Predicting and controlling entropy-wave-driven combustion instabilities is challenging, because the production, advection and dispersion of entropy waves in practical systems is difficult to model. The present paper aims to shed new light on this problem by considering a highly turbulent configuration with experiments and large eddy simulations. In this configuration, the decay of entropy waves is not only governed by the shear dispersion of an idealized turbulent pipe flow, as assumed in the recent studies on the topic, but also enhanced by the highly three-dimensional dispersion due to large-scale coherent structures. A novel post-processing approach for background-oriented schlieren (BOS) thermometry is proposed and enables the measurement of entropy waves with high spatio-temporal resolution for wide ranges of entropy wave amplitudes and frequencies. Instantaneous BOS snapshots are arranged in a panoramic coordinate frame using velocity data. This work, therefore, contributes to filling the knowledge gap in experimental data on entropy waves. The new dataset is accompanied with large eddy simulations to further elucidate the mechanisms dominating the amplitude decay of the entropy waves. It is shown that shear dispersion models that are only based on mean profiles of the axial velocity significantly underestimate the decay in the present configuration, where the entropy waves are formed by periodically injecting hot air pockets in the main stream. It is shown that the turbulent nature of the coherent hot pockets plays a key role in the dispersion enhancement.

A quantitative understanding of the dominant mechanisms that govern the generation and decay of the counter-rotating vortex pair (CVP) produced by yawed wind turbines is needed to fully realize the potential of yawing for wind farm power maximization and regulation. Observations from large eddy simulations (LES) of yawed wind turbines in the turbulent atmospheric boundary layer and concepts from the aircraft trailing vortex literature inform a model for the shed vorticity and circulation. The model is formed through analytical integration of simplified forms of the vorticity transport equation. Based on an eddy viscosity approach, it uses the boundary-layer friction velocity as the velocity scale and the width of the vorticity distribution itself as the length scale. As with the widely used Jensen model for wake deficit evolution in wind farms, our analytical expressions do not require costly numerical integration of differential equations. The predicted downstream decay of maximum vorticity and total circulation agree well with LES results. We also show that the vorticity length scale grows linearly with downstream distance and find several power laws for the decay of maximum vorticity. These results support the notion that the decay of the CVP is dominated by gradual cancellation of the vorticity at the line of symmetry of the wake through cross-diffusion.

In this note, we establish the applicability and effectiveness of a recently developed approach to the recovery of nonlinear water waves from pressure measurements by proving that it is applicable to the celebrated extreme Stokes wave.

Rayleigh–Bénard convection (RBC) and Taylor–Couette flow (TCF) are two paradigmatic fluid dynamical systems frequently discussed together because of their many similarities despite their different geometries and forcing. Often these analogies require approximations, but in the limit of large radii where TCF becomes rotating plane Couette flow (RPC) exact relations can be established. When the flows are restricted to two spatial independent variables, there is an exact specification that maps the three velocity components in RPC to the two velocity components and one temperature field in RBC. Using this, we deduce several relations between both flows: (i) heat and angular momentum transport differ by $(1-R_{\Omega })$, explaining why angular momentum transport is not symmetric around $R_{\Omega }=1/2$ even though the relation between $Ra$, the Rayleigh number, and $R_{\Omega }$, a non-dimensional measure of the rotation, has this symmetry. This relationship leads to a predicted value of $R_{\Omega }$ that maximizes the angular momentum transport that agrees remarkably well with existing numerical simulations of the full three-dimensional system. (ii) One variable in both flows satisfies a maximum principle, i.e. the fields’ extrema occur at the walls. Accordingly, backflow events in shear flow cannot occur in this quasi two-dimensional setting. (iii) For free-slip boundary conditions on the axial and radial velocity components, previous rigorous analysis for RBC implies that the azimuthal momentum transport in RPC is bounded from above by $Re_S^{5/6}$, where $Re_S$ is the shear Reynolds number, with a scaling exponent smaller than the anticipated $Re_S^1$.

