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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, ' There is strong evidence in support of the notion that the formation of vortex rings is mainly an inviscid process. This can be easily checked by comparing the relaxation time toward a steady vortex ring with the viscous diffusion time at high Reynolds numbers. The long-time behavior of an axisymmetric Euler equation is characterized by an infinite number of invariants. 1 Any accurate model for vortex ring formation must respect these invariants.In a recent paper, Gharib et al. 2 have demonstrated that a time scale with a narrow range of values characterizes the formation of axisymmetric vortex rings. This time scale is the time that the vortex ring pinches off from its generating axisymmetric jet. Here we refer to the nondimensional stroke length L/D as the ''formation number.'' The relation between this formation number and the corresponding time scale is given by L/DϭU p t/D, where U p is the average piston velocity. Gharib et al. 2 demonstrated that this observed time, beyond which larger rings are not possible, is a direct manifestation of the variational principle proposed by Kelvin and Benjamin for steady axis-touching rings. For predicting the actual value of the formation number, they used a nondimensional energy parameter obtained from the experimental results. The purpose of this paper is to show that the formation number can be predicted analytically, based on simple assumptions on the formation process.To develop our model we make two assumptions. First that the discharge process of the fluid out of the cylinder can be predicted by a slug model. Second, that at the pinch-off moment the invariants of the resulting vortex ring can be approximated by those of a vortex in the family of Norbury vortices. 3 While real physical vortex rings have peakier vorticity distributions ͑ ͒ than those of the uniform vorticity density ( /r) of the Norbury family of vortices, the quantitative behavior, in particular the stream lines, is very similar. 4 These considerations along with the fact that Norbury family of vortices are very well documented suggest that for the limiting process under consideration the Norbury vortices can be considered as a first approximation to the more general case of vorticity distributions. Peakiness of the vorticity profiles in this context is considered in more detail in Ref. 2. Furthermore the experimental results provided by Gharib et al. 2 for different velocity profiles of the piston and exit conditions suggest that the limiting formation number is not very sensitive to the vorticity distributions. This is physically manifeste...

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.

Numerical simulations are used to study the formation of vortex rings that are generated by applying a non-conservative force of long duration, simulating experimental vortex ring generation with large stroke ratio. For sufficiently long-duration forces, we investigate the extent to which properties of the leading vortex ring are invariant to the force distribution. The results confirm the existence of a universal 'formation number' defined by Gharib, Rambod & Shariff (1998), beyond which the leading vortex ring is separated from a trailing jet. We find that the formation process is governed by two non-dimensional parameters that are formed with three integrals of the motion (energy, circulation, and impulse) and the translation velocity of the leading vortex ring. Limiting values of the normalized energy and circulation of the leading vortex ring are found to be around 0.3 and 2.0, respectively, in agreement with the predictions of Mohseni & Gharib (1998). It is shown that under this normalization smaller variations in the circulation of the leading vortex ring are obtained than by scaling the circulation with parameters associated with the forcing. We show that by varying the spatial extent of the forcing or the forcing amplitude during the formation process, thicker rings with larger normalized circulation can be generated. Finally, the normalized energy and circulation of the leading vortex rings compare well with the same properties for vortices in the Norbury family with the same mean core radius.

Capabilities for turbulence calculations of the Lagrangian averaged Navier-Stokes ͑LANS-␣͒ equations are investigated in decaying and statistically stationary three-dimensional homogeneous and isotropic turbulence. Results of the LANS-␣ computations are analyzed by comparison with direct numerical simulation ͑DNS͒ data and large eddy simulations. Two different decaying turbulence cases at moderate and high Reynolds numbers are studied. In statistically stationary turbulence two different forcing techniques are implemented to model the energetics of the energy-containing scales. The resolved flows are examined by comparison of the energy spectra of the LANS-␣ with the DNS computations. The energy transfer and the capability of the LANS-␣ equations in representing the backscatter of energy is analyzed by comparison with the DNS data. Furthermore, the correlation between the vorticity and the eigenvectors of the rate of the resolved strain tensor is studied. We find that the LANS-␣ equations capture the gross features of the flow, while the wave activity below the scale ␣ is filtered by a nonlinear redistribution of energy.

