Numerical Solutions to Free Boundary Problems

Acta Numerica ◽  
1995 ◽  
Vol 4 ◽  
pp. 335-415 ◽  
Author(s):  
Thomas Y. Hou
Keyword(s):  
Surface Tension ◽  
Crystal Growth ◽  
Free Boundary ◽  
Water Waves ◽  
Numerical Solutions ◽  
Boundary Integral ◽  
Stability And Convergence ◽  
Instability Wave ◽  
Free Surfaces ◽  
Free Boundary Value Problems

Many physically interesting problems involve propagation of free surfaces. Vortex-sheet roll-up in hydrodynamic instability, wave interactions on the ocean's free surface, the solidification problem for crystal growth and Hele-Shaw cells for pattern formation are some of the significant examples. These problems present a great challenge to physicists and applied mathematicians because the underlying problem is very singular. The physical solution is sensitive to small perturbations. Naïve discretisations may lead to numerical instabilities. Other numerical difficulties include singularity formation and possible change of topology in the moving free surfaces, and the severe time-stepping stability constraint due to the stiffness of high-order regularisation effects, such as surface tension.This paper reviews some of the recent advances in developing stable and efficient numerical algorithms for solving free boundary-value problems arising from fluid dynamics and materials science. In particular, we will consider boundary integral methods and the level-set approach for water waves, general multi-fluid interfaces, Hele–Shaw cells, crystal growth and solidification. We will also consider the stabilising effect of surface tension and curvature regularisation. The issue of numerical stability and convergence will be discussed, and the related theoretical results for the continuum equations will be addressed. This paper is not intended to be a detailed survey and the discussion is limited by both the taste and expertise of the author.

Theory and experiment on the low-Reynolds-number expansion and contraction of a bubble pinned at a submerged tube tip

1998 ◽  
Vol 356 ◽  
pp. 93-124 ◽  
Author(s):  
HARRIS WONG ◽  
DAVID RUMSCHITZKI ◽  
CHARLES MALDARELLI
Keyword(s):  
Surface Tension ◽  
Reynolds Number ◽  
Numerical Solutions ◽  
Full Range ◽  
Boundary Integral ◽  
Gas Pressure ◽  
Boundary Integral Method ◽  
Liquid Density ◽  
Gravitational Acceleration ◽  
Dynamic Surface

The expansion and contraction of a bubble pinned at a submerged tube tip and driven by constant gas flow rate Q are studied both theoretically and experimentally for Reynolds number Re[Lt ]1. Bubble shape, gas pressure, surface velocities, and extrapolated detached bubble volume are determined by a boundary integral method for various Bond (Bo=ρga2/σ) and capillary (Ca=μQ/σa2) numbers, where a is the capillary radius, ρ and μ are the liquid density and viscosity, σ is the surface tension, and g is the gravitational acceleration.Bubble expansion from a flat interface to near detachment is simulated for a full range of Ca (0.01–100) and Bo (0.01–0.5). The maximum gas pressure is found to vary almost linearly with Ca for 0.01[les ]Ca[les ]100. This correlation allows the maximum bubble pressure method for measuring dynamic surface tension to be extended to viscous liquids. Simulated detached bubble volumes approach static values for Ca[Lt ]1, and asymptote as Q3/4 for Ca[Gt ]1, in agreement with analytic predictions. In the limit Ca→0, two singular time domains are identified near the beginning and the end of bubble growth during which viscous and capillary forces become comparable.Expansion and contraction experiments were conducted using a viscous silicone oil. Digitized video images of deforming bubbles compare well with numerical solutions. It is observed that a bubble contracting at high Ca snaps off.

scholarly journals The surface-tension-driven evolution of a two-dimensional annular viscous tube

2007 ◽  
Vol 593 ◽  
pp. 181-208 ◽  
Author(s):  
I. M. GRIFFITHS ◽  
P. D. HOWELL
Keyword(s):  
Surface Tension ◽  
Numerical Solutions ◽  
Applied Pressure ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Free Surfaces ◽  
Final Shape ◽  
Capillary Tubing ◽  
Well Posed

