arXiv subjects

Timothy J. Moroney

Publications and source records attributed to Timothy J. Moroney.

18 recordsLinked to original sources

arXiv2024

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.

physics.flu-dyn↗

arXiv2024

Simulation of high-speed impact of surfactant-laden drops

We develop a computational model to simulate the immediate post-impact spreading behaviour of surfactant-laden drops that impact a flat and solid surface. The model is built on the InterFoam solver (OpenFOAM software), which uses the volume-of-fluid method to solve the Navier-Stokes equations. In order to incorporate surfactant in the bulk and on the interface, we make numerous modifications and extensions, such as coupling the volume-of-fluid method with a level-set method. Simulations demonstrate the accumulation of surfactant in the vicinity of the moving contact line, especially during the formation of the rim. Gradients of surfactant at the liquid-air interface lead to Marangoni forces that oppose the drop spreading, while high-velocity impacts reduce the overall surface tension and increase the magnitude of Marangoni forces. Both of these phenomena, which tend to reduce the maximum spreading, are highly dependent on the surfactant properties. Our computational methodology expands the potential for utilising Computational Fluid Dynamics to model complex interfacial flows that involve surfactants, leading to various opportunities in the future.

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arXiv2023

Data fusion for a multi-scale model of a wheat leaf surface: a unifying approach using a radial basis function partition of unity method

Realistic digital models of plant leaves are crucial to fluid dynamics simulations of droplets for optimising agrochemical spray technologies. The presence and nature of small features (on the order of 100$\mathrm{μm}$) such as ridges and hairs on the surface have been shown to significantly affect the droplet evaporation, and thus the leaf's potential uptake of active ingredients. We show that these microstructures can be captured by implicit radial basis function partition of unity (RBFPU) surface reconstructions from micro-CT scan datasets. However, scanning a whole leaf ($20\mathrm{cm^2}$) at micron resolutions is infeasible due to both extremely large data storage requirements and scanner time constraints. Instead, we micro-CT scan only a small segment of a wheat leaf ($4\mathrm{mm^2}$). We fit a RBFPU implicit surface to this segment, and an explicit RBFPU surface to a lower resolution laser scan of the whole leaf. Parameterising the leaf using a locally orthogonal coordinate system, we then replicate the now resolved microstructure many times across a larger, coarser, representation of the leaf surface that captures important macroscale features, such as its size, shape, and orientation. The edge of one segment of the microstructure model is blended into its neighbour naturally by the partition of unity method. The result is one implicit surface reconstruction that captures the wheat leaf's features at both the micro- and macro-scales.

math.NA↗

arXiv2021

A review of one-phase Hele-Shaw flows and a level-set method for non-standard configurations

The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman-Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both standard and non-standard Hele-Shaw flows. Our method consists of using a modified finite difference stencil in conjunction with the level set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman-Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of non-standard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.

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arXiv2021

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.

math.NA↗

arXiv2021

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.

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arXiv2019

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.

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arXiv2019

Moving boundary problems for quasi-steady conduction limited melting

The problem of melting a crystal dendrite is modelled as a quasi-steady Stefan problem. By employing the Baiocchi transform, asymptotic results are derived in the limit that the crystal melts completely, extending previous results that hold for a special class of initial and boundary conditions. These new results, together with predictions for whether the crystal pinches off and breaks into two, are supported by numerical calculations using the level set method. The effects of surface tension are subsequently considered, leading to a canonical problem for near-complete-melting which is studied in linear stability terms and then solved numerically. Our study is motivated in part by experiments undertaken as part of the Isothermal Dendritic Growth Experiment, in which dendritic crystals of pivalic acid were melted in a microgravity environment: these crystals were found to be prolate spheroidal in shape, with an aspect ratio initially increasing with time then rather abruptly decreasing to unity. By including a kinetic undercooling-type boundary condition in addition to surface tension, our model suggests the aspect ratio of a melting crystal can reproduce the same non-monotonic behaviour as that which was observed experimentally.

