Dr Matthew Turner

The interaction of two line vortices of differing strengths in the presence of a circular cylinder is considered. Explicit criteria are derived, a function of vortex strengths (including strengths of opposite signs) and the cylinder radius, which separate different behaviors of the system. If the initial position of the vortices satisfies these criteria, they will undergo a periodic leapfrogging motion as they rotate around the cylinder; otherwise, the vortices still interact weakly with one another except without leapfrogging. This is in contrast to the planar wall case where if no periodic leapfrogging occurs, the vortices move apart and do not interact with each other. Numerical results for initial vortex positions which do and do not satisfy these criteria are presented to demonstrate the different motions available, as well as the robustness of the criteria. Published under license by AIP Publishing.

The Earth’s atmosphere is an aerosol, it contains suspended particles. When air flows over an obstacle such as an aircraft wing or tree branch, these particles may not follow the same paths as the air flowing around the obstacle. Instead the particles in the air may deviate from the path of the air and so collide with the surface of the obstacle. It is known that particle inertia can drive this deposition, and that there is a critical value of this inertia, below which no point particles deposit. Particle inertia is measured by the Stokes number, St. We show that near the critical value of the Stokes number, Stc, the amount of deposition has the unusual scaling law of exp(-1/(St-Stc)1/2). The scaling is controlled by the stagnation point of the flow. This scaling is determined by the time for the particle to reach the surface of the cylinder varying as 1/(St-Stc)1/2, together with the distance away from the stagnation point (perpendicular to the flow direction) increasing exponentially with time. The scaling law applies to inviscid flow, a model for flow at high Reynolds numbers. The unusual scaling means that the amount of particles deposited increases only very slowly above the critical Stokes number. This has consequences for applications ranging from rime formation and fog harvesting to pollination.

We elucidate the origin of the critical Stokes number $\mathrm{St}_\mathrm{c}$ for inertial particle capture by obstacles in flow fields, and explain the empirical observation made by Araujo et al. [Phys. Rev. Lett. 97, 138001 (2006)] that the capture efficiency grows as $(\mathrm{St}-\mathrm{St}_\mathrm{c})^\beta$ with $\beta=1/2$ for some critical Stokes number. This behaviour, which is inaccessible to classic perturbation theory, derives from the global structure of the phase space of particle trajectories from which viewpoint it is both generic and inevitable except in the limit of highly singular stagnation point flows which we example. In the context of airborne disease transmission, the phenomenon underlies the sharp decline in filtration efficiency of face coverings for micron-sized aerosol droplets.

The damping efficiency of vertical porous baffles is investigated for a dynamically coupled fluid-vessel system. The system comprises of a two-dimensional vessel, with a rectangular cross-section, partially filled with fluid, undergoing rectilinear motions with porous baffles obstructing the fluid motion. The baffles pierce the surface of the fluid, thus the problem can be considered as separate fluid filled regions of the vessel, connected by infinitely thin porous baffles, at which transmission conditions based on Darcy's law are applied. The fluid is assumed to be inviscid, incompressible and irrotational such that the flow in each region is governed by a velocity potential. The application of Darcy's law at the baffles is significant as it makes the system non-conservative, and thus the resulting characteristic equation for the normal modes leads to damped modes coupled to the moving vessel. Numerical evaluations of the characteristic equation show that the lowest frequency mode typically has the smallest decay rate, and hence will persist longest in an experimental setup. The maximum decay rate of the lowest frequency mode occurs when the baffles split the vessel into identically sized regions.

We formulate an effective numerical scheme that can readily, and accurately, calculate the dynamics of weakly interacting multi-pulse solutions of the quintic complex Ginzburg-Landau equation (QCGLE) in one space dimension. The scheme is based on a global centre-manifold reduction where one considers the solution of the QCGLE as the composition of individual pulses plus a remainder function, which is orthogonal to the adjoint eigenfunctions of the linearised operator about a single pulse. This centre-manifold projection overcomes the difficulties of other, more orthodox, numerical schemes, by yielding a fast-slow system describing 'slow' ordinary differential equations for the locations and phases of the individual pulses, and a 'fast' partial differential equation for the remainder function. With small parameter ϵ = e −λr d 0 where λr is a constant and d0 > 0 is the minimal pulse separation distance, we write the fast-slow system in terms of first-order and second-order correction terms only, a formulation which is solved more efficiently than the full system. This fast-slow system is integrated numerically using adaptive time-stepping. Results are presented here for two-and three-pulse interactions. For the two-pulse problem, cells of periodic behaviour, separated by an infinite set of heteroclinic orbits, are shown to 'split' under perturbation creating complex spiral behaviour. For the case of three pulse interaction a range of dynamics, including chaotic pulse interaction, are found. While results are presented for pulse interaction in the QCGLE, the numerical scheme can also be applied to a wider class of parabolic PDEs.

A spatiotemporal stability analysis is conducted on a flow representing both jets and wakes, subject to confinement by identical compliant walls. The walls are placed at equal distances from the fluid center line for a range of wall and flow parameters. By following the position of special saddle points (pinch points) of the dispersion relation in the complex wave-number plane, the absolute and convective instability stability properties of the flow are determined for various system parameters. The compliant walls are shown to modify the shear-induced instabilities, which exist in the rigid wall case, as well as introduce new additional instabilities originating from the presence of the wall itself. It is observed that under certain system parameters, these wall-induced modes become the dominant instability present in the system and can induce an absolute instability into flows which are only convectively unstable when confined by rigid walls, as well as extending the region of absolute instability to large confinement parameters. Results are presented for both a piecewise linear velocity profile and a smooth velocity profile, with the results of the two studies in qualitative agreement.

This paper focuses on the problem of inviscid, irrotational, incompressible fluid sloshing in a rectangular vessel with rigid, impermeable side-wall baffles, and investigates the feasibility of using time-dependent conformal mappings to numerically simulate the evolution of the unknown free-surface in fully-dynamic simulations. An algorithm which uses conformal mappings of a multiply-connected domain to relate the conjugate harmonic functions along the free-surface is documented, and kinematic results presented for a prescribed free-surface motion. The results show that the specific mapping for an infinite depth fluid has one free, within specific bounds, mapping parameter, while the mapping for finite depth fluids has two free mapping parameters. It is shown that having two free parameters gives a wider range of situations under which the conformal mapping can be computed, and it is concluded that the finite depth mapping should be used (in the appropriate limit) even for infinite depth simulations. Overall it is found that a computationally efficient algorithm can be devised to relate the conjugate harmonic functions along the free-surface of the flow domain.

