Offline accuracy is not enough: closed-loop instability and stabilisation of a wall-sensor neural estimator in opposition control

Technical Summary: Offline Accuracy is Not Enough: Closed-Loop Instability and Stabilisation of a Wall-Sensor Neural Estimator in Opposition Control

Problem Statement
Opposition control is a canonical active flow control strategy that reduces skin-friction drag by imposing a wall-normal blowing-and-suction velocity equal and opposite to the wall-normal velocity measured on a detection plane within the buffer layer. While effective, this strategy requires sensing quantities (velocity at $z^+ \approx 14$) that are not available from wall-mounted sensors. Although deep learning and linear stochastic estimation have demonstrated high-fidelity offline reconstruction of inner-flow quantities from wall-shear stress ($\tau_x, \tau_y$), it remains unproven whether such estimators can function as live surrogates within a closed-loop feedback system. The central question addressed is whether a neural estimator, trained on a fixed flow state, can maintain stability and performance when deployed to actively modify the very flow it senses.

Methodology
The authors employ Direct Numerical Simulation (DNS) of a turbulent channel flow at a friction Reynolds number $Re_\tau = 180$. The control loop operates as follows:

1. Sensing: The controller reads instantaneous wall-shear stress components ($\tau_x, \tau_y$).
2. Estimation: A Convolutional Gated Recurrent Unit (ConvGRU) network maps these wall signals to an estimate of the detection-plane wall-normal velocity ($\hat{w}$). The recurrent architecture provides temporal context, addressing the under-determined nature of inferring off-wall fields from a single wall snapshot.
3. Actuation: The estimated velocity is used in the opposition law ($w_{wall} = -\alpha \hat{w}$) to impose a zero-net-mass blowing-and-suction boundary condition.

The study proceeds through three distinct phases:

• Offline Validation: The estimator is trained on data from a statistically steady, ideal opposition-controlled flow (OC) and validated offline.
• Closed-Loop Failure Analysis: The estimator is deployed directly. The authors analyze the mechanism of instability, observing that small high-wavenumber errors in the estimate, when imposed as a boundary condition, accumulate and drive the flow off the training distribution (distribution shift).
• Stabilisation Strategy: Two corrective measures are introduced:
  1. Spectral Matching: A spectral low-pass filter and spanwise spectral rescaling are applied to the deployed actuation. This forces the energy spectrum of the estimated field to match the reference training distribution, removing unstable high-wavenumber modes that the wall signal cannot resolve.
  2. Closed-Loop Retraining: The estimator is retrained on data collected from the controlled flow itself (iterative data collection). This aligns the estimator's training distribution with the actual attractor of the controlled system.

Key Results

• Offline vs. Online Discrepancy: The estimator achieves near-perfect offline accuracy (correlation $\rho \approx 0.99$) but fails catastrophically in closed loop. Without intervention, the correlation drops to zero within 15–20 viscous time units, and the solution diverges.
• Mechanism of Failure: The instability is driven by high-wavenumber errors. While these errors are energetically negligible offline, they are amplified by the nonlinear feedback loop when imposed as a boundary condition. Standard remedies like exponential moving averages (EMA) or low-pass filtering delay divergence but accelerate the loss of correlation, failing to produce a functional controller.
• Stabilisation Success:
  • Applying spectral matching alone stabilizes the loop indefinitely but yields only 14.6% drag reduction (vs. 22.7% for ideal OC) because the estimator still operates on a distribution mismatch.
  • Retraining the estimator on the closed-loop data (iterative process) realigns the flow statistics. A second retraining iteration (retrain-2) produces a stable controller with a mean drag reduction of 19.7%, closely approaching the ideal OC performance.
• Flow Structure: The twice-retrained wall-only controller reproduces the near-wall turbulence structure of ideal opposition control. Mean velocity profiles, Reynolds shear stress, quadrant event fractions (ejections and sweeps), and two-dimensional energy spectra are nearly indistinguishable from the ideal OC case, differing only by a slight excess in near-wall activity consistent with the marginally lower drag reduction.

Significance and Claims
The paper claims that offline accuracy is insufficient for closed-loop flow control. The primary obstacle is not the ability to reconstruct the flow offline, but the dynamic consistency of that reconstruction when the controller modifies the flow.

The authors identify that the failure arises from a distribution shift induced by the controller itself, specifically in the high-wavenumber band where wall measurements carry insufficient information. The study demonstrates that stable wall-only opposition control is achievable through a two-part correction:

1. Spectral Consistency: Constraining the actuation to the spectral band resolvable by the wall sensors prevents the injection of destabilizing high-wavenumber energy.
2. In-Loop Retraining: Retraining the estimator on the data generated by the controlled flow ensures the estimator remains valid for the specific statistical state it creates.

The work establishes that a supervised estimator, trained offline, can serve as a live surrogate sensor if the deployment strategy accounts for the spectral limitations of wall sensing and iteratively adapts to the controlled flow's attractor. The residual gap in drag reduction (approx. 3 percentage points) is attributed to the fundamental limit of wall-to-detection-plane coherence in the high-wavenumber band, a physical constraint rather than a modeling failure.