In thermal convection for very large Rayleigh numbers ($Ra$), the thermal and viscous boundary layers are expected to undergo a transition from a classical state to an ultimate state. In the former state, the boundary-layer thicknesses follow a laminar-like Prandtl–Blasius–Polhausen scaling, whereas in the latter, the boundary layers are turbulent with logarithmic corrections in the sense of Prandtl and von Kármán. Here, we report evidence of this transition via changes in the boundary-layer structure of vertical natural convection (VC), which is a buoyancy-driven flow between differentially heated vertical walls. The numerical dataset spans $Ra$ values from $10^{5}$ to $10^{9}$ and a constant Prandtl number value of $0.709$. For this $Ra$ range, the VC flow has been previously found to exhibit classical state behaviour in a global sense. Yet, with increasing $Ra$, we observe that near-wall higher-shear patches occupy increasingly larger fractions of the wall areas, which suggest that the boundary layers are undergoing a transition from the classical state to the ultimate shear-dominated state. The presence of streaky structures – reminiscent of the near-wall streaks in canonical wall-bounded turbulence – further supports the notion of this transition. Within the higher-shear patches, conditionally averaged statistics yield a logarithmic variation in the local mean temperature profiles, in agreement with the log law of the wall for mean temperature, and an $Ra^{0.37}$ effective power-law scaling of the local Nusselt number. The scaling of the latter is consistent with the logarithmically corrected $1/2$ power-law scaling predicted for ultimate thermal convection for very large $Ra$. Collectively, the results from this study indicate that turbulent and laminar-like boundary layer coexist in VC at moderate to high $Ra$ and this transition from the classical state to the ultimate state manifests as increasingly larger shear-dominated patches, consistent with the findings reported for Rayleigh–Bénard convection and Taylor–Couette flows.

We present a systematic investigation of the effects of roughness geometry on turbulent Rayleigh–Bénard convection (RBC) over rough plates with pyramid-shaped and periodically distributed roughness elements. Using a parameter $\unicode[STIX]{x1D706}$ defined as the height of a roughness element over its base width, the heat transport, the flow dynamics and the local temperatures are measured for the Rayleigh number range $7.50\times 10^{7}\leqslant Ra\leqslant 1.31\times 10^{11}$ and Prandtl numbers $Pr$ from 3.57 to 23.34 at four values of $\unicode[STIX]{x1D706}$ (0.5, 1.0, 1.9 and 4.0). It is found that the heat transport scaling, i.e. $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ where $Nu$ is the Nusselt number, may be classified into three regimes in turbulent RBC over rough plates. In Regime I, the system is in a dynamically smooth state. The heat transport scaling is the same as that in a smooth cell. In Regimes II and III, the heat transport is enhanced. When $\unicode[STIX]{x1D706}$ is increased from 0.5 to 4.0, $\unicode[STIX]{x1D6FC}$ increases from 0.36 to 0.59 in Regime II and it increases from 0.30 to 0.50 in Regime III. The experiment thus clearly demonstrates that the heat transport scaling in turbulent RBC can be manipulated using $\unicode[STIX]{x1D706}$ in the heat transport enhanced regime. Previous studies suggest that the transition to heat transport enhanced regime, i.e. from Regime I to Regime II, occurs when the thermal boundary layer (BL) thickness becomes smaller than the roughness height. Direct measurements of the viscous BL in the present study suggest that the transition from Regime II to Regime III is likely a result of the viscous BL thickness becoming smaller than the roughness height. The scaling exponent of the Reynolds number $Re$ with respect to $Ra$ changes from 0.471 to 0.551 when $\unicode[STIX]{x1D706}$ is increased from 0.5 to 4.0, suggesting a change of the dynamics of the large-scale circulation. Interestingly, the transition from Regime II to Regime III in terms of the heat transport scaling is not reflected in the $Re$ scaling with $Ra$. It is also found that increasing $\unicode[STIX]{x1D706}$ increases the clustering of thermal plumes which effectively increases the plume lifetime. This leads to a great increase in the probability of observing large temperature fluctuations in the bulk flow, which corresponds to the formation of more coherent plumes or plume clusters that are ultimately responsible for the enhanced heat transport.

