arXiv subjects

Philippe H. Trinh

Publications and source records attributed to Philippe H. Trinh.

At least 19 recordsLinked to original sources

arXiv2025

Time-dependent nonlinear gravity-capillary surface waves with viscous dissipation and wind forcing

We develop a time-dependent conformal method to study the effect of viscosity on steep surface waves. When the effect of surface tension is included, numerical solutions are found that contain highly oscillatory parasitic capillary ripples. These small amplitude ripples are associated with the high curvature at the crest of the underlying viscous-gravity wave, and display asymmetry about the wave crest. Previous inviscid studies of steep surface waves have calculated intricate bifurcation structures that appear for small surface tension. We show numerically that viscosity suppresses these. While the discrete solution branches still appear, they collapse to form a single smooth branch in limit of small surface tension. These solutions are shown to be temporally stable, both to small superharmonic perturbations in a linear stability analysis, and to some larger amplitude perturbations in different initial-value problems. Our work provides a convenient method for the numerical computation and analysis of water waves with viscosity, without evaluating the free-boundary problem for the full Navier-Stokes equations which becomes increasingly challenging at larger Reynolds numbers.

physics.flu-dyn↗

arXiv2024

Exponential asymptotics and higher-order Stokes phenomenon in singularly perturbed ODEs

The higher-order Stokes phenomenon can emerge in the asymptotic analysis of many problems governed by singular perturbations. Indeed, over the last two decades, the phenomena has appeared in many physical applications, from acoustic and optical wave phenomena and gravity-capillary ripples, to models of crystal growth and equatorial Kelvin waves. It emerges in a generic fashion in the exponential asymptotics of higher-order ordinary and partial differential equations. The intention of this work is to highlight its importance, and develop further practical methodologies for the study of higher-order Stokes phenomena, primarily for general non-integrable problems. Our formal methodology is demonstrated through application to a second-order linear inhomogeneous ODE that exemplifies the simplest example of higher-order Stokes phenomena. In this model problem, the Borel transform can be derived explicitly, and this gives insight into the beyond-all-orders structure. We review and study additional examples, with physically-important connections, including higher-order ODEs and eigenvalue problems.

math.CA↗

arXiv2024

On the selection of Saffman-Taylor viscous fingers for divergent flow in a wedge

We study self-similar viscous fingering for the case of divergent flow within a wedge-shaped Hele-Shaw cell. Previous authors have conjectured the existence of a countably-infinite number of selected solutions, each distinguished by a different value of the relative finger angle. Interestingly, the associated solution branches have been posited to merge and disappear in pairs as the surface tension decreases. We demonstrate how exponential asymptotics is used to derive the selection mechanism. In addition, asymptotic predictions of the finger-to-wedge angle are given for different sized wedges and surface-tension values. The merging of solution branches is explained; this feature is qualitatively different to the case of classic Saffman-Taylor viscous fingering in a parallel channel configuration. The phenomena of branch merging in our self-similar problem relates to tip splitting instabilities in time-dependent flows in a circular geometry, where the viscous fingers destabilise and divide in two.

physics.flu-dyn↗

arXiv2024

Exponential asymptotics and Stokes surfaces in nonlinear three-dimensional flows

In this dissertation, we seek to expand the scope of work done by Lustri and Chapman (2013) in modelling 3D flow past a point source, in order to account for more general flows, where the strength of such a source, $δ$, is now $O(1)$. We find that in order to solve for the Stokes surfaces of this nonlinear system, we must develop a numerical scheme to shoot from the analytically continued free surface into real space. The principal contribution of this work is a demonstration of the disparity between the line of intersection found by Lustri and Chapman (2013) and that found by our own nonlinear numerical method, along with proposed extensions for further avenues of research e.g. including surface tension.

physics.flu-dyn↗

arXiv2024

Pathologies of low-Froude free-surface flows over smoothed bodies

In the study of low-speed or low-Froude flows of a potential gravity-driven fluid past a wave-generating object, the traditional asymptotic expansion in powers of the Froude number predicts a waveless free-surface at every order. This is due to the fact that the waves are, in fact, exponentially small and beyond-all-orders of the naive expansion. The theory of exponential asymptotics indicates that such exponentially-small water waves are switched-on across so-called Stokes lines -- these curves partition the fluid-domain into wave-free regions and regions with waves. In prior studies, Stokes lines are associated with singularities in the flow field such as stagnation points, or corners in submerged objects or rough beds. In this work, we present a smoothed geometry that was recently highlighted by Pethiyagoda et al. [Int. J. Numer. Meth. Fluids. 2018; 86:607--624] as capable of producing waves, yet paradoxically exhibiting no obvious Stokes line insofar as conventional exponential asymptotics theory. In this work, we demonstrate that the Stokes line for this smooth geometry originates from an essential singularity at infinity in the analytic continuation of free-surface quantities. We discuss some of the difficulties in extending the typical methodology of exponential asymptotics to general wave-structure interaction problems with smooth geometries.

