

Figure 12.

Instantaneous spanwise velocity fluctuations and error distributions in the $x{-}z$ plane at $y^{+}=5$ .

For pressure fluctuations, the reference case exhibits fine-scale pressure activity associated with an attached, transitional shear layer, whereas the coarse-grid baseline develops an open separation downstream of the leading edge. The separated shear layer in CLES is dominated by large coherent structures that do not break down, consistent with the fact that the present under-resolved mesh suppresses instability-wave growth and therefore delays laminar-to-turbulent transition under an adverse pressure gradient. This failure manifests directly in the pressure-fluctuation topology and in the large errors concentrated along the separated shear layer and the near-wake region. In the DA results, wall-pressure observations are incorporated every 25 LES time steps; between updates, the flow evolves under the incompressible NSEs. Both PAS and nudging imprint an immediate near-wall pressure response already at $t^{+}=0$ , indicating that assimilation is active from the start. By $t^{+}=1.35$ , both strategies markedly suppress the spurious leading-edge separation seen in CLES, and promote reattachment. The apparent delay between the onset of assimilation and the macroscopic reattachment is therefore not indicative of ‘inactive’ DA; rather, it reflects hysteresis associated with the upstream flow history and the finite convection time of the pre-existing separated structures.

Crucially, the two strategies diverge in the fidelity of the fluctuation field after the transient. The PAS yields a pressure-fluctuation pattern that rapidly approaches the reference across most of the domain, with error maps showing a pronounced reduction by $t^{+}=2.7$ , including in the near-wall $y^{+}=5$ plane. In contrast, while nudging can reduce the wall-pressure misfit, its localised feedback forcing produces steep local pressure gradients and an over-energised fluctuation response; as time advances, this manifests as visibly amplified and distorted $p\textrm{'}$ structures, and a persistently larger $\epsilon$ away from the wall, especially once the solution enters the statistically stationary regime. The LSE results are not reported for pressure here, since LSE/extended POD (EPOD) provides a direct estimate of velocity from wall pressure; the corresponding pressure field could be obtained by post-processing via a Poisson solve, but this is beyond the present scope.

Reconstructing $w\textrm{'}$ is more stringent than matching wall pressure because $w\textrm{'}$ is not directly observed and must be recovered through pressure–velocity coupling and the subsequent nonlinear evolution. The CLES snapshots again show separated-shear-layer dynamics with large-scale, weakly broken-down structures and correspondingly large velocity errors. Both PAS and nudging recover the emergence of near-wall multi-scale motions during the transition ( $t^{+}=1.35$ ), consistent with the rapid correction of leading-edge separation observed in the pressure fields. However, the error maps clarify that ‘multi-scale activity’ alone is not sufficient: nudging tends to generate overly intense fluctuations, leaving a larger wall-normal penetration error and a poorer match to the reference fluctuation organisation at $t^{+}=2.7$ . The PAS, by contrast, provides a more spatially coherent correction – consistent with the elliptic, domain-wide propagation of the pressure adjustment – and therefore yields smaller $\epsilon$ in both planes once the flow reaches statistical stationarity. The LSE/EPOD provides a complementary, purely statistical baseline for $w\textrm{'}$ : it can reproduce the wall-attached, strongly wall-correlated components reasonably well, yielding small near-wall errors. However, accuracy degrades with increasing wall distance and at small scales, because detached motions leave only weak pressure footprints, and the truncated POD basis suppresses weakly correlated, low-energy fluctuations.

Figure 13(a) shows the time evolution of the normalised observation residual, which quantifies the misfit between the predicted wall pressure and the assimilated wall-pressure measurements. Three distinct behaviours are observed. In the uncontrolled coarse-grid LES, the residual departs rapidly from the reference despite identical initialisation. This behaviour reflects the fact that on the present under-resolved mesh, a free Navier–Stokes evolution does not remain on the reference attractor: following an initial transient divergence, nonlinear dynamics dominates, and the estimation error saturates at a finite level. Introducing a proportional feedback term through the nudging formulation stabilises the computation and reduces the wall-pressure misfit. A gain sweep indicates that the feedback gain is strongly constrained by numerical stability on this mesh; we therefore report results at the largest stable value $4\times 10^{6}$ . With this choice, the residual decreases by approximately two decades, then reaches a plateau at $\mathcal{O}(10^{-2})$ . This plateau demonstrates convergence of the wall-pressure constraint, but also indicates an irreducible mismatch associated with coarse-grid modelling/discretisation errors and the inherently local nature of the nudging correction. In contrast, the proposed PAS strategy yields a markedly faster and stronger reduction of the residual: the misfit decreases quasi-exponentially by approximately three orders of magnitude over $t^{+}\lt 2.4$ , then levels off at a lower plateau. This rapid decay indicates that the adjoint-based pressure modulation efficiently removes the transient induced by the separated initial condition, and steers the solution towards the statistically stationary state.

Figure 13.

(a) Normalised residual trajectory. (b–d) Spanwise vorticity evolution at $y^{+}=5$ : (b) Reference, (c) CLES, (d) PAS. Dashed lines indicate mean convection ridges.

