In this study we propose a novel data-driven reduced-order model for complex dynamics, including nonlinear, multi-attractor, multi-frequency and multiscale behaviours. The starting point is a fully automatable cluster-based network model (CNM) (Li et al., J. Fluid Mech., vol. 906, 2021, A21) that kinematically coarse grains the state with clusters and dynamically predicts the transitions in a network model. In the proposed dynamics-augmented CNM (dCNM) the prediction error is reduced with trajectory-based clustering using the same number of centroids. The dCNM is first exemplified for the Lorenz system and then demonstrated for the three-dimensional sphere wake featuring periodic, quasi-periodic and chaotic flow regimes. For both plants, the dCNM significantly outperforms the CNM in resolving the multi-frequency and multiscale dynamics. This increased prediction accuracy is obtained by stratification of the state space aligned with the direction of the trajectories. Thus, the dCNM has numerous potential applications to a large spectrum of shear flows, even for complex dynamics.

The non-Oberbeck–Boussinesq effects on the stability of a vertical natural convection boundary layer are investigated using the linearised disturbance equations for air flows up to a temperature difference of $\Delta T=100\,{\rm K}$. Based on the linear stability results, the neutral curve is shown to be sensitive to the choice of reference temperature. When evaluated using the film temperature $T_f$, a lower film Grashof number is required to trigger the linear instability for larger $\Delta T$. The relative contributions of shear and buoyant production to the perturbation kinetic energy budget reveals that the marginally unstable modes are amplified based on different mechanisms: for lower wavenumbers at relatively small Grashof number, the instability is driven by buoyancy; whereas for higher wavenumbers and larger Grashof number, the flow becomes unstable due to a shear instability. The use of reference temperature is found to scale the shear- and buoyant-driven instabilities differently so that no single reference temperature definition would collapse the neutral curves. The linear stability result further demonstrates that at a given Grashof number a higher temperature difference would give a larger amplification rate of the perturbation, which then leads to an earlier onset of the nonlinearities when evaluated at $T_f$. Finally, by comparing the amplification rates obtained from direct numerical simulation and the linear stability results, the extent of the linear regime is determined for $\Delta T = 100\,{\rm K}$.

In this study, the asymptotic solutions of the pressure variations induced by two trains passing each other in a tunnel are theoretically investigated. The one-dimensional inviscid compressible airflow is analysed, and two methods to obtain numerically exact solutions and $M_{H}$ expansion formulas for approximate equations are presented, where $M_{H}$ is the Mach number of the high-speed train. The pressure coefficient, corresponding to the maximum value of the magnitude of the pressure, is expressed as $|c_{p}|_{max}=|c_{p,min}|=[({R}/({1-R}))$ $(1+\alpha )^{2}+({R(1-R)}/{(1-2R)^{2}})(1-\alpha )^{2}]+O[M_{H}]$, where $c_{p,min}<0$, $\alpha =U_{L}/U_{H}$ and $U_{L}$ and $U_{H}$ denote the speeds of the low- and high-speed trains, respectively, and $R$ is the cross-sectional area ratio of the train to the tunnel. The theoretical results indicate the dependence of the speeds of the two trains on the pressure distribution and that the maximum magnitude of the asymptotic pressure for a fixed value of $M_{H}$ is obtained for $\alpha =1$ and $\alpha =0$ when $R< R_{c}$ and $R>R_{c}$, respectively, where $R_{c}$ denotes the critical blockage ratio. Because the airflow along the side of the low-speed train, induced by the low-speed train, is along the running direction of the high-speed train and reduces the relative velocity of the high-speed train as the two trains pass each other, $|c_{p}|_{max}$ for $\alpha =0$ is larger than $|c_{p}|_{max}$ for $\alpha =1$ when $R>R_{c}$. It is theoretically demonstrated that, as conventional high-speed railway systems satisfy $R< R_{c}$, a conservative pressure estimation can be established assuming $\alpha =1$.

Vortex-induced vibrations and galloping of an elastically mounted square cylinder are investigated for cylinder mass ratio m* = 2–50, damping ratio ζ = 0–1.0, mass-damping ratio m*ζ = 0–50 and flow reduced velocity Ur = 1–80. We home in on the effects of m*, ζ, m*ζ, $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ and $({m^\ast } + m_{ae}^\ast )\zeta$ on the critical reduced velocity Urc marking the onset of galloping, where $m_{a\textrm{0}}^\ast $ is the quiescent-fluid added mass ratio and $m_{ae}^\ast $ is the effective added mass ratio. Vibration responses, forces, vibration frequencies and added mass ratios are studied and discussed. The different branches of vortex-induced vibrations have different dependencies of $m_{ae}^\ast $ on Ur. The $m_{ae}^\ast $ in the initial branch is positive and drops rapidly with Ur, but that in the lower branch is negative and declines gently. In the galloping regime, $m_{ae}^\ast $ jumps from negative to positive at the onset of galloping, declining slightly with increasing Ur. Our results and prediction equations show that when ζ = 0, Urc is independent of m* for m* ≥ 5, albeit slightly higher for m* = 3. The latter is ascribed to mode competition. When ζ > 0, Urc linearly increases with increasing ζ. Detailed analysis substantiates that m*ζ or $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ does not serve as the unique criterion to predict the galloping occurrence. Here, we propose a new combined mass-damping parameter $({m^\ast } + m_{ae}^\ast )\zeta$ in the relationship between galloping onsets and structural properties, which successfully scales all data of Urc at different m* and ζ values.

