Mathematical underpinnings of stratified turbulence

Turbulence is an everyday experience, from the water emerging from a tap to the wind on one's face. Despite its widespread importance, a mathematical description of turbulent flow remains elusive. In most turbulent flows of practical interest, the vast number of degrees of freedom makes it impossible to solve the equations of motion directly. Recently, however, significant progress has been made by taking the view point that turbulence can be treated as a large dynamical system in which the turbulence is represented by differential equations that, in principle, can be integrated forward in time to evolve from one state to the next. While previous work in this area has focused on uniform density fluids, the buoyancy forces associated with density variations play an important dynamical role in many applications (e.g. warm air is less dense than cold air, which is critical for efficient building heating and cooling strategies).

This project will develop the mathematical underpinnings of stratified turbulence. The project builds on the dynamical systems viewpoint, developed for a fluid which has a constant density and extends it to one in which the fluid density varies in space and time. In addition to extending the analysis to new practical applications, including buoyancy effects associated with density variations will allow us to probe universal aspects of turbulence in a way that would not be possible in a homogeneous fluid. Much of the progress in turbulence theory to date has been in flows which are transitional so the flow is in an intermediate state between being completely laminar and completely turbulent. In a stratified fluid, the buoyancy force inhibits vertical motions (e.g. warm air tends to stay near the ceiling) and makes laminar/turbulent transitions more prevalent, their structures more identifiable, and the dynamics richer so there are more phenomena to explore.

Our approach is based on tight coupling between mathematics, simulation and experimentation. Although stratified turbulence has been studied experimentally for many years, none of the existing flow geometries can address all three of our primary objectives. Consequently, we will build a new experiment to study stratified turbulence in a canonical geometry in which the turbulence is generated by shear and opposed by the stratification. This new geometry will allow us to observe and measure laminar/turbulent transitions, the flow structures that are involved and their distributions in time and space. We will also carry out new experiments in two other geometries to examine how the flow features change in different configurations. We will use numerical simulations of each experimental configuration to complement the laboratory experiments by allowing additional diagnostics, and providing a direct link between the mathematical tools and the flow geometries.

One of the important outcomes of combining mathematical analysis, laboratory experiments and numerical simulations will be to develop a simplified dynamical description of the system. This reduced dynamical system will capture the key physics of stratified turbulence, and provide a generic tool for understanding and modelling turbulence and mixing processes in diverse contexts of economic, environmental and societal importance. Eventually we hope to provide practical estimates of mixing and transport in a turbulent stratified fluid.

Stratified turbulence inevitably leads to irreversible mixing of the stratifying agent. Any industrial process involving fluid connects with this research if the fluid either contains different concentrations of scalars or varies in temperature. We give three examples to illustrate the likely beneficiaries of this research, all of which have a potential significant impact on the nation's economy.

1. Architects, designers and structural engineers. In the drive for low-energy (or zero-energy) buildings there is a demand for new methods of heating and cooling. These often involve hybrid (or mixed-mode) approaches that combine mechanical cooling and natural ventilation, and inevitably lead to flow and mixing of air of different temperatures and different pollutant concentrations. Estimates of mixing rates are critical to ventilation efficiency and energy consumption, indoor air quality and perceived comfort. Designers use software in which these mixing processes are ignored or represented in simplified inaccurate forms. The outcomes of our research will provide accurate estimates of mixing rates related to the ventilation flows that can be implemented into design codes. Such issues are often critical to conservation of artwork, or in heritage buildings (e.g. museums).

2. Engineers in the hydrocarbon and related industries. Most industrial fluids are mixtures with components of different densities (often different phases). Transporting such fluids in pipelines typically induces turbulence and shearing flows with many features in common with the flows we are considering. Coherent flow structures of the kind we will identify will be present in these practical situations with consequences for wear, corrosion and/or deposition within pipes. The results of our research will provide guidance on the properties of the coherent structures by identifying the circumstances under which they might occur and ultimately mitigating their unwanted effects. There are also many circumstances in chemical processing where thorough mixing is desired, and our research should point towards ways to enhance mixing in the presence of density gradients. Furthermore, our approach of solving technical challenges through a tightly-coupled and inter-related experimental, numerical and mathematical effort can serve as an exemplar.

3. Climate scientists, oceanographers and meteorologists. Irreversible mixing processes in the oceans and atmosphere take place on very small length and time scales (a few cm and seconds) that are far beyond the spatial and temporal resolution of even the most highly-resolved numerical weather forecast models. Consequently, they are parametrised by approximate models that relate these processes to the larger-scale resolved fields. The outcome of our research will provide new insights into the dynamics and new mathematical representations that will lead to improved parametrisations.

The scale of these impacts is potentially very large. Hydrocarbons are the largest component of global energy supply. Buildings contribute about 30% of the UK's carbon emissions and more energy-efficient ventilation will reduce that significantly. Similarly, improvements in weather and climate predictions have immense economic benefits. We expect that some of these impacts will be realised within the next 10 years. Demand reduction in the built environment is a hot topic and there are an increasing number of architects who claim 'green' credentials, some using the results of our work published over the last 20 years, and the general public will benefit from reduced energy bills. We also expect our work to guide the development of new parameterisations for climate models.

The staff working on the project will gain a wide range of skills including computational and laboratory expertise, team working, presentational skills and interpersonal skills.