Dr Matthew Turner

Research Interests

My research interests lie in the field of fluid dynamics, in particular:

  • Boundary layer receptivity
  • Vortex dynamics
  • Jet stability and breakup
  • General flow stability
  • Sloshing fluids and dynamical coupling

For more information on my research interests, see the research page of my personal web site.

For more information on my publications, including draft copies of papers, see the publications page of my personal web page.


Autumn: MAT3041 Fluid Mechanics & MAT2050 Inviscid Fluid Dynamics

Departmental Duties

PhD Programme Leader

Summer Research Studentships Coordinator

Contact Me

Phone: 01483 68 6183

Find me on campus
Room: 26A AA 04


Journal articles

  • Alemi Ardakani H, 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,
    [ Status: Accepted ]
  • Alemi Ardakani H, Turner MR. (2016) 'Numerical simulations of dynamic coupling between shallow-water sloshing and horizontal vessel motion with baffles'. Fluid Dynamics Research,
    [ Status: Accepted ]
  • 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'. ELSEVIER SCIENCE BV JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 296, pp. 462-479.
    [ Status: Accepted ]
  • Turner MR, Bridges TJ. (2016) 'Time-dependent conformal mapping of doubly-connected regions'. Advances in Computational Mathematics, , pp. 1-26.


    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. (2016) 'Liquid sloshing in a horizontally forced vessel with bottom topography'. Elsevier Journal of Fluids and Structures.,
    [ Status: Accepted ]


    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.

  • Turner MR, Bridges TJ, Ardakani HA. (2015) 'The pendulum-slosh problem: Simulation using a time-dependent conformal mapping'. ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD JOURNAL OF FLUIDS AND STRUCTURES, 59, pp. 202-223.
  • Ardakani HA, Bridges TJ, Turner MR. (2015) 'Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing'. ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD JOURNAL OF FLUIDS AND STRUCTURES, 59, pp. 432-460.
  • 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.
  • Turner MR. (2014) 'Temporal evolution of vorticity staircases in randomly strained two-dimensional vortices'. Physics of Fluids, 26 (11)


    © 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. (2013) 'Dynamic coupling in Cooker's sloshing experiment with baffles.'. American Institute of Physics Phys Fluids, 25 Article number 112102


    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.

  • 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, 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.

  • Alemi Ardakani H, Bridges TJ, Turner MR. (2012) 'Resonance in a model for Cooker's sloshing experiment'. Elsevier European Journal of Mechanics B - Fluids, 36 (November-December), pp. 25-38.


    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.

  • Turner MR, Healey JJ, Sazhin SS, Piazzesi R. (2012) 'Wave packet analysis and break--up length calculations for accelerating planar liquid jets.'. Institute of Physics Fluid Dyn Res, 44 (1) Article number 015503


    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.

  • Turner MR. (2012) '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'. Oxford University Press Quarterly Journal of Mechanics & Applied Maths, 65 (2), pp. 183-210.


    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.

  • Turner MR, Berger MA. (2011) 'A study of mixing in coherent vortices using braiding factors'. Institute of Physics Fluid Dynamics Research, 43 (3) Article number 035501


    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.

  • Turner MR, Healey JJ, Sazhin SS, Piazzesi R. (2011) 'Stability analysis and breakup length calculations for steady planar liquid jets'. CAMBRIDGE UNIV PRESS JOURNAL OF FLUID MECHANICS, 668, pp. 384-411.
  • 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'. American Institute of Physics Physics of Fluids, 21 (6)


    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, Bassom AP, Gilbert AD. (2009) 'Diffusion and the formation of vorticity staircases in randomly strained two-dimensional vortices'. CAMBRIDGE UNIV PRESS JOURNAL OF FLUID MECHANICS, 638, pp. 49-72.
  • Turner MR, Gilbert AD. (2009) 'Spreading of two-dimensional axisymmetric vortices exposed to a rotating strain field'. CAMBRIDGE UNIVERSITY PRESS Journal of Fluid Mechanics, 630, pp. 155-177.


    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, Hammerton PW. (2009) 'Analysis of the unstable Tollmien-Schlichting mode on bodies with a rounded leading edge using the parabolized stability equation'. CAMBRIDGE UNIV PRESS JOURNAL OF FLUID MECHANICS, 623, pp. 167-185.
  • Turner MR, Gilbert AD, Thuburn J. (2008) 'Effective diffusion of scalar fields in a chaotic flow'. AMER INST PHYSICS PHYSICS OF FLUIDS, 20 (10) Article number ARTN 107103
  • Turner MR, Gilbert AD. (2008) 'Thresholds for the formation of satellites in two-dimensional vortices'. CAMBRIDGE UNIV PRESS JOURNAL OF FLUID MECHANICS, 614, pp. 381-405.
  • Turner MR, Gilbert AD, Bassom AP. (2008) 'Neutral modes of a two-dimensional vortex and their link to persistent cat's eyes'. AMER INST PHYSICS PHYSICS OF FLUIDS, 20 (2) Article number ARTN 027101
  • Turner MR, Gilbert AD. (2007) 'Linear and nonlinear decay of cat's eyes in two-dimensional vortices, and the link to Landau poles'. Cambridge University Press Journal of Fluid Mechanics, 593, pp. 255-279.


    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. (2007) 'Far downstream analysis for the Blasius boundary-layer stability problem'. QJMAM, 60 (3), pp. 255-274.


    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, Hammerton PW. (2006) 'Asymptotic receptivity analysis and the Parabolized Stability Equation : a combined approach to boundary layer transition'. J. Fluid Mech, 562, pp. 355-382.


    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).

Theses and dissertations

  • Turner MR. (2005) Numerical and Asymptotic approaches to Boundary-Layer Receptivity and Transition. University of East Anglia

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