Consider the propagation of a gravity current (GC) released from a lock of length $x_0$ and height $h_0$ into an ambient fluid of height $H h_0$ and density $\rho _{o}$. The lock contains a layer of thickness $H_L h_0$ of density $\rho _L$ overlaid by a layer of thickness $(1-H_L)h_0$ and density $\rho _U$, where $\rho _{o} < \rho _U < \rho _L$ and $H_L \in (0, 1)$. Assume Boussinesq and large Reynolds-number flow. The internal stratification parameter is $\sigma = (\rho _L - \rho _U)/(\rho _L - \rho _{o})$, in the range $(0,1)$; the classical GC is $\sigma =0$. Such GCs were investigated experimentally (Gladstone et al., Sedimentology, vol. 51, 2004, pp. 767–789; Dai, Phys. Rev. Fluids, vol. 2, 2017, 073802; Wu & Dai, J. Hydraul. Res., 2019, pp. 1–14.); we present a new self-contained model for the prediction of the thickness $h$ and depth-averaged velocity $u$ as functions of distance $x$ and time $t$; the position and speed of the nose $x_N(t)$ and $u_N(t)$ follow. We derive a compact scaling upon which, for a given $H$ (height ratio of ambient to lock), the flows differ in only one parameter: $\varPsi = \{ [1 -\sigma (1 - H_L)]/[1 - \sigma (1 - H_L^2)] \} ^{1/2}$. The parameter $\varPsi$ equals $1$ for the classical GC and is larger in the presence of stratification; a larger $\varPsi$ means a faster and a thinner GC. The solution reveals an initial slumping phase with constant $u_N$, a self-similar phase $x_N \sim t^{2/3}$, and the transition at $x_V$ to the viscous regime. Comparisons with published experiments show good data collapse with the present scaling $\varPsi$, and fair-to-good quantitative agreement (the discrepancy and the stability conditions are discussed).

The attached eddy hypothesis of Townsend (The Structure of Turbulent Shear Flow, 1956, Cambridge University Press) states that the logarithmic mean velocity admits self-similar energy-containing eddies which scale with the distance from the wall. Over the past decade, there has been a significant amount of evidence supporting the hypothesis, placing it to be the central platform for the statistical description of the general organisation of coherent structures in wall-bounded turbulent shear flows. Nevertheless, the most fundamental question, namely why the hypothesis has to be true, has remained unanswered over many decades. Under the assumption that the integral length scale is proportional to the distance from the wall $y$, in the present study we analytically demonstrate that the mean velocity is a logarithmic function of $y$ if and only if the energy balance at the integral length scale is self-similar with respect to $y$, providing a theoretical basis for the attached eddy hypothesis. The analysis is subsequently verified with the data from a direct numerical simulation of incompressible channel flow at the friction Reynolds number $Re_\tau \simeq 5200$ (Lee & Moser, J. Fluid Mech., vol. 774, 2015, pp. 395–415).

Although the relative importance of airborne transmission of the SARS-CoV-2 virus is controversial, increasing evidence suggests that understanding airflows is important for estimation of the risk of contracting COVID-19. The data available so far indicate that indoor transmission of the virus far outstrips outdoor transmission, possibly due to longer exposure times and the decreased turbulence levels (and therefore dispersion) found indoors. In this paper we discuss the role of building ventilation on the possible pathways of airborne particles and examine the fluid mechanics of the processes involved.

We consider the dynamics of a vertically stratified, horizontally forced Kolmogorov flow. Motivated by astrophysical systems where the Prandtl number is often asymptotically small, our focus is the little-studied limit of high Reynolds number but low Péclet number (which is defined to be the product of the Reynolds number and the Prandtl number). Through a linear stability analysis, we demonstrate that the stability of two-dimensional modes to infinitesimal perturbations is independent of the stratification, whilst three-dimensional modes are always unstable in the limit of strong stratification and strong thermal diffusion. The subsequent nonlinear evolution and transition to turbulence are studied numerically using direct numerical simulations. For sufficiently large Reynolds numbers, four distinct dynamical regimes naturally emerge, depending upon the strength of the background stratification. By considering dominant balances in the governing equations, we derive scaling laws for each regime which explain the numerical data.