The flow of one Howarth stagnation-point flow impinging directly on another Howarth stagnation-point flow is studied, and an exact similarity solution to the Navier-Stokes equations is found. The upper layer fluid has density ρ1 and kinematic viscosity ʋ1 while the lower layer fluid has density ρ2 and kinematic viscosity ʋ2 and the two fluids are assumed to be immiscible. This problem has potentially five independent parameters to investigate, but application of the continuity of the normal stresses at the interface imposes restrictions which reduces the problem to one with three independent parameters, namely a ratio σ of strain rates and the fluid parameter ratios ρ = ρ1/ρ2 and ʋ = ʋ1/ʋ2. Numerical results are presented for selected values of ρ and ʋ for a range of σ and show that stable results exist for all values of σ > 0, and for a range of negative σ values. Sample stable velocity profiles are also presented.

The problem of dynamic coupling between a rectangular container undergoing planar pendular oscillations and its interior potential fluid sloshing is studied. The Neumann boundary-value problem for the fluid motion inside the container is deduced from the Bateman–Luke variational principle. The governing integro-differential equation for the motion of the suspended container, from a single rigid pivoting rod, is the Euler-Lagrange equation for a forced pendulum. The fluid and rigid-body partial differential equations are linearised, and the characteristic equation for the natural and resonant frequencies of the coupled dynamical system are presented. It is found that internal 1 : 1 resonances exist for an experimentally realistic setup, which has important physical implications. In addition, a new instability is found in the linearised coupled problem whereby instability occurs when the rod length is shorter than a critical length, and an explicit formula is given.

This paper examines the dynamic coupling between a sloshing fluid and the motion of the vessel containing the fluid. A mechanism is identified which leads to an energy exchange between the vessel dynamics and fluid motion. It is based on a 1:1 resonance in the linearized equations, but nonlinearity is essential for the energy transfer. For definiteness, the theory is developed for Cooker's pendulous sloshing experiment. The vessel has a rectangular cross section, is partially filled with a fluid, and is suspended by two cables. A nonlinear normal form is derived close to an internal 1:1 resonance, with the energy transfer manifested by a heteroclinic connection which connects the purely symmetric sloshing modes to the purely anti-symmetric sloshing modes. Parameter values where this pure energy transfer occurs are identified. In practice, this energy transfer can lead to sloshing-induced destabilization of fluid-carrying vessels.

A unified formulation for stagnation-point flows and linearly stretching plates is given wherein the two can occur separately or in unison. Reductions to known cases are given. It is noticed that previous work on stretching plates beneath planar and axisymmetric stagnation point flows have respectively aligned planar stretching and axisymmetric stretching. The general formulation reveals other combinations of stretching beneath stagnation-point flows exist and three new cases are studied in detail. The linear stability of dual and multiple solutions are calculated.

A numerical method is proposed to solve the two-layer inviscid, incompressible and immiscible 1D shallow-water equations in a moving vessel with a rigid-lid with different boundary conditions based on the high-resolution f-wave finite volume methods due to Bale et al. (2002). The method splits the jump in the fluxes and source terms including the pressure gradient at the rigid-lid into waves propagating away from each grid cell interface. For the influx–efflux boundary conditions the time dependent source terms are handled via a fractional step approach. In the linear case the numerical solutions are validated by comparison with the exact analytical solutions. Numerical solutions presented for the nonlinear case include shallow-water sloshing waves due to prescribed surge motion of the vessel.

A class of augmented approximate Riemann solvers due to George (2008) [12] is extended to solve the shallow-water equations in a moving vessel with variable bottom topography and variable cross-section with wetting and drying. A class of Roe-type upwind solvers for the system of balance laws is derived which respects the steady-state solutions. The numerical solutions of the new adapted augmented f-wave solvers are validated against the Roe-type solvers. The theory is extended to solve the shallow-water flows in moving vessels with arbitrary cross-section with influx–efflux boundary conditions motivated by the shallow-water sloshing in the ocean wave energy converter (WEC) proposed by Offshore Wave Energy Ltd. (OWEL) [1]. A fractional step approach is used to handle the time-dependent forcing functions. The numerical solutions are compared to an extended new Roe-type solver for the system of balance laws with a time-dependent source function. The shallow-water sloshing finite volume solver can be coupled to a Runge–Kutta integrator for the vessel motion.

The coupled motion of shallow-water sloshing in a horizontally translating upright annular vessel is considered. The vessel motion is restricted to a single space dimension, such as for Tuned Liquid Damper systems. For particular parameters, the system is shown to support an internal 1 : 1 resonance, where the frequency of coupled sloshing mode which generates the vessel motion is equal to the frequency of a sloshing mode which occurs in a static vessel. Using a Lagrangian Particle Path formation, the fully nonlinear motion of the system is simulated using an efficient numerical symplectic integration scheme. The scheme is based upon the implicit-midpoint-rule which conserves energy and preserves the energy partition between the fluid and the vessel over many timesteps. Linear and nonlinear results are presented, including those showing the system transitioning to higher frequency eigenmodes as the fluid depth is reduced.

The steady radial stretching of a disk beneath a rigidly rotating flow with constant angular velocity is considered. The steady base flow is determined numerically for both a stretching and a shrinking disk. The convective instability properties of the flow are examined using temporal stability analysis of the governing Rayleigh equation, and typically for small to moderate radial wave numbers, the range of azimuthal wave numbers β , over which the flow is unstable increases for both a stretched and a shrinking disk, compared to the unstretched case. The inviscid absolute instability properties of the resulting base flows are also examined using spatiotemporal stability analysis. For suitably large stretching rates, the flow is absolutely unstable in only a small range of positive β . For small stretching rates there exists a second region of absolute instability for a range of negative β values. In this region the “effective” two-dimensional base flow, comprised of a linear combination of the radial and azimuthal velocity profiles that enter the Rayleigh equation calculation, has a critical point (unlike for β > 0 ) that can dominate the absolute instability growth rate contribution compared to the shear layer component. A similar behavior is found to occur for a radially shrinking disk, except these profiles have a strong shear layer structure and hence are more unstable than the stretching disk profiles. We thus find for a suitably large shrinking rate the absolute instability contribution from the critical point becomes subdominant to the shear layer contribution.

