Understanding Vortex Reconnection
Progress has been made in understanding the notoriously difficult problem of vortex reconnection in a new paper published recently in Journal of Fluid Mechanics (JFM). The study looked at numerical simulations of a blade slicing through a vortex and understanding how this action affects the flow field could lead to the design of safer helicopters.
Professor Marshall and his team conducted numerical simulations of the cutting of a vortex by a perpendicular blade and found that the cutting motion resulted in waves propagating both upstream and downstream. He described the wave propagation as “analogous to blocking a garden hose, where the resultant waves will travel both up and down”.
As the blade cuts through the vortex, the difference in the radius of the vortex core on the upstream and downstream edges gives rise to a force acting on the blade. The team were able to calculate this unsteady force using a scaling analysis and found that the viscosity (thickness or stickiness) of the fluid does not affect the force on the blade, though it is essential to the problem of vortex reconnection.
Without the inclusion of viscosity in the problem (for example in an inviscid flow), the three Helmholtz laws for vortex motion are violated by the process of reconnection. In simple terms, the laws state that the strength of a vortex tube does not vary with time; vortex lines move with a fluid and they, along with vortex tubes, must appear as a closed loop, extend to infinity or terminate at a solid boundary; and fluid elements initially free of vorticity (swirling motion) will remain free of vorticity.
Vortex reconnection is notoriously one of the most difficult problems in fluid mechanics and for good reason. Professor Keith Moffatt explained to me in an interview in 2015 that “vortex reconnection was the key to being able to fully understand the Navier-Stokes equations” and with it claim the $1 million prize from the Clay Institute. Whilst the new study has improved our understanding of the processes present within a vortex, there is still a long way to go to fully understand the problem.
The applications of the work, however, remain far-reaching. From submarine propellers to wind turbines, the action of a blade slicing through a vortex is a surprisingly common occurrence. It is of particular relevance to helicopters, where the vehicle actually flies through the wake generated by the two propellers. The vortices shed by the main propeller travel downstream and hit the rear propeller which can generate a lift force and lead to a sudden jerking motion. This is not only unpleasant for the passengers and pilot but also increases the forces being applied to the tail arm, potentially causing damage and possibly leading to crashes.
Ideally, the solution would be to prevent vortices from interacting entirely, but this is unfortunately not practical in many of the situations described above. This study sought to improve our understanding of the physical processes that occur when a vortex interacts with a blade and the authors can certainly lay claim to have succeeded in their aim. The ultimate idea is to use the improved understanding to design better propellers for helicopters, wind turbines and submarines that are able to deal with the inevitable vortex interaction.
This paper is freely available for two weeks
Saunders, D., & Marshall, J. (2017). Transient lift force on a blade during cutting of a vortex with non-zero axial flow. Journal of Fluid Mechanics, 819, 258-284. doi:10.1017/jfm.2017.188