Self-similar Vortex Research Articles

Effect of secondary atomization on blood backspatter affected by muzzle gases

Blood backspatter pattern analysis provides important evidence in firearm-related crime scenes. The mechanisms behind particular patterns have attracted significant recent attention in forensic sciences, in general, and in forensic-science-motivated fluid dynamics, in particular. However, investigations on the secondary atomization of blood drops in flight and its effects on trajectories and the corresponding blood stain distributions were scarce. The present work is especially concerned with the effect of secondary atomization on the blood backspatter interaction with muzzle gases at short-range shooting, where it can be very significant. A secondary breakup model is incorporated into the blood backspatter model accounting for interaction with a self-similar vortex ring formed by muzzle gases and moving with high speed in the direction opposite to that of the initial motion of blood drops. The behavior of blood drops of different sizes is investigated, and different scenarios are identified. The secondary atomization stems from high relative velocities of drops and air/muzzle gases and results in the formation of small blood droplets, which are swept easier by muzzle gases and even turned around toward a target. Overall, the secondary atomization in the presence of muzzle gases results in blood stains deposited on the floor closer to the target or even behind the target. It is revealed that in the cases of short-range shooting, the predicted blood stain locations on the floor without accounting for the secondary atomization could be misleading for realistic drop sizes observed experimentally.

Self-similar vortex configurations: Collapse, expansion, and rigid-vortex motion

The motion of point vortices that preserves the initial geometry of the vortex arrangement is an important class of vortex motion associated with N-vortex systems, referred to as the self-similar motion. While the self-similar motion of three-point vortices is well understood, larger vortex systems still need to be explored. Here, we use simple concepts from linear algebra to numerically investigate the distribution of initial conditions that lead to the self-similar motion of point vortices.

Dipole dynamics in the point vortex model

At the very heart of turbulent fluid flows are many interacting vortices that produce a chaotic and seemingly unpredictable velocity field. Gaining new insight into the complex motion of vortices and how they can lead to topological changes of flows is of fundamental importance in our strive to understand turbulence. Our aim is form an understanding of vortex interactions by investigating the dynamics of point vortex dipoles interacting with a hierarchy of vortex structures using the idealized point vortex model. Motivated by its close analogy to the dynamics of quantum vortices in Bose–Einstein condensates, we present new results on dipole size evolution, stability properties of vortex clusters, and the role of dipole–cluster interactions in turbulent mixing in 2D quantum turbulence. In particular, we discover a mechanism of rapid cluster disintegration analogous to a time-reversed self-similar vortex collapse solution.

Estimating the non-dimensional energy of vortex rings by modelling their roll-up

The non-dimensional energy of starting vortex rings typically converges to values around 0.33 when they are created by a piston-cylinder or a bluff body translating at a constant speed. To explore the limits of the universality of this value and to analyse the variations that occur outside of those limits, we present an alternative approach to the slug-flow model to predict the non-dimensional energy of a vortex ring. Our approach is based on the self-similar vortex sheet roll-up described by Pullin (J. Fluid Mech., vol. 88, 1978, pp. 401–430). We derive the vorticity distribution for the vortex core resulting from a spiralling shear layer roll-up and compute the associated non-dimensional energy. To demonstrate the validity of our model for vortex rings generated through circular nozzles and in the wake of disks, we consider different velocity profiles of the vortex generator that follow a power law with a variable time exponent $m$ . Higher values of $m$ indicate a more uniform vorticity distribution. For a constant velocity ( $m=0$ ), our model yields a non-dimensional energy of ${{E}^{{*}}}=0.33$ . For a constant acceleration ( $m=1$ ), we find ${{E}^{{*}}}=0.19$ . The limiting value $m \rightarrow \infty$ corresponds to a uniform vorticity distribution and leads to ${{E}^{{*}}}=0.16$ , which is close to values found in the literature for Hill's spherical vortex. The radial diffusion of the vorticity within the vortex core results in the decrease of the non-dimensional energy. For a constant velocity, we obtain realistic vorticity distributions by radially diffusing the vorticity distribution of the Pullin spiral and predict a decrease of the non-dimensional energy from 0.33 to 0.28, in accordance with experimental results. Our proposed model offers a practical alternative to the existing slug flow model to predict the minimum non-dimensional energy of a vortex ring. The model is applicable to piston-generated and wake vortex rings and only requires the kinematics of the vortex generator as input.