This paper presents a single-column model of moist atmospheric convection. The problem is formulated in terms of conservation laws for mass, moist potential temperature and specific humidity of air parcels. A numerical adjustment algorithm is devised to model the convective adjustment of the column to a statically stable equilibrium state for a number of test cases. The algorithm is shown to converge to a weak solution with saturated and unsaturated parcels interleaved in the column as the vertical spatial grid size decreases. Such weak solutions would not be obtainable via discrete PDE methods, such as finite differences or finite volumes, from the governing Eulerian PDEs. An equivalent variational formulation of the problem is presented and numerical results show equivalence with those of the adjustment algorithm. Results are also presented for numerical simulations of an ascending atmospheric column as a series of steady states. The adjustment algorithm developed in this paper is advantageous over similar algorithms because first it includes the latent heating of parcels due to the condensation of water vapour, and secondly it is computationally efficient making it implementable into current weather and climate models.

This paper examines two key features of time-dependent conformal mappings in doubly-connected regions, the evolution of the conformal modulus Q(t) and the boundary transformation generalizing the Hilbert transform. It also applies the theory to an unsteady free surface flow. Focusing on inviscid, incompressible, irrotational fluid sloshing in a rectangular vessel, it is shown that the explicit calculation of the conformal modulus is essential to correctly predict features of the flow. Results are also presented for fully dynamic simulations which use a time-dependent conformal mapping and the Garrick generalization of the Hilbert transform to map the physical domain to a time-dependent rectangle in the computational domain. The results of this new approach are compared to the complementary numerical scheme of Frandsen (2004) (J. Comp. Phys. 196, 53-87) and it is shown that correct calculation of the conformal modulus is essential in order to obtain agreement between the two methods.

The steady laminar flow of two immiscible, uniformly rotating fluid layers is studied and exact similarity solutions of the axisymmetric Navier-Stokes equations in cylindrical polar coordinates are found. The similarity solutions occur with a flat interface at z = 0 under the parameter restriction that σ²ρ = 1 where σ is the ratio of the fluid angular velocities at z = ±∞ and ρ is the density ratio of the two fluids. Under this restriction the problem reduces to one with two independent parameters, σ and μ, which is the viscosity ratio of the fluids. Numerical results of the resulting system of ODEs are found for selected values of μ and σ, and it is shown that similarity solutions exist for σc(μ)≤ σ ≤ 1 where σc(μ) ˂ 0 (i.e. counter-rotating flows). For σ ˂ 0 the lower fluid can become divided into two distinct recirculation regions between which fluid cannot transfer.

In this paper, we examine the large Reynolds number (Re) asymptotic structure of the wave number in the Orr–Sommerfeld region for the Blasius boundary layer on a semi-infinite flat plate given by Goldstein (1983, J. Fluid Mech., 127, 59–81). We show that the inclusion of the term which contains the leading-order non-parallel effects, at O(Re− 1/2), leads to a non-uniform expansion. By considering the far downstream form of each term in the asymptotic expansion, we derive a length scale at which the non-uniformity appears, and compare this position with the position seen in plots of the wave number.

The spreading and diffusion of two-dimensional vortices subject to weak external random strain fields is examined. The response to such a field of given angular frequency depends on the profile of the vortex and can be calculated numerically. An effective diffusivity can be determined as a function of radius and may be used to evolve the profile over a long time scale, using a diffusion equation that is both nonlinear and non-local. This equation, containing an additional smoothing parameter, is simulated starting with a Gaussian vortex. Fine scale steps in the vorticity profile develop at the periphery of the vortex and these form a vorticity staircase. The effective diffusivity is high in the steps where the vorticity gradient is low: between the steps are barriers characterized by low effective diffusivity and high vorticity gradient. The steps then merge before the vorticity is finally swept out and this leaves a vortex with a compact core and a sharp edge. There is also an increase in the effective diffusion within an encircling surf zone. In order to understand the properties of the evolution of the Gaussian vortex, an asymptotic model first proposed by Balmforth, Llewellyn Smith & Young (J. Fluid Mech., vol. 426, 2001, p. 95) is employed. The model is based on a vorticity distribution that consists of a compact vortex core surrounded by a skirt of relatively weak vorticity. Again simulations show the formation of fine scale vorticity steps within the skirt, followed by merger. The diffusion equation we develop has a tendency to generate vorticity steps on arbitrarily fine scales; these are limited in our numerical simulations by smoothing the effective diffusivity over small spatial scales.

This paper studies the use of braiding fluid particles to quantify the amount of mixing within a fluid flow. We analyze the pros and cons of braid methods by considering the motion of three or more fluid particles in a coherent vortex structure. The relative motions of the particles, as seen in a space–time diagram, produce a braid pattern, which is correlated with mixing and measured by the braiding factor. The flow we consider is a Gaussian vortex within a rotating strain field that generates cat's eyes in the vortex. We also consider a modified version of this strain field that contains a resonance frequency effect that produces multiple sets of cat's eyes at different radii. As the thickness of the cat's eyes increases, they interact with one another and produce complex Lagrangian motion in the flow that increases the braiding of particles, hence implying more mixing within the vortex. It is found that calculating the braiding factor using only three fluid particles gives useful information about the flow, but only if all three particles lie in the same region of the flow, i.e. this gives good local information. We find that we only require one of the three particles to trace a chaotic path to give an exponentially growing braiding factor. i.e. a non-zero 'braiding exponent'. A modified braiding exponent is also introduced which removes the spurious effects caused by the rotation of the fluid. This analysis is extended to a more global approach by using multiple fluid particles that span larger regions of the fluid. Using these global results, we compare the braiding within a viscously spreading Gaussian vortex in the above strain fields, where the flow is determined both kinematically and dynamically. We show that the dynamic feedback of the strain field onto the flow field reduces the overall amount of braiding of the fluid particles.

