We introduce a projection-based model reduction method that systematically accounts for nonlinear interactions between the resolved and unresolved scales of the flow in a low-dimensional dynamical systems model. The proposed method uses a separation of time scales between the resolved and subscale variables to derive a reduced-order model with cubic closure terms for the truncated modes, generalizing the classic Stuart–Landau equation. The leading-order cubic terms are determined by averaging out fast variables through a perturbation series approximation of the action of a stochastic Koopman operator. We show analytically that this multiscale closure model can capture both the effects of mean-flow deformation and the energy cascade before demonstrating improved stability and accuracy in models of chaotic lid-driven cavity flow and vortex pairing in a mixing layer. This approach to closure modelling establishes a general theory for the origin and role of cubic nonlinearities in low-dimensional models of incompressible flows.

From biological tissues to microactuators and absorption of solvents into layers of paint, macroscopically non-porous materials with the capacity to swell when in contact with a solvent are ubiquitous. In these systems, owing to strong solid–fluid interactions, chemically driven flows can yield large geometric changes. We study experimentally and theoretically the canonical problem of the swelling of a thin hydrogel layer by a single water drop. Using a bespoke experimental set-up, we observe fast absorption leading to a radially spreading axisymmetric blister. We use a fully three-dimensional linear poroelastic framework with nonlinear kinematic equations to obtain governing equations, which we then reduce with thin-layer scalings to a one-dimensional nonlinear diffusion equation for the evolution of the blister geometry. In the limits of large and small deformations, the evolution of the blister characteristic height and radius are self-similar, following power laws in time. Our experimental measurements show that the evolution of the blister is broadly within the theoretical predictions in the large and small deformation regimes. In the general intermediate deformation regime, the measurements are well captured by our reduced one-dimensional diffusion model, which does not require the sophisticated and computationally expensive numerical approaches necessary for the original two-dimensional nonlinear coupled transport problem. By adapting modelling techniques from the fluid dynamics of thin porous elastic layers to a polymer swelling problem, our modelling framework extends the range of polymer swelling problems that can be treated with semi-analytical methods. Moreover, our detailed experimental data can serve as a test case for future nonlinear poroelastic frameworks of swelling polymer materials.

We study the inertial migration of a torque-free neutrally buoyant sphere in wall-bounded plane Couette flow over a wide range of channel Reynolds numbers $Re_c$ in the limit of small particle Reynolds number ($Re_p\ll 1$) and confinement ratio ($\lambda \ll 1$). Here, $Re_c = V_{wall}H/\nu$, where $H$ denotes the separation between the channel walls, $V_\text {wall}$ denotes the speed of the moving wall, and $\nu$ is the kinematic viscosity of the Newtonian suspending fluid. Also, $\lambda = a/H$, where $a$ is the sphere radius, with $Re_p=\lambda ^2 Re_c$. The channel centreline is found to be the only (stable) equilibrium below a critical $Re_c$ ($\approx 148$), consistent with the predictions of earlier small-$Re_c$ analyses. A supercritical pitchfork bifurcation at the critical $Re_c$ creates a pair of stable off-centre equilibria, located symmetrically with respect to the centreline, with the original centreline equilibrium becoming unstable simultaneously. The new equilibria migrate wall-ward with increasing $Re_c$. In contrast to the inference based on recent computations, the aforementioned bifurcation occurs for arbitrarily small $Re_p$ provided that $\lambda$ is sufficiently small. An analogous bifurcation occurs in the two-dimensional scenario, that is, for a circular cylinder suspended freely in plane Couette flow, with the critical $Re_c$ being approximately $110$.

