Reconstruction and Prediction of Three-Dimensional Transient Flow Field in a Draft Tube of Francis Turbine Using Sparse Sensors and a Proper Orthogonal Decomposition-Long Short-Term Memory Network

The accurate reconstruction and real-time prediction of transient three-dimensional flow fields in hydraulic turbines are critical for ensuring operational stability under renewable energy-driven variable-load conditions, yet conventional computational fluid dynamics (CFD) approaches remain too computationally expensive for digital twin applications. This paper proposes a hybrid framework that integrates Proper Orthogonal Decomposition (POD) with Long Short-Term Memory (LSTM) networks to reconstruct and predict the unsteady flow field within the draft tube of a Francis turbine using only four sparse wall-mounted pressure sensors. The methodology begins with high-fidelity Large Eddy Simulation (LES) to establish a comprehensive flow field database under Part Load (PL), Best Efficiency Point (BEP), and High Load (HL) conditions. POD is subsequently applied to extract dominant coherent structures and their temporal coefficients, achieving a low-dimensional representation of the high-dimensional flow field. A comparative analysis between standard POD and weighted POD reveals that under the PL condition characterized by a strong double-helical vortex rope, the weighting effect is significant—standard POD captures 90% of the total energy with the first 2 modes, while weighted POD requires up to 8 modes to reach the same threshold. Under the BEP and HL conditions, the energy distributions of the two methods are nearly identical, yet weighted POD still yields cleaner spatial modes with sharper vortex boundaries and fewer spurious wall-region vortices. An LSTM network is then trained to establish a mapping between time-series signals from the four sensors and the POD temporal coefficients. The results demonstrate that LSTM prediction performance is governed by the spatial correlation between each mode and the sensor locations rather than by temporal regularity. Modes that project strongly onto the sensor locations—PL Modes 1–2 (R² = 0.85 and 0.513), BEP Mode 1 (R² = 0.96), and HL Mode 1 (R² = 0.92)—are reliably predictable, while PL Mode 3 and HL Mode 2, despite their regular temporal oscillations, yield strongly negative R² values (−3.366 and −186.6) because their spatial structures are concentrated away from the wall. With a condition-adaptive strategy predicting only sensor-correlated, energetic modes, the reconstructed pressure fields achieve mean L2 relative errors of 17.01% (PL), 7.17% (BEP), and 12.91% (HL). Because the mean flow dominates total pressure energy (86.66–98.07%), the effective absolute error is substantially lower. The proposed POD-LSTM framework successfully bridges the gap between high-fidelity CFD and real-time monitoring, enabling full-field flow state estimation from sparse sensor measurements without the computational expense of online simulations. This capability is particularly valuable for digital twin applications in hydraulic turbines operating under rapidly varying renewable energy conditions.