We report on a numerical study of turbulent convection driven by a combination of internal heat sources and sinks. Motivated by a recent experimental realisation (Lepot etal., Proc. Natl Acad. Sci. USA, vol. 115 (36), 2018, pp. 8937–8941), we focus on the situation where the cooling is uniform, while the internal heating is localised near the bottom boundary, over approximately one tenth of the domain height. We obtain scaling laws ${Nu} \sim {Ra} ^{\gamma } {Pr}^{\chi }$ for the heat transfer as measured by the Nusselt number ${Nu}$ expressed as a function of the Rayleigh number ${Ra}$ and the Prandtl number ${Pr}$. After confirming the experimental value $\gamma \approx 1/2$ for the dependence on ${Ra}$, we identify several regimes of dependence on ${Pr}$. For a stress-free bottom surface and within a range as broad as ${Pr} \in [0.003, 10]$, we observe the exponent $\chi \approx 1/2$, in agreement with Spiegel's mixing-length theory. For a no-slip bottom surface we observe a transition from $\chi \approx 1/2$ for ${Pr} \leq 0.04$ to $\chi \approx 1/6$ for ${Pr} \geq 0.04$, in agreement with scaling predictions by Bouillaut etal. (J. Fluid Mech. vol. 861, 2019, R5). The latter scaling regime stems from heat accumulation in the stagnant layer adjacent to a no-slip bottom boundary, which we characterise by comparing the local contributions of diffusive and convective thermal fluxes.

We explore the near-resonant interaction of inertial waves with geostrophic modes in rotating fluids via numerical and theoretical analysis. When a single inertial wave is imposed, we find that some geostrophic modes are unstable above a threshold value of the Rossby number $kRo$ based on the wavenumber and wave amplitude. We show this instability to be caused by triadic interaction involving two inertial waves and a geostrophic mode such that the sum of their eigenfrequencies is non-zero. We derive theoretical scalings for the growth rate of this near-resonant instability. The growth rate scaled by the global rotation rate is proportional to $(kRo)^2$ at low $kRo$ and transitions to a $kRo$ scaling for larger $kRo$. These scalings are in excellent agreement with direct numerical simulations. This instability could explain recent experimental observations of geostrophic instability driven by waves.

The dynamics of a stably and thermally stratified fluid-filled cavity harmonically forced in the vertical direction, resulting in a periodic gravity modulation, is studied numerically. Prior simulations in a two-dimensional cavity showed a myriad of complex dynamic behaviours near the onset of instabilities, and here we address the extent to which these persist in three dimensions. Focusing on a parameter regime where the primary subharmonic mode is resonantly driven, we demonstrate comprehensive qualitative agreement between the dynamics in two and three dimensions; the quantitative difference is due to the larger forcing amplitudes needed in three dimensions to overcome the additional viscous damping from the spanwise walls. Using a small detuning of the forcing frequency, together with a relatively large forcing amplitude, leads to a wave-breaking regime where the qualitative agreement between two and three dimensions breaks down.

Reconnection plays a significant role in the dynamics of plasmas, polymers and macromolecules, as well as in numerous laminar and turbulent flow phenomena in both classical and quantum fluids. Extensive studies in quantum vortex reconnection show that the minimum separation distance $\delta$ between interacting vortices follows a $\delta(t) \sim t^{1/2}$ scaling. Due to the complex nature of the dynamics (e.g. the formation of bridges and threads as well as successive reconnections and avalanche), such scaling has never been reported for (classical) viscous vortex reconnection. Using direct numerical simulation of the Navier–Stokes equations, we study viscous reconnection of slender vortices, whose core size is much smaller than the radius of the vortex curvature. For separations that are large compared to the vortex core size, we discover that $\delta (t)$ between the two interacting viscous vortices surprisingly also follows the 1/2-power scaling for both pre- and post-reconnection events. The prefactors in this 1/2-power law are found to depend not only on the initial configuration but also on the vortex Reynolds number (or viscosity). Our finding in viscous reconnection, complementing numerous works on quantum vortex reconnection, suggests that there is indeed a universal route for reconnection – an essential result for understanding the various facets of the vortex reconnection phenomena and their potential modelling, as well as possibly explaining turbulence cascade physics.

