ColtBig: Compressible and thermal lattice Boltzmann methods on interpolation-based grids

我们的目标是研究、改进和应用新的格子Boltzmann方法(LBM)来计算可压缩流动。尽管LBM在模拟弱可压缩流动方面取得了广泛的成功，但根据文献，仍然缺乏一个可接受的框架来模拟热和完全耦合的可压缩流动，这是由于大量可能的扩展，以及对各种方法在再现变密度或固有可压缩性效应方面的优缺点缺乏了解。对再现这些影响的方法缺乏详细的分析。首先，LBM模型必须是能量守恒的，当标准LBM公式用于完全可压缩流动时，这一要求是不能满足的。其次，速度集必须适用于高速流动和较宽的温度范围，由具有不同粒子速度的大量能量壳表示。与用于弱可压缩流动的标准LBM相反，与笛卡尔网格重合的速度集大多不满足这些要求。最后，平流步长的离散化对方法的灵活性起着决定性的作用。标准格式受到固定时间步长和用于速度离散化的巨大速度集的影响，因为这些集必须与笛卡尔网格匹配，并且在形状上服从对称性。最近，提出了两种非常有前途的方法。第一个是由Frapolli等人撰写的。称为熵LBM(ELBM)，表示可压缩流动的格子上LBM求解器。我们的项目将比较ELBM和我们最近开发的表示基于非格子内插的半拉格朗日LBM求解器(SLLBM)的方法。它代表了LBM的一种新的推广形式，允许在不规则网格上进行有效的模拟。我们的新方法的优点包括区域的几何灵活性、高阶平流步长、可变的时间步长和易于应用复杂的速度集。这些优点将使SLLBM成为热流和可压缩流模拟的高潜力候选者。在本方案的第一部分对ELBM和SLLBM进行了大量的分析之后，在项目的第二部分进行了可压缩强迫各向同性湍流、可压缩时间混合层和超音速湍流通道流动的数值模拟，这是第一次在总体上使用可压缩LBM。这对于分析各自方法的差异和获得必要的洞察力是必要的，以便找到使用LBM的可压缩流动的公认和既定方法。测试案例允许分别研究固有和可变密度的压缩效应，包括激波，甚至允许(各向同性湍流)分裂为螺线管和膨胀部分，此外还与文献进行了详细的比较。

Our goal is to study, improve, and apply novel lattice Boltzmann methods (LBM) for compressible flows. Despite the widely acknowledged success of LBM for the simulation of weakly compressible flows, an accepted framework for the simulation of thermal and fully coupled compressible flows is still lacking according to the literature, which is due to the large number of possible extensions and a lack of understanding of the strengths and weaknesses of the various approaches in reproducing variable density or intrinsic compressibility effects. A detailed analysis of the approaches to reproduce those effects is lacking. Firstly, the LBM model has to be energy conserving, a requirement not met by the standard LBM formulation when adopted to fully compressible flows. Secondly, the velocity sets have to be suited to high-speed flows and to a broad temperature range, being represented by a large number of energy shells with different particle velocities. Contrary to the standard LBM for weakly compressible flows, velocity sets coinciding with the Cartesian grid mostly do not fulfill these requirements. Lastly, the discretization of the advection step plays a decisive role in the flexibility of the methods. Standard schemes suffer from the fixed time step and from the enormous velocity sets that are used for the velocity discretization, since the sets have to both match the Cartesian grid and to obey symmetry in their shape. Recently, two very promising approaches were presented. The first is by Frapolli et al. called the entropic LBM (ELBM), representing an on-lattice LBM solver for compressible flows. Our project will compare the ELBM to our recently developed approach representing an off-lattice interpolation based semi-Lagrangian LBM solver (SLLBM). It represents a new generalized formulation of the LBM that allows for efficient simulations on irregular grids. Advantages of our new method include the geometric flexibility of the domain, the high-order advection step, a variable time step size and the easy application of sophisticated velocity sets. These advantages will turn the SLLBM into a high-potential candidate for the simulation of thermal and compressible flows. Succeeding a substantial analysis of the ELBM and the SLLBM in the first part of this proposal, simulations of compressible forced isotropic turbulence, compressible temporal mixing layers, and supersonic turbulent channel flows are performed in the second part of the project, partly for the first time with compressible LBM in general. This is necessary to analyze differences in the respective approaches and to gain required insights into finding an accepted and established approach to compressible flows using LBM. The test cases allow investigating intrinsic and variable density compressibility effects seperately, include shocklets and even allow (isotropic turbulence) a splitting into solenoidal and dilatational parts, in addition to a detailed comparison with the literature.