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Oscillatory motion and wake instability of freely rising axisymmetric bodies. (English) Zbl 1108.76310
Summary: This paper reports on an experimental study of the motion of freely rising axisym- metric rigid bodies in a low-viscosity fluid. We consider flat cylinders with height $$h$$ smaller than the diameter $$d$$ and density $$\rho_{b}$$ close to the density $$\rho_{f}$$ of the fluid. We have investigated the role of the Reynolds number based on the mean rise velocity $$u_{m}$$ in the range $$80 \leqslant Re = u_{m}d/\nu \leqslant 330$$ and that of the aspect ratio in the range $$1.5 \leqslant \chi = d/h \leqslant 20$$. Beyond a critical Reynolds number, $$Re_{c}$$, which depends on the aspect ratio, both the body velocity and the orientation start to oscillate periodically. The body motion is observed to be essentially two-dimensional. Its description is particularly simple in the coordinate system rotating with the body and having its origin fixed in the laboratory; the axial velocity is then found to be constant whereas the rotation and the lateral velocity are described well by two harmonic functions of time having the same angular frequency, $$\omega$$. In parallel, direct numerical simulations of the flow around fixed bodies were carried out. They allowed us to determine (i) the threshold, $$Re_{cf^1}(\chi)$$, of the primary regular bifurcation that causes the breaking of the axial symmetry of the wake as well as (ii) the threshold, $$Re_{cf^2}(\chi)$$, and frequency, $$\omega _{f}$$, of the secondary Hopf bifurcation leading to wake oscillations. As $$\chi$$ increases, i.e. the body becomes thinner, the critical Reynolds numbers, $$Re_{cf^1}$$ and $$Re_{cf^2}$$, decrease. Introducing a Reynolds number $$Re^*$$ based on the velocity in the recirculating wake makes it possible to obtain thresholds and that are independent of $$\chi$$. Comparison with fixed bodies allowed us to clarify the role of the body shape. The oscillations of thick moving bodies $$(\chi < 6)$$ are essentially triggered by the wake instability observed for a fixed body: $$Re_{c}(\chi)$$ is equal to $$Re_{cf^1}(\chi)$$ and $$\omega$$ is close to $$\omega_{f}$$. However, in the range $$6 \leqslant \chi \leqslant 10$$ the flow corrections induced by the translation and rotation of freely moving bodies are found to be able to delay the onset of wake oscillations, causing $$Re_{c}$$ to increase strongly with $$\chi$$. An analysis of the evolution of the parameters characterizing the motion in the rotating frame reveals that the constant axial velocity scales with the gravitational velocity based on the body thickness, $$\sqrt{((\rho_f-\rho_b)/\rho_f)gh}$$, while the relevant length and velocity scales for the oscillations are the body diameter $$d$$ and the gravitational velocity based on $$d$$, $$\sqrt{((\rho_f-\rho_b)/\rho_f)gd}$$, respectively. Using this scaling, the dimensionless amplitudes and frequency of the body oscillations are found to depend only on the modified Reynolds number, $$Re^*$$; they no longer depend on the body shape.

##### MSC:
 76-05 Experimental work for problems pertaining to fluid mechanics 76D25 Wakes and jets 76E99 Hydrodynamic stability
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