Experimental characterization of the instability of the vortex rings. Part II: Non-linear phase

The first stage of the instability of a vortex ring is linear and characterized by the growth of an azimuthal stationary wave which develops around the ring. Theoretical works predict its origin, shape, number of waves and growth rate. Apart for the growth rate, experimental and numerical results in viscous fluids fit well with the predictions based on an ideal fluid hypothesis. On the other hand, the next stages of the development of the instability (which are non-linear) are not well known. Only few phenomena are described, in an isolated way, in various partial contributions. The aim of this paper is to report on a complete experimental investigation of the non-linear phase of the instability of the vortex ring. The vortices were produced in water and their Reynolds number Re _p was varied from 2,650 to 6,100. Visualizations were performed using planar laser induced fluorescence and measurements with 2D2C and 2D3C particle image velocimetry. Based on a Fourier analysis of the results, it appears that the non-linear phase begins with the development of harmonics of the linear modes (first unstable modes). But the growth of those harmonics is rapidly stopped by the development of low order modes. Then appears an m=0 mode, which corresponds to a mean azimuthal velocity around the vortex. Simultaneously, secondary vortical structures develop all around the vortex in its peripheral zone. These vortical structures are linked with the ejection of vorticity in the wake of the ring and they appear just before the transition towards turbulence. A tentative is made here to place all these phenomena chronologically, in order to propose a scenario for the transition from the linear phase to turbulence.     

Long-wave instability of a vortex ring

The instability of barrel-shaped vibrations of a vortex ring in an ideal fluid is investigated. These vibrations, stable for a vortex ring with a piecewise-uniform vorticity profile, appear to be unstable for a vortex ring with a smooth vorticity profile. The instability growth rate is found on the basis of the energy balance equation determining the energy transport from perturbations with negative energy in the critical layer to perturbations with positive energy in the rest of the flow. The curvature of the vortex ring, by virtue of which the perturbations with energies of different signs appear to be connected, plays a prominent role in the mechanism under consideration.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 72–78, November–December, 1995.     

Hierarchical instability of a vortex ring array in multipulse laser-matter interactions

In XeCl excimer Laser interactions with Co-coated steel surfaces, we have seemingly realized a quasi-linear array of vortex rings. These form from instabilities on an array of vortex filaments which emerge and then lead to a series of loops. Finally the collapse-and-reconnection process yields a cascade of nearby vortex rings. Since the filament array is subjected to randomly distributed local multipolar strains, unstable waves can develop on the vortex rings. These deformed shapes are frozen permanently by ultrafast cooling, following the last laser pulse. Using a modal analysis, an attempt is made to relate the wave structures to a parametric resonance instability which is caused by dipolar and a quadrupolar fields. Multipulse laser-matter interactions are thus capable of nonselective excitation of vortex ring instabilities of various sizes and various modal structures.     

Nonlinear friction dynamics on fibrous materials,application to the characterization of surface quality. Part I: global characterization of phase spaces

The diversity and the complexity of fibrous materials (woven fabrics, nonwovens or knitted fabrics) make the control of surface qualities very difficult. The control and the comprehension of the tribological phenomena generated by a dynamic friction are of major importance. Whereas the traditional investigations such as measurements of average parameters like roughness or friction coefficient quickly find their limits on such deformable, heterogeneous and flexible fibrous surfaces with very heterogeneous relief, this series of two publications proposes a new manner of observing the interactions of friction. Our laboratory has developed a patented method of measurements, called MODALSENS, which aims at rubbing a very fine and flexible blade on the analyzed surface. Dynamic friction with the contact generates nonlinear vibrations, and the response of the sensor is analyzed in its phase space using the tools of nonlinear analysis of time series. Then, when the evaluated surfaces change, this first part endeavors to characterize the portraits of phases related to the nonlinear vibrations of MODALSENS in a global way, by quantifying its invariants such as dimensions, Shannon entropies, diameters of attractors and largest Lyapunov exponents. From this first part, a classification of studied fibrous surfaces is built and is expected to give rise to new estimators for the surface quality analysis of fibrous media.     