We analyse acoustic streaming flows using an arbitrary Lagrangian Eulerian (ALE) perspective. The formulation stems from an explicit separation of time scales resulting in two subproblems: a first-order problem, formulated in terms of the fluid displacement at the fast scale, and a second-order problem, formulated in terms of the Lagrangian flow velocity at the slow time scale. Following a rigorous time-averaging procedure, the second-order problem is shown to be intrinsically steady, and with exact boundary conditions at the oscillating walls. Also, as the second-order problem is solved directly for the Lagrangian velocity, the formulation does not need to employ the notion of Stokes drift, or any associated post-processing, thus facilitating a direct comparison with experiments. Because the first-order problem is formulated in terms of the displacement field, our formulation is directly applicable to more complex fluid–structure interaction problems in microacoustofluidic devices. After the formulation’s exposition, we present numerical results that illustrate the advantages of the formulation with respect to current approaches.

The topology of the incompressible steady three-dimensional flow in a partially filled cylindrical rotating drum, infinitely extended along its axis, is investigated numerically for a ratio of pool depth to radius of 0.2. In the limit of vanishing Froude and capillary numbers, the liquid–gas interface remains flat and the two-dimensional flow becomes unstable to steady three-dimensional convection cells. The Lagrangian transport in the cellular flow is organised by periodic spiralling-in and spiralling-out saddle foci, and by saddle limit cycles. Chaotic advection is caused by a breakup of a degenerate heteroclinic connection between the two saddle foci when the flow becomes three-dimensional. On increasing the Reynolds number, chaotic streamlines invade the cells from the cell boundary and from the interior along the broken heteroclinic connection. This trend is made evident by computing the Kolmogorov–Arnold–Moser tori for five supercritical Reynolds numbers.

In order to make the extraction of tidal current energy economically viable, the power production per turbine must be optimised in each tidal array. Furthermore, the impact of power extraction on the marine flow environment must be understood. These two aims mean that designers must be able to model different configurations of a tidal array in order to create the most efficient, least invasive arrangement. In this paper, an analytical model is developed for array design in idealised rectangular tidal channels with idealised turbines. The model includes the effects of (i) local blockage, (ii) surface deformation and (iii) added drag due to the installation of the array. While these effects have been accounted for individually in past work, the model presented here is the first to include all three such that the interaction between different effects can be understood. Results are presented for optimal local blockage and turbine resistance as functions of inherent channel drag coefficient, channel length and Froude number at various global blockage values. It will be shown that it is necessary to model the effects of local blockage and added drag simultaneously in order to obtain the design parameters of a tidal array (global blockage, local blockage and turbine resistance), which will maximise the power extraction per turbine. Neglecting either effect will lead to an array design with lower power extraction than the optimum, the addition of unnecessary extra turbines and higher lost power from the array.

A hydrodynamic model and flow visualization experiments are used to understand the mechanisms through which soluble surfactants can influence the onset of dynamic wetting failure. In the model, a Newtonian liquid displaces air in a rectangular channel in the absence of inertia. A Navier-slip boundary condition and constant contact angle are used to describe the dynamic contact line, and surfactants are allowed to adsorb to the interface and moving channel wall (substrate). The Galerkin finite element method is used to calculate steady states and identify the critical capillary number $Ca^{crit}$ at which wetting failure occurs. It is found that surfactant solubility weakens the influence of Marangoni stresses, which tend to promote the onset of wetting failure. Adsorption of surfactants to the substrate can delay the onset of wetting failure due to the emergence of Marangoni stresses that thicken the air film near the dynamic contact line. The experiments indicate that $Ca^{crit}$ increases with surfactant concentration. For the more viscous solutions used, this behaviour can largely be explained by accounting for changes to the mean surface tension and static contact angle produced by surfactants. For the lowest-viscosity solution used, comparison between the model predictions and experimental observations suggests that other surfactant-induced phenomena such as Marangoni stresses may play a more important role.

