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.