The spatiotemporal map confirms this transition, with the shear flow’s downstream motion manifesting as inclined contour patterns. Figures 13(b) and 13(c) illustrate distinct flow behaviours: the reference case exhibits small-scale, positively inclined propagation typical of an attached boundary layer, while the CLES show large-scale, negatively inclined propagation indicative of separation. In contrast, the DA reconstruction displays an immediate transition from the separated state to a statistical steady state, reproducing the spectral complexity of the reference. Notably, an inclined dashed line in figure 13(d) marks a time lag of $t^{+}=2.4$ at $x/c=1$ that separates the large-scale separation from subsequent small-scale propagation. Previous work (Li et al. Reference Li, He, Ge and Liu2023a ) reported that vortical structures in the separated shear layer convect at $0.41U_{0}$ , corresponding to a convection time of 2.4 time units – matching the observed time lag. This agreement confirms that pressure modulation initiates immediately upon observation assimilation. Such instantaneous action stems from a direct viscous force linkage: unlike flow reconstruction achieving rigorous synchronisation from forward–backward sensitivity propagation (Wang & Zaki Reference Wang and Zaki2021; Wang et al. Reference Wang, Wang and Zaki2022a ), the present DA strategy serves as a physics-informed modulator, enabling simultaneous recovery of tangential pressure gradients along the aerofoil wall. To establish a more fundamental physical basis for the proposed PAS strategy, it is instructive to examine the process from the perspective of vorticity dynamics. The generation of vorticity at a solid boundary in an incompressible flow is governed by the wall-normal vorticity flux $J_{n}$ , which can be expressed as (Lighthill Reference Lighthill1963)

(4.3) \begin{align}J_{n}=-\hat{\boldsymbol{\tau }}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{V}_{\!w\textit{all}}}{\text{d}t}-\frac{1}{\rho }\left(\frac{\partial p}{\partial \tau }\right)_{w\textit{all}},\end{align}

where $\hat{\boldsymbol{\tau }}$ denotes the unit vector tangent to the wall. Equation (4.3) shows that the wall-normal vorticity flux consists of two distinct contributions: (i) the tangential acceleration of the wall, $\text{d}\boldsymbol{V}_{\!w\textit{all}}/\text{d}t$ , which represents the kinematic ‘rubbing’ of a moving boundary on the fluid; and (ii) the wall-tangential pressure gradient, $(\partial p/\partial \tau )_{w\textit{all}}$ , which is the sole mechanism for vorticity generation at a stationary wall. For the present study concerning stationary aerofoils, the wall velocity vanishes ( $\boldsymbol{V}_{\!w\textit{all}}=\mathbf{0}$ ), and (4.3) reduces to

(4.4) \begin{align}J_{n}=-\frac{1}{\rho }\left(\frac{\partial p}{\partial \tau }\right)_{w\textit{all}}.\end{align}

This simplified expression highlights a key physical insight: the wall-tangential pressure gradient is the unique source term responsible for the generation and flux of vorticity from a stationary boundary (Appendix C). The PAS framework acts directly on this fundamental mechanism. By introducing an adjoint-derived scalar source term into the pressure Poisson equation, the method modifies the near-wall pressure field, and in turn directly regulates the wall-normal vorticity flux. From this perspective, PAS does not merely correct a scalar pressure field; it intervenes at the physical origin of vorticity production. This interpretation explains how wall-pressure ‘footprints’ – the sole observational input in the present formulation – can exert a disproportionate influence on the regeneration of turbulence throughout the flow domain. While the current work focuses on stationary walls, the general form of (4.3) naturally provides a pathway for extending the methodology to flows involving moving or compliant boundaries.

Figure 14 further substantiates the above observations using spatiotemporal correlation between the reference solution and the reconstructed fields for pressure and spanwise-velocity fluctuations. For each fluctuation quantity $q\in \{p',w'\}$ , we compute a normalised similarity coefficient

(4.5) \begin{align}\mathcal{C}_{xz,q}\left(t^{+},y^{+}\right)=\left\langle q\left(t^{+},y^{+}\right),q_{o}\left(t^{+},y^{+}\right)\right\rangle _{\mathcal{S}}/\left\langle q_{o}\left(t^{+},y^{+}\right),q_{o}\left(t^{+},y^{+}\right)\right\rangle _{\mathcal{S}},\end{align}

where $q$ is the DA field, $q_{o}$ is the reference, and $\langle {\cdot },{\cdot }\rangle _{\mathcal{S}}$ denotes the inner product over the spatial region $\mathcal{S}$ ( $0\leq x/c\leq 1$ , $| z/c| \leq 0.15$ ). Figures 14(c) and 14(f) report the corresponding time-averaged profiles $\overline{\mathcal{C}_{q}}(y^{+})$ , evaluated over the statistically steady interval $2.5\lt t^{+}\lt 8.2$ . The pressure correlations highlight the fundamental difference between the two pressure-constrained strategies. The PAS reconstruction exhibits an immediate, domain-wide imprint in $\mathcal{C}_{p\textrm{'}}$ , consistent with the elliptic propagation of pressure adjustments. During the transient ( $t^{+}\lt 2.4$ ), the correlation is strongest in the inner layer ( $y^{+}\approx 10$ ), reflecting the fact that the flow is still dominated by the pre-existing, large-scale separated-shear-layer structure; once this erroneous large-scale organisation is suppressed, the correlated region extends to a broader wall-normal extent ( $y^{+}\approx 40$ ), indicating that the PAS correction penetrates beyond the viscous sublayer through non-local pressure–velocity coupling. By contrast, the nudging strategy displays a qualitatively similar transient/steady progression, but a much sharper wall-normal decay in $\mathcal{C}_{p\textrm{'}}$ : the feedback term acts locally, so correlation remains concentrated near the wall, and deteriorates rapidly with increasing $y^{+}$ . The spanwise-velocity correlations show the same overall spatial organisation, but attain substantially lower peak values (typically $\mathcal{C}_{w\textrm{'}}\sim 0.4$ ). This reduction is expected because the present objective is not to synchronise the full instantaneous three-dimensional turbulent state from wall pressure alone; rather, it is to correct the resolved dynamics sufficiently to recover accurate statistics and coherent structures at the resolved scales. Consistent with wall-based observability arguments, the reconstruction is most effective in the near-wall region where wall-attached motions leave strong pressure footprints, whereas farther from the wall, the recovery of smaller eddies (scales $\lt \mathcal{O}(y)$ ) is limited because detached motions imprint only weakly on wall pressure (Encinar & Jiménez Reference Encinar and Jiménez2019; Wang et al. Reference Wang, Wang and Zaki2022a ). Nevertheless, both PAS and nudging retain aspects of the large-scale organisation: the reconstructed $w\textrm{'}$ remains arranged in streak-like patterns above the inner layer, in agreement with the instantaneous fields in figures 11 and 12. Importantly, the mean profiles $\overline{\mathcal{C}_{q}}(y^{+})$ (figures 14 c,f) confirm that PAS maintains higher correlation over a thicker wall-normal region than nudging once statistical stationarity is reached. For reference, figure 14(f) also includes an EPOD/LSE baseline for $w\textrm{'}$ , evaluated both in-sample on the training segment used to identify the estimator (LSE-train) and out-of-sample on a temporally separated dataset (LSE-test). The marked degradation from LSE-train to LSE-test is expected because LSE/EPOD assumes a time-invariant linear wall-to-flow relationship; in the present flow, this relationship varies with the evolving mean state and the intermittency of wall footprints, and finite-sample covariance estimation together with POD truncation further limits out-of-sample transferability.

