Turbulent motions enhance the diffusion of large-scale flows and temperature gradients. Such diffusion is often parameterized by coefficients of turbulent viscosity ($\nu _{t}$) and turbulent thermal diffusivity ($\chi _{t}$) that are analogous to their microscopic counterparts. We compute the turbulent diffusion coefficients by imposing sinusoidal large-scale velocity and temperature gradients on a turbulent flow and measuring the response of the system. We also confirm our results using experiments where the imposed gradients are allowed to decay. To achieve this, we use weakly compressible three-dimensional hydrodynamic simulations of isotropically forced homogeneous turbulence. We find that the turbulent viscosity and thermal diffusion, as well as their ratio the turbulent Prandtl number, $\textit {Pr}_{t} = \nu _{t}/\chi _{t}$, approach asymptotic values at sufficiently high Reynolds and Péclet numbers. We also do not find a significant dependence of $\textit {Pr}_{t}$ on the microscopic Prandtl number $\textit {Pr} = \nu /\chi$. These findings are in stark contrast to results from the $k{-}\epsilon$ model, which suggests that $\textit {Pr}_{t}$ increases monotonically with decreasing $\textit {Pr}$. The current results are relevant for the ongoing debate on, for example, the nature of the turbulent flows in the very-low-$\textit {Pr}$ regimes of stellar convection zones.

We consider nonlinear dynamical systems driven by stochastic forcing. It has been largely evidenced in the literature that the linear response of non-normal systems (e.g. fluid flows) may exhibit a large variance amplification, even in a linearly stable regime. This linear response, however, is relevant only in the limit of vanishing forcing intensity. As the intensity increases, the frequency distribution and the variance of the response may be strongly modified due to nonlinear effects. To go beyond this limitation, we propose a theoretical approach to derive an amplitude equation governing the weakly nonlinear evolution of these systems. This approach does not rely on the presence of an eigenvalue close to the neutral axis, applying instead to any sufficiently non-normal operator, and the Fourier components of the response are allowed to be arbitrarily different from any eigenmode. The methodology is outlined for a generic nonlinear dynamical system, and the application case highlights a common non-normal mechanism in hydrodynamics: convective non-normal amplification in the flow past a backward-facing step.

An isolated charge-neutral droplet in a uniform electric field experiences no net force. However, a droplet pair can move in response to field-induced dipolar and hydrodynamic interactions. If the droplets are identical, the centre of mass of the pair remains fixed. Here, we show that if the droplets have different properties, the pair experiences a net motion due to non-reciprocal electrohydrodynamic interactions. We analyse the three-dimensional droplet trajectories using asymptotic theory, assuming spherical droplets and large separations, and numerical simulations based on a boundary integral method. The dynamics can be quite intricate depending on the initial orientation of the droplets line-of-centres relative to the applied field direction. Drops tend to migrate towards a configuration with line-of-centres either parallel or perpendicular to the applied field direction, while either coming into contact or indefinitely separating. We elucidate the conditions under which these different interaction scenarios take place. Intriguingly, we find that in some cases droplets can form a pair (tandem) that translates either parallel or perpendicular to the applied field direction.

We present an experimental study of Rayleigh–Bénard convection using liquid metal alloy gallium-indium-tin as the working fluid with a Prandtl number of $Pr=0.029$. The flow state and the heat transport were measured in a Rayleigh number range of $1.2\times 10^{4} \le Ra \le 1.3\times 10^{7}$. The temperature fluctuation at the cell centre is used as a proxy for the flow state. It is found that, as $Ra$ increases from the lower end of the parameter range, the flow evolves from a convection state to an oscillation state, a chaotic state and finally a turbulent state for $Ra>10^5$. The study suggests that the large-scale circulation in the turbulent state is a residual of the cell structure near the onset of convection, which is in contrast with the case of $Pr\sim 1$, where the cell structure is transiently replaced by high order flow modes before the emergence of the large-scale circulation in the turbulent state. The evolution of the flow state is also reflected by the heat transport characterised by the Nusselt number $Nu$ and the probability density function (p.d.f.) of the temperature fluctuation at the cell centre. It is found that the effective local heat transport scaling exponent $\gamma$, i.e. $Nu\sim Ra^{\gamma }$, changes continuously from $\gamma =0.49$ at $Ra\sim 10^4$ to $\gamma =0.25$ for $Ra>10^6$. Meanwhile, the p.d.f. at the cell centre gradually evolves from a Gaussian-like shape before the transition to turbulence to an exponential-like shape in the turbulent state. For $Ra>10^6$, the flow shows self-similar behaviour, which is revealed by the universal shape of the p.d.f. of the temperature fluctuation at the cell centre and a $Nu=0.19Ra^{0.25}$ scaling for the heat transport.

