Simple and efficient numerical methods for vortex sheet motion with surface tension

2013 ◽  
Vol 74 (6) ◽  
pp. 422-438 ◽  
Author(s):  
Suyeon Shin ◽  
Sung-Ik Sohn ◽  
Woonjae Hwang
Keyword(s):  
Surface Tension ◽  
Numerical Methods ◽  
Vortex Sheet

A numerical study of the descent of a vortex pair in a stably stratified atmosphere

1975 ◽  
Vol 71 (1) ◽  
pp. 1-13 ◽  
Author(s):  
F. M. Hill
Keyword(s):  
Numerical Methods ◽  
Numerical Study ◽  
Vortex Pair ◽  
Vortex Sheet ◽  
Discrete Line

Numerical methods are used to investigate the motion of a horizontal vortex pair through a stably stratified atmosphere. The vortices carry with them a mass of fluid whose density differs from that of the air through which it descends, and the surface of this accompanying fluid becomes a vortex sheet, which is modelled by a set of discrete line vortices.It is shown that, at first, the vortex pair slows down with the shape of the envelope of the accompanying fluid remaining constant. Later, vorticity concentrates at the rear, initiating detrainment and causing a downward acceleration of the vortex pair. Throughout the motion, the vortices approach each other.

Stable Methods for Vortex Sheet Motion in the Presence of Surface Tension

1998 ◽  
Vol 19 (5) ◽  
pp. 1737-1766 ◽  
Author(s):  
Gregory Baker ◽  
André Nachbin
Keyword(s):  
Surface Tension ◽  
Vortex Sheet

On the evolution of disturbances at an inviscid interface

1981 ◽  
Vol 108 ◽  
pp. 159-170 ◽  
Author(s):  
P. N. Shankar
Keyword(s):  
Surface Tension ◽  
Finite Time ◽  
Mixed Boundary ◽  
Vortex Sheet ◽  
Major Effect ◽  
Interface Shape ◽  
Initial Value ◽  
Initial Shape ◽  
Infinite Extent ◽  
Effect Of Surface

The initial-value problem for the evolution of the interface η(x, t) separating two unbounded, inviscid streams is considered in the framework of linearized analysis. Given the initial shape y = εη0(x) of the interface at t = 0 the objective is to calculate the interface shape η(x,t) for later times. First, it is shown that, if the vortex sheet is of infinite extent, if surface tension is absent and if the two streams are of the same density, the evolution is given by \[ \eta(x,t) = \epsilon(1-\alpha)^{-1}{\rm Re}[\{(1-\alpha)+(1+\alpha)i\}\eta_0\{x-{\textstyle\frac{1}{2}}((1+\alpha)+(1-\alpha)i)t\}], \] where α (≠ 1) is the ratio of the speeds of the streams, provided the initial interface shape εη0(x) is analytic and its Fourier transform decays sufficiently rapidly. An interesting consequence is that it is possible, under certain circumstances, for the interface to develop singularities after a finite time. Next it is shown that when the two streams move at the same speed (α = 1) the growth of η is given by \[ \eta(x,t) = \epsilon\eta_0(x-t)+\epsilon t\,d\eta_0(x-t)/dx \] with mild restrictions on η0(x). The major effect of surface tension, it is found, is to prevent the occurrence of singularities after a finite time, a distinct possibility in its absence. Finally the vortex sheet shed by a semi-infinite flat plate is considered. The unsteady mixed boundary-value problem is formally solved by using parabolic coordinates and Fourier-Laplace transforms.

Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability

1982 ◽  
Vol 119 ◽  
pp. 507-532 ◽  
Author(s):  
D. I. Pullin
Keyword(s):  
Surface Tension ◽  
Interfacial Tension ◽  
Vortex Sheet ◽  
Taylor Instability ◽  
Late Time ◽  
Initial Value ◽  
Linearized Stability ◽  
Highly Nonlinear ◽  
Rayleigh Taylor Instability ◽  
Tension Term

We consider the behaviour of an interface between two immiscible inviscid incompressible fluids of different density moving under the action of gravity, inertial and interfacial tension forces. A vortex-sheet model of the exact nonlinear two-dimensional motion of this interface is formulated which includes expressions for an appropriate set of integral invariants. A numerical method for solving the vortex-sheet initial-value equations is developed, and is used to study the nonlinear growth of finite-amplitude normal modes for both Kelvin-Helmholtz and Rayleigh-Taylor instability. In the absence of an interfacial or surface-tension term in the integral-differential equation that describes the evolution of the circulation distribution on the vortex sheet, it is found that chaotic motion of, or the appearance of curvature singularities in, the discretized interface profiles prevent the simulations from proceeding to the late-time highly nonlinear phase of the motion. This unphysical behaviour is interpreted as a numerical manifestation of possible ill-posedness in the initial-value equations equivalent to the infinite growth rate of infinitesimal-wavelength disturbances in the linearized stability theory. The inclusion of an interfacial tension term in the circulation equation (which stabilizes linearized short-wavelength perturbations) was found to smooth profile irregularities but only for finite times. While coherent interfacial motion could then be followed well into the nonlinear regime for both the Kelvin-Helmholtz and Rayleigh-Taylor modes, locally irregular behaviour eventually reappeared and resisted subsequent attempts at numerical smoothing or suppression. Although several numerical and/or physical mechanisms are discussed that might produce irregular behaviour of the discretized interface in the presence of an interfacial-tension term, the basic cause of this instability remains unknown. The final description of the nonlinear interface motion thus awaits further research.

scholarly journals An analytical and numerical study of capillary menisci

2021 ◽  
pp. 80-93
Author(s):  
А.А. Сокуров
Keyword(s):  
Surface Tension ◽  
Numerical Methods ◽  
Existence And Uniqueness ◽  
Approximate Calculation ◽  
Large Scale ◽  
Size Dependence ◽  
Numerical Study ◽  
Equilibrium Shape ◽  
Model Parameters ◽  
Uniqueness Theorems

В работе рассмотрены математические модели осесимметричных капиллярных менисков — лежащая и висящая капли, развернутый мениск, учитывающие размерную зависимость поверхностного натяжения. Доказаны теоремы существования и единственности решений задач, описывающих равновесные поверхности менисков. Разработаны и протестированы эффективные численные методы, предназначенные для приближенного расчета профилей менисков. На языке «Wolfram Language» написана компьютерная программа, с помощью которой проведены масштабные вычислительные эксперименты по выявлению степени и характера влияния параметров моделей на равновесную форму каждого из рассматриваемых менисков In the current paper we consider the mathematical models of axisymmetric capillary menisci — sessile and pendant drops, rolled out meniscus, taking into account the size dependence of surface tension. Existence and uniqueness theorems for solutions of problems describing equilibrium meniscus surfaces are proved. Effective numerical methods have been developed and tested for the approximate calculation of meniscus profiles. A computer program is written in the Wolfram Language, with the help of which large-scale computational experiments were carried out to reveal the degree and nature of the influence of the model parameters on the equilibrium shape of each type of menisci.

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