Vortex Flows with Particles and Droplets (A Review)

Abstract

Single-phase vortices are a classic example of objects characterized by symmetry in the distribution of all main parameters. The presence of inertial particles (or droplets) in such objects, even with their initial uniform distribution in space, leads to symmetry breaking due to the inverse effect of the dispersed phase on the characteristics of carrier vortices. A review of calculation-theoretical and experimental works devoted to the study of the motion of particles (or droplets) in various concentrated vortex structures, as well as their inverse effect on vortex characteristics, is conducted. The main characteristics (inertia, concentration) as well as dimensionless parameters (Reynolds, Stokes, Froude, Tachikawa numbers) determining the interaction between the dispersed phase and vortices are described. The results of available studies are analyzed in order to establish the peculiarities of particle (or droplet) behavior and stability of different vortex structures, including natural ones. The works analyzed in the review cover a wide range of inertia of the dispersed phase (Stk_f = 0.002 − 14.7) and vortex intensities (Re_Γ = 200 − 5000).

1. Introduction

There are many works in the scientific literature devoted to vortex flows, as well as many works on two-phase flows. Therefore, based on the topic of the review, only those articles where exactly these areas of research intersect were selected. Single-phase vortices are a classic example of objects characterized by symmetry in the distribution of all main parameters. The presence of inertial particles (or droplets) in such objects, even with their initial uniform distribution in space, leads to breaking of symmetry due to the inverse effect of the dispersed phase on the characteristics of carrier vortices. Vortex motion is one of the most common states of a moving continuous environment [1,2,3]. Vortex flows of a continuous medium carrying disperse impurities in the form of solid particles or droplets occur in a number of natural phenomena and are used in a huge number of technical devices [4,5,6]. Examples here are vortex furnace chambers, vortex burners, and cyclone separators. The exceptional complexity of two-phase flows is associated with a large variety of properties of dispersed particles or droplets (primarily inertia and concentration), which leads to the realization of numerous modes (classes) of these types of flows.

Inertia and particle concentrations can vary over a very wide range (by many orders of magnitude). Thus, practically inertia-free particles with negligible volumetric and mass concentrations are used to measure the velocity of the carrier phase using non-contact optical methods of flow diagnostics. If the condition of insignificance of the particle relaxation time compared to the characteristic time scales of the carrying media is observed, the instantaneous velocities of such passive particles (tracer particles) will be practically equal to the instantaneous velocities of the gas. Turbulent flows consisting of vortex structures are characterized by a wide range of spatial and corresponding temporal scales. From this point of view, knowledge of peculiarities of particle behavior in vortices characterized by different time scales appears to be relevant.

It is known that solid particles (or aerosol droplets) suspended in a turbulent flow have an uneven concentration distribution even under conditions of statistically homogeneous turbulence [7]. There are a significant number of papers which show that heavy particles, whose density is much higher than that of the carrier gas, are ejected from regions of high swirl to the region of high strain [8,9,10], forming regions of high concentration. This effect is called the clustering effect. As a rule, regions of heavy particle cluster formation (caustics) are associated with some special hyperbolic points (attractors). Given that periodicity is inherent to vortex structures, heavy particles will concentrate in the vicinity of elliptical fixed points in the rotating coordinate system.

Analysis of particle motion in various vortex structures is of great importance for solving the fundamental problem of studying the behavior of particles in turbulent flows consisting entirely of vortices of different scales.

Although increased attention is paid to vortex motion and various vortex effects in the hydrodynamics literature, there are virtually no monographs or reviews devoted directly to two-phase concentrated vortices.

The subject of this review is two-phase vortex flows. The purpose of the review is to describe and analyze the currently available calculation-theoretical and experimental studies devoted to the investigation of the behavior of particles and droplets in various concentrated vortex structures, as well as their inverse effect on the characteristics of vortices.

The review is structured as follows. The first section presents the basic parameters and criteria used both in the formulation of computational and experimental studies of two-phase vortex flows and in the generalization of the results obtained. The second section is devoted to the description and analysis of available research papers devoted to the study of the peculiarities of particle and droplet motion in various vortex structures (forced vortex, Rankin vortex, Burgers vortex, Lamb–Oseen vortex, etc.) and their inverse effect on the parameters of the carrying gas flow. The third and final section substantiates the necessity to take into account the two-phase nature of the tornado, and presents the results of studies on the behavior of particles and debris in natural eddies as well as the effect on the parameters of the latter.

2. The Motion of Particles and Droplets in Vortex Flows (Basic Characteristics)

The motion of particles and droplets in vortex flows, as well as their inverse effect on the carrier gas flow, is determined by their inertia and concentration. In order to analyze and generalize the results of studies of single-phase vortex flows, well-established criteria are used. In this section, characteristics and criteria specific to two-phase vortex flows, along with parameters of the carrier phase, are presented. Then, these characteristics will be used for the purpose of describing and analyzing the available research.

2.1. Concentration of Particles and Droplets. Modes of Interaction

An extensive physical characteristic of two-phase flows is the concentration of dispersed impurity. In [11] a classification of two-phase flows depending on the volume concentration of the dispersed phase $\mathsf{\Phi }$ was proposed.

In modeling the motion of particles in dilute two-phase flow, when $\mathsf{\Phi }\le O(10^{-6})$, i.e., at low volume concentration of the dispersed phase, the main attention is usually given to establishing the characteristics (behavior) of particles during their interaction with the turbulent vortices of the carrier flow. Such calculations are called “one-way coupling” [11], which means taking into account only the one-directional influence of the carrying current on the particles suspended in it, which fully determines the features of their behavior.

As the concentration of particles increases, when $O(10^{-6})<\mathsf{\Phi }\le O(10^{-3})$, they, in turn, begin to have an inverse effect on the characteristics (all without exception) of the carrier medium. Taking into consideration the mutual influence of the dispersed and carrier phases significantly complicates mathematical modeling of two-phase flow (such calculations are called “two-way coupling”) [11].

A further increase in the concentration of particles, when $\mathsf{\Phi }>O(10^{-3})$, leads to the necessity of taking into account the contribution of interparticle interactions to the process of momentum and energy transfer of the dispersed phase. In dense two-phase flows, it is interparticle collisions that play a decisive role in the formation of the statistical properties of the dispersed phase. Accounting for paired (binary) particle collisions further complicates mathematical modeling (in the literature, such calculations are called “four-way coupling”) [11].

2.2. Reynolds Numbers

The Reynolds numbers in single-phase flows are well known. For their construction, the radius (diameter) of the channel or jet, the longitudinal coordinate in the boundary layer, etc., are used as characteristic geometric dimensions.

In two-phase flows, the main criterion determining the flow pattern is the Reynolds number of the particle (drop), calculated from the relative velocity between the phases and the diameter of the impurity (particle, drop) $d_p$

$${\mathrm{Re}}_p=\frac{\left|\mathbf{u}-\mathbf{v}\right|d_p}{\nu }$$

(1)

where $\mathbf{u}$—velocity vector of carrier gas, $\mathbf{v}$—velocity vector of particles (droplets), and $\nu$—kinematic viscosity of carrier gas.

Considering that velocity of carrying gas is a function of spatial coordinates and time, it is obvious that the value of the adjusted Reynolds number can undergo significant changes along the trajectory of particle motion. This circumstance should be taken into consideration when carrying out calculations of particle motion.

When analyzing vortex flows, the so-called circulation (or vortex) Reynolds number is used, which has the following form:

$${\mathrm{Re}}_{\mathsf{\Gamma }}=\frac{\mathsf{\Gamma }}{\nu }$$

(2)

where $\mathsf{\Gamma }$—characteristic value of vortex circulation.

Often researchers use a different from (2) definition of the circulation Reynolds number:

$${\mathrm{Re}}_{\mathsf{\Gamma }}=\frac{\mathsf{\Gamma }}{2\pi \nu }$$

(3)

Considering that the expression for circulation usually contains a multiplier $2\pi$, the use of Expression (3) seems somewhat more correct.

2.3. Stokes Numbers

The velocity of a particle (drop) caught in the vortex begins to “adjust” to the local velocity of the vortex structure. The process of relaxation of time-averaged particle and gas velocities can be characterized by a dimensionless parameter—the Stokes number, defined as follows:

$${\mathrm{Stk}}_f=\frac{{\tau }_p}{{\tau }_f}$$

(4)

where ${\tau }_p$—time of dynamic relaxation of particle (droplet) and ${\tau }_f$—characteristic time of vortex.

Expression for the dynamic relaxation time ${\tau }_p$ is defined as follows:

$${\tau }_p=\frac{{\tau }_{p0}}{C}=\frac{{\rho }_p^{}{d}_p^2}{18\mu C}=\frac{{\rho }_p^{}{d}_p^2}{18\mu (1+0.15{\mathrm{Re}}_p^{0.687})}$$

(5)

where ${\rho }_p$—particle density and $\mu$—dynamic viscosity of the carrier gas.

The dynamic relaxation time of a particle is a complex characteristic of its inertia, since in addition to the particle’s material density and size, it also contains the viscosity of the carrier phase in which the motion occurs.

