We model filtration of a feed solution, containing both small and large foulant particles, by a membrane filter. The membrane interior is modelled as a network of pores, allowing for the simultaneous adsorption of small particles and sieving of large particles, two fouling mechanisms typically observed during the early stages of commercial filtration applications. In our model, first-principles continuum partial differential equations model transport of the small particles and adsorptive fouling in each pore, while sieving particles are assumed to follow a discrete Poisson arrival process with a biased random walk through the pore network. Our goals are to understand the relative influences of each fouling mode and highlight the effect of their coupling on the performance of filters with a pore-size gradient (specifically, we consider a banded filter with different pore sizes in each band). Our results suggest that, due to the discrete nature of pore blockage, sieving alters qualitatively the rate of the flux decline. Moreover, the difference between sieving-particle sizes and the initial pore size (radius) in each band plays a crucial role in indicating the onset and disappearance of sieving–adsorption competition. Lastly, we demonstrate a phase transition in the filter lifetime as the arrival frequency of sieving particles increases.

We investigate the oscillation of the large-scale circulation (LSC) in turbulent Rayleigh–Bénard convection by combining laboratory experiments and numerical simulations. The experiments are conducted mainly in a laterally confined rectangular cell, and the flow fields are measured by particle image velocimetry in the vertical mid-plane. It is found that the velocity field exhibits not only the commonly observed oscillation frequencies $(f\tau _o=1,2)$ but also higher-order harmonic frequencies $(f\tau _o\gt 2)$, where $\tau _o$ is the turnover time of the LSC. Spectral proper orthogonal decomposition (SPOD) reveals that the coherent structures underlying these frequencies display an approximate azimuthal periodicity when viewed in polar coordinates. These structures can be interpreted as travelling waves, consistent with the advected-oscillation picture proposed by Brown & Ahlers (2009 J. Fluid Mech., vol. 638, pp. 383–400). A data-driven resolvent analysis, used here as an effective input–output model inferred from the experimental data, yields response modes and gain peaks that are consistent with the SPOD results, and support the interpretation that the dominant oscillations are selected through linear amplification about the mean circulation. This perspective is further supported by direct numerical simulation in a rectangular cell and by velocity measurements in a cylindrical cell. These findings provide an additional viewpoint for understanding the oscillation mechanism of the LSC in turbulent Rayleigh–Bénard convection.

Obtaining accurate field statistics continues to be one of the major challenges in turbulence theory and modelling. From the various existing modelling approaches, multifractal models have been successful in capturing intermittency in velocity gradient and increment distributions. Moreover, superstatistical models from non-equilibrium statistical mechanics have shown the capacity to model probability density functions (p.d.f.s) of various statistical turbulent quantities as ensembles of simpler stochastic processes. Here, we present an approach that generates field statistics in the form of a characteristic functional by promoting a model for multifractal increment statistics to an ensemble of Gaussian fields. By carefully designing the correlation function and the corresponding weight of each subensemble, we are able to define a functional that exhibits multifractal two-point inertial-range and dissipation-range statistics, and that blends into realistic large-scale behaviour. Additionally, the method is capable of producing multifractal statistics with any of the widely used singularity spectra. We characterise the fidelity of our approach through comparisons to literature results from direct numerical simulations. Overall, our framework thereby bridges between three different perspectives: superstatistics, multifractals and functional approaches to turbulence.

In this paper, we investigate the influences of wall temperature on compressible turbulent boundary layers at free-stream Mach number 6.0 and moderate Reynolds numbers. The findings demonstrate that turbulent statistics, including the logarithmic scaling of Reynolds stresses in the overlap region, the presence of very-large-scale motions in the outer layer, and their superposition on near-wall turbulence, exhibit qualitative invariance across varying wall temperatures. However, the reduced scale separation between near-wall small-scale motions and outer-layer large-scale motions leads to a contraction in the vertical extent of Reynolds stresses adhering to the logarithmic law. Very-large-scale motions are attenuated in viscous units but intensified in global scalings due to the lower free-stream Reynolds number, following approximately the power law. Their superposition effects on near-wall turbulence, when weighted by density, show only weak dependence on wall temperature. Conversely, the modulation of near-wall velocity and temperature fluctuations by very-large-scale motions diminishes with decreasing wall temperature. Through detailed analysis of spanwise spectra in the outer region, a distinct inertial subrange obeying the classical $-5/3$ scaling law is identified, alongside a universal scaling over the dissipative range. These observations suggest that the mechanisms governing energy cascade processes and the associated small-scale turbulent fluctuations remain independent of wall temperature variations.