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arXiv2018

Computational modelling of cardiac ischaemia using a variable-order fractional Laplacian

Heart failure is one of the most common causes of death in the western world. Many heart problems are linked to disturbances in cardiac electrical activity, such as wave re-entry caused by ischaemia. In terms of mathematical modelling, the monodomain equation is widely used to model electrical activity in the heart. Recently, Bueno-Orovio et al. [J. R. Soc. Interface 11: 20140352, 2014] pioneered the use of a fractional Laplacian operator in the monodomain equation to account for the complex heterogeneous structures in heart tissue. In this work we consider how to extend this approach to apply to hearts with regions of damaged tissue. This requires the use of a fractional Laplacian operator whose fractional order varies spatially. We develop efficient numerical methods capable of solving this challenging problem on domains ranging from simple one-dimensional intervals with uniform meshes, through to full three-dimensional geometries on unstructured meshes. Results are presented for several test problems in one dimension, demonstrating the effects of different fractional orders in regions of healthy and damaged tissue. Then we showcase some new results for a three-dimensional fractional monodomain equation with a Beeler-Reuter ionic current model on a rabbit heart mesh. These simulation results are found to exhibit wave re-entry behaviour, brought about only by varying the value of the fractional order in a region representing damaged tissue.

math.NA↗

arXiv2018

Three dimensional free-surface flow over arbitrary bottom topography

We consider steady nonlinear free surface flow past an arbitrary bottom topography in three dimensions, concentrating on the shape of the wave pattern that forms on the surface of the fluid. Assuming ideal fluid flow, the problem is formulated using a boundary integral method and discretised to produce a nonlinear system of algebraic equations. The Jacobian of this system is dense due to integrals being evaluated over the entire free surface. To overcome the computational difficulty and large memory requirements, a Jacobian-free Newton Krylov (JFNK) method is utilised. Using a block-banded approximation of the Jacobian from the linearised system as a preconditioner for the JFNK scheme, we find significant reductions in computational time and memory required for generating numerical solutions. These improvements also allow for a larger number of mesh points over the free surface and the bottom topography. We present a range of numerical solutions for both subcritical and supercritical regimes, and for a variety of bottom configurations. We discuss nonlinear features of the wave patterns as well as their relationship to ship wakes.

physics.flu-dyn↗

arXiv2018

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.

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arXiv2017

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.

physics.flu-dyn↗

arXiv2017

Extending fields in a level set method by solving a biharmonic equation

We present an approach for computing extensions of velocities or other fields in level set methods by solving a biharmonic equation. The approach differs from other commonly used approaches to velocity extension because it deals with the interface fully implicitly through the level set function. No explicit properties of the interface, such as its location or the velocity on the interface, are required in computing the extension. These features lead to a particularly simple implementation using either a sparse direct solver or a matrix-free conjugate gradient solver. Furthermore, we propose a fast Poisson preconditioner that can be used to accelerate the convergence of the latter. We demonstrate the biharmonic extension on a number of test problems that serve to illustrate its effectiveness at producing smooth and accurate extensions near interfaces. A further feature of the method is the natural way in which it deals with symmetry and periodicity, ensuring through its construction that the extension field also respects these symmetries.

math.NA↗

arXiv2016

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.

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arXiv2015

Wake angle for surface gravity waves on a finite depth fluid

Linear water wave theory suggests that wave patterns caused by a steadily moving disturbance are contained within a wedge whose half-angle depends on the depth-based Froude number $F_H$. For the problem of flow past an axisymmetric pressure distribution in a finite-depth channel, we report on the apparent angle of the wake, which is the angle of maximum peaks. For moderately deep channels, the dependence of the apparent wake angle on the Froude number is very different to the wedge angle, and varies smoothly as $F_H$ passes through the critical value $F_H=1$. For shallow water, the two angles tend to follow each other more closely, which leads to very large apparent wake angles for certain regimes.

physics.flu-dyn↗

arXiv2015

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.

physics.flu-dyn↗

arXiv2014

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.

physics.flu-dyn↗

arXiv2014

Jacobian-free Newton-Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

The nonlinear problem of steady free-surface flow past a submerged source is considered as a case study for three-dimensional ship wave problems. Of particular interest is the distinctive wedge-shaped wave pattern that forms on the surface of the fluid. By reformulating the governing equations with a standard boundary-integral method, we derive a system of nonlinear algebraic equations that enforce a singular integro-differential equation at each midpoint on a two-dimensional mesh. Our contribution is to solve the system of equations with a Jacobian-free Newton-Krylov method together with a banded preconditioner that is carefully constructed with entries taken from the Jacobian of the linearised problem. Further, we are able to utilise graphics processing unit acceleration to significantly increase the grid refinement and decrease the run-time of our solutions in comparison to schemes that are presently employed in the literature. Our approach provides opportunities to explore the nonlinear features of three-dimensional ship wave patterns, such as the shape of steep waves close to their limiting configuration, in a manner that has been possible in the two-dimensional analogue for some time.

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