Linearisation of the Navier–Stokes equations about the mean of a turbulent flow forms the foundation of popular models for energy amplification and coherent structures, including resolvent analysis. While the Navier–Stokes equations can be equivalently written using many different sets of dependent variables, we show that the properties of the linear operator obtained via linearisation about the mean depend on the variables in which the equations are written prior to linearisation, and can be modified under nonlinear transformation of variables. For example, we show that using primitive and conservative variables leads to differences in the singular values and modes of the resolvent operator for turbulent jets, and that the differences become more severe as variable-density effects increase. This lack of uniqueness of mean-flow-based linear analysis provides new opportunities for optimising models by specific choice of variables while also highlighting the importance of carefully accounting for the nonlinear terms that act as a forcing on the resolvent operator.

In the preparation of café latte, spectacular layer formation can occur between the espresso shot in a glass of milk and the milk itself. Xue et al. (Nat. Commun., vol. 8, 2017, pp. 1–6) showed that the injection velocity of espresso determines the depth of coffee–milk mixture. After a while, when a stable stratification forms in the mixture, the layering process can be modelled as a double diffusive convection system with a stably stratified coffee–milk mixture cooled from the side. More specifically, we perform (two-dimensional) direct numerical simulations of laterally cooled double diffusive convection for a wide parameter range, where the convective flow is driven by a lateral temperature gradient while stabilized by a vertical concentration gradient. Depending on the strength of stabilization as compared to the thermal driving, the system exhibits different flow regimes. When the thermal driving force dominates over the stabilizing force, the flow behaves like vertical convection in which a large-scale circulation develops. However, with increasing strength of the stabilizing force, a meta-stable layered regime emerges. Initially, several vertically-stacked convection rolls develop, and these well-mixed layers are separated by sharp interfaces with large concentration gradients. The initial thickness of these emerging layers can be estimated by balancing the work exerted by thermal driving and the required potential energy to bring fluid out of its equilibrium position in the stably stratified fluid. In the layered regime, we further observe successive layer merging, and eventually only a single convection roll remains. We elucidate the following merging mechanism: as weakened circulation leads to accumulation of hot fluid adjacent to the hot sidewall, larger buoyancy forces associated with hotter fluid eventually break the layer interface. Then two layers merge into a larger layer, and circulation establishes again within the merged structure.

A roughness scaling behaviour is tested by performing the direct numerical simulation (DNS) of a turbulent channel flow over three-dimensional sinusoidal rough walls. By systematically varying the roughness height ${{k}^{+}}$ and the roughness steepness S, the results for three groups of cases are considered and compared with those for flat-wall turbulence. The results show that the mean velocity and Reynolds stresses are highly dependent on both ${{k}^{+}}$ and S. To describe these specific relationships, we define a coupling scale ${{k}^{+}} S$. With this coupling scale, all the simulated data for the roughness function (${\rm \Delta} {{U}^{+}}$), the ratio of the pressure drag to the total wall resistance (${{\gamma }_{p}}$), the normalized bulk mean velocity ($U_{b}^{+}$) and the peak of the streamwise turbulent velocity fluctuations ($\overline {u_{p}^{\prime +}}$) collapse onto single curves, which shows that there is a strong direct correlation between them, i.e. ${\rm \Delta} U^{+}, \gamma _{p}, U_{b}^{+}, \overline {u_{p}^{\prime +}} \propto f(k^{+} S)$. Furthermore, a model for the prediction of wall resistance based on the roughness function can be established by defining a drag increasing ratio (DI). Accordingly, the wall resistance coefficient ${{C}_{f}}$ can be estimated directly from ${{k}^{+}}S$ of a given rough surface. These results suggest that this coupling scale provides a useful alternative to the equivalent sand grain roughness ${{k}_{s}}$.

Fluid deformable surfaces show a solid–fluid duality which establishes a tight interplay between tangential flow and surface deformation. We derive the governing equations as a thin film limit and provide a general numerical approach for their solution. The simulation results demonstrate the rich dynamics resulting from this interplay, where, in the presence of curvature, any shape change is accompanied by a tangential flow and, vice versa, the surface deforms due to tangential flow. However, they also show that the only possible stable stationary state in the considered setting is a sphere with zero velocity.

Inclining a fluid layer subjected to a temperature gradient introduces a profusion of fascinating patterns and regimes. Previous experimental and computational studies form the starting point for an extensive numerical bifurcation study by Reetz & Schneider (J. Fluid Mech., vol. 898, 2020, A22) and Reetz et al. (J. Fluid Mech., vol. 898, 2020, A23). Intricate trajectories passing through multiple steady and periodic states organize the dynamics. The consequences for chaotic patterns in large geometries is discussed.

