A wall-based, interpretable reduced-order model for real-time estimation of near-wall turbulence

Technical Summary: A Wall-Based, Interpretable Reduced-Order Model for Real-Time Estimation of Near-Wall Turbulence

Problem Statement
The near-wall region of turbulent flows sustains a self-sustaining cycle involving the formation, instability, breakdown, and regeneration of low-speed streaks and quasi-streamwise vortices. While this cycle leaves distinct signatures on wall shear stress, it remains an open question whether the wall stress acts merely as a passive footprint or if it contains sufficient information to fully reconstruct the underlying dynamical system. Existing approaches to reconstructing near-wall dynamics typically rely on off-wall measurements (impractical at high Reynolds numbers) or function as state estimators for external solvers rather than autonomous dynamical systems. The challenge is to determine if a reduced-order model (ROM) can be constructed solely from wall-based variables that is both autonomous (capable of free integration) and capable of tracking the true flow trajectory when constrained by sparse, noisy observations.

Methodology
The authors propose a non-intrusive, interpretable reduced-order modelling framework applied to a minimal turbulent channel flow at a friction Reynolds number of $Re_\tau = 180$. The methodology proceeds through the following steps:

1. Data Generation and POD: Direct Numerical Simulation (DNS) data is generated for the streamwise ($\tau_x$) and spanwise ($\tau_y$) wall shear stresses. Proper Orthogonal Decomposition (POD) is applied to extract the dominant modes. A truncation of $d_o = 45$ modes is selected, capturing approximately 90% of the variance and the dynamically relevant scales of the near-wall cycle.
2. Clustering and Manifold Adaptation: The latent trajectory in the reduced space is partitioned using K-means clustering. The number of clusters ($K=16$) is determined via Bayesian Information Criterion (BIC) and geometric diagnostics (participation ratio and flow speed). Crucially, the clustering is adapted to the local geometry of the attractor using a Mahalanobis metric derived from the covariance of each cluster.
3. Vector Field Construction: A global, differentiable vector field $\dot{a} = f(a)$ is constructed using Radial Basis Functions (RBFs). The RBFs are anisotropic, with kernel widths and orientations adapted to the local tangent and normal directions of the attractor within each cluster. The weights are determined via a single global ridge regression, ensuring a continuous field without switching between local models.
4. Autonomous Integration and Analysis: The model is tested via free integration (autonomous rollouts) to assess its ability to recover the invariant measure (long-term statistics) and predictability. The differentiable nature of the field allows for direct dynamical systems analysis, including the location of slow points (equilibria) and the computation of the Jacobian to analyze non-normal growth and phase-volume contraction.
5. Data Assimilation: An Ensemble Kalman Filter (EnKF) is employed to assimilate sparse, noisy wall shear measurements into the ROM. This framework is used to track the true trajectory and quantify the "closure deficit"—the error introduced by the model's inability to resolve off-wall interior motions.

Key Contributions

• Autonomous Wall-Driven ROM: The paper establishes a reduced-order model that evolves the latent wall state autonomously using only wall-based inputs, without requiring off-wall data or explicit governing equations.
• Interpretability and Dynamical Analysis: Unlike "black-box" neural networks, the RBF-based vector field is differentiable. This allows for the direct identification of weakly unstable near-equilibrium points (slow points) and the resolution of non-normal growth and phase-volume contraction directly from the fitted coefficients.
• Quantification of Wall-Only Sufficiency: The study rigorously tests the hypothesis that wall shear stress alone can constrain the near-wall cycle. It distinguishes between the model's statistical autonomy (recovering the invariant measure) and its deterministic tracking capability.

Results

• Phase Separation: The unsupervised clustering naturally separates the near-wall cycle into three distinct dynamical phases: a quiescent state, a vortex-dominated phase, and a mature streak phase. These correspond to the known physics of the self-sustaining cycle.
• Statistical Fidelity: In unforced long-term rollouts, the ROM stays bounded and successfully recovers the invariant measure of the wall shear stress, including the correct power spectral density and skin-friction distribution. This confirms that the wall observable contains sufficient information to sustain the statistical properties of the cycle.
• Predictability Limits: While the model reproduces long-term statistics, short-term trajectory prediction is limited. The valid prediction time (VPT) is constrained, and the model fails to track individual trajectories deterministically over long horizons without correction.
• Dynamical Insights:
  • Slow Points: Equilibrium points of the fitted field are located exclusively in the quiescent half of the cycle, acting as weakly unstable anchors where trajectories linger.
  • Non-Normal Growth: Disturbance amplification peaks during the transition from the mature streak to the burst phase, while phase-space volume contraction is strongest during the decay of the mature streak.
• Data Assimilation and Closure Deficit: When assimilating sparse, noisy wall measurements, the EnKF successfully tracks the true flow trajectory. However, to maintain consistency with observations, an additive inflation (process noise) of approximately $q^* \approx 0.25$ (relative to the climatological standard deviation) is required. This inflation represents the "closure deficit"—the portion of the dynamics driven by unobserved interior motions (e.g., lift-up and streak breakdown mechanisms occurring off the wall). This deficit is a fundamental property of the wall-only model, remaining invariant across different sensor noise levels and spatial strides.

Significance and Claims
The paper claims that the near-wall cycle is statistically autonomous with respect to wall shear stress but not deterministically autonomous. The wall observable is sufficient to recover the invariant measure and the statistical "climate" of the turbulence, but it cannot deterministically predict the instantaneous trajectory due to the influence of unobserved off-wall dynamics.

The proposed framework demonstrates that a wall-based, interpretable ROM can serve as a well-posed forecast operator within an ensemble filter. It effectively distinguishes between the model's ability to reproduce long-term statistics and its inability to resolve instantaneous interior dynamics without correction. The authors posit that this approach provides a direct pathway for real-time flow control strategies using sparse wall sensors, provided that the closure deficit is managed through data assimilation or future closure models. The findings are currently established for a minimal channel at $Re_\tau = 180$, and the authors note that the sufficiency of the wall coordinate system at higher Reynolds numbers and larger domains remains to be tested.