New Approaches and Concepts in Turbulence

I would like to welcome you to Monte Verità, the Mount of Truth. At the beginning of the century this place attracted a number of painters, sculptors, performing artists, writers, but also psychologists, medical doctors, businessmen and socialites, who were seeking a new way of life. They became anarchists, dadaists, nudists, etc., and broke with all traditions and morals of society hoping to establish a new, utopian society which would make human beings free and happy. They were looking for a chimera. If Saffman is right, there is no better place to talk about turbulence!

Similarity laws for simple turbulent wall flows are usually justified by dimensional arguments. However, the dimensional analysis is always based on idealization of the problem, which allows the number of relevant physical parameters to be severely restricted. The idealization, in turn, is based on some model for the mechanism of wall tubulence. The classical model developed by L. Prandtl, T. von Karman and G. I. Taylor in the twenties and thirties implies in particular the so-called Reynolds number similarity principles (i.e., independence of turbulent regime far from solid walls on the molecular viscosity) and the vanishing of the effect of the external length scale in the proximity of the wall. However, recent discoveries related to an important part played by the large-scale organized structures everywhere within turbulent flows make one doubt about the correctness of the formulated statements. Some suggestions and experimental data which contradict the classical similarity laws are considered. The difficulties related to the applicability of simple similarity laws for turbulence in a thermally stratified medium are noted.

It is shown that the multifractal model of fully developed turbulence predicts a new form of universality for the energy spectrum E(k), which can be tested experimentally. Denoting by R the Reynolds number, log E/logR should be a universal function of log k/log R. This includes an intermediate dissipation range in which a continuous range of multifractal scaling exponents are successively turned off by viscosity.

In this paper, we present a detailed statistical analysis of selected experimental data which leads to a new model of the energy cascade process. At fixed inertial scale r, experiments are not inconsistent with a Gaussian distribution of lnε(r) where ε(r) is the local transfert rate. The same experiments suggest, however, that the variance >δ(lnε(r))²< has a power law behaviour r^-β and not a logarithmic one as assumed in the log-normal model. From this point of view βlnr appears as a universal variable. This is confirmed by energy spectra E(k), measured during the past thirty years, which merge on a single curve when plotted as ln(E(k))/lnRe versus ln(k)/lnRe, suggesting that β is inversely proportional to lnRe. An extremum principle approach gives all these results and suggests that β would be the co-dimension of the dissipative structures.

Using direct or large-eddy simulations, we study coherent vortices in free-shear or wall-bounded flows. In free-shear flows, we show that the Kelvin-Helmholtz instability is responsible for the condensation of vorticity into large tubes corresponding to strong depressions. These tubes may be quasi two-dimensional, or strongly distorted due to helical pairings. Smaller-scale hairpin vortices may also be stretched between the large vortices. In wall-bounded flows, hairpin vortices are also stretched by the basic shear. Kelvin-Helmholtz and hairpin-type eddies seem then to be the basic coherent ingredients of turbulent-shear flows. In 3D decaying isotropic turbulence, we show again the existence of large coherent vortices corresponding to strong depressions, which stretch intense passive temperature fluctuations inbetween. The temperature spectrum has an anomalous k ^-1 range in the large scales. The temperature pdf is exponential in the wings and behaves like the vorticity and the low pressures. An attempt is made for explaining these observations in terms of superposition of a quasi 2D slow turbulence and a Kolmogorov rapid turbulence.

A dynamical mechanism is considered which connects the cascade with nongaussian statistics of velocity gradients. Turbulence is characterized by the continuous excitation of all scales, but in the Fourier space of the velocity field, the excited amplitude decreases rapidly with increasing wave numbers so that contribution to the total kinetic energy from the small scale components is negligibly small. Roughly speaking by the central limit theorem, the sum of a large number of Fourier modes is distributed normally when the Fourier amplitudes of different wave numbers are independent in the energy-containing eddies (Batchelor 1953).

