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Isaac Newton Institute for Mathematical Sciences

Turbulence

January - June 1999

Organisers: GF Hewitt (Imperial College), PA Monkewitz (Lausanne), N Sandham (QMW), JC Vassilicos (Cambridge)

Scientific Basis

Though there is no doubt whatsoever about the importance of turbulent flow phenomena in engineering systems, the reliable prediction of turbulent flows remains an elusive goal. Although direct numerical simulation of turbulent flows is possible at sufficiently low Reynolds numbers, the range attainable in such calculations is well below the higher Reynolds number range of engineering importance. The advent of CFD has allowed complex systems to be calculated provided that some model can be assumed to describe relevant aspects of the turbulence and to bypass the intractability associated with the nonlinearity of the Navier Stokes equations. This has led to the development of a whole variety of ingenious intuitive models at different levels of complexity and generality. All of them raise fundamental questions that need further examination. The proposed research programme is organised to bring together experts from different communities (in mathematical sciences, fundamental fluid mechanics, turbulence modelling) to explain their different approaches and to develop new understandings of practical and fundamental problems. Though it seems unlikely that specific prediction methods will emerge, it is reasonable to hope that a better understanding of turbulent flow will result. This should lead to useful perspectives on current practices in modelling turbulence and provide new approaches for future research.

In what follows, the main issues to be addressed are discussed in turn, namely universality, transition and control, analysis and numerical simulation, closure strategies and applications. The overall issue is, of course, the need to increase knowledge of the various phenomena and hence to develop improved prediction methods.

Universality

The issue of existence and extent of those aspects of turbulent flows that can be described as universal will provide the backbone for the program on which all other issues, both engineering and mathematical, will connect. From a statistical standpoint there are arguments and evidence to show that high order, and sometimes even second order, moments of the turbulent velocity field are not universal. Is it, for example, the case that scaling exponents are universal for high enough Reynolds numbers, whereas multiplicative coefficients are not? In some instances, even scaling exponents may depend on Reynolds number, at least for scalar statistics in the range of Reynolds numbers currently attainable in the laboratory. Even so, are some of the topological features of realisations of turbulent flows universal, or do they depend on Reynolds number - an important point because this would cause scaling exponents to vary between flows? The issue of universality has also another aspect: what features of fluid turbulence are common even to systems outside the realm of the Navier-Stokes equations? Concepts of a topological nature used to study turbulence are intermittency and space-filling properties. What are the differences and similarities between intermittency in turbulence and intermittency in other dynamical systems? Comparisons with simpler dynamical systems could, for example, indicate how intermittency connected to the large-scale motions. Many engineering and closure models depend on constants that need to be adapted to different geometries, thus again raising the issue of universality. Do we need different models for different turbulent flows? And if we do, what aspects are universal, i.e., common to all such models?

If this Research Programme could produce answers to the above problems, it would have a profound effect on the directions of both basic and applied research and on the appropriate exploitation and application of the existing range of methods of calculating and simulating in different turbulent models. In particular, the answer to the basic question of universality should indicate for what range of flows or universality class it is likely to be fruitful to attempt developing general models for turbulent statistics. Mathematical issues for the six-month program will involve studies centred around the Navier-Stokes equation, different statistical and geometrical approaches to intermittency using, for example, the theory of large deviations and wavelets, the mathematical properties of different turbulence closures, and a review and assessment of existing codes and models.

Transition and Control.

There are many systems where the maintenance of laminar flow can lead to great economies in design and operation. An example here is that of transition to turbulence in turbomachinery; in this case, natural transition and the influences of laminar separation, free stream turbulence and incoming periodic disturbances are all of significance. Modelling the smaller range of motions characteristic of transition flows using the methods of non-linear dynamical systems is already yielding new insights into transition and fully turbulent flows. How far can this approach be taken and for what type of flows? A new application is the use of methods of adaptive control allied to transitional fluid mechanics to design systems for controlling the turbulence.

Analysis and Numerical Simulation.

The turbulence is characterised by a large number of weakly correlated motions interacting on a wide range of length-and time-scales of motion and appears disordered and chaotic but also containing long-lived characteristic elements. Indeed, recent physical experiments and Direct Numerical Simulations (DNS) based on spectral computations have revealed the spontaneous formation of little shear layers and vortex tubes which are surrounded by a "random and shapeless sea" in the small scales of the turbulence. Fundamentally different approaches ensue from stressing either the space-filling disorder or the localised order in the flow. To what extent can turbulence be represented in terms of space-filling functions such as Fourier or Chebychev basis functions or rather in terms of localised functions such as wavelets. Such different representations lead to different kinematical descriptions and dynamical analyses and computations. Many important ideas are coming from the interplay between the 'classical' approaches coming from statistical physics or Fourier analysis and spectral dynamics (DIA, RNG, EDQNM) and the local fluid dynamical approach of considering individual vortices and 'coherent' flow structures. The programme will stress and enhance this interplay and will include systematic comparisons between and evaluation of different approaches and prediction methods including analytical approaches, DNS for low Reynolds numbers and Large Eddy Simulation (LES) for high Reynolds numbers.

