Small-scale topography can significantly influence large-scale motions in geophysical flows, but the dominant mechanisms underlying this complicated process are poorly understood. Here, we present a systematic experimental study of the effect of small-scale topography on zonal jets. The jet flows form under the conditions of fast rotation, a uniform background $\beta$-effect, and sink–source forcing. The small-scale topography is produced by attaching numerous small cones on the curved bottom plate, and the height of the cones is much smaller than the water depth. It is found that for all tested cases, the energy fraction in the zonal mean flow consistently follows a scaling $E_{uZ}/E_{uT}=C_1 l_f^2\epsilon _{\textit{up}}^{-2/5}\beta _{\textit{eff}}^{6/5}$, where $l_f$ is the forcing scale, $\epsilon _{\textit{up}}$ is the upscale energy transfer rate, and $\beta _{\textit{eff}}$ measures the effective $\beta$-effect in the presence of topography. The presence of the small-scale topography weakens the jet strength notably. Moreover, the effect of topography on energy transfers depends on the topography magnitude $\beta _\eta$, and there exist three regimes. At small $\beta _\eta$, the inverse energy transfers are remarkably diminished while the jet pattern remains unchanged. When $\beta _\eta$ increases, a blocked flow pattern forms, and the jet width reaches saturation, becoming independent of the forcing magnitude and $\beta$. At moderate $\beta _\eta$, the inverse energy fluxes are surprisingly enhanced. A further increase of $\beta _\eta$ leads to a greater reduction of the energy fluxes. We finally examine the effect of topography from the perspective of turbulence–topography interaction.

If a body of inviscid fluid is disturbed, it will typically eject a jet of fluid. If the effects of gravity and surface tension are negligible, these jets travel in straight lines, with the tips approaching a constant velocity. Earlier works have concentrated upon jets which result from the occurrence of shocks or singularities in the fluid flow. In this paper, by contrast, we describe the simplest case, in two dimensions: an infinitely deep body of inviscid fluid, with no surface tension or gravitational forces acting, responds to a generic impulsive disturbance. We find that, contrary to some earlier suggestions, the jet has a hyperbolic profile (away from its tip and its base).

The propagation of linear waves in non-ideal compressible fluids plays a crucial role in numerous physical and engineering applications, particularly in the study of instabilities, aeroacoustics and turbulence modelling. This work investigates linear waves in viscous and heat-conducting non-ideal compressible fluids, modelled by the Navier–Stokes–Fourier equations and a fully arbitrary equation of state (EOS). The linearised governing equations are derived to analyse the dispersion relations when the EOS differs from that of an ideal gas. Special attention is given to the influence of non-ideal effects and various dimensionless numbers on wave propagation speed and attenuation. By extending classical results from Kovásznay (1953 J. Aeronaut. Sci. vol. 20, no. 10, pp. 657–674) and Chu (1965 Acta Mech. vol. 1, no. 3, pp. 215–234) obtained under the ideal gas assumption, this study highlights the modifications introduced by arbitrary EOSs to the linear wave dynamics in non-ideal compressible flows. This work paves the path for an improved understanding and modelling of wave propagation, turbulence and linear stability in arbitrary viscous and heat-conducting fluids.

In air-entraining flows, there is often strong turbulence beneath the free surface. We consider the entrainment of bubbles at the free surface by this strong free-surface turbulence (FST). Our interest is the entrainment size distribution (per unit free surface area) $I(a)/A_{\textit{FS}}$, for bubbles with radius $a$ greater than the capillary scale ($\approx 1.3\ \mathrm{mm}$ for air–water on Earth), where gravity dominates surface tension. We develop a mechanistic model based on entrained bubble size being proportional to the minimum radius of curvature of the initial surface deformation. Using direct numerical simulation of a flow that isolates entrainment by FST, we show that, consistent with our mechanism, $I(a)/A_{\textit{FS}} = C_I \, g^{-3} \varepsilon ^{7/3} (2 a)^{-14/3}$, where $g$ is gravity, and $\varepsilon$ is the turbulence dissipation rate. In the limit of negligible surface tension, $C_I\approx 3.62$, and we describe how $C_I$ decreases with increasing surface tension. This scaling holds for sufficiently strong FST such that near-surface turbulence is nearly isotropic, which we show is true for turbulent Froude number ${\textit{Fr}}^2_T = \varepsilon /u_{\textit{rms}} g \gt 0.1$. While we study FST entrainment in isolation, our model corroborates previous numerical results from shear-driven flow, and experimental results from open-channel flow, showing that the FST entrainment mechanism that we elucidate can be important in broad classes of air-entraining flows.