We consider the evolution of an annular two-dimensional region occupied by viscous fluid driven by surface tension and applied pressure at the free surfaces. We assume that the thickness of the domain is small compared with its circumference, so that it may be described as a thin viscous sheet whose ends are joined to form a closed loop. Analytical and numerical solutions of the resulting model are obtained and we show that it is well posed whether run forwards or backwards in time. This enables us to determine, in many cases explicitly, which initial shapes will evolve into a desired final shape. We also show how the application of an internal pressure may be used to control the evolution.This work is motivated by the production of non-axisymmetric capillary tubing via the Vello process. Molten glass is fed through a die and drawn off vertically, while the shape of the cross-section evolves under surface tension and any applied pressure as it flows downstream. Here the goal is to determine the die shape required to achieve a given desired final shape, typically square or rectangular. We conclude by discussing the role of our two-dimensional model in describing the three-dimensional tube-drawing process.

Highly nonlinear standing water waves with small capillary effect

1998 ◽  
Vol 369 ◽  
pp. 253-272 ◽  
Author(s):  
WILLIAM W. SCHULTZ ◽  
JEAN-MARC VANDEN-BROECK ◽  
LEI JIANG ◽  
MARC PERLIN
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Standing Waves ◽  
High Amplitude ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Ripple Formation ◽  
Node Distribution ◽  
Highly Nonlinear ◽  
Periodic Standing Waves

We calculate spatially and temporally periodic standing waves using a spectral boundary integral method combined with Newton iteration. When surface tension is neglected, the non-monotonic behaviour of global wave properties agrees with previous computations by Mercer & Roberts (1992). New accurate results near the limiting form of gravity waves are obtained by using a non-uniform node distribution. It is shown that the crest angle is smaller than 90° at the largest calculated crest curvature. When a small amount of surface tension is included, the crest form is changed significantly. It is necessary to include surface tension to numerically reproduce the steep standing waves in Taylor's (1953) experiments. Faraday-wave experiments in a large-aspect-ratio rectangular container agree with our computations. This is the first time such high-amplitude, periodic waves appear to have been observed in laboratory conditions. Ripple formation and temporal symmetry breaking in the experiments are discussed.

The dynamics of strong turbulence at free surfaces. Part 1. Description

2001 ◽  
Vol 449 ◽  
pp. 225-254 ◽  
Author(s):  
M. BROCCHINI ◽  
D. H. PEREGRINE
Keyword(s):  
Surface Tension ◽  
Surface Roughness ◽  
Free Surface ◽  
Water Waves ◽  
Theoretical Studies ◽  
Free Surfaces ◽  
Submerged Jets ◽  
Wide Range ◽  
Experimental And Theoretical Studies ◽  
Two Parameter

A free surface may be deformed by fluid motions; such deformation may lead to surface roughness, breakup, or disintegration. This paper describes the wide range of free-surface deformations that occur when there is turbulence at the surface, and focuses on turbulence in the denser, liquid, medium. This turbulence may be generated at the surface as in breaking water waves, or may reach the surface from other sources such as bed boundary layers or submerged jets. The discussion is structured by consideration of the stabilizing influences of gravity and surface tension against the disrupting effect of the turbulent kinetic energy. This leads to a two-parameter description of the surface behaviour which gives a framework for further experimental and theoretical studies. Much of the discussion is necessarily heuristic, and is often limited by a lack of appropriate experimental observations. It is intended that such experiments be stimulated, to test the value or otherwise of our two-parameter description.

scholarly journals Radiation of water waves by a heaving submerged horizontal disc

1997 ◽  
Vol 337 ◽  
pp. 365-379 ◽  
Author(s):  
P. A. MARTIN ◽  
L. FARINA
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Water Waves ◽  
Numerical Solutions ◽  
Boundary Integral ◽  
Fredholm Integral Equations ◽  
Mass Coefficient ◽  
Added Mass Coefficient ◽  
Thin Rigid Plate ◽  
Hypersingular Boundary Integral Equation