Spreading rate of a toroidal air bubble in water

The rise of a toroidal bubble in water obtained by injecting a pulsed jet of air into the water through a nozzle vertically upwards has been experimentally investigated. It was found that, despite the decrease in the cross-section of the toroidal bubble during the ascent, its radius grows linearly depending on the distance traveled, as in the case of self-similar vortex rings. The coefficient of expansion of the bubble at different volumes of injected air is determined. A comparison was made with toroidal bubbles obtained when balloons burst under water. A qualitative explanation of the observed differences is given. Keywords: vortex ring, toroidal bubble.

Коэффициент расширения тороидального воздушного пузыря в воде

The rise of a toroidal bubble in water obtained by injecting a pulsed jet of air into the water through a nozzle vertically upwards has been experimentally investigated. It was found that despite the decrease in the cross-section of the toroidal bubble during the ascent, its radius, depending on the distance traveled, grows linearly, as in the case of self-similar vortex rings. The coefficient of expansion of the bubble at different volumes of injected air is determined. A comparison was made with toroidal bubbles obtained when balloons burst in water. A qualitative explanation of the observed differences is given.

Uniqueness criteria for the Oseen vortex in the 3d Navier-Stokes equations

In this paper, we consider the uniqueness of solutions to the 3d Navier-Stokes equations with initial vorticity given by where is the one dimensional Hausdorff measure of an infinite, vertical line and is an arbitrary circulation. This initial data corresponds to an idealized, infinite vortex filament. One smooth, mild solution is given by the self-similar Oseen vortex column, which coincides with the heat evolution. Previous work by Germain, Harrop-Griffiths, and the first author implies that this solution is unique within a class of mild solutions that converge to the Oseen vortex in suitable self-similar weighted spaces. In this paper, the uniqueness class of the Oseen vortex is expanded to include any solution that converges to the initial data in a sufficiently strong sense. This gives further evidence in support of the expectation that the Oseen vortex is the only possible mild solution that is identifiable as a vortex filament. The proof is a 3d variation of a 2d compactness/rigidity argument in originally due to Gallagher and Gallay.

Self-similar turbulent vortex rings: interaction of propellant gases with blood backspatter and the transport of gunshot residue

Self-similar turbulent vortex rings are investigated theoretically in the framework of the semi-empirical turbulence theory for the modified Helmholtz equation. The velocity and vorticity fields are established, as well as the transport of passive admixture by turbulent vortex rings. Turbulent vortex rings of propellant gases originating from the muzzle of a gun after a gunshot are an important phenomenon to consider in crime scene reconstruction. In this work, it is shown that this has a significant repercussion on the outcome of backward blood spatter resulting from a gunshot. Turbulent vortex rings of propellant gases skew the distribution of bloodstains on the ground and can either propel blood droplets further from the target, or even turn them backwards towards the target. This is revealed through the final bloodstain locations and the respective distributions of the number of stains and their area as a function of distance from the target for two different shooter-to-target distances. An image of the propagating muzzle gases after bullet ejection is overlaid with the predicted flow field, which reveals satisfactory agreement. Gunshot residue is an important factor in determining the events of a violent crime due to a gunshot and are considered to be entrained and transported by the propellant gases. The self-similar solutions for the flow, vorticity and concentration of gunpowder particles are predicted and the results are shown to be within the measured range of a limited set of experimental data.

Self-similar vortex reconnection

As shown by Crow in 1970, the evolution of two almost parallel vortex filaments with opposite circulation exhibits a long-wave instability. Ultimately, the symmetric mode increases its amplitude reconnecting both filaments and ending into the formation of an almost periodic structure of vortex rings. This is a universal process, which appears in a wide range of scales: from the vortex trails behind an airplane to a microscopic scale of superfluids and Bose–Einstein condensates. In this paper, I will focus on the vortex reconnection for the latter case by employing Gross–Pitaevskii theory. Essentially, I focus on the well-known laws of interaction and motion of vortex filaments. By means of numerical simulations, as well as theoretically, I show that a self-similar finite-time dynamics manifests near the reconnection time. A self-similar profile is selected showing excellent agreement with numerical simulations.

