We extend the resolvent framework to two-phase flows with low-inertia particles. The particle velocities are modelled using the equilibrium Eulerian model. We analyse the turbulent flow in a vertical pipe with Reynolds number of $5300$ (based on diameter and bulk velocity), for Stokes numbers $St^+=0-1$ , Froude numbers $Fr_z=-4,-0.4,0.4,4$ and $1/Fr_z = 0$ (gravity omitted). The governing equations are written in input–output form and a singular value decomposition is performed on the resolvent operator. As for single-phase flows, the operator is low rank around the critical layer, and the true response can be approximated using one singular vector. Even with a crude forcing model, the formulation can predict physical phenomena observed in Lagrangian simulations, such as particle clustering and gravitational effects. Increasing the Stokes number shifts the predicted concentration spectra to lower wavelengths; this shift also appears in the direct numerical simulation spectra and is due to particle clustering. When gravity is present, there are two critical layers, one for the concentration field, and one for the velocity field. For upward flow, the peak of concentration fluctuations shifts closer to the wall, in agreement with the literature. We explain this with the aid of the different locations of the two critical layers. Finally, the model correctly predicts the interaction of near-wall vortices with particle clusters. Overall, the resolvent operator provides a useful framework to explain and interpret many features observed in Lagrangian simulations. The application of the resolvent framework to higher $St^+$ flows in combination with Lagrangian simulations is also discussed.

中文翻译：

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含有低惯性颗粒的湍流的溶解分析

我们将解析框架扩展到具有低惯性颗粒的两相流。使用平衡欧拉模型对粒子速度进行建模。我们分析了垂直管道中的湍流，雷诺数为 $5300$ （基于直径和体积速度），对于斯托克斯数 $St^+=0-1$ , 弗劳德数 $Fr_z=-4,-0.4,0.4,4$ 和 $1/Fr_z = 0$ （省略重力）。控制方程以输入输出形式编写，并对求解算子进行奇异值分解。对于单相流，算子在关键层周围的等级较低，并且可以使用一个奇异向量来近似真实响应。即使使用粗略的强迫模型，该公式也可以预测拉格朗日模拟中观察到的物理现象，例如粒子聚类和引力效应。增加斯托克斯数会将预测的浓度光谱移动到较低的波长；这种转变也出现在直接数值模拟光谱中，并且是由于粒子聚类造成的。当重力存在时，有两个临界层，一层用于浓度场，一层用于速度场。对于向上流动，浓度波动的峰值移近壁面，与文献一致。我们借助两个关键层的不同位置来解释这一点。最后，该模型正确预测了近壁涡旋与粒子簇的相互作用。总的来说，解析算子提供了一个有用的框架来解释和解释拉格朗日模拟中观察到的许多特征。解决框架在更高层次上的应用 $圣^+$ 还讨论了与拉格朗日模拟相结合的流动。