Silva Goncalo, Semiao Viriato
LAETA, IDMEC, Mechanical Engineering Department, Instituto Superior Tecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal.
Phys Rev E. 2017 Jul;96(1-1):013311. doi: 10.1103/PhysRevE.96.013311. Epub 2017 Jul 18.
The first nonequilibrium effect experienced by gaseous flows in contact with solid surfaces is the slip-flow regime. While the classical hydrodynamic description holds valid in bulk, at boundaries the fluid-wall interactions must consider slip. In comparison to the standard no-slip Dirichlet condition, the case of slip formulates as a Robin-type condition for the fluid tangential velocity. This makes its numerical modeling a challenging task, particularly in complex geometries. In this work, this issue is handled with the lattice Boltzmann method (LBM), motivated by the similarities between the closure relations of the reflection-type boundary schemes equipping the LBM equation and the slip velocity condition established by slip-flow theory. Based on this analogy, we derive, as central result, the structure of the LBM boundary closure relation that is consistent with the second-order slip velocity condition, applicable to planar walls. Subsequently, three tasks are performed. First, we clarify the limitations of existing slip velocity LBM schemes, based on discrete analogs of kinetic theory fluid-wall interaction models. Second, we present improved slip velocity LBM boundary schemes, constructed directly at discrete level, by extending the multireflection framework to the slip-flow regime. Here, two classes of slip velocity LBM boundary schemes are considered: (i) linear slip schemes, which are local but retain some calibration requirements and/or operation limitations, (ii) parabolic slip schemes, which use a two-point implementation but guarantee the consistent prescription of the intended slip velocity condition, at arbitrary plane wall discretizations, further dispensing any numerical calibration procedure. Third and final, we verify the improvements of our proposed slip velocity LBM boundary schemes against existing ones. The numerical tests evaluate the ability of the slip schemes to exactly accommodate the steady Poiseuille channel flow solution, over distinct wall slippage conditions, namely, no-slip, first-order slip, and second-order slip. The modeling of channel walls is discussed at both lattice-aligned and non-mesh-aligned configurations: the first case illustrates the numerical slip due to the incorrect modeling of slippage coefficients, whereas the second case adds the effect of spurious boundary layers created by the deficient accommodation of bulk solution. Finally, the slip-flow solutions predicted by LBM schemes are further evaluated for the Knudsen's paradox problem. As conclusion, this work establishes the parabolic accuracy of slip velocity schemes as the necessary condition for the consistent LBM modeling of the slip-flow regime.
与固体表面接触的气流所经历的第一个非平衡效应是滑移流态。虽然经典流体动力学描述在流体主体中有效，但在边界处，流体与壁面的相互作用必须考虑滑移。与标准的无滑移狄利克雷条件相比，滑移情况被表述为流体切向速度的罗宾型条件。这使得其数值建模成为一项具有挑战性的任务，尤其是在复杂几何形状中。在这项工作中，利用配备晶格玻尔兹曼方法（LBM）方程的反射型边界格式的封闭关系与滑移流理论建立的滑移速度条件之间的相似性，用晶格玻尔兹曼方法来处理这个问题。基于这种类比，我们得出了与二阶滑移速度条件一致、适用于平面壁的LBM边界封闭关系的结构这一核心结果。随后，进行了三项任务。首先，基于动力学理论流体 - 壁面相互作用模型的离散类似物，我们阐明了现有滑移速度LBM格式的局限性。其次，我们提出了改进的滑移速度LBM边界格式，通过将多重反射框架扩展到滑移流态，直接在离散层面构建。这里考虑了两类滑移速度LBM边界格式：（i）线性滑移格式，它是局部的，但保留了一些校准要求和/或操作限制；（ii）抛物线滑移格式，它采用两点实现方式，但在任意平面壁离散化时保证了预期滑移速度条件的一致规定，进一步免除了任何数值校准过程。第三也是最后，我们针对现有格式验证了我们提出的滑移速度LBM边界格式的改进效果。数值测试评估了滑移格式在不同壁面滑移条件下，即无滑移、一阶滑移和二阶滑移，精确容纳稳态泊肃叶通道流解的能力。在晶格对齐和非网格对齐配置下都讨论了通道壁的建模：第一种情况说明了由于滑移系数建模不正确导致的数值滑移，而第二种情况增加了由主体解的不充分适配产生的虚假边界层的影响。最后，针对克努森悖论问题进一步评估了LBM格式预测的滑移流解。作为结论，这项工作确立了滑移速度格式的抛物线精度是滑移流态一致LBM建模的必要条件。