In Expression (5), ${\tau }_{p0}$ characterizes the dynamic relaxation time of Stokesian particle (${\mathrm{Re}}_p<<1$). Adjustment function $C=C({\mathrm{Re}}_p)$ [12,13] takes into account the effect of inertial forces on the relaxation time of a non-Stokesian particle. Thus, in the case of non-Stokesian particle motion, its inertia also depends on the Reynolds number of the particle ${\mathrm{Re}}_p$.

The characteristic time of a free vortex (time of one “turn”) can be represented as

$${\tau }_f=\frac{\pi d_0}{u_{\phi }}$$

(6)

where $d_0$—diameter of vortex core and $u_{\phi }$—characteristic value of the azimuthal velocity of the vortex.

Taking into account (6), Expression (4) for the Stokes number becomes

$${\mathrm{Stk}}_f=\frac{u_{\phi }{\tau }_{p0}}{\pi d_0}$$

(7)

The value of the found Stokes number determines the speed of relaxation (convergence) of velocities of solid particles and gas. Obviously, if the dynamic relaxation time of the particles is short compared to the characteristic time of the vortex structure (${\mathrm{Stk}}_f<<1$), then the particles almost instantly reach the velocity of the carrier gas.

Researchers of vortex flows often use the following expression to estimate the characteristic time of the carrier phase:

$${\tau }_f=\frac{2\pi r_0^2}{\mathsf{\Gamma }}$$

(8)

where $r_0$—vortex core radius and $\mathsf{\Gamma }$—vortex circulation.

Taking into account (8), expression (4) for the Stokes number becomes

$${\mathrm{Stk}}_f=\frac{{\tau }_{p0}\mathsf{\Gamma }}{2\pi r_0^2}$$

(9)

Comparing expressions (7) and (9) and assuming that $\mathsf{\Gamma }=2\pi r_0{u}_{\phi }$ and $d_0=2r_0$, one can conclude that they are equal up to $2\pi$.

There are studies in which the Stokes number is determined in a slightly different way:

$${\mathrm{Stk}}_f=\frac{{\rho }_p{d}_p^2{\mathrm{Re}}_{\mathsf{\Gamma }}}{18\rho R^2}=\frac{{\tau }_{p0}\mathsf{\Gamma }}{R^2}$$

(10)

where ${\mathrm{Re}}_{\mathsf{\Gamma }}$—vortex Reynolds number calculated using (2) and $R$—radius of the channel in which the vortex current is formed.

When considering the motion of particles in shear gas flows, the characteristic time of the carrier phase often is

$${\tau }_f=\frac{\nu }{\mathsf{\Gamma }\sigma }$$

(11)

where $\mathsf{\Gamma }$—vortex circulation and $\sigma$—strain parameter.

Taking into account (11), the expression for the Stokes number (4) takes the following form:

$${\mathrm{Stk}}_{f\sigma }=\frac{{\rho }_p{d}_p^2\sigma \mathsf{\Gamma }}{18\rho {\nu }^2}$$

(12)

2.4. Froude Numbers

The dimensionless criteria characterizing the relative importance of the influence of inertial and gravitational forces on the behavior of particles is the Froude number:

$$\mathrm{Fr}=\frac{u}{\sqrt{gl}}$$

(13)

where $u$—characteristic velocity, $g$—gravitational acceleration, and $l$—characteristic geometric size.

Large values of the Froude number are realized in the case when the forces of inertia exert a dominant influence on the motion of particles, and gravity does not play a special role. As the Froude number decreases, the gravitational forces begin to have a significant influence on the behavior of the particles.

When considering the motion of particles in shear gas flows, it is convenient to use the following expression to determine the Froude number:

$${\mathrm{Fr}}_{\sigma }=\frac{{\sigma }^{3/4}\mathsf{\Gamma }}{{\nu }^{3/4}\sqrt{g}}$$

(14)

where $\sigma$—strain parameter and $\mathsf{\Gamma }$—vortex circulation.

2.5. Tachikawa Number

To calculate the motion of debris of different shapes (compact, sheet, rod, etc.) produced and carried by wind, the Tachikawa number is often used [14]. This number expresses the ratio of the aerodynamic force to the gravitational force and has the following form:

$$\mathrm{Ta}=\frac{\rho S{u}^2}{2m_{p}g}$$

(15)

where $\rho$—density of carrier gas, $S$—characteristic area (midsection) of the flying debris, $u$—certain characteristic vortex velocity, $m_p$—mass of moving debris, and $g$—gravitational acceleration.

3. The Motion of Particles and Droplets in Various Vortex Structures

This section describes and analyzes the available papers on the peculiarities of motion of particles and droplets in various vortex structures (forced vortex, Rankin vortex, Burger’s vortex, Lamb–Oseen vortex, etc.) and their inverse effect on the parameters of the carrying gas flow.

3.1. Particle Motion in a Forced Vortex

The simplest case of vortex motion of gas is its rotation as a solid body with constant angular velocity $\mathsf{\Omega }=\mathrm{const}$. Such vortices are called forced vortices. The azimuthal velocity in these vortices grows linearly with increasing distance from the axis of rotation. Circulation is directly proportional to the square of the distance from the axis of rotation. The swirl has a non-zero value.

In [15], an analytical study of the motion of inertial particles in the simplest solid-body rotation gas flow (with constant angular velocity) according to the solid-body law (Figure 1) was carried out. The purpose of this study was to develop a simple technique for estimating the inertia of particles used to visualize and diagnose vortex structures of various intensities.

The two-dimensional field of gas velocities was set as follows:

$$u_x=-\mathsf{\Omega }r\sin \phi =-\mathsf{\Omega }y$$

(16)

$$u_y=\mathsf{\Omega }r\cos \phi =\mathsf{\Omega }x$$

(17)

where $u_x$ and $u_y$—projections of gas velocity on the axes $x$ and $y$ in the Cartesian coordinate system ($x$, $y$, $z$), $\mathsf{\Omega }={\mathsf{\Omega }}_z$—the projection of the angular velocity vector on the axis $z$ in the Cartesian coordinate system, and $r$, $\phi$—radius–vector and angle in the polar coordinate system ($r$, $\phi$), respectively.

The simplest case of the motion of single particles, taking into account only the aerodynamic drag force, defined according to Stokes law (${\mathrm{Re}}_p<<1$) was considered.

The equations of motion of the particle, taking into account (16) and (17), look like

$$\frac{d{v}_x}{d\tau }\equiv \frac{d^{2}x}{d{\tau }^2}=\frac{1}{{\tau }_{p0}}\left(-\mathsf{\Omega }y-\frac{dx}{d\tau }\right)$$

(18)

$$\frac{d{v}_y}{d\tau }\equiv \frac{d^{2}y}{d{\tau }^2}=\frac{1}{{\tau }_{p0}}\left(\mathsf{\Omega }x-\frac{dy}{d\tau }\right)$$

(19)

where $v_x$ and $v_y$—projections of particle velocity on the axes $x$ and $y$ in the Cartesian coordinate system ($x$, $y$, $z$), $\tau$—time, ${\tau }_{p0}={\rho }_p{d}_p^2/18\mu$—Stokesian particle dynamic relaxation time, ${\rho }_p$—the physical density of the particle material, $d_p$—particle diameter, and $\mu$—dynamic viscosity of the carrier phase.

In order to obtain an analytical solution to the problem in question, an assumption of equality of angular velocities of particles and carrier gas was made $d\phi /d\tau =\mathsf{\Omega }$, strictly applicable only to inertia-free particles (${\tau }_{p0}=0$). Equations (18) and (19), using the adopted assumption and substitution $d/d\tau =\mathsf{\Omega }(d/d\phi )$, are rewritten as

$$\frac{d^{2}x}{d{\phi }^2}+\frac{1}{A}\frac{dx}{d\phi }+\frac{y}{A}=0$$

(20)

$$\frac{d^{2}y}{d{\phi }^2}+\frac{1}{A}\frac{dy}{d\phi }-\frac{x}{A}=0$$

(21)

where $A=\mathsf{\Omega }{\tau }_{p0}$—a parameter determining the behavior of particles in a rotating gas.

Note that there is a simple relation between the inertia parameter and the Stokes number ${\mathrm{Stk}}_f$, determined according to (7), in the case of Stokes particles (${\tau }_p={\tau }_{p0}$):

$${\mathrm{Stk}}_f=\frac{A}{2\pi }$$

(22)

Solutions of the system of Equations (20) and (21), taking into account the initial conditions accepted in [15], were obtained using the Laplace transform [16]. Due to their cumbersome nature, they are presented here in the functional form

$$x=f_1(x_0,\psi ,A,B)$$

(23)

$$y=f_2(x_0,\psi ,A,B)$$

(24)

where $\psi =\frac{\phi }{2A}$ and $B=\frac{{(dx/d\tau )}_{\tau =0}}{\mathsf{\Omega }{x}_0}$.

Expressions (23) and (24) determine the types of trajectories of particles of different inertia ($A=\mathrm{var}$) and different initial velocities ($B=\mathrm{var}$). Note that the inertia of the particles in the case under consideration is determined not only by the value of their relaxation time, but also by the angular velocity of the rotating gas.

