The 15th FDR prize

Theoretical treatment of the motion of a slender vortex tube, in most cases, assumes that the cross-section of the vortical core is uniform along the tube (in the z direction), though variation in the vortex core cross-section is crucial for the vortex breakdown phenomena. In the late 20th century, two heuristic theories were proposed for the problem of axisymmetric area waves on the core of a vortex tube embedded in an incompressible inviscid fluid. Lundgren and Ashurst (1989) derived a model equation, being referred to as ‘momentum wave model’, for the axial velocity W (z, t) by taking the spatial average of the z component of the Euler equation for axial flow, assuming that the axial flow is uniform over the cross-section of the core (‘slug flow’). Leonard (1994) derived an evolution equation (‘vorticity wave model’) for the ratio of W (z, t) to the cross section A(z, t) from the spatial average of the azimuthal component of the vorticity equation. Although these two partial differential equations look alike, there is a decisive difference in the dispersion relation, namely in the travelling speed of the core-area waves. For the momentum wave model, the area wave propagates at the same speed upstream and downstream in a coordinate system that moves with the axial flow, while for the vorticity wave model, it propagates asymmetrically, faster downstream than upstream. This paper systematically derives a system of evolution equations for axisymmetric deformation of the vortex core using differential geometric techniques, and supports for the vortex wave model with respect to the dispersion relation. Moreover, this approach makes it possible to calculate the time evolution of the cross-section subject to general irrotational external straining flows, whereby an insight is gained into the Leray scaling for the stretching of vortex tubes in a turbulent flow. The theoretical tool is the hybrid Euler–Lagrange approach, which has been devised to describe generalized Lagrangian mean theory, and the authors themselves developed a differential geometric method. The basic flow, with time dependence being admitted, is treated in the Lagrangian way, and the disturbance is treated in the Eulerian way. Two coordinate systems, the physical system (x̃) and the reference system (x), are introduced. In the reference system, the vortex tube is taken to be of circular cylindrical shape, with allowance being made for axial vorticity and axial flow in the core. A time-dependent mapping x̃ = φ (x, t) from the reference system to the physical system represents the axisymmetric nonlinear deformation of the vortex tube, whose velocity field Ũ= ∂tφ ◦φ−1 has radial and axial components only. This part is treated in the Lagrangian way and the axisymmetric Euler equations are written down in the non-orthogonal coordinate system moving with the velocity Ũ. Supposing a uniform vorticity distribution for the basic field and taking the ‘slender limit’ of the vortex region in the Lagrangian space, the scaling that its axial size is much longer than its radial size, give rise to a coupled system of the evolution equations of the azimuthal velocity and the vortex