The evidence on small compact vortex structures in turbulent flows is summarised for various experimental and numerical flow fields. It is consistent with a model of strained almost two dimensional vortices with radii of the order of the Kolmogorov scale, and circulation Reynolds numbers of a few hundred. The known alignment properties of the strain tensor are also consistent with the kinematics of this model. A possible scenario for the generation of these structures within the “turbulent cascade is offered. The compact vortices are postulated to be essentially passive from the point of view of energy transfer, connected to the coherent structures observed in two dimensional turbulence

Spiral structures are natural candidates for the role of the ‘generic structures’ of turbulent flow, because they are the eventual outcome of Kelvin-Helmholtz instability, an all-pervasive phenomenon associated with nearly all shear flows at very high Reynolds number. Such structures were proposed by Lundgren (1982) in a model of the fine structure of turbulence in which axial stretching of rolled-up spiral vortices played an essential role. This model could be viewed as a natural development of Townsend’s (1951) model of the dissipative structures of turbulence in terms of a random distribution of vortex sheets and/or tubes, each such structure being subjected to the local rate of strain (assumed uniform and constant) associated with all other vortex structures (see Batchelor 1982, § is known that compressive strain (with two positive principle rates of strain) is more likely in isotropic turbulence than extensive strain, so that sheets form with higher probability than tubes. However these are immediately subject to the Kelvin-Helmholtz instability which may be impeded, but not entirely suppressed, by the stretching process.

A new approach to the problem of turbulence is described. This approach is based on conditional averaging of equation for a local characteristic of motion, which has a mechanism of self-amplification. An exact closed equation for conditionally averaged 3-D vorticity field (with fixed vorticity at a certain point) is obtained from the Navier-Stokes equation. Corresponding closed equation is derived for conditionally averaged vorticity gradient (vg) in 2-D turbulence. Solutions of these equations are presented. The conclusion is made that the local structure of turbulent flows is not unique. The nonuniqueness is due to the phenomenon of intermittency, which depends on the large-scale structure of turbulent flows. The high order two-point moments of vorticity and vg are presented. The developed method is quite general and can be applied to a variety of physical systems with strong interaction, including magnetized plasma.

The above question is closely related to other questions like the following ones:

Some problems of dynamical chaos are discussed in their relation to the problem of turbulent motion. They include the symmetry of patterns in preturbulent state, space-time chaos, Lagrangian turbulence, anomalous transport, intermittency and others. All of them are considered from a dynamical background.

In the first announcement for this meeting the conveners, Th. Dracos and A. Tsinober, asked four questions. Their second and third question were:

How important are kinematical properties (topology, geometry) for the dynamic behavior of turbulent flows?

Are there any prospects that a dynamical systems/dynamical chaos approach will allow describing real turbulence?

We present a discussion of the role of finite-dimensional dynamics and chaos in interpreting nonlinear fluid mechanical motion. The discussion will be restricted to two examples of fluid flows which have been studied by the author and which appear to be understandable in terms of ideas based in modern thinking in dynamical systems. The specific examples we have chosen are the flow between concentric rotating cylinders commonly called the Taylor-Couette problem and the flow through a nominally two dimensional sudden symmetric expansion. The first of these is a so called ‘closed flow’ problem and the second is an example of an ‘open flow’ where disturbances can grow as they are carried down stream. The aim of the present article is to focus attention on the practicalities of applying the abstract concepts of finite-dimensional dynamics to the experimental study of fluid flows. We will show how a careful consideration of the important symmetries of these problems can lead to the uncovering of structurally stable local organising centres for the global low-dimensional dynamical behaviour.

The process of transition from a laminar to a turbulent boundary layer flow is discussed. From detailed experimental measurements and dye filament observations it appears that transition often occurs in a very localised way involving the formation of distinct coherent structures. Mostly experiments as well as analysis are carried out on purely periodic disturbances and these features are not apparent. It turns out that naturally occurring waves, excited by a random turbulence field, for example, behave in a quite different manner from periodic excitation. This aspect of the transition process is explored.

One of the major challenges in turbulence research has been to explain how the turbulence is created and to describe and analyze the space-time behavior of the turbulent eddies. For wallbounded shear flows the pioneering experiments by Kline and his group (Kline et al. 1967, Kim et al., 1971) and by Corino and Brodkey (1969) brought out the special significance of the strongly intermittent processes in the near-wall region during turbulent bursting and identified alternating low- and high-speed flow regions, “streaks”, to be the predominant flow structures there. The oscillations and lift-up of a low-speed streak were seen to initiate a rapid outflow (ejection) of fluid from the wall region and a subsequent breakup of the streak.

