A multiscale asymptotic analysis of time evolution equations on the complex plane

2016 ◽  
Vol 57 (7) ◽  
pp. 073504 ◽  
Author(s):  
Gastão A. Braga ◽  
William R. P. Conti
Keyword(s):  
Asymptotic Analysis ◽  
Time Evolution ◽  
Complex Plane ◽  
Evolution Equations

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.

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