Generalised high-order Eulerian Smoothed Particle Hydrodynamics for internal flows applied to flow-induced vibration and nuclear tube banks

计算流体动力学（CFD）已经成熟，有几个通用的商业代码可供使用，这些代码基于有限体积或有限元（基于网格）方法，具有各种湍流建模选项。然而，CFD在工业设计中的成功促进了对流动分辨率和更细粒度物理学的需求不断增加，需要在计算和人力方面使用越来越多的资源。在核反应堆、涡轮机械、燃烧室、热交换器、船用涡轮机、车辆空气动力学、航空、近海工程等领域取得了显著的成功。商业企业往往集中在两个关键方面，以提高效率和精度的模拟越来越复杂和苛刻的实际问题：性能的大规模并行计算和优化网格生成。商业CFD的最新技术表明，网格可能包含数亿个单元，核反应堆模拟超过10亿个（CD-Adapco，2016），并通过大规模并行计算（通常使用数千个处理器）运行，需要数天或数周。高阶（HO）方法的实现得到了相对较少的关注，但可以提供灵活性，并在效率和准确性方面的收益超出了通过优化网格划分和并行化单独实现的。HO方法在非定常涡主导和湍流建模中是有益的，甚至是必要的，其中许多问题仍然超出了最先进的二阶CFD的范围，即使在超级计算机上。来自学术界的重要开源代码在提高HO方法的使用率方面取得了进展，但在复杂的3-D几何形状（可能包含任意移动的边界）和自适应性中的最佳实现仍然是高阶框架中具有挑战性的问题。我们建议通过一种替代的数值方法来解决这些问题，该方法简单而有吸引力，适合复杂域中的高阶空间近似，同时保留对新兴架构的并行化的天然亲和力。我们提供这种改进的能力，放弃网格和使用粒子，其中，拉格朗日形式，已被广泛用于建模的高度扭曲的流动，涉及接口和多物理。研究人员一直积极发展光滑粒子流体动力学（SPH），特别是在发散自由不可压缩的形式，并在开发节能硬件的算法。最近的欧拉形式已被测试的研究人员与高阶高斯插值内核（高达6阶），证明空间收敛到机器的模型周期性问题的精度。在二阶时间步进的粘性瞬态流中，所获得的精度与谱方法相似。这一新办法特别为内部流动提供了大量机会。这种方法的一个缺点是，复杂系统需要数十亿个粒子，并且SPH中每个粒子每秒的浮点运算（FLOPS）通常比有限体积/hp元素等效值大一个数量级。由于SPH公式的局部插值（无网格）性质和易于在包括大多数GPU在内的新兴硬件上实现，因此SPH公式非常适合并行处理。

Computational fluid dynamics or CFD is mature with several general-purpose commercial codes available based on the finite-volume or finite-element (mesh-based) approaches with various options for turbulence modelling. The success of CFD in industrial design has however encouraged increasing demands to be made in terms of the resolution of the flow and finer grain physics, requiring ever increasing resources to be employed, both in terms of computation and manpower. Notable successes are in nuclear reactors, turbo-machinery, combustion chambers, heat exchangers, marine turbines, vehicle aerodynamics, aeronautics, offshore engineering amongst many others. Commercial enterprise tends to focus on two key aspects for improving the efficiency and accuracy of simulations in increasingly complex and demanding practical problems: performance on massively parallel computing and optimal mesh generation. State-of-the-art in commercial CFD suggests meshes may comprise several hundred million cells, over a billion for a nuclear reactor simulation (CD-Adapco, 2016), and runs with massively parallel computing (often with thousands of processors) taking days or weeks. Implementation of High-Order (HO) methods has received comparatively less attention, but can offer flexibility and gains in efficiency and accuracy beyond what can be achieved through optimal meshing and parallelisation alone. HO methods are known to be beneficial, even necessary, in unsteady vortex-dominated and turbulent flow modelling where many problems remain beyond the reach of state-of-the-art second-order CFD even on supercomputers. Important open-source codes from academia are making headway in increasing uptake of HO methods, but optimal implementation within complex 3-D geometries (that may contain arbitrarily moving boundaries) and adaptivity remain challenging problems in a high-order framework. We propose to address these problems through an alternative numerical method that is attractive in its simplicity, amenable to high-order spatial approximations in complex domains while retaining a natural affinity for parallelisation on emerging architectures. We provide this improvement in capability by abandoning the mesh and using particles, which, in Lagrangian form, have been used widely for the modelling of highly distorted flows involving interfaces and multi-physics. The investigators have been active in the development of smoothed particle hydrodynamics (SPH) particularly in divergence-free incompressible form and in developing algorithms for energy efficient hardware. Recently an Eulerian form has been tested by the investigators with high order Gaussian interpolating kernels (up to 6th order) demonstrating spatial convergence to machine accuracy in model periodic problems. In viscous transient flow with second-order time stepping, the accuracy obtained is similar to spectral methods. This new approach opens up considerable opportunities particularly for internal flows. One downside of this approach is that several billion particles will be required for complex systems, and the floating point operations per second (FLOPS) per particle in SPH are typically an order of magnitude greater than the finite volume/hp-element equivalent. This is compensated by the SPH formulation being ideally suited for parallel processing due to its locally interpolative (meshless) nature and ease of implementation on emerging hardware including most GPUs.