Complex ineractions of two vortex sheets
Abstract:
We give a numerical computation of the motion of two, nearly parallel vortex sheet. The motion of two vortex sheet is described by an integral equation which has a singular kernel. This fact makes numerical computation difficult. In order to overcome this difficulty, Chorin's vortex blob method and Krasny's Fourier filtering techniques is adopted as a numerical method. It is known that in vortex sheet a singularity emerges in finite time. Therefore, the time and the place of appearance of the singularity are important index for the motion of vortex sheet. We clarify the interaction of two vortex sheets, by studying these indices of singularity when some numerical parameters are given. In addition, since we are interested in pattern formation which is constructed by two vortex sheets, various complex pattern will be reported, too.
Conclusion:
We investigated numerically the motion of two vortex sheets with various initial locations.
Many interesting patterns of spirals are discovered.
- ( Gamma_1,Gamma_2 )=(1,1)
(Large initial distance H:) Both vortex sheets roll up independently and move in the opposite directions. The critical time of both sheets is very close to that of single vortex sheet with the same disturbance. It is almost independent of the change of initial phase difference alpha.
The influence of two vortex sheets weaken rapidly as H becomes large.
(Small initial distance H:) The vortical core structure is constructed by two vortex sheets. The core structure changes sensitively by varying initial phase difference alpha. The right and left movement of two sheets, which is seen when H is large, is not observed in this case (Vortex core locking).
The critical time of upper vortex sheet is changed by the existence of lower vortex sheet, although its initial disturbance is the same. Lower vortex sheet blows up earlier than single vortex sheet with the same vorticity and initial disturbance for all alpha.
- ( Gamma_1,Gamma_2 )=(1,-1)
(Large initial distance H:) One roll emerges on each vortex sheet and moves to the right. The change of initial phase difference is only connected with the position of spirals. H has little influence on the critical time, too.
(Small initial distance H:) When alpha=0.5, there is no qualitative difference between numerical solutions with various distance. However, the speed of both vortex sheets to the right quicken when H becomes small.
When alpha varies, various vortical patterns are observed.
The critical time gets earlier as H becomes smaller.
It depends on alpha, too.
The influence of initial distance H plays an important role in the interaction of two vortex sheets. Since two vortex sheets roll up almost independently for large H, the only interaction we can observe is movement to the right or to the left direction. For smaller H, a vortex core structure appears and vorticity concentrates to the center. The critical time of two vortex sheets changes sensitively with the change of the initial distance H. Initial phase difference $\alpha$ has an influence on the location and the critical time of two vortex sheets. Vortex patterns changes sensitively when alpha changes.
Parameters:
- Gamma1: the strength of upper vortex sheet.
- Gamma2: the strength of lower vortex sheet.
- H: The initial average distance of two vortex sheets.
- alpha: The initial phase difference of two vortex sheets.
Pictures:
- Gamma1=+1,Gamma2=+1,alpha=0.5pi,H=0.2, t= 0.0 to 1.4
- Gamma1=+1,Gamma2=+1,alpha=0.0pi,H=0.2, t= 0.0 to 1.4
- Gamma1=+1,Gamam2=-1,alpha=0.0pi,H=0.2, t=0.0 to 1.4 , t=1.2 to 2.6
- Gamma1=+1,Gamma2=-1,alpha=0.5pi,H=0.2, t=0.0 to 1.4 , t=1.2 to 2.6
Reference:
T. Sakajo, The interaction of two vortex sheets, Adv. Math. Sci. Apl., vol. 8 no. 2, (1998), pp. 631-662.
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