This project aims at developing new methods of analysis of the stability of fluid flows and flow control. Flow control is among the most promising routes for reducing drag, thus reducing carbon emissions, which is the strongest challenge for aviation today. However, the stability analysis of fluid flows poses significant mathematical and computational challenges. The project is based on a recent major breakthrough in mathematics related to positive-definiteness of polynomials. Positive-definiteness is important in stability and control theory because it is an essential property of a Lyapunov function, which is a powerful tool for establishing stability of a given system. For more than a century since their introduction in 1892 constructing Lyapunov functions was dependent on ingenuity and creativity of the researcher. In 2000 a systematic and numerically tractable way of constructing polynomials that are sums of squares and that satisfy a set of linear constraints was discovered. If a polynomial is a sum of squares of other polynomials then it is positive-definite. Thus, systematic, computer-aided construction of Lyapunov functions became possible for systems described by equations with polynomial non-linearity. In the last decade the Sum-of-Squares approach became widely used with significant impact in several research areas. The Navier-Stokes equations governing motion of incompressible fluid have a polynomial nonlinearity. This project will achieve its goals by applying sum-of-squares approach to stability and control of the fluid flows governed by these equations. This will require development of new advanced analytical techniques combined with extensive numerical calculations. The project has a fundamental nature, with main expected outcomes being applicable to a large variety of fluid flows. The rotating Taylor-Couette flow will be the first object to which the developed methods will be applied. Taylor-Couette flow, encountered in a wide range of industrial application, for a variety of reasons has an iconic status in the stability theory, traditionally serving as a test-bench for new methods. In order to maximise the impact of the research, the project collaborators will conduct targeted dissemination activities for industry and academia in the form of informal and formal workshops, in addition to traditional dissemination routes of journal papers and conferences. Selected representatives from industry will be invited to attend the workshops. Wider audience will be reached via a specially created and continuously maintained web page.

The world's oil supply is decreasing rapidly and over the next 10 or 20 years the price per barrel will spiral inexorably. Aviation is a significant consumer of oil and is also implicated in global warming through its generation of massive quantities of carbon dioxide and nitrogen oxide. Aircraft noise continues to be an increasingly important problem as airports expand. For these reasons aviation as we know it now will rapidly become unviable. There is no single solution to the problem and enormous changes to engines, airframe design, scheduling and indeed people's expectations of unlimited air travel are inevitable. Here we address one of the most important issues, improved aerodynamics, and develop the underpinning technology for Laminar Flow Control (LFC), the technology of drag reduction on aircraft. This will become the cornerstone of aircraft design. Even modest savings in drag of the order of 10% translate into huge savings in fuel costs and huge reductions in atmospheric pollution. Applications of the technology to military aircraft where range is often the main requirement and marine applications are similarly important. The development of viable LFC designs requires sophisticated mathematical, computational and experimental investigations of the onset of transition to turbulence and its control. Existing tools are too crude to be useful and contain little input from the flow physics. Major hurdles to be overcome concern: a) How do we specify generic input disturbances for flow past a wing in a messy atmosphere in the presence of surface imperfections, flexing, rain, insects and a host of other complicating features b) How do we solve the mathematical problems associated with linear and nonlinear disturbance growth in complex 3D flows c) How do we find a criterion for the onset of transition based on flow physics which is accurate enough to avoid the massive over-design associated with existing LFC strategies yet efficient enough to be useable in the design office d) How can we use experiments in the laboratory to predict what happens in flight experiments e) How can we devise control strategies robust enough to be used on civilian aircraft f) How can we quantify the manufacturing tolerances such as say surface waviness or bumps needed to maintain laminar flow The above challenges are huge and can only be overcome by innovative research based on the mathematical, computational and experimental excellence of a team like the one we have assembled. The solution of these problems will lead to a giant leap in our understanding of transition prediction and enable LFC to be deployed. The programme is based around a unique team of researchers covering all theoretical, computational, and experimental aspects of the problem together with the necessary expertise to make sure the work can be deployed by industry. Indeed our partnership with most notably EADS and Airbus UK will put the UK aeronautics industry in the lead to develop the new generation of LFC wings. The programme is focussed primarily on aerodynamics but the tools we develop are relevant in a wide range of problems. In Chemical Engineering there has long been an interest in how to pump fluids efficiently in pipelines and how flow instabilities associated with interfaces can compromise certain manufacturing processes. In Earth Sciences the formation of river bed patterns behind topology or man-made obstructions is governed by the same process that describes the initiation of disturbances on wings. Likewise surface patterns on Mars can be explained by the instability mechanisms of sediment carrying rivers. In Atmospheric Dynamics and Oceanography a host of crucial flow phenomena are intimately related to the basic instabilities of a 3D flow over a curved aerofoil. Our visitor programme will ensure that our work impinges on these and other closely related areas and that likewise we are aware of ideas which can be profitably be used in aerodynamics.