On two-dimensional self-similar axisymmetric vortices

Superpositions of Lamb-Oseen axisymmetric vortices with Gaussian vorticity having zero net circulation and finite kinetic energy in unbounded domain are considered. Their evolution is described by self-similar solutions depending on a certain combination of space, time, and viscous diffusion. It is shown that the structure of a popular self-similar solution for a shielded vortex with Gaussian fluid rotation rate corresponds to two Lamb-Oseen vortices with opposite sign and nearly the same spatial scale. The radial structure of a combined vortex with different spatial scales is well suited for characterization of realistic vortical structures like atmospheric hurricanes. The linear stability of the combined vortex is investigated. The results have important implications for better understanding of vortex structures in two-dimensional flow.

Uniqueness of axisymmetric viscous flows originating from circular vortex filaments

The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [10]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogues of the self-similar Lamb-Oseen vortices in two-dimensional flows. Our method provides the leading term in a fixed-viscosity short-time asymptotic expansion of the solution, and may in principle be extended so as to give a rigorous justification, in the axisymmetric situation, of higher-order formal asymptotic expansions that can be found in the literature [7].

Self-similar point vortices and confinement of vorticity

ABSTRACTThis papers deals with the large time behavior of solutions of the incompressible Euler equations in dimension 2. We consider a self-similar configuration of point vortices which grows like the square root of the time. We study the confinement properties of a blob of vorticity initially located around the first point vortex and moving in the velocity field produced by itself and by the other point vortices. We find a sufficient condition on the point vortices such that the vorticity stays confined around the first point vortex at a rate better than the square root of the time. The relevance to the large time behavior of the Euler equations is discussed.

Formation and characterization of the vortices generated by a DBD plasma actuator in burst mode

The present study reports the formation and evolution characteristics of the continuously generated vortical structure and resulting flow field in quiescent air induced by a dielectric-barrier-discharge (DBD) plasma actuator in burst mode operation. A starting vortex is formed during the initial actuation period, which disappears after a small time interval for continuous mode operation of the DBD plasma actuator. A burst input signal to the actuator generates a train of self-similar vortices. The behaviour of vortices and the average flow field induced by the actuator has been studied using high speed schlieren visualization and particle image velocimetry technique for different actuation amplitude and duty cycle parameters. These repeating vortices travel faster than the starting vortex, and the vortex core velocity of these repeating vortices increases with increase in duty cycle parameter. Fuller u-velocity profile, higher v-velocity near the edge of the outer shear layer region, and higher growth of the wall jet thickness is observed due to enhanced entrainment by repeating vortices for burst mode operation. The repeating vortices travel at an angle of 21° relative to the wall surface for duty cycle parameter of 90.9% in comparison to 31° for the starting vortex. Self-similarity of the velocity profile is delayed in the streamwise direction for burst mode operation in comparison to that for the continuous mode of operation. This can be attributed to delay in attaining the maximum velocity of the wall jet profile and presence of coherent structures for the burst mode operation. The non-dimensional vortex core location and size for repeating vortices follow power law fit similar to the starting vortex with difference in value of the power law exponent. The phase difference between the input voltage and current drawn is in the range of π12 to π9 (in radians) for both continuous and burst mode operation indicating identical electrical behaviour of the actuator in both modes of operation. Overall, the present study illustrates the effectiveness of a DBD plasma actuator as a vortex generator while operating in the burst mode. The train of vortices generated during the burst mode operation shows overall similarity with the starting vortex, and the difference in behavior is attributed to the interaction between the vortices shed during consecutive bursting cycle, which is a function of the duty cycle parameter.

Self-similar vortex filament motion under the non-local Biot–Savart model

One type of thin vortex filament structure that has attracted interest in recent years is that which obeys self-similar scaling. Among various applications, these filaments have been used to model the motion of quantized vortex filaments in superfluid helium after reconnection events. While similarity solutions have been described analytically and numerically using the local induction approximation (LIA), they have not been studied (or even shown to exist) under the non-local Biot–Savart model. In this present paper, we show not only that self-similar vortex filament solutions exist for the non-local Biot–Savart model, but that such solutions are qualitatively similar to their LIA counterparts. This suggests that the various LIA solutions found previously should be valid physically (at least in the small amplitude regime), since they agree well with the more accurate Biot–Savart model.