This paper considers the evolution of smooth, two-dimensional vortices subject to a rotating external strain field, which generates regions of recirculating, cat's eye stream line topology within a vortex. When the external strain field is smoothly switched off, the cat's eyes may persist, or they may disappear as the vortex relaxes back to axisymmetry. A numerical study obtains criteria for the persistence of cat's eyes as a function of the strength and time scale of the imposed strain field, for a Gaussian vortex profile.In the limit of a weak external strain field and high Reynolds number, the disturbance decays exponentially, with a rate that is linked to a Landau pole of the linear inviscid problem. For stronger strain fields, but not strong enough to give persistent cat's eyes, the exponential decay of the disturbance varies: as time increases the decay slows down, because of the nonlinear feedback on the mean profile of the vortex. This is confirmed by determining the decay rate given by the Landau pole for these modified profiles. For strain fields strong enough to generate persistent cat's eyes, their location and rotation rate are determined for a range of angular velocities of the external strain field, and are again linked to Landau poles of the mean profiles, modified through nonlinear effects.

We investigate a moist atmospheric column convection model by considering the atmosphere as a single vertical column of air parcels, each of which contains water vapour. The moist convective adjustment of both air and water mass in the column is considered from an (unstable) initial state to a statically stable final configuration of parcels. Two variations of an algorithm based upon swapping neighbouring parcels are compared: after swapping, no parcels remain supersaturated. The results of these algorithms are compared directly with those of the adjustment algorithm of Cheng et al. 2017 (Q. J. R. Meteorol. Soc. 143(708) p. 2925-2939) which adjusts an atmospheric column to achieve the global maximum of a relevant cost functional. Two examples are considered: in the first,the algorithms adjust to similar arrangements, showing that the global maximum of the functional is the dynamically preferred state, while in the second, the algorithms adjust to significantly different states. Thus we identify a non-uniqueness to the solution to the adjustment problem in terms of local and global cost functional maximisers. We then discuss the relevance of this nonuniqueness to numerical prediction in weather and climate models.

The synchronous interaction between a sloshing fluid and its container may take place when the motion of the container is not prescribed a priori. When a partially- filled container is suspended as a bifilar pendulum, the fluid contributes to the restoring force by its weight through the wire suspensions and may either augment or diminish the restoring force through hydrodynamic interaction with the container. Here we report on linearized shallow-water theory and experiments for multi-chamber box containers, a cylinder and an upright 90 degree wedge. The nature of system response at low fluid mass is analyzed in detail using linear theory and shallow-water simulations. These results show that the system frequency smoothly transitions through successive higher frequency eigenmodes so that, in the limit where the fluid mass tends to zero, the system frequency is that of the dry container.

This paper examines the evolution of an axisymmetric two-dimensional vortex in a steadily rotating strain field and the dynamical interactions that can enhance vortex spreading through resonant behaviour. Starting with a point vortex localized at the origin, the applied strain field generates a cat's eye topology in the co-rotating stream function, localized around a radius r(ext). Now the vortex is allowed to spread viscously: initially r(ext) lies outside the vortex, but as it spreads, vorticity is advected into the cat's eyes, leading to a local flattening of the mean profile of the vortex and so to enhanced mixing and spreading of the vortex. Together with this is a feedback: the response of the vortex to the external strain depends on the modified profile. The feedback is particularly strong when r(ext) coincides with the radius r(cat) at which the vortex can support cat's eyes of infinitesimal width. There is a particular time at which this occurs, as these radii change with the viscous spread of the vortex: r(ext) moves inwards and r(cat) outwards. This resonance behaviour leads to increased mixing of vorticity, along with a rapid stretching of vorticity contours and a sharp increase in the amplitude of the non-axisymmetric components. The dynamical feedback and enhanced diffusion are studied for viscously spreading vortices by means of numerical simulations of their time evolution, parameterized only by the Reynolds number R and the dimensionless strength A of the external strain field.

Suspending a rectangular vessel partially filled with an inviscid fluid from a single rigid pivoting rod produces an interesting physical model for investigating the dynamic coupling between the fluid and vessel motion. The fluid motion is governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion is given by a modified forced pendulum equation. The fully nonlinear, two-dimensional, equations of motion are derived and linearised for small-amplitude vessel and free-surface motions, and the natural frequencies of the system analysed. It is found that the linear problem exhibits an unstable solution if the rod length is shorter than a critical length which depends on the length of the vessel, the fluid height and the ratio of the fluid and vessel masses. In addition, we identify the existence of 1:1 resonances in the system where the symmetric sloshing modes oscillate with the same frequency as the coupled fluid/vessel motion. The implications of instability and resonance on the nonlinear problem are also briefly discussed.

The coupled interaction between a sloshing fluid in a partially- filled container suspended as a bifilar pendulum is investigated. The sloshing fluid has a free-surface upon which waves are generated this fluid contributes a restoring force to the container motion by its weight through the wire suspensions and the free-surface waves may either enhance or diminish the restoring force through hydrodynamic interaction with the container walls. Results are presented for inviscid, irrotational sloshing in both a two-dimensional hyperbolic container and a three dimensional hyperboloid container. Frequency results for the coupled system are presented for various pendulum lengths and fluid fill heights. It is found that for long pendulum lengths the container and the fluid oscillate in a synchronous motion when the vessel is released with typical experimental initial conditions, but for pendulum lengths below a given threshold the container and fluid oscillate asynchronously from the same initial condition.

The coupled motion, between multiple inviscid, incompressible, immiscible fluid layers in a rectangular vessel with a rigid lid and the vessel dynamics, is considered. The fluid layers are assumed to be thin and the shallow-water assumption is applied. The governing form of the Lagrangian functional in the Lagrangian Particle Path (LPP) framework is derived for an arbitrary number of layers, while the corresponding Hamiltonian is explicitly derived in the case of two- and three-layer fluids. The Hamiltonian formulation has nice properties for numerical simulations, and a fast, effective and symplectic numerical scheme is presented in the two- and three-layer cases, based upon the implicit-midpoint rule. Results of the simulations are compared with linear solutions and with the existing results of Alemi Ardakani, Bridges & Turner [1] (J. Fluid Struct. 59 432-460) which were obtained using a finite volume approach in the Eulerian representation. The latter results are extended to non-Boussinesq regimes. The advantages and limitations of the LPP formulation and variational discretization are highlighted.

