Towards a finite-time singularity of the Navier–Stokes equations. Part 3. Maximal vorticity amplification

Abstract

An exact analytical solution is obtained for the dynamical system derived in Part 1 of this series (Moffatt & Kimura, J. Fluid Mech, vol. 861, 2019a, pp. 930–967), which describes the approach of two initially circular vortices of finite but small cross-section symmetrically located on inclined planes. This exact solution, applicable in the inviscid limit, allows determination of the amplification $\mathcal {A}_{\omega }$ of the axial vorticity within the finite time $T$ during which the basic assumptions of the model continue to apply. It is first shown that, for arbitrarily prescribed $\mathcal {A}_{\omega }$ , it is possible to specify smooth initial conditions of finite energy such that, in the inviscid limit, this amplification is achieved within the time $T$ . When viscosity is included, an estimate is provided for the minimum vortex Reynolds number that is sufficient for the same result to hold. The predictions are broadly compatible with results from direct numerical simulations at moderate Reynolds numbers. Moreover, it is shown that one may come arbitrarily close to a finite-time singularity of the Navier–Stokes equation by appropriate choice of an initial, smooth, finite-energy velocity field; however, this approach to a singularity is ultimately thwarted through breach of the assumptions on which the dynamical system is based. Thus we make no claim here concerning realisation of a Navier–Stokes singularity. Moreover, we find that the conditions required to attain a large amplification $\mathcal {A}_{\omega }\gg 1$ during the time $T$ are far beyond those that can be realised in either experiment or direct numerical simulation.