Reduced-order models have been widely adopted in fluid mechanics, particularly in the context of Newtonian fluid flows. These models offer the ability to predict complex dynamics, such as instabilities and oscillations, at a considerably reduced computational cost. In contrast, the reduced-order modeling of non-Newtonian viscoelastic fluid flows remains relatively unexplored. This work leverages the sparse identification of nonlinear dynamics algorithm to develop interpretable reduced-order models for viscoelastic flows. In particular, we explore a benchmark oscillatory viscoelastic flow on the four-roll mill geometry using the classical Oldroyd-B fluid. This flow exemplifies many canonical challenges associated with non-Newtonian flows, including transitions, asymmetries, instabilities, and bifurcations arising from the interplay of viscous and elastic forces, all of which require expensive computations in order to resolve the fast timescales and long transients characteristic of such flows. First, we demonstrate the effectiveness of our data-driven surrogate model to predict the transient evolution and accurately reconstruct the spatial flow field for fixed flow parameters. We then develop a fully parametric, nonlinear model capable of capturing the dynamic variations as a function of the Weissenberg number. While the training data is predominantly concentrated on a limit cycle regime for moderate Wi, we show that the parameterized model can be used to extrapolate, accurately predicting the dominant dynamics in the case of high Weissenberg numbers. The proposed methodology represents an initial step in the field of reduced-order modeling for viscoelastic flows with the potential to be further refined and enhanced for the design, optimization, and control of a wide range of non-Newtonian fluid flows using machine learning and reduced-order modeling techniques.

翻译：降阶模型已广泛应用于流体力学，特别是在牛顿流体流动中。这些模型能够以显著降低的计算成本预测复杂动态行为，例如不稳定性和振荡。相比之下，非牛顿粘弹性流体流动的降阶建模仍相对未被探索。本研究利用非线性动力学的稀疏识别算法，为粘弹性流动开发可解释的降阶模型。具体而言，我们以经典Oldroyd-B流体为对象，探索了四辊轧机几何结构上的基准振荡粘弹性流动。该流动体现了与非牛顿流动相关的许多典型挑战，包括由粘性和弹性力相互作用引起的转变、不对称性、不稳定性和分岔，所有这些都需要昂贵的计算来解决此类流动特有的快速时间尺度和长瞬态过程。首先，我们展示了数据驱动代理模型在预测固定流动参数下的瞬态演化及准确重构空间流场方面的有效性。随后，我们开发了一个完全参数化的非线性模型，该模型能够捕获随韦森伯格数变化的动态特性。尽管训练数据主要集中于中等Wi下的极限环区域，但研究表明，参数化模型可用于外推，准确预测高韦森伯格数情况下的主导动力学行为。所提出的方法代表了粘弹性流降阶建模领域的初步探索，有望通过机器学习和降阶建模技术进一步改进和增强，应用于广泛非牛顿流体流动的设计、优化与控制。