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This book offers a guide to understanding models of vortex rings, starting from classical ones (circular vortex filament, Hill and Norbury-Fraenkel inviscid models) to very recent models incorporating viscous effects and realistic shapes of the vortex core. Unconfined and confined viscous vortex rings are described by closed formulae for vorticity, stream function, translational velocity, energy, impulse and circulation. Models are applied to predict the formation number of optimal vortex rings and to describe two-phase vortex ring-like structures generated in internal combustion engines. The book provides a detailed presentation of analytical developments of models, backed up by illustrations and systematic comparisons with results of direct numerical simulations. The book is useful for graduate students in applied mathematics, engineering and physical sciences. It is a useful reference for researchers and practising engineers interested in modelling flows with vortex rings.

### Chapter 1. The Vortex Ring Problem

Abstract
This introductory chapter offers the physical and mathematical bases for the description of laminar vortex rings. We start by presenting the incompressible Navier–Stokes equations of motion and the vorticity equation in three dimensions. We use the cylindrical coordinate system which best fits the geometry of the problem. We subsequently simplify the vorticity equation for axisymmetric flows, with or without swirl. We then characterise the structure of vortex rings without swirl by presenting the space distribution of vorticity and the Stokes stream function in both laboratory and vortex frames of reference. The concepts of vortex bubble, core and inner core are introduced to describe the geometry of vortex rings. Circulation, hydrodynamic impulse and energy of the vortex ring are defined as main integral characteristics. We finally derive the Helmholtz–Lamb formula used to calculate the translational velocity of the vortex ring. This chapter is self-contained. However, Appendix A, containing all the details of the derivation of equations in cylindrical coordinates, could be a valuable companion for a reader who is not familiar with the form of equations in this particular coordinate system.
Ionut Danaila, Felix Kaplanski, Sergei S. Sazhin

### Chapter 2. Steady Inviscid Vortex Rings

Abstract
Most of the vortex ring models developed so far are based on the hypothesis of an inviscid and steady fluid flow. Deriving these classical and idealistic models is a useful first exercise for anyone interested in the mathematical modelling of vortex rings. We first derive a fundamental solution to the vorticity equation using the Green function. This solution is then used to describe the circular vortex filament, characterised by an infinitely small core and singular vorticity distribution. A closed-form solution for the vortex filament is also provided using the Fourier–Haenkel transform. We subsequently introduce the general solution for the steady inviscid vortex ring with finite core size. As a particular case, we present Hill’s spherical vortex ring, which is the only known closed analytical solution for this problem. For a general shape of the vortex core, the Norbury–Fraenkel family of inviscid vortex rings is obtained by numerical calculation. This very popular model is described in detail. The existence of solutions for inviscid vortex rings with fixed elliptical cross section is also addressed. Finally, we derive the famous Kelvin’s formula for the translational velocity of the vortex ring. As a practical application of this theory, we finally address the problem of the reconstruction of the velocity field of a vortex ring using theoretical models or optimisation approaches.
Ionut Danaila, Felix Kaplanski, Sergei S. Sazhin

### Chapter 3. Viscous Vortex Rings

Abstract
We present a viscous vortex ring model based on a solution to the time-dependent Stokes-flow equation. The solution has a quasi-isotropic Gaussian vorticity distribution, which is more realistic compared to the one assumed in steady inviscid vortex rings models discussed in the previous chapter. We derive closed formulae for the stream function, translational velocity of the vorticity centroid and integral characteristics (impulse, circulation, kinetic energy) of the viscous vortex ring. Well-known results for the translational velocity are recovered for the limiting regimes of short-time evolution (Saffman) and long-time evolution (Rott and Cantwell). The model is then generalised to include a time-evolving eddy viscosity scale, offering a unified framework to describe both laminar and turbulent vortex rings. A Reynolds-number correction of the model is derived using the method of matched asymptotic expansions. A comparison with experimental results and direct numerical simulations shows that the model describes well the integral quantities and provides a more accurate description of the vortex ring geometry than inviscid models.
Ionut Danaila, Felix Kaplanski, Sergei S. Sazhin