A thin rigid plate is submerged beneath the free surface of deep water. The plate performs small-amplitude oscillations. The problem of calculating the radiated waves can be reduced to solving a hypersingular boundary integral equation. In the special case of a horizontal circular plate, this equation can be reduced further to one-dimensional Fredholm integral equations of the second kind. If the plate is heaving, the problem becomes axisymmetric, and the resulting integral equation has a very simple structure; it is a generalization of Love's integral equation for the electrostatic field of a parallel-plate capacitor. Numerical solutions of the new integral equation are presented. It is found that the added-mass coefficient becomes negative for a range of frequencies when the disc is sufficiently close to the free surface.

Open capillary siphons

2021 ◽  
Vol 932 ◽  
Author(s):  
Kaizhe Wang ◽  
Pejman Sanaei ◽  
Jun Zhang ◽  
Leif Ristroph
Keyword(s):  
Surface Tension ◽  
Fluid Transport ◽  
Numerical Solutions ◽  
Strong Dependence ◽  
Viscous Friction ◽  
Geometrical Parameters ◽  
Controlled System ◽  
Free Surfaces ◽  
Biological Phenomena ◽  
Shaped Tube

Flow in the inverted U-shaped tube of a conventional siphon can be established and maintained only if the tube is filled and closed, so that air does not enter. We report on siphons that operate entirely open to the atmosphere by exploiting surface tension effects. Such capillary siphoning is demonstrated by paper tissue that bridges two containers and conveys water from the upper to the lower. We introduce a more controlled system consisting of grooves in a wetting solid, formed here by pressing together hook-shaped metallic rods. The dependence of flux on siphon geometry is systematically measured, revealing behaviour different from the conventional siphon. The flux saturates when the height difference between the two container's free surfaces is large; it also has a strong dependence on the climbing height from the source container's free surface to the apex. A one-dimensional theoretical model is developed, taking into account the capillary pressure due to surface tension, pressure loss due to viscous friction, and driving by gravity. Numerical solutions are in good agreement with experiments, and the model suggests hydraulic interpretations for the observed flux dependence on geometrical parameters. The operating principle and characteristics of capillary siphoning revealed here can inform biological phenomena and engineering applications related to directional fluid transport.

On uniqueness in the problem of gravity–capillary water waves above submerged bodies

2009 ◽  
Vol 465 (2106) ◽  
pp. 1743-1761 ◽  
Author(s):  
Oleg V. Motygin ◽  
Philip McIver
Keyword(s):  
Surface Tension ◽  
Unique Solvability ◽  
Water Waves ◽  
Linear Problem ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Trapped Modes ◽  
Capillary Water ◽  
Simple Bounds

In this paper, we consider the two-dimensional linear problem of wave–body interaction with surface tension effects being taken into account. We suggest a criterion for unique solvability of the problem based on symmetrization of boundary integral equations. The criterion allows us to develop an algorithm for detecting non-uniqueness (finding trapped modes) for given geometries of bodies; examples of numerical computation of trapped modes are given. We also prove a uniqueness theorem that provides simple bounds for the possible non-uniqueness parameters.

scholarly journals An Existence Theory for Gravity–Capillary Solitary Water Waves

Water Waves ◽  
2021 ◽  
Author(s):  
M. D. Groves
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Applied Mathematics ◽  
Spatial Dynamics ◽  
Existence Theory ◽  
Water Wave Problem ◽  
Reduction Methods ◽  
Centre Manifold Reduction ◽  
Weak Surface ◽  
The One

AbstractIn the applied mathematics literature solitary gravity–capillary water waves are modelled by approximating the standard governing equations for water waves by a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). These formal arguments have been justified by sophisticated techniques such as spatial dynamics and centre-manifold reduction methods on the one hand and variational methods on the other. This article presents a complete, self-contained account of an alternative, simpler approach in which one works directly with the Zakharov–Craig–Sulem formulation of the water-wave problem and uses only rudimentary fixed-point arguments and Fourier analysis.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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