In this paper we examine the influence of periodic islands within a time periodic chaotic flow on the evolution of a scalar tracer. The passive scalar tracer is injected into the flow field by means of a steady source term. We examine the distribution of the tracer once a periodic state is reached, in which the rate of injected scalar balances advection and diffusion with the molecular diffusion kappa. We study the two-dimensional velocity field u(x, y, t) = 2 cos(2)(omega t)(0, sin x) + 2 sin(2)(omega t)(sin y, 0). As omega is reduced from an O(1) value the flow alternates through a sequence of states which are either globally chaotic, or contain islands embedded in a chaotic sea. The evolution of the scalar is examined numerically using a semi-Lagrangian advection scheme. By time-averaging diagnostics measured from the scalar field we find that the time-averaged lengths of the scalar contours in the chaotic region grow like kappa(-1/2) for small kappa, for all values of omega, while the behavior of the time-averaged maximum scalar value, (C-max) over bar, for small kappa depends strongly on omega. In the presence of islands (C-max) over bar similar to kappa(-alpha) for some alpha between 0 and 1 and with kappa small, and we demonstrate that there is a correlation between alpha and the area of the periodic islands, at least for large omega. The limit of small omega is studied by considering a flow field that switches from u=(0, 2 sin x) to u=(2 sin y, 0) at periodic intervals. The small kappa limit for this flow is examined using the method of matched asymptotic expansions. Finally the role of islands in the flow is investigated by considering the time-averaged effective diffusion of the scalar field. This diagnostic can distinguish between regions where the scalar is well mixed and regions where the scalar builds up. c 2009 American Institute of Physics. [DOI: 10.1063/1.3159615]

The coupled motion between shallow water sloshing in a moving vessel with baffles and the vessel dynamics is considered. Here the vessel dynamics is restricted to horizontal motion such as in Tuned Liquid Dampers. It was shown by Turner {____it et al.} ____cite{tbaa} (Phys. Fluids (2013) {____bf 25}(10) 112102) that partitioning a moving vessel into $n$ separate compartments leads to an interesting dynamical behaviour of the system. Also, under particular input parameter values an internal $(n+1)$-fold $1:____cdots:1$ resonance can be generated, where the frequency of the sloshing fluid in each compartment is equal, and equal to the frequency of the vessel itself. Here the form of the sloshing eigenmodes at this resonance are derived in the shallow-water limit. Using the Lagrangian formulation of the problem, an efficient numerical algorithm is implemented to solve the fully nonlinear system of equation based on the implicit midpoint rule. This algorithm is simple, fast and maintains the energy partition between the vessel and the fluid over long times. In this work numerical results are presented for dynamical vessel/sloshing motion attached to a nonlinear spring.

Suspending a rectangular vessel which is partially filled with fluid from a single rigid pivoting pole produces an interesting theoretical model with which to investigate the dynamic coupling between fluid motion and vessel rotation. The exact equations for this coupled system are derived with the fluid motion governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion governed by a modified forced pendulum equation. The nonlinear equations of motion for the fluid are solved numerically via a time-dependent conformal mapping, which maps the physical domain to a rectangle in the computational domain with a time dependent conformal modulus. The numerical scheme expresses the implicit free-surface boundary conditions as two explicit partial differential equations which are then solved via a pseudo-spectral method in space. The coupled system is integrated in time with a fourth-order Runge-Kutta method. The starting point for the simulations is the linear neutral stability contour discovered by Turner, Alemi Ardakani ____& Bridges (2014, {____it J. Fluid Struct.} {____bf 52}, 166-180). Near the contour the nonlinear results confirm the instability boundary, and far from the neutral curve (parameterised by longer pole lengths) nonlinearity is found to significantly alter the vessel response. Results are also presented for an initial condition given by a superposition of two sloshing modes with approximately the same frequency from the linear characteristic equation. In this case the fluid initial conditions generate large nonlinear vessel motions, which may have implications for systems designed to oscillate in a confined space or on the slosh-induced-rolling of a ship.

This paper investigates the coupled motion between the dynamics of N vessels coupled together in a one-dimensional array by springs, and the motion of the inviscid fluid sloshing within each vessel. We develop a fully-nonlinear model for the system relative to a moving frame such that the fluid in each vessel is governed by the Euler equations and the motion of each vessel is modelled by a forced spring equation. By considering a linearization of the model, the characteristic equation for the natural frequencies of the system is derived, and analysed for a variety of non-dimensional parameter regimes. It is found that the problem can exhibit a variety of resonance situations from the 1 : 1 resonance to (N + 1)-fold 1 : · · · : 1 resonance, as well as more general r : s : · · · : t resonances for natural numbers r, s, t. This paper focuses in particular on determining the existence of regions of parameter space where the (N + 1)-fold 1 : · · · : 1 resonance can be found.

This paper considers the relaxation of a smooth two-dimensional vortex to axisymmetry after the application of an instantaneous, weak external strain field. In this limit the disturbance decays exponentially in time at a rate that is linked to a pole of the associated linear inviscid problem (known as a Landau pole). As a model of a typical vortex distribution that can give rise to cat’s eyes, here distributions are examined that have a basic Gaussian shape but whose profiles have been artificially flattened about some radius rc. A numerical study of the Landau poles for this family of vortices shows that as rc is varied so the decay rate of the disturbance moves smoothly between poles as the decay rates of two Landau poles cross. Cat’s eyes that occur in the nonlinear evolution of a vortex lead to an axisymmetric azimuthally averaged profile with an annulus of approximately uniform vorticity, rather like the artificially flattened profiles investigated. Based on the stability of such profiles it is found that finite thickness cat’s eyes can persist (i.e., the mean profile has a neutral mode) at two distinct radii, and in the limit of a thin flattened region the result that vanishingly thin cat’s eyes only persist at a single radius is recovered. The decay of nonaxisymmetric perturbations to these flattened profiles for larger times is investigated and a comparison made with the result for a Gaussian profile.

