Two cases of symmetry breaking of free surface flows

The present thesis consists of two parts; both are devoted to two celebrated old problems in fluid dynamics. The first deals with symmetry breaking in a liquid layer flowing down an inclined plane. The second problem concerns the equilibrium and symmetry breaking of interfacial polygonal patterns generated by a system of vortices arranged on a circular ring. The first problem dates back to Nusselt (1916) who obtained the solution for the basic flow. Since then, thin layers of liquid falling down inclined plane continues to be the subject of extensive studies for both their practical applications and theoretical value. In this thesis, the problem is approached analytically. Three new mathematical models are proposed. The first two involve three and four equations respectively. These produce linear stability results that agree fairly with past experimental outcomes and results obtained with similar models. For a deeper and qualitative analysis a lower dimension model that retains the physics is needed. Hence, a two-equation model (involving only two fundamental flow parameters namely the film thickness and flow rate) is derived. The new model taking account of the shear stress at the free surface is shown to be superior to the existing two-equation model of Usha and Uma in Phys Fluid (2004). The influence of electrical and magnetic fields on the stability of falling film of an electrically conductor fluid is also investigated. In comparison with the model of Korsunsky ( Eur.J.F.M.1999 ) for higher Reynolds numbers. The proposed model takes account of the inertia terms, which are of second order with respect to a small parameter namely the long wave parameter. As shown through the chapter four of the part one, the proposed two-equation model improves significantly Korsunsky's model. The second problem dates back to Kelvin (1867) who hypothesized atoms to be point vortices arranged in circular ring forming symmetrical polygonal patterns. Although, the atomic vortex model is long abandoned, the problem of system of point vortices has become of great interest in superfluidity and by analogy in plasma physics. Moreover, polygonal patterns, which are the signature of the presence of vortices, equally distributed in) rings were also observed in several engineering problems and geophysical flows in nature. In fluid dynamics, polygonal patterns become clearly visible in swirling flows where the vortex core is hollow. The empty core can eventually support polygonal shapes (up to hexagonal). The first experimental report on the phenomenon was by Vatistas in 1990. In this thesis the phenomenon is revisited using image-processing technique that allows a deeper and more precise investigation. The dynamics of the patterns is investigated and for the first time the transition from one pattern to another is explored in detail. The stability condition for a system of point vortices on circular ring derived first by J.J Thomson (1897) and generalized later by Havelock (1931) for N point vortices including the influence of circular boundaries surrounding the equilibrium is confirmed. Frequency locking between the pattern and the disk frequencies which are suspected in the previous experiments is established and quantified. Moreover, the transition from the elliptical to the hexagonal pattern is found that it follows a "devil's staircase" scenario. Due to the similarity between the problem under the scope and other fields of physics, the present experimental results are anticipated to go beyond the field of fluid mechanics

Mixing Within Patterned Vortex Core

The video shows the flow dynamics within inner and outer regions of a vortex core. The observed phenomena mimic a transport process occurring within the Antarctic vortex. The video shows two distinct regions: a strongly mixed core and broad ring of weakly mixed region extending out the vortex core boundaries. The two regions are separated by a thin layer that isolates the weakly and strongly mixed regions; this thin layer behaves as barrier to the mixing of the two regions. The video shows that the barriers deplete when a swirl of the vortex core increases and the vortex core espouses a triangular pattern.Comment: 62nd Annual Meeting of the APS Division of Fluid Dynamics, Fluid Dynamics Vide

Flow visualization and numerical simulation of a two-dimensional fluid flow over a foil

This paper deals with a simple, fast and economical visualization method to validate two-dimensional large eddy simulations (LES) of the flow over a foil. This technique exploits the optical properties of soap film and relies on the wake patterns and the frequency at which these are shed at the trailing edge of the foil

The effect of viscosity on the rotating waves and polygonal patterns within a hollow vortex core

The question of how viscosity influences the development of instabilities within a rotating shallow layer of liquid, which gives rise to polygonal patterns, has been investigated experimentally. A phase diagram of the existence regions of these polygonal patterns is constructed in plane, where Fr is the Froude number and Ta is the Taylor number. The results show that the effect of the viscosity on the domain of existence of the patterns depends on the initial fluid height above the rotating disc. The results also show that the variation of viscosity does not affect the locking ratio between the rotational frequencies of the pattern to the disc; the two frequencies remain locked at approximately 1/3

Transitions between systems of satellite vortices in a rotating fluid

We investigate experimentally the transitions between systems of two and three satellite vortices found within a shallow layer of water above a rotating disk at the bottom of an open stationary cylindrical tank. We found that these systems of two and three satellite vortices are associated with two fundamental multipolar modes, namely, the quadrupole and hexapole, respectively. The forward and backward transitions between the two systems of satellite vortices, which occur at critical disk speeds, involve growth and decay of the fundamental modes as well as the excitation of their common harmonic (the dodecapole) and symmetric radial oscillations

Transitions between systems of satellite vortices in a rotating fluid

ABSTRACT: We investigate experimentally the transitions between systems of two and three satellite vortices found within a shallow layer of water above a rotating disk at the bottom of an open stationary cylindrical tank. We found that these systems of two and three satellite vortices are associated with two fundamental multipolar modes, namely, the quadrupole and hexapole, respectively. The forward and backward transitions between the two systems of satellite vortices, which occur at critical disk speeds, involve growth and decay of the fundamental modes as well as the excitation of their common harmonic (the dodecapole) and symmetric radial oscillations

A note on relative equilibria in a rotating shallow water layer

AbstractRelative equilibria of two and three satellite vortices in a rotating shallow water layer have been recorded via particle image velocimetry (PIV) and their autorotation speed was estimated. This study shows that these equilibria retain the fundamental characteristics of Kelvin’s equilibria, and could be adequately described by the classical idealized point vortex theory. The same conclusion can also be inferred using the experimental dataset of Bergmann et al. (J. Fluid Mech., vol. 679, 2011, pp. 415–431; J. Fluid Mech., vol. 691, 2012, pp. 605–606) if the assigned field’s contribution to pattern rotation is included.</jats:p