Dr Matthew Turner

Senior Lecturer
PhD (UEA), MMath (UEA)
+44 (0)1483 686183
26A AA 04

Academic and research departments

Department of Mathematics.


Areas of specialism

Fluid Mechanics

University roles and responsibilities

  • Mathematics PGR Director


Research interests

Courses I teach on

My publications


Turner MR, Bridges TJ (2013) Nonlinear energy transfer between fluid sloshing and vessel motion, Journal of Fluid Mechanics 719 pp. 606-636
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.
Turner MR, Gilbert AD, Bassom AP (2008) Neutral modes of a two-dimensional vortex and their link to persistent cat's eyes, PHYSICS OF FLUIDS 20 (2) ARTN 027101 AMER INST PHYSICS
© 2014 AIP Publishing LLC.The evolution of a Gaussian vortex subject to a weak-external-random n-fold multipolar strain field is examined using fully nonlinear simulations. The simulations show that at large Reynolds numbers, fine scale steps form at the periphery of the vortex, before merging, generally leaving one large step, which acts as a barrier between the vorticity within the coherent core and the surrounding, well mixed, "surf zone." It is shown for n = 2 that the width and the number of fine scale steps which initially form at the periphery of the vortex is dependent on the strain parameters, but that the range of radial values for which steps initially occur is only dependent on n and the amplitude of the strain field. A criteria is developed which can predict this range of radial values using the linear stability results of Le Dizès ["Non-axisymmetric vortices in two-dimensional flows," J. Fluid Mech. 406, 175 (2000)]. This criteria is based upon the perturbation vorticity needing to be larger than some fraction of the vorticity gradient to flatten the vortex profile. For n = 3 and 4, the radial step range is again predicted, and it is observed that for these higher wavenumbers the long lasting steps are narrower than the n = 2 case. For n = 4 the steps which form are so narrow that they do not persist very long before they are destroyed by the strain field and viscosity.
Turner MR, Bridges TJ, Alemi Ardakani H (2015) The pendulum-slosh problem: simulation using a time-dependent conformal mapping, Journal of Fluids and Structures 59 pp. 202-223 Elsevier
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.
Turner MR, Berger MA (2011) A study of mixing in coherent vortices using braiding factors, Fluid Dynamics Research 43 (3) 035501 Institute of Physics
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.

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.
Turner MR, Bassom AP, Gilbert AD (2009) Diffusion and the formation of vorticity staircases in randomly strained two-dimensional vortices, JOURNAL OF FLUID MECHANICS 638 pp. 49-72 CAMBRIDGE UNIV PRESS
Turner MR, Bridges TJ Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion dataset, University of Surrey
Turner MR (2005) Numerical and Asymptotic approaches to Boundary-Layer Receptivity and Transition,
We consider the interaction of a uniformly pulsating free-stream with the leading edge
of a body, and consider its effect on transition. The free-stream is assumed to be incompressible, high Reynolds number flow parallel to the chord of the body, with a small,
unsteady, perturbation of a single harmonic frequency. We present a method which calculates
Tollmien-Schlichting (T-S) wave amplitudes downstream of the leading edge, by a combination of an asymptotic receptivity approach in the leading edge region and a numerical method which marches through the Orr-Sommerfeld region. The asymptotic receptivity analysis produces a three deck eigenmode which, in its far downstream limiting form, produces an upstream initial condition for our numerical Parabolized Stability
Equation (PSE).

Downstream T-S wave amplitudes are calculated for the flat plate, and good comparisons are found with the Orr-Sommerfeld asymptotics available in this region. The importance of the O(Re?{1/2}) term of the asymptotics is discussed, and, due to the complexity
in calculating this term, we show the importance of numerical methods in the Orr-Sommerfeld region to give accurate results.

We also discuss the initial transients present for certain parameter ranges, and show that their presence appears to be due to the existence of higher T-S modes in the initial
upstream boundary condition.

Extensions of the receptivity/PSE method to the parabola and the Rankine body are considered, and a drop in T-S wave amplitudes at lower branch is observed for both bodies, as the nose radius increases. The only exception to this trend occurs for the Rankine body at very large Reynolds numbers, which are not accessible in experiments, where a double maximum of the T-S wave amplitude at lower branch is observed.

