2. Overview

Stevens et al. (Reference Stevens, Wilczek and Meneveau2014) exploit the recently discovered log-law for the variance of the streamwise velocity in wall-bounded neutral (no buoyancy) flows:

(2.1) $$\begin{eqnarray}\frac{\overline{{u^{\prime }}^{2}}}{u_{\ast }^{2}}=B_{1}-A_{1}\log \left(\frac{y}{{\it\delta}}\right),\end{eqnarray}$$

where the prime denotes the perturbation relative to the mean velocity; $u_{\ast }$ is the friction velocity, $y$ the vertical distance from the wall and ${\it\delta}$ the depth of the boundary layer (see experimental results in Marusic & Kunkel Reference Marusic and Kunkel2003, and theoretical foundation in Perry & Chong Reference Perry and Chong1982 and Hultmark Reference Hultmark2012). Here $A_{1}\approx 1.25$ is a universal constant for wall-bounded flows, while $B_{1}$ varies with flow conditions. If the velocity statistics are assumed to be Gaussian, one can also relate the higher even-order statistics to those of the second order and derive log-laws for them in which the universal coefficients $A_{p}$ , equivalent to $A_{1}$ , can then be theoretically related to $A_{1}$ .

Using wind tunnel data, Meneveau & Marusic (Reference Meneveau and Marusic2013) had established that although log-laws are indeed observed for even-order statistics up to order 10, the measured coefficients indicate departure from Gaussianity, as expected. These findings provide the motivation for Stevens et al. (Reference Stevens, Wilczek and Meneveau2014) to attempt to reproduce them with LES. As the authors point out, a well-established paradigm in LES is that the first-order means are largely resolved and not very sensitive to the SGS contribution and thus a code should aim to reproduce the energy cascade from resolved to subgrid scales to accurately capture the second-order turbulent energy (see Meneveau & Katz Reference Meneveau and Katz2000). However, no conditions are explicitly imposed to ensure that statistics of order three or higher are correctly reproduced, and indeed there are studies that document that LES could match first- and second-order statistics accurately while simultaneously failing to capture higher orders (Pan, Chamecki & Isard Reference Pan, Chamecki and Isard2014). Stevens et al. apply a well-tested LES code with advanced numerical schemes and dynamic SGS modelling (Bou-Zeid, Meneveau & Parlange Reference Bou-Zeid, Meneveau and Parlange2005). They perform 20 simulations with increasing grid resolutions, up to a massive $2048\times 1024\times 577$ grid points. The two main parameters they vary are the roughness length $z_{0}$ and the grid scale ${\it\Delta}$ ; these are the most obvious candidates to serve as an inner scale in wall-modelled LES where the fluid viscosity does not appear in the equations and no inner viscous scale exists. Here ${\it\Delta}$ also controls the fraction of the resolved turbulence.

The authors demonstrate that it is critical to resolve well over 90 % of the variance (their highest resolutions resolve over 98 %) for the LES to adequately reproduce the log-law for higher even-order moments. Since the resolved fraction decreases as the wall is approached, low resolutions were not able to reproduce a log-region or erroneously overestimated the height of its lower limit. This lower limit is then shown to depend on ${\it\Delta}$ but not on $z_{0}$ , indicating that ${\it\Delta}$ is the effective inner scale of turbulence in these simulations. The central and major contribution of their paper is linked to the success of the LES in capturing remarkably well the log-regions for the streamwise velocity variance and even-order statistics up to order 10, as measured experimentally. Even more interesting perhaps is the fact that the universal coefficients $A_{p}$ deduced from LES also match the wind tunnel data. This is far from being a trivial success since it requires the LES to accurately capture the departure of the velocity statistics from a Gaussian distribution.

3. Future

One can at present almost take for granted that in many flows that have been adequately investigated using LES, the mean and root-mean-square velocities produced by proper simulations will be in good agreement with theory and measurements (compressible or reacting flows remain more challenging). However, various studies have already documented some limitations of LES results, for example when considering structure functions and space–time correlations (Kang, Chester & Meneveau Reference Kang, Chester and Meneveau2003; Yang, He & Wang Reference Yang, He and Wang2008). Matching some lower-order statistics is thus no guarantee that all other statistics are also accurately simulated. The study of Stevens et al. therefore makes a noteworthy contribution by establishing a robust test case that can allow us to push LES outside its ‘comfort zone’. An appealing feature of this test case is that the results can be expressed as universal laws. Further similar generalized benchmarks are critically needed.

Another important lesson from Stevens et al. is that even higher resolutions are still needed. The author of this article collected data on the typical resolutions used in LES by searching the Journal of Fluid Mechanics for papers with ‘large eddy simulation’ in the title and randomly selecting data from papers published after 1980 (excluding papers that purposely use low resolution to test the limits of LES). Considering the reduction in time step needed when the total number of grid points $N$ increases in LES, along with the rise in available computing power predicted (and realized) by Moore’s law, one would expect $N$ to scale ${\sim}2^{Y/2}$ in LES, where $Y$ is the year. However, figure 1 suggests that LES grid resolutions have been increasing more slowly, roughly ${\sim}2^{Y/3}$ . The densest grid in Stevens et al. lies closer to Moore’s law and demonstrates the feasibility of such high resolutions. Tests from another high-resolution paper (Sullivan & Patton Reference Sullivan and Patton2011), considering up to third-order moments, also confirm the importance of further higher resolutions in LES. This underlines the need to raise the bar not only for LES testing, but also for its grid resolutions.

Figure 1. The increase in the number of grid nodes in LES in J. Fluid Mech. (list of article dois available as supplementary material at http://dx.doi.org/10.1017/jfm.2014.616), data from a study by Voller & Porte-Agel (Reference Voller and Porte-Agel2002) of computational fields that follow Moore’s law more closely, and the grids of Stevens et al. and Sullivan & Patton (Reference Sullivan and Patton2011).

Supplementary material

Supplementary material is available at http://dx.doi.org/10.1017/jfm.2014.616.