We investigate the transport of inertial particles by cellular flows when advection dominates over inertia and diffusion, that is, for Stokes and Péclet numbers satisfying $St \ll 1$ and $Pe \gg 1$. Starting from the Maxey–Riley model, we consider the distinguished scaling $St \, Pe = O(1)$ and derive an effective Brownian dynamics approximating the full Langevin dynamics. We then apply homogenisation and matched-asymptotics techniques to obtain an explicit expression for the effective diffusivity $\bar {D}$ characterising long-time dispersion. This expression quantifies how $\bar {D}$, proportional to $Pe^{-1/2}$ when inertia is neglected, increases for particles heavier than the fluid and decreases for lighter particles. In particular, when $St \gg Pe^{-1}$, we find that $\bar {D}$ is proportional to $St^{1/2}/(\log ( St \, Pe))^{1/2}$ for heavy particles and exponentially small in $St \, Pe$ for light particles. We verify our asymptotic predictions against numerical simulations of the particle dynamics.

We study concentration-driven natural convection boundary layers on horizontal surfaces, subjected to a weak, surface normal, uniform blowing velocity $V_i$ for three orders of range of the dimensionless blowing parameter $10^{-8}\le J=Re_x^3/Gr_x\le 10^{-5}$, where $Re_x$ and $Gr_x$ are the local Reynolds and Grashof numbers at the horizontal location $x$, based respectively on $V_i$ and ${\rm \Delta} C$, the concentration difference across the boundary layer. We formulate the integral boundary layer equations, with the assumption of no concentration drop within the species boundary layer, which is valid for weak blowing into the thin species boundary layers that occur at the high Schmidt number ($Sc \simeq 600$) of concentration-driven convection. The equations are then numerically solved to show that the species boundary layer thickness $\delta _d = 1.6\,x(Re_x/Gr_x)^{1/4}$, the velocity boundary layer thickness $\delta _v=\delta _d Sc^{1/5}$, the horizontal velocity $u = V_i(Gr_x/Re_x)^{1/4}f(\eta )$, where $\eta =y/\delta _v$, and the drag coefficient based on $V_i$, $C_D = 2.32/\sqrt {J}$. We find that the vertical profile of the horizontally averaged dimensionless concentration across the boundary layer becomes, surprisingly, independent of the blowing and the species diffusion effects to follow a $Gr_y^{2/3}$ scaling, where $Gr_y$ is the Grashof number based on the vertical location $y$ within the boundary layer. We then show that the above profile matches the experimentally observed mean concentration profile within the boundary layers that form on the top surface of a membrane, when a weak flow is forced gravitationally from below the horizontal membrane that has brine above it and water below it. A similar match between the theoretical scaling of the species boundary layer thickness and its experimentally observed variation is also shown to occur.

Our previous study (J. Fluid Mech., vol. 874, 2019, pp. 677–698) confirmed that two different types of $-5/3$ energy spectra (i.e. non-Kolmogorov and quasi-Kolmogorov $-5/3$ spectra) can be found in turbulent flows behind two side-by-side square cylinders. In the upstream region (i.e. $X/T_0=6$ with $T_0$ being the cylinder thickness), albeit the turbulent flow is highly inhomogeneous and intermittent and Kolmogorov's hypothesis does not hold, the energy spectrum exhibits a well-defined $-5/3$ power-law range for over one decade. Meanwhile, the power-law exponent of the corresponding second-order structure function is 1, which is significantly larger than the expected value, i.e. $2/3$. At the downstream location, i.e. $X/T_0=26$, in contrast, the quasi-Kolmogorov $-5/3$ energy spectrum (and also the 2/3 scaling of the second-order structure) can be identified. Through decomposing the streamwise velocity fluctuations into the spanwise average of instantaneous velocity and the turbulent residual, we demonstrate that the non-Kolmogorov $-5/3$ spectrum at $X/T_0=6$ is caused by the turbulent residual part. To shed light on the physics of the scale-by-scale energy transfer, we resort to the Kármán–Howarth–Monin–Hill equation. At $X/T_0=6$, the expected balance between the nonlinear term and the dissipation term cannot be detected. Instead, the contributions from the non-local pressure, advection, nonlinear transport and turbulent transport terms are dominant. Moreover, because the corresponding flow field is highly intermittent, the magnitudes of the non-local pressure, advection, nonlinear transport and turbulent transport terms are significantly larger than that of the dissipation term. At a far downstream location, i.e. $X/T_0=26$, where the dual-wake flow is fully turbulent and becomes much more homogeneous and isotropic, within a short intermediate range the two dominant terms in the two-point turbulent kinetic energy budget are the nonlinear transport term and the dissipation term, which to some extent echoes Kolmogorov's scenario, albeit the contribution from the large-scale advection term cannot be ignored. By comparing the behaviour of the one-point and two-point energy transfer, it can be seen that the two different energy transfer processes are actually closely related, that is, the similar relative importance of the viscous dissipation and the same role of the non-negligible terms in terms of being a source or sink term.