physics.flu-dyn↗

arXiv2023

On the development and analysis of coupled surface-subsurface models of catchments. Part 1. Analysis of dimensions and parameters for UK catchments

The objective of this three-part work is to formulate and rigorously analyse a number of reduced mathematical models that are nevertheless capable of describing the hydrology at the scale of a river basin (i.e. catchment). Coupled surface and subsurface flows are considered. In this first part, we identify and analyse the key physical parameters that appear in the governing formulations used within hydrodynamic rainfall-runoff models. Such parameters include those related to catchment dimensions, topography, soil and rock properties, rainfall intensities, Manning's coefficients, and river channel dimensions. Despite the abundance of research that has produced data sets describing properties of specific river basins, there have been few studies that have investigated the ensemble of typical scaling of key physical properties; these estimates are needed to perform a proper dimensional analysis of rainfall-runoff models. Therefore, in this work, we perform an extensive analysis of the parameters; our results form a benchmark and provide guidance to practitioners on the typical parameter sizes and interdependencies. Crucially, the analysis is presented in a fashion that can be reproduced and extended by other researchers, and wherever possible, uses publicly available data sets for catchments in the United Kingdom.

physics.flu-dyn↗

arXiv2023

On the development and analysis of coupled surface-subsurface models of catchments. Part 3. Analytical solutions and scaling laws

The objective of this three-part work is to formulate and rigorously analyse a number of reduced mathematical models that are nevertheless capable of describing the hydrology at the scale of a river basin (i.e. catchment). Coupled surface and subsurface flows are considered. In this third part, we focus on the development of analytical solutions and scaling laws for a benchmark catchment model that models the river flow (runoff) generated during a single rainfall. We demonstrate that for catchments characterised by a shallow impenetrable bedrock, the shallow-water approximation allows a reduction of the governing formulation to a coupled system of one-dimensional time-dependent equations for the surface and subsurface flows. Asymptotic analysis is used to derive semi-analytical solutions for the model. We provide simple asymptotic scaling laws describing the peak flow formation, and demonstrate its accuracy through a comparison with the two-dimensional model developed in Part 2. These scaling laws can be used as an analytical benchmark for assessing the validity of other physical, conceptual, or statistical models of catchments.

physics.flu-dyn↗

arXiv2023

On the development and analysis of coupled surface-subsurface models of catchments. Part 2. A three-dimensional benchmark model and its properties

The objective of this three-part work is to formulate and rigorously analyse a number of reduced mathematical models that are nevertheless capable of describing the hydrology at the scale of a river basin (i.e. catchment). Coupled surface and subsurface flows are considered. In this second part, we construct a benchmark catchment scenario and investigate the effects of parameters within their typical ranges. Previous research on coupled surface-subsurface models have focused on numerical simulations of site-specific catchments. Here, our focus is broad, emphasising the study of general solutions to the mathematical models, and their dependencies on dimensionless parameters. This study provides a foundation based on the examination of a geometrically simple three-dimensional benchmark scenario. We develop a nondimensional coupled surface-subsurface model and extract the key dimensionless parameters. Asymptotic methods demonstrate under what conditions the model can be reduced to a two-dimensional form, where the principal groundwater and overland flows occur in the hillslope direction. Numerical solutions provide guidance on the validity of such reductions, and demonstrate the parametric dependencies corresponding to a strong rainfall event.

physics.flu-dyn↗

arXiv2023

Asymptotic differences between a lumped probability-distributed rainfall-runoff model and a physical benchmark model

Typically, physical and conceptual rainfall-runoff models are developed independently and are based on different, though not entirely incompatible, governing principles. In this work, we perform a systematic asymptotic analysis of a typical conceptual rainfall-runoff model and compare its results to a physical model. The setting for this experiment is a geometrically simplified benchmark catchment consisting of a two-dimensional hillslope. For demonstration purposes, we analyse the lumped Probability Distributed Model (PDM) formulated by Lamb (1999). Our analysis reveals fundamental differences between the PDM and physical models, both in qualitative and quantitative behaviours, particularly when studying overland-flow-dominated catchments. For example, our study highlights the fact that the PDM model overestimates the rate of river flow growth during highly intensive rainfalls. Additionally, we observe that the PDM model cannot accurately predict peak flows in different seasons, characterised by a wide range of mean precipitation/evapotranspiration rates. We argue that this analytical approach of identifying fundamental differences between models can serve to understand model limitations and develop more theoretically-justified rainfall-runoff models.

physics.flu-dyn↗

arXiv2023

On the evaluation of grid and grid-to-grid rainfall-runoff models and their differences with physical benchmarks