We investigate the formation of subaqueous transverse bedforms in turbulent open channel flow by means of direct numerical simulations with fully resolved particles. The main goal of the present analysis is to address the question whether the initial pattern wavelength scales with the particle diameter or with the mean fluid height. A previous study (Kidanemariam & Uhlmann, J. Fluid Mech., vol. 818, 2017, pp. 716–743) has observed a lower bound for the most unstable pattern wavelength in the range 75–100 times the particle diameter, which was equivalent to 3–4 times the mean fluid height. In the current paper, we vary the streamwise box length in terms of the particle diameter and of the mean fluid height independently in order to distinguish between the two possible scaling relations. For the chosen parameter range, the obtained results clearly exhibit a scaling of the initial pattern wavelength with the particle diameter, with a lower bound around a streamwise extent of approximately $80$ particle diameters. In longer domains, on the other hand, patterns are observed at initial wavelengths in the range 150–180 times the particle diameter, which is in good agreement with experimental measurements. Variations of the mean fluid height, on the other hand, seem to have no significant influence on the most unstable initial pattern wavelength. We argue that the observed scaling with the particle diameter is due to the wake effect induced by the seeds which are formed by initially dislodged lumps of particles, in accordance with the ideas of Coleman & Nikora (Water Resour. Res., vol. 45, 2009, W04402). Finally, for the cases with the largest relative submergence, we observe spanwise and streamwise sediment waves of similar amplitude to evolve and superimpose, leading to three-dimensional sediment patterns.

We extend the resolvent-based estimation approach recently introduced by Towne etal. (J. Fluid Mech., vol. 883, 2020, A17) to obtain optimal, non-causal estimates of time-varying flow quantities from low-rank measurements. We derive optimal transfer functions between the measurements and certain nonlinear terms that act as a forcing on the linearised Navier–Stokes equations, and show that the resulting transfer function to the flow state is equivalent to a multiple-input, multiple-output Wiener filter if the colour of the forcing statistics is known. A matrix-free implementation is developed based on integration of the direct and adjoint linearised Navier–Stokes operators, enabling application to the large systems encountered for transitional and turbulent flows without the need for a priori model reduction. Using a linearised Ginzburg–Landau problem, we show that the non-casual resolvent-based method outperforms a casual Kalman filter for general sensor configurations and recovers the Kalman filter transfer function in specific cases, leading to causal estimates at a significantly reduced computational cost. Additionally, our method is shown to be more accurate and robust than popular approaches based on truncation of the resolvent operator to its leading modes. The applicability of the method to transitional and turbulent flows is demonstrated via application to a (linearised) transitional boundary layer and a (nonlinear) turbulent channel flow. Errors on the order of 2 % are achieved for the boundary layer, and the channel flow case highlights the need to account for the forcing colour to achieve accurate flow estimates. In practice, our method can be used as a post-processing tool to reconstruct unmeasured quantities from limited experimental data, and, in cases where the transfer function can be accurately truncated to its causal components, as a low-cost estimator for flow control.

A new theory is presented for the generation of two-dimensional internal wave beams, including the effects of viscosity and unsteadiness on the propagation of the waves, and extending to the near field the classical theory of Lighthill for the far field. For this, the forcing is assumed to be of compact support. Several equivalent expressions of the waves are obtained, each associated with the choice of a support of simple shape embedding the actual support of the forcing. When the two match, the expression of the waves is valid everywhere in the fluid. For an oscillating body, the existence of critical points where the waves rays are tangential to the body is correctly accounted for, an essential requirement with regard to later inclusion of nonlinear effects and boundary layer eruption into the analysis, both of which take their origin at the critical points. Embedding supports in the shape of a circle, an ellipse and a strip are considered. Line forcing is also considered, on a weaker assumption of rapid decrease at infinity. The analysis reduces to the classical analysis of Hurley & Keady in the isotropic case of an oscillating circular cylinder, and is otherwise applied to four anisotropic oscillating bodies: an elliptic cylinder, a vertical plate, a vertical wave generator and a thin Gaussian bump.