Experimental investigations of large coherent structures in turbulent shear flows bypassed the boundary layer in the presence of a strong, adverse-pressure-gradient and the wall-jet. Both are wall bounded flows having one characteristic in common: their mean velocity profile is inviscidly unstable to two dimensional perturbations. Thus the identification of the large coherent structures with the predominant instability modes might be extended to this class of flows and could be quantitatively analyzed. The apparent similarity between the wall-jet and a combination of a free jet and a boundary layer is explored was and the relevance of the solid surface to the evolution of the large coherent structures was assessed. From this point of view, the boundary layer in a strong adverse pressure gradient, is regarded as a wake evolving in the vicinity of a solid surface. In both flows the significance- of the outer region is accentuated while the no-slip conditions at the solid surface are maintained.

The response of these flows to external, two dimensional excitation is currently being investigated in an attempt to further the understanding of the interactions between the inner and the outer structures in a turbulent boundary layer. While doing so, two technologically important effects were discovered. The mean flow in the wall-wake remained attached in spite of the strong adverse pressure gradient which caused separation in the absence of the excitation while the skin friction in the wall-jet was reduced as a consequence of the excitation.

This is a review of how, when linear distortions are applied to turbulent velocity fields, certain changes to some or all components of the turbulence can be calculated using linear theory. Important examples of such distortions are mean and random straining motions, body forces, interactions with other flows (eg. waves). This theory is usually known as Rapid Distortion Theory (RDT) because it is valid for all kinds of rapidly changing turbulent flows (RCT), when the distortion is applied for a time (defined in a Lagrangian frame) T _D that is short compared to the ‘turn-over’ or decorrelation time scales T _L or τ _D (k) of the energy containing eddies or smaller eddies of scale k ^-l, respectively. However, for certain kinds of distortion the theory is also applicable to slowly changing turbulence (SCT) where T _D is of the order or greater than

New insight about the structure of slowly changing turbulence has been derived from RDT by considering different strain rates, initial conditions and time scales. RDT calculations show that in shear flows, whatever the initial form of the energy spectrum E(k) (provided it decreases with wavenumberk k faster than k ^-2 ) or of the anisotropy, for a long enough period of strain, E(k) always tends to the limiting form where it is proportional to k ^-2 for the small scales. Other statistical properties, such as ratios of Reynolds stresses, are also insensitive to the initial conditions. By contrast RDT shows how turbulent flows without mean strain or with irrotational strain are significantly more sensitive to initial and external conditions. These conclusions are consistent with those drawn from Direct Numerical Simulations (DNS) and experiments at moderate Reynolds numbers.

The eddy structure of small scale turbulence can be studied by calculating the distortion of random Fourier components in a velocity field caused by different kinds of large scale straining motions. This method is used here in conjunction with an analysis of how the different types of straining motions in different regions of the small-scale turbulence are affected by the distortion. Interestingly, linear analysis shows that the vorticity vector tends to become aligned with the middle eigenvector of the rate of strain tensor, which is consistent with DNS for turbulent flows having a continuous spectrum.

At sufficiently high values of the Reynolds number, even in homogeneous and in-homogeneous distorted turbulence (mean straining flows, body forces and boundaries), the nonlinear processes act over a wide enough spectrum to ensure that at small scales the energy spectrum E(k) has an approximately universal form (as proposed by Kolmogorov). However, at the same time different components of the spectrum have an anisotropic structure. This can be estimated by applying RDT to each wavenumber component of the spectrum and taking the time of distortion to be approximately equal to the turnover time t(k) appropriate to the value of k.

The basic philosophy of large eddy simulation (LES) is to explicitly compute only those large-scale motions that are directly affected by the boundary conditions and to model the small scales. LES has always been described as attractive for engineering computations because it is significantly less computer-intensive than direct numerical simulation (DNS), yet promises to be more accurate and robust than single point closures.

Experimental methods such as planar laser induced fluorescence and particle image velocimetry, offer the ability to measure flow properties, scalar or vector, at many points simultaneously. The manner in which such experimental methods are likely to advance the frontiers of turbulence research, and the challenges that must be surmounted are discussed.