Linear viscoelastic models: Part I. Relaxation modulus and melt calibration

Constitutive models for the linear viscoelasticity of polymers are presented for the relation between the relaxation modulus and the molecular weight distribution (MWD). We also compute the MWD from a simulated relaxation modulus curve by first obtaining the rheologically effective distribution (RED) as a function of time, and converting this into the MWD by melt calibration; that is, the relation between timescale and the molecular weight. This procedure has similarities with the widely used universal calibration with solved polymers. The main principles of our technique are applied here to familiar relaxation modulus data, for which we present two models: (1) an analytical model derived from control theory, which is known capable of modelling partially observed system and (2) a practical characteristic model for obtaining usable results. No relaxation time or spectrum procedures are used to model the process of linear viscoelastic relaxation. The use of relative calculations and melt calibration dispenses with the need to know the real chain structures such as branching and entangled chain dynamics, and the model remains useful for future investigations of polymer chain structures and dynamics, such as using tube theory.     

Explosive instability of geostrophic vortices. Part 1: baroclinic instability

In a quasi-geostrophic model, we study the baroclinic instability of a two-layer vortex. The singular unstable modes for potential vorticity anomalies are compared with the classical normal modes. Short-time singular modes are explosively unstable and, at short times, depend only on the baroclinic component of the flow. As time progresses, they evolve towards the normal modes and their sensitivity to flow parameters is explored. Asymptotic solutions are provided.     

Explosive instability of geostrophic vortices. Part 2: parametric instability

In a two-layer quasi-geostrophic model, a baroclinic vortex is submitted to a periodic forcing of its mean baroclinic azimuthal velocity. It is shown that parametric effects could stabilize a vortex which is baroclinically unstable in the absence of forcing. Conversely, parametric resonance can destabilize a baroclinically stable vortex, under conditions on the vortex parameters, on the ratio of layer thicknesses or on the forcing frequency.     

The evolution of an initially circular vortex near an escarpment. Part I: analytical results

The motion of a quasigeostrophic, equivalent-barotropic, initially circular vortex patch near an infinitely long topographic escarpment is studied using f-plane dynamics. There are two time scales in the problem: the advective time scale associated with the vortex, and the time scale for topographic vortex stretching. Analytical progress is possible when these two time scales are well-separated and results are presented here.If topographic vortex stretching dominates advection by the vortex the vortex is said to be ‘weak’. The vortex patch remains circular on the topographic time scale, and dispersive topographic waves rapidly propagate the initial disturbance away from the vicinity of the vortex. Subsequently cross-isobath motion is inhibited, and the vortex moves as though the escarpment were a plane wall. The same behaviour was observed for the motion of a weak singular vortex near an escarpment by Dunn, McDonald and Johnson [7], who named the phenomenon the ‘pseudoimage’ of the vortex.If advection dominates over topographic effects, the vortex is said to be ‘intense’. The vortex also remains circular to leading order, but the relative vorticity produced by the swirl of the vortex is less able to escape the vicinity of the vortex. The vortex follows a similar curved trajectory to those observed for intense vortices on the β-plane. The dipolar mechanism for this behaviour is described. Large time solutions are inhibited by the form of the escarpment topography, but examination of the equations leads to the conclusion that the leading order solution may be predict the motion for times beyond its formal range of validity.     