Figure 14.

Spatiotemporal correlations for (a,b) pressure and (d,e) spanwise velocity fluctuations between the reference and two strategies. (c,f) The time-averaged correlation profiles over the statistically steady period $2.5\lt t^{+}\lt 8.2$ .

Figure 14. Long description

The image contains six panels of heatmaps and line graphs. Panel A and B show heatmaps of spatiotemporal correlations for pressure fluctuations using PAS and Nudging strategies, respectively. The y-axis is labeled y+ and ranges from 10^0 to 10^2, while the x-axis is labeled t+ and ranges from 0 to 5. Panel D and E display heatmaps of spatiotemporal correlations for spanwise velocity fluctuations using PAS and Nudging strategies, respectively, with the same axes labels and ranges. Panel C and F present line graphs of time-averaged correlation profiles over the statistically steady period for pressure and spanwise velocity fluctuations, respectively. The y-axis is labeled y+ and ranges from 10^0 to 10^2, while the x-axis is labeled with the correlation values ranging from 0 to 1. Different lines represent PAS, Nudging, LSE-train, and LSE-test strategies. The heatmaps show varying levels of correlation, with higher correlations indicated by warmer colors and lower correlations by cooler colors. The line graphs show how the correlation profiles differ among the strategies.

Figure 15 compares mean streamwise velocity and resolved TKE across strategies. With observational constraints, the DA scheme reproduces both quantities in close agreement with the reference, demonstrating statistical validity. By contrast, CLES and SST predict excessive separation and a large recirculation bubble in the wake. As indicated in figures 11 and 12, the CLES solution exhibits a large-scale flapping motion in the separation region, producing intense fluctuations that appear as elevated TKE. This discrepancy stems from insufficient mesh resolution, which prevents vortex breakdown to smaller scales, and promotes coherent vortex shedding. Nudging provides only a partial correction: while it improves the near-wall mean profile relative to CLES, it retains a noticeable outer-layer velocity deficit and elevated TKE, consistent with its predominantly local forcing and limited wall-normal penetration on the present coarse grid. Instead, the DA procedure dynamically modulates individual flow snapshots by introducing a scalar source into the pressure equation, thereby enabling targeted energy transfer to resolved modes, and controlled initiation and growth of instabilities. Nevertheless, a slight decay in the second-order statistics remains due to the wavenumber cut-off imposed by the coarse-grid resolution.

Figure 15.

Profiles of (a) mean streamwise velocity, and (b) resolved TKE.

Figure 16 further illustrates the distributions of mean wall pressure and pressure fluctuations. Mean pressure coefficients from instantaneous DA fields agree closely with the reference, whereas CLES and SST substantially under-predict near the leading edge due to excessive separation. For wall-pressure fluctuations, both the DA and reference exhibit a rapid increase in the fluctuation coefficient $C_{p'}$ beginning near the leading edge and persisting beyond the mean saturation point during late transition; the peak attains approximately 20 % of the freestream dynamic pressure at $x/c=0.08$ , marking the onset of transition. This is consistent with previous experimental observation (Kurelek et al. Reference Kurelek, Kotsonis and Yarusevych2018). In contrast, CLES exhibits separation without reattachment, and shows a markedly reduced fluctuation amplitude with no distinct peak.

Figure 16.

Comparison of the (a) mean pressure and (b) RMS wall-pressure fluctuation distributions along the chordwise direction.

4.3. Algorithm performance: pressure modulation

The previous subsection demonstrated a statistical reconstruction of the flow field from pressure modulation, highlighting the role of shear-layer instability excitation in flow evolution. This motivates two questions. First, do the observed improvements reflect Navier–Stokes-consistent dynamics, or are they primarily a consequence of the assimilation operator? Second, in a genuinely assimilation-driven transition, which spectral signatures in wall-pressure fluctuations mark the excitation of critical shear-layer modes, and through what vortex-mediated pathways do these fluctuations reorganise the flow? To address these questions, this subsection examines the transition physics during DA. We begin with a time–frequency analysis of the DA fields to present an overall picture of the frequency distribution and to highlight shear-layer-dominated features. Subsequently, we analyse the pressure-modulated flow fields to correlate evolving flow structures with changes in scalar source and its fluctuation magnitude. Finally, spatial modes analysis of scalar sources, pressure and velocity fields elucidates how targeted pressure perturbations govern vorticity transport via phase-locked resonant forcing.

