The reshocked turbulent Richtmyer–Meshkov (RM) mixing of two media is the most representative problem of more general and complex turbulent mixing induced by interfacial instabilities, broadly occurring in both nature and engineering applications. An accurate prediction of its evolving of spatial structure and mixing width (MW) is of fundamental importance. However, satisfactory prediction with the large-eddy simulation (LES) has not yet been achieved, even for the most important MW. In this paper, we innovatively solve this problem by combining the idea of the constrained large-eddy simulation (CLES), which succeeded previously only in classical single-medium turbulence, and our recently developed Reynolds averaged Navier–Stokes (RANS) model, which realized a satisfactory prediction of MW. Specifically, in our currently developed CLES model, with the aid of Reynolds decomposition, the unclosed subgrid scale (SGS) LES model is decomposed into two parts, i.e. the averaged and the fluctuating. The averaged part is dominated and modelled by the counterpart of our recently developed RANS model to accurately predict the MW, while the fluctuating part is modelled with the classical Smagorinsky model. Consequently, besides successfully capturing the three-dimensional large-scale structure of turbulence and the evolution of the (normalized) mixed mass, our newly proposed CLES also predicts a satisfactory MW with a very coarse grid. To the best of our knowledge, this is the first time that the LES can yield such a comparable result with experiment.

We investigate evanescent inertial waves, both theoretically and experimentally, in a fluid subject to a background rotation of $\varOmega$. We predict that there is a smooth transition from conventional inertial waves to evanescent disturbances at a frequency of $\varpi = 2\varOmega$, and that at this cross-over frequency the evanescent disturbances are spatially extensive, having a horizontal extent which is limited only by viscosity, or by the size of the domain. These findings are confirmed by our experiments, which, to the best of our knowledge, represent the first quantitative experimental investigation of evanescent inertial waves.

The ability of vortices to induce aerodynamic loads on proximal surfaces plays a significant role in a wide variety of flows. However, most studies of vortex-induced effects primarily focus on analysing the influence of the rotation-dominated cores of vortices. In this work, we show that not only are vortices in viscous flows surrounded by strain-dominated regions, but these regions are dynamically important and can sometimes even dictate the induced aerodynamic loads. We demonstrate this for a pitching airfoil, which exhibits dynamic stall and generates several force-inducing vortices. Using a data-driven force partitioning method (FPM), we quantify the influence of vortices as well as vortex-associated strain to show that our current understanding of vortex-dominated phenomena, such as dynamic stall, is incomplete without considering the substantial effect of strain-dominated regions that are associated with vortices.

A Boussinesq fluid inside a stably thermally stratified square container whose walls are inclined ${45}^{\circ }$ with respect to gravity, with two opposite walls kept at constant temperatures and the other two insulated is nearly isothermal in the regions above and below the horizontal diagonal. The flow is concentrated in the wall boundary layers and a shear layer centred about the horizontal diagonal. The equilibrium is maintained by the balance between dissipation in the shear and boundary layers, the heat fluxes at the constant temperature walls, and the induced flow resulting from the no-flux condition at the inclined insulated walls. The dynamical response of the fluid to vertical oscillations of the container is studied over a range of forcing frequencies. For a small forcing amplitude and below a viscosity-dependent cutoff forcing frequency, this response exhibits a modal cellular structure localized about the shear layer. With increasing forcing amplitude, the response experiences instabilities, studied here numerically at a forcing frequency above the cutoff frequency, that are similar to those encountered in the Faraday wave problem, such as parametric subharmonic instability, triadic resonance and resonant collapse.

Turbulent wall-flow research often includes documenting and describing mean flow and turbulence properties under variations in Reynolds number or other relevant parameters. Concomitant efforts are associated with clarifying the flow dynamics and energetics that accompany the observed statistical behaviours. Relative to these aims, the so-called canonical wall flows (pipe, channel and flat plate boundary layer flows) comprise an important baseline. Additionally, flows that add complexity relative to the canonical baseline while retaining analytically idealized properties provide unique opportunities to clarify the nature of wall-flow turbulence and its response to modifying influences. In this regard, the meticulously conducted study by Ferro et al. (J. Fluid Mech., vol. 915, 2021, A80) provides significant and remarkable contributions by both demonstrating that the idealized flow state of the turbulent asymptotic suction boundary layer (TASBL) is realizable in a laboratory experiment, and by providing a high fidelity documentation of the mean flow and streamwise velocity fluctuation properties in the TASBL flow.