We consider the interaction of free-stream disturbances with the leading edge of a body and its effect on the transition point. We present a method which combines an asymptotic receptivity approach, and a numerical method which marches through the Orr–Sommerfeld region. The asymptotic receptivity analysis produces a three-deck eigensolution which in its far downstream limiting form produces an upstream boundary condition for our numerical parabolized stability equation (PSE). We discuss the advantages of this method compared to existing numerical and asymptotic analysis and present results which justify this method for the case of a semi-infinite flat plate, where asymptotic results exist in the Orr–Sommerfeld region. We also discuss the limitations of the PSE and comment on the validity of the upstream boundary conditions. Good agreement is found between the present results and the numerical results of Haddad & Corke (1998).

This paper examines the evolution of a two-dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m=2 added to it. If the perturbation is weak, then the vortex returns to an axisymmetric state and the non-zero Fourier harmonics generated by the perturbation decay to zero. However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it. The paper considers the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine-scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative diagnostics, the appearance of an inflection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical-layer equations numerically. These calculations are supported by simulations of the full Navier–Stokes equations using a family of profiles based on the tanh function.

We consider the interaction of free-stream disturbances with the leading edge of a body and its effect on the transition point. We present a method which combines an asymptotic receptivity approach, and a numerical method which marches through the Orr–Sommerfeld region. The asymptotic receptivity analysis produces a three-deck eigensolution which in its far downstream limiting form produces an upstream boundary condition for our numerical parabolized stability equation (PSE). We discuss the advantages of this method compared to existing numerical and asymptotic analysis and present results which justify this method for the case of a semi-infinite flat plate, where asymptotic results exist in the Orr–Sommerfeld region. We also discuss the limitations of the PSE and comment on the validity of the upstream boundary conditions. Good agreement is found between the present results and the numerical results of Haddad & Corke (1998).

We study a single column model of moist convection in the atmosphere. We state the conditions for it to represent a stable steady state. We then evolve the column by subjecting it to an upward displacement which can release instability, leading to a time dependent sequence of stable steady states. We propose a definition of measure valued solution to describe the time dependence and prove its existence.

The interaction between free-stream disturbances and the boundary layer on a body with a rounded leading edge is considered in this paper. A method which incorporates calculations using the parabolized stability equation in the Orr-Sommerfeld region, along with an upstream boundary condition derived from asymptotic theory in the vicinity of the leading edge, is generalized to bodies with an inviscid slip velocity which tends to a constant far downstream. We present results for the position of the lower branch neutral stability point and the magnitude of the unstable Tollmien-Schlichting (T-S) mode at this point for both a parabolic body and the Rankine body. For the Rankine body, which has an adverse pressure gradient along its surface far from the nose, we find a double maximum in the T-S wave amplitude for sufficiently large Reynolds numbers.

The sloshing motion of two inviscid, incompressible, immiscible, shallowwater fluid layers in a rectangular vessel with a rigid lid is considered. The vessel is forced to oscillate in a horizontal rectilinear motion, at the lowest resonant frequency of the system, such that the lower fluid hits the rigid lid causing a wetting-drying scenario. The two-layer shallow-water equations are solved using a conservative quasi-monotone semi-Lagrangian scheme which conserves the mass of the system. Two examples are considered: a low-fill example where the lower fluid does not interact with the upper-rigid lid, in order to validate the code against an existing Lagrangian particle path scheme, and a high-fill example where there is wetting and drying of the upper rigid lid. In both examples, results of the semi-Lagrangian simulations are compared to existing experimental data and good agreement is obtained.

This paper examines the process of transition to turbulence within an accelerating planar liquid jet. By calculating the propagation and spatial evolution of disturbance wave packets generated at a nozzle where the jet emerges, we are able to estimate break-up lengths and break-up times for different magnitudes of acceleration and different liquid to air density ratios. This study uses a basic jet velocity profile that has shear layers in both air and the liquid either side of the fluid interface. The shear layers are constructed as functions of velocity which behave in line with our CFD simulations of injecting diesel jets. The non-dimensional velocity of the jet along the jet centre-line axis is assumed to take the form V (t) = tanh(at), where the parameter a determines the magnitude of the acceleration. We compare the fully unsteady results obtained by solving the unsteady Rayleigh equation to those of a quasi-steady jet to determine when the unsteady effects are significant and whether the jet can be regarded as quasi-steady in typical operating conditions for diesel engines. For a heavy fluid injecting into a lighter fluid (density ratio ρair/ρjet = q < 1), it is found that unsteady effects are mainly significant at early injection times where the jet velocity profile is changing fastest. When the shear layers in the jet thin with time, the unsteady effects cause the growth rate of the wave packet to be smaller than the corresponding quasi-steady jet, whereas for thickening shear layers the unsteady growth rate is larger than that of the quasi-steady jet. For large accelerations (large a), the unsteady effect remains at later times but its effect on the growth rate of the wave packet decreases as the time after injection increases. As the rate of acceleration is reduced, the range of velocity values for which the jet can be considered as quasi-steady increases until eventually the whole jet can be considered quasi-steady. For a homogeneous jet (q = 1), the range of values of a for which the jet can be considered completely quasi-steady increases to larger values of a. Finally, we investigate approximating the wave packet break-up length calculations with a method that follows the most unstable disturbance wave as the jet accelerates. This approach is similar to that used in CFD simulations as it greatly reduces computational time. We investigate whether or not this is a good approximation for the parameter values typically used in diesel engines.

In this paper a breakup model for analysing the evolution of transient fuel sprays characterised by a coherent liquid core emerging from the injection nozzle, throughout the injection process, is proposed. The coherent liquid core is modelled as a liquid jet and a breakup model is formulated. The spray breakup is described using a composite model that separately addresses the disintegration of the liquid core into droplets and their further aerodynamic breakup. The jet breakup model uses the results of hydrodynamic stability theory to de ne the breakup length of the jet, and downstream of this point, the spray breakup process is modelled for droplets only. The composite breakup model is incorporated into the KIVA II Computational Fluid Dynamics (CFD) code and its results are compared with existing breakup models, including the classic WAVE model and a previously developed composite WAVE model (modi ed WAVE model) and in{house experimental observations of transient Diesel fuel sprays. The hydrodynamic stability results used in both the jet breakup model and the WAVE droplet breakup model are also investigated. A new velocity pro le is considered for these models which consists of a jet with a linear shear layer in the gas phase surrounding the liquid core to model the e ect of a viscous gas on the breakup process. This velocity pro le changes the driving instability mechanism of the jet from a surface tension driven instability for the currently used plug ow jet with no shear layers, to an instability driven by the thickness of the shear layer. In particular, it is shown that appreciation of the shear layer instability mechanism in the composite model allows larger droplets to be predicted at jet breakup, and gives droplet sizes which are more consistent with the experimental observations. The inclusion of the shear layer into the jet velocity pro le is supported by previous experimental studies, and further extends the inviscid ow theory used in the formulation of the classic WAVE breakup model.

