A Volume-Averaged Hyperbolic System of Governing Equations for Granular Flow Modeling with Phase Change

Typical granular flow simulations are based upon governing equations developed by local volume-averaging of flow properties, and then discretizing and solving the equations using numerical schemes. The formulation presented herein uses the basics of volume-averaging to derive a hyperbolic system of governing equations for modeling turbulent, dense granular flows. The Large Eddy Simulations (LES) framework is used for the fluid phase, whereas the solid phase equations are based on the Kinetic Theory. To obtain the LES equations, the volume-averaged equations are further manipulated by filtering the small scale fluctuations; these filtered terms must be modeled to recover their activity in the LES equations. Volume-averaging of the continuity, momentum and energy equations result in many integrals that are used to rigorously define the meaning of terms that have only been included in heuristic models in existing formulations. The concept of added mass, representing fluid that closely follows each particle in its boundary layer and wake and that travels with the particle velocity is used to develop a hyperbolic model for granular flows in the LES framework. Additionally, the pseudo-turbulent kinetic energy (PTKE), which is the turbulent kinetic energy in the fluid phase that arises due to the presence of the solid particles, is included in the formulation and it is shown how its contribution is distinct from turbulence and leads to different terms that must be modeled in the conservation equations. A test case represented by the interaction of a turbulent supersonic jet with a bed of solid particles is simulated and the results are analyzed to demonstrate hyperbolicity. This demonstration is performed by inspecting the eigenvalues of the governing system of equations at different spatial locations of the flow-field. By simulating an additional test case where the PTKE is set to be null and assessing the hyperbolicity of the results, it is demonstrated that the inclusion of PTKE has no role in bestowing hyperbolicity to the model.

经典的颗粒流数值模拟通常基于对流动特性进行局部体积平均得到的控制方程，随后采用数值格式对方程进行离散与求解。本文提出的建模框架以体积平均理论为基础，推导得到用于模拟湍流稠密颗粒流的双曲型控制方程组。其中流体相采用大涡模拟（Large Eddy Simulations, LES）框架，而固相方程则基于动理论（Kinetic Theory）构建。为推导LES方程，需对体积平均后的方程进行进一步处理，通过滤波剔除小尺度脉动；为还原这些滤波项在LES方程中的作用，需对其进行建模。对连续性方程、动量方程与能量方程进行体积平均后，会得到大量积分项，这些积分项可用于严格定义现有建模框架中仅通过启发式模型引入的物理量的物理意义。附加质量（added mass）的概念指的是在颗粒边界层与尾流中随颗粒速度同步运动的流体，利用该概念可在LES框架下构建颗粒流的双曲型模型。此外，本建模框架还引入了伪湍动能（pseudo-turbulent kinetic energy, PTKE）——即由固相颗粒存在而引发的流体相湍动能，研究表明其贡献与自然湍流存在本质区别，会在守恒方程中产生需单独建模的独特项。针对湍流超音速射流与固体颗粒床相互作用的典型测试算例开展数值模拟，并通过分析模拟结果验证模型的双曲性，该验证过程通过考察流场不同空间位置处控制方程组的特征值完成。通过增设一组将PTKE置零的测试算例并评估其结果的双曲性，可证明引入PTKE并不会为模型赋予双曲特性。