Pendant drops appear in many engineering applications, such as inkjet printing and optical tensiometry, and they have also been the subject of studies of droplet–particle interaction. While the hydrostatics of pendant drops has been studied extensively, the influence of external flow disturbances has received limited attention. This research aims to incorporate aerodynamic factors into the understanding of pendant drop behaviour. Employing a simplified model, an irrotational flow aligned with the drop’s axis is derived from a distribution of singularity elements within the drop. The drop’s equilibrium shape is then determined using a numerical model that couples the flow field with the Young–Laplace equation. The model’s predictions are compared to droplet images captured via high-speed shadowgraph in a vertical wind tunnel, showing good agreement with the experimentally observed shapes. Additionally, under certain flow conditions, the drop exhibits instability in the form of periodic pendulum-like motion. This instability was linked to two distinct critical drop heights, and the corresponding stability criterion was mathematically derived from the numerical model. Our theoretical and experimental findings provide the first quantitative description of the equilibrium shape and stability criterion of pendant drops under the influence of external flow.

We investigate interactions between two like-signed vortices over either an isolated seamount or a basin (a depression in the bathymetry), using a quasi-geostrophic, two-layer model on the $f$-plane. When the vortex pair is centred over the seamount, the vortices are pushed together by the secondary flow generated in the bottom layer, facilitating their merger. Over a basin, the deep anomalies are much stronger and their interaction strains out the surface vortices. The results are supported by an analytical estimation of the initial potential vorticity anomalies in the lower layer and by analysis of the linear stability of a single vortex over the bathymetry. Similar phenomena are observed when the vortex pair is displaced from the bathymetric centre and when the initial vortices are initially compensated. Sub-deformation-scale vortices are less influenced by bathymetry than larger vortices. The results help explain asymmetries noted previously in turbulence simulations over bathymetry.

Experiments have shown that ultrasound-stimulated microbubbles can translate through gel phantoms and tissues, leaving behind tunnel-like degraded regions. A computational model is used to examine the tunnelling mechanisms in a model material with well-defined properties. The high strain rates motivate the neglect of weak elasticity in favour of viscosity, which is taken to degrade above a strain threshold. The reference parameters are motivated by a 1 $\unicode{x03BC}$m diameter bubble in a polysaccharide gel tissue phantom. This is a reduced model and data are scarce, so close quantitative agreement is not expected, but tunnels matching observations do form at realistic rates, which provides validation sufficient to analyse potential mechanisms. Simulations of up to 100 acoustic cycles are used to track tunnelling over 10 bubble diameters, including a steady tunnelling phase during which tunnels extend each forcing cycle in two steps: strain degrades the tunnel front during the bubble expansion, and then the bubble is drawn further along the tunnel during its subsequent inertial collapse. Bubble collapse jetting is damaging, though it is only observed during a transient for some initial conditions. There is a threshold behaviour when the viscosity of the undamaged material changes the character of the inertial bubble oscillation. Apart from that, the tunnel growth rate is relatively insensitive to the high viscosity of the material. Higher excitation amplitudes and lower frequencies accelerate tunnelling. That acoustic radiation force, elasticity and bubble jetting are not required is a principal conclusion.

Large numbers of relative periodic orbits (RPOs) have been found recently in doubly periodic, two-dimensional Kolmogorov flow at moderate Reynolds numbers ${\textit{Re}} \in \{40, 100\}$. While these solutions lead to robust statistical reconstructions at the ${\textit{Re}}$ values where they were obtained, it is unclear how their dynamical importance changes with ${\textit{Re}}$. Arclength continuation on this library of solutions reveals that large numbers of RPOs quickly become dynamically irrelevant, reaching dissipation values either much larger or smaller than the values typical of the turbulent attractor at high ${\textit{Re}}$. The scaling of the high-dissipation RPOs is shown to be consistent with a direct connection to solutions of the unforced Euler equation, and is observed for a wide variety of states beyond the ‘unimodal’ solutions considered in previous work (Kim & Okamoto, Nonlinearity vol. 28, 2015, p. 3219). However, the weakly dissipative states have properties indicating a connection to exact solutions of a forced Euler equation. The dynamical irrelevance of many solutions leads to poor statistical reconstruction at higher ${\textit{Re}}$, raising serious questions for the future use of RPOs for estimating probability densities. Motivated by the Euler connection of some of our RPOs, we also show that many of these states can be well described by exact relative periodic solutions in a system of point vortices. The point vortex RPOs are converged via gradient-based optimisation of a scalar loss function which (i) matches the dynamics of the point vortices to the turbulent vortex cores and (ii) insists the point vortex evolution is itself time-periodic.

Compliant walls made from homogeneous viscoelastic materials may attenuate the amplification of Tollmien–Schlichting waves (TSWs) in a two-dimensional boundary-layer flow, but they also amplify travelling-wave flutter (TWF) instabilities at the interface between the fluid and the solid, which may lead to a premature laminar-to-turbulent transition. To mitigate the detrimental amplification of TWF, we propose to design compliant surfaces using phononic structures that aim at avoiding the propagation of elastic waves in the solid in the frequency range corresponding to the TWF. Thus, stiff inserts are periodically incorporated into the viscoelastic wall in order to create a band gap in the frequency spectrum of the purely solid modes. Fluid–structural resolvent analysis shows that a significant reduction in the amplification peak related to TWF is achieved while only marginal deterioration in the control of TSWs is observed. This observation suggests that the control of TSWs is still achieved by the overall compliance of the wall, while the periodic inserts inhibit the amplification of TWF. Bloch analysis is employed to discuss the propagation of elastic waves in the phononic surface to deduce design principles, accounting for the interaction with the flow.

