We consider the Cauchy problem for the two-dimensional vorticity equation. We show that the solution ω behaves like a constant multiple of the Gauss kernel having the same total vorticity as time tends to infinity. No particular structure of initial data ω 0 = ω(x, 0) is assumed except the restriction that the Reynolds number R -^\ω Q \dx/v is small, where v is the kinematic viscosity. Applying a time-dependent scale transformation, we show a stability of Burgers' vortex, which physically implies formation of a concentrated vortex.
This paper investigates the propulsive performance of the lunate tails of aquatic animals achieving high propulsive efficiency (the hydromechanical efficiency being defined as the ratio of the work done by the mean forward thrust to the mean rate at which work is done by the tail movements on the surrounding fluid). Small amplitude heaving and pitching motions of a finite flat-plate wing of general planform with a rounded leading edge and a sharp trailing edge are considered. This is a generalization of Chopra's (1974) work on model rectangular tails. This motion characterizes vertical oscillations of the horizontal tail flukes of some cetacean mammals. The same oscillations, turned through a right angle to become horizontal motions of side-slip and yaw, characterize the caudal fins of certain fast-swimming fishes; viz. wahoo, tunny, wavyback skipjack, etc., from the Percomorphi and whale shark, porbeagle, etc., from the Selachii. Davies’ (1963, 1976) method of finding the loading distribution on the wing and generalized force coefficients, through approximate solution of an integral equation relating the loading and the upwash (lifting-surface theory), is used to find the total thrust and the rate of working of the tail, which in turn specify the hydromechanical swimming performance of the animals. The physical parameters concerned are the tail aspect ratio ((span)2/planform area), the reduced frequency (angular frequency x typical length/forward speed), the feathering parameter (the ratio of the tail slope to the slope of the path of the pitching axis), the position of the pitching axis, and the curved shapes of the leading and trailing edges. The variation of the thrust and the propulsive efficiency with these parameters has been discussed to indicate the optimum shape of the tail. It is found that, compared with a rectangular tail, a curved leading edge as in lunate tails gives a reduced thrust contribution from the leading-edge suction for the same total thrust; however, a sweep angle of the leading edge exceeding about 30° leads to a marked reduction of efficiency. Another implication of the present analysis is that no negative work is involved in the actual oscillation of the tail.The present results are used to obtain an estimate of the drag coefficient for the motion of the animals, based on observed data and the computed thrust. The results show some evidence of differences between the CD's for cetacean mammals and scombroid fish respectively. Some discussion of this difference is also given.
Fundamental aspects of the acoustic emission by vortex motions are considered by summarizing our recent work. Three typical cases are presented as illustrative examples: (i) head-on collision of two vortex rings, (ii) a vortex ring moving near a circular cylinder, and (iii) a vortex ring moving near a sharp edge of a semi-infinite plate. The theory of aerodynamic sound for low-Mach-number motion of an inviscid fluid predicts that the amplitude of the acoustic pressure in the far field is proportional to U4, U3 and U2.5 for (i)-(iii) respectively, where U is the translation velocity of a single vortex ring. Therefore the vortex-edge interaction generates the most powerful sound among the three cases at low Mach numbers. Our observations have confirmed these scaling laws. In addition to the scaling properties, we show the wave profiles of the emission as well as the directionality pattern. The head-on collision radiates waves of quadrupole directionality, whereas waves of dipole property are originated by the vortex-cylinder interaction. The third, vortex-edge, interaction generates waves of a cardioid directionality pattern. The wave profiles of all three cases are related to the time derivatives of the volume flux (through the vortex ring) of an imaginary potential flow which is characteristic of each configuration, although the orders of the time derivatives are different for each case. The observed profiles are surprisingly well fitted to the curves predicted by the theory, except the final period of the first case, in which viscosity is assumed to play an important role. The observed wave profiles are shown in a perspective diagram.
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