Absolute and convective instabilities of, and signalling in, a flow in a porous medium with inclined temperature gradient and vertical throughflow

具有倾斜温度梯度和垂直通流的多孔介质中流动的绝对和对流不稳定性以及信号传输

• 批准号: EP/G002835/1
• 负责人: Michael Ruderman
• 金额: $ 0.73万
• 依托单位: University of Sheffield
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2008
• 资助国家: 英国
• 起止时间: 2008 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG002835%2F1
• 关键词: Absolute; convective; instabilities; signalling; flow

Flows in saturated porous media, such as ground water flows, saturated contaminant flows in a contaminanted soil and magma flows in the Earth's mantle are considerably influenced by an inhomogeneous temperature distribution in the media. In a tyipical model geometry of such flows, the porous medium is assumed to be a horizontal layer. The temperature of the bottom of the layer is supposed to be greater that that of its top resulting in the vertical variation of temperature within the layer. Also, according to observations one imposes a horizontal variation of temperature and allows for a vertical throughflow in the model. If strong enough, the variability of temperature can trigger a motion of the fluid deviating form the base state, i.e. a convection. Such a motion emerges as a consequense of the destabilisation of the basic state. The destabilisations occurs owing to the localised perturbations of the flow that are present under all conditions. The purpose of the proposed research is to use the modern methods of the linear stability theory in order to extend the existing treatments of convection in a porous medium with inclined temperature gradient, that analyse only spatially sinusoidal perturbations, to treating realistic localised disturbances. An analysis of localised disturbances would allow us to distinguish between two different destabilisation scenarios. In the first one the localised disturbances grow indefinitly at every location of the flow, thus destroying the base state throughout. This is the scenario of absolute instability. In the second case, the localised unstable perturbations move away from the place of their origin leaving behind an unperturbed state. Such a case is defined as the case of convective instability. In this case the base state flow, though being unstable, can be viewed as representing a physical end state in a certain portion of space. In other words, the absolute instability is a catastrophic instability as it results in the destruction of the base state throughout, whereas a convectively unstable, but absolutely stable state can be viewed as unperturbed in a certain portion of space despite the instability. This distinction means that the emergence of a secondary motion in the model and the characteristics of this motion depend on whether the unstable state is absolutely unstable or absolutely stable, but convectively unstable.In the proposed research, we first extend the existing analysis of monochromatic disturbances for a variety of the control parameter values (the vertical and horizontal Rayleigh numbers and the Peclet number) and obtain neutral curves. Further, localised disturbances will be treated by using the methods of the theory of absolute and convective instabilities. The numerical procedure in the treatment uses a Chebyshev collocation method for discretising the stability problem and a software implemention of the QZ-algorithm for treating the resulting generalised algebraic eigenvalue problem.

饱和多孔介质中的流动，如地下水流、污染土壤中的饱和污染物流和地幔中的岩浆流，都受到介质中温度分布不均匀的影响。在这种流动的典型几何模型中，多孔介质被假定为水平层。层底温度高于层顶温度，导致层内温度的垂直变化。此外，根据观察，一个强加的温度水平变化，并允许在模型中的垂直通流。如果足够强，温度的变化性可以触发流体偏离基本状态的运动，即对流。这种运动是作为基本状态不稳定的结果而出现的。由于在所有条件下都存在的流动的局部扰动而发生不稳定。拟议的研究的目的是使用现代方法的线性稳定性理论，以延长现有的治疗对流在多孔介质中的倾斜温度梯度，分析只有空间正弦扰动，治疗现实的局部扰动。对局部扰动的分析将使我们能够区分两种不同的不稳定情景。在第一种情况下，局部扰动在流动的每一个位置都不确定地增长，从而破坏了整个基态。这是绝对不稳定的情况。在第二种情况下，局部的不稳定扰动远离它们的起源，留下一个未扰动的状态。这种情况被定义为对流不稳定的情况。在这种情况下，基态流虽然是不稳定的，但可以被视为代表空间某一部分中的物理结束状态。换句话说，绝对不稳定性是一种灾难性的不稳定性，因为它会导致整个基态的破坏，而对流不稳定，但绝对稳定的状态可以被视为在空间的某一部分，尽管不稳定。这种区别意味着二次运动在模型中的出现和这种运动的特点取决于是否不稳定的状态是绝对不稳定或绝对稳定，但对流unstable.在拟议的研究中，我们首先扩展了现有的分析单色干扰的各种控制参数值（垂直和水平的瑞利数和Peclet数），并获得中性曲线。此外，局部扰动将通过使用绝对和对流不稳定性理论的方法来处理。在治疗中的数值程序使用Chebyshev配置法离散化的稳定性问题和软件实现的QZ算法治疗所产生的广义代数特征值问题。