To depict the velocity spectral response to pressure modulation, figure 17 presents continuous wavelet transform (CWT) time–frequency diagrams of wall-normal velocity fluctuations at three critical locations: the leading edge, separation region and separated shear layer. The CWT provides a localised time–frequency representation of a signal. For a time series $f(t)$ , it is defined as

(4.6) \begin{align}\mathcal{K}_{f}\left(a,b\right)=\frac{1}{\sqrt{a}}{\int }_{-\infty }^{+\infty }f(t)\,\psi ^{*}\left(\frac{t-b}{a}\right)\text{d}t,\end{align}

where $\mathcal{K}_{f}(a,b)$ are the wavelet coefficients, $a\gt0$ is the scale parameter (related to frequency), and $b$ is the translation parameter. The function $\psi (t)$ is the mother wavelet, and $\psi ^{*}$ denotes its complex conjugate. In this analysis, the signal $f(t)$ is the scalar source time history at a fixed streamwise location. The Morlet wavelet is employed as the mother wavelet $\psi (t)$ for its balanced resolution in time and frequency (Akamine et al. Reference Akamine, Tsutsumi, Okamoto, Teramoto and Nonaka2022). The CWT reveals distinct spatiotemporal dynamics in each region. At the leading edge ( $x/c=0.1$ , $y^{+}=5$ ), both DA and reference cases exhibit intense high-frequency activity when observations are incorporated, confirming that pressure modulation immediately excites transition onset. In the downstream separation region ( $x/c=0.8$ , $y^{+}=20$ ), pressure modulation induces a hysteretic response characterised by a phase delay of 1.5 time units before the solution asymptotically converges to the reference state. Two mechanisms explain this behaviour. First, as shown in figure 17(b), pressure fluctuations and the streamwise pressure gradient decay substantially. According to the Lighthill relation (4.3), this induces a sharp decay in scalar source energy, a trend corroborated by figure 18. Consequently, pre-existing large-scale structures continue to dominate the separation region. Nevertheless, near-wall flows continue receiving energy injections from residual scalar sources despite weakening pressure modulation. Figures 11(c) and 12(c) show the break-up of these large scales into finer structures, suggesting the onset of an energy cascade. Second, instantaneous observations can fail to capture the dynamics that form large-scale structures because those structures evolve on much longer time scales and are therefore less sensitive to transient features. Sustained pressure modulation over time can amplify such structures through the inherent flow dynamics. This is supported by the signature of large-scale structures in figure 14, as well as by previous findings on wall-pressure-driven state estimation using 4DVar methods (Wang et al. Reference Wang, Wang and Zaki2022a ). In the separated shear layer ( $x/c=0.8$ , $y^{+}=90$ ; figure 17), the DA spectrum exhibits a banded structure similar to CLES for $t^{+}\lt 2.3$ . Convection analysis reveals scale-dependent kinematics: high-frequency perturbations propagate at $0.7U_{0}$ , while low-frequency structures undergo slower advection by mean convection – manifesting hysteresis that aligns with scale-separated convection theory (Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020). Subsequent evolution yields asymptotic convergence towards the reference statistics. Another noteworthy observation is that despite the separated shear being located far from the wall at $x/c=0.8$ , small-scale, orderly wrinkle-like structures emerge during the transition period (not shown here for brevity). This suggests the global propagation property of the elliptic adjoint equation, and thus the scalar correction. Crucially, the DA framework captures these multi-scale phenomena with fidelity comparable to the reference.

Figure 17.

The CWT of wall-normal velocity fluctuation at three wall-normal positions.

Figure 18.

Scalar source dynamics: (a) spatiotemporal evolution at $z/c=0,\ y^{+}=5$ ; (b) RMS at $y^{+}=5$ ; (c) CWT of scalar source at selected chordwise location.

We now examine the pressure-modulated flow to link the underlying physics with variations in the scalar source and its fluctuation amplitude. Figure 18 shows the spatiotemporal evolution of scalar source dynamics at $y^{+}=5$ . In figure 18(a), the downstream modulation appears as an inclined contour pattern, which indicates unsteady excitation concentrated around a dominant frequency. Near the leading edge, three-dimensional modulations from the Lighthill relation directly initialise the roll-up of low-speed parcels (see figures 11 and 12). This roll-up introduces spanwise and wall-normal inflection points, which in turn activate Kelvin–Helmholtz (KH) instabilities. The onset of KH instability exhibits two distinct regimes: for $x/c\lt 0.08$ , the flow is characterised by low convection speed associated with linear lift-up (Brandt Reference Brandt2014), whereas beyond $x/c\approx 0.08$ , the flow becomes KH-unstable, resulting in an increased convection speed. This behaviour mirrors the transition growth documented by Mao et al. (Reference Mao, Zaki, Sherwin and Blackburn2017). As the rollers approach saturation (coincident with the development of three-dimensional instabilities), the shear layer develops quasi-periodic vortex shedding that enhances mixing between inner and outer regions. Saturation of vortex shedding leads to amplified pressure fluctuations, with scalar and wall-pressure fluctuations peaking at $x/c=0.08$ . After transition, streamwise streaks emerge. Unlike classical KH transition where streaks form via secondary instabilities in braid regions (Ho & Huerre Reference Ho and Huerre1984), here the streaks primarily originate from the sustained action of local strain and linear lift-up initiated by Lighthill-driven modulation. Further downstream, two nearly ‘white’ bands appear for $x/c\gt 0.4$ , indicating local minima in the scalar modulation, flanked upstream and downstream by two distinct peak regions. This is also observed in the root mean square (RMS) value of scalar modulation. We attribute this bimodal RMS signature to the emergence of subharmonic resonance in the shear layer. Vortex merging is triggered by the nonlinear evolution of KH subharmonic modes, with its dynamic manifestations realised via Orr-type energy redistribution mechanisms (Jiménez Reference Jiménez2018; Li et al. Reference Li, He, Ge and Liu2023a ). In the coarse-grid LES, the inhibited nonlinear cascade fails to redistribute energy from the KH mode to the Orr mode, resulting in pronounced large-scale oscillations and a single broad peak. In contrast, scalar modulation selectively damps these large-scale instabilities and restores a more realistic cascade: energy is transferred from the primary band into the subharmonic band, yielding the upstream and downstream peaks and the intervening trough. Figure 18(c) presents the CWT of scalar source fluctuations at three chordwise locations $x/c=0.8, 0.6, 0.1$ at $y^{+}=5$ . Relative to figure 17, the scalar modulation and spanwise velocity fluctuations exhibit an approximately linear relationship, characterised by a similar frequency band and comparable energy intensity.