The first bifurcation of the flow around a spheroid is analysed using global stability analysis to understand the development of flow asymmetry despite the symmetry of the configuration. The base flow, perturbations, adjoint modes, vorticity fields, structural sensitivity and skin friction lines are analysed to characterise the flow. The study of aspect ratio effects at zero angle of attack establishes that the structure of the asymmetry is the same whether the axisymmetric body is bluff or more streamlined. Stability of the flow field around the 6 : 1 spheroid is then investigated for angles of attack $\alpha \in [0{-}90]^\circ$ as a function of Reynolds number. Comparison of the low angle of attack results with the DARPA SUBOFF experiments of Ashok et al. (J. Fluid Mech., 2015, vol. 774, pp. 416–442) shows that the asymmetry observed in the experiments is similar to the global mode predicted by the stability calculations. It is conjectured that the experimental asymmetry is triggered by the weak cross-stream circulating flow. The leading eigenmode is a stationary asymmetric mode in the angle of attack $\alpha \in [0{-}65]^\circ$ range, while above $\alpha =65^\circ$, the leading mode is an oscillatory shedding mode. The rapid decrease in the critical Reynolds number between cases $\alpha =49.25^\circ$ and $49.5^\circ$ is attributed to coexisting symmetric flow states that have different susceptibilities to asymmetry; there exists a symmetric stationary mode that does not become unstable first, but appears to be the difference between the base flows at the same Reynolds number at the two angles of attack. The change from a stationary to an oscillatory instability between $\alpha =65^\circ$ and $70^\circ$ is linked to the ability/inability of the vortex sheets to roll up and reattach to the body in the former/latter cases, respectively. The difference in the separation patterns and the similarity between the eigenmodes indicate that asymmetry of the flow field is governed by the same mechanism across a wide angle of attack range, regardless of whether the flow is like a bluff body wake, a streamlined body wake or a vortex wake. Previous studies have argued that the asymmetry of vortex pairs emerges either because of a vortex instability or a separation/reattachment related mechanism; since the development of the asymmetry cannot be linked to specific features in the separation pattern in the investigated configurations, our results support the former argument.

The canonical scenario of two-degree-of-freedom vortex-induced vibration (VIV) of a circular cylinder is re-examined in this study through high-fidelity large-eddy simulations (LES) at a Reynolds number of 10 000. The in-line and cross-flow vibration amplitudes, frequency responses and hydrodynamic coefficients predicted by the present LES match classical experimental results better than previous numerical attempts. In particular, motivated by an inadequate study yet vital importance of the small-amplitude in-line response in offshore engineering design, we present the first numerical evidence for the existence of three in-line response regions. Furthermore, the present in-line response agrees well with the design guideline DNV-RP-F105. After validating the present results against DNV-RP-F105 and published experiments, the detailed LES datasets enable further analysis of new VIV characteristics and physical mechanisms that have not been explored previously. For example, we identify and explain (i) the existence of twin governing frequencies for several VIV branches with partial synchronisation, (ii) decoupled in-line and cross-flow vibrations in the first in-line branch with symmetric vortex-shedding pattern, where an in-line resonance may not induce a cross-flow resonance, (iii) existence of a new elliptic vibration trajectory for a perfectly in-line resonant condition, (iv) gradualness in the 2S ↔ 2T transition of the vortex-shedding pattern and thus a continuous variation in the vibration amplitudes and hydrodynamic coefficients amid this transition and (v) lowest spanwise correlation of vortex shedding in the super-upper and lower branches, which is induced by complex interactions among ≥4 shed vortices over a cylinder vibration period.