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 (SINDy) algorithm to develop interpretable reduced-order models for a broad class of 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 in predicting the transient evolution on a simplified representation of the dynamical system. We then describe the ability of the reduced-order model to accurately reconstruct spatial flow field in a basis obtained via proper orthogonal decomposition. Finally, we develop a fully parametric, nonlinear model that captures the dominant variations of the dynamics with the relevant nondimensional Weissenberg number. This work illustrates the potential to reduce computational costs and improve design, optimization, and control of a large class of non-Newtonian fluid flows with modern 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力学行为，如不稳定性与振荡。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。本研究利用稀疏非线性动力学识别（SINDy）算法，为一大类粘弹性流动开发了可解释的降阶模型。具体而言，我们采用经典Oldroyd-B流体，在四辊轧机几何结构上研究了一种基准振荡粘弹性流动。该流动体现了非牛顿流动中许多典型挑战，包括由粘弹性力相互作用产生的过渡、不对称、不稳定性与分岔，所有这些都需要高昂的计算成本来解析此类流动特有的快速时间尺度与长瞬态过程。首先，我们展示了数据驱动代理模型在预测动力学系统简化表征上的瞬态演变中的有效性。接着，我们描述了降阶模型通过本征正交分解获得的基函数精确重构空间流场的能力。最后，我们开发了一个全参数化的非线性模型，该模型捕捉了动力学随相关无量纲Weissenberg数的主导变化。本研究展示了利用现代机器学习与降阶建模技术降低计算成本、改进非牛顿流体流动的设计、优化与控制方面的潜力。