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Tham khảo tài liệu 'kundu fluid mechanics 2 episode 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 5. Oijfunion of a Vortex Sheri 289 8. Diffusion of a Vortex Sheet Consider the case in which the initial velocity field is in the form of a vortex sheet with u u for y 0 and u u for y 0. We want to investigate how the vortex sheet decays by viscous diffusion. The governing equation is du _ d2u dt dy2 subject to u y 0 t sgn y u oo r u u oo t 17 where sgn y is the sign function defined as 1 for positive y and 1 for negative y. As in the previous section the parameter u can be eliminated from the governing set by regarding u u as the dependent variable. Then u u must be a function of y t v and a dimensional analysis reveals that there must exist a similarity solution in the form ri UTt The detailed arguments for the existence of a solution in this form are given in the preceding section. Substitution of the similarity form into the governing set transforms it into the ordinary differential equation F 2rjF F oo l F -oo -1 whose solution is F rj erf ty . The velocity distribution is therefore u i erf 9.38 A plot of the velocity distribution is shown in Figure 9.11. If we define the width of die transition layer as the distance between the points where u 0.9517 then the corresponding value of r is 1.38 and consequently the width of the fransition layer is 5.52yfvt. It is clear that the flow is essentially identical to that due to the impulsive Stan of a flat plate discussed in the preceding section. In fact each half of Figure 9.11 is identical to Figure 9.10 within an additive constant of 1 . In both problems 290 Laminar Flow Figure 9.11 Viscous decay of a vortex sheet. The right panel shows the nondimcnsional solution and he left panel indicates Ike vorticity distribution at two times. the initial delta-fimction-like vorticity is diffused away. In the present problem the magnitude of vorticity at any time is 1 du 9.39 This is a Gaussian distribution whose width increases with lime as -y t while the maximum value decreases as 1 y t. The total amount of vorticity is OC .