Mathematical Sciences: Propagating Fronts By A Consistent Primitive Algorithm With Application To Bubble Dynamics

研究人员将开发新的数值方法来模拟源自一组守恒定律的非线性双曲型微分方程的弱解。该系统的守恒形式被广泛认为是持续的激波数值捕捉的必要条件。在实践中，虽然这对于以强激波为主的流动可能是正确的，但其他流动现象可能更好地模拟不同的、不一定保守的流动描述。在这种情况下，人们可能准备牺牲严格守恒，转而采用非严格保守的模型，该模型更好地描述了主要的流动特征。为了可靠，该模型应该产生可接受的微小守恒误差。研究人员指出，传播锋面就是这样一种现象。用保守模型捕捉传播波阵面会引起物质阵面附近的振荡和其他不准确，通常是不能令人满意的。这些不是与高阶格式有关的常见振荡，而是保守系统简单波动模型(即本征结构)的结果。一种补救方法是使用非保守的流描述，它提供了不同的特征结构。它可以精确地靠近接触面，并产生干净、无振荡的界面。在激波附近，模型并不准确，守恒也犯了错误。这位研究人员最近开发的一种技术为这个问题提供了解决办法。它使用粘性摄动，使算法保守到数值近似的数量级，并产生可接受的微小守恒误差。该技术已成功地应用于一维传播波阵面的计算。研究人员将研究这种非严格保守算法的几个理论方面，以及它在几个空间维度上的气泡动力学计算中的应用。该项目的工作包括流动的理论分析和计算机模拟。这些现象发生在涉及流体混合、涡旋的启动和演化、流体中化学反应的点火和演化以及火焰传播的物理现象中。该项目的目的是开发分析和计算工具，以加强对所涉及的复杂物理现象的理解。通过这种方式获得的洞察力将补充通常昂贵、困难甚至危险的实验数据。例如，对平流层上层空间条件的风洞模拟既困难又昂贵，可能会得出不可靠的结果；研究枪支弹道需要进行强烈爆炸试验，这既昂贵又危险。该项目的潜在影响涉及多个行业，包括航天和航空业(航天器和飞行器机翼的设计)、海运业(改进水动力性能的船的设计、气泡对海底水动力的影响等)。和武器工业(例如，枪支的内弹道)。

The investigator will develop new numerical methods to model weak solutions to nonlinear hyperbolic differential equations derived from a set of conservation laws. Conservation form of the system is widely held as imperative for consistent numerical capturing of shocks. In practice, while this may be true for flows dominated by strong shocks, other flow phenomena may be better modelled by different, not necessarily conservative flow descriptions. In such cases, one may be prepared to sacrifice strict conservation in favor of a nonstrictly conservative model that describes better the dominant flow features. To be reliable, the model should generate acceptably small conservation errors. The investigator shows that propagating fronts is one such phenomenon. Capturing propagating fronts by conservative models gives rise to oscillations and other inaccuracies near material fronts, and is in general unsatisfactory. These are not the common oscillations associated with high order schemes, but rather a result of the simple wave model (i.e. eigenstructure) of the conservative systems. A remedy is to use a nonconservative flow description, which offers a different eigenstructure. It becomes exact near contact surfaces and produces clean oscillation-free interfaces. Near shocks, the model is not exact and conservation errors are committed. A technique recently developed by the investigator offers a cure to this problem. It uses viscous perturbations that render the algorithm conservative to the order of numerical approximation and generate acceptably small conservation errors. The technique has been successfully applied to computing propagating fronts in 1D. The investigator will study several theoretical aspects of this nonstrictly conservative algorithm and its application to the computation of bubble dynamics in several space dimensions. The work of the project involves theoretical analysis and computer simulation of flows. These arise in physical phenomena involving mixing of fluids, the initiation and evolution of vortices, the ignition and evolution of chemical reactions in fluids, and the propagation of flames. The aim of the project is to develop analytical and computational tools to enhance the understanding of the complicated physical phenomena involved. Insight gained this way will supplement experimental data, which are often expensive, difficult or even dangerous to obtain. For example, wind tunnel simulations of space conditions in the upper stratosphere are difficult and expensive to perform and may give unreliable results; studying gun ballistics requires experimenting with strong explosions, which are both expensive and dangerous. Potential impact of this project extends to a variety of industries including the space and aircraft industries (design of space vehicles and of aicraft wings), marine industry (design of boats with improved hydrodynamic properties, the effect of bubbles on submarine hydrodynamics, etc.) and the weapon industries (e.g., internal ballistics of guns).