Singularity formation in a model for the vortex sheet with surface tension

2009 ◽  
Vol 80 (1) ◽  
pp. 102-111 ◽  
Author(s):  
David M. Ambrose
Keyword(s):  
Surface Tension ◽  
Vortex Sheet ◽  
Singularity Formation

Singularity formation in three-dimensional motion of a vortex sheet

1995 ◽  
Vol 300 ◽  
pp. 339-366 ◽  
Author(s):  
Takashi Ishihara ◽  
Yukio Kaneda
Keyword(s):  
Asymptotic Analysis ◽  
Numerical Integration ◽  
Finite Time ◽  
Fourier Coefficients ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Vortex Sheet ◽  
Three Dimensions ◽  
Leading Order ◽  
Singularity Formation

The evolution of a small but finite three-dimensional disturbance on a flat uniform vortex sheet is analysed on the basis of a Lagrangian representation of the motion. The sheet at time t is expanded in a double periodic Fourier series: R(λ1, λ2, t) = (λ1, λ2, 0) + Σn,mAn,m exp[i(nλ1 + δmλ2)], where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. By generalizing Moore's analysis for two-dimensional motion to three dimensions, we derive evolution equations for the Fourier coefficients An,m. The behaviour of An,m is investigated by both numerical integration of a set of truncated equations and a leading-order asymptotic analysis valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time tc = O(lnε−1) where ε is the amplitude of the initial disturbance. The singularity is such that An,0 = O(tc−1) behaves like n−5/2, while An,±1 = O(εtc) behaves like n−3/2 for large n. The evolution of A0,m(spanwise mode) is also studied by an asymptotic analysis valid at large t. The analysis shows that a singularity appears at a finite time t = O(ε−1) and the singularity is characterized by A0,2k ∝ k−5/2 for large k.

On the formation of Moore curvature singularities in vortex sheets

1999 ◽  
Vol 378 ◽  
pp. 233-267 ◽  
Author(s):  
STEPHEN J. COWLEY ◽  
GREG R. BAKER ◽  
SALEH TANVEER
Keyword(s):  
Complex Plane ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Effects ◽  
Vortex Sheet ◽  
Finite Amplitude ◽  
Analytical Continuation ◽  
Curvature Singularity ◽  
Singularity Formation ◽  
Vortex Sheets

Moore (1979) demonstrated that the cumulative influence of small nonlinear effects on the evolution of a slightly perturbed vortex sheet is such that a curvature singularity can develop at a large, but finite, time. By means of an analytical continuation of the problem into the complex spatial plane, we find a consistent asymptotic solution to the problem posed by Moore. Our solution includes the shape of the vortex sheet as the curvature singularity forms. Analytic results are confirmed by comparison with numerical solutions. Further, for a wide class of initial conditions (including perturbations of finite amplitude), we demonstrate that 3/2-power singularities can spontaneously form at t=0+ in the complex plane. We show that these singularities propagate around the complex plane. If two singularities collide on the real axis, then a point of infinite curvature develops on the vortex sheet. For such an occurrence we give an asymptotic description of the vortex-sheet shape at times close to singularity formation.

On singularity formation in three-dimensional vortex sheet evolution

1999 ◽  
Vol 11 (11) ◽  
pp. 3198-3200 ◽  
Author(s):  
M. Brady ◽  
D. I. Pullin
Keyword(s):  
Three Dimensional ◽  
Vortex Sheet ◽  
Singularity Formation
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