Figure 2 shows trajectories of particles of different inertia ($A=\mathrm{var}$) in a solid-body rotating gas ($\mathsf{\Omega }=\mathrm{const}$). During the calculations, the initial velocity of the particles was assumed to be equal to zero ($B=0$). The angular velocity of the carrier flow was assumed to be $\mathsf{\Omega }=10$ s⁻¹; the particle entry point had coordinates $x_0=0.05$ m, $y_0=0$; the physical density of the particle material was ${\rho }_p=3900$ kg/m³; and the gas dynamic viscosity was $\mu =18.1\cdot 10^{-6}$ kg/(m/s). The inertia of the particles was changed by varying their diameters. Calculations were performed for three different diameters—$d_p=10$, 32, and 100 μm.

It follows from the data in Figure 2 that particles of small sizes ($d_p=10$ µm, $A=0.012$) track the motion of the rotating gas almost completely (Figure 2a). As the size ($d_p=32$ μm, $A=0.12$) increases, the radial component of the particle velocity, which leads to their removal from the vortex center, increases sharply (Figure 2b). This leads to the fact that in this case, the distance from the particle to the center of the vortex structure doubles for each revolution. A further increase in the particle size ($d_p=100$ µm, $A=1.2$) leads to an even greater increase in the radial velocity, so that in the first revolution the distance of the particle from the center of the vortex increases several times (Figure 2c). Thus, the value of the parameter $A=0.01$ can be taken as a boundary value in the conditional division of particles into low-inertia, and in large ones when they move in vortex flows.

3.2. Particle Motion in the Rankine Vortex

The Rankin vortex is a combined vortex consisting of a forced vortex (rotating with a constant angular velocity) and a potential vortex. In this vortex, there is a core of radius $r=r_0$. As the radial coordinate increases, the azimuthal velocity increases inside the vortex core and decreases outside of it. All characteristics of the combined vortex are defined by expressions for the forced vortex at $r\to 0$ and expressions for the potential vortex at $r\to \infty$.

The distribution of the azimuthal velocity in the vortex has the form

$$u_{\phi }(r)=\frac{\mathsf{\Gamma }r}{2\pi r_0^2}\mathrm{when}r<r_0$$

(25)

$$u_{\phi }(r)=\frac{\mathsf{\Gamma }}{2\pi r_{}^{}}\mathrm{when}r>r_0$$

(26)

where $\mathsf{\Gamma }=\mathsf{\Gamma }(r_0)=\mathrm{const}$—circulation along any contour that encompasses the entire vortex core once.

The maximum speed is reached at the boundary of the vortex core. The azimuthal velocity distribution (25) and (26) can be represented in an even simpler form:

$$u_{\phi }(r)=u_{\phi max}\frac{r}{r_0^{}}\mathrm{when}r<r_0$$

(27)

$$u_{\phi }(r)=u_{\phi max}\frac{r_0}{r_{}^{}}\mathrm{when}r>r_0$$

(28)

where $u_{\phi max}$—maximum azimuthal velocity at $r=r_0$.

The simple velocity distribution makes the Rankin vortex a convenient object for studying its stability by analytical methods. There are a number of papers that have studied the stability of the Rankin vortex. A monograph [17] analyzes the linear stability of an incompressible Rankine vortex. It is shown that for the case of two-dimensional perturbations characterized by the eigenmode number $m$, there is only one valid intrinsic value. This intrinsic value indicates that the perturbation will propagate without growth or decay, so that neutral stability is maintained. In [18] it was found that, in addition to these neutrally stable eigenmodes, there are combinations of eigenmodes that lead to short-time perturbation growth.

Acoustic waves, compressibility effects, possible stratification—all of these factors can destabilize the vortex column. For example, the configuration of a radially stratified vortex is of particular importance. In [19], it is shown that if the density increases monotonically with the radius, such a flow is stable to both axisymmetric and non-axisymmetric modes. Otherwise [20], a vortex with a heavy core may become unstable due to centrifugal instability caused by short-wave modes in the axial direction and Rayleigh-Taylor instability due to two-dimensional modes. Similar conclusions were made in [21], where the authors found that two-dimensional vortices with a heavy core are subject to Rayleigh–Taylor instability. The main mechanism of baroclinic vortex generation is the mismatch between the density gradient and the centripetal acceleration. Using numerical simulations [21], it was found that unstable modes lead to the formation of spiral arms in the density and swirl distributions, which eventually collapse due to the appearance of nonlinearity effects.

The presence of particles in the flows cannot lead to a violation of their stability, in the case of not taking into consideration the mutual influence of the dispersed and carrier phases («one-way coupling»). In recent years, there has been a growing interest in the problem of flow stability due to the presence of small heavy particles («two-way coupling»). In [22], the effect of particle inertia in dusty Rayleigh–Taylor turbulence was considered. It is shown that the system with low-inertia particles behaves similarly to a single-phase liquid of equivalent (higher) density. As the inertia of the particles increases, turbulent mixing slows down. The nonmonotonic role of the inertia of the dispersed phase is shown in [23], where the stability of the two-phase Kolmogorov flow was studied. The «two-way coupling» simulation results revealed increased instability of the flow in the case of low inertia of the particles. However, with increasing inertia (large Stokes numbers) the particles can both stabilize and destabilize the considered flow with a non-monotonic dependence on their mass concentration.

In [24], the effect of the presence of particles injected into a circular region of finite size on the stability of a two-dimensional Rankine vortex for the case of a medium dense two-phase flow was studied.

Monodisperse heavy (${\rho }_p/\rho =830$) particles were placed randomly within the vortex core with velocities equal to those of the carrier gas at their location. The mass concentration of particles equaled $\mathrm{M}=1$, which corresponds to $\mathsf{\Phi }=1.2\cdot 10^{-3}$ and predetermines the presence of the inverse effect of particles on the carrier phase (see Section 2.1).

The Stokes and Reynolds numbers were chosen to equal ${\mathrm{Stk}}_f={\tau }_{p0}/{\tau }_f=0.025$ and ${\mathrm{Re}}_{\mathsf{\Gamma }}=\mathsf{\Gamma }/(2\pi \nu )=1000$, respectively. Here, ${\tau }_{p0}={\rho }_p^{}{d}_p^2/(18\mu )$—particle dynamic relaxation time, ${\tau }_f=2\pi r_0^2/\mathsf{\Gamma }$—vortex characteristic timescale, and $r_0^{}$—initial radius of the vortex core with circulation $\mathsf{\Gamma }$.

Numerical studies used a very large number of particles, $N=376,612$. This was made possible by their relatively small size—$r_0^{}/d_p=1400$. Calculations were performed using one-way coupling and two-way coupling methods. In the first case, the component in the equation of motion of the gas, responsible for the interphase momentum exchange, was taken as equal to zero. Thus, the results obtained allowed the researchers to reach a conclusion regarding the influence of the presence of particles on the flow dynamics.

Further, in [24], the isocontours of the axial vorticity, normalized to its initial value at the vortex center ${\omega }_0={\omega }_z(r=0,\tau =0)$, and the isocontours of the volume concentration of particles, normalized to its initial value calculated for different moments of time ($\tau /{\tau }_f=0-48$), were analyzed.

The analysis showed (Figure 3) that in the case of one-way coupling, the vorticity field always remains axisymmetric. The vorticity magnitude within the vortex core remains practically the same, except for the region near the vorticity spike, since the viscous effects are too small to cause significant diffusion over the entire time interval $0\le \tau /{\tau }_f\le 48$. In the case of two-way coupling, a significant distortion of the flow field was observed. The vorticity rapidly loses cylindrical symmetry due to the emergence of azimuthal perturbations. Over time ($\tau /{\tau }_f\approx 12$) perturbations grow into vortex threads, which gradually move away from the vortex core. By the time $\tau /{\tau }_f\approx 48$, several clearly distinguishable spiral arms originating from the vortex core were observed. In contrast with the almost homogenous profile in the case of one-way coupling, the vorticity profile in this area obtains a diffusive nature.

Particle dispersion is also significantly affected by the interaction between the two phases. In the case of one-way coupling, particles were assembled into a ring-shaped cluster with a diameter of approximately $1.5 r_0^{}$. Similar dynamics have been observed in [25,26,27]. The emission of particles from areas with high vorticity leads to an increase in their concentration in narrow areas. This behavior of inertial particles is well known as the clustering effect [8,9,10]. As for the carrier phase flow, the particle distribution has axial symmetry only in the case of one-way coupling. In contrast to the aforementioned studies, in the case of two-way coupling, a loss of symmetry with a rapid and wide blurring of the particle cloud was observed. Calculations revealed the presence of azimuthal perturbations at $\tau /{\tau }_f\approx 12$, affecting the particle cloud and leading to the formation of spiral threads of particles. This particle behavior resembles the effect of the formation of spiral arms observed earlier in [21], in the case of Rayleigh-Taylor instability in the case of density stratification in the radial direction.

3.3. Particle Motion in the Burgers Vortex

The Burgers vortex is a vortex structure which is simple, but implementable in real experiments. This is because taking viscosity into account allows for the smoothing of the features arising in the vortex vicinity in infinitely thin vortex threads and Rankin vortex models [28]. The Burgers vortex, first proposed in [29,30,31] to describe twisted turbulent flows, belongs to the class of axisymmetric exact solutions of the Navier–Stokes equations of the form

$$u_r=u_r(r),u_{\phi }=u_{\phi }(r),u_z=u_z(r)=zf(r)$$

(29)

where $u_r$, $u_{\phi }$, $u_z$—radial, azimuthal, and axial velocity components, respectively. A family of solutions of the type (29) is described in [32].