We report a direct numerical simulation (DNS) study of the mean velocity and temperature profiles in turbulent Rayleigh–Bénard convection (RBC) at low Prandtl numbers ($Pr$). The numerical study is conducted in a vertical thin disk with $Pr$ varied in the range $0.17\leq Pr\leq 4.4$ and the Rayleigh number ($Ra$) varied in the range $5\times 10^8\leq Ra \leq 1\times 10^{10}$. By varying $Pr$ from 4.4 to 0.17, we find a sharp change of flow patterns for the large-scale circulation (LSC) from a rigid-body rotation to a near-wall turbulent jet. We numerically examine the mean velocity equation in the bulk region and find that the mean horizontal velocity profile $u(z)$ can be determined by a balance equation between the mean convection and turbulent diffusion with a constant turbulent viscosity $\nu _t$. This balance equation admits a self-similarity jet solution, which fits the DNS data well. In the boundary-layer region, we find that both the mean temperature profile $T(z)$ and $u(z)$ can be determined by a balance equation between the molecular diffusion and turbulent diffusion. Within the viscous boundary layer, both $u(z)$ and $T(z)$ can be solved analytically and the analytical results agree well with the DNS data. Our careful characterisation of the mean velocity and temperature profiles in low-$Pr$ RBC provides a further understanding of the intricate interplay between the LSC, plume emission and boundary-layer dynamics, and pinpoints the physical mechanism for the emergence of a pronounced LSC in low-$Pr$ RBC.

Two new flows, a tilted-vortex flow and a periodic-vortex flow, are found numerically at the onset of instability in rotating plane Couette flow (RPCF). It has hitherto been believed that the first few bifurcation sequences from the laminar state are quite well understood, namely, that a streamwise-independent flow with a two-dimensional roll-cell (2dRC) structure bifurcates first, followed by the bifurcation of a wavy-vortex flow (WVF). The present study shows that the 2dRC is, in fact, unstable from its onset, and that the newly found tilted-vortex flow takes over in place of 2dRC. It is further found that the periodic-vortex flow takes part in the early stages of laminar–turbulent transition. The existence of the new flows can be verified by careful examinations of the experimental observations in Hiwatashi et al. (Phys. Fluids, vol. 19, 2007, 048103) and Kawata & Alfredsson (J. Fluid Mech., vol. 791, 2016, 191–213).

Turbulent flows are often bounded by regions of irrotational or non-turbulent flow, where the magnitude of the potential velocity fluctuations can be surprisingly high. This includes virtually all turbulent free-shear flows and also turbulent boundary layers, and is particularly true near the so-called turbulent/non-turbulent interface (TNTI) layer, which separates the regions of turbulent and non-turbulent fluid motion. In the present work, we show that in the non-turbulent region and for distances $x_2$ sufficiently far from the TNTI layer, the asymptotic variation laws for the variance of the velocity fluctuations $\langle u_i^{2} \rangle$ ($i=1,2,3$), Taylor micro-scale $\lambda$ and viscous dissipation rate $\varepsilon$ depend on the shape of the kinetic energy spectrum in the infrared region $E(k) \sim k^{n}$. Specifically, by using rapid distortion theory (RDT), we show that for Saffman turbulence ($E(k) \sim k^{2}$), we obtain the asymptotic laws $\langle u_i^{2} \rangle \sim x_2^{-3}$ ($i=1,2,3$), $\lambda \sim x_2$ and $\varepsilon \sim x_2^{-5}$. Additionally, we confirm the classical results obtained by Phillips (Proc. Camb. Phil. Soc., vol. 51, 1955, p. 220) for Batchelor turbulence ($E(k) \sim k^{4}$), with $\langle u_i^{2} \rangle \sim x_2^{-4}$ ($i=1,2,3$), $\lambda \sim x_2$ and $\varepsilon \sim x_2^{-6}$. The new theoretical results are confirmed by direct numerical simulations (DNS) of shear-free turbulence and are shown to be independent of the Reynolds number. Therefore, these results are expected to be valid in other flow configurations, such as in turbulent planar jets or wakes, provided the kinetic energy spectra in the turbulence region can be described by a Batchelor or a Saffman spectrum.