Microorganisms are commonly found swimming in complex biological fluids such as mucus and these fluids respond elastically to deformation. These viscoelastic fluids have been previously shown to affect the swimming kinematics of these microorganisms in non-trivial ways depending on the rheology of the fluid, the particular swimming gait and the structural properties of the immersed body. In this report we put forth a previously unmentioned mechanism by which swimming organisms can experience a speed increase in a viscoelastic fluid. Using numerical simulations and asymptotic theory we find that significant swirling flow around a microscopic swimmer couples with the elasticity of the fluid to generate a marked increase in the swimming speed. We show that the speed enhancement is related to the introduction of mixed flow behind the swimmer and the presence of hoop stresses along its body. Furthermore, this effect persists when varying the fluid rheology and when considering different swimming gaits. This, combined with the generality of the phenomenon (i.e. the coupling of vortical flow with fluid elasticity near a microscopic swimmer), leads us to believe that this method of speed enhancement could be present for a wide range of microorganisms moving through complex fluids.

In this study, a comprehensive description of the adjoint formulation based on a systematic use of Lagrange's identity is proposed to compute acoustic propagation effects induced by the presence of a mean flow. The adjoint method is a clever approach introduced by Tam & Auriault (J. Fluid Mech., vol. 370, 1998, pp. 149–174) in aeroacoustics to predict noise of distributed stochastic sources in a complex environment. A clear statement is also provided about the application of the flow reversal theorem, and its restriction to self-adjoint wave equations. As an illustration, sound propagation is computed numerically over a sheared and stratified mean flow for Lilley's and Pierce's wave equations. Acoustic solutions obtained with the adjoint approach are then compared with predictions obtained with the flow reversal theorem. Additionally Pierce's wave equation for potential acoustics is identified as an outstanding candidate to compute accurately acoustic propagation while removing possible instability waves.

The background method has been a successful tool in obtaining strict bounds on global quantities such as the rate of energy dissipation and heat transfer in turbulent flows. However, all applications of this method until now have focused on flows confined between solid boundaries. An important class of problems that, by contrast, has received no attention is the class of external flows, i.e. flow past a body. In this context, obtaining the dependence of the drag coefficient on the Reynolds number is of crucial relevance for many engineering applications. In this paper, we consider the classical problem of flow past a flat plate of finite length at zero angle of incidence and use the background method to obtain a bound on the drag coefficient. Assuming a statistically steady state and appropriate far-field decay rates for the flow variables, we show that at large Reynolds numbers, the drag coefficient ($C_D$) is bounded by a constant, a bound that is within a logarithmic factor of experimental data.

The nonlinear stability of an inertialess two-layer surfactant-laden Couette flow is considered. The two fluids are immiscible and have different thicknesses, viscosities and densities. One of the fluids is contaminated with a soluble surfactant whose concentration may be above the critical micelle concentration, in which case micelles are formed in the bulk of the fluid. A surfactant kinetic model is adopted that includes the adsorption and desorption of molecules to and from the interface, and the formation and breakup of micelles in the bulk. The lubrication approximation is applied and a strongly nonlinear system of equations is derived for the evolution of the interface and surfactant concentration at the interface, as well as the vertically averaged monomer and micelle concentrations in the bulk (as a result of fast vertical diffusion). The primary aim of this study is to determine the influence of surfactant solubility on the nonlinear dynamics. The nonlinear lubrication model is solved numerically in periodic domains and saturated travelling waves are obtained at large times. It is found that a sufficiently soluble surfactant can either destabilise or stabilise the interface depending on certain fluid properties. The stability behaviour of the system depends crucially on the values of the fluid viscosity ratio $m$ and thickness ratio $n$ in reference to the boundary $m=n^{2}$. If the surfactant exists at large concentrations that exceed the critical micelle concentration, then long waves are stable at large times, unless density stratification effects overcome the stabilising influence of micelles. Travelling wave bifurcation branches are also calculated and the impact of various parameters (such as the domain length or fluid thickness ratio) on the wave shapes, amplitudes and speeds is examined. The mechanism responsible for interfacial (in)stability is explained in terms of the phase difference between the interface deformation and concentration waves, which is shifted according to the sign of the crucial factor $(m-n^{2})$ and the strength of the surfactant solubility.

Most of our understanding of moving contact lines relies on the limit of small capillary number ${Ca}$. This means the contact line speed is small compared to the capillary speed $\gamma /\eta$, where $\gamma$ is the surface tension and $\eta$ the viscosity, so that the interface is only weakly curved. The majority of recent analytical work has assumed in addition that the angle between the free surface and the substrate is also small, so that lubrication theory can be used. Here, we calculate the shape of the interface near a slip surface for arbitrary angles, and for two phases of arbitrary viscosities, thereby removing a key restriction in being able to apply small capillary number theory. Comparing with full numerical simulations of the viscous flow equation, we show that the resulting theory provides an accurate description up to $Ca \approx 0.1$ in the dip coating geometry, and a major improvement over theories proposed previously.