In the monograph [28] the solution for the Burgers vortex for the special case (29) is obtained:

$$u_r=u_r(r),u_{\phi }=u_{\phi }(r),u_z=\alpha z,\alpha =\mathrm{const}$$

(30)

The solution of the Navier–Stokes equations and the continuity equation in a cylindrical coordinate system leads to the following velocity distribution [28]:

$$u_r(r)=-\alpha r/2$$

(31)

$$u_{\phi }(r)=\frac{\mathsf{\Gamma }}{2\pi r}\left[1-\exp \left(-\frac{\alpha r^2}{4\nu }\right)\right]$$

(32)

In [33], an important assumption is made that much of the physics of complex turbulent flows can be understood by treating them as a set of Burgers vortices. In addition, ref. [33] states that for a particle moving in a Burgers vortex, the drag force created by the inwardly directed radial motion of the fluid can counterbalance the centrifugal force, thereby preventing the particles from being ejected. A similar assumption about the possible balance between the centrifugal force and the aerodynamic drag force was put forward in [34].

Thus, the study of particle dynamics in “model” Burgers vortices, which are somewhat analogous to small-scale turbulent structures, is of great importance for the study of two-phase turbulent flows in general.

In [35], a detailed study of the behavior of heavy particles in a stationary Burgers vortex is conducted. It was assumed that the drag of the particles would follow Stokes’s law. The inverse effect of particles on the gas flow parameters and interparticle collisions was not taken into account. The analysis was performed for the two-dimensional case ($u_z=0$). The distributions of the radial and azimuthal gas velocities for the case in question were

$$u_r(r)=-\sigma r$$

(33)

$$u_{\phi }(r)=\frac{\mathsf{\Gamma }}{2\pi r}\left[1-\exp \left(-\frac{r^2}{2{\delta }^2}\right)\right]$$

(34)

where $\mathsf{\Gamma }$—vortex circulation, $r$—distance from the vortex center, $\delta =\sqrt{\nu /\sigma }$—vortex core size, $\nu$—kinematic viscosity, and $\sigma$—strain parameter.

Comparing the azimuthal velocity distributions (34) and (32), we can conclude that both are completely equal.

Then, in [35], a transition to non-dimensional equations was made by using $\delta$ as the characteristic length and $\mathsf{\Gamma }/\delta$ as the characteristic velocity, i.e.,

$${\overline{u}}_r(r)=-A\overline{r}$$

(35)

$${\overline{u}}_{\phi }(r)=\frac{1}{2\pi \overline{r}}\left[1-\exp \left(-\frac{{\overline{r}}^2}{2}\right)\right]$$

(36)

where $A=\sigma {\delta }^2/\mathsf{\Gamma }=\nu /\mathsf{\Gamma }$—dimensionless strain parameter, inverse of the vortex Reynolds number (see (2)), i.e., $A^{-1}={\mathrm{Re}}_{\mathsf{\Gamma }}$.

Velocity of a single spherical particle in a gas medium (${\rho }_p/\rho \approx O(10^3)$) depends on the carrier gas velocity and time, and is described by the following non-dimensional equations [35]:

$$\overline{\mathbf{v}}=\frac{d{\overline{\mathbf{x}}}_p}{d\overline{\tau }}$$

(37)

$$\frac{d\overline{\mathbf{v}}}{d\overline{\tau }}=\frac{1}{{\mathrm{Stk}}_{f\sigma }}(\overline{\mathbf{u}}-\overline{\mathbf{v}})+\frac{1}{{\mathrm{Fr}}_{\sigma }^2}{\mathbf{e}}_{\mathbf{g}}$$

(38)

where $\overline{\mathbf{v}}$—vector of the dimensionless velocity of the particle, $\overline{\mathbf{u}}$—vector of the dimensionless velocity of the carrier gas flow, ${\overline{\mathbf{x}}}_p$—radius-vector of the instantaneous position of the particle, $\overline{\tau }$—dimensionless time, and ${\mathbf{e}}_{\mathbf{g}}$—singular vector in the direction of the projection of gravity. ${\mathrm{Stk}}_{f\sigma }$ and ${\mathrm{Fr}}_{\sigma }^{}$—Stokes and Froude numbers calculated by relations (12) and (14), respectively.

We can conclude that the non-dimensional equations of particle motion contain two criteria determining the behavior of particles, namely, the Stokes number ${\mathrm{Stk}}_{f\sigma }$ and the Froude number ${\mathrm{Fr}}_{\sigma }^{}$.

Small values of Stokes numbers are realized in the case of motion of either very light (low-inertia) particles, or very viscous fluids. In both of these cases, the viscous force (viscosity) has a dominant influence on the behavior of the particles. On the contrary, large values of Stokes numbers are realized when heavy (inertial) particles move, or when they move in a less viscous fluid. In these cases, the forces of inertia have a dominant influence on the motion of particles.

The Froude number characterizes the comparative importance of the influence of inertial and gravitational forces on the behavior of particles. Large values of Froude numbers are realized in the case when the inertial forces have a dominant influence on the motion of particles, gravity does not play a special role, and the behavior of the particle is determined only by the ratio of viscous and inertial forces, i.e., characterized only by the Stokes number. As the Froude number decreases, gravitational forces begin to have a dominant influence on the behavior of particles.

Further, in [35], the stability of heavy particles in the Burgers vortex is analyzed with and without taking into account the gravitational force.

A simple criterion of particle stability is obtained. If the Stokes number is less than some critical value, i.e.,

$${\mathrm{Stk}}_{f\sigma }<{\mathrm{Stk}}_{f\sigma cr}=16{\pi }^{2}A$$

(39)

then the centrifugal force acting on the particle is always less than the radial component of the drag force directed toward the center of the vortex. In this case, the particle will move to the center of the vortex. Otherwise, the centrifugal force exceeds the radial drag force at short distances, and the particle will move away from the center of the vortex and asymptotically approach a circular trajectory, where the balance of forces is restored due to an increase in the radial force from the carrier gas.

In [35], by equating the centrifugal force and the drag force, a transcendental equation for the radius of a stable particle trajectory was obtained.

$${\overline{r}}^2\sqrt{\frac{A}{{\mathrm{Stk}}_{f\sigma }}}-\frac{1}{2\pi }\left[1-\exp \left(-\frac{{\overline{r}}^2}{2}\right)\right]=0$$

(40)

It follows from (40) that a particle with a Stokes number greater than the critical value rotates around the vortex center along a circular trajectory, whose radius is a function of Stokes number ${\mathrm{Stk}}_{f\sigma }$ and strain parameter $A$.

Special attention was paid in [35] to the search of singular points (equilibrium points) for particles, i.e., points where velocity and acceleration of particles are equal to zero and there is an absolute balance of all forces acting on a particle. Without taking into account the effect of gravity, there is only one point of equilibrium—it is the center of the vortex.

Further, in [35], it was shown that taking gravitation into account leads to a qualitatively different behavior of particles. For this case, the transcendental equation for the radius of a stable trajectory was also obtained:

$$\overline{r}=\frac{{\mathrm{Stk}}_{f\sigma }}{{\mathrm{Fr}}_{\sigma }^{2}A}\frac{1}{\sqrt{1+{\chi }^2(\overline{r})}}$$

(41)

where $\chi (\overline{r})=\frac{1-\exp (-{\overline{r}}^2/2)}{2\pi A{\overline{r}}^2}$.

It follows from (41) that the particle motion depends not on two, as was the case earlier according to (40), but on three dimensionless parameters, namely, on Stokes number ${\mathrm{Stk}}_{f\sigma }$, strain parameter $A$, and Froude number ${\mathrm{Fr}}_{\sigma }$.

The stability analysis has shown that if the gravitational force is taken into account, the center of the vortex is no longer an equilibrium point, but instead there appear either one or three equilibrium points away from the center. The location of these equilibrium points depends on the maximum sedimentation rate ${\mathrm{Stk}}_{f\sigma }/{\mathrm{Fr}}_{\sigma }^2$ and strain parameter $A$. If the strain parameter is less than the critical value, i.e., $A<A_{cr}$, then the three equilibrium points exist within a certain range of variation ${\mathrm{Stk}}_{f\sigma }/{\mathrm{Fr}}_{\sigma }^2$. Otherwise, if $A>A_{cr}$, then there is only one equilibrium point.

3.4. Particle Motion in the Lamb–Oseen Vortex

The distribution of azimuthal velocity in the Lamb–Oseen vortex has the form [28]

$$u_{\phi }(r,\tau )=\frac{\mathsf{\Gamma }}{2\pi r}\left[1-\exp \left(-\frac{r^2}{4\nu \tau }\right)\right]$$

(42)

where $\mathsf{\Gamma }$—vortex circulation, $\nu$—kinematic viscosity, and $\tau$—time.

In its structure, the azimuthal velocity distribution (42) coincides with the corresponding distribution for a three-dimensional stationary Burgers vortex, but to obtain a time-dependent velocity profile, we introduce the scale $\sqrt{4\nu \tau }$, which is a linear measurement of the vortex core at the time $\tau$.

