Abstract Over the past decade, filtered Two Fluid Models (fTFMs) have emerged as a promising approach for enabling fluidized bed simulations at industrially relevant scales. In these models, the filtered drag force is considered to be the most important quantity that requires closure. To date, such closures have typically relied on an isotropic interphase momentum exchange coefficient by applying a drag correction factor to the microscopic drag closures commonly used in resolved simulations. In the present study, both isotropic and anisotropic closures are developed for predicting the filtered interphase forces. The relative performance of these two approaches is then evaluated by means of an a priori assessment, considering data obtained from simulations in which all flow variables are resolved, which were also used for closure derivation. Also, an a posteriori assessment, which compares coarse grid simulation results to a benchmark resolved simulation of a bubbling fluidized bed, is presented. The primary conclusion from the present study is that it is essential to account for the anisotropy of the filtered momentum exchange coefficient. It is shown that this can be done by employing a drift velocity formulation of the filtered drag force and by considering a gravitational contribution that only acts in the vertical direction. Furthermore, it is found that for the large computational grid sizes that are typically required in industrial scale fluidized bed simulations, a closure for the meso-scale interphase force is essential. Finally, also for coarse grids, a non-linearity correction factor, which accounts for assumptions in deriving the drift velocity-based form of the filtered drag force, requires closure. The present study therefore highlights multiple avenues for improving drag closures used in fTFMs. Hence, these results may critically strengthen the predictive capabilities of fTFMs, as well as guide future modelling efforts.

中文翻译：

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过滤两种流体模型的各向异性阻力闭合的开发和验证

摘要 在过去的十年中，过滤的双流体模型 (fTFM) 已成为一种有前途的方法，可用于在工业相关规模上进行流化床模拟。在这些模型中，过滤阻力被认为是需要闭合的最重要的量。迄今为止，这种封闭通常依赖于各向同性的相间动量交换系数，方法是将阻力校正因子应用于解析模拟中常用的微观阻力封闭。在本研究中，开发了各向同性和各向异性闭包来预测过滤后的相间力。然后通过先验评估来评估这两种方法的相对性能，考虑从模拟中获得的数据，其中解析了所有流量变量，这些数据也用于闭包推导。还，后验评估将粗网格模拟结果与鼓泡流化床的基准解析模拟进行比较。本研究的主要结论是，必须考虑滤波动量交换系数的各向异性。结果表明，这可以通过采用过滤阻力的漂移速度公式并考虑仅作用于垂直方向的重力贡献来实现。此外，发现对于工业规模流化床模拟中通常需要的大型计算网格，中尺度界面力的闭合是必不可少的。最后，对于粗网格，还有一个非线性校正因子，考虑了推导基于漂移速度的过滤阻力形式的假设，需要闭合。因此，本研究强调了改进 fTFM 中使用的阻力闭合的多种途径。因此，这些结果可能会严重加强 fTFM 的预测能力，并指导未来的建模工作。