In this paper we study the perturbations produced in the boundary layer by an impinging oblique shock wave or Prandtl–Meyer expansion fan. The flow outside the boundary layer is assumed supersonic, and we also assume that the point, where the shock wave/expansion fan impinges on the boundary layer, moves downstream. To study the flow, it is convenient to use the coordinate frame moving with the shock; in this frame, the body surface moves upstream. We first study numerically the case when the shock velocity $V_{sh} = O (Re^{-1/8})$. In this case the interaction of the boundary layer with the shock can be described by the classical equations of the triple-deck theory. We find that, as $V_{sh}$ increases, the boundary layer proves to be more prone to separation when exposed to the expansion fan, not the compression shock. Then we assume $V_{sh}$ to be in the range $1 \gg V_{sh} \gg Re^{-1/8}$. Under these conditions, the process of the interaction between the boundary layer and the shock/expansion fan can be treated as inviscid and quasi-steady if considered in the reference frame moving with the shock/expansion fan. The inviscid analysis allows us to determine the pressure distribution in the interaction region. We then turn our attention to a thin viscous sublayer that lies closer to the body surface. In this sublayer the flow is described by classical Prandtl's equations. The solution to these equations develops a singularity provided that the expansion fan is strong enough. The flow analysis in a small vicinity of the singular point shows an accelerated ‘expansion’ of the flow similar to the one reported by Neiland (Izv. Akad. Nauk SSSR, Mech. Zhidk. Gaza, vol. 5, 1969a, pp. 53–60) in his analysis of supersonic flow separation from a convex corner.

Considering channel flow at Reynolds numbers below the linear stability threshold of the laminar profile as a generic example system showing a subcritical transition to turbulence connected with the existence of simple invariant solutions, we here discuss issues that arise in the application of linear feedback control of invariant solutions of the Navier–Stokes equations. We focus on the simplest possible problem, that is, travelling waves with one unstable direction. In view of potential experimental applicability we construct a pressure-based feedback strategy and study its effect on the stable, marginal and unstable directions of these solutions in different periodic cells. Even though the original instability can be removed, new instabilities emerge as the feedback procedure affects not only the unstable but also the stable directions. We quantify these adverse effects and discuss their implications for the design of successful control strategies. In order to highlight the challenges that arise in the application of feedback control methods in principle and concerning potential applications in the search for simple invariant solutions of the Navier–Stokes equations in particular, we consider an explicitly constructed analogue to closed-loop linear optimal control that leaves the stable directions unaffected.

Direct numerical simulations, performed with a high-order spectral-element method, are used to study coherent structures in turbulent pipe flow at friction Reynolds numbers $Re_{\tau } = 180$ and $550$. The database was analysed using spectral proper orthogonal decomposition (SPOD) to identify energetically dominant coherent structures, most of which turn out to be streaks and quasi-streamwise vortices. To understand how such structures can be modelled, the linear flow responses to harmonic forcing were computed using the singular value decomposition of the resolvent operator, using the mean field as a base flow. The SPOD and resolvent analysis were calculated for several combinations of frequencies and wavenumbers, allowing the mapping out of similarities between SPOD modes and optimal responses for a wide range of relevant scales in turbulent pipe flows. In order to explore physical reasons behind the agreement between both methods, an indicator of lift-up mechanism in the resolvent analysis was introduced, activated when optimal forcing is dominated by the wall-normal and azimuthal components, and associated response corresponds to streaks of streamwise velocity. Good agreement between leading SPOD and resolvent modes is observed in a large region of parameter space. In this region, a significant gain separation is found in resolvent analysis, which may be attributed to the strong amplification associated with the lift-up mechanism, here understood as nonlinear forcing terms leading to the appearance of streamwise vortices, which in turn form high-amplitude streaks. For both Reynolds numbers, the observed concordances were generally for structures with large energy in the buffer layer. The results highlight resolvent analysis as a pertinent reduced-order model for coherent structures in wall-bounded turbulence, particularly for streamwise elongated structures corresponding to near-wall streamwise vortices and streaks.