A slip asymmetry can break the fore–aft symmetry of the local hydrodynamic force distribution on the surface of an otherwise no-slip or uniform-slip particle. Here, we use the Lorentz reciprocal theorem to demonstrate that such asymmetry, even in a fractional amount, can qualitatively alter the swimming characteristics of a self-propelled spherical squirmer, markedly different from those of no-slip or uniform-slip squirmers. Unlike the usual tangential squirming by the thrust-providing B1 mode and the type-determining B2 mode, we discover two unique features for a stick-slip squirmer. First, the squirmer can acquire a swimming velocity U without the B1 mode but simply by a symmetric extensile/contractile squirming from the B2 mode, which is able to reverse the swimming direction of the squirmer. Second, a stresslet $\boldsymbol{\mathsf{S}}$ can also be induced by a unidirectional squirming from the B1 mode, capable of inverting the squirmer's stresslet from extensile type to contractile type or vice versa to change the squirmer from puller to pusher or in a reverse manner. We further show that the two squirming modes can reinforce or compete with each other to enhance or diminish U and $\boldsymbol{\mathsf{S}}$ due to interplays between the asymmetric squirming forces on the stick and the slip faces. A phase diagram is also established to categorize a variety of newly emerging swimming states, such as an enhanced/degraded puller/pusher and a backward puller/pusher, depending on the relative strength of the squirming modes β = B2/B1, the direction of the stick-slip polarity and the degree of the slip disparity. As a result of such cooperative and competitive natures, a stick-slip squirmer can swim more or less efficiently than no-slip and uniform-slip ones. These distinctive features arising from stick-slip disparity can not only be made geometrically tuneable for steering the motion of a squirmer, but also provide new means for making efficient artificial microswimmers using amphiphilic Janus particles.

We show that the energy required by a turbulent flow to displace a given amount of fluid through a straight duct in a given time interval can be reduced by modulating in time the pumping power. The control strategy is hybrid: it is passive, as it requires neither a control system nor control energy, but it manipulates how pumping energy is delivered to the system (as in active techniques) to increase the pumping efficiency. Our control employs a temporally periodic pumping pattern, where a short and intense acceleration (in which the pumping system is on) followed by a longer deceleration (in which the pumping system is off) makes the flow alternately visit a quasi-laminar and a turbulent state. The computational study is for a plane channel flow, and employs direct numerical simulations, which present specific computational challenges, for example the highly varying instantaneous value of the Reynolds number, and the importance of discretisation effects. Particular care is devoted to a meaningful definition of drag reduction in the present context. The ability of the forcing to yield significant savings is demonstrated. Since only a small portion of the parameter space is investigated, the best performance of the control technique remains to be assessed.

Natural flyers are capable of producing excessive lift via a stabilized leading-edge vortex (LEV), which appears to linger above the wing for a longer duration than it could in an equivalent two-dimensional flow. Previous studies found this stabilization behaviour closely related to a spanwise flow along the LEV axis; however, it is still debatable how the spanwise flow influences the LEV stability. In this work, potential flow theory is adopted to model an LEV attached to a flat-plate wing. To account for the spanwise flow effect, we propose a finite-area sink (FAS) model which allows the dynamical interaction between co-located LEV and spanwise flow. Through linear stability analysis of the dynamical system associated with the LEV movement, we arrive at a stable spiral-sink type of equilibrium, which is the first mathematical evidence supporting LEV stabilization by spanwise flow. It is further concluded that the LEV stability can be enhanced by either increasing the strength or decreasing the cross-section area of the spanwise flow.

In this study, the transport equation for scalar iso-surface area density ($\varSigma$) in a turbulent, temporally developing mixing layer is examined. Exploring the spatial and temporal evolution of the terms in the $\varSigma$ transport equation is vital to improving our understanding of turbulent flows characterized by distinct interfaces, e.g. the flame surface or the turbulent/non-turbulent interface. Previous work reported by the authors identified that $\varSigma$ exhibits self-similar behaviour consistent with the development of the temporal mixing layer (Blakeley et al., J. Fluid Mech., vol. 951, 2022, A44). Accordingly, each of the terms in the $\varSigma$ transport equation is found to behave in a self-similar manner, though there are notable differences in the self-similar behaviours for each term. Based on the results presented herein, it is suggested that the rate of change of $\varSigma$ ($\partial \varSigma /\partial t$) and the advection term scale with $h\lambda _\varPhi /\Delta U$, where $h$ is the width of the mixing layer, $\lambda _\varPhi$ is the scalar Taylor length scale and $\Delta U$ is the velocity difference. The production and destruction terms are found to scale with an additional factor $({Re}\,Sc)^{1/2}$. In contrast, the molecular diffusion term is found to scale with a factor $({Re}\,Sc)^{-1/2}$ compared to $\partial \varSigma /\partial t$. Importantly, it is found that the difference between the production and destruction terms, or net surface ‘stretch’, scales with the same factor as $\partial \varSigma /\partial t$ and the advection term, which may have a significant impact on how the evolution of $\varSigma$ is understood and modelled in turbulent flows.