This relation is further elucidated by spectral POD (SPOD), which provides a frequency-based decomposition that identifies spatial structures and their dynamic behaviour. For a detailed derivation and discussion of its mathematical properties, refer to Towne et al. (Reference Towne, Schmidt and Colonius2018) and Schmidt & Colonius (Reference Schmidt and Colonius2020). For the analysis, 5000 snapshots are used per case, with each segment containing 512 successive instantaneous scalar fields for discrete Fourier transform (DFT) computation, overlapping by 384. This yields 36 segments over which SPOD is performed. The choice of these parameters influences the energy distribution, but the dynamics of the leading energetic modes and the main conclusions remain consistent. Figure 19 shows the energy spectra and SPOD modes at two representative frequencies for the scalar source, pressure and wall-normal velocity. The blue shaded area highlights the separation between the first and second modes. A large separation indicates that the leading mode dominates the flow, exhibiting low-rank behaviour (Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018). Two frequencies are selected based on low-rank behaviour observed in DA results, corresponding to two distinct flow structures: the KH type and the Orr-type wavepacket (Tissot et al. Reference Tissot, Zhang, Lajús, Cavalieri and Jordan2017). At $St=20$ , the most energetic mode forms a compact wavepacket in the initial shear-layer region of the aerofoil, resembling the KH shear-layer instability. As the flow propagates downstream, the dominant mode transitions from KH to the Orr-type wavepacket at $St=7$ (Kurelek et al. Reference Kurelek, Kotsonis and Yarusevych2018). The Orr-type mode contains two wavepackets: one in the initial shear layer, similar to the KH wavepackets, and another downstream in the developing region. This can also be distinguished by their phase speed in figure 18(a). The KH-type convect with phase velocity $0.8U_{0}$ , while Orr-type waves convect at a lower speed in accordance with the recovery velocity rate.

Figure 19.

The SPOD of PAS flow structures: (a) scalar source, (b) pressure, and (c) wall-normal velocity. (a) normalised energy spectra. (b,c) spatial contours of modes 1 and 2 at $St=7$ and $20$ .

To elucidate how scalar modulation feeds back into the pressure and velocity fields, we revisit the filtered pressure Poisson equation (2.13) and transform it into spectral space. Noting that $\mathcal{F}\{\partial _{i}\partial _{i}p\}=-| k| ^{2}\,P(k)$ and that the nonlinear inertial term becomes a convolution, we obtain

(4.7) \begin{align}-\left| k\right| ^{2}P\left(k\right)=\rho k_{i}\int {k}_{\kern-1pt j}'\,U_{\kern-1pt j}\left(k'\right)U_{i}\left(k-k'\right)\text{d}k'+\mathcal{S}\left(k\right),\end{align}

so that the pressure spectrum follows:

(4.8) \begin{align}P\left(k\right)=-\frac{1}{\left| k\right| ^{2}}\left(\rho k_{i}\int U_{i}\left(k-k'\right)U_{\kern-1pt j}\left(k'\right)\,{k}_{\kern-1pt j}'\,\text{d}k'\right)-\frac{\mathcal{S}\left(k\right)}{\left| k\right| ^{2}}.\end{align}

From this form, we see two routes by which the scalar-source spectrum $\mathcal{S}(k)$ influences $P(k)$ : linear modulation via $-\mathcal{S}(k)/| k| ^{2}$ , and nonlinear coupling through the convolution integral, the latter enabling cross-scale energy transfer analogous to a turbulence cascade seeded by $\mathcal{S}(k)$ . Phase analysis in figures 19(a) and 19(b) demonstrates a consistent $180^{\circ}$ phase difference between pressure and scalar modes for both first- and second-order modes, confirming linear dominance in low-order pressure modulation. As for the Orr-type wavepacket, no clear phase relationship exists between the scalar source and pressure, nor is there a distinct spatial distribution. This suggests that nonlinear coupling may play a significant role alongside linear effects in pressure modulation for the Orr-type wavepacket. From the perspective of local stability theory, this distinction mirrors the difference between modal (KH-type) and non-modal (Orr-type) mechanisms, as discussed by Marquet et al. (Reference Marquet, Lombardi, Chomaz, Sipp and Jacquin2009), Jordan & Colonius (Reference Jordan and Colonius2013) and Tissot et al. (Reference Tissot, Zhang, Lajús, Cavalieri and Jordan2017). Supporting evidence arises from the pressure/velocity mode at $St=7$ , which displays a double-wavepacket structure comprising an upstream KH-type wavepacket and a downstream component that Schmidt et al. (Reference Schmidt, Towne, Rigas, Colonius and Brès2018) attribute to non-modal Orr-type growth. Finally, the pressure gradient enters the momentum equation:

