Abstract

An inner-scaled, shear stress-driven flow is considered as a model of independent near-wall turbulence as the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$ . In this limit, the model is applicable to the near-wall region and the lower part of the logarithmic layer of various parallel shear flows, including turbulent Couette flow, Poiseuille flow and Hagen–Poiseuille flow. The model is validated against damped Couette flow and there is excellent agreement between the velocity statistics and spectra for the wall-normal height  $y^{+}<40$ . A near-wall flow domain of similar size to the minimal unit is analysed from a dynamical systems perspective. The edge and fifteen invariant solutions are computed, the first discovered for this flow configuration. Through continuation in the spanwise width $L_{z}^{+}$ , the bifurcation behaviour of the solutions over the domain size is investigated. The physical properties of the solutions are explored through phase portraits, including the energy input and dissipation plane, and streak, roll and wave energy space. Finally, a Reynolds number is defined in outer units and the high- $Re$ asymptotic behaviour of the equilibria is studied. Three lower branch solutions are found to scale consistently with vortex–wave interaction (VWI) theory, with wave forcing localising around the critical layer.

Vaccines have played a central role in mitigating severe disease and death from COVID-19 in the past 12 months. However, efficacy wanes over time and this loss of protection is being compounded by the emergence of the Omicron variant. By fitting an immunological model to population-level vaccine effectiveness data, we estimate that neutralizing antibody titres for Omicron are reduced by 3.9 fold (95% CrI 2.9 to 5.5) compared to the Delta variant. Under this model, we predict that 90 days after boosting with the Pfizer-BioNTech vaccine, efficacy against severe disease (admission to hospital) declines to 95.9% (95% CrI 95.4% to 96.3%) against the Delta variant and 78.8% (95% CrI 75.0% to 85.1%) against the Omicron variant. Integrating this immunological model within a model of SARS-CoV-2 transmission, we demonstrate that the size of the Omicron wave will depend on the degree of past exposure to infection across the population, with relatively small Omicron waves in countries that previously experienced a large Delta wave. We show that booster doses can have a major impact in mitigating the epidemic peak, although in many settings it remains possible that healthcare capacity could still be challenged. This is particularly the case in zero-COVID countries where there is little prior infection-induced immunity and therefore epidemic peaks will be higher. Where dose supply is limited, targeting boosters to the highest risk groups to ensure continued high protection in the face of waning immunity is of greater benefit than giving these doses as primary vaccination to younger age-groups. In many settings it is likely that health systems will be stretched, and it may therefore be necessary to maintain and/or reintroduce some level of NPIs to mitigate the worst impacts of the Omicron variant as it replaces the Delta variant.

No abstract

The computation of invariant solutions and the visualisation of the associated state space have played a key role in the understanding of transition and the self-sustaining process in wall-bounded shear flows. In this study, an extension of this approach is sought for a turbulent flow which explicitly exhibits multi-scale behaviour. The minimal unit of multi-scale near-wall turbulence, which resolves two adjacent spanwise integral length scales of motion, is considered using a shear stress-driven flow model (Doohan, Willis & Hwang J. Fluid Mech., vol. 913, 2021, A8). The edge state, 26 travelling waves and two periodic orbits are computed, which represent either the large- or small-scale self-sustaining processes. Given that the spanwise length scales are not widely separated here, it could be envisaged that turbulent trajectories visit these solutions in the state space. Considering the intra- and inter-scale dynamics of the flow, numerous phase portraits are examined, but the turbulent state is not found to approach any of these solutions. A detailed analysis reveals that this is due to the lack of scale interaction processes captured by the invariant solutions, including the mean–fluctuation interaction, the energy cascade in the streamwise wavenumber space and the cascade-driven energy production discovered recently. There is a single solution that resembles turbulence much more than the others, which captures two-scale energetics and a scale interaction process involving energy feeding from small to large spanwise scales through the subharmonic sinuous streak instability mode. Based on these observations, it is conjectured that the state space view of turbulent trajectories wandering between solutions would need suitable refinement to model multi-scale turbulence, when each solution does not represent multi-scale processes of turbulence. In particular, invariant solutions that are inherently multi-scale would be required.