Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Xueyuan Rd 1088, Shenzhen, China.
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR 999077, China.
Phys Rev E. 2023 Jan;107(1-2):015104. doi: 10.1103/PhysRevE.107.015104.
Finding extended hydrodynamics equations valid from the dense gas region to the rarefied gas region remains a great challenge. The key to success is to obtain accurate constitutive relations for stress and heat flux. Data-driven models offer a new phenomenological approach to learning constitutive relations from data. Such models enable complex constitutive relations that extend Newton's law of viscosity and Fourier's law of heat conduction by regression on higher derivatives. However, the choices of derivatives in these models are ad hoc without a clear physical explanation. We investigated data-driven models theoretically on a linear system. We argue that these models are equivalent to nonlinear length scale scaling laws of transport coefficients. The equivalence to scaling laws justified the physical plausibility and revealed the limitation of data-driven models. Our argument also points out that modeling the scaling law could avoid practical difficulties in data-driven models like derivative estimation and variable selection on noisy data. We further proposed a constitutive relation model based on scaling law and tested it on the calculation of Rayleigh scattering spectra. The result shows our data-driven model has a clear advantage over the Chapman-Enskog expansion and moment methods.
从稠密气体区域到稀薄气体区域寻找扩展的流体动力学方程仍然是一个巨大的挑战。成功的关键是获得准确的应力和热通量本构关系。数据驱动模型为从数据中学习本构关系提供了一种新的唯象方法。这些模型通过对更高阶导数进行回归，使复杂的本构关系扩展了牛顿粘度定律和傅立叶热传导定律。然而，这些模型中导数的选择是任意的，没有明确的物理解释。我们在线性系统上对数据驱动模型进行了理论研究。我们认为这些模型等效于输运系数的非线性长度标度律。这种等效性证明了数据驱动模型的物理合理性，并揭示了其局限性。我们的论证还指出，对尺度律进行建模可以避免数据驱动模型中导数估计和对噪声数据的变量选择等实际困难。我们进一步提出了一个基于尺度律的本构关系模型，并将其应用于瑞利散射谱的计算。结果表明，我们的数据驱动模型比 Chapman-Enskog 展开和矩方法具有明显的优势。