This paper investigates the dynamic coupling between fluid sloshing and the motion of the vessel containing the fluid, for the case when the vessel is partitioned using non-porous baffles. The vessel is modelled using Cooker's sloshing configuration [M. J. Cooker, “Water waves in a suspended container,” Wave Motion20, 385–395 (1994)]. Cooker's configuration is extended to include n − 1 non-porous baffles which divide the vessel into n separate fluid compartments each with a characteristic length scale. The problem is analysed for arbitrary fill depth in each compartment, and it is found that a multitude of resonance situations can occur in the system, from 1 : 1 resonances to (n + 1)−fold 1 : 1: ⋯ : 1 resonances, as well as ℓ: m: ⋯ : n for natural numbers ℓ, m, n, depending upon the system parameter values. The conventional wisdom is that the principle role of baffles is to damp the fluid motion. Our results show that in fact without special consideration, the baffles can lead to enhancement of the fluid motion through resonance.

This study uses spatio-temporal stability analysis to investigate the convective and absolute instability properties of a steady unconfined planar liquid jet. The approach uses a piecewise linear velocity profile with a finite-thickness shear layer at the edge of the jet. This study investigates how properties such as the thickness of the shear layer and the value of the fluid velocity at the interface within the shear layer affect the stability properties of the jet. It is found that the presence of a finite-thickness shear layer can lead to an absolute instability for a range of density ratios, not seen when a simpler plug flow velocity profile is considered. It is also found that the inclusion of surface tension has a stabilizing effect on the convective instability but a destabilizing effect on the absolute instability. The stability results are used to obtain estimates for the breakup length of a planar liquid jet as the jet velocity varies. It is found that reducing the shear layer thickness within the jet causes the breakup length to decrease, while increasing the fluid velocity at the fluid interface within the shear layer causes the breakup length to increase. Combining these two effects into a profile, which evolves realistically with velocity, gives results in which the breakup length increases for small velocities and decreases for larger velocities. This behaviour agrees qualitatively with existing experiments on the breakup length of axisymmetric jets.

Cooker's sloshing experiment is a prototype for studying the dynamic coupling between fluid sloshing and vessel motion. It involves a container, partially filled with fluid, suspended by two cables and constrained to remain horizontal while undergoing a pendulum-like motion. In this paper the fully-nonlinear equations are taken as a starting point, including a new derivation of the coupled equation for vessel motion, which is a forced nonlinear pendulum equation. The equations are then linearized and the natural frequencies studied. The coupling leads to a highly nonlinear transcendental characteristic equation for the frequencies. Two derivations of the characteristic equation are given, one based on a cosine expansion and the other based on a class of vertical eigenfunctions. These two characteristic equations are compared with previous results in the literature. Although the two derivations lead to dramatically different forms for the characteristic equation, we prove that they are equivalent. The most important observation is the discovery of an internal $1:1$ resonance in the fully two-dimensional finite depth model, where symmetric fluid modes are coupled to the vessel motion. Numerical evaluation of the resonant and nonresonant modes are presented. The implications of the resonance for the fluid dynamics, and for the nonlinear coupled dynamics near the resonance are also briefly discussed.

The boundary layer flow above a disk in torsional motion with azimuthal velocity proportional to rm is investigated for all values of m ≥ 1; here r is the radial coordinate measured from the center of the disk. The resulting flow is a fully three-dimensional exact solution to the steady, axisymmetric boundary layer equations in the form of similarity solutions. We compute wall shear stresses in the radial and azimuthal directions as a function of the torsional exponent m, as well as the flow induced in the far field. The large m asymptotics of the problem are computed and compared with numerical solutions. The induced radial velocity profiles have a ‘wall-jet’ structure and it is found that both the radial and azimuthal velocity components become confined close to the surface of the disk as m increases.

This paper presents a numerical study of the free-surface evolution for inviscid, incompressible, irrotational, horizontally forced sloshing in a twodimensional rectangular vessel with an inhomogeneous bottom topography. The numerical scheme uses a time-dependent conformal mapping to map the physical fluid domain to a rectangle in the computational domain with a time-dependent aspect ratio Q(t), known as the conformal modulus. The advantage of this approach over conventional potential flow solvers is the solution automatically satisfies Laplace’s equation for all time, hence only the integration of the two free-surface boundary conditions is required. This makes the scheme computationally fast, and as grid points are required only along the free-surface, high resolution simulations can be performed which allows for simulations for mean fluid depths close to the shallow water water regime. The scheme is robust and can simulate both resonate and non-resonate cases, where in the latter, the large amplitude waves are well predicted. Results of nonlinear simulations are presented in the case of non-breaking waves for both an asymmetrical ‘step’ and a symmetric ‘hump’ bottom topography. The natural free-sloshing mode frequencies are compared with the small topography asymptotic results of Faltinsen and Timokha (2009) (Sloshing, Cambridge University Press (Cambridge)), and are found to be lower than this asymptotic prediction for moderate and large topography magnitudes. For forced periodic oscillations it is shown that the hump profile is the most effective topography for minimising the nonlinear response of the fluid, and hence this topography would reduce the stresses on the vessel walls generated by the fluid. Results also show that varying the width of the step or hump has a less significant effect than varying its magnitude.

In this paper, the interaction of free-stream acoustic waves with the leading edge of an aerodynamic body is investigated and two different methods for analysing this interaction are considered. Results are compared for a method which incorporates Orr–Sommerfeld calculations using the parabolized stability equation to those of direct numerical simulations. By comparing the streamwise amplitude of the Tollmien–Schlichting wave, it is found that non-modal components of the boundary layer response to an acoustic wave can persist some distance downstream of the lower branch. The effect of nose curvature on the persisting non-modal eigenmodes is also considered, with a larger nose radius allowing the non-modal eigenmodes to persist farther downstream.

