Inviscid swirling flows are modeled, for analytical studies, using axisymmetric azimuthal, V(r), and axial, W(r), velocity profiles (r is the distance from the axis). The asymptotic analysis procedure (large wave numbers, k axial and m azimuthal) developed by Leibovich and Stewartson [J. Fluid Mech. 126, 335 (1983)], and used by many authors, breaks down if kW$'${}(r)+m$\Omega${}$'${}(r)$\ne${}0,$\forall${}r or if kW$'${}(r)+m$\Omega${}$'${}(r)=0,$\forall${}r, $\Omega${}=V/r. This latter case occurs if W is constant with m=0, if $\Omega${} is constant with k=0, or if both W and $\Omega${} are constant with arbitrary wave-number vector. These particular cases are considered by Leblanc and LeDuc [J. Fluid Mech. 537, 433 (2005)]. Thus, the case where W and $\Omega${} both vary and the Leibovich and Stewartson asymptotics breaks down remains. It is addressed in the present paper for weak variations of axial and azimuthal velocities. The asymptotic results are checked using numerically computed growth rates of the linearized Euler equations for a family of variable-density Batchelor-like vortices as base flows. Good agreement is found even for low values of m and k.

We report a self-similar behaviour of solutions (obtained numerically) of the Navier–Stokes equations behind a single-blade rotor. That is, the helical vortex core in the wake of a rotating blade is self-similar as a function of its age. Profiles of vorticity and azimuthal velocity in the vortex core are characterized, their similarity variables are identified and scaling laws of these variables are given. Solutions of incompressible three-dimensional Navier–Stokes equations for Reynolds numbers up to are considered.

Using spectrally accurate, direct numerical simulations, we show that vortices with axial flows (without an imposed strain) generate small scales, i.e., breakdown, when they are randomly perturbed. We show that the breakdown that occurs without the necessity of a stagnation point in the flow is nonlinear and that energy spectra of the breakdown present a range of a $k^{-5/3}$ scaling ($k$ being a wavenumber). These features are the same for centrifugally stable and unstable azimuthal velocities. However, centrifugally unstable azimuthal velocities promote mixing in vortices with an axial flow. We also find that vortex breakdown could be a physical mechanism against a finite time singularity in Navier–Stokes equations. Contrarily, we do not observe a breakdown using spectrally truncated Euler equations. For these equations we find a scaling of the energy spectra compatible with the existence of a finite time singularity.

The generation of wind waves at the surface of a pre-existing underlying vertically sheared water flow of constant vorticity is considered. Emphasis is put on the role of the vorticity in water on wind-wave generation. The amplitude growth rate increases with the vorticity except for quite old waves. A limit to the wave energy growth is found in the case of negative vorticity, corresponding to the vanishing of the growth rate.