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          1)  Boundary layer receptivity
          2)  Vortex dynamics and kinematics
          3)  Mixing in fluids
          4)  Stability of jets
          5)  Sloshing
          6)  Mathematical biology

Boundary layer receptivity

Asymptotic receptivity analysis and the Parabolized Stability Equation : a combined approach to boundary layer transition

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 against existing numerical and asymptotic analysis and present results which justifies 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).

Far downstream analysis for the Blasius boundary-layer stability problem

In this paper we examine the large Reynolds number, Re, asymptotic structure of the wavenumber in the Orr-Sommerfeld region, for the Blasius boundary-layer on a semi-infinite flat plate given by Goldstein (1983). 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 wavenumber.

Analysis of the unstable Tollmien--Schlichting mode on bodies with a rounded leading edge using the parabolized stability equation

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 (PSE) in the Orr--Sommerfeld region along with an upstream boundary condition derived from asymptotic theory in the vicinity of the leading edge, is generalised 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.

Tollmien--Schlichting wave amplitudes on a semi--infinite flat plate and a parabolic body: comparison of a parabolized stability equation method and direct numerical simulations

In this paper the interaction of free--stream acoustic waves with the leading edge of an aerodynamic body is investigated and we compare two different methods for analysing this interaction. Results are compared for a method which incorporates Orr--Sommerfeld calculations using the parabolized stability equation (PSE) to those of direct numerical simulations (DNS). By comparing the streamwise amplitude of the Tollmien--Schlichting (T--S) 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.

Vortex dynamics and kinematics

Linear and nonlinear decay of cat's eyes in two-dimensional vortices, and the link to Landau poles

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.

Neutral modes of a two-dimensional vortex and their link to persistent cat's eyes

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 r_c. A numerical study of the Landau poles for this family of vortices shows that as r_c 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 non--axisymmetric perturbations to these flattened profiles for larger times is investigated and a comparison made with the result for a Gaussian profile.

Thresholds for the formation of satellites in two--dimensional vortices

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.

Spreading of two--dimensional axisymmetric vortices exposed to a rotating strain field

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 localised at the origin, the applied strain field generates a cat's eye topology in the co--rotating stream function, localised 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 $\rcat$ 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, parameterised only by the Reynolds number Re and the dimensionless strength A of the external strain field.

Diffusion and the formation of vorticity staircases in randomly strained two-dimensional vortices

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.

Mixing in fluids

Effective diffusion of scalar fields in a chaotic flow

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.

The influence of periodic islands in the flow on a scalar tracer in the presence of a steady source

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 the molecular diffusion $\kappa$. We study the two--dimensional velocity field u(x,y,t)=2cos^2(omega t)(0,sin x)+2sin^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 behaviour of the time--averaged maximum scalar value, c_max, for small kappa depends strongly on omega. In the presence of islands c_max=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=(0,2sin x) to u=(2sin 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.

A study of mixing in coherent vortices using braiding factors

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, produces 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 which generates cat's eyes in the vortex. We also consider a modified version of this strain field which contains a resonance frequency effect that produces multiple sets of cat's eyes at different radii. As the thickness of the cat's eyes increase they interact with one another and produce complex Lagrangian motion in the flow which 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.

Stability of jets

Stability analysis and break--up length calculations for steady planar liquid jets.

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 affects 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 break--up 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 break--up length to decrease, while increasing the fluid velocity at the fluid interface within the shear layer causes the break--up length to increase. Combining these two effects into a profile, which evolves realistically with velocity, gives results in which the break--up length increases for small velocities and decreases for larger velocities. This behaviour agrees qualitatively with existing experiments on the break--up length of axisymmetric jets.

Wave packet analysis and break--up length calculations for an accelerating planar liquid jet

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 which has shear layers in both the 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 if 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 rho_air/rho_jet=q<) 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, while 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 \newpage \noindent 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 that 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 which 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 typical used Diesel engines.


Resonance in a model for Cooker's sloshing experiment

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.

Nonlinear energy transfer between fluid sloshing and vessel motion

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.

Dynamic coupling in Cooker's sloshing experiment with baffles.

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, Wave Motion 20, 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.