We propose an aerodynamic force model associated with a Galerkin model for the unforced fluidic pinball, the two-dimensional flow around three equal cylinders with one radius distance to each other. The starting point is a Galerkin model of a bluff-body flow. The force on this body is derived as a constant-linear-quadratic function of the mode amplitudes from first principles following the pioneering work of Noca (On the evaluation of time-dependent fluid-dynamic forces on bluff bodies. PhD thesis, California Institute of Technology, 1997), Noca et al. (J. Fluids Struct., vol. 13, issue 5, 1999, pp. 551–578) and Liang & Dong (Virtual force measurement of POD modes for a flat plate in low Reynolds number flows. AIAA Paper 2014-0054). The force model is simplified for the mean-field model of the unforced fluidic pinball (Deng et al., J. Fluid Mech., vol. 884, 2020, p. A37) using symmetry properties and sparse calibration. The model is successfully applied to transient and post-transient dynamics in different Reynolds number regimes: the periodic vortex shedding after the Hopf bifurcation and the asymmetric vortex shedding after the pitchfork bifurcation comprising six different Navier–Stokes solutions. We foresee many applications of the Galerkin force model for other bluff bodies and flow control.

We study non-Newtonian effects associated with power-law rheology of behaviour index $n$ on the propagation of horizontal gravity currents. Two different set-ups are examined: (i) converging flow toward the origin in a channel of gap thickness $b(x) \propto x^k$ and $k<1$ and (ii) converging flow along $r$ toward the centre in a cylinder. The front of the current propagates in the negative $x$ or $r$ direction reaching the origin in a finite touch-down time $t_c$ during the pre-closure phase; in the post-closure phase, the current flows back in the positive direction and progressively levels out. Under the classical viscous-buoyancy balance, the current propagation is described by a differential problem amenable to a self-similar solution of the second-kind coupling space and reduced time $t_r=t_c-t$. The problem formulation in the phase plane leads to an autonomous system of differential equations which requires numerical integration and yields the shape of the current and its front as $\xi _f\propto t_r^{\delta _c}$, $\xi _f$ being the self-similar variable value at the front and $\delta _c$ being the critical eigenvalue. The latter is a function of fluid rheology $n$ and of channel geometry $k$ for the first set-up; it is a function only of $n$ for the second set-up. The dependency on $n$ is modest. The theoretical formulation is validated through experiments conducted during both pre- and post-closure phases and aimed at measuring the front position and the profile of the current. Experimental results are in fairly good agreement with theory and allow quantitative determination of the time interval of validity of the intermediate asymptotics regime, when self-similarity is achieved and when it is lost.

We investigate the convective cooling of a fluid with a quadratic equation of state (EOS) by performing three-dimensional direct numerical simulations of a flow with a fixed top-boundary temperature, which is lower than the initial fluid temperature. We consider fluid temperatures near the density maximum, where the nonlinearity is expected to be important. When the EOS is nonlinear, the resultant vertical transport of heat is fundamentally different and significantly lower than the predictions derived for a linear EOS. Further, three dimensionless groups parameterise the convective system: the Rayleigh number (${Ra}_0$), the Prandtl number (Pr) and the dimensionless bottom water temperature $(T_B)$. We further define an effective Rayleigh number (${Ra}_{eff} = {Ra}_0 \ T_B^2$), which is equivalent to the traditional Rayleigh number used with a linear EOS. We present a predictive model for the vertical heat flux, the top boundary-layer thickness, and the turbulent kinetic energy (TKE) of the system. We show that this model agrees well with the direct numerical simulations. This model could be used to understand how quickly freshwater lakes cool in high-latitude environments.