Abstracts are not currently available in GtR for all funded research. This is normally because the abstract was not required at the time of proposal submission, but may be because it included sensitive information such as personal details.

This project will develop a systematic approach to flight control system (FCS) design for very flexible or very large aircraft, of the type being considered for low-environmental-impact air transport and for long-endurance unmanned operations. It will create a virtual flight test environment that will support the design of advanced nonlinear FCS that fully account for the vehicle structural flexibility. To model the flight dynamics of flexible aircraft, it is necessary to develop analytical methods for generating Reduced Order Models (ROMs) via reduction of the full-order nonlinear equations of motion, and to do this in such a way that the essential nonlinear behaviour is preserved. The key issues addressed by our approach are that:1. The usual separation of flight dynamics and aeroelasticity is not appropriate for flight control when very low structural frequencies (which are also often associated with large amplitude motions) are present. Modelling and design methods based on a fully coupled system analysis are therefore necessary.2. Large wing deformations bring nonlinear dynamic behaviour, but current model reduction methods assume linearity. The development of nonlinear ROMs is an area that urgently needs advances, in general, and is necessary for control applications of flexible aircraft, in particular.3. Standard linear control design methods are inadequate for highly flexible aircraft, since their dynamic behaviour is intrinsically nonlinear. Fresh approaches to nonlinear FCS design are then required to control these systems in a provably robust way.The technical and scientific challenges to be overcome then include the simulation of significant aerodynamic and structural nonlinearities in full aircraft dynamics through the systematic development of a hierarchy of fully coupled large-order models, the reduction of these models to small-order nonlinear systems suitable for control development, and the development of robust control laws based on these reduced nonlinear models for gust load alleviation, trajectory control and stability augmentation. These methods will be exemplified in next-generation aircraft concepts that will be defined in discussion with end users. In fact, the project will benefit from a strong collaboration with major UK industrial partners, which will provide substantial technical inputs and support to the planned research activities.

The aim of this proposal is to transform the design and manufacture of structural systems by relieving the bottleneck caused by the current practice of restricting designs to a linear dynamic regime. Our ambition is to not only address the challenge of dealing with nonlinearity, but to unlock the huge potential which can be gained from exploiting its positive attributes. The outputs will be a suite of novel modelling and control techniques which can be used directly in the design processes for structural systems, which we will demonstrate on a series of industry based experimental demonstrators. These design tools will enable a transformation in the performance of engineering structural systems which are under rapidly increasing demands from technological, economic and environmental pressures. The performance of engineering structures and systems is governed by how well they behave in their operating environment. For a significant number of engineering sectors, such as wind power generation, automotive, medical robotics, aerospace and large civil infrastructure, dynamic effects dominate the operational regime. As a result, understanding structural dynamics is crucial for ensuring that we have safe, reliable and efficient structures. In fact, the related mathematical problems extend to other modelling problems encountered in other important research areas such as systems biology, physiological modelling and information technology. So what exactly is the problem we are seeking to address in this proposal? Typically, when the behaviour of an engineering system is linear, computer simulations can be used to make very accurate predictions of its dynamic behaviour. The concept of end-to-end simulation and virtual prototyping, verification and testing has become a key paradigm across many sectors. The problem with this simulation based approach is that it is built on implicit assumptions of repeatability and linearity. For example, many structural analysis methods are based on the concept of a frequency domain charaterisation, which assumes that response of the system can be characterised by linear superposition of the response to each frequency seperately. But, the response of nonlinear systems is known to display amplitude dependence, sensitivity to transient effects in the forcing, and potential bistability or multiplicity of outcome for the same input frequency. As a result, when the system is nonlinear (which is nearly always the case for a large number of important industrial problems) it is almost impossible to make dynamic predictions without introducing very limiting approximations and simplifications. For example, throughout recent history, there have been many examples of unwanted vibrations; Failure of the Tacoma Narrows bridge (1940); cable-deck coupled vibrations on the DongTing Lake Bridge (1999); human induced vibration on the Millennium Bridge (2000); NASA Helios failure (2003); Coupling between thrusters and natural frequencies of the flexible structure on the International Space Station (2009); Landing gear shimmy. In many cases, the complexity of modern designs has outstripped our ability to understand their dynamic behaviour in detail. Even with the benefit of high power computing, which has enabled engineers to carry out detailed simulations, interpreting results from these simulations is a fundamental bottleneck, and it would seem that our ability to match experimental results is not improving, due primarily to the combination of random and uncertain effects and the failure of the linear superposition approach. As a result a new type of structural dynamics, which fully embraces nonlinearity, is urgently needed to enable the most efficient design and manufacture of the next generation of engineering structures.