In the first part of our study, we demonstrated how a simple physical benchmark model can be used to assess assumptions of the conceptual models, based on a lumped Probability Distributed Model (PDM) formulated by Lamb (1999). In this second part, we extend the scope of our study to distributed models, which aim to represent the spatial variability of model's elements (e.g. input precipitation, soil moisture levels, flow components etc.). For demonstration purposes, we assess the assumptions of the Grid and Grid-to-Grid models, commonly used for flood real-time forecasting in the UK. While the distributed character of these models is conceptually closer to the physical model, we demonstrate that its exact implementation leads to many qualitative and quantitative differences in the model behaviour. For example, we show that the main assumption, namely that the speed of surface and subsurface flow is constant, causes the Grid-to-Grid model to significantly misrepresent scenarios with no rainfall, leading to too fast river flow decay, and scenarios with upstream rainfall, failing to capture characteristic flash flood formation. We argue that this analytical approach of finding fundamental differences between models may help us to develop more theoretically-justified rainfall-runoff models, e.g. models that can better handle the two aforementioned scenarios and other scenarios in which the spatial dependence is crucial to properly represent the catchment dynamics.

physics.flu-dyn↗

arXiv2023

A calibration-free physicality-based model for predicting peak river flows

Many simple hydrologic models are based on parametric statistical relations between the river flow and catchment properties such as its area, precipitation rates, soil properties, etc., fitted to the available data. The main objective of this work is to explain how these statistical relations emerge from the physical laws governing surface and subsurface flow at a catchment scale. The main achievement of this work is the derivation of an analytic formula for predicting peak monthly and annual river flows. It does not require any parameter calibration, but requires a measurement or estimation of the mean flow at the given catchment's outlet. We found that this model 1) has a simple physical interpretation, 2) provides more precise estimates than the median maximum annual flow (QMED) estimation method from the Flood Estimation Handbook (FEH), commonly used to estimate flood risk in the ungauged catchments in the UK, and 3) is highly accurate for all types of catchments, including the small catchments, for which the standard FEH method is the least accurate.

physics.geo-ph↗

arXiv2023

Pathological exponential asymptotics for a model problem of an equatorially trapped Rossby wave

We examine a misleadingly simple linear second-order eigenvalue problem (the Hermite-with-pole equation) that was previously proposed as a model problem of an equatorially-trapped Rossby wave. In the singularly perturbed limit representing small latitudinal shear, the eigenvalue contains an exponentially-small imaginary part; the derivation of this component requires exponential asymptotics. In this work, we demonstrate that the problem contains a number of pathological elements in exponential asymptotics that were not remarked upon in the original studies. This includes the presence of dominant divergent eigenvalues, non-standard divergence of the eigenfunctions, and inactive Stokes lines due to the higher-order Stokes phenomenon. The techniques developed in this work can be generalised to other linear or nonlinear eigenvalue problems involving asymptotics beyond-all-orders where such pathologies are present.

physics.flu-dyn↗

arXiv2023

Exponential asymptotics and the generation of free-surface flows by submerged line vortices

There has been significant recent interest in the study of water waves coupled with non-zero vorticity. We derive analytical approximations for the exponentially-small free-surface waves generated in two-dimensions by one or several submerged point vortices when driven at low Froude numbers. The vortices are fixed in place, and a boundary-integral formulation in the arclength along the surface allows the study of nonlinear waves and strong point vortices. We demonstrate that for a single point vortex, techniques in exponential asymptotics prescribe the formation of waves in connection with the presence of Stokes lines originating from the vortex. When multiple point vortices are placed within the fluid, trapped waves may occur, which are confined to lie between the vortices. We also demonstrate that for the two-vortex problem, the phenomenon of trapped waves occurs for a countably infinite set of values of the Froude number. This work will form a basis for other asymptotic investigations of wave-structure interactions where vorticity plays a key role in the formation of surface waves.

physics.flu-dyn↗

arXiv2022

Resurgent aspects of applied exponential asymptotics

In many physical problems, it is important to capture exponentially-small effects that lie beyond-all-orders of a typical asymptotic expansion; when collected, the full expansion is known as the trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically in the context of singular non-linear perturbative differential or integral equations. Separate to applied exponential asymptotics, there exists a closely related line of development known as Écalle's theory of resurgence, which describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. This connection is realised through the process of Borel resummation. However, in contrast to singularly perturbed problems, Borel resummation and Écalle's resurgence theory have mainly focused on non-parametric asymptotic expansions (i.e. differential equations without a parameter). The relationships between these latter areas and applied exponential asymptotics has not been thoroughly examined, partially due to differences in language and emphasis. In this work, we explore these connections by developing an alternative framework for the factorial-over-power ansatz in exponential asymptotics that is centred on the Borel plane. Our work clarifies a number of elements used in applied exponential asymptotics, such as the heuristic use of Van Dyke's rule and the universality of factorial-over-power ansatzes. Along the way, we provide a number of useful tools for probing more pathological problems in exponential asymptotics known to arise in applications; this includes problems with coalescing singularities, nested boundary layers, and more general late-term behaviours.