The extension of the receptivity/PSE method to experimentally realistic bodies is also considered, by using slender body theory to model the inviscid flow around a modified
super ellipse to compare with numerical studies.

Turner MR, Gilbert AD (2009) Spreading of two-dimensional axisymmetric vortices exposed to a rotating strain field, Journal of Fluid Mechanics 630 pp. 155-177 CAMBRIDGE UNIVERSITY PRESS
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.
Turner MR, Sazhin SS, Healey JJ, Crua C, Martynov SB (2012) A breakup model for transient Diesel fuel sprays, Fuel 97 pp. 288-305
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 define 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 (modified 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 profile 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 effect of a viscous gas on the breakup process. This velocity profile changes the driving instability mechanism of the jet from a surface tension driven instability for the currently used plug flow 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 profile is supported by previous experimental studies, and further extends the inviscid flow theory used in the formulation of the classic WAVE breakup model. © 2012 Elsevier Ltd. All rights reserved.
Turner MR, Gilbert AD (2008) Thresholds for the formation of satellites in two-dimensional vortices, JOURNAL OF FLUID MECHANICS 614 pp. 381-405 CAMBRIDGE UNIV PRESS
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.
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).
Turner MR, Alemi Ardakani H, Bridges TJ (2015) Instability of sloshing motion in a vessel undergoing pivoted oscillations, Journal of Fluids and Structures 52 pp. 166-180
© 2014 Elsevier Ltd.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.
Ardakani HA, Bridges TJ, Turner MR (2015) Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing, JOURNAL OF FLUIDS AND STRUCTURES 59 pp. 432-460 ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
Turner MR, Gilbert AD (2007) Linear and nonlinear decay of cat's eyes in two-dimensional vortices, and the link to Landau poles, Journal of Fluid Mechanics 593 pp. 255-279 Cambridge University Press
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.
Turner MR, Thuburn J, Gilbert AD (2009) The influence of periodic islands in the flow on a scalar tracer in the presence of a steady source, Physics of Fluids 21 (6) American Institute of Physics
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]
Turner MR, Gilbert AD, Thuburn J (2008) Effective diffusion of scalar fields in a chaotic flow, PHYSICS OF FLUIDS 20 (10) ARTN 107103 AMER INST PHYSICS
Ardakani HA, Bridges TJ, Turner MR (2016) Adaptation of f-wave finite volume methods to the two-layer shallow-water equations in a moving vessel with a rigid-lid, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 296 pp. 462-479 ELSEVIER SCIENCE BV
Ardakani HA, Bridges TJ, Turner MR (2016) Shallow-water sloshing in a moving vessel with variable cross-section and wetting-drying using an extension of George's well-balanced finite volume solver, JOURNAL OF COMPUTATIONAL PHYSICS 314 pp. 590-617 ACADEMIC PRESS INC ELSEVIER SCIENCE
Turner M, Bridges T (2015) Time-dependent conformal mapping of doubly-connected regions, Advances in Computational Mathematics 42 (4) pp. 947-972 Springer
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.
Turner MR, Bridges TJ, Alemi Ardakani H (2013) Dynamic coupling in Cooker's sloshing experiment with baffles., Phys Fluids 25 112102 American Institute of Physics
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: ï : 1 resonances, as well as ?: m: ï : n for natural numbers ?, 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.
Turner MR, Healey JJ, Sazhin SS, Piazzesi R (2011) Stability analysis and breakup length calculations for steady planar liquid jets, JOURNAL OF FLUID MECHANICS 668 pp. 384-411 CAMBRIDGE UNIV PRESS
Turner MR, Weidman P (2016) Coupled sloshing in hyperbolic containers suspended as a bifilar pendulum., Physical Review Fluids 1 (4)
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.
Turner MR, Healey JJ, Sazhin SS, Piazzesi R (2012) Wave packet analysis and break--up length calculations for accelerating planar liquid jets., Fluid Dyn Res 44 (1) 015503 Institute of Physics
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 Áair/Ájet = q