Analysis (42) allows us to draw the following conclusions. At $\tau =0$ there is a velocity distribution induced by an infinitely thin vortex thread, i.e., $u_{\phi }=\mathsf{\Gamma }/(2\pi r)$. When $\tau >0$ a local maximum appears on the profiles $u_{\phi }(r)$, which shifts over time to infinity with a simultaneous decrease in the value of the maximum. At $r<<\sqrt{4\nu \tau }$ the velocity is $u_{\phi }=\mathsf{\Gamma }r/(8\pi \nu \tau )$, i.e., the gas rotates as a solid body with angular velocity $\mathsf{\Gamma }/(8\pi \nu \tau )$. Over time, due to diffusion, the vorticity spreads to the entire space occupied by the gas.

The results of direct numerical simulations of turbulent flows (e.g., [36] and later) indicate that stretched and concentrated tube-like vortices play a prominent role in fully developed turbulence.

In [37], the effect of inertial particles on the stability and decay of a certain prototype vortex tube represented by a two-dimensional Lamb–Oseen vortex was studied. Such a vortex is characterized by a Gaussian distribution of the vorticity.

It is known that, in the absence of particles, there is a strong stability of the Lamb–Oseen vortex to perturbations, with the result that the vorticity and enstrophy decay at a slow rate, which is controlled by viscosity. Mass concentration of particles is $\mathrm{M}=1$ ($\mathrm{M}=\mathsf{\Phi }{\rho }_p/\rho$), which predetermines the presence of the inverse effect of particles on the carrier phase (see Section 2.1). Particle inertia and vortex intensity were varied to obtain Stokes and circulatory Reynolds numbers in the following ranges: ${\mathrm{Stk}}_f={\tau }_{p0}/{\tau }_f=0.1-0.4$ and ${\mathrm{Re}}_{\mathsf{\Gamma }}=\mathsf{\Gamma }/2\pi \nu =800-5000$. Here, ${\tau }_{p0}$—dynamic relaxation time of a Stokes particle; ${\tau }_f=2\pi r_0^2/\mathsf{\Gamma }$—characteristic time of the vortex.

The authors’ simulations using the combined Euler–Lagrange approach showed that the dispersion of inertial particles at relatively low concentrations accelerates the decay of the vortex tube by orders of magnitude. The clusterization process causes inertial particles to be ejected from the vortex core, forming a ring-shaped area of elevated concentration and a “bubble” of hollow fraction, which expands outward. Outward migration of particles causes flattening of the vortex profile, which accelerates vortex decay. This effect is further enhanced by the clustering of particles at small scales, which causes the growth of enstrophy in contrast to its monotonic damping, which takes place in one-phase two-dimensional vortices. The processes described occur at small time scales which are determined by the clustering process, and are two orders of magnitude lower than the time scale determined by viscosity. It is shown that an increase in particle inertia leads to faster vortex decay. The results [37] show that introducing inertial particles into the vortex flow is an effective way to control and suppress vortex tubes.

3.5. Particle Motion in a Flow Induced by Two or More Vortices

The behavior of particles in a flow formed by two or more vortices will be qualitatively different from the case of their motion in a single vortex structure. The case of several vortices or the case of a multi-vortex structure is a kind of next approximation to real turbulence and is of great interest for practical use.

In [38], it was studied how heavy particles characterized by small Reynolds numbers move in a two-dimensional flow induced by two identical point vortices of the same sign, rotating relative to the axis. The axis of rotation passed through the middle of the segment connecting the centers of the vortex. Such flow is the simplest unsteady flow with natural periodicity due to mutual influence of vortices. Particular emphasis was placed on the analysis of the effect of gravity on the behavior of particles. For the purpose of simplicity, the paper did not take into account the inverse effect of particles on the carrier gas and interparticle collisions.

The main characteristic of particle sedimentation is the rate of sedimentation:

$$v_T={\tau }_{p}g/{\mathsf{\Omega }}^{2}d$$

(43)

where $g$—free fall acceleration, $\mathsf{\Omega }=\mathrm{const}$—angular velocity of rotation of vortices relative to the origin of coordinates, and $d$—half of the distance between vortex centers.

The angular velocity is expressed through the vortex intensity as

$$\mathsf{\Omega }=\mathsf{\Gamma }/4\pi d^2$$

(44)

where $\mathsf{\Gamma }$—circulation of each of the vortices in consideration.

The results led to two main conclusions. The first conclusion is that inertia-free heavy particles (${\tau }_p\to 0$, $0<v_T<<1$) injected into the considered current may have chaotic (undermined) trajectories, due to the combined effect of gravity and vortex rotation. This circumstance leads to an increase in the mixing of particles. The second conclusion is that some of the inertial particles with small gravity effects ($v_T<<{\tau }_p=O\left(1\right)$) can accumulate in two special points (attractors) of attraction, rotating with vortices. The noted circumstance, on the contrary, leads to the effect of accumulation (clustering) of particles. It has been shown that the behavior of inertia-free sedimenting particles is chaotic due to the combined action of gravity and circular displacement of vortices. This phenomenon is very sensitive to particle inertia, if it is present. Using the Hamiltonian dynamics of the system theory for the equation of motion of particles written in a rotating coordinate system, it is found that the small inertial terms of the equation of motion of particles strongly modify the Melnikov function for homoclinic trajectories of the unperturbed system. This happens as soon as the relaxation time of the particle becomes of the order of the setup time (the Froude number is of the order of one).

Particles with finite inertia, and in the absence of gravity, are not necessarily centrifuged away from the vortex system. It was found that such particles can have different equilibrium positions in a rotating coordinate system (like the Lagrangian points of celestial bodies) depending on whether their Stokes number is smaller or larger than some critical value. Taking into account the viscosity of the carrier gas led to a qualitative restructuring of the flow. Two vortices merged, forming one vortex. After the coalescence of the vortices, the particles were centrifuged to its periphery.

The authors [39,40] studied the formation of caustics (areas of increased concentration) of droplets when they move in vortices.

In [39], an investigation of the behavior of heavy inertial particles in the flow generated by a pair of vortices of the same sign was carried out. In the system of coordinates rotating together with two vortices, it is found that for heavy inertial particles, the formation of stable stationary points (equilibrium points), forming only in a certain range of Stokes numbers, takes place ${\mathrm{Stk}}_f<{\mathrm{Stk}}_{fcr}$. Here, ${\mathrm{Stk}}_f={\tau }_p\mathsf{\Gamma }/r_{cr}^2$ and is equal in value to the Stokes number determined according to (7). Estimates have been made of this critical Stokes number, ${\mathrm{Stk}}_{fcr}$, and it has been shown that taking viscosity into account slightly increases its value. It has also been found that the rate of particles reaching stationary points increases until the stationary points disappear at ${\mathrm{Stk}}_f={\mathrm{Stk}}_{fcr}$.

The study [39] was further elaborated on in [40], which studied the formation of caustics (clusters) in flows with the predominance of vortices, i.e., for the case of a multi-vortex structure. The calculations have shown that only the particles that start their motion within a certain critical distance from the center of the vortex form caustics. This critical distance is defined as the square root of the product of particle circulation and inertia, i.e., $r_{cr}^{}\approx \sqrt{\mathsf{\Gamma }{\tau }_p}$. It was found that the particles that start their movement in the circular region around this critical radius form the densest clusters in the flow. It was concluded that a significant increase in the concentration of particles with even a small inertia will lead to a significant increase in interparticle collisions.

3.6. Particle Motion in a Closed Vortex Back-Step Gas Flow

In [41], the dispersion of polydisperse particles in a wall-limited swirled flow was studied based on the Euler–Lagrange approach. Such a flow is also called a back-step flow, and is geometrically close to the flows realized in real combustion chambers.

The most well-known models describing particle dispersion have been tested: the Sommerfeld model [42], the MOB model [43], the PDF model of the probability density function [44,45], and the model based on the notion of vortex lifetime ELT [46]. In this work, the emphasis was placed on resolving two features of the flow in question by comparing it with the available experimental data [47]: (1) penetration of medium-sized particles into the corner recirculation zone; (2) distribution of small particles downstream behind the internal recirculation zone.

In accordance with the available experiments [47], the trajectories of eight classes of particles characterized by different diameters, from $d_p=12.5$ were calculated.  m to $d_p=105$ μm, were calculated.

For the considered flow, the inertia of the particles was estimated using Expression (4), where the characteristic timescale of the carrier phase in the averaged motion is ${\tau }_f=l_f/u_f$. According to the recommendations of [48] as a characteristic length, the distance from the entrance to the coordinate of the braking point of the internal recirculation zone ($l_f=73$ mm), as well as the speed of the main jet, were chosen as characteristic velocities ($u_f=12.89$ m/s). The Stokes numbers thus obtained for the eight particle classes described above were in the range of ${\mathrm{Stk}}_f=0.21-14.7$.

Calculations have shown that medium-sized particles (${\mathrm{Stk}}_f\approx 4.8$) have a high tendency to migrate to the corner recirculation zone, compared to other classes of particles. It has also been shown that this migration takes place through a fairly narrow region located between the boundary of the large-scale vortex and the side walls.

The purpose of [41] is to study the interaction of particles with vortex structures and the ability of sophisticated dispersion models to describe the dynamics of particles, including the values of their time-averaged and fluctuation (rms) velocities. The calculation results have shown that the frequently used ELT model leads to the most significant errors (up to 60%) in determining the fluctuation (rms) velocities of particles among all the models tested. However, calculations without models led to even higher errors (up to 80%). The paper clearly shows that particle concentration is the most sensitive parameter compared to time-averaged and fluctuation particle velocities, and can be recommended for testing models.

