Measuring Features of Complex Systems

This chapter offers a guide to quantifying complexity based on the fruits of the analysis of the previous chapters. Many measures of complexity have been proposed since scientists first began to study complex systems, and the list is still growing. If complexity is a collection of features rather than a single phenomenon, then all quantitative measures of complexity can quantify only aspects of complexity rather than complexity as such. The chapter demonstrates the truism of complexity science that it is computational and probabilistic. It also further explains some of the new kinds of invariance and forms of universal behaviour that emerge when complex systems are modelled as networks and information-processing systems. The chapter then looks at a few, by now classic, measures of complexity from the 1980s and 1990s, including effective complexity, effective measure complexity, statistical complexity, and logical depth.

Shock–turbulence interactions at high turbulence intensities

Shock–turbulence interactions are investigated using well-resolved direct numerical simulations (DNS) and analysis at a range of Reynolds, mean and turbulent Mach numbers ($R_{\unicode[STIX]{x1D706}}$, $M$ and $M_{t}$, respectively). The simulations are shock and turbulence resolving with $R_{\unicode[STIX]{x1D706}}$ up to 65, $M_{t}$ up to 0.54 and $M$ up to 1.4. The focus is on the effect of strong turbulence on the jumps of mean thermodynamic variables across the shock, the shock structure and the amplification of turbulence as it moves through the shock. Theoretical results under the so-called quasi-equilibrium (QE) assumption provide explicit laws for a number of statistics of interests which are in agreement with the new DNS data presented here as well as all the data available in the literature. While in previous studies turbulence was found to weaken jumps, it is shown here that stronger jumps are also observed depending on the regime of the interaction. Statistics of the dilatation at the shock are also investigated and found to be well represented by QE for weak turbulence but saturate at high turbulence intensities with a Reynolds number dependence also captured by the analysis. Finally, amplification factors are found to present a universal behaviour with two limiting asymptotic regimes governed by $(M-1)$ and $K=M_{t}/R_{\unicode[STIX]{x1D706}}^{1/2}(M-1)$, for weak and strong turbulence, respectively. Effect of anisotropy in the incoming flow is also assessed by utilizing two different forcing mechanisms to generate turbulence.

Statistics of turbulence in the energy-containing range of Taylor–Couette compared to canonical wall-bounded flows

Considering structure functions of the streamwise velocity component in a framework akin to the extended self-similarity hypothesis (ESS), de Silva et al. (J. Fluid Mech., vol. 823, 2017, pp. 498–510) observed that remarkably the large-scale (energy-containing range) statistics in canonical wall-bounded flows exhibit universal behaviour. In the present study, we extend this universality, which was seen to encompass also flows at moderate Reynolds number, to Taylor–Couette flow. In doing so, we find that also the transversal structure function of the spanwise velocity component exhibits the same universal behaviour across all flow types considered. We further demonstrate that these observations are consistent with predictions developed based on an attached-eddy hypothesis. These considerations also yield a possible explanation for the efficacy of the ESS framework by showing that it relaxes the self-similarity assumption for the attached-eddy contributions. By taking the effect of streamwise alignment into account, the attached-eddy model predicts different behaviour for structure functions in the streamwise and in the spanwise directions and that this effect cancels in the ESS framework – both consistent with the data. Moreover, it is demonstrated here that also the additive constants, which were previously believed to be flow dependent, are indeed universal at least in turbulent boundary layers and pipe flow where high Reynolds number data are currently available.

Universal behaviour of the glass and the jamming transitions in finite dimensions for hard spheres

We investigate the glass and the jamming transitions of hard spheres in finite dimensionsd, through a revised cell theory, that combines the free volume and the Random First Order Theory (RFOT).