(4.9) \begin{align}\partial _{t}\tilde{u}_{i}\sim -\frac{1}{\rho }\partial _{i}\tilde{p}\quad\Rightarrow\quad\frac{\text{d}U_{i}\left(k\right)}{\text{d}t}\sim -\frac{ik}{\rho }P\left(k\right).\end{align}

The source term $\mathcal{S}(k)$ governs the evolutionary trajectory of the velocity spectrum through temporal integration, while the instantaneous velocity spectrum $U_{i}(k)$ reflects the cumulative effect of past source-term excitations and nonlinear feedback.

Within the DA framework, we assess spectral and structural fidelity using SPOD: agreement requires recovery of both the frequency-wise energy distribution and the spatial organisation of the leading modes. Figures 20 and 21 show that CLES exhibit the largest discrepancy: beyond $St\gt 4$ , their spectra collapse relative to the reference, consistent with insufficient shear-layer amplification on the coarse grid. This broadband energy shortfall is mirrored by the mode-level diagnostics, where the dominant structures lose coherence and no longer represent the reference instability topology, indicating a dynamical under-resolution rather than a simple amplitude bias. The PAS, by contrast, yields the smallest spectral error over the dynamically relevant band. The SPOD spectra recover both the low- $St$ content and the shear-layer-dominated range, including the KH peak for $St\approx 20$ , and the associated leading modes retain the correct spatial coherence and localisation at the key frequencies. This consistency across energy and mode structure indicates recovery of the correct low-rank statistical dynamics, not merely a tuned power spectrum. Figure 22 further substantiates this by showing high spatial correlation between PAS and LES SPOD modes at the energetic peaks; reduced correlation at the very lowest frequencies is consistent with finite-window effects, since the longest time scales approach or exceed the SPOD segment length, and DFT-based segmentation can decorrelate extremely low- $St$ structures.

Figure 20.

Normalised SPOD energy spectra of pressure and wall-normal velocity.

Figure 21.

Spatial contours of first modes and corresponding errors of wall-normal velocity for $St=7$ and $20$ across different methods.

Figure 22.

Spatial correlation of SPOD modes: (a) pressure at $z/c=0$ ; (b) wall-normal velocity at $z/c=0$ .

Nudging produces a qualitatively different error signature. Although its spectra show an overall decay reminiscent of LES, the dominant frequency content is shifted: peak energy for $St\approx 16$ rather than the reference $St\approx 20$ , and a narrowband spike appears near $St\approx 180$ . The latter is consistent with periodic observation injection ( $\Delta {t}_{0}^{+}=0.0055$ ), i.e. an assimilation-imposed time scale that contaminates both pressure and wall-normal velocity despite Poisson filtering. The SPOD mode errors corroborate this interpretation: near-wall features may appear plausible, but the leading coherent structures at the physically relevant instability frequencies are not recovered with the correct phase-locked organisation. For LSE/EPOD, the dominant error is concentrated in the high-frequency tail. While the reconstruction reproduces the leading energetic content and yields SPOD modes close to the reference, the spectrum decays too rapidly at large $St$ , reflecting the low-rank truncation intrinsic to POD-based estimators, and the weakening of wall–outer correlations at small scales. Consequently, LSE delivers a strong statistical recovery of the dominant dynamics but under-represents high- $St$ fluctuations.

To summarise these trends, we introduce two aggregate measures computed from figures 20 and 21: (i) a spectral energy recovery ratio, defined as the fraction of the reference SPOD energy recovered by the reconstruction when integrating the PSD over the dynamically relevant band; and (ii) a dominant-mode similarity, defined from the spatial inner product of the leading SPOD modes at the representative instability frequencies (here $St=7$ and $St=20$ ), averaged over the transitional and statistically stationary windows. With these metrics, PAS recovers $\approx 99\,\%$ of the reference spectral energy and achieves $\approx 90 \, \textrm{% }$ dominant-mode similarity; EPOD/LSE attains $\approx 80 \, \textrm{% }$ energy recovery and $\approx 60 \, \textrm{% }$ mode similarity; and nudging shows degraded spectral/mode agreement due to peak shifting and update-frequency contamination.

4.4. Localised pressure inputs on vorticity flux modulation

Spectral resolvent analysis has established that the Orr mechanism – a fundamental process of non-modal linear transient growth – is ubiquitously present across the aerofoil (Mao et al. Reference Mao, Zaki, Sherwin and Blackburn2017). This mechanism, especially in its role in turbulence and the self-sustaining cycle, forms the theoretical basis for spatially localised pressure modulation. The double wavepacket in figure 19 further substantiates this behaviour. This raises two key questions for our DA framework. (i) Can spatially localised pressure modulation, using only a subset of wall-pressure sensors, still recover the correct flow statistics? (ii) If so, through which physical pathways do these localised pressure inputs reshape the fluid dynamics? In this subsection, we address these questions by varying the wall-pressure sensor coverage and assessing the resulting accuracy of statistical correction. By comparing the reconstructed spectrum redistribution and coherent structure regeneration across each sensor layout, we aim to identify the spatial footprint required to capture the pressure modulation that underpins the self-sustaining cycle of wall turbulence.