Numerical simulations have been conducted to identify the dominant mechanism responsible for driving secondary flow motions in horizontal particle-laden pipe flows, based on an analysis of the forces acting on each phase. A four-way coupling Euler–Lagrangian approach was employed, using direct numerical simulations for the gas phase and Lagrangian particle tracking to account for the drag, gravitational and lift forces, together with the interactions that occur for both particle–wall and inter-particle collisions. The four different flow regimes, which had been identified previously as depending on various combinations of flow parameters and are characterised by the secondary flow structures of both the fluid and particle phases, were identified via varying the mass loading alone from $\varPhi _m=0.4$ to $\varPhi _m=1.8$. The distribution of the divergence of Reynolds stresses was used to help characterise the classes of the secondary fluid flow. This shows that secondary fluid flows of both the first and second kinds can either exist separately or co-exist in such flows. The forces exerted on the fluid phase by the pressure gradient and fluid–particle interactions were examined qualitatively and quantitatively to identify their contribution to the secondary fluid flow motions. A similar study was also applied to the drag, lift and gravitational forces exerted on the particle phase for the secondary particle flow motions. These were found to explain the secondary flows of both the fluid and particle phases with regard to both the flow direction and magnitude, together with the interaction between the two phases.

Kirby et al. (J. Fluid Mech., vol. 953, 2022, A39) adapted the two-scale momentum theory (Nishino & Dunstan, J. Fluid Mech., vol. 894, A2) to large finite-sized farms. They demonstrated that analytical estimates agree excellently with large eddy simulations, and that the model provides a good upper limit of the power production for a given array density. Crucially, they introduced the concepts of farm-scale losses, caused by the atmospheric response to the whole farm, and turbine-scale losses, owing to internal flow interactions in the wind farm. These two new theoretical concepts offer a novel way to analyse the performance of extended wind farms. For large offshore wind farms, losses at the wind-farm scale are typically twice as high as at the turbine scale. This demonstrates that there is limited potential for layout optimizations of extended arrays. Instead, optimization strategies should focus on developing methods to increase the energy entrainment into the wind farm. This work provides an exciting roadmap for analysing the effective efficiency of large wind farms.

A new resolvent-based method is developed to predict the space–time properties of the flow field. To overcome the deterioration of the prediction accuracy with increasing distance between the measurements and predictions in the resolvent-based estimation (RBE), the newly proposed method utilizes the RBE to estimate the relative energy distribution near the wall rather than the absolute energy directly estimated from the measurements. Using this extra information from RBE, the new method modifies the energy distribution of the spatially uniform and uncorrelated forcing that drives the flow system by minimizing the norm of the cross-spectral density tensor of the error matrix in the near-wall region in comparison with the RBE-estimated one, and therefore it is named as the resolvent-informed white-noise-based estimation (RWE) method. For validation, three time-resolved direct numerical simulation (DNS) datasets with the friction Reynolds numbers $Re_\tau = 180$, 550 and 950 are generated, with various locations of measurements ranging from the near-wall region ($y^+ = 40$) to the upper bound of the logarithmic region ($y/h \approx 0.2$, where h is the half-channel height) for the predictions. Besides the RWE, three existing methods, i.e. the RBE, the $\lambda$-model and the white-noise-based estimation (WBE), are also included for the validation. The performance of the RBE and scale-dependent model ($\lambda$-model) in predicting the energy spectra shows a strong dependence on the measurement locations. The newly proposed RWE shows a low sensitivity on $Re_{\tau }$ and the measurement locations, which may range from the near-wall region to the upper bound of the logarithmic region, and has a high accuracy in predicting the energy spectra. The RWE also performs well in predicting the space–time properties in terms of the correlation magnitude and the convection velocity. We further utilize the new method to reconstruct the instantaneous large-scale structures with measurements from the logarithmic region. Both the RWE and RBE perform well in estimating the instantaneous large-scale structure, and the RWE has smaller errors in the estimations near the wall. The structural inclination angles around $15^\circ$ are predicted by the RWE and WBE, which generally recover the DNS results.