Some of the flow structures mentioned above are long-lived and seem to have a 'life of their own'. Can they be thought of as 'eigen-structures' and if so in what sense? First of all we still need to establish the importance of these structures. Then, what are their Lagrangian and Eulerian properties, and how do their conditional statistics relate to the well-established unconditional Eulerian and Lagrangian statistics (e.g. spectra and cascades of energy up-and down-scale) and the scaling properties of the entire flow? Are these 'eigen-structures' related to the conjectured finite-time singularities or near-singularities that emerge from some theories of turbulence? Since the mean rate of dissipation of energy is the variable that determines (in a scaling sense) the statistics of small-scale turbulence, one also needs to understand the dissipative properties of these structures.

Can answers to the above questions, in particular those relating to energy transfer and dissipation, be used for novel turbulence modelling, in particular new LES that can incorporate the intermittency of coarse-grid energy dissipation? What are the best methods for analysing and describing these structures and their interactions? Can localised analytical techniques such as wavelets be used for dynamical as well as kinematical analysis? A wavelet decomposition of the turbulence may be appropriate for the study of the interaction and energy transfer between different spatially localised length-scales and 'eigen-structures' in the flow. How can this wavelet decomposition be chosen and operate, and how does it compare and complement Fourier decompositions and the ensuing study of triadic interactions between different scales of motion?

In all these statistical and dynamical problems we need to extend to and understand more realistic problems where the turbulence is inhomogeneous and affected by boundaries. A practical goal for DNS modellers and mathematicians might be the generation of an improved near-wall turbulence model.

Closure Strategies.

Numerical solution of the averaged Navier-Stokes equations (the Reynolds equations) is often straightforward provided a suitable model of turbulence can be provided to determine the turbulent stresses and scalar fluxes. While the first example of models of this type possessed a rather narrow breadth of applicability and was heavily reliant on empirically tuned coefficients, a wide range of newer models is becoming available that is more fundamentally based and to which mathematical analysis can make a major contribution. The most interesting level in practice is that of second-moment closure which leads, using the momentum and scalar transport equations, together with assumptions about tensorial invariance and local homogeneity, to a formally soluble set of equations for the Reynolds stresses and other second moments. Such models pose challenging problems of accounting for inhomogeneous 'pressure-reflection' effects from boundaries while ensuring that the turbulence field is infallibly realisable. Analogous problems arise in the fine scales near boundaries and also in combustion phenomena. Such problems need the interplay of the mathematician and the physicist for satisfactory resolution. There are also problems of numerical solution associated with the strong intercoupling of the turbulence equations especially in flows where buoyancy or other force fields are strong or where combustion is present.

Applications.

It is important to see the fundamentals of turbulence in the context of its applications. If, indeed, the nature of the turbulence is strongly affected by the particular system and there is no general modelling framework, then consideration of individual systems becomes even more important. The Research Programme should include studies of applied topics such as high-speed flows, turbo-machinery, external aerodynamic flows, turbulent diffusion, multiphase flows, discrete particle flows, interactions with chemical behaviour (chemical reactors and combustion) and non-Newtonian flows. Unsteady turbulent flows are often of importance. Theories of turbulent diffusion, fundamentally based on the analysis of how turbulence advects and distorts vector and scalar fields, have contributed to the development of practical models for heat and mass transfer and environmental dispersion.

There is also a whole class of problems in the interaction of fluid turbulence with dispersed elements within the fluids. Such dispersed elements can range from solid particles to deformable liquid droplets and compressible deformable bubbles. The interaction with such dispersed systems is highly complex and sensitive to different geometrical features of the turbulence.

The unique feature of this programme is that it should provide insights into how to define, measure and study limitations of models and simulations. These aspects should be underpinned by theoretical and experimental studies to develop an understanding of the concepts and approximations inherent in existing models.

For some applications (especially in geophysical flows) broad estimates of sensitivity, levels of accuracy and possible types of solution are as valuable as detailed computations. The Research Programme should help establish a better methodology for examining these issues.

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