## Abstract

Tomographic particle image velocimetry measurements of homogeneous isotropic turbulence that have been made in a large mixing tank facility at Cambridge are analysed in order to characterize thin highly sheared regions that have been observed. The results indicate that such regions coincide with regions of high enstrophy, dissipation and stretching. Large velocity jumps are observed across the width of these regions. The thickness of the shear layers seems to scale with the Taylor microscale, as has been suggested previously. The results discussed here concentrate on examining individual realizations rather than statistics of these regions.

## 1. Introduction

Most flows of practical interest are turbulent and hence the understanding of these flows is important from both an engineering and a scientific viewpoint. Turbulence consists of a wide range of three-dimensional motions, from large and slow to small and fast. The smallest (and most rapid) motions dissipate the kinetic energy of the flow and determine drag on bodies, dispersion of pollutants and chemical mixing. Unfortunately, the very smallness of these motions has, until recently, made them inaccessible to both experiments and computations in flows of practical importance. Predictions of turbulent flows have thus been based on uncertain theories and models of these ‘fine scales’ that are assumed to be the same for all flows, i.e. universal. Since these fine scales dissipate the kinetic energy they play a crucial role in controlling turbulent flows. There have been a number of studies of the nature of these fine structures in turbulence (e.g. [1–4]) mostly using direct numerical simulation owing to the difficulty of making experimental measurements of three-dimensional structures with sufficient resolution, although some experimental measurements are now becoming available (e.g. [5]).

Although observations suggest that these motions come in a wide variety of shapes and sizes (e.g. [6]), recent research has emphasized the importance of one particular morphology, which here will be referred to as thin shear layers [7]. It has been known for some time that the fine-scale motions are spatially intermittent, in the sense that large variations of important quantities occur in small, well-separated regions in the flow. This means that large contributions to mean properties come from intense motions which occupy only a small fraction of the overall volume of the flow. To date, though, the form and nature of these spatially compact, intense regions have been the subject of some debate. Thin shear layers consist of ribbon-like features, with a large velocity jump occurring over their smallest dimension, leading to very large velocity gradients. Two quantities of particular interest for the turbulent motion, particularly at the small scales, are the dissipation of kinetic energy (usually just called ‘the dissipation’) and the enstrophy. In the equation for the kinetic energy of the turbulence the dissipation is a sink of energy corresponding to the conversion of the kinetic energy of velocity fluctuations into internal energy (heat) of the fluid through the action of viscosity. In tensor notation the mean dissipation, *ϵ*, is given by
(1.1)
where
(1.2)
is the strain-rate tensor. The 〈〉 notation refers to a volume average over the flow of interest. Apart from its important dynamical role, the dissipation may also be used to define possible appropriate length and velocity (and time) scales for the small motions. The usual argument is that the scaling of these motions depends only on the viscosity and mean dissipation, which leads to the Kolmogorov length scale
(1.3)
and the Kolmogorov velocity scale
(1.4)
One further length scale of interest is known as the Taylor microscale, *λ*, which may be estimated in homogeneous isotropic turbulence from
(1.5)
where *u*′ is a velocity fluctuation.

The other term of particular interest in turbulence theory is the enstrophy, (where *ω*_{i} is a component of the vorticity that is directly related to the angular velocity of fluid elements along the particular axis). This is a measure of the intensity of the rotational motion of the fluid and is a scalar quantity. The enstrophy transport equation may be written as
(1.6)
where the terms from right to left are the rate of change of the enstrophy with time, the advection of the enstrophy by the velocity field, the change in enstrophy owing to stretching (or compression), the diffusion of enstrophy by viscosity and the destruction of enstrophy known as enstrophy dissipation.

In this paper, we examine some thin shear-layer regions in a turbulent flow and consider the importance of the dissipation and enstrophy balance terms in these regions.

## 2. Experimental method

The experiments were conducted in the large mixing tank facility in the Department of Engineering, University of Cambridge. This facility was specially designed for making well-resolved measurements in homogeneous, isotropic turbulence at high Reynolds numbers. It consists of a large Perspex tank with 12 sides (approx. cylindrical) of 2 m in diameter and 2 m in height, filled with water. Two impellers with eight straight, radial blades are located in the top and bottom of this tank and are contra-rotated in this experiment. This creates a highly sheared flow between the impellers that generates the turbulence. Near the centre of the tank the shear is very small, and in this region the turbulence is homogeneous and isotropic to a good approximation. The large size of the tank results in fine-scale structures that are measurable even at reasonably high Reynolds numbers. In the experiments discussed here the Kolmogorov length scale is approximately 1 mm and the Kolmogorov velocity scale is approximately 1 s at a Reynolds number (based on the Taylor microscale) *Re*_{λ}=162 (more precise details will be given later in the paper).

