Towards Streaming Prediction of Oscillatory Flows: A Data Assimilation and Machine Learning Approach

TL;DR

This work addresses the challenge of predicting time-varying oscillatory flows from limited measurements by integrating data assimilation with machine learning. It couples an Ensemble Kalman Filter (EnKF) data-assimilation loop to estimate the flow state and a moving cylinder's parameters with a low-fidelity penalisation Immersed Boundary Method, augmented by phase-resolved, sliding-window machine learning (Random Forest Regression) to learn local corrections for each oscillation phase. The study demonstrates, on a 2D oscillating cylinder in quiescent flow at $Re=100$ and $KC=5$, that DA–IBM reproduces high-fidelity-like velocity and vorticity fields and accurately captures resistive-force spectra, while ML–IBM with sliding windows closely tracks DA corrections and offers online adaptability. The results highlight a viable pathway toward real-time, digital-twin–like CFD for unsteady flows and motivate future extensions to more complex oscillatory regimes via a temporal mixture-of-experts framework and broader parameter spaces.

Abstract

Data-driven methods have demonstrated strong predictive capabilities in fluid mechanics, yet most current applications still focus on simplified configurations, often characterised by statistical stationarity or limited temporal variability. This work proposes a methodology that combines Data Assimilation (DA) and Machine Learning (ML) to predict flow configurations that exhibit cyclic behaviour over time. Starting from limited, sparse high-fidelity measurements and a low-fidelity numerical model, the DA approach performs data fusion to obtain complete and accurate flow state estimations in time. This complete dataset is used to train multiple ML tools, which are applied across different phases of the flow cycle to augment the model's predictions when high-fidelity data might not be available for the DA application. The methodology is applied to the analysis of an oscillating cylinder in a laminar regime using a sliding-window approach, in which separate models are trained for specific flow conditions to ensure each model specialises in flow dynamics representative of a phase of the oscillation period. This phase-resolved learning enables the efficient capture of transient features that would be challenging for a single global model. The results highlight the potential of this method to study complex flow configurations with oscillatory features in which neither the flow nor the cycle is known a priori, in particular by exploiting real-time training and updates, as is commonly done in digital twins, which require continuous model correction and adaptation.

Figures (16)

• Figure 1: Ensemble Kalman Filter algorithm, highlighting the forecast and analysis steps
• Figure 2: Sliding-window approach for a generic signal $s(t)$ from an initial time $t_0$ up to the current time $t_\textrm{now}$. Each green rectangle denotes a window of size $T_W$, while $p_{SW}$ indicates the uniform sliding period between consecutive windows
• Figure 3: Illustration of the test case, where (a) denotes the computational domain employed for the penalty IBM simulation, and (b) represents the initial configuration of the cylinder, highlighting the discretised immersed boundary and penalised cells (the coloured ones) where $\boldsymbol{f}_P$ is applied
• Figure 4: Velocity profiles when $\Delta\phi = \pi$ at three different planes $x/D \in [ -0.6, 0, 0.6]$ for () $\alpha_p = 40$, () $\alpha_p = 4$, and () $\alpha_p = 0.4$. The three simulations are performed with a resolution $\Delta \boldsymbol{x} = D/25$. () represents the data from the experiments conducted in Dutsch1998_jfm
• Figure 5: (a) Resistive force $F_x$ over one period $T$, and (b) frequency resistive force spectra. The simulations represented are () $\alpha_p = 40$, () $\alpha_p = 4$, and () $\alpha_p = 0.4$. The three simulations are performed with a grid resolution $\Delta \boldsymbol{x} = D/25$. () represents the data from the experiments conducted in Dutsch1998_jfm and ($\blacksquare$) is a body-fitted simulation run in Tsetoglou2024_nmf