2. Overview

After a brief introduction to the arguments leading to the classical axisymmetric wake scalings in which ${\it\lambda}=1/3$ , not least their reliance on the assumption that $C_{{\it\epsilon}}=\text{const}.$ , Dairay et al. (Reference Dairay, Obligado and Vassilicos2015) introduce their alternative non-equilibrium dissipation law (discussed more fully by Vassilicos (Reference Vassilicos2015)), which states that $C_{{\it\epsilon}}\sim \mathit{Re}_{G}^{m}/\mathit{Re}_{l}^{n}$ , where $\mathit{Re}_{G}$ is a global Reynolds number set by the initial conditions and $\mathit{Re}_{l}$ is a local one, which for an axisymmetric wake (as for grid turbulence) falls with distance downstream. From the equations of motion and on the basis of the similarity arguments by George (Reference George, George and Arndt1989) (and see also Johansson et al. (Reference Johansson, George and Gourlay2003)), this leads to the wake scaling laws derived by Nedić, Vassilicos & Ganapathisubramani (Reference Nedić, Vassilicos and Ganapathisubramani2013), which can essentially be expressed as ${\it\delta}\sim (x-x_{o})^{1/(3-n)}$ and $u_{o}\sim (x-x_{o})^{-2/(3-n)}$ , with $n=m=1$ (cf. $n=m=0$ for ${\it\lambda}=1/3$ ). Note that this value of $n$ and $m$ is the same as for a laminar wake ( ${\it\lambda}=1/2$ ), although it arises for different reasons. However, it can also arise (and does) when the classical Kolmogorov law is assumed for a fully turbulent, high-Reynolds-number wake characterised by large ratios of turbulence to viscous stress, provided only that the eddy viscosity is constant (Redford et al. Reference Redford, Castro and Coleman2012). The presence of $n=1$ scaling is thus not necessarily associated with low Reynolds number.

Dairay et al. (Reference Dairay, Obligado and Vassilicos2015) examine their scalings by exploring the wake of an irregular (fractal-type) bluff plate with sharp edges, using both a wind tunnel experiment and a matching DNS. This is unusual, not only in that both approaches are used in the same work but also because the DNS is of a spatially developing wake (as in the experiment), rather than a time-developing one, with the generating body included in the computational domain. Whilst this allows capture of the near wake it has the inevitable consequence that the available downstream extent of the wake is somewhat limited. It is also unusual because of the use of a fractal-type wake-generating plate, which leads, in the very near field, to a (‘multi-scale’) flow having a mixture of wake-like and jet-like character. For the DNS, great care is taken to ensure that there is sufficient domain size, grid resolution and statistical convergence. For the experiments, the plate is placed in a low-turbulence wind tunnel at $\mathit{Re}_{G}=U_{\infty }L_{b}/{\it\nu}=40\,000$ , where $L_{b}=\sqrt{A}$ , with $A$ the plate area. The measurements are made using hot-wire anemometry and extend to $x\approx 50L_{b}$ , whereas the DNS has $\mathit{Re}_{G}=5000$ and reaches approximately $x=100L_{b}$ .

The results suggest that, over most of the extent of the wind tunnel wake ( $15\leqslant x/L_{b}\leqslant 50$ ), $C_{{\it\epsilon}}\mathit{Re}_{l}^{n}$ with $n=1$ is more closely constant than is $C_{{\it\epsilon}}$ . Actually, they show that a better fit requires $n\approx 0.77$ . Further downstream ( $55\leqslant x/L_{b}\leqslant 100$ ) the DNS data suggest a change to $n\approx 0.5$ . To derive theoretical scalings for wake width and velocity deficit that have $n\neq 1$ , the authors make a ‘constant anisotropy’ assumption – that the Reynolds shear stress and the turbulent kinetic energy profiles scale in the same way (but not with $u_{o}^{2}$ ). This is essentially a revised Townsend–George theory (Townsend Reference Townsend1976; George Reference George, George and Arndt1989), but includes the new ‘non-equilibrium dissipation’ law. (Note that only the latter is necessary for $n=1$ .) The variations of $u_{o}$ and ${\it\delta}$ along the wake are shown to conform quite well to the new scalings, $u_{o}\sim (x-x_{o})^{{\it\alpha}}$ and ${\it\delta}\sim (x-x_{o})^{{\it\beta}}$ , with ${\it\alpha}=-2{\it\beta}=-2(1+n)/(3+n)$ , albeit with different $n$ ( ${<}1$ ) for the upstream and downstream halves of the $x$ region studied. Given that the local Reynolds number ( $\sqrt{k}{\it\delta}/{\it\nu}$ ) only falls to around 230 by $x=100L_{b}$ in the DNS, and is very much higher in the wind tunnel, it is arguably difficult to claim that it is too small to expect the classical scaling to hold.