The measurements presented here are made using a tomographic particle image velocimetry (PIV) system developed in-house. This allows for measurements of the full three-dimensional velocity field in a small volume with a spatial resolution of one Kolmogorov length and less than one-tenth of the Kolmogorov time scale (for this Reynolds number). The size of the volume is 60×60×12 mm. A dual-head 120 mJ PIV laser is used to illuminate the volume. Four high-speed Photron APX cameras are used to view the volume. The volume reconstruction and correlation is carried out using in-house tomographic PIV software, which results in 51×51×8 vectors in the volume. Since all velocity components are available within a volume it is possible to calculate the full velocity gradient tensor at points within the volume and hence to calculate a variety of useful quantities such as the dissipation, enstrophy and all the terms in the enstrophy transport equation. The measurements to be presented here were made as part of the PhD project of the first author and further details may be found in his thesis [8]. They have been re-analysed here in order to investigate thin shear-layer motions.

## 3. Results

In this section, results are presented showing particular examples of thin shear-layer regions in the homogeneous turbulence experiment. A number of examples are considered and the characteristics of each are shown and discussed. In particular the enstrophy, dissipation and the main terms in the enstrophy transport equation are presented. These terms in the enstrophy equation have been scaled by the global r.m.s. enstrophy and hence have the dimensions of inverse time. They are all scaled similarly so the importance of each term can be evaluated by direct comparison with the other terms. Also considered later are the velocity differences across the thin shear layers in comparison with the r.m.s. velocity fluctuation for the whole flow region. The three-dimensional measurements of the structures are shown in figure 2. The first four images are from a single time series of the flow and have been chosen to illustrate the nature of the thin shear-layer regions. The Reynolds number for this case is *Re*_{λ}=162. The last two are from a higher Reynolds number case, *Re*_{λ}=448. A few simple observations can be made immediately. The regions shown are very long in comparison with their thickness. Although these measurements only show a thick ‘slice’, and so we cannot know their extent beyond the measurement volume, they appear to extend unchanged through the volume and hence it is, at least, plausible that their third dimension is also large in comparison with the thickness (it seems rather unlikely that the tubes of enstrophy terminate very quickly or change direction very suddenly outside the region of interest). Hence they appear to be ‘ribbon like’. Combined enstrophy and dissipation plots (not shown here) show that they consist of sheets of dissipation with embedded vortex structures. Their thickness can be estimated from the images (and those shown later) and appears to be about 10–12*η* for the low Reynolds number case. This corresponds to approximately 0.5*λ* (*λ*/*η*=25). Estimates of the thickness for the higher Reynolds number case suggest that it is about 20–25*η*, which again corresponds roughly to 0.5*λ* as for this case *λ*/*η*=42. The estimates are based on the following figures—in particular traces of the velocity change across the sheet. These limited examples then are consistent with a thickness that scales with the Taylor microscale.

Figures 3–6 show the same four examples for the lower Reynolds number case (*Re*_{λ}=162) in more detail. The results shown are from a two-dimensional plane in the vertical centre of the measurement volume. Since the volume is thin these are representative of the full three-dimensional results but are somewhat easier to visualize and understand. It should be noted that the full three-dimensional velocity field in the volume was used to calculate the quantities presented in the plane. The dashed lines shown on the figures illustrate a line across the structure along which the variations of velocity will be shown later. The figures show, for the four cases, the dissipation, the enstrophy, enstrophy stretching/compression, enstrophy diffusion, enstrophy dissipation and advection.

Although there is some overlap between the high-enstrophy regions and the high-dissipation regions they are not entirely co-incident. There are regions of high dissipation where the enstrophy is relatively small. This is particularly noticeable by comparing figure 4*a* and 4*b*. The high-dissipation region shown in figure 4*a* covers a larger region than the enstrophy region shown in figure 4*b*. This is also clearly apparent in figure 5 (again comparing figure 5*a* and 5*b*).