We study the spreading of Newtonian viscous (aqueous glycerin solution) and viscoelastic (aqueous polymer solution) drops on solid substrates with different wettabilities. For drops of the same zero-shear viscosity, we find in the early stages of spreading that viscoelastic drops (i) spread faster and (ii) their contact radius shows a different power law vs time than Newtonian drops. We argue that the effect of viscoelasticity is only observable for experimental time scales of the order of or larger than the internal relaxation time of the viscoelastic polymer solution. We attribute this behaviour to the shear thinning of the viscoelastic polymer solution. When approaching the contact line, the shear rate increases and the steady-state viscosity of the viscoelastic drop is lower than that of the Newtonian drop. We support our experimental findings with a simple (first-order) perturbation model that qualitatively agrees with our findings.

The evolution of the water-entry cavity affects the impact load and the motion of the body. This paper adopts the Eulerian finite element method for multiphase flow for simulations of the high-speed water-entry process. The accuracy and convergence of the numerical method are verified by comparing it with the experimental data and the results of the transient cavity dynamics theory. Based on the results, the representative characteristics of the cavity are discussed from the perspective of the cavity cross-section. It is found that the asymmetry of the cavity expansion and contraction durations is related to the motion of the free surface and the closure of the cavity. The uplift of the free surface suppresses cavity expansion, while the jet generated from free surface closure accelerates cavity contraction. The duration of the contraction of the cavity near the free surface is shorter than the expansion duration due to the change in the velocity distribution caused by the free surface motion. The necking phenomenon during deep closure leads to an increase in the internal pressure of the cavity, prolonging cavity contraction near the deep closure area. This work provides new insights into the cavity dynamics in high-speed water entry.

Emptying–filling boxes have been studied in a wide range of configurations for decades, but the flow created in the box by two plumes rising from sources of arbitrary strength and elevation was previously unsolved. Guided by experiments and simplified analytical modelling, we reveal a rich array of two- and three-layer stratifications across seven possible flow regimes. The governing equations for these regimes show how the prevailing regime and stratification properties vary with three key parameters: the relative strength of the plumes, the height difference between their sources and a parameter characterising the resistance of the box to emptying. We observe and explain new behaviours not described in previous studies that are crucial to understanding emptying–filling boxes with multiple plumes. In particular, we demonstrate that the oft-assumed premise that $n$ plumes leads to a stratification with $n+1$ layers is not necessarily true, even in the absence of mixing. Two emptying–filling box models are developed: an analytical model addressing all combinations of the governing parameters and an extended model for three-layer stratifications that incorporates two mixing processes observed in the experiments. The predictions of these two models are in generally excellent agreement with measurements from the experimental campaign covering 69 combinations of the governing parameters. This study improves our understanding of emptying–filling boxes and could facilitate improvements to natural ventilation building design, as demonstrated by an example scenario in which occupants feel cooler upon the addition of a second source of heat.

A hybrid lattice Boltzmann and finite difference method is applied to study the influence of soluble surfactants on droplet deformation and breakup in simple shear flow. First, the influence of bulk surfactant parameters on droplet deformation in two-dimensional shear flow is investigated, and the surfactant solubility is found to influence droplet deformation by changing average interface surfactant concentration and non-uniform effects induced by non-uniform interfacial tension and Marangoni forces. In addition, the droplet deformation first increases and then decreases with Biot number, increases significantly with adsorption number $k$ and decreases with Péclet number or adsorption depth; and among the parameters, $k$ is the most influential one. Then, we consider three-dimensional shear flow and investigate the roles of surfactants on droplet deformation and breakup for different capillary numbers and viscosity ratios. Results show that in the soluble case with $k=0.429$, the droplet exhibits nearly the same deformation as in the insoluble case due to the balance between surfactant adsorption and desorption; upon increasing $k$ from 0.429 to 1, the average interface surfactant concentration is greatly enhanced, leading to significant increase in droplet deformation. The critical capillary number of droplet breakup $C{{a}_{cr}}$ is identified for varying viscosity ratios in clean, insoluble and soluble ($k=0.429$ and 1) systems. As the viscosity ratio increases, $C{{a}_{cr}}$ first decreases and then increases rapidly in all systems. The addition of surfactants always favours droplet breakup, and increasing solubility or $k$ could further reduce $C{{a}_{cr}}$ by increasing average interface surfactant concentration and local surfactant concentration near the neck during the necking stage.