In addition to the increasing body of evidence that the value of $C_{{\it\epsilon}}$ can depend on initial conditions, Dairay et al. (Reference Dairay, Obligado and Vassilicos2015) (following Nedić et al. (Reference Nedić, Vassilicos and Ganapathisubramani2013), along with the same group’s work on grid turbulence) thus go much further and question the universality of Reynolds-number independence of $C_{{\it\epsilon}}$ . No physical explanation is offered for why the $C_{{\it\epsilon}}\mathit{Re}_{l}^{n}=\text{const}.$ dissipation law might apply, but it does provide revised scaling laws for axisymmetric wakes that fit the present data.

3. Future

Although there is considerable evidence for the adequacy of the classical dissipation relation, it is apparent that it may be too simplistic, at more than one level. (Actually, it has long been recognised that $C_{{\it\epsilon}}$ is unlikely to be universal (Taylor Reference Taylor1935).) It is already clear that, whilst at sufficiently large Reynolds number $C_{{\it\epsilon}}$ may become constant, its precise value can depend on initial conditions (e.g. Sreenivasan Reference Sreenivasan1998; Antonia & Pearson Reference Antonia and Pearson2000). Dairay et al.’s data seem even more revealing, in that, using only one wake-generating body, they show that $C_{{\it\epsilon}}$ is not even constant, but rather varies with local Reynolds number. The Kolmogorov law ${\it\epsilon}=C_{{\it\epsilon}}k^{3/2}/L$ , with $C_{{\it\epsilon}}=\text{const}.$ , is often seen as ‘one of the cornerstone assumptions of turbulence theory’ but Lumley (Reference Lumley1992) remarked that the ‘mechanism that sets the level of dissipation in a turbulent flow, particularly in changing circumstances’, is worthy of further study. If the results of the Dairay et al. (Reference Dairay, Obligado and Vassilicos2015) study can be shown to be typical of other high-Reynolds-number turbulent shear flows ‘in changing circumstances’, or indeed in axisymmetric wakes at significantly higher $\mathit{Re}_{l}$ than they could reach, the premonition implied by Lumley’s remark will prove to have been prophetic. The matter is certainly worthy of more extensive study, not least because the non-equilibrium dissipation law seems to break the link between the presence of $-5/3$ spectra and classical cascade arguments. This new law seems to hold over much of any wake region that is likely to exist in real applications, so, even without any physical explanation, it would seem to be important. One can expect further experiments exploring the issue, aimed not least at finding whether (and if so, why) there is a final transition to a more classical scaling at some greater distance downstream, as Dairay et al. (Reference Dairay, Obligado and Vassilicos2015) suggest, even though $\mathit{Re}_{l}$ must continue to fall.

Finally, note that little is known about how the very-near-wake flow transitions to the region explored by Dairay et al. (Reference Dairay, Obligado and Vassilicos2015). The process must surely be very dependent on the geometry of the generating body. The near-wake usually contains interesting and complex dynamics (e.g. the recent work of Rigas et al. (Reference Rigas, Morgans, Brackston and Morrison2015) and references therein). There would seem to be much scope for exploration of the various transition regions. And there remains the need for a physical explanation for the new non-equilibrium dissipation relation.