50 research outputs found
Efficient computation of two-dimensional steady free-surface flows
We consider a family of steady free-surface flow problems in two dimensions,
concentrating on the effect of nonlinearity on the train of gravity waves that
appear downstream of a disturbance. By exploiting standard complex variable
techniques, these problems are formulated in terms of a coupled system of
Bernoulli's equation and an integral equation. When applying a numerical
collocation scheme, the Jacobian for the system is dense, as the integral
equation forces each of the algebraic equations to depend on each of the
unknowns. We present here a strategy for overcoming this challenge, which leads
to a numerical scheme that is much more efficient than what is normally
employed for these types of problems, allowing for many more grid points over
the free surface. In particular, we provide a simple recipe for constructing a
sparse approximation to the Jacobian that is used as a preconditioner in a
Jacobian-free Newton-Krylov method for solving the nonlinear system. We use
this approach to compute numerical results for a variety of prototype problems
including flows past pressure distributions, a surface-piercing object and
bottom topographies.Comment: 20 pages, 13 figures, under revie
Spectrograms of ship wakes: identifying linear and nonlinear wave signals
A spectrogram is a useful way of using short-time discrete Fourier transforms
to visualise surface height measurements taken of ship wakes in real world
conditions. For a steadily moving ship that leaves behind small-amplitude
waves, the spectrogram is known to have two clear linear components, a
sliding-frequency mode caused by the divergent waves and a constant-frequency
mode for the transverse waves. However, recent observations of high speed ferry
data have identified additional components of the spectrograms that are not yet
explained. We use computer simulations of linear and nonlinear ship wave
patterns and apply time-frequency analysis to generate spectrograms for an
idealised ship. We clarify the role of the linear dispersion relation and ship
speed on the two linear components. We use a simple weakly nonlinear theory to
identify higher order effects in a spectrogram and, while the high speed ferry
data is very noisy, we propose that certain additional features in the
experimental data are caused by nonlinearity. Finally, we provide a possible
explanation for a further discrepancy between the high speed ferry spectrograms
and linear theory by accounting for ship acceleration.Comment: 21 pages, 10 figures, submitte
Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations
Viscous fingering experiments in Hele-Shaw cells lead to striking pattern
formations which have been the subject of intense focus among the physics and
applied mathematics community for many years. In recent times, much attention
has been devoted to devising strategies for controlling such patterns and
reducing the growth of the interfacial fingers. We continue this research by
reporting on numerical simulations, based on the level set method, of a
generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is
altered. First, we investigate how imposing constant and time-dependent
injection rates in a Hele-Shaw cell that is either standard, tapered or
rotating can be used to reduce the development of viscous fingering when an
inviscid fluid is injected into a viscous fluid over a finite time period. We
perform a series of numerical experiments comparing the effectiveness of each
strategy to determine how these non-standard Hele-Shaw configurations influence
the morphological features of the inviscid-viscous fluid interface. Tapering
plates in either converging or diverging directions leads to reduced metrics of
viscous fingering at the final time when compared to the standard parallel
configuration, especially with carefully chosen injection rates; for the
rotating plate case, the effect is even more dramatic, with sufficiently large
rotation rates completely stabilising the interface. Next, we illustrate how
the number of non-splitting fingers can be controlled by injecting the inviscid
fluid at a time-dependent rate while increasing the gap between the plates.
Simulations compare well with previous experimental results for various
injection rates and geometric configurations. Further, we demonstrate how the
fully nonlinear dynamics of the problem affect the number of fingers that
emerge and how well this number agrees with predictions from linear stability
analysis
Saffman-Taylor fingers with kinetic undercooling
The mathematical model of a steadily propagating Saffman-Taylor finger in a
Hele-Shaw channel has applications to two-dimensional interacting streamer
discharges which are aligned in a periodic array. In the streamer context, the
relevant regularisation on the interface is not provided by surface tension,
but instead has been postulated to involve a mechanism equivalent to kinetic
undercooling, which acts to penalise high velocities and prevent blow-up of the
unregularised solution. Previous asymptotic results for the Hele-Shaw finger
problem with kinetic undercooling suggest that for a given value of the kinetic
undercooling parameter, there is a discrete set of possible finger shapes, each
analytic at the nose and occupying a different fraction of the channel width.
In the limit in which the kinetic undercooling parameter vanishes, the fraction
for each family approaches 1/2, suggesting that this 'selection' of 1/2 by
kinetic undercooling is qualitatively similar to the well-known analogue with
surface tension. We treat the numerical problem of computing these
Saffman-Taylor fingers with kinetic undercooling, which turns out to be more
subtle than the analogue with surface tension, since kinetic undercooling
permits finger shapes which are corner-free but not analytic. We provide
numerical evidence for the selection mechanism by setting up a problem with
both kinetic undercooling and surface tension, and numerically taking the limit
that the surface tension vanishes.Comment: 10 pages, 6 figures, accepted for publication by Physical Review
Time-frequency analysis of ship wave patterns in shallow water: modelling and experiments
A spectrogram of a ship wake is a heat map that visualises the time-dependent
frequency spectrum of surface height measurements taken at a single point as
the ship travels by. Spectrograms are easy to compute and, if properly
interpreted, have the potential to provide crucial information about various
properties of the ship in question. Here we use geometrical arguments and
analysis of an idealised mathematical model to identify features of
spectrograms, concentrating on the effects of a finite-depth channel. Our
results depend heavily on whether the flow regime is subcritical or
supercritical. To support our theoretical predictions, we compare with data
taken from experiments we conducted in a model test basin using a variety of
realistic ship hulls. Finally, we note that vessels with a high aspect ratio
appear to produce spectrogram data that contains periodic patterns. We can
reproduce this behaviour in our mathematical model by using a so-called
two-point wavemaker. These results highlight the role of wave interference
effects in spectrograms of ship wakes.Comment: 14 pages, 7 figure
What is the apparent angle of a Kelvin ship wave pattern?