The shape of a sessile drop on a horizontal substrate depends upon the Bond number $Bo$ and the contact angle $\alpha$. Inspired by puddle approximations at large $Bo$ (Quéré, Rep. Prog. Phys., vol. 68, 2005, p. 2495), we address here the limit of small contact angles at fixed drop volume and arbitrary $Bo$. It readily leads to a pancake shape approximation, where the drop height and radius scale as $\alpha$ and $\alpha ^{-1/2}$, respectively, with capillary forces being appreciable only near the edge. The pancake approximation breaks down for $Bo=\textrm {ord}(\alpha ^{2/3})$. In that distinguished limit, capillary and gravitational forces are comparable throughout, and the drop height and radius scale as $\alpha ^{2/3}$ and $\alpha ^{-1/3}$, respectively. For $Bo\ll \alpha ^{2/3}$ these scalings remain, with the drop shape turning into a spherical cap. The asymptotic results are compared with a numerical solution of the exact problem.

It has been reported that a fully localized turbulent band in channel flow becomes sustained when the Reynolds number is above a threshold. Here we show evidence that turbulent bands are of a transient nature instead. When the band length is controlled to be fixed, the lifetime of turbulent bands appears to be stochastic and exponentially distributed, a sign of a memoryless transient nature. Besides increasing with the Reynolds number, the mean lifetime also strongly increases with the band length. Given that the band length always changes over time in real channel flow, this size dependence may translate into a time dependence, which needs to be taken into account when clarifying the relationship between channel flow transition and the directed percolation universality class.

We investigate the modulation of turbulence caused by the presence of finite-size dispersed particles. Bluff (isotropic) spheres versus slender (anisotropic) fibres are considered to understand the influence of the shape of the objects on altering the carrier flow. While at a fixed mass fraction – but different Stokes number – both objects provide a similar bulk effect characterized by a large-scale energy depletion, a scale-by-scale analysis of the energy transfer reveals that the alteration of the whole spectrum is intrinsically different. For bluff objects, the classical energy cascade shrinks in its extension but is unaltered in the energy content and its typical features, while for slender ones we find an alternative energy flux which is essentially mediated by the fluid–solid coupling.

Pattern-forming with externally imposed symmetry is ubiquitous in nature but little studied. We present experimental studies of pattern formation and selection by spatial periodic forcing in rapidly rotating convection. When periodic topographic structures are constructed on the heated boundary, they modulate the local temperature and velocity fields. Symmetric convection patterns in the form of regular vortex lattices are observed near the onset of convection, when the periodicity of the external forcing is set close to the intrinsic vortex spacing. We show that the new patterns arise as a dynamical process of imperfect bifurcation which is well described by a Ginzburg–Landau-like model. We explore the phase diagram of buoyancy strength and periodicity of external forcing to find the optimal experimental settings for which the vortex patterns best match that of the external forcing.

We consider direct statistical simulation (DSS) of a paradigm system of convection interacting with mean flows. In the Busse annulus model, zonal jets are generated through the interaction of convectively driven turbulence and rotation; non-trivial dynamics including the emergence of multiple jets and bursting ‘predator–prey’ type dynamics can be found. We formulate the DSS by expanding around the mean flow in terms of equal-time cumulants and arrive at a closed set of equations of motion for the cumulants. Here, we present results using an expansion terminated at the second cumulant (CE2); it is fundamentally a quasilinear theory. We focus on particular cases including bursting and bistable multiple jets and demonstrate that CE2 can reproduce the results of direct numerical simulation if particular attention is given to symmetry considerations.