Vortices in Noncompact Blanks During Twist Extrusion

The causes and consequences of the formation of multifractal vortices during severe plastic deformation of powder and porous bodies by twist extrusion are analyzed. It is demonstrated that the developed pore space in noncompact blanks leads to the formation of self-similar vortices at different scales. It is established that the multifractal vortices occurring during simple shear in twist extrusion, are responsible for the differences in the mechanisms of hardening of compact and noncompact materials. Being significantly increased (in comparison with the concentric streamlines during deformation of compacted materials), the integral deformation path intensifies the accommodation of particles of a mixture of powder components. In addition, the vortices cause intensive mixing and mass transfer at both micro and macro scales. The intensive mass transfer leads to chemical and structural homogenization and elimination of porosity.

Compression and stretching of ring-vortex solitons in a bulk nonlinear medium

Compression and stretching of ring-vortex solitons, which is a novel self-similar solution of (2+1)-dimensional diffraction decreasing waveguide, is investigated analytically and numerically. We obtain the ring-vortex solitons via the similarity transformation method. The distance modulation for the width, the diffraction, and the nonlinear response, strongly affects the form and the behavior of the self-similar vortex, and facilitates the efficient compression of optical waves. This approximate ring-vortex solitons can reflect the real properties of self-similar optical vortex beams during propagation under certain parameter window selection. Specific examples and figures are given to illustrate discussed features. The results obtained in this paper may have potential values for all-optical data-processing schemes and the design of beam compressors and amplifiers.

Localized convective flows in a nonuniformly heated liquid layer

Within the class of exact solutions of the thermal-convection equations in the Oberbeck-Boussinesq approximation, which assumes a linear dependence of the temperature and the vertical velocity component on the height, a non-self-similar behavior of localized disturbances of a special type in a nonuniformly heated liquid layer is studied. It is shown that in an unstably stratified medium these disturbances can evolve to isothermal vortex structures of Burgers type. In the conditions of stable stratification or uniform heating of the layer, the disturbances considered tend to the state of rest in an oscillating or monotonic manner. New solutions describing self-similar convective vortices are found.

Self-similar vortex dynamics in superfluid 4He under the Cartesian representation of the Hall-Vinen model including superfluid friction

We determine conditions under which the fully nonlinear form of the local induction approximation (LIA) governing the motion of a vortex filament in superfluid 4He (that is, the Hall-Vinen model) in the Cartesian frame of reference permits the existence of self-similar solutions, even in the presence of superfluid friction parameters. Writing the Cartesian Hall-Vinen LIA in potential form for the motion of a vortex filament, we find that a necessary condition for self-similarity is that the normal-fluid component vanishes (which makes sense in the low temperature limit), and we reduce the potential form of the Hall-Vinen LIA to a complex nonlinear ordinary differential equation governing the behavior of a similarity solution. In the limit where superfluid friction parameters are negligible, we provide some analytical and asymptotic results for various regimes. While such analytical results are useful for determining the qualitative behavior of the vortex filament in the limit where superfluid friction parameters vanish, numerical simulations are needed to determine the true behavior of the filaments in the case of non-zero superfluid friction parameters. While the superfluid friction parameters are small, the numerical results demonstrate that the influence of the superfluid friction parameters on the self-similar vortex structures can be strong. We classify two types of filaments from the numerical results: singular filaments (which demonstrate growing oscillations and hold kink-type solutions as a special case) and bounded filaments (whose behavior is a bounded function of x). We also comment on how to include the case where there is a non-zero normal fluid, and we find a transformation of the self-similar solutions into non-similar solutions that can account for this.

Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity

We consider the incompressible Navier–Stokes equations in a two-dimensional exterior domain Ω, with no-slip boundary conditions. Our initial data are of the form u0=αΘ0+v0, where Θ0 is the Oseen vortex with unit circulation at infinity and v0 is a solenoidal perturbation belonging to L2(Ω)2∩Lq(Ω)2 for some q∈(1,2). If α∈ℝ is sufficiently small, we show that the solution behaves asymptotically in time like the self-similar Oseen vortex with circulation α. This is a global stability result, in the sense that the perturbation v0 can be arbitrarily large, and our smallness assumption on the circulation α is independent of the domain Ω.

Self-Similar Vortex Solitons for the Distributed-Coefficient Nonlinear Schrödinger Equation

Via the similarity transformation method, we obtain an approximate vortex solution to the generalized (2+1)-dimensional nonlinear Schrödinger equation with space-dependent diffraction, nonlinearity and gain/loss coefficients. Under certain parametric conditions, we investigate the propagation dynamics of self-similar vortex solitons in optical media.