While the half-angle which encloses a Kelvin ship wave pattern is commonly
accepted to be 19.47 degrees, recent observations and calculations for
sufficiently fast-moving ships suggest that the apparent wake angle decreases
with ship speed. One explanation for this decrease in angle relies on the
assumption that a ship cannot generate wavelengths much greater than its hull
length. An alternative interpretation is that the wave pattern that is observed
in practice is defined by the location of the highest peaks; for wakes created
by sufficiently fast-moving objects, these highest peaks no longer lie on the
outermost divergent waves, resulting in a smaller apparent angle. In this
paper, we focus on the problems of free surface flow past a single submerged
point source and past a submerged source doublet. In the linear version of
these problems, we measure the apparent wake angle formed by the highest peaks,
and observe the following three regimes: a small Froude number pattern, in
which the divergent waves are not visible; standard wave patterns for which the
maximum peaks occur on the outermost divergent waves; and a third regime in
which the highest peaks form a V-shape with an angle much less than the Kelvin
angle. For nonlinear flows, we demonstrate that nonlinearity has the effect of
increasing the apparent wake angle so that some highly nonlinear solutions have
apparent wake angles that are greater than Kelvin's angle. For large Froude
numbers, the effect on apparent wake angle can be more dramatic, with the
possibility of strong nonlinearity shifting the wave pattern from the third
regime to the second. We expect our nonlinear results will translate to other
more complicated flow configurations, such as flow due to a steadily moving
closed body such as a submarine.Comment: 19 pages, 9 figures, accepted for publication by Journal of Fluid
Mechanic
Kelvin wake pattern at small Froude numbers
The surface gravity wave pattern that forms behind a steadily moving
disturbance is well known to comprise divergent waves and transverse waves,
contained within a distinctive V-shaped wake. In this paper, we are concerned
with a theoretical study of the limit of a slow-moving disturbance (small
Froude numbers), for which the wake is dominated by transverse waves. We
consider three configurations: flow past a submerged source singularity, a
submerged doublet, and a pressure distribution applied to the surface. We treat
the linearised version of these problems and use the method of stationary phase
and exponential asymptotics to demonstrate that the apparent wake angle is less
than the classical Kelvin angle and to quantify the decrease in apparent wake
angle as the Froude number decreases. These results complement a number of
recent studies for sufficiently fast-moving disturbances (large Froude numbers)
where the apparent wake angle has been also less than the classical Kelvin
angle. As well as shedding light on the wake angle, we also study the fully
nonlinear problems for our three configurations under various limits to
demonstrate the unique and interesting features of Kelvin wake patterns at
small Froude numbers
Implicit reconstructions of thin leaf surfaces from large, noisy point clouds
Thin surfaces, such as the leaves of a plant, pose a significant challenge
for implicit surface reconstruction techniques, which typically assume a
closed, orientable surface. We show that by approximately interpolating a point
cloud of the surface (augmented with off-surface points) and restricting the
evaluation of the interpolant to a tight domain around the point cloud, we need
only require an orientable surface for the reconstruction. We use polyharmonic
smoothing splines to fit approximate interpolants to noisy data, and a
partition of unity method with an octree-like strategy for choosing subdomains.
This method enables us to interpolate an N-point dataset in O(N) operations. We
present results for point clouds of capsicum and tomato plants, scanned with a
handheld device. An important outcome of the work is that sufficiently smooth
leaf surfaces are generated that are amenable for droplet spreading
simulations
Mathematical models and time-frequency heat maps for surface gravity waves generated by thin ships
Recent research suggests that studying the time-frequency response of ship
wave signals has potential to shed light on a range of applications, such as
inferring the dynamical and geometric properties of a moving vessel based on
the surface elevation data detected at a single point in space. We continue
this line of research here with a study of mathematical models for thin ships
using standard Wigley hulls and Wigley transom-stern hulls as examples.
Mathematical models of varying sophistication are considered. These include
basic minimal models which use applied pressure distributions as proxies for
the ship hull. The more complicated models are Michell's thin ship theory and
the Hogner model, both of which explicitly take into account the shape of the
hull. We outline a methodology for carefully choosing the form and parameter
values in the minimal models such that they reproduce the key features of the
more complicated models in the time-frequency domain. For example, we find that
a two-pressure model is capable of producing wave elevation signals that have a
similar time-frequency profile as that for Michell's thin ship theory applied
to the Wigley hull, including the crucially important features caused by
interference between waves created at the bow and stern of the ship. One of the
key tools in our analysis is the spectrogram, which is a heat-map visualisation
in the time-frequency domain. Our work here extends the existing knowledge on
the topic of spectrograms of ship waves. The theoretical results in this study
are supported by experimental data collected in a towing tank at the Australian
Maritime College using model versions of the standard Wigley hulls and Wigley
transom-stern hulls