2. Overview

As applied to particulate flows, the basic idea of the immersed boundary method (originally invented in the 1970s for the simulation of blood flow in the heart, see Peskin Reference Peskin2002) is, once again, to disregard the particle boundary, but in a subtle way: the continuity and momentum equations are solved both outside and inside the particles but, in the latter domains, fictitious forces are imposed which cause the fluid to move like a solid so as to restore the no-slip boundary condition.

While the general idea is simple, its accurate and efficient implementation has proved far from straightforward and it is still a topic of current research. After some limited early work in the late 1990s, a major breakthrough was initiated by Uhlmann (Reference Uhlmann2008) who combined the direct-forcing method of Fadlun et al. (Reference Fadlun, Verzicco, Orlandi and Mohd-Yusof2000) with the regularized ${\it\delta}$ -function approach of the original method of Peskin. In this way, he was able to simulate thousands of particles in a channel flow with a Reynolds number large enough for spontaneous turbulence to set in (Uhlmann Reference Uhlmann2008). The mismatch between the particle boundary and the underlying fixed grid required interpolation back and forth between one and the other. These steps make it difficult to satisfy the no-slip condition precisely on the particle surface, and this is one of several areas where recent work has focused (see e.g. Kempe & Frölich Reference Kempe and Frölich2012 and, in particular, earlier work by Breugem, one of the authors of the paper under discussion).

Picano et al. (Reference Picano, Breugem and Brandt2015) have carried out simulations of pressure-driven channel flow with a bulk Reynolds number of 5600 individually resolving up to 10 000 neutrally buoyant spherical particles for three values of the particle volume fraction ${\it\Phi}$ (i.e. the fraction of the total computational volume occupied by the particles). Even after the recent improvements of the immersed boundary method, simulations of this type still required a major effort, each one running for approximately $10^{6}$ CPU hours on a Cray XE6 with 2048 cores, i.e. of the order of a month of real time.

The results give a striking sense of the profound and complex ways in which the addition of particles affects fluid flow. While a log layer can still be identified, the corresponding von Kármán constant decreases by nearly 50 % as ${\it\Phi}$ increases from 0 % to 20 %, suggesting a decrease of the turbulent drag, while the additive constant decreases from 5.5 to approximately $-6.3$ , which normally corresponds to an increased drag. The net result is still an increase in the overall drag, but not because turbulence intensity is increased. As a matter of fact, the authors stress the point that, at ${\it\Phi}=20\,\%$ , turbulent fluctuations are actually decreased by the presence of the particles. They attribute their drag increase to the counterbalancing and ultimately prevailing effect of the particle contribution to the mixture stress. In this connection they offer the interesting speculation that a neighbouring region of parameter space may exist in which the reduced turbulence intensity may be stronger than the particle stress effect, so much so that particles might even result in an overall drag reduction.

Other interesting observations concern the wall region where, unlike the fluid, the particle velocity does not have to vanish. Thus, while the mean velocity of the particles is fairly close to that of the fluid over most of the channel, near the wall they are markedly different, and this is the only region of the flow where the particles are found to increase, rather than dampen, the fluid velocity fluctuations. The strong effect of the particles on the wall layer causes a considerable modification of the well-known low-velocity streaks of single-phase wall-bounded turbulence which can be clearly seen in figure 1. As ${\it\Phi}$ becomes larger there is a gradual increase of the separation between the streaks and a progressive fading of the low/high velocity contrast. The conclusion is that the structure of the wall turbulence is significantly altered by the particles with a much weaker organization into coherent structures.

From the numerical point of view, a method stable enough to handle particles with the same density as the fluid is quite an achievement but, because of the peculiarities of this situation, the results of this paper cannot be easily extrapolated to other cases. A different particle inertia, for example, would promote a more developed relative motion with respect to the fluid, very likely resulting in an increase of the fluid velocity fluctuations, as Picano et al. (Reference Picano, Breugem and Brandt2015) have found in the only region of their flow where such a difference exists, namely near the wall. This was indeed the conclusion of the simulations of Lucci et al. (Reference Lucci, Ferrante and Elghobashi2010) who studied decaying homogeneous turbulence.

3. Future

The ability to perform direct numerical simulations of single-phase turbulent flows has had an enormous impact on the understanding of single-phase turbulence and the development of reduced descriptions such as large-eddy simulation. It may well be expected that the gradual emergence of a comparable possibility for particulate, or more general, multi-phase flows will prove equally beneficial. The addition of the particle parameters (size, number, density, shape, etc.) to those of the fluid generates a huge parameter space and so little work of this type has been carried out that this vast territory is still essentially virgin. We are really at the beginning of a golden age of science in this area. In addition to that of physicists, particulate flows must hold the interest of computational scientists. In spite of continuous progress, the available hardware will always fall short of what would be desirable. Thus, work on new algorithms and the further improvement of the existing ones is necessary. Adapting them to new computational platforms, such as massively parallel GPUs, may be expected to bring handsome rewards.

All this computing will of course shed light on the microphysics of the flow, but it will enable us to pursue an even more important practical objective. It is evident that the fully resolved simulation of systems large enough to approach the actual size of a fluidized bed, for example, will not be possible for a very long time, if ever. We must rely on some sort of averaged description. The impetus in this direction has been very strong over more than half a century, but the results have been somewhat meagre. Unlike single-phase turbulence, where even simple Reynolds closure models have proven of some usefulness, in the case of particulate flows simple closures by and large have failed. Restoring the necessary amount of information lost on averaging has proven a major obstacle, with adverse effects not only on the physical realism, but in some cases on the very mathematical structure of the equations. Basing closure relations on the results of fully resolved simulations requires the development of tools to interrogate and suitably average the simulation results. This is not a simple task as the amount of information is massive and convergence of averaging slow. Furthermore, while there is a general idea about the important quantities that need to be calculated, a large amount of work is still necessary to understand how to incorporate them in robust and physically relevant closure relations. With a few others, Picano and his colleagues belong to the vanguard of what can well be expected to represent a major component of fluid dynamics research for years to come.