We simulate the head-on collision between vortex rings with circulation Reynolds numbers of 4000 using an adaptive, multiresolution solver based on the lattice Green's function. The simulation fidelity is established with integral metrics representing symmetries and discretization errors. Using the velocity gradient tensor and structural features of local streamlines, we characterize the evolution of the flow with a particular focus on its transition and turbulent decay. Transition is excited by the development of the elliptic instability, which grows during the mutual interaction of the rings as they expand radially at the collision plane. The development of antiparallel secondary vortex filaments along the circumference mediates the proliferation of small-scale turbulence. During turbulent decay, the partitioning of the velocity gradients approaches an equilibrium that is dominated by shearing and agrees well with previous results for forced isotropic turbulence. We also introduce new phase spaces for the velocity gradients that reflect the interplay between shearing and rigid rotation and highlight geometric features of local streamlines. In conjunction with our other analyses, these phase spaces suggest that, while the elliptic instability is the predominant mechanism driving the initial transition, its interplay with other mechanisms, e.g. the Crow instability, becomes more important during turbulent decay. Our analysis also suggests that the geometry-based phase space may be promising for identifying the effects of the elliptic instability and other mechanisms using the structure of local streamlines. Moving forward, characterizing the organization of these mechanisms within vortices and universal features of velocity gradients may aid in modelling turbulent flows.

We examined the influences of slip parameters on the velocity and thermal characteristics of a ferrofluid film of fixed thickness. The flow is generated on a rough and inclined whirling surface that is positioned in an external magnetic (dipole) field. The similarity transformation reduces the model equations (continuity, momentum, energy and concentration), and the solution of the normalized coupled ordinary differential equations is carried out through the finite element process. The influences of slip effects, Brownian motion, thermophoresis and a heat source on the velocity (radial, tangential and axial), gravity (drainage, induced), temperature profile and concentration profile are determined. The tangential flow and temperature are both decreased by an increase in the velocity slip parameter, whereas drainage, induced, radial and axial flows are increased. Enlarging the thermal slip parameter decreases the temperature. Improving slip parameters (velocity and thermal) also improves the concentration profile. Both Nusselt and Sherwood numbers are found to improve on improving the velocity slip parameter, while they decrease on decreasing the thermal slip parameter. The results and insights from this work could be applied to a wide range of medicinal fields, such as targeted medication therapy and delivery, tissue engineering, etc. as well as different industrial processes including coating, lubrication, heat transfer, etc.

Under adverse pressure gradient (APG) conditions, the outer regions of turbulent boundary layers (TBLs) are characterized by an increased velocity defect $U_{e}-U$, an outwards shift of the peak value of the Reynolds shear stress $-\langle uv\rangle$ and an appearance of the outer peak value of the Reynolds normal stress $\langle uu\rangle$. Here $U_{e}$ is the TBL edge velocity. Scaling APG TBLs is challenging due to the non-equilibrium effects caused by changes in the APG. To address this, the response distance of TBLs to non-equilibrium conditions is utilized to extend the Zagarola–Smits scaling $U_{zs} = U_{e}({\delta ^{*} }/{\delta })$ and ensure that the original properties of the Zagarola–Smits scaling are maintained as $Re \to \infty$. Here $\delta ^{*}$ is the displacement thickness and $\delta$ is the boundary layer thickness. Based on the established correlation between $U_{e}-U$ and $-\langle uv\rangle$, the scaling is extended to $-\langle uv\rangle$. Furthermore, considering the coupling relationship between Reynolds stress components, the scaling is extended to encompass each Reynolds stress component. The proposed consistent scaling is verified using five non-equilibrium databases and five near-equilibrium databases, successfully collapsing the data of the TBL outer region. The pressure gradient parameter $\beta =({\delta ^{*} }/{\rho u_{\tau }^{2} }) ({\mathrm {d} P_{e} }/{\mathrm {d}\kern0.7pt x})$ of these databases spans two orders of magnitude. Here $P_{e}$ is the boundary layer edge pressure, $u_{\tau }$ is the friction velocity and $\rho$ is the density. Finally, the influence of the APG on the inner and outer regions of TBLs is analysed using the mean momentum balance equation. The analysis suggests that the shift of the $-\langle uv\rangle$ peak to the outer region under APG conditions is due to an insufficient inertia term near the inner region to balance the APG. It is observed that the APG promotes interaction between the inner and outer regions of TBLs, but the inner and outer regions still retain distinctive properties.