Studies of the flow of the same configuration were continued in [49,50], where the dispersion of particles in a swirling coaxial flow with four-way coupling was studied using the Eulerian–Eulerian consideration. In these works, a new model has been developed, which describes, on a sub-cell scale, the transfer process of kinetic energy, both of the carrying gas and of the floating particles.

The results of this study showed that the presence in the flow of particles with sizes ranging from $d_p=12.5$ μm to $d_p=105$ μm (as shown above, the Stokes numbers varied from 0.21 to 14.7) has a significant impact on the flow structure and coherent vortex structures. It was also found that interparticle collisions lead to additional dissipation of the kinetic energy of particles.

The proposed model adequately described two-phase coherent structures, the motion of particles of different inertia, and the processes of vortex stretching and decay topologies. Particle accumulation was observed in regions characterized by low vorticity and high shear tensions. Most of the low-inertia particles ($d_p<45$ μm, ${\mathrm{Stk}}_f<2.7$) on the walls in the corner recirculation zone followed the downward lines of the carrier gas current. Larger particles ($45<d_p<60$ μm, $2.7<{\mathrm{Stk}}_f<4.81$) are characterized by greater penetration downstream and interaction with the vortex structures of the internal recirculation zone.

3.7. Particle Motion in a Closed Vortex Gas Flow Induced by Rotating Cylinders

In [51], the accumulation of heavy particles (${\rho }_p/\rho >>1$) in a circular wall-limited vortex was studied, taking viscosity into account. The flow field is induced by a small cylinder of radius $r_S$, rotating not only around its axis with speed $\mathsf{\Gamma }/2\pi r_S^2$, but also around the vertical axis of the considering circular region of radius $R$ with constant angular speed:

$$\mathsf{\Omega }=\frac{\mathsf{\Gamma }}{2\pi }\frac{1}{R^2-r_1^2}$$

(45)

where $r_1$—relative distance from the center of the vortex to the center of the circular region, $r_1/R=0.5$. The relative radius of the small cylinder was selected to be $r_S/R=0.1$. The diameter of the injected particles and the vortex Reynolds number, determined according to (2), varied in the ranges $d_p/R=75\cdot {10}^{-5}-187.5\cdot {10}^{-5}$ and ${\mathrm{Re}}_{\mathsf{\Gamma }}=200-600$, respectively.

Numerical simulations were performed using the Euler–Lagrange consideration based on the discrete element method (DEM) and the immersed boundary method (IBM). In contrast to the vast majority of other studies, this work took into account not only the inverse effect of particles on the vortex flow of the carrier gas (“two-way coupling”), but also the collisions of particles within the walls and among themselves (“four-way coupling”). All of the mentioned collision processes were calculated on the basis of the solid-sphere model.

Calculations demonstrated that in the first case (“one-way coupling”), most of the particles move along spiral trajectories and accumulate in the accumulation point located in the vicinity of the braking point of the vortex flow. This accumulation point represents a stable equilibrium point, since it is at this point that the resistance created by the velocity field of the carrier flow balances the destabilizing centrifugal force acting on heavy particles.

For the second case (“two-way coupling”), the absence of a stable point of particle accumulation due to the strong influence of particles on the dynamics of the carrier gas was revealed. For this calculation case, most of the particles are ejected from the circular region of the rotating gas and accumulate on the wall that limits the flow. It is also shown that the number of particles accumulated on the wall increases with an increasing Reynolds number and an increasing in their diameter. All calculations were performed using three well-known models [52,53,54], which take into account the influence of the increased concentration of particles on their resistance in the carrier gas. It was found that all models describe the effect of particle accumulation well, although there are small quantitative discrepancies.

The study [51] was continued in [55,56]. In [55], the effect of the accumulation of heavy particles in a circular wall-limited viscous vortex flow was studied. The calculations were performed using the Euler–Lagrange consideration. Interparticle and particle–wall collisions were calculated based on the solid-sphere model. The influences of the Stokes number, size, and position of a small rotating cylinder for the case of “one-way coupling” and “two-way coupling” particles on the gas were studied.

The diameter of the injected particles was chosen to be $d_p=500$ μm. The Stokes number determined according to (10) varied in the following range: ${\mathrm{Stk}}_f=0.25-1$. Note that the Stokes number was changed by changing the density of the particle material, i.e., ${\rho }_p=\mathrm{var}$. The relative radius of the small cylinder was varied in the range $r_S/R=0.075-0.2$. The relative distance from the center of the vortex to the center of the circular region was varied in the range $r_1/R=0.25-0.75$.

The calculation results have shown that, for the case of “one-way coupling,” most of the particles gather at the accumulation point, which moves away from the point of the vortex flow braking with increasing Stokes number. However, for the case of “two-way coupling,” it leads to the disappearance of the accumulation point. The higher the Stokes number, the more particles accumulate on the wall in both “one-way coupling” and “two-way coupling.” A small change in the size of the rotating cylinder has little effect on the results, indicating that the rotational flow created by the cylinder has the necessary stability. However, changing the position of the rotating cylinder has a significant impact on the results. Thus, in the case of the removal of the cylinder from the center of the circular region, the accumulation point becomes an ever-increasing accumulation region whose size is comparable with the size of the circular region of the vortex flow. This happens in the case of “one-way coupling.” In both cases, removal of the cylinder leads to weakening of the vortex flow and reduction in particle ejection onto the channel walls.

In [56], the process of accumulation of heavy particles in a circular wall-limited viscous vortex flow created by two small cylinders rotating in one direction and located at the same distance from the center of the circular region under consideration is studied. The effect of Stokes and Reynolds numbers, the intercenter distance between cylinders, and the position of a small rotating cylinder for the case of “one-way coupling” and “two-way coupling” have been studied.

It was found that in “one-way coupling,” there is always a stable accumulation of particles trapped inside the circular region due to the balance of the pressure gradient force, the drag force, and the centrifugal force. In addition, five different particle concentration fields are identified as the two rotating cylinders move away from the center of the circular region in question. However, in “two-way coupling,” the motion of particles in the considered vortex flow becomes unstable, stable enclosed trajectories cease to exist, and accumulation points disappear. The calculations also show that the percentage of particles accumulating on the wall increases with increasing Stokes and Reynolds numbers, and decreases as the intercenter distance between the circular domain and each rotating cylinder increases, for both “one-way coupling” and “two-way coupling.”

3.8. Motion of Polydisperse Droplets in Vortices

At the very beginning of this subsection, it should be noted that calculation-theoretical studies often make no differentiation when carrying out calculations of flows with solid particles and droplets. This is reasonable for small Weber numbers, when there is no difference in the behavior of particles and droplets, and in the case of low concentrations, when there is no hydrodynamic (in traces) and mechanical (collisions) interaction between particles (droplets). Real two-phase flows (in experiments and technical devices) are accompanied by phase and chemical transformations and, as a rule, are polydisperse, i.e., contain particles (drops) of different sizes. These circumstances complicate matters greatly, because the difference in size leads to a difference in velocity, which leads to a tremendous increase in the collision cross section.

The velocity of a single spherical particle $\mathbf{v}$ in a carrier medium (${\rho }_p/\rho \approx O(10^3)$) depends on the velocity of the carrying gas $\mathbf{u}$ and time $\tau$, and is described by the following equation:

$$\frac{d\mathbf{v}}{d\tau }=\frac{3\rho }{4{\rho }_p{d}_p}{C}_D\left|\mathbf{u}-\mathbf{v}\right|(\mathbf{u}-\mathbf{v})+\mathbf{g}$$

(46)

where $\mathbf{v}$—velocity vector of a particle, $\mathbf{u}$—velocity vector of carrier gas, $C_D$—aerodynamic drag coefficient, ${\rho }_{}$ and ${\rho }_p$—densities of the carrier gas and particles (droplets), $d_p$—particle (droplet) diameter, and $\mathbf{g}$—gravitational acceleration. The terms on the right side of Equation (46) describe the aerodynamic drag and gravity forces, respectively.

Substitution of the relation for the aerodynamic drag coefficient into Equation (46) simplifies the latter, in the case of Stokesian particle motion, to the following form:

$$\frac{d\mathbf{v}}{d\tau }=\frac{\mathbf{u}-\mathbf{v}}{{\tau }_{p0}}+\mathbf{g}$$

(47)

where ${\tau }_{p0}$—dynamic relaxation time of a Stokesian particle.

In the case of motion of even initially monodisperse droplets that change their size due to, for example, evaporation and coalescence processes, the dynamic relaxation time will be a function of time, i.e.,

$${\tau }_{p0}(\tau )=\frac{{\rho }_p{d}_p^2(\tau )}{18\mu }$$

(48)

Obviously, this circumstance greatly complicates the study of the motion of droplets in vortex flows.

The behavior of evaporating liquid droplets in vortex media plays an important role in many practical applications. Examples here are, for example, gas turbines and vortex combustion chambers. The interaction of single-phase gas-diffusion flames with the vortex flow field has been thoroughly studied, for example, in [57,58,59].

