Solution of the equations of rotational motion for a class of torque-free gyrostats

Abstract

with the same cylinders fitted with end plates to simulate the case of infinite length. The cylinders were mounted sufficiently rigid to prevent oscillation. The results are presented in Fig. 2. In accordance with earlier findings the Strouhal number of the infinite cylinder is approximately 0.2, computed from the frequency peak protruding from the otherwise turbulent spectrum. In case of the finite cylinder with aspect ratio 4, a regular motion indicated by a distinct frequency in the spectral distribution with S « 0.45 was found at Rer> ~ 2.4 X 10s. Apparently the frequency peak tended to become weaker with decreasing Reynolds number and at RCD ~ 5 X 104 a distinct frequency could no longer be discerned. The spectra obtained with the cylinder of aspect ratio 8 showed a shedding frequency with S = 0.2 at subcritical Reynolds numbers with the indication to become stronger with ReD increasing. The evaluation of Roshko's wake Strouhal number/' Sw = fDw/Uw showed for all measurements Sw « 0.2. Naumann2 found that due to the three dimensionality of the flow the separation line is no longer straight and thus the circulation of a separating vortex is no longer constant, which leads to a break up. This seems to be confirmed by some surface patterns presented by Gould, Raymer, and Ponsford.3 The surface patterns which were observed during this work, however, exhibit a very straight separation line and at the same time there was no indication of vortex shedding. It appears therefore that some additional explanations may be necessary. For any blunt body part of the flow going over the top of the body is being dragged into the wake. The strength of this downwash obviously decreases with increasing Reynolds number and is restricted to a certain length of the cylinder from top downwards, i.e., its over-all influence decreases with increasing aspect ratio. One may consider the downwash flow as a kind of air curtain acting to some extent like a splitter plate. Depending on its over-all strength it dampens the feedback from both sides of the cylinder arid thus the shedding mechanism. Two tests were made to check on this model. In the first case the cylinder with end plate was provided with a slit in the base open to atmosphere, so that by action of the low base pressure a weak jet was blown into the wake. This largely damped the shedding which was clearly observed when the slit was blocked. In the second test the downwash in the near region of the finite cylinder was prevented by a small plate behind the cylinder between half height and top. The presence of this plate accounted for the occurrence of regular shedding. The main observations may be summarized as follows: 1) the pressure distribution around a blunt cylinder, being similar to that of the infinite cylinder has the same Ren — dependence; 2) the region of highly three-dimensional flow in the wake is restricted to the first 3—4 cylinder diameters; 3) turbulence levels in the near wake reach values of approximate 80% of the local mean velocity; 4) macro scales in the nonperiodic wake are almost an order of magnitude smaller than in the periodic, i.e., shedding case. Their mean values over the cylinder height increase linearly with downstream distance after the first 2 to 3 cylinder diameters; 5) the universal wake Strouhal number appears to be a constant for finite and infinite cylinders alike;