Abstract
A vortex ring in a viscous, incompressible fluid rises obliquely at an angle of 45° toward a deformable free surface. The mathematical description of this flow situation is a time-dependent nonlinear free-surface problem that has been solved numerically for a three-dimensional laminar flow with the aid of the Navier-Stokes equations. Boundary-fitted coordinates were used that include adaptive gridding and grid refinement. For two different Froude numbers at Reynolds number 100, results are presented on the encounter of the vortex ring with the free surface, the deformation of the free surface, the decay of the primary vortices, the generation of surface vorticity and secondary vortices, and reconnection with the free surface. These results are presented in the form of streamlines and equivorticity lines in the plane of symmetry, contour lines of surface elevation, and streamlines and equivorticity lines on the free surface. The two cases reveal a novel ring reconnection at the free surface in the form of a cylindrical sheet.
Sommario
Un vortice ad anello in un fluido incomprimibile e viscoso si avvicina obliquamente, con un angolo di 45°, ad una superficie libera deformabile. La descrizione matematica di questa situazione di flusso è un problema di superficie libera non-lineare e dipendente dal tempo, che è stato risolto numericamente per un flusso laminare tridimensionale con l'ausilio delle equazioni di Navier-Stokes. A tal fine sono state usate opportune coordinate al contorno che includono la possibilità di adattare il reticolo ed il raffinamento del reticolo stesso. Per due differenti numeri di Froude e per un numero di Reynolds pari a 100, vengono presentati i risultati sull'incontro del vortice ad anello con la superficie libera, la deformazione di quest'ultima, il decadimento dei vortici primari, la generazione di una vorticità superficiale e di vortici secondari e la riconnessione con la superficie libera. Questi risultati sono presentati sotto forma di linee di corrente e linee di equivorticità nel piano di simmetria, linee di contorno della proiezione verticale della superficie, linee di corrente e linee di equivorticità sulla superficie libera. I due casi rivelano una nuova riconnessione ad anello sulla superficie libera nella forma di un foglio cilindrico.
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Lugt, H.J., Ohring, S. The oblique rise of a viscous vortex ring toward a deformable free surface. Meccanica 29, 313–329 (1994). https://doi.org/10.1007/BF00987567
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DOI: https://doi.org/10.1007/BF00987567