Direct numerical simulations have contributed significantly to the study of the physics of turbulence and to turbulence modelling, albeit at low Reynolds numbers. Hot wire experiments should continue to play a useful part in providing data at higher Reynolds numbers but it is important that their limitations are realised. To this end, hot wire measurements in a fully developed turbulent duct flow have been compared with other types of measurements as well as direct numerical simulations for the same flow and comparable Reynolds numbers. Preliminary hot wire measurements of spatial velocity derivatives in a turbulent channel flow show encouraging agreement with the simulations.

We compare high-resolution direct numerical simulations (DNS) to the predictions of moment closure for two- and three-dimensional turbulence. Our goal is to assess the impact of vortical structures on two-point covariances, through a comparison of DNS to simple closures, which make no explicit reference to structures at all. In two dimensions, we note that there is a profound effect, and this is associated with the development of isolated vortex structures. We interpret their presence in terms of a shutting down of the enstrophy transfer process, and give phenomenological estimates in terms of a modified closure. In three dimensions, the DNS spectra at R _λ ~ 100 is in good agreement with the Direct Interaction Approximation (DIA). The agreement is poorer for Markovian theories, such as the Eddy Damped Quasinormal Markovian (EDQNM) approximation. We also comment on the possible development of finite time singularities in the vorticity for inviscid flows.

Detailed measurements of the velocity field were made in the wall boundary layer of the 80’ by 120’ facility at NASA Ames. The Reynolds Number R_λ, based on the Taylor microscale λ at the measurement location, was approximately 1450, one the largest attained in laboratory flows. The data indicate that to within measurement accuracy, the w-spectrum follows, but the v-spectrum deviates from, the isotropic relation in the inertial subrange. No definite statement can be made regarding local isotropy for the dissipating scales because the spectral measurements were contaminated by high-frequency electrical noise, but it appears that the inertial-subrange anisotropy persist in the dissipation region.

The potential of numerical and physical experiments to address important questions in turbulence research has increased dramatically over the last decade. Both are now able to provide multidimensional data on scalar and vector fields, although the capabilities of physical experiments are somewhat different from those of numerical experiments. These two approaches will be discussed, and compared, followed by a discussion of the experiments that should be done with them, and the important issues that might be answered by these means.

We decided that there were essentially five areas in which chaotic behaviour in dynamical systems might contribute to turbulence research.

This is a subject that was not actually covered explicitly in the discussions of the last three days. I think I should explain first what we mean by reduction of non-linearity because there are different interpretations of this term. If we adopt a dynamical systems viewpoint, then a turbulent flow can be thought of in terms of a trajectory in the function space of all solenoidal vector fields satisfying certain weak constraints (e.g. boudedness of |u|). The fixed points in this function space are then fields ^uE(x) that are steady solutions of the Euler equations where ω= curl u, P =p/< + (1/2) ^u2. By solving the Poisson equation ∇²P=∇ ·(u x ω) for P, and substitutinig back in (1), this may equally be written where the suffix S represents solenoidal projection. Just as Beltrami flows satisfy the condition u x ω ≡ 0, so Euler flows satisfy the weaker conditon (_{u x ω)S} ≡ 0.

I would like to share with you some things that I heard at this meeting, and I heard a great deal. Maybe you can correct me if you did not hear them, that might be useful. We started off talking about intuition and I found this to be a delightful statement and it really sets a tone for the whole meeting, and I think there has been a lot of progress on that. What surprised me most was that I was unaware of the extent of the data and the difference of opinions with respect to what goes on in the cascade-process with all the small scales, etc. I didn’t realize how widely shared some of these speculations were. The subject which I noted to be of considerable interest was the difference in views on spectral representations. As I understood what was said earlier in the meeting was that the concept of infinite Reynolds number as a limit may not anymore be a meaningful concept, but the viscosity may act at all scales. I think that is what this graph means, which we saw in a couple of the early papers. Now along comes Keith Moffat with a reminder that spiral structures can also produce something which looks pretty similar to the fractal structure. Does it take a multifractal set of scales, a wide range of scales to produce turbulence or can you do it with a single structure and a single scale? These are the different set of views.