A hyperbolic two-fluid model for gas–particle flow derived using the Boltzmann–Enskog kinetic theory is generalized to include added mass. In place of the virtual-mass force, to guarantee indifference to an accelerating frame of reference, the added mass is included in the mass, momentum and energy balances for the particle phase, augmented to include the portion of the particle wake moving with the particle velocity. The resulting compressible two-fluid model contains seven balance equations (mass, momentum and energy for each phase, plus added mass) and employs a stiffened-gas model for the equation of state for the fluid. Using Sturm's theorem, the model is shown to be globally hyperbolic for arbitrary ratios of the material densities $Z = \rho _f / \rho _p$ (where $\rho _f$ and $\rho _p$ are the fluid and particle material densities, respectively). An eight-equation extension to include the pseudo-turbulent kinetic energy (PTKE) in the fluid phase is also proposed; however, PTKE has no effect on hyperbolicity. In addition to the added mass, the key physics needed to ensure hyperbolicity for arbitrary $Z$ is a fluid-mediated contribution to the particle-phase pressure tensor that is taken to be proportional to the volume fraction of the added mass. A numerical solver for hyperbolic equations is developed for the one-dimensional model, and numerical examples are employed to illustrate the behaviour of solutions to Riemann problems for different material-density ratios. The relation between the proposed two-fluid model and prior work on effective-field models is discussed, as well as possible extensions to include viscous stresses and the formulation of the model in the limit of an incompressible continuous phase.

A study of low-speed streaks (LSSs) embedded in the near-wall region of a turbulent boundary layer is performed using selective visualization and analysis of time-resolved tomographic particle image velocimetry (tomo-PIV). First, a three-dimensional velocity field database is acquired using time-resolved tomo-PIV for an early turbulent boundary layer. Second, detailed time-line flow patterns are obtained from the low-order reconstructed database using ‘tomographic visualizations’ by Lagrangian tracking. These time-line patterns compare remarkably well with previously observed patterns using hydrogen bubble flow visualization, and allow local identification of LSSs within the database. Third, the flow behaviour in proximity to selected LSSs is examined at varying wall distances ($10 < y^+ < 100$) and assessed using time-line and material surface evolution, to reveal the flow structure and evolution of a streak, and the flow structure evolving from streak development. It is observed that three-dimensional wave behaviour of the detected LSSs appears to develop into associated near-wall vortex flow structures, in a process somewhat similar to transitional boundary layer behaviour. Fourth, the presence of Lagrangian coherent structures is assessed in proximity to the LSSs using a Lagrangian-averaged vorticity deviation process. It is observed that quasi-streamwise vortices, adjacent to the sides of the streak-associated three-dimensional wave, precipitate an interaction with the streak. Finally, a hypothesis based on the behaviour of soliton-like coherent structures is made which explains the process of LSS formation, bursting behaviour and the generation of hairpin vortices. Comparison with other models is also discussed.

This study provides a detailed account of stochastic approaches that may be utilized in Eulerian–Lagrangian simulations to account for neighbour-induced drag force fluctuations. The frameworks examined here correspond to Langevin equations for the particle position (PL), particle velocity (VL) and fluctuating drag force (FL). Rigorous derivations of the particle velocity variance (granular temperature) and dispersion resulting from each method are presented. The solutions derived herein provide a basis for comparison with particle-resolved direct numerical simulation. The FL method allows for the most complex behaviour, enabling control of both the granular temperature and dispersion. A Stokes number $St_F$ is defined for the fluctuating force that relates the integral time scale of the force to the Stokes response time. Formal convergence of the FL scheme to the VL scheme is shown for $St_F \gg 1$. In the opposite limit, $St_F \ll 1$, the fluctuating drag forces are highly inertial and the FL scheme departs significantly from the VL scheme.

