Accurate Prediction of Fluid Motion

准确预测流体运动和由此输送的材料对于许多关键的工程和科学应用是至关重要的。举两个例子，流量预测是限制飓风对人的生命和经济的损害(后者在2017年估计达到数千亿美元)和能源效率优化(美国85%的能源来自燃烧，对湍流混合的准确模拟至关重要)的关键。不幸的是，在这些和拟议研究中涉及的其他应用中，存在着准确、高效和可靠地预测流体流动的根本障碍。在数据不确定的情况下，准确的预测需要可靠和有效的集成模拟。当前方法的成本限制了集合大小，从而限制了预测精度。进一步的改进需要新的计算工具，从根本上降低模拟成本和内存需求。解决这些需求的算法将在这个项目中开发。到目前为止，人工压缩方法是每时间步长最有效的方法，但由于时间精度低、时间步长条件受限、稳定性问题、病态和非物理声波等原因，人工压缩方法很少使用。它们的解决方案将使人工压缩方法重新成为预测流体运动的准确、可靠和有效的方法，扩展系综模拟和耦合流动预测明显超出了它们目前的限制。人工压缩方法显示出寄生压力波，这种压力波在较高的雷诺数时变得共振。这项研究将发展一种依赖于方法的莱特希尔流动产生声理论，并将其应用于设计抑制寄生声学的时间过滤器。时间精度将通过开发一系列新的变步长、变阶方法来实现。变步长、变阶数方法已被证明是求解较小的常微分方程组的最有效、最准确、最可靠的方法。然而，以前的变步长、变阶数方法由于其实现的复杂性和每一步的成本增加而限制了对计算流体力学实践的渗透。新方法具有与完全隐式方法相同的认知和计算复杂性(达到数量级)。在人工压缩方法中，速度和压力的解耦在速度求解中引入了额外的梯度项，减少了稀疏性，增加了病理性。因此，随着每一步存储和解算器成本的增加，人工压缩方法的效率会降低。这项研究将开发一种新的实现方式，模块化Grad-Div，在初步测试中将存储和周转时间减少30倍。虽然每个开发都有独立的兴趣，但它们将被集成到一个整体，人工压缩方法中，并在引人注目的问题上进行测试。这项拟议的研究发展了博士生在分析、数值分析和应用领域的专业知识，同时致力于研究具有广泛影响的引人注目的数学问题，促进流体运动的准确预测。它与PI的博士生和本科生研究人员的发展精心结合。在该项目中，每个博士生可以制定他们自己的研究议程，并在研究问题的接触点进行合作。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Accurate prediction of fluid motion and materials thereby transported is essential for many critical engineering and scientific applications. As two examples, flow predictions are key to limiting damage of hurricanes to human life and to the economy (the latter estimated to be hundreds of billions of dollars in 2017) and to energy efficiency optimization (85% of US energy is generated by combustion for which accurate simulation of turbulent mixing is critical). Unfortunately, fundamental barriers to accurate, efficient and reliable prediction of fluid flow exist in these and other applications addressed in the proposed research. Accurate prediction with uncertain data requires reliable and efficient ensemble simulations. The cost of current methods limits prediction accuracy by limiting ensemble sizes. Further improvement requires new computational tools with a fundamental decrease in simulation cost and memory requirements. Algorithms which address these needs will be developed in this project. Artificial compression methods are by far the most efficient per time step but little used due to low time accuracy, restrictive time step conditions, stability problems, ill-conditioning and nonphysical acoustic waves. Their resolution will resurrect artificial compression methods into accurate, reliable and efficient methods for the prediction of fluid motion, expanding ensemble simulations and coupled flow prediction markedly beyond their current limitations. Artificial compression methods exhibit parasitic pressure waves that become resonant at higher Reynolds numbers. This research will develop a method dependent Lighthill theory of flow generated sound and apply it to design time filters to suppress parasitic acoustics. Time accuracy will be achieved by development of a new family of variable step, variable order methods. Variable step, variable order method have proven to be the most efficient, accurate and reliable methods to solve smaller systems of ordinary differential equations. However, previous variable step, variable order methods have limited penetration into computational fluid dynamics practice due partially to their implementation complexity and increased cost per step. The new methods have (to leading order) the same cognitive and computational complexity as the fully implicit method. Uncoupling of velocity and pressure in artificial compression methods introduces an extra grad-div term in the velocity solve, decreasing sparsity and increasing ill conditioning. Thus, the efficiency of artificial compression methods is lost with increased storage and solver cost per step. The research will develop a new realization, modular Grad-Div, reducing storage and turnaround time by a factor of 30 in preliminary tests. While each development has independent interest, they will be integrated into an ensemble, artificial compression method and tested on problems of compelling interest. The proposed research develops expertise of PhD students in analysis, numerical analysis and application areas while working on compelling mathematics problems of broad impact advancing the accurate prediction of fluid motion. It is carefully integrated with the development of the PI's PhD students and undergraduate researchers. Within the project, each PhD student can develop their own research agenda and collaborate at the points of contact among the research problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.