Axial and radial dynamics of compressible vortex rings

Abstract

Vortex rings are typically generated by a piston within a tube for the incompressible case and a bursting membrane in a shock tube for the compressible case. Despite these different origins, they show a common general behaviour which allows us to compare both with a single numerical setup. The two main control parameters are the non-dimensional mass supply which can be expressed as the ratio of length and diameter of the tube ( L ∕ D ) and the pressure ratio from the reservoir to ambient ( p 0 ∕ p ∞ ) . They basically control the blowing duration and the compressibility. We performed Direct Numerical Simulations and studied the influence of variations in ( L ∕ D ) as well as compressibility by keeping the discharge-time constant and appropriate combination of both parameters on four aspects: torus radius R , core radius r , axial position and the axial velocity of the vortex rings. Comparison with a large number of available data from the literature shows good agreement and we were able to reconcile an apparent contradiction in the community about the exponent of the power law describing the axial dynamics of the vortex ring being 1 ∕ 2 or 3 ∕ 2 . In particular, an increase in L ∕ D leads to faster and longer growth of the vortex ring, while the maximum is reached earlier. The transition to the ( t ∗ ) 1 ∕ 2 power law commences also earlier and the axial velocity is higher for larger values of L ∕ D . Keeping the discharge time constant and increasing the Mach number, we found increased torus radii R , the maximum core radius r is reached later, and lower propagation velocities.