Topology of two-dimensional flow associated with degenerate dividing streamline on a free surface

Deliceoğlu, Ali

doi:10.1017/s0956792512000319

Cited by 3 publications

(2 citation statements)

References 20 publications

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“…A topological approach has been used to characterize two-dimensional flows [5,6,13] and axisymmetric flows [7,8]. For an incompressible helical flow a stream function can be defined, and we discuss as to what extent the topology of the streamlines can be inferred from this.…”

Section: Introductionmentioning

confidence: 99%

Topology of helical fluid flow

Andersen¹,

Brøns²

2014

Eur. J. Appl. Math

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Considering a coordinate-free formulation of helical symmetry rather than more traditional definitions based on coordinates, we discuss basic properties of helical vector fields and compare results from the literature obtained with other approaches. In particular, we discuss the role of the stream function for the topology of the streamline pattern in incompressible flows. On this basis, we perform a comprehensive study of the topology of the flow field generated by a helical vortex filament in an ideal fluid. The classical expression for the stream function obtained by Hardin (Hardin, J. C. 1982 Phys. Fluids 25, 1949-1952 contains an infinite sum of modified Bessel functions. Using the approach by Okulov (Okulov, V. L. 1995 Russ. J. Eng. Thermophys. 5, 63-75) we obtain a closed-form approximation which is considerably easier to analyse. Critical points of the stream function can be found from the zeroes of a single real function of one variable, and we show that three different flow topologies can occur, depending on a single dimensionless parameter. By including the selfinduced velocity on the vortex filament by a localised induction approximation, the stream function is slightly modified and an extra parameter is introduced. In this setting two new flow topologies arise, but not more than two critical points occur for any combination of parameters.

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Section: Introductionmentioning

confidence: 99%

Topology of helical fluid flow

Andersen¹,

Brøns²

2014

Eur. J. Appl. Math

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“…For unsteady flows time can also be considered a parameter in the system (1). A general bifurcation theory for streamline patterns has been developed by several authors [28,29,30,31,32,33] and many applications to specific flow problems such as vortex breakdown [34,35], driven cavities [36,37,38], the cylinder wake [39,40] and peristaltic flows [41] are available. The analysis of topological bifurcations consists in identifying degenerate streamline patterns and their unfoldings, that is, parametrized families of velocity fields which can represent all possible perturbations of the given degenerate pattern.…”

Section: Introductionmentioning

confidence: 99%

Codimension three bifurcation of streamline patterns close to a no-slip wall: A topological description of boundary layer eruption

et al. 2015

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A vortex close to a no-slip wall gives rise to the creation of new vorticity at the wall. This vorticity may organize itself into vortices that erupt from the separated boundary layer. We study how the eruption process in terms of the streamline topology is initiated and varies in dependence of the Reynolds number Re. We show that vortex structures are created in the boundary layer for Re around 600, but that these disappear again without eruption unless Re > 1000. The eruption process is topologically unaltered for Re up to 5000. Using bifurcation theory, we obtain a topological phase space for the eruption process, which can account for all observed changes in the Reynolds number range we consider. The bifurcation diagram complements previously analyzes such that the classification of topological bifurcations of flows close to no-slip walls with up to three parameters is now complete.

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Cusps and cuspidal edges at fluid interfaces: Existence and application

Krechetnikov

2015

Phys. Rev. E

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One of the intriguing questions in fluid dynamics is on the interrelation between dynamic singularities in the solutions of fluid dynamic equations - unboundedness of the velocity field in an appropriate norm - and the geometric ones - divergence of curvature at fluid interfaces. The present work focuses on two generic interfacial singularities - genuine cusps and cuspidal edges - found here in both two and three dimensions thus establishing a relation between real fluid interfaces and geometric singularity theory. The key finding is the necessary condition for the existence of geometric singularities, which is a variation of surface tension. It is also established here that the dynamic and geometric singularities entail each other only in the case of three-dimensional cusps. Explicit asymptotic solutions for the flow field and interface shape near steady-state singularities at fluid interfaces are developed as well. The practical motivation for the present study comes from the fundamental role interfacial singularities play in sustaining self-driven conversion of chemical into mechanical energy.

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Topology of two-dimensional flow associated with degenerate dividing streamline on a free surface

Cited by 3 publications

References 20 publications

Topology of helical fluid flow

Topology of helical fluid flow

Codimension three bifurcation of streamline patterns close to a no-slip wall: A topological description of boundary layer eruption

Cusps and cuspidal edges at fluid interfaces: Existence and application

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