Motivated by the explosive launch of Sphagnum spores, this experimental study investigates how the vortices generated from two different sources, a piston–cylinder apparatus and a translating bluff body, interact with each other. While there have been numerous studies on the formation of a single vortex ring or multiple vortex rings, little is known about the effect of a translating bluff body on the formation of the coupled vortices. By varying the stroke ratio of the piston and the velocity ratio of the body to the piston, three distinct modes are identified for the mutual interaction between the starting jet from the piston and the wake from the cap: spill mode, attached mode and detached mode. The transitions between the vortex modes are predicted with simple analytical models. For the attached mode that appears at a velocity ratio intermediate between those of the spill mode and the detached mode, the merged flow structure becomes similar to a single vortex ring in the absence of the bluff body. By virtue of stable propagation following the cap, the vortex of the attached mode is capable of transporting a significant fluid volume initially inside the cylinder over a long distance, which shows its effectiveness in transport using a ballistic mechanism.

Symmetric flapping foils are known to produce deflected jets at high frequency–amplitude combinations even at a zero mean angle of attack. This reduces the frequency range of useful propulsive configurations without side force. In this study, we numerically analyse the interaction of these deflected jets for tandem flapping foils undergoing coupled heave-to-pitch motion in a two-dimensional domain. The impact of the flapping Strouhal number, foil spacing and phasing on wake interaction is investigated. Our primary finding is that the back foil is capable of cancelling the wake deflection and mean side force of the front foil, even when located up to five chord lengths downstream. This is achieved by attracting the incoming dipoles and disturbing their cohesion within the limits of the back foil's range of flapping motion. We also show that the impact on cycle-averaged thrust varies from high augmentation to drag generation depending on the wake patterns downstream of the back foil. These findings provide new insights towards the design of biomimetic tandem propulsors, as they expand their working envelope and ability to rapidly increase or decrease the forward speed by manipulating the size of the shed vortices.

Orbits tracked by ellipsoids immersed in inviscid and viscous environments are studied by means of Kirchhoff's equations and high resolution numerical simulations using a variant of the immersed boundary method. We explore the consequences of Kozlov and Onishchenko's theorem of non-integrability of Kirchhoff's equations to show how the fraction of phase space in chaotic orbits is sensitively determined by the body shape, fluid/solid density ratio and the fraction of initial energy in rotational motion. We show how the added mass tensor of the system is an important player in both viscous and inviscid flow, in causing chaos in a triaxial ellipsoid while acting to suppress it in a spheroid. We identify a new integral of motion for a spheroid in inviscid fluid: one component of the generalised angular momentum. A spheroid, which can never execute chaotic dynamics in inviscid flow, is shown to display chaos in viscous flow due to irregular vortex shedding. But the dynamics of the spheroid is restricted whether in viscous or in inviscid flow, unlike in the triaxial ellipsoid, due to our extra integral of motion.

Biflagellate algal cells of the genus Volvox form spherical colonies that propel themselves, vertically upwards in still fluid, by the coordinated beating of thousands of flagella, that also cause the colonies to rotate about their vertical axes. When they are swimming in a chamber of finite depth, pairs (or more) of Volvox carteri colonies were observed by Drescher et al. (Phys. Rev. Lett., vol. 102, 2009, 168101) to exhibit hydrodynamic bound states when they are close to a rigid horizontal boundary. When the boundary is above, the colonies are attracted to each other and orbit around each other in a ‘waltz’; when the boundary is below they perform more complex ‘minuet’ motions. These dances are simulated in the present paper, using a novel ‘spherical squirmer’ model of a colony in which, instead of a time-independent but $\theta$-dependent tangential velocity being imposed on the spherical surface (radius $a$; $\theta$ is the polar angle), a time-independent and uniform tangential shear stress is applied to the fluid on a sphere of radius $(1+\epsilon )a, \epsilon \ll 1$, where $\epsilon a$ represents the length of the flagella. The fluid must satisfy the no-slip condition on the sphere at radius $a$. In addition to the shear stress, the motions depend on two dimensionless parameters that describe the effect of gravity on a colony: $F_g$, proportional to the ratio of the sedimentation speed of a non-swimming colony to its swimming speed, and $G_{bh}$, that represents the fact that colonies are bottom heavy; $G_{bh}$ is the ratio of the time scale to swim a distance equal to the radius, to the time scale for gravitational reorientation of the colony's axis to the vertical when it is disturbed. In addition to reproducing both of the dancing modes, the simulations are able to determine values of $F_g$ and $G_{bh}$ for which they are stable (or not); there is reasonable agreement with the experiments. A far-field model for the minuet motions is also shown to have qualitative agreement, but does not describe some features that are reproduced in the full simulations.

