Anomalous transport has been commonly observed in the fluid flow through a complex porous medium, where the evolution exhibits a complex memory-like behavior. Classical integer-order models fail to depict anomalous transport phenomena, while fractional calculus has been proved effective in describing such behavior due to its ability to characterize long memory processes. In this paper, the time-fractional generalized Navier-Stokes (N-S) equations are formulated to model anomalous transport in porous media at the representative elementary volume (REV) scale by incorporating the Caputo fractional derivative of time into the widely employed generalized N-S equations. An innovative lattice Boltzmann (LB) model is presented for solving the time-fractional generalized N-S equations. We validate the proposed LB model by conducting a numerical example with analytical solutions, and observe a good agreement between the LB results and the analytical solutions. The proposed LB model is utilized for simulating Poiseuille flow, Couette flow, and cavity flow in porous media. Unlike the previous studies, which focus on the steady state of these flows, the present work focuses on the unsteady process from the initial state to the final steady state. It is found that time-fractional effects play a significant role in the unsteady processes of these flows. As the fractional order decreases, the flows undergo a slower evolution process and the impact of Darcy number on the unsteady process becomes increasingly noticeable. The Reynolds number has a significant impact on the velocity profile in the steady state, but it has relatively little effect on the duration of the unsteady process.

中文翻译：

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具有时间分数效应的多孔介质不可压缩流动的格子玻尔兹曼模型

在流经复杂多孔介质的流体中经常观察到异常传输，其中的演化表现出复杂的类似记忆的行为。经典的整数阶模型无法描述异常传输现象，而分数阶微积分由于其能够描述长记忆过程，因此已被证明可以有效地描述这种行为。在本文中，通过将时间的 Caputo 分数阶导数纳入广泛使用的广义纳维斯托克斯 (NS) 方程，制定了时间分数广义纳维斯托克斯 (NS) 方程，以模拟代表性单元体积 (REV) 尺度下多孔介质中的反常输运。提出了一种创新的格子玻尔兹曼 (LB) 模型来求解时间分数广义 NS 方程。我们通过使用解析解进行数值示例来验证所提出的 LB 模型，并观察到 ​​LB 结果与解析解之间具有良好的一致性。所提出的 LB 模型用于模拟多孔介质中的泊肃叶流、库埃特流和空腔流。与以前的研究重点关注这些流动的稳态不同，目前的工作重点关注从初始状态到最终稳态的非稳态过程。研究发现，时间分数效应在这些流动的不稳定过程中起着重要作用。随着分数阶的降低，流动的演化过程变慢，达西数对非稳态过程的影响变得越来越明显。雷诺数对稳态下的速度剖面有显着影响，但对非稳态过程的持续时间影响相对较小。