The motion of a gaseous Taylor bubble in a capillary tube is typical of many biological and engineering systems, such as small-scale reactors and microfluidic devices. Although the dynamics of a bubble in a Newtonian liquid has been the subject of several studies since the seminal works of Taylor (J. Fluid Mech., vol. 10, issue 2, 1961, pp. 161–165) and Bretherton (J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188), the case where the fluid exhibits a shear-thinning behaviour is much less understood. To fill this gap, we study the dynamics of a bubble that moves in a shear-thinning fluid whose viscosity is described by the Ellis viscosity model. With this aim, we derive a lubrication model in the film region to identify the scaling laws for the bubble speed, the film thickness and the pressure drop as a function of the Ellis number and the degree of shear thinning. Our model generalizes Bretherton's results to shear-thinning fluids by identification of a universal scaling law for the effective viscosity that accounts for the interplay of the zero-shear-rate and shear-thinning effects. The film thickness follows a $2/3$ scaling law with respect to the capillary number based on the proposed effective viscosity. The ratio between the bubble speed and the average velocity of the fluid ahead of the bubble is a function of the effective capillary number only. We show that some portions of the bubble are dominated by the zero-shear-rate effect discussing the extent to which the use of the power-law viscosity model can be legitimized. Finally, we study the location of the recirculating vortices ahead of the bubble.

This experimental study examined the effects of strategic positioning of tabs in the periphery of the exit plane of a transverse jet as a means of controlling shear layer instabilities and vorticity as well as jet structure and mixing. Conditions corresponding to a naturally absolutely unstable upstream shear layer (USL), with low jet-to-crossflow momentum flux ratios $J$ (below 8–9) and to a convectively unstable USL ($J\geq 20$) were explored for different jet Reynolds numbers, 1900 and 2300. Acetone planar laser-induced fluorescence imaging and stereo particle image velocimetry were utilized to explore the influence of the tab. Placement of the tab in the upstream region of the jet exit caused significant weakening of the USL instability at lower $J$, but with only marginal weakening for larger $J$ values. Yet tab placement was observed to affect cross-sectional structure and vorticity dynamics much more significantly at high $J$ values. For all flow conditions, tab placement in the upstream region improved molecular mixing to a greater degree than at other locations. Vorticity fields, proper orthogonal decomposition modes, and coefficient plots extracted from centreplane velocity field measurements showed significant influence of tab placement on jet dynamical characteristics depending on $J$. Tab locations with the greatest influence were consistent with wavemaker regions predicted in numerical simulations of the transverse jet (Regan & Mahesh, J. Fluid Mech., vol. 877, 2019, pp. 330–372), providing evidence for the potential to tailor shear layer rollup, jet structure and mixing via simple passive geometrical alterations.

We present a two-point model to investigate the underlying source mechanisms for broadband shock-associated noise (BBSAN) in shock-containing supersonic jets. In the model presented, the generation of BBSAN is assumed to arise from the nonlinear interaction between downstream-propagating coherent structures with the quasi-periodic shock cells in the jet plume. The turbulent perturbations are represented as axially extended wavepackets and the shock cells are modelled as a set of stationary waveguide modes. Unlike previous BBSAN models, the physical parameters describing the hydrodynamic components are not scaled using the acoustic field. Instead, the source characteristics of both the turbulent and shock components are extracted from the hydrodynamic region of large-eddy simulation and particle image velocimetry datasets. Apart from using extracted data, a reduced-order description of the wavepacket structure is obtained using parabolised stability equations. The validity of the model is tested by comparing far-field sound pressure level predictions to azimuthally decomposed experimental acoustic data from a cold Mach 1.5 underexpanded jet. At polar angles and frequencies where BBSAN dominates, encouraging comparisons of the radiated noise spectra for the first three azimuthal modes, in both frequency and amplitude (${\pm }2\ \textrm {dB}\,\textrm {St}^{-1}$ at peak frequency), reinforce the suitability of using reduced-order wavepacket sources for predicting BBSAN peaks. On the other hand, wavepacket jitter is found to have a critical role in recovering sound amplitude at interpeak frequencies. The paper presents a quantitative demonstration that the wavepacket–shock interaction, carefully reconstructed by extracting components from data or linearised models, contains the correct essential flow physics that accounts for most features of the far-field BBSAN spectra.

