Gianni Pedrizzetti

Unsteady tube flow over an expansion

Unsteady flow in a circular conduit with a smooth expansion is studied in detail by numerical integration of the equation of motion in the axisymmetric approximation. The values of governing parameters are chosen to be relevant to medical problems, and the geometry corresponds to a scenario of post-surgical conditions. The flow determined by an oscillatory volume is characterized by a sequence of vortex rings moving in the expanded part of the tube. The development of wall shear stress is governed by the separated translating vorticity which induces an evolving band of large intensity for about a complete oscillation cycle. This influences the dynamics of unsteady separation whose space-time development has revealed features of some generality which have been classified. The time variation of the pressure jump is dominated by inertial effects. The dependence of the details of the flow on the dimensionless parameters has been investigated systematically. The results obtained here have been compared with experimental and numerical studies of similar problems, similarities have been pointed out and differences discussed.

On Markov modelling of turbulence

We consider Lagrangian stochastic modelling of the relative motion of two fluid particles in the inertial range of a turbulent flow. Eulerian analysis of such modelling corresponds to an equation for the Eulerian probability distribution of velocity-vector increments which introduces a hierarchy of constraints for making the model consistent with results from the theory of locally isotropic turbulence. A nonlinear Markov process is presented, which is able to satisfy exactly, in the statistical sense, incompressibility, the exact results on the third-order structure function, and the experimental second-order statistics. The corresponding equation for the Eulerian probability density of velocity-vector increments is solved numerically. Numerical results show non-Gaussian statistics of the one-dimensional Lagrangian probability distributions, and a complex shape of the three-dimensional Eulerian probability density function. The latter is then compared with existing experimental data.

Fluid flow in a tube with an elastic membrane insertion

The unsteady flow of a viscous incompressible fluid in a circular tube with an elastic insertion is studied numerically. The deformation of the elastic membrane is obtained by the theory of nite elasticity whose equations are solved simultaneously with the fluid equations in the axisymmetric approximation. The elastic wall expands outwards due to the positive transmural pressure and represents an idealized model for the response of pathologies in large arteries. It is found that if either the fluid discharge or the reference pressure are imposed downstream of the insertion, the fluid{wall interaction develops travelling waves along the membrane whose period depends on membrane elasticity; these are unstable in a perfectly elastic membrane and are stabilized by viscoelasticity. In the reversed system, when the fluid discharge is imposed on the opposite side, the stable propagation phenomenon remains the same because of symmetry arguments. Such arguments do not apply to the originally unstable behaviour. In this case, even when the membrane is perfectly elastic, propagation is damped and two natural fluctuations appear in the form of stationary waves. In all cases the resonance of the fluid{wall interaction has been analysed. Comparisons with previously observed phenomena and with results of analogous studies are discussed.

Insight into singular vortex flows

The method of three-dimensional vortex singularities is analyzed. The fact that it does not represent a weak solution of the Euler equations has little bearing on its validity as an operative model. Other evolution inconsistencies are a major problem. The non-solenoidality of the vortex field is analyzed and a linear filtering feedback procedure, requiring no additional computations, is introduced to allow alignment with the reconstructed solenoidal vorticity field. A rough estimation of the dissipative effect during vortex reconnection has shown a finite viscous effect that is implicitly present in the model, growing with stretching, representing the mechanism of shifting to reconnected topologies without observing strong velocity gradients. The dividing vorton method is rearranged in order to allow the quantized reproduction of a predefined core evolution law, an error estimation, due to the corpuscularity of the field, is given. Two different numerical computations are performed corresponding to a weak and a strong vortex interaction. The improvements are tested, and confrontations with existing numerical and experimental results are performed showing good agreement. The possibility of the singular vortex flow serving as a rough, but simple, model for very high Reynolds number complex flows is discussed.

Controlled capture of a continuous vorticity distribution

We develop a chaotic control and targeting scheme to capture and stabilize a concentrated vortex around a cylindrical body embedded in an open fluid flow. We demonstrate that the point vortex-based control model is also valid in a continuum fluid framework, by simulating the system using a Navier-Stokes-like dynamics.

