Generalization Capability of Deep Learning for Predicting Drag Reduction in Pulsating Turbulent Pipe Flow with Arbitrary Acceleration and Deceleration

TL;DR

The paper tackles the generalization challenge of deep learning for predicting drag reduction in pulsating turbulent pipe flow driven by arbitrary acceleration and deceleration. It extends a CNN-LSTM Seq2Seq framework with a TDNN and introduces physics-informed losses that penalize wall shear stress, achieving better RD predictions. The authors demonstrate that training on a limited set of sinusoidal pulsations can generalize to unseen non-sinusoidal waveforms if the training data sufficiently cover local flow-state space, and they introduce the pulsating trajectory difference (PTD) to quantify state similarity and predictability. They also show that incorporating representative intermittent and relaminarizing regimes in training is essential to accurately predict high-drag or relaminarizing flows, highlighting the importance of data selection in generalized flow prediction.

Abstract

The spatiotemporal evolution of pulsating turbulent pipe flow was predicted by deep learning. A convolutional neural network (CNN) and long short-term memory (LSTM) were employed for long-term prediction by recursively predicting the local temporal evolution. To enhance prediction, physical components such as wall shear stress were informed into the training process. The datasets were obtained from direct numerical simulation (DNS). The model was trained exclusively on a limited set of sinusoidal pulsating flows driven by pressure gradients defined by their period and amplitude. Subsequently, 36 pulsating flows with arbitrary non-sinusoidal acceleration and deceleration were predicted to evaluate the generalization capability, defined as the predictive performance on unseen data during training. The model successfully predicted drag reduction rates ranging from $-1\%$ to $86\%$, with a mean absolute error of 9.2. This predictive performance for unseen pulsations indicates that local temporal prediction plays a central role, rather than learning the global profile of the pulsating waveforms. This implication was quantitatively verified by analyzing the differences in periodic $C_f$--$Re_b$ trajectories between the training and test datasets, demonstrating that flows exhibiting local similarity to the training data are more predictable. Furthermore, it was demonstrated that flows exhibiting intermittent laminar--turbulent transition and relaminarization become predictable when such regimes are incorporated into the training data. The results indicate that accurate prediction is achievable provided that the training data sufficiently cover the local flow-state space, highlighting the importance of appropriate training data selection for generalized flow prediction.

Figures (18)

• Figure 1: Schematic of the computational domain and coordinate system.
• Figure 2: Example of generating an arbitrary non-sinusoidal pressure gradient waveform using cubic spline interpolation with control points.
• Figure 3: Map of drag reduction rates ($R_D$) for sinusoidal pulsating flows in the $(T^*, A^*)$ parameter space obtained from DNS, where $T^*$ and $A^*$ are defined in Eq. (\ref{['eq:sine']}). Points (a), (b), and (c) correspond to $(T^*, A^*) = (5, 5)$, $(9, 5)$, and $(5, 9)$, respectively, and are utilized as training data candidates in Section \ref{['sine']}. The symbols indicate flow regimes: $+$, turbulent flow; $\blacksquare$, relaminarizing flow ($R_D = 86\%$).
• Figure 4: Map of drag reduction rates ($R_D$) for arbitrary non-sinusoidal pulsating flows in the $(T^*, A_{\mathrm{acc}})$ parameter space obtained from DNS. Here, $A_{\mathrm{acc}}$ denotes the effective amplitude for arbitrary non-sinusoidal waveforms, as defined in Eq. (\ref{['eq:a_acc']}). The symbols indicate flow regimes: $+$, turbulent flow; $\blacksquare$, relaminarizing flow ($R_D = 86\%$).
• Figure 5: Architecture of the physics-informed CNN LSTM Seq2Seq with TDNN model. (a) Training phase: the time series of latent space vectors, dimensionally compressed by the CNN encoder, is time-evolved by the LSTM and subsequently reconstructed to the original spatial resolution by the CNN decoder. (b) Test phase: starting from a sequence of DNS data as the initial input, the predicted output sequence is recursively fed back as the input to enable long-term prediction.