A model for universal time scale of vortex ring formation

An analytical model for predicting the universal time scale for formation of vortex rings generated through impulsively started jets is considered. The model is based on two assumptions, namely the validity of the slug model in simulating the discharge process of the fluid out of the cylinder and the approximation of the vortex at the pinch off moment by a vortex in the Norbury family. The nondimensional stroke length L/D (referred to as "formation number," following Gharib et al. [J. Fluid Mech. 360, 121 (1998)]) predicted by the model satisfactorily matches the experimental observation of Gharib et al. The model introduces two nondimensional parameters that govern the limiting formation number: nondimensional energy End and circulation Gammand. The predicted value of End matches very well with the experimental data. It is also predicted that there is a limiting value for the nondimensional circulation in the range 1.77 <~ Gammand <~ 2.07

Symplectic Model Reduction of Hamiltonian Systems

In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while preserving the symplectic structure. As an analogy to the classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is designed to build a symplectic subspace to fit empirical data, while the symplectic Galerkin projection constructs a reduced Hamiltonian system on the symplectic subspace. For practical use, we introduce three algorithms for PSD, which are based upon: the cotangent lift, complex singular value decomposition, and nonlinear programming. The proposed technique has been proven to preserve system energy and stability. Moreover, PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems. Owing to these properties, the proposed technique is better suited than the classical POD-Galerkin approach for model reduction of Hamiltonian systems, especially when long-time integration is required. The stability, accuracy, and efficiency of the proposed technique are illustrated through numerical simulations of linear and nonlinear wave equations.Comment: 25 pages, 13 figure