Figure 23 compares the spectral response to localised wall-pressure inputs across the different leeward sensor coverage configurations. The labels LNN denote the streamwise extent of wall-pressure sensing on the leeward surface, where NN is a two-digit number specifying the starting chordwise location: sensors are placed over $x/c\in [\text{NN}/100,1]$ . Thus L00 corresponds to full leeward coverage $(0 \leq x/c \leq 1)$ , whereas L50 restricts sensors to the downstream half $(0.5 \leq x/c \leq 1)$ . The CWT frequency–time diagrams report the response of wall-normal velocity fluctuations at three representative locations. At the leading edge ( $x/c=0.1$ , $y^{+}=5$ ), L00 shows an immediate onset of high-frequency, KH-type activity. In contrast, L50 exhibits an orderly transition absent in L00: characterised by an initial stage dominated by low-frequency energy ( $St\approx 7$ , $t^{+}\lt 0.3$ ), followed by a transition through intermediate frequencies ( $St\approx 20$ and 10), and finally in turbulent broadband structures that later ( $t^{+}\gt 4$ ) asymptotically approach L00 statistics. This difference suggests that whereas local modulation in L00 directly excites transition, the L50 response reveals the global, acoustic-like propagation properties associated with the elliptic adjoint field. In the separation region ( $x/c=0.8$ , $y^{+}=20$ ), the modulation follows a similar transition pathway to L00 but with an approximate two-time-unit phase lag, indicating hysteresis coupled to upstream flow history. In the far separated shear layer ( $x/c=0.8$ , $y^{+}=90$ ), L50 remains at low-frequency dominance until $t^{+}\lt 4$ , again reflecting delayed receptivity due to the slower leading-edge transition. Eventually, the flow develops and converges towards the reference state. Overall, L50 reproduces the multi-scale dynamics seen in L00, and demonstrates comparable fidelity in representing the underlying flow physics.

Figure 23.

The CWT of wall-normal velocity fluctuations at three wall-normal positions. Left- and right-hand columns correspond to cases L00 and L50, respectively.

Figure 24 shows the evolution of the near-wall streamwise velocity ( $y^{+}=5$ ) following pressure-modulation onset at $t^{+}=0.3$ , $0.65$ and $2.1$ . In L00, the flow responds immediately: perturbations amplify rapidly and by $t^{+}=0.65$ exhibit clear spanwise-periodic signatures consistent with streamwise streaks. By contrast, the leading-edge response in L50 follows a more orderly, classical transition route (Zaki et al. Reference Zaki, Wissink, Rodi and Durbin2010). There, two-dimensional Tollmien–Schlichting (TS) waves first grow linearly – a process described by the Orr–Sommerfeld equation – once the boundary-layer profile becomes sufficiently inflectional under the imposed adverse pressure gradient. When the primary TS amplitude reaches a threshold ( $u'/U_{0}\approx 0.1$ ), secondary instabilities emerge, leading to three-dimensional breakdown. This breakdown is marked by sinuous deformations of the TS wavefront (depicted as blue/red interlaced stripes in figure 24 b) and the breaking of spanwise symmetry. Subsequent vortex stretching by mean shear amplifies streamwise vorticity, saturating into spatially localised turbulent structures (figure 24 b). This deterministic cascade – where viscous instability transitions to nonlinear TS self-interaction and ultimately vortex dynamics – completes the laminar-to-turbulent transition.

Figure 24.

Evolution of the streamwise velocity near the wall ( $y^{+}=5$ ) following pressure-modulation onset at wall units: $t^{+}=0.3$ , $0.65$ , $2.1$ . Data from (a) case L00, (b) case L50.

The emergence of this orderly TS–streamwise-vortex pathway in L50 suggests that the rear-edge pressure modulation excites specific instability modes of the upstream boundary layer, rather than generating broadband disturbances. To identify the perturbations preferentially amplified under such modulation, and quantify their growth, we employ resolvent analysis – a global input–output framework introduced by McKeon & Sharma (Reference McKeon and Sharma2010). Theoretical foundations and numerical implementation details are provided in Appendix F; here, we summarise the key elements. By sweeping through the frequency–spanwise wavenumber $(St{-}k_{z}c)$ parameter space, we compute the resolvent operator’s leading singular value and associated forcing/response modes via singular value decomposition following Martini & Schmidt (Reference Martini and Schmidt2024). This sweep produces a Bode-like gain map that identifies parameter combinations for which small perturbations are strongly amplified by the linear dynamics (Yeh & Taira Reference Yeh and Taira2019). When the resolvent gain is large, modest modulation projects onto the operator’s dominant singular mode, and produces a response shaped by that mode; if the response grows sufficiently, then nonlinear interactions alter the mean flow, which is precisely the intended outcome of pressure modulation. Following Kurelek et al. (Reference Kurelek, Kotsonis and Yarusevych2018), our parametric study spans $k_{z}c\in [0,40\pi ]$ with $\Delta (k_{z}c)=\pi$ , and $St=\omega c/2\pi U_{0}\in [0,50]$ with $\Delta St=1$ . Figure 25 shows the gain distribution for the CLES base flow, along with the streamwise components of the optimal forcing and response at $St=20$ , $k_{z}c=5\pi$ . The peak gain at this combination indicates resonant amplification: the resolvent’s leading singular response matches the CWT observations at the leading edge, and its spatial structure peaks near the displacement thickness (dashed lines in figures 25 b,c), consistent with energy extraction from the shear layer’s inflectional profile. This confirms that pressure modulation at rear edge selectively targets the optimal forcing mode of the global eigenmode. The perturbation then propagates upstream via the adjoint pressure field, and amplifies linearly at the leading edge – precisely where the inflectional profile enables rapid growth. Upon exceeding the amplitude threshold, this deterministic pathway catalyses streamwise-vortex formation that triggers intended mean-flow alteration.