Modeling of the metal powder compaction process using the cap model. Part I. Experimental material characterization and validation

In order to produce crack free metal powder compacts that respect both the dimensional tolerances and the mechanical strength requirements, both tooling design and compaction sequence have to be adequately determined. The finite element method, through the use of an appropriate constitutive model of the powder medium, has recently been used as an efficient design tool. The accuracy of this method highly depends on the faithfulness of the constitutive model and the quality of the material parameter set. Furthermore, in order for the simulation results to be reliable, they should be experimentally validated on real parts featuring density variations. Hence, the main concerns of this paper are the development of a standard calibration procedure for the cap material model as well as the development of a reliable technique for the experimental validation of the powder compaction simulation results.The developed calibration procedure, applied for the case of 316L stainless steel powders, is based on a series of isostatic, triaxial and uniaxial compaction tests as well as resonant frequency tests. In addition, a sensitivity study was performed in order to determine the relative importance of each factor and basic simulations served to validate the parameter set extraction procedure.On the other hand, a local density measurement technique was developed for the experimental validation of the model results. This technique is based on correlation with Vickers macro-hardness. Finally, an application featuring the compaction of a 316L stainless steel cylindrical component is presented to illustrate the predictive capabilities of the cap material model as well as the accuracy of the acquired material parameter set.     

Model of vortex ring formation

A semi-empirical model of vortex ring formation during exhaustion of a pulsed submerged jet from a circular orifice is presented. Formulas for determining the parameters of the vortex ring, depending on the conditions of formation of the latter, are derived. The calculated characteristics of the vortex ring as functions of criteria determining the vortex formation process are demonstrated to be in good agreement with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 25–36, November–December, 2008.     

Diffusion of a ring vortex

The diffusion of a ring vortex is investigated in the present paper with allowance for the influence of the initial radius of the toroidal vorticity distribution on the flow structure. The statement of the problem in such a formulation makes it possible to classify and reinterpret results obtained previously. A vortex pair is studied together with a vortex ring. The toroidal vorticity and stream function distributions are obtained analytically. The self-induced lift velocity of the ring vortex is found. The influence of inertial terms is investigated numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 10–15, November–December, 1987.     

On the linear instability of the Hall-Stewartson vortex

It is demonstrated that the Hall-Stewartson leading-edge vortex is linearly unstable to viscous perturbations of the center-mode type. Center modes are found to occur in two reigons of Reynolds-number-wave-number space, in limits in which the axial wave number is large. The appropriate center-mode equations in these neighborhoods are established, and it emerges that the two sets are identical. The single system of equations, which depends on the azimuthal wave number m and a distance parameter only, is solved numerically for various values of m and . Highly unstable modes are found for large positive , and the results are shown to be in good agreement with proposed asymptotic expansions when >1. To lowest order, unstable modes have phase surfaces that rotate with the fluid: in addition constant phase surfaces propagate upstream but the group velocity is directed downstream. The growth rate of the instability increases faster than Reynolds number to the quarter power. This, together with the finding that the length scale of the unstable modes found goes to zero as the Reynolds number tends to infinity, makes this instability an unusual one.This work was supported by the Air Force Office of Scientific Research under contract AFOSR-89-0346 monitored by Dr. L. Sakell, and by the U.S. Army Research Office at the Mathematical Sciences Institute of Cornell University.     

Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory

A model of evolving martensitic microstructures is formulated that incorporates the interfacial energy and dissipation on three different scales corresponding to the grain boundaries attained by martensite plates, the interfaces between austenite and martensite plates, and the twin interfaces within martensite plates. Three different time scales are also considered in order to clarify the meaning of rate-independent dissipation related to instabilities at more refined temporal and spatial scales. On the slowest time scale, the process of deformation and martensitic phase transformation is investigated as being composed of segments of smooth quasi-static evolution separated by sudden jumps associated with creation or annihilation of interfaces. A general evolution rule is used in the form of minimization of the incremental energy supply to the whole compound thermodynamic system, including the rate-independent dissipation. Close relationship is shown between the evolution rule and the thermodynamic condition for stability of equilibrium, with no a priori assumption on convexity of the dissipation function. It is demonstrated that the extension of the minimum principle from the first-order rates to small but finite increments requires a separate symmetry restriction imposed on the state derivative of the dissipation function. Formulae for the dissipation associated with annihilation of interfaces are proposed that exhibit limited path-independence and satisfy that symmetry requirement. By exploiting the incremental energy minimization rule with the help of the transport theorems, local propagation conditions are derived for both planar and curved phase transformation fronts. The theory serves as a basis for the algorithm for calculation of the stress-induced evolution of martensitic microstructures along with their characteristic dimensions and related hysteresis loops in shape memory alloys; the examples are given in Part II of the paper.     