### Chapter 4. Viscous Vortex Rings with Elliptical Cores

Abstract
The viscous vortex ring model presented in the previous chapter is modified to take into account the elliptic shape of the vortex core. The new model is based on a modified functional form of the solution to the Stokes equations presented in the previous chapter. Two new parameters are introduced in the expression of the vorticity: the axial elongation $$\beta$$ and the radial elongation or compression $$\lambda$$. Based on this modification, new expressions for the translational velocity $$U_e$$, energy $$E_e$$, circulation $$\varGamma _e$$ and stream function $$\varPsi _e$$ are derived for a wide range of ellipticity parameters, typical for actual vortices. The estimates of $$\beta$$ and $$\lambda$$, based on the asymptotic behaviour of the ring’s translational velocity, are also presented. The new model is used to describe numerical (DNS) vortex rings with elongated elliptic-shape cores. The fitting procedure, using normalised values of energy and circulation, is presented in detail. Theoretical estimates based on the new model are then compared to experimental and numerical results.
Ionut Danaila, Felix Kaplanski, Sergei S. Sazhin

### Chapter 5. Confined Vortex Rings

Abstract
We derive a model for an axisymmetric vortex ring confined in a tube. We start by assuming that the vorticity distribution in the vortex ring is described by models for unconfined viscous vortex rings presented in Chaps. 3 and 4. The Stokes stream function of the confined vortex ring is then presented as the difference between the stream function of the unconfined vortex ring and a wall-induced correction. Based on the asymptotic development of the vorticity in the vicinity of the tube wall, we generalise Brasseur’s approach (Brasseur 1979) to derive the wall-induced correction. The model takes into account vortex ring cores with quasi-circular or elliptical shapes. For the confined vortex ring, closed formulae for the stream function and vorticity distribution are derived. The predictions of the model are shown to be in agreement with direct numerical simulations of confined vortex rings generated by a piston–cylinder mechanism. A simplified procedure for fitting experimental and numerical data with the predictions of the model is described. This opens the way for applying the model to realistic confined vortex rings in various applications.
Ionut Danaila, Felix Kaplanski, Sergei S. Sazhin

### Chapter 6. Formation Number of Vortex Rings

Abstract
Theoretical models developed in previous chapters are used to predict the values of the formation time and formation number of a vortex ring. These concepts introduced by Gharib et al. (1998) quantify the observation that the vortex ring carries only a part of the circulation produced by the vortex generator during the injection phase. We first illustrate these concepts using Direct Numerical Simulations (DNS) of axisymmetric vortex rings. Then, we present a unified frame for different versions of the model suggested by Shusser and Gharib (2000a) to describe the separation (or pinch-off) of the vortex ring from the injection slug. The model is based on the match of the slug-flow model describing the vortex generator and models for viscous vortex rings presented in this book. Theoretical predictions are compared to DNS results obtained for unconfined or confined vortex rings.
Ionut Danaila, Felix Kaplanski, Sergei S. Sazhin

### Chapter 7. Applications of the Models

Abstract
Recent developments in modelling of two-phase flows in vortex rings and vortex ring-like structures in gasoline engines are summarised. For the two-phase flows, the carrier phase parameters are assumed to be the ones predicted by the viscous vortex ring model presented in Chap. 3. The mixing of inertial droplets can be accompanied by crossing of droplet trajectories. Their number densities are calculated based on the Fully Lagrangian Approach. Two flow regimes corresponding to two initial conditions are investigated. These are injection of a two-phase jet and propagation of a vortex ring through a cloud of droplets. The ranges of governing parameters leading to the formation of mushroom-like clouds of droplets are identified. The caps of the mushrooms contain caustics or edges of folds of the dispersed media, which correspond to particle accumulation zones. The values of velocities in the regions of maximal vorticity, predicted by the generalised vortex ring model, are compared with the results of experimental studies of vortex ring-like structures in gasoline engine conditions. It is shown that most of the observed values of these velocities are compatible with the predictions of the model.
Ionut Danaila, Felix Kaplanski, Sergei S. Sazhin