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In the last ten years there has been a surge of interest in non-modal analysis applied to standard problems in fundamental fluid mechanics. Even in simple flows, the behaviour predicted by these non-modal analyses can be completely different from - and far more accurate than - that predicted by conventional analyses, particularly for the types of flows found in industrial situations.The successful application of non-modal analysis to standard problems sets the scene for step changes in engineering practice. Nevertheless, some very significant challenges must be overcome. Firstly, the standard approach cannot handle the non-linear problems often found in engineering. Secondly, the standard approach is computationally expensive and cannot handle problems with many degrees of freedom. Thirdly, the standard approach deals with simple measures, such as kinetic energy density, while other measures are usually more pertinent for industrial situations. Encouragingly, applied mathematicians and engineers have made significant progress in all of these areas. This progress has revealed that a generalized formulation of the problem in terms of constrained optimization and variational methods, adapting and applying methods from the control and computational communities, will bridge the gap between standard flows and engineering problems.Our vision is that future generations of engineering Computational Fluid Dynamics (CFD) tools will contain modules that can perform non-modal analysis. If and when such analyses can be made practicable they are certain to change the way that engineers design fluid mechanical systems, such as combustion chambers, turbine blades, reaction chambers and ink jet printers. Furthermore, they can readily deal with transient effects and non-periodic time-varying base flows, which are often particularly relevant in engineering situations.This research will benefit UK industries that rely on the modelling and control of fluid mechanics and thermoacoustics. For example, the pharmaceutical industry will benefit from a better understanding of transition to turbulence and relaminarization in physiological flows, which is important for the application of drugs via the nose and upper airways; The gas turbine industry will benefit from being able to perform instant sensitivity analyses of their fuel injectors and to combine this with greater understanding of the thermo-acoustics that leads to combustion instability; and the wind turbine industry will benefit from an improved prediction of the sensitivity of an aerofoil to turbulence transition and results of exposure to a gust or to the wake of the preceding aerofoil.The investigators in this proposal are all founder members of the EPSRC-funded Advanced Instability Methods (AIM) Network, which was set up in January 2009 to explore the relevance of non-normal analysis to industrial problems. Through masterclasses and workshops in academia and industry and an increasing number of web-based resources, the network provides a route for dissemination and exploitation of this research.In summary, the objectives of this proposal are to bridge the gap between fundamental work and engineering practice, to embed these techniques in the engineering design cycle and to reinforce a growing centre of excellence within the UK in this area. The generalized framework proposed here, combined with two challenging engineering examples and the resources of the AIM Network, will make this possible and demonstrate it to a wider engineering community.

In many scientific and industrial situations, it is important to predict whether a small perturbation in a flow will grow (unstable flow) or decay (stable flow). Industrial applications of stability theory include: the break-up of the jet in an ink-jet printer; large scale mixing in a combustion chamber; thermo-acoustic oscillation in a gas turbine; coupled mode flutter of a wind turbine and mixing in small channels for pharmaceutical applications. The conventional technique is to decompose the perturbation into modes that are normal (i.e. orthogonal) in two spatial dimensions and to study the growth of each mode separately. This, however, often gives inaccurate results. As a simple example, this technique predicts that the flow in a pipe will be stable at all Reynolds (Re) numbers (i.e. at all velocities). In reality, however, the flow becomes turbulent at Re ~ 2000, depending on external noise and the pipe's roughness.This discrepancy arises because, in the third spatial dimension, the modes are non-normal (i.e. non-orthogonal). This means that they can feed energy into each other and should not be considered separately. This non-normal behaviour often causes strong transient growth at the intermediate times that are of most interest to scientists and engineers. For instance, in pipe flow, a non-normal analysis predicts that tiny perturbations will rapidly develop into stream-wise streaks at Re ~ 2000, agreeing with experimental evidence. In the last decade, there has been a surge of interest in non-normal stability analysis within the applied maths community. It is widely thought that non-normality is the root cause of the transient behaviour of the simple flows they have analysed. The aim of this network is to accelerate its exploitation in more complex flows, particularly those with industrial relevance. Conventional stability analyses are currently applied to many industrial situations and, as for simple flows, could miss some of the most significant behaviour.Non-normal analyses, as well as being more accurate, also predict the regions of a flow that are most influential in creating a desired result, such as good mixing. With development, this information will allow engineers to design 'backwards' from an end result, rather than 'forwards' by trial and error. Our long term vision is that the next generation of Computational Fluid Dynamics tools will contain modules that can perform non-normal stability analysis. An important goal is to distinguish between the situations in which a non-normal analysis is required and those in which a conventional analysis is sufficient. We will do this both by reviewing the canonical flows, such as jets/wakes, pipe flow, boundary layers and thermo-acoustic oscillations in a Rijke tube, and by accelerating work on a number of industrial case studies.To achieve this, we will create a multi-disciplinary international network with both academic and industrial partners. The technical goals will require a broad range of expertise: mathematical, to retain the understanding developed for the canonical flows; numerical, to perform the high order computations that will be necessary when moving from simple to complicated flows; experimental, to assemble a catalogue of evidence that will demonstrate when the technique is more relevant than normal mode analysis. The network will expand to a broader industrial community as the ranges of applicability becomes clearer. Currently, several groups are working in this area but, in this relatively young field, there is little formal interaction between them. The network will build on the UK's traditional strength in flow instability and incorporate partners from India, where there has recently been some outstanding work in non-normal analysis. The network will start with one very significant overseas partner (Peter Schmid from Ecole Polytechnique, France) and expand internationally during the two year start-up period.