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Waves from a Supersonically-Revolving Disturbance in Relation to Over-Reflection Instability of an Annular Flow: In-Lab Shallow-Water Simulations

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Wave generation due to curvilinear supersonic motion of a disturbance came first to the attention of researchers in practical and theoretical aeronautics because of non- trivial behaviour of the sonic boom fronts from smoothly-turning jet planes. Having passed some distance to the centre of curvature of the trajectory, the ‘inner’ one of the two fronts suffered a sharp kink, which looked like a kind of reflection, and thence went backwards, gradually receding from the centre to the trajectory [1, 2]. Later these aeronautical data were not confirmed in a comprehensive theoretical study based on Lighthill's kinematic wave method in application to the general case of dispersive media and combined with hydrodynamic experiments on soap films [3, 4]. This time the front was shown to experience not reversion but termination in its point closest to the centre.
Meanwhile the phenomenon can play an important role in dynamics of non-straight supersonic flows. If the wave from such a disturbance on a streamline in a boundary or shear zone of the flow does return back to the same streamline, it can trigger generation of noise or even destabilization of the flow due to the mechanism of wave over-reflection [5-7]. Processes of the type are proved to be key elements in developing the over-reflection instability of differentially rotating compressible media [8, 9], which is of remarkable significance for understanding dynamics of astrophysical discs [10].
The confirmation to the aeronautical fact of the front reversion has been obtained in the presented laboratory simulations with shallow-water waves in a thin free-surface layer of liquid as a model of sound in 2D gas. The waves from a small body (artificial disturbance), which is dipped into the layer and revolves there in a circular orbit, behave indeed exactly like the fronts from the supersonic planes. The loci of the waves observed (Huygens-Mach fronts) turn out to be excellently described both qualitatively and quantitatively by the equations of a circle involute. The radius of the corresponding circular evolute equals the linear velocity of the (‘sonic’) shallow-water waves divided by the angular velocity of the disturbance. The evolute itself is like a ‘guideway’ for the reversion point that moves along the circle at the sonic speed.
Among unordinary features of the phenomenon emerged in the simulations, there is wave generation by the disturbance at its sonic motion. In this peculiar case, when the evolute coincides with the orbit, the inner wave degrades into nothing, but the wave outside the orbit remains to be generated, appearing well-pronounced and ‘full in amplitude’. Nothing of the kind could obviously happen if the disturbance moved rectilinearly.
The results of the simulations have been used for analysing and interpreting results of the earlier experiments on over-reflection instability of a ‘supersonic’ annular flow in shallow water [9]. The flow was maintained around a circular ‘core’, i.e. around an initially-still central part of a free-surface liquid layer.
Due to the instability, coherent wave structures were exited in the core, varying in their order of axial symmetry (mode number) and velocity of rotation. The structures can be considered as resonant superpositions of the above kind of waves reflected multiply from the edge of the core. In this approach, the core acts as an acoustic resonator independent, in principle, on whether the waves are amplified at the reflection or are not. Parameters of the self-modes calculated for such a resonator correlate very well with those of the wave structures observed in the experiments (design, mode numbers, and rotation frequencies).
Also this simple approach allows one to clarify easily the origin of some significant features of the structures that were revealed but explained in the experiment and general numerical study [8, 9]. Among them, there are increase in the perturbation amplitude with approaching to the evolute and absence of any perturbation behind the evolute in the very central area of the core.


References

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8. Fridman A.M., Polyachenko E.V., Torgashin Yu.M., Yanchenko S.G., Snezhkin E.N., Phys. Lett. A 349, 198 (2006).
9. Fridman A.M., Snezhkin E.N., Chernikov G.P., Rylov A.Yu., Titishov K.B., Torgashin Yu.M., Phys. Lett. A 372, 4822 (2008).
10. Fridman A.M., Bisikalo D.V., Boyarchuk A.A., Torgashin Yu.M., Pustil’nik L.A., Astron. Rep. 53, no. 8, 750 (2009).

Author(s):

Evgeny Snezhkin    
National Research Centre ‘Kurchatov Institute’
Russian Federation