We measure the Kelvin–Helmholtz instability in between a layer of a diamagnetic fluid flowing in a channel and a layer of ferrofluid resting on top. When the diamagnetic fluid exceeds a critical flow velocity the interface in between both fluids becomes unstable and waves develop. It has been predicted by Sutyrin & Taktarov (J. Appl. Math. Mech., vol. 39, 1975, pp. 520–524) that a homogeneous magnetic field, oriented horizontally, stabilizes the liquid interface. To test this prediction we apply in a closed flow channel a local periodic perturbation of the interface by magnetic or mechanic means. From the measured growth and decay rates of the interface undulations we determine the critical flow velocity for various driving frequencies and applied magnetic fields. In this way we confirm quantitatively the stabilizing effect of the horizontal field. Moreover we measure the dispersion relation of the interfacial waves.

We study the propagation of water waves in a three-dimensional device alternating open canals and resonant canals with subwavelength resonances. The dispersion of water waves in such a medium is obtained by analysing the full three-dimensional problem and combining Bloch–Floquet analysis with an asymptotic technique. We obtain the closed forms of the dispersions for resonant canals containing one or two resonators, which depend on only two functions associated with symmetric and antisymmetric modes, and on a geometric parameter analogous to the hopping parameter in topological systems. The analysis of the complete band structure reveals frequency ranges alternating between elliptical and hyperbolic dispersions; in particular, the hyperbolic regime gives rise to a negative effective water depth with a consequent negative refraction. Throughout the course of our study, our theoretical results are validated by comparison with numerical calculations of the full three-dimensional problem.

The objective of the present study is to provide a numerical database of thermal boundary layers and to contribute to the understanding of the dynamics of passive scalars at different Prandtl numbers. In this regard, a direct numerical simulation (DNS) of an incompressible zero-pressure-gradient turbulent boundary layer is performed with the Reynolds number based on momentum thickness ${\textit {Re}}_{\theta }$ ranging up to $1080$. Four passive scalars, characterized by the Prandtl numbers ${\textit {Pr}} = 1,2,4,6$ are simulated using the pseudo-spectral code SIMSON (Chevalier et al., SIMSON : a pseudo-spectral solver for incompressible boundary layer flows. Tech. Rep. TRITA-MEK 2007:07. KTH Mechanics, Stockholm, Sweden, 2007). To the best of our knowledge, the present DNS provides the thermal boundary layer with the highest Prandtl number available in the literature. It corresponds to that of water at ${\sim }24\,^{\circ }{\rm C}$, when the fluid temperature is considered as a passive scalar. Turbulence statistics for the flow and thermal fields are computed and compared with available numerical simulations at similar Reynolds numbers. The mean flow and scalar profiles, root-mean-squared velocity and scalar fluctuations, turbulent heat flux, turbulent Prandtl number and higher-order statistics agree well with the numerical data reported in the literature. Furthermore, the pre-multiplied two-dimensional spectra of the velocity and of the passive scalars are computed, providing a quantitative description of the energy distribution at the different length scales for various wall-normal locations. The energy distribution of the heat-flux fields at the wall is concentrated on longer temporal structures with increasing Prandtl number. This is due to the thinner thermal boundary layer as thermal diffusivity decreases and, thereby, the longer temporal structures exhibit a different footprint at the wall.