Enstrophy stretching (part (*c*) of figures 3–6) seems to correlate better than enstrophy with the regions of high dissipation in these examples in the lower Reynolds number cases. The diffusion of enstrophy is also significant in these regions, particularly near the middle of the thickness where it is acting to diffuse the enstrophy towards the edge of the sheet. The white regions in the middle show a reduction of the enstrophy by diffusion, while the black regions near the edge indicate the increase associated with the enstrophy diffusing out from the middle. This is perhaps not surprising since the enstrophy stretching term leads to increased gradients in enstrophy then leading to increased diffusion. A consideration of the magnitude of these terms (given by the scale bar) suggests that enstrophy stretching and diffusion are approximately in balance in these cases.

In the higher Reynolds number (*Re*_{λ}=448) case (figures 7 and 8), however, the viscous diffusion is fairly small in these regions, suggesting that a balance has not been reached for these high Reynolds number examples. It is perhaps not surprising that viscous diffusion plays a lesser role at the higher Reynolds number. The stretching (compression) in these cases is also much more complicated with both stretching and compression evident in different parts of the shear layers. This is because of more complex three-dimensional motions through the plane. In these cases then a simple correlation of enstrophy stretching with dissipation fails completely.

The enstrophy dissipation term (*e*) is quite small in all cases, indicating that the main dynamics are governed by stretching and redistribution of the enstrophy by diffusion. In the last term (*f*), the advection merely shows the direction of movement of the sheets as they are advected by the turbulence. Video sequences show that the structures maintain their overall shear-layer structure as they advect through the volume, although they may become more convoluted. Their longevity is consistent with the low values of the enstrophy dissipation.

Figure 9 shows the change in the three instantaneous velocity components along a line through the shear layer. The line along which this is examined is shown in the previous figures for each case. The instantaneous velocities have been normalized by the r.m.s. value of that component for the whole flow. In all cases it is obvious that there are large velocity jumps across the shear layer with magnitudes of approximately two to three times the r.m.s. Given the thickness of the layers discussed earlier it is obvious that very large gradients occur across these regions. Given that the velocity gradients in these regions then must scale with the r.m.s. velocity divided by the Taylor microscale, the dissipation must also scale with these values, a scaling which is consistent with the usual definition of the Taylor microscale. It should be noted that in the low Reynolds number case the ratio of the global r.m.s. velocity to the Kolmogorov velocity scale, *υ*, is 6 and in the higher Reynolds number case it is 11. This suggests that the appropriate scale for the velocity jumps is indeed the r.m.s. rather than the Kolmogorov velocity as the jump is very large in terms of the Kolmogorov velocity and also the two Reynolds number cases show approximately the same jump in terms of the r.m.s.—if scaled with the Kolmogorov velocity there would be a factor of 2 difference. The unusual shapes of the velocity trace in the higher Reynolds number case ((*e*) and (*f*)) is due to the fact that the line used cuts through more than one sheet. In the case of (*e*) the vorticity of the sheets is of opposite sign (though that is not apparent from the enstrophy). As such, the velocity jump across the first sheet is offset by that through the second, leading to no net jump across the pair—though a region of high velocity exists between them. In the last case there are two sheets of the same sign leading to an increase, a drop and then another increase, giving an overall jump for the pair.

## 4. Conclusion

Observations of thin shear-layer structures in well-resolved experimental results from homogeneous, isotropic turbulence have been presented. A number of instances have been analysed in order to throw light on their detailed structure. The limited number of observations presented suggest that the thickness of these regions is approximately equal to the Taylor microscale in the flow, and the enstrophy, dissipation and local velocity increases across the layers show values several times higher than the r.m.s. values for the flow. Hence, these regions are very intense and may make large contributions to the statistics of the flow despite their small size. The local velocity jump across the sheets seems to be of the order of the r.m.s. velocity fluctuation for the entire flow (rather than the Kolmogorov velocity scale). These facts seem broadly consistent with the ‘cartoon’ of thin shear layers presented in Hunt *et al.* [7]. Terms in the enstrophy equation have been examined and show that these regions are, for the most part, being stretching significantly.

## Footnotes

One contribution of 9 to a Theme Issue ‘Dynamical barriers and interfaces in turbulent flows’.

- This journal is © 2011 The Royal Society