Multimoment Hydrodynamics in Problem on Flow around a Sphere: Entropy Interpretation of the Appearance and Development of Instability

Igor V. Lebed

Zhukovsky Central Institute of Aerohydrodynamics, Moscow, Russia.

DOI: 10.4236/ojfd.2014.42015                

Abstract

Multimoment hydrodynamics equations are applied to investigate the phenomena of appearance and development of instability in problem on a flow around a solid sphere at rest. The simplest solution to the multimoment hydrodynamics equations coincides with the Stokes solution to the classic hydrodynamics equations in the limit of small Reynolds number values, . Solution  to the multimoment hydrodynamics equations reproduces recirculating zone in the wake behind the sphere having the form of an axisymmetric toroidal vortex ring. The  solution remains stable while the entropy production in the system exceeds the entropy outflow through the surface confining the system. The passage of the first critical value  is accompanied by the  solution stability loss. The  solution, when loses its stability, reproduces periodic pulsations of the periphery of the recirculating zone in the wake behind the sphere. The  and  solutions to the multimoment hydrodynamics equations interpret a vortex shedding. After the second critical value  is reached, the  solution at the periphery of the recirculating zone and in the far wake is replaced by the  solution. In accordance with the  solution, the periphery of the recirculating zone periodically detached from the core and moves downstream in the form of a vortex ring. After the attainment of the third critical value , the  solution at the periphery of the recirculating zone and in the far wake is replaced by the  solution. In accordance with the  solution, vortex rings penetrate into each other and form the continuous vortex sheet in the wake behind the sphere. The replacement of one unstable flow regime by another unstable regime is governed the tendency of the system to discover the fastest path to depart from the state of statistical equilibrium. Having lost the stability, the system does not reach a new stable position. Such a scenario differs from the ideas of classic hydrodynamics, which interprets the development of instability in terms of bifurcations from one stable state to another stable state. Solutions to the multimoment hydrodynamics equations indicate the direction of instability development, which qualitatively reproduces the experimental data in a wide range of Re values. The problems encountered by classic hydrodynamics when interpreting the observed instability development process are solved on the way toward an increase in the number of principle hydrodynamic values.

Keywords

Multimoment Hydrodynamics, Instability, Entropy, Vortex Shedding

Conflicts of Interest

The authors declare no conflicts of interest.

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