math.CA↗

arXiv2022

Exponential asymptotics for steady parasitic capillary ripples on steep gravity waves

In this paper we develop an asymptotic theory for steadily travelling gravity-capillary waves under the small-surface tension limit. In an accompanying work [Shelton et al. (2021), J. Fluid Mech., vol 922] it was demonstrated that solutions associated with a perturbation about a leading-order gravity wave (a Stokes wave) contain surface-tension-driven parasitic ripples with an exponentially-small amplitude. Thus a naive Poincaré expansion is insufficient for their description. Here, we shall develop specialised methodologies in exponential asymptotics for derivation of the parasitic ripples on periodic domains. The ripples are shown to arise in conjunction with Stokes lines and the Stokes phenomenon. The analysis relies crucially upon the derivation and analysis of singularities in the analytic continuation of the classic Stokes wave. A solvability condition is derived, showing that solutions of this type do not exist at certain values of the Bond number. The asymptotic results are compared to full numerical solutions and show excellent agreement. The work provides corrections and insight of a seminal theory on parasitic capillary waves first proposed by Longuet-Higgins [J. Fluid Mech., vol. 16 (1), 1963, pp. 138-159].

physics.flu-dyn↗

arXiv2020

On the structure of parasitic gravity-capillary waves in the small surface tension limit

In this paper, we examine the formation of small capillary waves (parasitic ripples) on the surface of steep steadily-travelling gravity waves. Previously, authors have developed ad-hoc analytical procedures for describing the formation of such parasitic ripples in potential flows; however, it has not been clear whether the small-surface tension limit is well-posed -- that is, whether it is possible for an appropriate travelling gravity-capillary wave to be continuously deformed to the classic Stokes wave in the limit of vanishing surface tension. The work of Chen & Saffman (1980) had suggested smooth continuation was not possible. In this paper, we numerically explore the low surface tension limit of the steep gravity-capillary travelling-wave problem. Our results allow for a classification of the bifurcation structure that arises, and serve to unify a number of previous numerical studies. Crucially, we demonstrate that different choices of solution amplitude can lead to subtle restrictions on the continuation procedure; the use of wave energy as an amplitude condition allows solution branches to be continuously deformed to the zero surface tension limit.

physics.flu-dyn↗

arXiv2020

Three-dimensional exponential asymptotics and Stokes surfaces for flows past a submerged point source

When studying fluid-body interactions in the low-Froude limit, traditional asymptotic theory predicts a waveless free-surface at every order. This is due to the fact that the waves are in fact exponentially small---that is, beyond all algebraic orders of the Froude number. Solutions containing exponentially small terms exhibit a peculiarity known as the Stokes phenomenon, whereby waves can 'switch-on' seemingly instantaneously across so-called Stokes lines, partitioning the fluid domain into wave-free regions and regions with waves. In three dimensions, the Stokes line concept must extend to what are analogously known as 'Stokes-surfaces'. This paper is concerned with the archetypal problem of uniform flow over a point source---reminiscent of, but separate to, the famous Kelvin wave problem. In theory, there exist Stokes surfaces i.e. manifolds in space that divide wave-free regions from regions with waves. Previously, in Lustri & Chapman (2013) the intersection of the Stokes surface with the free-surface, z=0, was found for the case of a linearised point-source obstruction. Here we demonstrate how the Stokes surface can be computed in three-dimensional space, particularly in a manner that can be extended to the case of nonlinear bodies.

physics.flu-dyn↗

arXiv2020

Gravity-capillary waves in reduced models for wave-structure interactions

In order to determine the steady-state subcritical gravity-capillary waves that are produced by potential flow past a wave-making body, it is typically necessary to impose a radiation condition that allows for capillary waves upstream, but disallows those corresponding to gravity. However, this radiation condition is not known a priori and consequently, the computation of accurate numerical solutions to the steady-state problem remains problematic. Although the physical model might be modified (e.g. with viscosity), recovery of the original problem is not always possible. In this work, we discuss the above radiation problem, and show how, in the low-speed limit, the steady gravity-capillary waves can be resolved using a Sommerfeld-type boundary condition applied to an asymptotically reduced set of water-wave equations. These results allow us to validate the specialized classes of low-speed waves theoretically predicted by Trinh & Chapman (2013) using methods in exponential asymptotics [J. Fluid Mech. 724, pp. 392--424]. The issues of numerically solving the full set of gravity-capillary equations for a potential flow are discussed, and the sensitivity to errors in the boundary conditions is clearly demonstrated.

physics.flu-dyn↗