A.A.Gourjii Interaction of axisymmetric vortex rings in an infinite pipe filled by an ideal uncompressible fluid |
Applied hydromechanics, Vol. 10 (82) ¹ 4, (2008) |
| The problem on interaction of the system of axisymmetrical vortex rings with the small circular cross section (Dyson's vortex rings) in an unbounded rectilinear pipe with the circular cross section, which is filled by an ideal incompressible fluid, is considered. The method of discrete singularities is proposed for numeral-analytical solution, which adapted to the axisymmetrical problems. To satisfy boundary conditions on an internal surface one introduces either the sequence of imaginary vortex filaments or imaginary vortex layer. Distributing of intensity of imaginary vortex structures is determined from either equalities to the zero the radial velocity components of flow or equality to the constant the value of stream function. To detect the best solution one introduce ``failure function'' of boundary condition on velocity and analyses its maximal and overage values on an internal surface of the pipe. Researches shows that the solution based on an introduction the system of imaginary vortex filaments of identical radius with boundary condition for stream function is the best both from point of local satisfaction of boundary condition for velosity and from counting time interval. The equations of motion of the system of thin vortex rings are given. Hamiltonian form of equations of motion coincides with equation for coaxial vortex rings in unbounded space with hamiltonian, which takes into account the influence of boundaries. It is shown that these equations have two invariants of motion, which correspond to the momentum conservation law along the axis of the pipe and kinetic energy conservation law of vortex rings. |
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KEY WORDS: *** |
| TEXT LANGUAGE: Russian |