Accurate Prediction of Fluid Motion

流体运动和由此输送的材料的准确预测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年将造成数千亿美元的损失），也是优化能源效率的关键（美国85%的能源来自燃烧，因此精确模拟湍流混合至关重要）。不幸的是，在这些以及本研究所涉及的其他应用中，存在着准确、高效和可靠的流体流动预测的基本障碍。不确定数据下的准确预测需要可靠、高效的集合模拟。当前方法的成本限制了集合大小，从而限制了预测的准确性。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本项目中开发。目前为止，人工压缩方法在时间步长上是最有效的，但由于时间精度低、时间步长条件有限、稳定性问题、不适应和非物理声波等问题，很少使用。他们的解决方案将使人工压缩方法复活，成为准确、可靠和有效的流体运动预测方法，扩展了集合模拟和耦合流动预测，显着超越了它们目前的局限性。人工压缩方法表现出寄生压力波，在较高的雷诺数下产生共振。本研究将发展一种基于流动产生声音的Lighthill理论的方法，并将其应用于设计时间滤波器以抑制寄生声学。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、变阶方法已被证明是求解小型常微分方程组最有效、最准确、最可靠的方法。然而，先前的变步长、变阶方法在计算流体力学实践中的渗透有限，部分原因是它们的实现复杂性和每步成本的增加。新方法具有与全隐式方法相同的认知复杂度和计算复杂度。人工压缩方法中速度和压力的不耦合在速度解中引入了一个额外的梯度项，降低了稀疏性，增加了病态。因此，人工压缩方法的效率随着每一步的存储和求解器成本的增加而降低。该研究将开发一种新的实现，模块化的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，但它们将被整合到一个整体中，人工压缩方法，并在引人注目的兴趣问题上进行测试。拟议的研究发展博士生在分析，数值分析和应用领域的专业知识，同时致力于广泛影响的引人注目的数学问题，推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紧密结合。在项目中，每个博士生都可以制定自己的研究议程，并在研究问题之间的接触点进行合作。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.