Flow complexity is related to a number of phenomena in science and engineering and has been approached from the perspective of chaotic dynamical systems, ergodic processes or mixing of fluids, just to name a few. To the best of our knowledge, all existing methods to quantify flow complexity are only valid for infinite time evolution, for closed systems or for mixing of two substances. We introduce an index of flow complexity coined interlacing complexity index (ICI), valid for a single-phase flow in an open system with inlet and outlet regions, involving finite times. ICI is based on Shannon’s mutual information (MI), and inspired by an analogy between inlet–outlet open flow systems and communication systems in communication theory. The roles of transmitter, receiver and communication channel are played, respectively, by the inlet, the outlet and the flow transport between them. A perfectly laminar flow in a straight tube can be compared to an ideal communication channel where the transmitted and received messages are identical and hence the MI between input and output is maximal. For more complex flows, generated by more intricate conditions or geometries, the ability to discriminate the outlet position by knowing the inlet position is decreased, reducing the corresponding MI. The behaviour of the ICI has been tested with numerical experiments on diverse flows cases. The results indicate that the ICI provides a sensitive complexity measure with intuitive interpretation in a diversity of conditions and in agreement with other observations, such as Dean vortices and subjective visual assessments. As a crucial component of the ICI formulation, we also introduce the natural distribution of streamlines and the natural distribution of world-lines, with invariance properties with respect to the cross-section used to parameterize them, valid for any type of mass-preserving flow.

We study the use of small counter-rotating cylinders to control the streaming flow past a larger main cylinder for drag reduction. In a water tunnel experiment at a Reynolds number of 47 000 with a three-dimensional and turbulent wake, particle image velocimetry (PIV) measurements show that rotating cylinders narrow the mean wake and shorten the recirculation length. The drag of the main cylinder was measured to reduce by up to 45 %. To examine the physical mechanism of the flow control in detail, a series of two-dimensional numerical simulations at a Reynolds number equal to 500 were conducted. These simulations investigated a range of control cylinder diameters in addition to rotation rates and gaps to the main cylinder. Effectively controlled simulated flows present a streamline that separates from the main cylinder, passes around the control cylinder, and reattaches to the main cylinder at a higher pressure. The computed pressure recovery from the separation to reattachment points collapses with respect to a new scaling, which indicates that the control mechanism is viscous.

Rapid distortion of turbulence (RDT) theory is applied to homogeneous, isotropic turbulence incident on a horizontal axis turbine rotor such as a wind turbine or tidal-stream turbine. The mean flow field of the rotor which distorts the turbulence is represented by the commonly used axisymmetric actuator disc model due to Betz and Joukowski. The fluctuating streamwise component of the turbulence distorted by this field is calculated at the actuator disc plane. Turbulence velocity intensities and spectra are evaluated for general ratios of turbulence integral length scale to the rotor diameter, including the small-scale limit for which the original homogeneous strain analysis of Batchelor and Proudman may be applied. The distortion of the mean velocity profile of an incident rotor wake which may be considered a zero frequency disturbance relevant to wind and tidal turbine operation in large arrays is also analysed by the same method, treating it as a deterministic disturbance in the incident flow.

Direct numerical simulations (DNS) of subsonic and supersonic impinging jets with Reynolds numbers of 3300 and 8000 are carried out to analyse their statistical properties with respect to heat transfer. The Reynolds number range is at low or moderate values in terms of practical applications, but very high regarding the technical possibilities of DNS. A Reynolds number of 8000 is technically relevant for the cooling of turbine blades. In this case, the flow is dominated by primary and secondary vortex rings. Statistics of turbulent heat fluxes and Reynolds stresses as well as the Nusselt number are provided and brought into accordance with these vortices. Velocity and temperature fluctuations were found to have a positive influence on cooling of the impinging plate. Beside the description of the flow, a second aim of this article is the provision of data for improvement of turbulence models. Modern large eddy simulations are still not able to precisely predict impingement heat transfer (Dairay et al., Intl J. Heat Fluid Flow, vol. 50 (0), 2014, pp. 177–187). Common relations between heat and mass transfer respectively temperature and velocity fields are applied to the impinging jet. These relations include the Reynolds and Chilton Colburn analogy, the Crocco–Busemann relation and the generalised Reynolds analogy (GRA). It was found that the first two deliver useful values if the distance to the jet axis is larger than one diameter, away from the strong pressure gradient around the stagnation point. The GRA, in contrast, precisely predicts the mean temperature field if no axial velocity gradient is present. The estimation of temperature fluctuations according to the GRA fails. As third main topic of this article, the influence of the Mach number on heat transfer and the flow field, is studied. Against the common practise of neglecting compressibility effects in experimental Nusselt correlations, we observed that higher Mach numbers (up to 1.1) have a positive influence on heat transfer in the deflection zone due to higher flow fluctuations.