In [60], the peculiarities of combustion of two-phase fuel (spray with droplets) in an axisymmetric combustion chamber were studied by the Large Eddy Simulation method (LES). Calculations revealed the effect of droplet accumulation in the presence of eddies in large recirculation zones. It was shown that this effect has a determining role on the characteristics of large-scale turbulent flames and predetermines their difference from single-phase gaseous flames.

In [61], using the direct numerical simulation (DNS) method, combustion features near the recirculation zone in a diesel engine were studied. Calculations have shown that droplets larger than the Kolmogorov spatial microscale ($d_p>{\eta }_K$) do form clusters. High density (concentration) of droplets leads to the formation of fuel vapor clusters, which accelerates the formation of the fuel–air mixture.

Droplet clusters, as has long been known [62], are formed in vortex flows, as droplets possessing inertia are accelerated and ejected to the outer region of the vortex. The influence of the droplet clustering process on evaporation has been studied in detail in [63,64]. In [65], a numerical simulation of the interaction of vortex structures with droplets was performed, and a significant influence of the latter on the structure of the evaporating two-phase jet was revealed. The influence of the presence of droplets on the characteristics of the Carman vortex track, using the direct numerical simulation (DNS) method and the original theoretical approach related to the imposition of harmonic oscillations on the initial field of the carrier gas, was studied in [66].

All real gas-droplet streams contain drops of different sizes, i.e., they are polydisperse. Taking this circumstance into account is a nontrivial task. The first works on polydisperse sprays used a simplifying fractional approach. The polydisperse droplets were divided into several fractions, each represented by droplets of the same size. This approach has been used to model both non-reactive and reactive media. There are a number of papers, for example, [67,68,69], in which it has been shown that the fractional approach gives excellent concordance with the available experimental data of various sputtering systems.

Later, however, several papers [70,71,72] clearly showed the limitations of the fractional approach in correctly reproducing the dynamics of inertial particles characterized by high Stokes numbers.

In [73], the evaporation of polydisperse droplets introduced into a two-dimensional unsteady axisymmetric vortex flow was studied. Modeling was based on the Eulerian–Eulerian consideration. The inverse effect of droplets on the vortex flow of the gas phase was taken into account by introducing the corresponding source term into the right parts of the equations, which takes into account the inter-phase momentum exchange as well as the term responsible for the transfer of additional momentum from the vapor generated by evaporation. The equations for droplets of different fractions were written in the Stokes approximation, taking into account losses of linear momentum due to evaporation and growth of impulse due to transition of larger droplets to a smaller fraction, also due to evaporation.

As a result of the analysis in [73], new analytical solutions for the dynamics of the spraying of monodisperse droplets were obtained. Then, the obtained solution was extended to the case of evaporation of polydisperse droplets in a vortex flow. It was shown that the vortex pushes the drops outward, and the mass fraction of liquid decreases in the region close to the vortex core. The droplets acquire a radial velocity that does not exist in the original vortex flow of the carrier gas. It is shown that, for a given initial radial position of the droplets, the maximum radial velocity is reached. A simple model that accurately predicts this maximum velocity for each fraction as a function of vorticity, kinematic viscosity, droplet radial position, and droplet relaxation time was developed.

In [73], it was also found that smaller droplets, which have low inertia, accelerate faster than larger droplets until they reach their maximum radial velocity. However, after the initial stage of acceleration, the larger droplets are pushed further away from the vortex core and eventually acquire a higher radial velocity.

4. Natural Vortex Structures

Below are the results of works that show the need to take into account the two-phase nature of the tornado. The features of particle motion in natural vortex structures and their inverse effect on their characteristics are studied.

4.1. Accounting for the Two-Phase Nature of the Tornado

The study of the motion of the dispersed phase (droplets, particles, debris) in tornado-like vortices is of considerable interest, due to several circumstances given below.

The first circumstance is related to the fact that the presence of the dispersed phase visualizes (makes visible) atmospheric vortices [74,75,76]. Secondly, by measuring the velocity of suspended dispersed inclusions, one can obtain the necessary information about the dynamics (velocity fields) of the air vortex [15,77]. The third circumstance is related to the fact that the latent heat of phase transformations (primarily, condensation and evaporation) during the formation (disappearance) of droplets has a significant impact on the generation process, dynamics, and stability of tornado-like vortices [78,79,80]. Fourth, at certain concentrations, the dispersed phase can have a significant effect on the characteristics of the atmospheric vortex and its behavior (up to disintegration) [81,82,83,84]. The fifth and last circumstance is that the presence of debris and other dispersed inclusions can make a decisive contribution to the negative consequences (destruction and casualties) of a tornado [85,86,87,88,89].

4.2. Particle Motion in a Tornado

In addition to the classic single-cell tornado (radial air inflow and central upward flow) with circular circulation, there are also multi-cell tornadoes. Some of these tornadoes have downward air currents in the core, which may extend to ground level. Multicellular tornadoes are more common than single-cell ones. Tornadoes are characterized by considerable non-stationarity, and their characteristics undergo significant changes during their lifetime. The maximum recorded tangential component wind speeds are 90–110 m/s, but for most tornadoes, the wind speed is much lower. The core radius (distance from the center to the maximum tangential velocity) is in the range of 50–200 m, and the progressive velocities of the tornado funnel are of the order of 5–15 m/s.

There are a significant number of studies that have investigated the behavior of debris from wind-damaged buildings. These works calculated trajectories of debris of different types (“compact” type, “sheet” type, “rod” type) when they move in the flow created by the wind, but did not consider their behavior in vortex atmospheric structures (“dust devil,” tornado, etc.). The results of these studies are systematized in the recently published review [90].

In [91], based on numerical calculations using the LES and the Lagrangian trajectory approach, the features of motion of solid particles of varying inertia in the “dust devil” were studied. Calculations of the motion of dust particles (density 2560 kg/m³) of three different sizes (100 μm, 200 μm, and 300 μm) were performed in the pre-calculated, unsteady field of velocities of vortex air flow. The calculations analyzed the trajectories of 20,000 particles of each size, which were thrown in equal portions (400 particles each) into the lower part of the vortex every 0.1 s. As a result, pictures of the spatial arrangement of all dust particles introduced into the “dust devil” were obtained after 5 s. Analysis of the calculated distributions allows one to conclude that the distribution of particles in space is significantly heterogeneous. The particles with the lowest inertia (100 μm), rising to a height of 25–30 m, are most uniformly located in space. A decrease in the vertical velocity of air with increasing distance from the ground and an increase in inertia leads to the fact that the maximum lift of larger particles is much smaller: 15–20 m and 10–12 m for particles of 200 μm and 300 μm in size, respectively. The uneven distribution of large particles in space is determined by lower values of their velocities due to their intense ejection by centrifugal forces into the area of stationary air, as well as by their processes of turning and beginning to move towards the surface.

In [92], based on numerical calculations, the behavior of trajectories and concentration fields of particles of varying inertia in an axisymmetric flow of a viscous, incompressible fluid, simulating the interaction of a vertical vortex thread with a horizontal plane, was studied. In order to describe the motion of the carrier gas phase, an automodel solution of the Navier–Stokes equations obtained by M.A. Goldshtik was used. Dispersed phase parameters, including concentration, are calculated along selected trajectories using the full Lagrangian consideration. The calculations assumed that the dispersed phase consists of identical spherical particles. The volume fraction and mass concentration of the impurity were assumed to be small, which allowed the researchers not to take into account the influence of the particles on the parameters of the carrier gas.

As a result of calculations in [92], the possibility of numerous intersections of heavy particle current tubes and formation of “folds” in the disperse phase concentration field was shown (Figure 4). For heavy particles (exceeding the density of the carrier phase) $\eta =\rho /{\rho }_p<1$, the formation of a “bowl-shaped” surface of the dispersed phase accumulation and a zone of particle precipitation near the vortex base was detected (Figure 5).

The effect of gravity on the motion of particles was characterized by the Froude number, defined according to (11), where the geometric size is represented by the characteristic length of velocity relaxation of particles under the Stokes law of resistance, defined as

$$l={\left(\frac{m_p\mathsf{\Gamma }}{3\pi d_p\mu }\right)}^{1/2}$$

(49)

where $m_p$—particle mass, $\mathsf{\Gamma }$—azimuthal velocity circulation assumed constant, and $\mu$—dynamic viscosity.

When gravity is taken into account ($\mathrm{Fr}\ne \infty$) the edge of the cup-shaped particle accumulation surface spirals around a circle (Figure 6). The position of this circle is determined by the zero point balance of forces of hydrodynamic nature (drag force), gravitational (gravity force), and inertial (centrifugal force), acting on the particles in the vortex flow. In the case of light particles ($\eta >1$) the action of Archimedean forces leads to the accumulation of particles near the vortex axis. For neutral buoyancy particles ($\eta =1$) the current tubes oscillate with decaying amplitude, touching a narrow, upwardly-expanding axisymmetric surface, inside which the dispersed phase is absent.

Peculiarities of visualization and diagnosis of free concentrated unsteady laboratory vortices were studied in [93]. Dense atmospheric tornadoes are characterized by the presence of a distinct funnel, which has a considerable length, a small diameter and a more or less vertical position [94]. A vortex is the main component of a tornado. It consists of an inner cavity characterized by low air velocity (downward flow) and a rapidly rotating wall (upward flow).