Turner MR, Weidman P (2017) The stability of a radially stretching disc beneath a uniformly rotating fluid., Physical Review Fluids 2 (7) 073904 American Physical Society
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
, 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
. For small stretching rates there exists a second region of absolute instability for a range of negative
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
) 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.
Turner MR (2016) Liquid sloshing in a horizontally forced vessel with bottom topography, Journal of Fluids and Structures. 64 pp. 1-26 Elsevier
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.
Alemi Ardakani H, Bridges T, Turner M (2012) Resonance in a model for Cooker's sloshing experiment, European Journal of Mechanics B - Fluids 36 (Nov) pp. 25-38 Elsevier
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.
Cheng B, Cullen M, Esler J, Norbury J, Turner M, Vanneste J, Cheng J (2017) A Model for Moist Convection in an Ascending Atmospheric Column, Quarterly Journal of the Royal Meteorological Society 143 (708) pp. 2925-2939 Wiley
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.
Turner Matthew, Bridges Thomas, Alemi Ardakani H (2017) Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion, Journal of Engineering Mathematics 106 (1) pp. 75-106 Springer Verlag
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.
Cheng B, Cheng J, Cullen M, Norbury J, Turner MR (2017) A rigorous treatment of moist convection in a single column, SIAM Journal on Mathematical Analysis 49 (5) pp. 3854-3892 Society for Industrial and Applied Mathematics
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.
Huang Y, Turner M (2017) Dynamic fluid sloshing in a one-dimensional
array of coupled vessels.,
Physical Review Fluids 2 (12) 124801 American Physical Society
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.
Turner Matthew (2018) Fluid sloshing in rectangular vessels with side-wall baffles using conformal mappings of multiply-connected domains, Quarterly Journal of Mechanics and Applied Mathematics 71 (3) hby004 pp. 245-272 Oxford University Press
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.
Turner M, Gilbert A (2008) Thresholds for the formation of satellites
in two-dimensional vortices,
Journal of Fluid Mechanics 614 pp. 381-405 Cambridge University Press
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.

Turner M, Gilbert A, Bassom A (2008) Neutral modes of a two-dimensional vortex and their link to persistent
cat?s eyes,
Physics of Fluids 20 (2) 027101 American Institute of Physics
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.
Turner M, Bassom A, Gilbert A (2009) Diffusion and the formation of vorticity staircases in randomly strained two-dimensional vortices, Journal of Fluid Mechanics 638 pp. 49-72 Cambridge University Press
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.

Turner M, Gilbert A Spreading of two-dimensional axisymmetric vortices exposed to a rotating strain field, Journal of Fluid Mechanics 630 pp. 155-177 Cambridge University Press
This paper examines the evolution of an axisymmetric two{dimensional vortex in a
steadily rotating strain eld, and the dynamical interactions that can enhance vortex
spreading through resonant behaviour.

Starting with a point vortex localised at the origin, the applied strain eld generates
a cat's eye topology in the co{rotating stream function, localised around a radius rext.
Now the vortex is allowed to spread viscously: initially rext lies outside the vortex but
as it spreads, vorticity is advected into the cat's eyes, leading to a local
attening of
the mean prole 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 modied prole. The feedback is particularly strong when rext coincides with the
radius rcat at which the vortex can support cat's eyes of innitesimal width. There is
a particular time at which this occurs, as these radii change with the viscous spread
of the vortex: rext 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 diusion are studied for viscously spreading vortices by means of numerical simulations of their time evolution, parameterised only
by the Reynolds number R and the dimensionless strength A of the external strain eld.

Turner M. R., Weidman P. D. (2018) Impinging Howarth stagnation-point flows, European Journal of Mechanics - B/Fluids Elsevier
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 Á1 and
kinematic viscosity ‹1 while the lower layer fluid has density Á2 and kinematic
viscosity ‹2 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 Ã
of strain rates and the fluid parameter ratios Á = Á1/Á2 and ‹ = ‹1/‹2. Numerical
results are presented for selected values of Á and ‹ for a range of à and show that
stable results exist for all values of à > 0, and for a range of negative à values.
Sample stable velocity profiles are also presented.