Statistically homogeneous flows obey exact kinematic relations. The Betchov homogeneity constraints (Betchov, J. Fluid Mech., vol. 1, 1956, pp. 497–504) for the average principal invariants of the velocity gradient are among the most well-known and extensively employed homogeneity relations. These homogeneity relations have far-reaching implications for the coupled dynamics of strain and vorticity, as well as for the turbulent energy cascade. Whether the Betchov homogeneity constraints are the only possible ones or whether additional homogeneity relations exist has not been proven yet. Here we show that the Betchov homogeneity constraints are the only homogeneity constraints for incompressible and statistically isotropic velocity gradient fields. Our analysis also applies to compressible/perceived velocity gradients, and it allows the derivation of homogeneity relations involving the velocity gradient and other dynamically relevant quantities, such as the pressure Hessian and viscous stresses.

Symmetry relations are derived for the added mass and damping of structures where the shape is unchanged by rotation about the vertical axis through an angle $\theta = 2 {\rm \pi}/N$ with the integer $N\ge 3$. For this type of structure, the added mass and damping for horizontal translation are the same for all directions, as in the case of axisymmetric structures. The same symmetry applies to rotations about horizontal axes. The principal application is to offshore structures and other bodies floating on the free surface or submerged, but the same symmetry relations apply more generally to unsteady body motions in an ideal fluid.

A nonlinear ensemble-variational data assimilation is performed in order to estimate the unknown flow field over a slender cone at Mach 6, from isolated wall-pressure measurements. The cost functional accounts for discrepancies in wall-pressure spectra and total intensity between the experiment and the prediction using direct numerical simulations, as well as our relative confidence in the measurements and the estimated state. We demonstrate the robustness of the predicted flow by direct propagation of posterior statistics. The approach provides a unique first look at the flow beyond the sensor data, and rigorously accounts for the role of nonlinearity, unlike previous efforts that adopted ad hoc inflow syntheses. Away from the wall, two- and three-dimensional assimilated states both show rope-like structures, qualitatively similar to independent schlieren visualizations. Despite this resemblance, and even though the planar second modes are the most unstable upstream, three-dimensional waves must be included in the assimilation in order to accurately reproduce the wall-pressure measurements recorded in the AFRL Ludwieg Tube facility. The results highlight the importance of three-dimensionality of the field and of the base-state distortion on the instability waves in this experiment, and motivate future measurements that probe the three-dimensional nature of the flow field.

We derive from first principles analytic relations for the second- and third-order moments of $\boldsymbol{\mathsf{m}}$, the spatial gradient of fluid velocity $\boldsymbol{u}$, $\boldsymbol{\mathsf{m}} = \nabla \boldsymbol{u}$, in compressible turbulence, which generalize known relations in incompressible flows. These relations, although derived for homogeneous flows, hold approximately for a mixing layer. We also discuss how to apply these relations to determine all the second- and third-order moments of the velocity gradient experimentally for isotropic compressible turbulence.

We report on the phenomenon of detonation synchronization and demonstrate the existence of the Arnold tongues and devil's staircases in the problem of gaseous detonation in a periodically inhomogeneous reactive medium. Universal properties of these dynamical structures – the Farey tree, fractal dimension and period-doubling bifurcations – are revealed.

The effect of damping the longest streaks in wall-bounded turbulence is explored using numerical experiments. It is found that long streaks are not required for the self-sustenance of the bursting process, which is relatively little affected by their absence. In particular, there are turbulence states in which the fluctuations of the streamwise velocity have approximately the same length as the bursts, and are thus presumably associated with the bursts themselves, while the burst structure is essentially indistinguishable from flows in which longer velocity fluctuations are present. This suggests that the long streaks found in unmodified flows may be by-products, rather than active parts of the process.