Figure 25.

(a) Gain distribution ( $\log _{10}(\sigma )$ ) in frequency–wavenumber space for the CLES base flow. Streamwise component of (b) the optimal forcing $f$ and (c) the response mode $r$ at $St=20$ and $k_{z}c=5\pi$ (peak in (a)). The black dashed line marks the displacement thickness of the separated boundary layer.

Building on the resolvent result, we now examine how trailing-edge modulation projects to the leading edge and redistributes energy across frequencies. Figure 26 shows the magnitude of the initial pressure response at $x/c = 0.1$ , $y^{+}=5$ , computed via CWT immediately following modulation onset. The analysis spans three Strouhal numbers ( $St=7, 10, 20$ ) and varying wall-pressure sensor coverages (L30–L70). Two features emerge from the data. First, the response amplitude is strongly frequency-dependent: the low-frequency band contains approximately two orders of magnitude more energy than the high-frequency band across all sensor coverages. Second, spatial sensitivity is frequency-dependent – the low-frequency resonance shows minimal decay as sensor coverage shifts downstream, whereas high-frequency energy decays rapidly when coverage moves away from the leading edge. Both observations align with the dispersive behaviour of the bounded-domain Green’s function (Hejlesen et al. Reference Hejlesen, Winckelmans and Walther2019): trailing-edge modulation excites long- and short-wave modes, but transmission efficiency differs. High-frequency short waves are strongly attenuated by viscous and geometric dissipation, and therefore have limited propagation distance; low-frequency long waves couple to resonant, low-dissipation pathways and thus reach the leading edge more efficiently. The bias of the trailing-edge forcing towards low frequencies further amplifies this effect. It is important to note that this modulation effect depends crucially on spatiotemporal coherence: reference wall-pressure fluctuations carry phase information associated with quasi-ordered boundary-layer vortex structures, and obey the nonlinear phase relationships imposed by the NSEs, enabling phase-locking and ordered spectral transfer.

Figure 26.

The CWT magnitude of the initial pressure response at $x/c=0.1$ , $y^{+}=5$ following pressure-modulation onset, evaluated at $St=7$ , $10$ and $20$ , with wall-pressure sensor coverage ranging from L30 to L70.

Figure 27 presents the temporal evolution of the CWT magnitude of the pressure response at $x/c=0.1$ , $y^{+}=5$ following the pressure modulation onset. Immediately after onset, the low-frequency band experiences a rapid energy decay, whereas high-frequency TS waves undergo strong amplification. This dual process arises as Reynolds stress induced by Orr-type perturbations ( $\partial \langle -u'v'\rangle/\partial y$ ) tilts the mean shear profile, lowering the threshold for linear TS amplification (Pier Reference Pier2002). Crucially, pre-existing TS waves – intrinsic to the inflectional base flow – are selectively enhanced via this modified shear, producing immediate high-frequency growth that facilitates secondary instabilities (figure 24 b). Åkervik et al. (Reference Åkervik, Ehrenstein, Gallaire and Henningson2008) describe a similar low-frequency Orr-type receptivity, which operates through catalytic modulation of the boundary-layer profile – accelerating TS growth while dissipating its own structure through vortex stretching and viscous decay. Notably, although the pressure resonance level varies across different sensor coverages, the saturated TS energy in the leading edge is essentially invariant across configurations. This observation aligns with Mao et al. (Reference Mao, Zaki, Sherwin and Blackburn2017), who report that critical TS amplitudes trigger secondary instability, leading to the formation of three-dimensional vortex structures. This, in turn, results in a burst in the high-frequency turbulence energy spectrum (also shown in figure 24). A further observation is the asymptotic recovery of the low-frequency features. After the initial phase ( $t^{+}\gt 1$ ), these features gradually amplify and return to the reference baseline state (figure 27 a). This recovered state agrees with the SPOD results in figure 19, which identifies the Orr mechanism as a fundamental mode with ubiquitous presence across the aerofoil surface. This is consistent with the results of Pickering et al. (Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020).

Figure 27.

The CWT magnitude of the pressure response at $x/c=0.1$ , $y^{+}=5$ for $St=7$ , $10$ and $20$ , with wall-pressure sensor coverage ranging from L30 to L70.

Figure 28 compares the premultiplied spectra of streamwise velocity fluctuations induced by localised pressure modulation on the windward ( $x/c\lt 0.5$ ) and leeward ( $x/c\gt 0.5$ ) sides, evaluated at $y^{+}\approx 10$ , where velocity fluctuations reach their maximum. All spectra are normalised by the maximum energy in L00 for consistency. The energy of streamwise-elongated structures ( $\lambda _{x}\gt \lambda _{z}$ , figure 28 a) is nearly fully recovered by the DA model, whereas ‘fat’ motions ( $\lambda _{x}\lt \lambda _{z}$ , figure 28 a) are less accurately captured, particularly when sensors are positioned farther from the leading edge (case L70). This indicates a scale separation in DA flow modulation: streamwise-elongated structures can maintain their features even with sensors positioned further downstream, whereas spanwise-dominated motions are more directly driven by local transient vortices. Similar reconstruction behaviour has been reported by Ying et al. (Reference Ying, Liang, Li and Fu2023). Nevertheless, it is clear that the DA model successfully captures the majority of the energetic motions within the buffer layer. This indicates that the DA model can still recover accurate flow statistics using only a subset of wall-pressure sensors.

Figure 28.

Premultiplied energy spectra of streamwise velocity fluctuations at $y^{+}\approx 10$ for L00, L30 and L70, normalised by $P_{\textit{LES},\textit{max}}$ .