Systematic search for extreme and singular behaviour in some fundamental models of fluid mechanics.

This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behaviour in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier-Stokes system. Inspired by the seminal work of Lu & Doering (2008 Ind. Univ. Math. 57, 2693-2727), we sought such extreme behaviour by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier-Stokes flows constructed in this manner in three dimensions. We also discuss the results obtained for one-dimensional Burgers and two-dimensional Navier-Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Alignments of triad phases in extreme one-dimensional Burgers flows.

We analyze the fine structure of nonlinear modal interactions in inviscid and viscous Burgers flows in 1D, which serve as toy models for the Euler and Navier-Stokes dynamics. This analysis is focused on preferential alignments characterizing the phases of Fourier modes participating in triadic interactions, which are key to determining the nature of energy fluxes between different scales. We develop diagnostic tools designed to probe the level of coherence among triadic interactions and apply them to Burgers flows corresponding to different initial conditions, including unimodal, extreme (in the sense of maximizing the growth of enstrophy in finite time), and generic. We find that in all cases triads involving energy-containing Fourier modes align their phases so as to maximize the energy flux toward small scales, and most of this flux is realized by only a handful of triads revealing a universal statistical distribution. We then identify individual triads making the largest contributions to the flux at different wave numbers and show that they represent a mixture of local and nonlocal interactions, with the latter becoming dominant at later times. These results point to the possibility of constructing a strongly reduced modal representation of Burgers flows that would require a much smaller number of degrees of freedom. Another interesting observation is that removing the spatial coherence from the extreme initial data (by randomizing the phases while retaining the magnitudes of the Fourier coefficients) does not profoundly change the nature of triadic interactions and synchronization as well as the resulting fluxes in these flows.

Data-driven optimal closures for mean-cluster models: Beyond the classical pair approximation.

This study concerns the mean-clustering approach to modeling the evolution of lattice dynamics. Instead of tracking the state of individual lattice sites, this approach describes the time evolution of the concentrations of different cluster types. It leads to an infinite hierarchy of ordinary differential equations which must be closed by truncation using a so-called closure condition. This condition approximates the concentrations of higher-order clusters in terms of the concentrations of lower-order ones. The pair approximation is the most common form of closure. Here, we consider its generalization, termed the "optimal approximation," which we calibrate using a robust data-driven strategy. To fix attention, we focus on a recently proposed structured lattice model for a nickel-based oxide, similar to that used as cathode material in modern commercial Li-ion batteries. The form of the obtained optimal approximation allows us to deduce a simple sparse closure model. In addition to being more accurate than the classical pair approximation, this "sparse approximation" is also physically interpretable which allows us to a posteriori refine the hypotheses underlying construction of this class of closure models. Moreover, the mean-cluster model closed with this sparse approximation is linear and hence analytically solvable such that its parametrization is straightforward, although it offers a good approximation of the actual time evolution of the cluster concentrations on short timescales only. On the other hand, parametrization of the mean-cluster model closed with the pair approximation is shown to lead to an ill-posed inverse problem.

Accurate Characterization of Ion Transport Properties in Binary Symmetric Electrolytes Using In Situ NMR Imaging and Inverse Modeling.

We used NMR imaging (MRI) combined with data analysis based on inverse modeling of the mass transport problem to determine ionic diffusion coefficients and transference numbers in electrolyte solutions of interest for Li-ion batteries. Sensitivity analyses have shown that accurate estimates of these parameters (as a function of concentration) are critical to the reliability of the predictions provided by models of porous electrodes. The inverse modeling (IM) solution was generated with an extension of the Planck-Nernst model for the transport of ionic species in electrolyte solutions. Concentration-dependent diffusion coefficients and transference numbers were derived using concentration profiles obtained from in situ (19)F MRI measurements. Material properties were reconstructed under minimal assumptions using methods of variational optimization to minimize the least-squares deviation between experimental and simulated concentration values with uncertainty of the reconstructions quantified using a Monte Carlo analysis. The diffusion coefficients obtained by pulsed field gradient NMR (PFG-NMR) fall within the 95% confidence bounds for the diffusion coefficient values obtained by the MRI+IM method. The MRI+IM method also yields the concentration dependence of the Li(+) transference number in agreement with trends obtained by electrochemical methods for similar systems and with predictions of theoretical models for concentrated electrolyte solutions, in marked contrast to the salt concentration dependence of transport numbers determined from PFG-NMR data.

Geometrical alignment properties in Fourier- and wavelet-filtered statistically stationary two-dimensional turbulence.

In this paper we compare the geometrical alignment properties of Fourier- and wavelet-filtered statistically stationary two-dimensional turbulence. The goal is to study the preferential alignment angle of vorticity gradient with respect to the compressing eigenvector of the rate-of-strain tensor, and use this quantity as a measure of how the two filtering methods affect the small scale geometric structure of the flow. The principal result is that for the case of the incoherent part obtained through wavelet filtering the probability density function of this angle is flat, meaning that this field is effectively unstructured and therefore dynamically passive. On the contrary, the corresponding field obtained through Fourier filtering reveals a bump at the angle pi/4, which indicates the presence of dynamically active filament-type structures. These results provide evidence that, unlike the wavelet filtering, the Fourier filtering does remove dynamically important information from the flow.