The evolution of starting jet circulation, impulse and kinetic energy are derived in terms of kinematics at the entrance boundary of a semi-infinite axisymmetric domain. This analysis is not limited to the case of parallel jet flows; and the effect of non-zero radial velocity is specifically identified. The pressure distribution along the entrance boundary is also derived as it is required for kinetic energy modelling. This is done without reliance on an approximated potential function (i.e. translating flat plate), making it a powerful analytical tool for any axisymmetric jet flow. The pressure model indicates that a non-zero radial velocity is required for any ‘over-pressure’ at the nozzle exit. Jet flows are created from multiple nozzle configurations to validate this model. The jet is illuminated in cross-section, and velocity and vorticity fields are determined using digital particle image velocimetry (DPIV) techniques and circulation, impulse and kinetic energy of the jet are calculated from the DPIV data. A non-zero radial velocity at the entrance boundary has a drastic effect on the final jet. Experimental data showed that a specific configuration resulting in a jet with a converging radial velocity, with a magnitude close to 40 % of the axial velocity at its maximum, attains a final circulation which is 90–100 % larger than a parallel starting jet with identical volume flux and nozzle diameter, depending on the stroke ratio. The converging jet also attains a final impulse which is 70–75 % larger than the equivalent parallel jet and a final kinetic energy 105–135 % larger.

Several previous experimental and theoretical studies have shown that a leading edge vortex (LEV) on an airfoil or wing can provide lift enhancement. In this paper, unsteady two-dimensional (2D) potential flow theory is employed to model the flow field of a pitching flat plate wing. A multi-vortices model is developed to model both the leading edge and trailing edge vortices (TEVs), which offers improved accuracy compared with using only single vortex at each separation location. The lift is obtained by integrating the unsteady Blasius equation. It is found that the motion of vortices contributes significantly to the overall aerodynamic force on the flat plate. A Kuttalike condition is used to determine the vortex intensity and location at the leading edge for large angle of attack cases; however, it is proposed to relax this condition for small angle of attack cases and apply a 2D shear layer model to calculate the circulation of the new added vortex. The results of the simulation are then compared with classical numerical, theoretical, and experimental data for canonical unsteady flat plat problems. Good agreement with these data is observed. Moreover, these results suggested that the leading edge vortex shedding for small angles of attack should be modeled differently than that for large angles of attack. Finally, the results of vortex motion vs. lift indicate that the slow convection of the LEV creates less negative lift while the rapid shedding of the TEV creates more positive lift. The difference between these two contributions of lift results in a total positive lift that lasts for about two chord-length travel of the plate. It is therefore concluded that the lift enhancement during the LEV "stabilization" above the wing is a combined effect of both the LEV and TEV motion. This also provides the insights for future active flow control of micro aerial vehicles (MAVs) that the formation and shedding process of LEVs and TEVs can be manipulated to provide lift enhancement. C 2013 AIP Publishing LLC. [http://dx.

SUMMARYThe thrust-generating mechanism of a prolate hydromedusa Sarsia tubulosa and an oblate hydromedusa Aequorea victoria was investigated by solving the incompressible Navier-Stokes equations in the swirl-free cylindrical coordinates. The calculations clearly show the vortex dynamics related to the thrust-generating mechanism, which is very important for understanding the underlying propulsion mechanism. The calculations for the prolate jetting hydromedusa S. tubulosa indicate the formation of a single starting vortex ring for each pulse cycle with a relatively high vortex formation number. However, the calculations for the oblate jet-paddling hydromedusa A. victoria indicate shedding of the opposite-signed vortex rings very close to each other and the formation of large induced velocities along the line of interaction as the vortices move away from the hydromedusa in the wake. In addition to this jet propulsion mechanism, the hydromedusa's bell margin acts like a paddle and the highly flexible bell margin deforms in such a way that the low pressure leeward side of the bell margin has a projected area in the direction of motion. This thrust is particularly important during refilling of the subumbrella cavity where the stopping vortex causes significant pressure drag. The swimming performances based on our numerical simulations, such as swimming velocity, thrust, power requirement and efficiency, were computed and support the idea that jet propulsion is very effective for rapid body movement but is energetically costly and less efficient compared with the jet-paddling propulsion mechanism.