The present work focuses on the effect of rough horizontal boundaries on the heat transfer in rotating Rayleigh–Bénard convection. We measure the non-dimensional heat transfer, the Nusselt number $Nu$, for various strengths of the buoyancy forcing characterized by the Rayleigh number $Ra$ (${10^5}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }Ra\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }5 \times {10^8}$), and rotation rates characterized by the Ekman number E ($1.4 \times {10^{ - 5}}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }E\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }7.6 \times {10^{ - 4}}$) for aspect ratios $\varGamma \approx 1$, $2.8$ and $6.7$. Similar to rotating convection with smooth horizontal boundaries, the so-called rotationally constrained (RC), rotation-affected (RA) and rotation-unaffected (RuA) regimes of heat transfer seem to persist for rough horizontal boundaries. However, the transition from the RC regime to RA regime occurs at a lower Rayleigh number for rough boundaries. For all experiments with rough boundaries in this study, the thermal and Ekman boundary layers are in a perturbed state, leading to a significant enhancement in the heat transfer as compared with that for smooth walls. However, the enhancement in heat transfer due to wall roughness is observed to attain a maximum in the RC regime. We perform companion direct numerical simulations of rotating convection over smooth walls to suggest a phenomenology explaining this observation. We propose that the heat transfer enhancement due to wall roughness reaches a maximum when the strength and coherence of the columnar structures are both significant, which enables efficient vertical transport of the additional thermal anomalies generated by the roughness at the top and bottom walls.

Subgrid-scale (SGS) modelling is formulated using a local transport of spectral kinetic energy estimated by a wavelet multiresolution analysis. Using a spectrally and spatially local decomposition by wavelet, the unresolved inter-scale energy transfer and modelled SGS dissipation are evaluated to enforce explicitly and optimally their balance a priori over a range of large-eddy simulation (LES) filter widths. The formulation determines SGS model constants that optimally describe the spectral energy balance between the resolved and unresolved scales at a given cutoff scale. The formulation is tested for incompressible homogeneous isotropic turbulence (HIT). One-parameter Smagorinsky- and Vreman-type eddy-viscosity closures are optimised for their model constants. The algorithm discovers the theoretical prediction of Lilly (The representation of small-scale turbulence in numerical simulation experiments. In Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences, pp. 195–210) at a filter cutoff scale in the inertial subrange, whereas the discovered constants deviate from the theoretical value at other cutoff scales so that the spectral optimum is achieved. The dynamic Smagorinsky model used a posteriori shows a suboptimal behaviour at filter scales larger than those in the inertial subrange. A two-parameter Clark-type closure model is optimised. The optimised constants provide evidence that the nonlinear gradient model of Clark et al. (J. Fluid Mech., vol. 91, issue 1, 1979, pp. 1–16) is prone to numerical instability due to its model form, and combining the pure gradient model with a dissipative model such as the classic Smagorinsky model enhances numerical stability but the standard mixed model is not optimal in terms of spectral energy transfer. A posteriori analysis shows that the optimised SGS models produce accurate LES results.

To investigate the influence of inertia and slip on the instability of a liquid film on a fibre, a theoretical framework based on the axisymmetric Navier–Stokes equations is proposed via linear instability analysis. The model reveals that slip significantly enhances perturbation growth in viscous film flows, whereas it exerts minimal influence on flows dominated by inertia. Moreover, under no-slip boundary conditions, the dominant instability mode of thin films remains unaltered by inertia, closely aligning with predictions from a no-slip lubrication model. Conversely, when slip is introduced, the dominant wavenumber experiences a noticeable reduction as inertia decreases. This trend is captured by an introduced lubrication model with giant slip. Direct numerical simulations of the Navier–Stokes equations are then performed to further confirm the theoretical findings at the linear stage. For the nonlinear dynamics, no-slip simulations show complex vortical structures within films, driven by fluid inertia near surfaces. Additionally, in scenarios with weak inertia, a reduction in the volume of satellite droplets is observed due to slip, following a power-law relationship.

We develop a theoretical and experimental framework for generating slip underneath thin-film flows of viscous fluids in the laboratory, with the ability to control slip as desired. Such a framework is useful for large-scale fluid-mechanical experiments in which basal sliding is important. In particular, we consider the flow of a thin film of viscous fluid spreading over a structured, slippery substrate, involving a sequence of two-dimensional cavities that are prewetted with a fluid of smaller viscosity. By averaging over small-scale inhomogeneities, we demonstrate that such a substrate gives rise to a macroscopic linear sliding law, or Navier slip condition, that is effectively homogeneous on the large scale. The slip length, determining the slipperiness of the substrate, is proportional to the viscosity ratio and width of each cavity. As such, the slipperiness of the substrate can be controlled by altering the viscosity ratio, as desired. Two asymptotic regimes arise, describing flow over very slippery substrates and flow over no-slip substrates. The former regime is valid for early times, when the depth of the overlying fluid is much less than the slip length, and the latter is valid for late times, when the depth is much greater than the slip length. Solutions to the full model approach similarity solutions describing the two regimes for early and late times. We confirm our theoretical predictions by conducting a series of analogue laboratory experiments.

