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，并起着至关重要的作用。例如，海洋中地球物理涡旋的湍流是流体动量和盐度或浮游生物等标量混合的原因。旋转流在工业过程中也很重要，通过有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）。流体流动的旋转轮廓通常是微分的，即角速度随旋转轴的半径而变化。当压力梯度和离心力之间存在不平衡时，即角动量随半径减小时，这种差转流就会变得离心不稳定。这种离心不稳定性是非常具有破坏性的，因此是湍流的一个重要来源。大多数关于离心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足够小，可以忽略控制方程中的非线性项。另一方面，不稳定性的非线性发展过程，如饱和或层流-湍流过渡，尚未得到深入的研究。特别是在热扩散和分层共同作用下的非线性离心不稳定性还没有完全认识。带传热的流体流动是各种自然系统和工程系统中非常常见的配置，因此揭示这种热效应对湍流的作用可以大大有助于我们了解物理科学和工程中的多物理流动系统。这种情况促使目前的研究方案有两个主要目标：(i)研究在热扩散和分层作用下离心不稳定性的非线性发展过程；（ii）开发一种新的湍流模型，用于多物理场模拟。在节目的第一部分，我们将研究著名的旋转剪切流的线性和非线性离心不稳定性，称为泰勒-库埃特（TC）流，两个独立旋转的同心圆柱体之间的流动。我们将首先使用WKBJ （Wentzel-Kramers-Brillouin-Jeffreys）方法分析热扩散和分层流体中TC流的线性离心不稳定性。线性分析将揭示热效应如何影响小振幅扰动的初始增长，WKBJ方法将使我们能够推导出不稳定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证明扰动和基流之间的非线性相互作用如何导致饱和或层流-湍流过渡过程。该计划的第二部分将侧重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用于构建一个适用于多物理场模拟的湍流粘度。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。更新后的代码将模拟恒星演化，并产生诸如质量、角动量或恒星内部化学物质的径向分布等结果。这些结果将与其他恒星演化模拟和观测的结果进行比较。通过实现提出的研究的主要目标，我们将推进我们对不稳定引起的湍流及其在恒星演化的多物理过程中的作用的理解，这只是一个例子。这样的湍流模型也将有利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.