Stratified turbulence has a horizontally layered structure with quasi-two-dimensional vortices due to buoyancy forces that suppress vertical motion. The Prandtl number $\textit {Pr}$ quantifies the relative strengths of viscosity and buoyancy diffusivity, which damp small-scale velocity and buoyancy fluctuations at different microscales. Direct numerical simulations (DNS) require high resolution to resolve the smallest flow features for large $\textit {Pr}$. To reduce computational demand, $\textit {Pr}$ is often set to 1. In this paper, we explore how varying $\textit {Pr}$ affects stratified turbulence. DNS of homogeneous forced stratified turbulence with $0.7 \le \textit {Pr} \le 8$ are performed for four stratification strengths and buoyancy Reynolds numbers $\textit {Re}_b$ between 0.5 and 60. Energy spectra, buoyancy flux spectra, spectral energy flux and physical space fields are compared for scale-specific $\textit {Pr}$-sensitivity. For $\textit {Re}_b \gtrsim 10$, $\textit {Pr}$-dependence in the kinetic energy is mainly found at scales around and below the Kolmogorov scale. The potential energy and flux exhibit more prominent $\textit {Pr}$-sensitivity. As $\textit {Re}_b$ decreases, this $\textit {Pr}$-dependence extends upscale. With increasing $\textit {Pr}$, the spectra suggest eventual convergence to a limiting spectrum shape at large, finite $\textit {Pr}$, at least at scales at and above the Ozmidov scale. The $\textit {Pr}$-sensitivity of the spectra in the most strongly stratified $\textit {Re}_b<1$ case differed from the rest, since large horizontal scales are affected by viscosity and diffusion. These findings suggest that $\textit {Pr}=1$ DNS reasonably approximate $\textit {Pr} > 1$ DNS with large $\textit {Re}_b$, as long as the focus is on kinetic energy at scales much larger than the Kolmogorov scale, but otherwise stray from $\textit {Pr} > 1$ spectra around and below the Kolmogorov scale, and even upscale when $\textit {Re}_b \lesssim 1$.

A turbulence enrichment model for subfilter-scale motions in large eddy simulations (LES) is comprehensively evaluated in the context of a posteriori analysis. The paper further develops the Gabor mode enrichment model first introduced in Ghate & Lele (J. Fluid Mech., vol. 819, 2017, pp. 494–539) by analysing three key requisites of LES enrichment using solenoidal small-scale velocity fields: (a) consistent spectral extrapolation and improvement of resolved single- and two-point second-order correlations; (b) ability to accurately capture the flow physics responsible for temporal decorrelation at small scales; and (c) accurate representation of spatially localized and intermittent interscale energy transfer between scales resolved by the coarse-grid LES and subfilter scales. We argue that the spatially and spectrally localized Gabor wavepackets offer an optimal basis to represent small-scale turbulence within quasi-homogeneous regions, although the alignment of fine-scale vorticity with large-scale strain appears to be somewhat overemphasized. Consequently, we interpret the resulting subfilter scales as those induced by a set of spatially dispersed Burgers–Townsend vortices with orientations determined by the larger scale velocity gradients resolved by the coarse-grid LES. Enrichment of coarse-grid simulations of two high Reynolds number flow configurations, homogeneous isotropic turbulence and a rough-wall turbulent boundary layer show promising results.