This paper performs a numerical study on the interaction of a planar shock wave with a water column embedded with/without a cavity of different sizes at high Weber numbers. The conservative-type Euler and non-conservative scalar two-equations representing the transportation of two-phase properties consist of the diffusion interface capture models. The numerical fluxes are computed by the Godunov-type Harten-Lax–van Leer contact Riemann solver coupled with an incremental fifth-order weighted essentially non-oscillatory (WENO) scheme. A third-order total variation diminishing (TVD) Runge–Kutta scheme is used to advance the solution in time. The morphology and dynamical characteristics are analysed qualitatively and quantitatively to demonstrate the breakup mechanism of the water column and formation of transverse jets under different incident shock intensities and embedded-cavity sizes. The jet tip velocities are extracted by analysing the interface evolution. The liquid column is prone to aerodynamic breakup with the formation of micro-mist at later stages instead of liquid evaporation because of the weakly heating effects of the surrounding air. It is numerically confirmed that the liquid-phase pressure will drop below the saturated vapour pressure, and the low pressure can be sustained for a certain time because of the focusing of the expansion wave, which accounts for the cavitation inside the liquid water column. The geometrical parameters of the deformed water column are identified, showing that the centreline width decreases but the transverse height increases nonlinearly with time. The deformation rates are nonlinearly correlated under different Mach numbers. The first transverse jet is found for a water column with an embedded cavity, whereas the water hammer shock and second jet do not occur under the impact of low intensity incident shock waves. The $x$-velocity component recorded at the rear stagnation point can remain unchanged for a comparable time after a declined evolution, which indicates that the downstream wall of the shocked water ring somehow moves uniformly. It can be explained that the acceleration of the downstream wall is balanced by the trailing shedding vortex, and this effect is more evident under higher Mach numbers. The increased enstrophy, mainly generated at the interface, demonstrates the competition of the baroclinic effects of the shock wave impact over dilatation.

We consider finite-amplitude convection in a mushy layer during the primary solidification of a ternary alloy. Previous linear stability theories applied to ternary alloy primary-solidification models have identified an exceptional class of direct convective instability when all the individual stratifying agencies (one thermal and two solutal) were statically stabilizing. A reduced model, in which the effects of latent heat, solute rejection and background solidification are neglected, contains the essential interactions that admit qualitatively the same instability. We examine pattern selection for steady convection in this model. We find that roll, square or hexagonal convection patterns can be nonlinearly stable, depending on the relative importance of a number of physical effects, namely the solutal diffusion rates, the liquidus slopes and the background thermal and solutal density stratifications. The results for a special case are found to isolate a purely double-diffusive phase-change mechanism of pattern selection. Subcritical behaviour is identified inside the domain of individual static stability. A physical system is proposed that may be a promising one in which to experimentally identify these novel instabilities.

Two sets of experiments are performed to study variable-density effects in turbulent round jets with coflow at density ratios, $s=4.2$ and $s=1.2$. Ten thousand instantaneous realisations of simultaneous two-dimensional particle image velocimetry and planar laser-induced fluorescence at three axial locations in the momentum-dominated region of the jet allow us to calculate the full turbulent kinetic energy (t.k.e.) budgets, providing insights into the mechanisms of density fluctuation correlations both axially and radially in a non-Boussinesq flow. The strongest variable-density effects are observed within the velocity half-width of the jet, $r_{\tilde{u} _{1/2}}$. Variable-density effects decrease the Reynolds stresses via increased turbulent mass flux in the heavy jet, as shown by previous jet centreline measurements. Radial profiles of turbulent flux show that in the lighter jet t.k.e. is moving away from the centreline, while in the heavy jet it is being transported both inwards towards the centreline and radially outwards. Negative t.k.e. production is observed in the heavy jet, and we demonstrate that this is caused by both reduced gradient stretching in the axial direction and increased turbulent mass fluxes. Large differences in advection are also observed between the two jets. The air jet has higher total advection caused by strong axial components, while density fluctuations in the heavy jet reduce the axial advection significantly. The budget mechanisms in the non-Boussinesq regime are best understood using effective density and velocity half-width, $\unicode[STIX]{x1D70C}_{eff}\bar{u}_{1,CL}^{3}/r_{\tilde{u} _{1/2,eff}}$, a modification of previous scaling.