Bifurcation of an elliptic vortex ring

Time-dependent vorticity fields of elliptic vortex rings of aspect ratios 2, 3 and 4 were measured by means of hot-wire anemometry. The time evolution of their vorticity fields was analyzed and the processes of vortex ring formation, advection, interaction and decay, and the mechanism of vortex bifurcation are studied. The following crosslinking model is proposed: A thick vortical region composed of many equivalent vortex filaments with distributed cores is initially formed at the orifice and they behave as inviscid filaments. The elliptic ring deforms and the end parts of its major axis get closer. Then, the vortex filaments interact at the touching point and the ring partially bifurcates. Almost simultaneously, turbulent spot appears at this point, and propagates around the ring cross section, thus preventing further bifurcation. And it becomes a turbulent blob. This model is also supported by numerical simulation by a high-order vortex method and the Navier-Stokes solution.     

Experimental Investigation of the Nonlinear Response of a Hanging Cable. Part I: Local Analysis

An experimental model of an elastic cable carrying eight concentrated masses and hanging at in-phase or out-of-phase vertically moving supports is considered. The system parameters are adjusted to approximately realize multiple 1:1 and 2:1 internal resonance conditions involving planar and nonplanar, symmetric and antisymmetric modes. Response measurements are made in various frequency ranges including meaningful external resonance conditions. A local analysis of the system response is made on the basis of numerous amplitude-frequency and amplitude-forcing plots obtained in different ranges of the control parameter space. Attention is mainly devoted to the detection of the main features of the regular motions exhibited by the system, and to the analysis of the relevant phenomena of nonlinear modal interaction, competition, and local bifurcation between planar and nonplanar regular responses. The resulting picture appears very rich and varied.     

Hydrodynamic structure of a vortex ring

Hot-wire anemometry and high-speed motion-picture photography have been used in an experimental study of the structure of a vortex ring in air. The velocity field and streamline pattern have been determined, together with the vorticity distribution. It has been shown that the vorticity is almost entirely localized in the region of the core of the vortex ring and quickly diminishes with distance from the core. Analysis of the experimental results permits a conclusion concerning the nonstationarity of vortex rings.The authors thank V. K. Sheremetov and V. A. Kosinov for their assistance.     

Experimental characterization of the 3D dynamics of a laminar shallow vortex dipole

Experimental results on the dynamics of a vortex dipole evolving in a shallow fluid layer are presented. In particular, the generation of a spanwise vortex at the front of the dipole is observed in agreement with previous experiments at larger Reynolds numbers. The results show that this secondary vortex is of comparable strength to the dipole. The present physical analysis suggests that the origin of this structure involves the stretching induced by the dipole of the boundary-layer vorticity generated by the dipole’s advection over the no-slip bottom.     

Numerical realization of helical vortices: application to vortex instability

Theoretical and Computational Fluid Dynamics - The need to numerically represent a free vortex system arises frequently in fundamental and applied research. Many possible techniques for realizing...     

Three-dimensional instability of an elliptic Kirchhoff vortex

The instability of a Kirchhoff vortex [1–3] with respect to three-dimensional perturbations is considered in the linear approximation. The method of successive approximations is applied in the form described in [4–6]. The eccentricity of the core is used as a small parameter. The analysis is restricted to the calculation of the first two approximations. It is shown that exponentially increasing perturbations of the same type as previously predicted and observed in rotating flows in vessels of elliptic cross section [4–9] appear even in the first approximation. As distinct from the case of plane perturbations [1-3], where there is a critical value of the core eccentricity separating the stable and unstable flow regimes, instability is predicted for arbitrarily small eccentricity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1988.