Hitching the subcritical branch of convection
连接对流的亚临界分支
基本信息
- 批准号:EP/X010937/1
- 负责人:
- 金额:$ 9.1万
- 依托单位:
- 依托单位国家:英国
- 项目类别:Research Grant
- 财政年份:2023
- 资助国家:英国
- 起止时间:2023 至 无数据
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
Thermal convection due to temperature differences in fluids drives multiple processes from planetary and stellar interiors, to the casting of metals and heat extraction. In its simplest form, convection sets in when the ratio of buoyancy to viscous forces, the Rayleigh number Ra, exceeds the critical value Rac at which an infinitesimal perturbation to the non-convective equilibrium becomes amplified by the flow dynamics. In more complex examples, subcritical convection may exist, i.e. for Ra below Rac. This not only happens in the ducts of heat exchangers, but also was recently discovered during the continuous casting of alloys, where it could lead to unwanted segregation defects in solidified alloys. Recently, subcritical convection also appeared in numerical simulations of planetary interiors, where a hot solid core is surrounded by a colder liquid metal extracting heat from it through a complex convective process controlled by the interplay between buoyancy, the Coriolis force induced by the planet's rotation and the Lorentz force due to its magnetic field.In all three cases, the subcritical nature of convection is crucial: igniting convection below Rac significantly enhances heat transfer: this is the holy grail of cooling applications and one of the technological deadlocks in the design of nuclear fusion reactors. The challenge is to perturb a non-convective flow into a subcritical convective state. Conversely, defects incurred by subcritical convection must be avoided in continuous casting. Convection is also the beating heart of planets, driving amongst other processes, the dynamo action that sustains their magnetic field. An excursion away from a potentially subcritical convective state could shut down convection in planetary cores, one of the ways planets may "die".In these problems the central questions are "how far below criticality can convection exist ?" and "what perturbation either ignites or extinguishes subcritical convection ?". Furthermore, whether subcritical convection even subsists in the presence of planetary magnetic fields is not even known. Straight simulations of the governing equations cannot answer these questions because they cannot reliably tell if convection is stable. Continuation methods can capture convective states regardless of their stability, but do not directly apply as reaching or leaving the convective state requires a discontinuous 'jump', as sought here.This project will answer these mathematical questions in all three examples, by taking advantage of recent developments in stability theory. For the first question, exact solutions on disconnected branches will be captured from either simulations or distant states by adapting the hook step and Time Delay Control methods currently used to study the transition to turbulence in shear flows. These states can then be traced back to the origin of the subcritical branch using continuation methods. For the second, we will use perturbations with optimal transient energy growth to destabilise the non-convective equilibrium into the subcritical branch (or the reverse) and find paths to the extinction or the ignition of convection.While the importance of subcritical convection in geophysical and casting problems only came to light very recently, so did the techniques to elucidate its true role. So too did the opportunity to exploit them in industry, as metallurgists increasingly turn to rigorous mathematics to control their processes. Ongoing collaboration with metallurgists and this work's relevance to nuclear fusion reactors offer direct opportunities for these new methods to start replacing current trial-and-error practice in design by tailored optimisation methods in these industries and potentially others. To this end, we will implement these methods into an open-source numerical package capable of finding or igniting the full range of subcritical convective flows in the widest possible range of problems.
由于流体中的温差引起的热对流驱动了从行星和恒星内部到金属铸造和热量提取的多个过程。在其最简单的形式中,当浮力与粘性力的比率,即瑞利数Ra,超过临界值Rac时,对流开始,在该临界值Rac处,对非对流平衡的无穷小扰动被流动动力学放大。在更复杂的例子中,可能存在亚临界对流,即Ra低于Rac。这不仅发生在热交换器的管道中,而且最近在合金的连续铸造期间也被发现,其中它可能导致凝固合金中的不希望的偏析缺陷。最近,亚临界对流也出现在行星内部的数值模拟中,在这种情况下,热的固体核心被较冷的液态金属包围,通过浮力、行星旋转引起的科里奥利力和磁场引起的洛伦兹力之间的相互作用控制的复杂对流过程从其中提取热量。在所有三种情况下,亚临界对流的性质是至关重要的:在Rac以下点燃对流显著增强了热传递:这是冷却应用的圣杯,也是核聚变反应堆设计中的技术僵局之一。挑战在于扰动非对流流进入亚临界对流状态。相反,在连续铸造中必须避免由亚临界对流引起的缺陷。对流也是行星跳动的心脏,在其他过程中驱动着维持其磁场的发电机作用。偏离潜在的亚临界对流状态可能会关闭行星核心的对流,这是行星可能“死亡”的方式之一。在这些问题中,核心问题是“对流在临界状态下能存在多远?”什么样的扰动会点燃或破坏亚临界对流?".此外,亚临界对流是否在行星磁场存在的情况下存在甚至还不知道。控制方程的直接模拟不能回答这些问题,因为它们不能可靠地判断对流是否稳定。连续方法可以捕捉对流状态,而不管它们的稳定性,但不直接应用,因为到达或离开对流状态需要一个不连续的“跳跃”,如这里所寻求的。本项目将回答所有三个例子中的这些数学问题,利用稳定性理论的最新发展。对于第一个问题,断开的分支上的精确解将被捕获的模拟或遥远的状态,通过调整钩步骤和时间延迟控制方法,目前用于研究过渡到湍流剪切流。然后,可以使用连续方法将这些状态追溯到亚临界分支的起源。对于第二个,我们将使用扰动与最佳的瞬态能量增长不稳定的非对流平衡到亚临界分支(或相反),并找到路径的灭绝或点燃convention.While的重要性,亚临界对流在地球物理和铸造问题只是最近才被曝光,所以没有技术来阐明其真正的作用。在工业中利用它们的机会也越来越多,因为生物学家越来越多地转向严格的数学来控制他们的过程。正在进行的与生物学家的合作以及这项工作与核聚变反应堆的相关性为这些新方法提供了直接的机会,可以开始在这些行业和其他行业中通过定制的优化方法来取代当前的试错设计实践。为此,我们将实施这些方法到一个开源的数值包能够找到或点燃的亚临界对流的全方位的问题。
项目成果
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