We examine the stability of laminar high-Reynolds-number flow through an asymmetric flexible-walled channel driven by a fixed upstream flux and subject to a (large) uniform external pressure. We construct a long-wavelength, spatially one-dimensional model using a flow profile assumption, modelling the flexible wall as a thin tensioned membrane subject to a large axial pre-stress. We numerically construct the non-uniform static shape of the flexible wall and consider its stability using both a global eigensolver and numerical simulation of the nonlinear governing equations. The system admits multiple static solutions, including a highly collapsed steady state where the membrane has a single constriction which increases with increasing external pressure. We demonstrate that the non-uniform static state is unstable to two distinct (infinite) families of normal modes which we characterise in the limit of large external pressure. In particular, there is a family of low-frequency oscillatory modes which each persist to low membrane tensions, where the most unstable mode has an oscillating membrane profile which is outwardly bulged at the centre of the domain with a narrow constriction at the downstream end. In addition, there is a family of high-frequency oscillatory modes which are each unstable beyond a critical value of the tension within a two-branch neutral curve. Unstable modes along the lower branch of the neutral curve are sustained by a leading-order balance between unsteady inertia and the restoring force of membrane tension along the channel. In addition, we elucidate the mechanism of energy transfer to sustain the self-excited oscillations: oscillations decrease the mean maximal constriction of the channel over a period, which reduces the overall dissipation of the mean flow and releases energy to sustain the instability. Fully nonlinear simulations indicate that as the Reynolds number increases these unstable normal modes can grow supercritically into sustained large-amplitude ‘slamming’ oscillations, where the membrane is periodically drawn very close to the opposite rigid wall before recovering.

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

When a large body of fluid is heated from below at a horizontal surface, the heat diffuses into the fluid, giving rise to a gravitationally unstable layer adjacent to the boundary. A consideration of the instantaneous Rayleigh number using the thickness of this buoyant layer as a length scale would lead one to expect that the heated fluid is initially stable, and only becomes unstable after a finite time. This transition would also apply to other situations, such as heating a large body of fluid from the side, where a buoyant upward flow develops near the boundary. In such cases, when the evolving thermal boundary layer first becomes unstable, the time scale for the growth of the instabilities may be comparable to the time scale of the evolution of the background temperature profile, and so analytical approximations such as the quasi-static approximation, where the time evolution of the background state is ignored, are not strictly appropriate. We develop a numerical scheme where we find the optimal growth of linear perturbations to the background flow over a given time interval. Part of this problem is to determine an appropriate measure of the amplitude to the disturbances, as inappropriate choices can lead to apparent growth of disturbances over finite time intervals even when the fluid is stable. By considering the Rayleigh–Bénard problem, we show that these problems can be avoided by choosing a measure of the amplitude that uses both the velocity and temperature perturbations, and which minimizes the maximum growth. We apply our analysis to the problems of heating a semi-infinite body of fluid from horizontal and vertical boundaries. We will show that for heating from a vertical boundary there are large- and small-Prandtl-number modes. For some Prandtl numbers, both modes may play a role in the growth of instabilities. In some cases there is transition during the evolution of the most unstable instabilities in fluids such as water, where initially the instabilities are large-Prandtl-number modes and then morph into small-Prandtl-number modes part of the way through their evolution.

Diffusion-driven flow occurs when any insulated sloping surface is in contact with a quiescent stratified and viscous fluid. This startling fluid motion is generally very slow, and is caused by a hydrostatic pressure imbalance due to bending of isotherms near the surface. In contrast to previous studies of the phenomenon, the low-Reynolds-number case is considered here, for which the induced steady motion is influenced by viscous and density diffusion over a much larger length scale than the size of the insulated object. The relevant linear equations of steady two-dimensional motion in a linearly stratified fluid are solved for a circular cylinder using a matched two-region approach that yields analytical solutions for the streamfunction and the density variations both close to and far from the object. The exact analytical expressions for the solutions in the ‘outer-flow region’ are new, and after matching enable accurate solutions to be evaluated easily at any point. Similar qualitative behaviour is expected under similar conditions near isolated objects of other shapes, including for a sphere. Implications for multiple objects are also discussed.