We investigate the Faraday instabilities of a three-layer fluid system in a cylindrical container containing low-viscosity liquid metal, sodium hydroxide solution and air by establishing the Mathieu equations with considering the viscous model derived by Labrador et al. (J. Phys.: Conf. Ser., vol. 2090, 2021, 012088). The Floquet analysis, asymptotic analysis, direct numerical simulation and experimental method are adopted in the present study. We obtain the dispersion relations and critical oscillation amplitudes of zigzag and varicose modes from the analysis of the Mathieu equations, which agree well with the experimental result. Furthermore, considering the coupling strength of two interfaces, besides zigzag and varicose modes, we find a beating instability mode that contains two primary frequencies, with its average frequency equalling half of the external excitation frequency in the strongly coupled system. In the weakly coupled system, the $A$-interface instability, $B$-interface instability and $A$&$B$-interface instability are defined. Finally, we obtain a critical wavenumber $k_c$ that can determine the transition from zigzag or varicose modes to the corresponding $A$-interface or $B$-interface instability.

Over the last decade, substantial progress has been made in understanding the topology of quasi-two-dimensional (2-D) non-equilibrium fluid flows driven by ATP-powered microtubules and microorganisms. By contrast, the topology of three-dimensional (3-D) active fluid flows still poses interesting open questions. Here, we study the topology of a spherically confined active flow using 3-D direct numerical simulations of generalized Navier–Stokes (GNS) equations at the scale of typical microfluidic experiments. Consistent with earlier results for unbounded periodic domains, our simulations confirm the formation of Beltrami-like bulk flows with spontaneously broken chiral symmetry in this model. Furthermore, by leveraging fast methods to compute linking numbers, we explicitly connect this chiral symmetry breaking to the entanglement statistics of vortex lines. We observe that the mean of linking number distribution converges to the global helicity, consistent with the asymptotic result by Arnold [In Vladimir I. Arnold – Collected Works (ed. A.B. Givental, B.A. Khesin, A.N. Varchenko, V.A. Vassiliev & O.Y. Viro), pp. 357–375. Springer]. Additionally, we characterize the rate of convergence of this measure with respect to the number and length of observed vortex lines, and examine higher moments of the distribution. We find that the full distribution is well described by a k-Gamma distribution, in agreement with an entropic argument. Beyond active suspensions, the tools for the topological characterization of 3-D vector fields developed here are applicable to any solenoidal field whose curl is tangent to or cancels at the boundaries of a simply connected domain.

We experimentally investigate the forced synchronization of a self-excited chaotic thermoacoustic oscillator with two natural frequencies, $f_1$ and $f_2$. On increasing the forcing amplitude, $\epsilon _f$, at a fixed forcing frequency, $f_f$, we find two different types of synchronization: (i) $f_f/f_1 = 1:1$ or $2:1$ chaos-destroying synchronization (CDS), and (ii) phase synchronization of chaos (PSC). En route to $1:1$ CDS, the system transitions from an unforced chaotic state (${\rm {CH}}_{1,2}$) to a forced chaotic state (${\rm {CH}}_{1,2,f}$), then to a two-frequency quasiperiodic state where chaos is destroyed ($\mathbb {T}^2_{2,f}$), and finally to a phase-locked period-1 state (${\rm {P1}}_f$). The route to $2:1$ CDS is similar, but the quasiperiodic state hosts a doubled torus $(2\mathbb {T}^2_{2,f})$ that transforms into a phase-locked period-2 orbit $({\rm {P2}}_f)$ when CDS occurs. En route to PSC, the system transitions to a forced chaotic state (${\rm {CH}}_{1,2,f}$) followed by a phase-locked chaotic state, where $f_1$, $f_2$ and $f_f$ still coexist but their phase difference remains bounded. We find that the maximum reduction in thermoacoustic amplitude occurs near the onset of CDS, and that the critical $\epsilon _f$ required for the onset of CDS does not vary significantly with $f_f$. We then use two unidirectionally coupled Anishchenko–Astakhov oscillators to phenomenologically model the experimental synchronization dynamics, including (i) the route to $1:1$ CDS, (ii) various phase dynamics, such as phase drifting, slipping and locking, and (iii) the thermoacoustic amplitude variations in the $f_f/f_1$–$\epsilon _f$ plane. This study extends the applicability of open-loop control further to a chaotic thermoacoustic system, demonstrating (i) the feasibility of using an existing actuation strategy to weaken aperiodic thermoacoustic oscillations, and (ii) the possibility of developing new active suppression strategies based on both established and emerging methods of chaos control.