Resolvent analysis identifies the most responsive forcings and most receptive states of a dynamical system, in an input–output sense, based on its governing equations. Interest in the method has continued to grow during the past decade due to its potential to reveal structures in turbulent flows, to guide sensor/actuator placement and for flow control applications. However, resolvent analysis requires access to high-fidelity numerical solvers to produce the linearized dynamics operator. In this work, we develop a purely data-driven algorithm to perform resolvent analysis to obtain the leading forcing and response modes, without recourse to the governing equations, but instead based on snapshots of the transient evolution of linearly stable flows. The formulation of our method follows from two established facts: (i) dynamic mode decomposition can approximate eigenvalues and eigenvectors of the underlying operator governing the evolution of a system from measurement data, and (ii) a projection of the resolvent operator onto an invariant subspace can be built from this learned eigendecomposition. We demonstrate the method on numerical data of the linearized complex Ginzburg–Landau equation and of three-dimensional transitional channel flow, and discuss data requirements. Presently, the method is suitable for the analysis of laminar equilibria, and its application to turbulent flows would require disambiguation between the linear and nonlinear dynamics driving the flow. The ability to perform resolvent analysis in a completely equation-free and adjoint-free manner will play a significant role in lowering the barrier of entry to resolvent research and applications.

The linear stability characteristics of pressure-driven core-annular pipe flow of two immiscible fluids are considered to investigate the effects of the density and viscosity ratios, the Reynolds number, the interface location and the interfacial tension. Both liquid–liquid and gas–liquid systems are examined. A new type of interfacial mode associated with the axisymmetric and corkscrew perturbations is discovered for certain ranges of the viscosity and density ratios in the immiscible liquid–liquid system. Two distinct unstable regions at long and short wavelengths are observed. The long-wavelength unstable region forms a close loop, indicating that it is not a Tollmien–Schlichting mode. The new interfacial mode observed in the present study is similar to that discovered by Mohammadi & Smits (J. Fluid Mech., vol. 826, 2017, pp. 128–157) in two-layer Couette flow for low viscosity ratios. In contrast to the two distinct unstable regions found in the immiscible configuration, the corresponding miscible system contains only one unstable mode. It is found that, in the liquid–liquid systems, the corkscrew (axisymmetric) perturbation is dominant when the annular fluid is less (more) viscous than the core fluid. On the other hand, the axisymmetric perturbation is always the dominant one in the gas–liquid system. In gas–liquid systems, the interfacial tension stabilises the short-wave and destabilises the long-wave perturbations, while increasing the interface radius stabilises the flow due to the presence of a plug region in the pipe.

The coating of rotating discrete objects with surface topography is a problem commonly encountered in manufacturing processes. Surface topography may induce undesired disturbances in the coating, leading to coatings of non-uniform thickness. To study this problem, we model the flow of thin liquid coatings in three dimensions on topographically patterned cylinders that rotate about their horizontal axes. An evolution equation describing variations in the coating thickness as a function of the axial coordinate, the angular coordinate, and time is solved numerically using a variable time-step finite-difference scheme. In the limit of a rapidly rotating cylinder, we neglect the effects of gravity and find that liquid accumulates at either pattern crests or pattern troughs. Using a long-wave analysis, we derive an expression for the critical Weber number that separates these regimes. If gravity is reincorporated, the accumulation of liquid at crests or troughs may cause the coating to sag under its weight, leading to the formation of droplets or rings whose spacing at large rotation rates is controlled by the balance between centrifugal and surface-tension forces. At lower rotation rates, where gravitational forces dominate, simulation results indicate that cylinder topography tends to alter the rate at which droplets form, but does not necessarily systematically affect the spacing between droplets. Flow visualization experiments yield results that agree quantitatively with predictions of the simulations and long-wave analysis. We observe the most uniform coatings in experiments at moderate rotation rates, where disturbances in the coating thickness develop slowly. This indicates that to obtain nearly uniform coatings in practice, the coating must be solidified faster than disturbances can develop.

