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必不可少的。举两个例子，流量预测是限制飓风对人类生命和经济造成损害的关键（后者估计在2017年将达到数千亿美元），也是能源效率优化的关键（美国85%的能源是由燃烧产生的，准确模拟湍流混合至关重要）。不幸的是，在提出的研究中解决的这些和其他应用中存在对流体流动的准确、有效和可靠预测的根本障碍。不确定数据的准确预测需要可靠和有效的集成模拟。当前方法的成本通过限制系综大小来限制预测准确性。进一步的改进需要新的计算工具，从根本上降低仿真成本和内存需求。解决这些需求的算法将在本项目中开发。人工压缩方法是迄今为止最有效的每一个时间步长，但很少使用，由于低的时间精度，限制性的时间步长条件，稳定性问题，病态和非物理声波。它们的解决方案将使人工压缩方法恢复为准确，可靠和有效的方法，用于预测流体运动，扩展集合模拟和耦合流预测，显着超出其目前的局限性。人工压缩方法表现出寄生压力波，成为共振在较高的雷诺数。本研究将发展一种依赖于莱特希尔流生声理论的方法，并将其应用于设计时间滤波器以抑制寄生声。时间精度将通过发展一系列新的变步长、变阶方法来实现。变步长、变阶方法已被证明是求解较小常微分方程组的最有效、最精确和最可靠的方法。 然而，以前的变步长，变阶数的方法有有限的渗透到计算流体动力学的实践，部分是由于其实现的复杂性和增加的成本，每一步。新的方法有相同的认知和计算的复杂性，全隐式方法（以领先的顺序）。人工压缩方法中速度和压力的解耦合在速度求解中引入了额外的梯度项，从而降低了稀疏性并增加了病态。因此，人工压缩方法的效率随着每步的存储和求解器成本的增加而损失。该研究将开发一种新的实现方式，即模块化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集成的人工压缩方法中，并在引人注目的问题上进行测试。拟议的研究开发博士生在分析，数值分析和应用领域的专业知识，同时致力于推进流体运动的准确预测的广泛影响的引人注目的数学问题。它与PI的博士生和本科研究人员的发展精心整合。在该项目中，每个博士生都可以制定自己的研究议程，并在研究问题的联系点上进行合作。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值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.