The flow of a viscous liquid film coating the interior of a vertical tube is studied for the case when the film is driven upwards against gravity by a constant volume flux of air through the centre of the tube. A nonlinear model exploiting the slowly varying liquid–air interface is first developed to estimate the interfacial stresses created by the airflow. A comparison of the model with both experiments and previously developed theoretical results is conducted for two geometrical settings: channel and pipe flow. In both geometries, the model compares reasonably well with previous experiments. A long-wave asymptotic theory is then developed for the air–liquid interface taking into account the estimated free-surface stresses created by the airflow. The stability of small interfacial disturbances is studied analytically, and it is shown that the modelled free-surface stresses contribute to both an increased upwards disturbance velocity and a more rapid instability growth than those of a previously developed ‘locally Poiseuille’ model. Numerical solutions to the long-wave model exhibit saturated waves, whose profiles and velocities show substantial improvement with respect to the previous model predictions. The theoretical results are compared with new experiments for a modified version of the set-up described in Part 1.

This work reports an experimental characterisation of the flow properties in a homogeneous bubble swarm rising at high Reynolds numbers within a homogeneous and isotropic turbulent flow. Both the gas volume fraction $\unicode[STIX]{x1D6FC}$ and the velocity fluctuations $u_{0}^{\prime }$ of the carrier flow before bubble injection are varied, respectively, in the ranges $0\,\%<\unicode[STIX]{x1D6FC}<0.93\,\%$ and $2.3~\text{cm}~\text{s}^{-1}<u_{0}^{\prime }<5.5~\text{cm}~\text{s}^{-1}$. The so-called bubblance parameter ($b=V_{r}^{2}\unicode[STIX]{x1D6FC}/u_{0}^{\prime 2}$, where $V_{r}$ is the bubble relative rise velocity) is used to compare the ratio of the kinetic energy generated by the bubbles to the one produced by the incident turbulence, and is varied from 0 to 1.3. Conditional measurements of the velocity field downstream of the bubbles in the vertical direction allow us to disentangle three regions that have specific statistical properties, namely the primary wake, the secondary wake and the far field. While the fluctuations in the primary wake are similar to that of a single bubble rising in a liquid at rest, the statistics of the velocity fluctuations in the far field follow a Gaussian distribution, similar to that produced by the homogenous and isotropic turbulence at the largest scales. In the secondary wake region, the conditional probability density function of the velocity fluctuations is asymmetric and shows an exponential tail for the positive fluctuations and a Gaussian one for the negative fluctuations. The overall agitation thus results from the combination of these three contributions and depends mainly on the bubblance parameter. For $0<b<0.7$, the overall velocity fluctuations in the vertical direction evolve as $b^{0.4}$ and are mostly driven by the far-field agitation, whereas the fluctuations increase as $b^{1.3}$ for larger values of the bubblance parameter ($b>0.7$), in which the significant contributions come both from the secondary wake and the far field. Thus, the bubblance parameter is a suitable parameter to characterise the evolution of liquid agitation in bubbly turbulent flows.

The purpose of this paper is to characterize and model waves that are observed within the potential core of subsonic jets and relate them to previously observed tones in the near-nozzle region. The waves are detected in data from a large-eddy simulation of a Mach 0.9 isothermal jet and modelled using parallel and weakly non-parallel linear modal analysis of the Euler equations linearized about the turbulent mean flow, as well as simplified models based on a cylindrical vortex sheet and the acoustic modes of a cylindrical soft duct. In addition to the Kelvin–Helmholtz instability waves, three types of waves with negative phase velocities are identified in the potential core: upstream- and downstream-propagating duct-like acoustic modes that experience the shear layer as a pressure-release surface and are therefore radially confined to the potential core, and upstream-propagating acoustic modes that represent a weak coupling between the jet core and the free stream. The slow streamwise contraction of the potential core imposes a frequency-dependent end condition on the waves that is modelled as the turning points of a weakly non-parallel approximation of the waves. These turning points provide a mechanism by which the upstream- and downstream-travelling waves can interact and exchange energy through reflection and transmission processes. Paired with a second end condition provided by the nozzle, this leads to the possibility of resonance in limited frequency bands that are bound by two saddle points in the complex wavenumber plane. The predicted frequencies closely match the observed tones detected outside of the jet. The vortex-sheet model is then used to systematically explore the Mach number and temperature ratio dependence of the phenomenon. For isothermal jets, the model suggests that resonance is likely to occur in a narrow range of Mach number, $0.82<M<1$.