Scale interaction is studied in wall-bounded turbulence by focusing on the frequency modulation (FM) mechanism of the large scales on small-scale velocity fluctuations. Different from the analysis of amplitude modulation (AM), FM has been less investigated owing to the difficulty of developing robust tools for broadband signals. To tackle this issue, the natural visibility graph approach is proposed in this work to map the full velocity signals into complex networks. We show that the network degree centrality is able to capture the signal structure at local scales directly from the full signal, thereby quantifying FM. Velocity signals from numerically-simulated turbulent channel flows and an experimental turbulent boundary layer are investigated at different Reynolds numbers. A correction of Taylor's hypothesis for time series is proposed to overcome the overprediction of near-wall FM obtained when local mean velocity is used as the convective velocity. Results provide network-based evidence of the large-to-small FM features for all the three velocity components in the near-wall region, with a reversal mechanism emerging far from the wall. Additionally, scaling arguments, in view of the quasi-steady quasi-homogeneous hypothesis, are discussed, and a delay time between the large and small scales is detected that is very close to the characteristic time of the near-wall cycle. Results show that the visibility graph is a parameter-free tool that turns out to be effective and robust to detect FM in different configurations of wall-bounded turbulent flows. Based on the present findings, the visibility network-based approach can represent a reliable tool to systematically investigate scale interaction mechanisms in wall-bounded turbulence.

The turbulent boundary layer over a Gaussian-shaped bump is computed by direct numerical simulation of the incompressible Navier–Stokes equations. The two-dimensional bump causes a series of strong pressure gradients alternating in rapid succession. At the inflow, the momentum thickness Reynolds number is approximately $1000$ and the boundary layer thickness is $1/8$ of the bump height. Direct numerical simulation results show that the strong favourable pressure gradient (FPG) causes the boundary layer to enter a relaminarization process. The near-wall turbulence is significantly weakened and becomes intermittent, however, relaminarization does not complete. The streamwise velocity profiles deviate above the standard logarithmic law and the Reynolds shear stress is reduced. The strong acceleration also suppresses the wall-shear normalized turbulent kinetic energy production rate. At the bump peak, where the FPG switches to an adverse gradient (APG), the near-wall turbulence is suddenly enhanced through a partial retransition process. This results in a new highly energized internal layer which is more resilient to the strong APG and only produces incipient flow separation on the downstream side. In the strong FPG and APG regions, the inner and outer layers become largely independent of each other. The near-wall region responds to the pressure gradients and determines the skin friction. The outer layer behaves similarly to a free shear layer subject to pressure gradients and mean streamline curvature effects. Results from a RANS simulation of the bump are also discussed and clearly show the lack of predictive capacity of the near-wall pressure gradient effects on the mean flow.

This paper investigates the motion of a liquid blister, trapped between an elastic sheet and a rigid substrate. The blister is driven by a frictionless blade moving at a constant velocity, forcing a constant gap that causes fluid to bleed from the blister. The sheet adheres to the substrate ahead of the blister. The main goal of the study is to assess the magnitude and orientation of the force applied by the blade on the moving blister. The solution is constructed for the asymptotic case of a long blister. Thanks to a separation of scales, the asymptotic solution is obtained by matching the boundary layers at the front end and at the back end of the blister to an outer solution characterised by a uniform pressure in the bulk. Both boundary layers are formulated as travelling-wave equations for the gap between the sheet and the substrate. The formulation accounts for a moving fluid front, distinct from the separation edge, and for a tail with a gap tending to an a priori unknown value far behind the blister. Scaling of the governing equations indicates that the solution depends on two numbers: a dimensionless toughness $\mathcal {K}$ and a scaled gap $\mathcal {W}$ imposed by the moving blade. The key result concerns the dependence of the scaled force on the two numbers controlling the solution of the moving liquid blister. There are two asymptotic solutions: for small gaps at the blade, the force on the blade is dominated by viscous dissipation at the back end and only depends on aperture $\mathcal {W}$; for large gaps, the horizontal force $H$ only depends on toughness $\mathcal {K}$, a function of both fluid viscosity and energy of separation at the front end, whereas the vertical force $V$ depends on both $\mathcal {K}$ and $\mathcal {W}$.