Modelling turbulence induced by hydrodynamic instability in differentially-rotating flow

旋转流体流动在许多自然发生和工程系统中普遍存在，并且起着至关重要的作用。例如，海洋中地球物理漩涡的湍流导致了流体动量和盐度或浮游生物等标量的混合。旋转流动在工业过程中也很重要，通过高效的湍流混合来生产均质产品(例如玻璃或聚合物制造过程)。流体流动的旋转剖面通常是不同的，即角速度随旋转轴的半径变化。当压力梯度和离心力之间存在不平衡时，当角动量随半径减小时，这种差动流动可能会变得离心力不稳定。这种离心力的不稳定性具有很强的破坏性，因此是湍流的一个重要来源。大多数关于离心机不稳定性的研究都考虑了线性分析，其中假定驱动不稳定性的扰动小到足以忽略控制方程中的非线性项。另一方面，不稳定的非线性发展过程，如饱和或层流-湍流转变，还没有得到深入的研究。特别是，在热扩散和层结的共同作用下，对非线性离心力不稳定性的认识还不够充分。流体流动与换热在各种自然和工程系统中是一种非常常见的构型，因此揭示这种热效应对湍流的作用将有助于我们在物理科学和工程中对多物理流动系统的认识。这种情况促使目前的研究计划有两个主要目标：(I)研究热扩散和层结影响下离心机不稳定的非线性发展过程；(Ii)发展一种新的湍流模型，用于多物理模拟。在程序的第一部分，我们将研究一种名为Taylor-Couette(TC)流的著名旋转剪切流的线性和非线性离心不稳定性，即两个独立旋转的同心圆柱体之间的流动。我们将首先使用Wentzel-Kramers-Brillouin-Jeffreys(WKBJ)方法分析热扩散和分层流体中TC流动的线性离心不稳定性。线性分析将揭示热效应如何影响小幅度微扰的初始增长，而WKBJ方法将允许我们推导出不稳定性增长的显式数学表达式。然后将通过直接数值模拟和半线性模型来研究非线性不稳定性。这种非线性分析可以说明扰动和底流之间的非线性相互作用如何导致饱和或层流-湍流转变过程。该计划的第二部分将集中于开发一种新的湍流模型。线性和非线性稳定性分析的结果将被用来构造湍流粘性，以应用于多物理模拟。更具体地说，我们将把新模型应用到最先进的恒星物理模拟旋转恒星演化的代码中。更新的代码将模拟恒星的演化，并产生诸如恒星内部质量、角动量或化学物质的径向分布等结果。结果将与其他恒星演化模拟和观测的结果进行比较。通过实现拟议研究的主要目标，我们将推进我们对不稳定性诱导的湍流及其在恒星演化的多物理过程中的作用的理解，仅举一个例子。这种湍流模型也将有益于其他物理科学和工程领域的研究人员。

Rotating fluid flow is ubiquitous in many naturally occurring and engineering systems and plays a crucial role. For instance, the turbulence of geophysical vortices in the oceans is responsible for the mixing of fluid momentum and scalars such as salinity or planktons. Rotating flow is also important in industrial processes to produce homogenised products by efficient turbulent mixing (e.g. glass or polymer manufacturing processes). Rotation profiles of fluid flow are often differential, i.e. the angular speed varies with radius from the rotation axis. Such differentially-rotating flow can become centrifugally unstable when an imbalance exists between the pressure gradient and centrifugal force, a situation arising when the angular momentum decreases with the radius. This centrifugal instability is very destructive and thus an important source of turbulence. Most of the studies on centrifugal instability have considered linear analyses in which perturbations that drive the instability are assumed to be small enough to neglect nonlinear terms in the governing equations. On the other hand, nonlinear development processes of the instability, such as saturation or laminar-turbulent transition, have not been thoroughly investigated. In particular, the nonlinear centrifugal instability is not fully understood under the combined effects of thermal diffusion and stratification. Fluid flow with heat transfer is a very common configuration in various natural and engineering systems, thus revealing the role of such thermal effects on turbulence can significantly contribute to our knowledge of multi-physical flow systems in physical sciences and engineering. This situation motivates the current research programme with two main objectives: (i) Investigate nonlinear development processes of the centrifugal instability under the effects of thermal diffusion and stratification, and; (ii) Develop a new turbulence model to apply to multi-physics simulations. In the first part of the programme, we will examine linear and nonlinear centrifugal instability of a famous rotating shear flow called Taylor-Couette (TC) flow, the flow between two concentric cylinders that rotate independently. We will first analyse linear centrifugal instability of the TC flow in thermally diffusive and stratified fluids using the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) method. The linear analysis will reveal how the thermal effects affect the initial growth of small-amplitude perturbations and the WKBJ method will allow us to derive explicit mathematical expressions of the instability growth. Nonlinear instability will then be investigated by both direct numerical simulations and a semi-linear model. Such nonlinear analyses can demonstrate how nonlinear interactions between perturbations and base flow lead to the saturation or laminar-turbulent transition processes. The second part of the programme will focus on developing a new turbulence model. Results from linear and nonlinear stability analyses will be used to construct a turbulent viscosity to apply to multi-physics simulations. More specifically, we will apply the new model to the state-of-the-art code for stellar physics simulations of the evolution of rotating stars. The updated code will simulate the stellar evolution and produce results such as radial distributions of mass, angular momentum or chemicals in the stellar interior. The outcomes will be compared with those from other stellar evolution simulations and observations. By achieving the main objectives of the proposed research, we will advance our understanding of instability-induced turbulence and its role in the multi-physics processes of the evolution of star, as just one example. Such turbulence modelling will also be beneficial for researchers in other fields of physical sciences and engineering.