The advection of a tracer field in a fluid flow can create complex scalar structures and increase the effect of weak diffusion by orders of magnitude. One tool to quantify this is to measure the flux of scalar across contour lines of constant scalar. This gives a diffusion equation in area coordinates with an effective diffusion that depends on the structure of the scalar field and, in particular, takes large values when scalar contours become very extended. The present paper studies the properties of this effective diffusion using a mixture of analytical and numerical tools. First the presence of hyperbolic stationary points, that is, saddles, in the scalar concentration field is investigated analytically, and it is shown that these give rise to singular spikes in the effective diffusion. This is confirmed in numerical simulations in which complex scalar fields are generated using a time-periodic flow. Issues of numerical resolution are discussed and results are given on the dependence of the effective diffusion on grid resolution and discretization in area or scalar values. These simulations show complex dependence of the effective diffusion on time as saddle points appear and disappear in the scalar field. It is found that time averaging (in the presence of an additional scalar source term) removes this dependence to leave robust results for the effective diffusion. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.2998461]

The flow generated by a biorthogonally stretched membrane below a steadily rotating flow at infinity is examined. The flow's velocity field is shown to be an exact, self-similar, solution of the fully three-dimensional Navier-Stokes equations with the solution governed by a set of four ordinary differential equations. It is demonstrated that dual solutions exist when the membrane is stretched in both directions (except in the radially symmetric case), as well as for a range of parameters where the membrane is stretched in one direction and allowed to shrink in the other. For stretching rates close to the radially stretched symmetric case, four solutions exist, including one which has a large wall-jet velocity profile close to the membrane. The linear stability of each solution is also examined, and it is found that only a single solution is stable (where one exists) for a given stretching and rotation rate.

This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern's centre (the core region) and decay exponentially away from the pattern's centre (the far-field region). The results are presented for a finite-depth, infinite expanse of ferrofluid equipped with a linear magnetisation law. These patterns bifurcate at the Rosensweig instability, where the applied magnetic field strength reaches a critical threshold. Techniques for finding localised solutions to a non-autonomous PDE system are established; solutions are decomposed onto a basis which is independent of the radius, reducing the problem to an infinite set of nonlinear, non-autonomous ODEs. Using radial centre manifold theory, local manifolds of small-amplitude solutions are constructed in the core and far-field regions, respectively. Finally, using geometric blow-up coordinates, we match the core and far-field manifolds; any solution that lies on this intersection is a localised radial pattern. Three distinct classes of stationary radial solutions are found: spot A and spot B solutions, which are equipped with two different amplitude scaling laws and achieve their maximum amplitudes at the core, and ring solutions, which achieve their maximum amplitudes away from the core. These solutions correspond exactly to the classes of localised radial solutions found for the Swift-Hohenberg equation. Different values of the linear magnetisation and depth of the ferrofluid are investigated and parameter regions in which the various localised radial solutions emerge are identified. The approach taken in this paper outlines a route to rigorously establishing the existence of axisymmetric localised patterns in the future.

The normal impingement of axisymmetric Homann stagnation-point flow on a surface executing perpendicular, planar, biaxial stretching is studied. The flow field generated is an exact solution of the steady, three-dimensional Navier–Stokes equations in the form of a similarity solution. It is shown that two sets of dual solutions exist, forming four different branches of steady solutions. For sufficiently small stretches (including compressions of the surface) the four branches exhibit a multi-branch spiralling behaviour in the surface shear stress parameter plane. The linear stability of the solutions are also examined, identifying only one stable solution for each set of parameters.

The advection of a tracer field in a fluid flow can create complex scalar structures and increase the effect of weak diffusion by orders of magnitude. One tool to quantify this is to measure the flux of scalar across contour lines of constant scalar. This gives a diffusionequation in area coordinates with an effective diffusion that depends on the structure of the scalar field and, in particular, takes large values when scalar contours become very extended. The present paper studies the properties of this effective diffusion using a mixture of analytical and numerical tools. First the presence of hyperbolic stationary points, that is, saddles, in the scalar concentration field is investigated analytically, and it is shown that these give rise to singular spikes in the effective diffusion. This is confirmed in numerical simulations in which complex scalar fields are generated using a time-periodic flow. Issues of numerical resolution are discussed and results are given on the dependence of the effective diffusion on grid resolution and discretization in area or scalar values. These simulations show complex dependence of the effective diffusion on time as saddle points appear and disappear in the scalar field. It is found that time averaging (in the presence of an additional scalar source term) removes this dependence to leave robust results for the effective diffusion.

This paper examines the evolution of a two-dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m = 2 added to it. If the perturbation is weak, then the vortex returns to an axisymmetric state and the non-zero Fourier harmonics generated by the perturbation decay to zero. However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it. The paper considers the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine-scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative diagnostics, the appearance of an inflection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical-layer equations numerically. These calculations are supported by simulations of the full Navier–Stokes equations using a family of profiles based on the tanh function.

Data files needed in order to recreate all the figures in the paper `Coupled shallow-water fluid sloshing in an upright annular vessel' by Turner and Rowe

This paper considers the relaxation of a smooth two-dimensional vortex to axisymmetry after the application of an instantaneous, weak external strain field. In this limit the disturbance decays exponentially in time at a rate that is linked to a pole of the associated linear inviscid problem (known as a Landau pole). As a model of a typical vortex distribution that can give rise to cat’s eyes, here distributions are examined that have a basic Gaussian shape but whose profiles have been artificially flattened about some radius rc. A numerical study of the Landau poles for this family of vortices shows that as rc is varied so the decay rate of the disturbance moves smoothly between poles as the decay rates of two Landau poles cross. Cat’s eyes that occur in the nonlinear evolution of a vortex lead to an axisymmetric azimuthally averaged profile with an annulus of approximately uniform vorticity, rather like the artificially flattened profiles investigated. Based on the stability of such profiles it is found that finite thickness cat’s eyes can persist (i.e., the mean profile has a neutral mode) at two distinct radii, and in the limit of a thin flattened region the result that vanishingly thin cat’s eyes only persist at a single radius is recovered. The decay of nonaxisymmetric perturbations to these flattened profiles for larger times is investigated and a comparison made with the result for a Gaussian profile.