# Dr Matthew Turner

### Biography

I obtained an MMath degree at the UEA in 2002 before completing my PhD in Fluid Dynamics in 2006, also at the UEA, under the supervision of Dr Paul Hammerton. After this, I spent 3 years at the University of Exeter working with Prof. Andrew Gilbert, before moving to the Sir Harry Ricardo Laboratories at the University of Brighton, where I worked with Prof. Sergei Sazhin, Prof. Jonathan Healey (Keele), Dr Renzo Piazzesi (ANSYS UK Ltd) and Dr Cyril Crua.

I started as a Lecturer at Surrey in May 2011 and was promoted to Senior Lecturer in 2016. I'm currently enjoying fruitful research collaborations with Prof. Tom Bridges on sloshing and dynamic coupling of fluid systems and Prof. Ian Roulstone, Dr Bin Cheng and Prof. John Norbury (Oxford) on convection in a moist atmosphere.

For more information, visit my personal web page.

### Areas of specialism

### University roles and responsibilities

- Mathematics PGR Director

### Research

### 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
- Moist convection

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.

### My teaching

Currently teaching MAT3041 (Mathematical Fluid Mechanics) and MAT2001 (Numerical and Computational Methods).

### Courses I teach on

## Undergraduate

### My publications

### Publications

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.

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.

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.

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.

Parabolized Stability Equation : a combined

approach to boundary layer transition, J. Fluid Mech 562 pp. 355-382

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.

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.

For a heavy fluid injecting into a lighter fluid (density ratio Áair/Ájet = q

²

, 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

²

. For small stretching rates there exists a second region of absolute instability for a range of negative

²

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

²

>

0

) that can dominate the absolute instability growth rate contribution compared to the shear layer component. A similar behavior is found to occur for a radially shrinking disk, except these profiles have a strong shear layer structure and hence are more unstable than the stretching disk profiles. We thus find for a suitably large shrinking rate the absolute instability contribution from the critical point becomes subdominant to the shear layer contribution.

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.

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.

array of coupled vessels., Physical Review Fluids 2 (12) 124801 American Physical Society

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.

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.

in two-dimensional vortices, Journal of Fluid Mechanics 614 pp. 381-405 Cambridge University Press

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.

cat?s eyes, Physics of Fluids 20 (2) 027101 American Institute of Physics

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.

Howarth stagnation-point flow is studied, and an exact similarity solution to

the Navier-Stokes equations is found. The upper layer fluid has density

*Á*1 and

kinematic viscosity

**1 while the lower layer fluid has density

*Á*2 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*=

*Á*1/

*Á*2 and

**=

**1/

**2. Numerical

results are presented for selected values of

*Á*and

**for a range of

*Ã*and show that

stable results exist for all values of

*Ã*> 0, and for a range of negative

*Ã*values.

Sample stable velocity profiles are also presented.

fluid layers in a rectangular vessel with a rigid lid is considered. The

vessel is forced to oscillate in a horizontal rectilinear motion, at the lowest

resonant frequency of the system, such that the lower fluid hits the rigid

lid causing a wetting-drying scenario. The two-layer shallow-water equations

are solved using a conservative quasi-monotone semi-Lagrangian

scheme which conserves the mass of the system. Two examples are considered:

a low-fill example where the lower fluid does not interact with

the upper-rigid lid, in order to validate the code against an existing Lagrangian

particle path scheme, and a high-fill example where there is

wetting and drying of the upper rigid lid. In both examples, results of

the semi-Lagrangian simulations are compared to existing experimental

data and good agreement is obtained.

*z*= 0 under the parameter restriction that Ã²Á = 1 where Ã is the ratio of the fluid angular velocities at

*z*= ±? and Á is the density ratio of the two fluids. Under this restriction the problem reduces to one with two independent parameters, Ã and ¼, which is the viscosity ratio of the fluids. Numerical results of the resulting system of ODEs are found for selected values of ¼ and Ã, and it is shown that similarity solutions exist for Ã

_{c}(¼)d Ã d 1 where Ã

_{c}(¼) Â 0 (i.e. counter-rotating flows). For Ã Â 0 the lower fluid can become divided into two distinct recirculation regions between which fluid cannot transfer.

translating upright annular vessel is considered. The vessel motion

is restricted to a single space dimension, such as for Tuned

Liquid Damper systems. For particular parameters, the system

is shown to support an internal 1 : 1 resonance, where the frequency

of coupled sloshing mode which generates the vessel motion

is equal to the frequency of a sloshing mode which occurs in a

static vessel. Using a Lagrangian Particle Path formation, the fully

nonlinear motion of the system is simulated using an efficient numerical

symplectic integration scheme. The scheme is based upon

the implicit-midpoint-rule which conserves energy and preserves the

energy partition between the fluid and the vessel over many timesteps.

Linear and nonlinear results are presented, including those

showing the system transitioning to higher frequency eigenmodes

as the fluid depth is reduced.

vertical column of air parcels, each of which contains water vapour. The moist convective adjustment of

both air and water mass in the column is considered from an (unstable) initial state to a statically stable

final configuration of parcels. Two variations of an algorithm based upon swapping neighbouring parcels are

compared: after swapping, no parcels remain supersaturated. The results of these algorithms are compared

directly with those of the adjustment algorithm of Cheng et al. 2017 (Q. J. R. Meteorol. Soc. 143(708) p.

2925-2939) which adjusts an atmospheric column to achieve the global maximum of a relevant cost functional.

Two examples are considered: in the first,the algorithms adjust to similar arrangements, showing that the

global maximum of the functional is the dynamically preferred state, while in the second, the algorithms

adjust to significantly different states. Thus we identify a non-uniqueness to the solution to the adjustment

problem in terms of local and global cost functional maximisers. We then discuss the relevance of this nonuniqueness

to numerical prediction in weather and climate models.

rectilinear vessel motion in a system of

connected vessels,

of a one-dimensional array of vessels, partially filled with fluid, being connected together

by nonlinear springs. The fluid motion induces a hydrodynamic force on the side walls of

the vessel, which induces the vessel to move. The vessel movement is also controlled via a

restoring force of the attached springs, which in turn causes the fluid to alter its motion.

The simplest coupled liquid sloshing system consists of one vessel connected to a side wall via

a spring. We review this example and investigate the techniques documented in the literature

to study the linear problem of such a system. Then we extend this linear theory to multivessel systems, in particular focussing on the 2-vessel system which introduces the notion of

modes being in-phase or out-of-phase with each other. In the general N-vessel system we

also identify the existence in parameter space of internal resonances, where different modes

oscillates at the same frequency. Such a resonance provides a mechanism for energy exchange

between modes in the nonlinear system.

The nonlinear dynamics of the coupled liquid sloshing system are studied by employing a

symplectic integration scheme based on a variational principle, to the shallow-water form of

the governing equations. We present results for the shallow-water scheme in the linear, weakly

nonlinear and fully nonlinear regimes. The shallow-water numerical scheme has trouble

dealing with breaking waves, so we perform a feasibility study where we attempt to use a

similar approach for a non-shallow fluid system. The governing equations are derived and

results presented in the linear amplitude regime for the 1-vessel system.

Finally, we develop variational principles for the coupled liquid sloshing system in the Eulerian

framework based on the principle of constrained variations to derive the governing fluid

equations and free-surface boundary conditions, from a natural Lagrangian functional. We

use this constrained variational approach to derive the fully rotational 2D Euler equations

and its stream function formulation.