Chaotic capture of vortices by a moving body. II. Bound pair model.

Previously, we have presented a simple model for the interaction of a fluid vortex structure with a moving bluff body, and demonstrated the existence of a trapping mechanism related to chaotic scattering. This single point vortex model required explicit perturbation to generate chaos and the subsequent complex dynamics. Here, we present a model which attempts to introduce internal degrees-of-freedom in the vortex structure in the simplest manner, by replacing the single vortex with a like-signed pair. We show that this model exhibits chaotic trapping without the need of explicit perturbation, however, the region of parameter space for which trapping occurs is exceedingly small due to the spatially dependent form of the perturbation. We claim that this result explains some the behavior observed in Navier-Stokes simulations of the same vortex-body system, where we find close correspondence between the dynamics of an extended vorticity distribution and the single vortex model. Finally, we generalize the model to unequal strength vortex pairs, and find more complex behavior which includes partial capture of the weaker vortex by the body. (c) 1994 American Institute of Physics.

Impulsive and pressure-driven transient flows in closed ducts

The transient incompressible flow starting from rest inside an irregular circular duct is here analyzed when a finite discharge or a finite pressure jump is instantaneously imposed. The flow field is solved numerically in the axisymmetric approximation and a new technique to impose a prescribed pressure drop in the vorticity-stream-function formulation is introduced. The differences in the unsteady regimes leading to the same asymptotic flow are outlined.

Instability and chaos in axisymmetric vortex-body interactions

The mixing of fluid produced by bouncing of a vortex ring inside a sphere is considered. The quasiperiodic, transitional, and chaotic regimes of the motion of particles are studied by using spectral analysis and Poincare sections. The one-particle and two-particles statistical characteristics are presented and discussed. In particular, it is concluded that vortex circulation plays the role of the effective coefficient of chaotic diffusion. The external interaction of a vortex ring with a moving sphere and corresponding regimes of instability and temporal capture of the ring by the sphere are also studied.

Vortex Ring — Moving Sphere Chaotic Interaction

Some kinds of turbulent phenomena in fluid motion, like intermittency, can be explained in terms of vortex structures evolution. Simple interactions between a vortex structure and a bluff body can exhibit complex underlying characteristics, and an unpredictable behavior. In the present work the interaction between a vortex ring and a moving rigid sphere is analyzed, in an ideal flow. We start by assuming the motion perfectly axisymmetric to define the main features of the flow; then an analysis of a full three-dimensional evolution is taken into account. The motion can be written in Hamiltonian form, and exhibits definite singular points. When the sphere oscillates there can be a ‘stochastization’ of the hyperbolic point and the behavior becomes unpredictable. The Melnikov technique is used to observe if this is the case, and proves that the vortex ring trajectory can be unpredictable and chaotic. An axisymmetric numerical simulation is performed in correspondence of unpredictable evolution, and chaotic ring trajectories are examined. A fully three-dimensional simulation is also performed in order to highlight the limitations of three-dimensional computational vortex methods.

Space temporal maps for vortical flow field construction

A class of models to generate a two-dimensional complex flow field, deriving from the coupled map lattice dynamics, is presented here. It automatically satisfies continuity equation for an incompressible fluid. The method is numerically implemented on a square lattice and some results relatively to a fully deterministic and a semi stochastic evolution are presented here. The qualitative similarity with two dimensional hydrodynamic turbulence is encouraging. In view of these first results, directions for advances are proposed.SommarioSi presenta una classe di modelli per generare un campo di moto complesso, derivante dai modelli dinamici a mappe accoppiate, in modo tale che lequazione di continuita per un fluido incomprimibile sia automaticamente soddisfatta. Il metodo eimplementato numericamente su una griglia quadrata. Si presentano alcuni risultati relativi ad unevoluzione deterministica e semi stocastica. La somiglianza qualitativa con campi di moto idrodinamici turbolenti bidimensionali eincoraggiante. Alla luce dei risultati ottenuti si propongono inoltre possibili direzioni per ulteriori sviluppi.