Laminar–turbulent transition in shear flow is complicated and follows many possible routes. In this study, we seek to examine a scenario based on three-dimensional (3-D) waves (Jiang et al., 2020, J. Fluid Mech., vol. 890, A11) in compressible mixing layers, and elucidate the role of 3-D waves in generating streamwise vorticity. The Eulerian–Lagrangian coupled method is used to track the evolution of flow structures. Qualitative evidence shows that localised 3-D waves travel coherently with vortex structures at the early transition stage, which is consistent with the behaviours of 3-D waves in boundary layer transitions. To examine the local flow events surrounding 3-D waves and investigate the cause and effect relationships inherent in wave–vortex interaction, the finite-time Lyapunov exponent and components of the strain rate tensor are integrated into evolving Lagrangian material surfaces. The formation of high-shear layers in the flanks of the 3-D waves is observed, driven by fluid ejection and sweep motions induced by the amplification of 3-D waves. The $\Lambda$-shaped vortices are found born in the vicinity of high-shear regions and then stretched into hairpin-shaped vortices farther downstream. Statistical findings reveal that streamwise vorticity develops concurrently with the significant growth of the oblique mode, while the normal motion of wave structures induces a high strain rate layer in the surrounding region. In addition, conditional statistics underscore the significance of high shear in enstrophy generation. Finally, a conceptual model is proposed to depict the evolution of coherent structures based on the relationship among the 3-D waves, high-shear/strain layers, and $\varLambda$-vortices, providing insights into their collective dynamics within transitional mixing layers.

Finite-amplitude spiral vortex flows are obtained numerically for the Taylor–Couette system in the narrow limit of the gap between two concentric rotating cylinders. These spiral vortex flows bifurcate from circular Couette flow before axisymmetric Taylor vortex flow sets in when the ratio $\mu$ of the angular velocities of the outer to the inner cylinder is less than −0.78, consistent with the results of linear stability analysis by Krueger et al. (J. Fluid Mech., vol. 24, 1966, pp. 521–538), while the boundary of existence of spiral vortex flows is determined not by the linear critical point, but by the saddle-node point of the subcritical spiral vortex flow branch for $\mu \lessapprox -0.75$, when the axial wavenumber $\beta =2.0$. It is found that the nonlinear spiral vortex flows exhibit the mean flow in the axial direction as well as in the azimuthal direction, and that the profiles of both mean-flow components are asymmetric about the centre plane between the gap.

In this work the fascinating dynamics of a two-layered channel flow characterised by the dispersion in composite media within its layers is investigated in depth. The top layer comprises of a fluid zone that allows the fluid to travel along its surface easily (with relatively higher velocity), while the bottom layer is packed with porous media. The primary objective of this research is to do an in-depth investigation of the complex two-dimensional concentration distribution of a passive solute discharged from the inflow region. A multi-scale perturbation analysis approach has been implemented to address the system’s inherent complexity. This accurate determination of the dispersion coefficient, mean concentration distribution and two-dimensional concentration distribution is accomplished deftly using Mei’s homogenisation approach up to second-order approximation, which satisfactorily capture the minor variations in the solute dynamics also. The influence of various flow and porous media elements on these basic parameters is thoroughly investigated, expanding our comprehension of the complex interaction between flow dynamics and porous media’s properties. The effect of Darcy number and the ratio of two viscosities ($M$) on the dispersion coefficient depends on the height of the porous layer. As the Péclet number ratio increases, the dispersion coefficient experiences a concurrent increase, resulting in a decline in the concentration peak. The results of the analytical studies have also been compared with those results obtained using a purely computational method to establish the validity of our studies. Both the sets of results show quite good agreement with each other. In this study, alternate flow models have been used for the porous region, and the outcomes are compared to determine which approach yields more suitable results under different conditions.

The paper discusses the stochastic dynamics of the vortex shedding process in the presence of external harmonic excitation and coloured multiplicative noise. The situation is encountered in a turbulent practical combustor experiencing combustion instability. Acoustic feedback and turbulent flow are imitated by the harmonic and stochastic excitations, respectively. The Ornstein–Uhlenbeck process is used to generate the noise. A low-order model for vortex shedding is used. The Fokker–Planck framework is used to obtain the evolution of the probability density function of the shedding time period. Stochastic lock-in and resonance characteristics are studied for various parameters associated with the harmonic (amplitude, frequency) and noise (amplitude, correlation time, multiplicative noise factor) excitations. We observed that: (i) the stochastic lock-in (s-lock-in) boundary strongly depends on the noise correlation time; (ii) the parameter sites for s-lock-in can be approximately identified from the noise-induced shedding statistics; and (iii) stochastic resonance is significant for some intermediate correlation times. The effects of the above-mentioned observations are discussed in the context of combustion instability.