Compressible mixing layer instabilities are of importance to a wide range of environmental and industrial applications. Past studies have focused on either ideal-gas or real-fluid thermodynamic regimes of single-species mixing layers. However, mixing layers of binary mixtures at supercritical conditions, commonly encountered in fuel injection systems, introduce additional complexities due to the added compositional degree-of-freedom. Moreover, the effect of strong variations in thermodynamic response functions across the Widom line on the binary mixing layer stability remains poorly understood. Thus, the objective of this study is to examine the coupling between the hydrodynamic instability and the real-fluid thermodynamics across the Widom line and its effects on the overall binary mixing layer dynamics. To this end, we develop a linear stability analysis of the full binary-species compressible transport equations coupled with the PC-SAFT equation of state. Analysis shows the existence of a novel instability mechanism that arises from juxtapositioning of the Widom-line transition and the hydrodynamic inflection point. This novel thermodynamically induced instability mechanism has the net effect of destabilizing the binary mixing layer at lowering supercritical conditions towards the critical pressure point. This is in contrast to previous stability analyses of supercritical single-species mixing layers, where increasing pressure destabilizes the flow due to its effect on reducing the density stratification. The discovered thermodynamically induced instability mechanism of binary mixing flows highlights the need for an extension of classical instability criteria to incorporate the effect of strong variations in the thermodynamic response functions across the Widom line on mixing layer instability.

Viscous streaming is an efficient mechanism to exploit inertia at the microscale for flow control. While streaming from rigid features has been thoroughly investigated, when body compliance is involved, as in biological settings, little is known. Here, we investigate body elasticity effects on streaming in the minimal case of an immersed soft cylinder. Our study reveals an additional streaming process, available even in Stokes flows. Paving the way for advanced forms of flow manipulation, we illustrate how gained insights may translate to complex geometries beyond circular cylinders.

Motivated by the variation of local shear produced by internal waves in the ocean, we use direct numerical simulations to investigate the effect of a time-dependent shear forcing on the evolution and mixing of turbulence produced by Kelvin–Helmholtz instability (KHI) at high Reynolds number. The forcing is implemented using a tilting coordinate system which causes the background shear to accelerate and decelerate periodically. We demonstrate that, with suitable timing between development of instability and the shear oscillation cycle, turbulence produced by KHI with a decelerating shear mixes in a distinctly different way from the flow with constant background shear, specifically with the energy for turbulent motions extracted from alternative sources. As a result, the total amount of mixing as measured by the change in background potential energy can in fact be significantly larger for flows in which the shear is decelerated, despite the fact that the total kinetic energy in the flow is significantly smaller. The mixing has characteristics more in common with convectively driven rather than shear-driven flows, supporting the argument for an underlying change in the mechanisms triggering the turbulence.

A simple model for the motion of shape-changing swimmers in Poiseuille flow was recently proposed and numerically explored by Omori et al. (J. Fluid Mech., vol. 930, 2022, A30). These explorations hinted that a small number of interacting mechanics can drive long-time behaviours in this model, cast in the context of the well-studied alga Chlamydomonas and its rheotactic behaviours in such flows. Here, we explore this model analytically via a multiple-scale asymptotic analysis, seeking to formally identify the causal factors that shape the behaviour of these swimmers in Poiseuille flow. By capturing the evolution of a Hamiltonian-like quantity, we reveal the origins of the long-term drift in a single swimmer-dependent constant, whose sign determines the eventual behaviour of the swimmer. This constant captures the nonlinear interaction between the oscillatory speed and effective hydrodynamic shape of deforming swimmers, driving drift either towards or away from rheotaxis.

Drop-on-demand printing applications involve a drop connected to a fluid reservoir between which volume can be exchanged, a situation that can be idealized as a sessile drop with prescribed volume flux across the drop/reservoir boundary. Here we compute the frequency spectrum for these pressure disturbances, as it depends upon the static contact-angle $\alpha$ (CA) and an empirical constant $\chi$ relating the reservoir pressure to volume exchanged, for either (i) pinned or (ii) free contact-lines. Mode shapes are characterized by the mode number pair $[k,\ell ]$ with property $k+\ell =\mathbb {Z}^{+}_{odd}$ that can be associated with the symmetry properties of the Rayleigh drop modes for the free sphere. We report instabilities to the axisymmetric $[1,0]$ and non-axisymmetric rocking $[2,1]$ modes that are related to centre-of-mass motions, and show how the spectral degeneracy of the Rayleigh drop modes breaks with the model parameters.