In addition to a funnel, a tornado often also has a cascade [94]. A cascade is a column of dust or a cloud of water spray formed when the tornado funnel touches the ground or water surface, respectively. Wide and low cascades are the most common, although there are narrow and very high cascades, the so-called cascade-folds. The funnel walls, rotating at high speed, suck in sand or water particles, which, due to their high inertia, are ejected beyond the funnel into the area of almost stationary air due to the action of centrifugal forces. Subsequently, these particles sink down due to gravity.

In physical modeling of air tornadoes, it is necessary to make “visible” (visualize) both the vortex and the cascade of laboratory vortex-analogs. Note that the tracer particles that visualize the vortex funnel and almost completely track the motion of the carrying gas must have different inertia from the particles that compose its cascade.

The vortex structures were generated by creating an unstable stratification of air over the underlying surface of an aluminum sheet (diameter—1100 mm, thickness—1.5 mm), heated from below by a gas burner (maximum heat power—3.5 kW). The schematic of the experimental setup and the heating modes that were used are described in detail in [75,76,95,96,97]. In order to visualize the formed vortex structures, magnesia particles (physical density—3900 kg/m³) were used, which were applied by a thin layer to the underlying surface prior to the experiments.

The analysis of the particle size distribution showed that their average diameter is $d_{ps}=6$ μm. The vast majority of particles are smaller than 5 μm, but there are also large conglomerates of particles with sizes of $d_{pl}=50$ μm and more. Prior to the experiments, the inertia of magnesia particles was estimated in the Stokes approximation using Expression (7) in Section 2.3. The following values of characteristic velocity and funnel diameter of laboratory vortices were accepted for the evaluations: $u_{\phi }=1$ m/s and $d_0=0.1$ m. As a result, two values for the Stokes number (${\mathrm{Stk}}_{fs}$, ${\mathrm{Stk}}_{fl}$) for low-inertia ($d_{ps}$) and large particles ($d_{pl}$), respectively, were obtained. They are equal to ${\mathrm{Stk}}_{fs}=0.0014$ and ${\mathrm{Stk}}_{fl}=0.10$.

It follows from the evaluations that low-inertia particles will almost instantly “adjust” to the air speed in the vortex and move in the vortex funnel, while large particles should be “ejected” from the vortex due to the action of centrifugal forces. Figure 7 shows a typical picture of a recorded laboratory vortex. The photo clearly shows the vortex funnel and cascade visualized by low-inertia and large particles, respectively.

The second way to perform visualization of generated vortices is by the use of tiny droplets obtained as a result of steam condensation [74]. Steam is formed by boiling a special liquid (VDLSL5, Velleman, Belgium), which is pre-applied to the underlying surface.

The development of vortex structures, which was observed many times in the experiments, occurred as follows. Due to air rotation, a low-pressure region emerges in which droplets “gather.” Thus, a vortex thread is formed first (see Figure 8a), which is analogous to a dense tornado with a continuous funnel-wall (no inner cavity). Acceleration of rotation of such a vortex leads to an even greater pressure drop in its center. This contributes to penetration of cold air from above into the funnel, which leads to formation of the internal cavity of the vortex—the “eye of the vortex,” which is an analogue of the “eye of the storm.”

Unfortunately, the low resolution of individual frames of the video did not allow for their citation in the article, but the video clearly shows the formation of the inner cavity of the vortex, in which there are no droplets. The authors made the assumption that in this area, there is a downward movement of cold air penetrating from above. The air velocity in this region is low, so the pressure there is higher than in the surrounding wall of the vortex. The high pressure is exactly what forms the inner cavity. In the vortex wall, upward air motion is realized, characterized by high velocity values. High velocity predetermines the presence of a low pressure region, contributing to the involvement of successive portions of warm air originating from the underlying surface.

As the inflow of warm air weakens, the rate of upward flow decreases, and the pressure in the wall increases, resulting in the closure of the inner cavity. Visually, this is expressed by thinning of the vortex thread, its warping (see Figure 8b), and the subsequent breaking.

A two-liquid model of the tornado was developed in [98,99]. The first (primary) fluid is water vapor, which condenses during a sharp pressure jump. The other (secondary) liquid is made up of solid particles picked up by the vortex. Note that [98,99] does not give an extensive description of the mathematical model; however, the results of a numerical simulation of the tornado life cycle and the effect of a tornado on a moving car and a small house are given.

In [100], a mathematical simulation of tornado-like vortices was performed using the Large Eddy Simulation (LES-method). The features of throwing «shells» of two types by a vortex are studied: a wooden board with a mass of 14 kg and a car with a mass of 1810 kg were investigated. As a result of the Lagrangian simulation of the motion of these objects in a three-dimensional unsteady velocity field of a simulated tornado-like vortex, statistical distributions of the maximum values of the horizontal components of their velocities were obtained. These data are of great practical importance for predicting the negative effects of destructive atmospheric vortices on strategic energy facilities, including nuclear power plants.

In [101,102], a simple engineering model was developed that succeeds in describing the distribution of velocities and pressures in a real tornado. It is shown that the new model is able to provide a method for predicting changes in wind speed and barometric pressure over time. The model also allows the calculation of debris impact energy, which can be used to develop a methodology for protecting buildings from tornado wind loads. First, a stationary single-cell tornado vortex was considered. Then, the obtained solutions were generalized to the tornado model consisting of two cells. As a result, the trajectories of debris inside the tornado were calculated and the main dimensionless parameters of the problem under consideration were determined: the tornado twisting parameter, the buoyancy parameter, the Tachikawa number, and the inverse Froude number.

4.3. Influence of Particles on Tornado Characteristics

In [103], a detailed analysis of the effect of solid low-inertia debris on the tornado characteristics was performed. The simulation is based on the Eulerian–Eulerian (two-liquid) model, which implies using equations of the same type to describe the solid and dispersed phases within the framework of interpenetrating media mechanics, taking into account the inverse effect of particles (“two-way coupling”).

Three dimensionless parameters determining the dynamics of debris in the tornado were found: (1) the swirling parameter of the corner flow defining the “type” (structure) of the tornado, $S_c\equiv r_c{\mathsf{\Gamma }}_{\infty }^2/\gamma$ ($r_c$, ${\mathsf{\Gamma }}_{\infty }^2$ and $\gamma$—the characteristic radius of the core over the region of corner flow, the rotation momentum at a large distance from the core, and the loss of flow momentum in the “surface–corner–core” region, respectively); (2) a parameter that is a measure of the relative importance of “centrifuging” debris and which is defined as the ratio of radial acceleration in corner flow to free-fall acceleration, ${\mathrm{A}}_a\equiv u_{\phi c}^2/(r_{c}g)$, where $u_{\phi c}^{}\equiv {\mathsf{\Gamma }}_{\infty }/r_c$—characteristic azimuthal velocity; (3) a parameter that is a measure of the ease of ascent of fragments and is defined as the ratio of the characteristic speed of the tornado to the rate of sedimentation (hovering) of debris, ${\mathrm{A}}_v\equiv u_{\phi c}^{}/w$.

The calculations were performed for a strong tornado, characterized by a peak value of tangential velocity near the Earth’s surface $u_{\phi max}^{}=126$ m/s and a tangential velocity in the funnel $u_{\phi c}^{}=92.8$ m/s.

At first, the tornado parameters were calculated in the absence of particles, and then with the presence of particles. Particle size was varied in the calculations ($d_p=200-2000$ μm), relation of phase densities (${\rho }_p/\rho =2000–8000$), as were some characteristics of the surface. It was found that the sand cloud (tornado cascade) reaches a quasi-stationary state when the mass of airborne particles becomes equal to the mass of depositing one.

As a result of the calculations, the following characteristics of the vortex and tornado cascade were obtained: (1) time of formation of the cascade of debris ${\tau }_D^{}$; (2) total mass of windborne debris $m_{\Sigma D}^{}$; (3) debris cascade height $H_D^{}$; (4) height of the center of mass of the debris cascade $H_{DC}^{}$; (5) distance from the center of mass of the cascade to the tornado axis $R_{DC}^{}$; (6) the value of the maximum tangential velocity of the vortex $u_{\phi max}^{}$; (7) the value of the maximum tangential velocity in the upper part of the tornado $u_{\phi c}^{}$.

The calculations show that for a tornado with relatively low inertia ($d_p=200$ μm, ${\rho }_p/\rho =2000$, $w=1.24$ m/s, ${\mathrm{A}}_v=74.8$) sand particles, the above characteristics took the following values: ${\tau }_D^{}=402$ s, $m_{\Sigma D}^{}=34.7\cdot {10}^6$ kg, $H_D^{}=1440$ m, $H_{DC}^{}=322$ m, $R_{DC}^{}=279$ m, $u_{\phi max}^{}=66.1$ m/s, and $u_{\phi c}^{}=64.9$ m/s.

An important conclusion can be drawn regarding a significant decrease in the maximum vortex velocity due to an intense momentum exchange between the air and the suspended sand particles composing the tornado cascade. Thus, the calculations demonstrated that the accumulation of low-inertia windborne debris near the corner flow in the surface layer can have a significant effect on the strength of the tornado.

As the inertia of particles increased, the time of formation of the tornado cascade, the total mass of particles involved, and the geometric dimensions of a cascade decreased significantly. Thus, for example, in the case of the presence of the largest ($d_p=2000$ μm, ${\rho }_p/\rho =2000$, $w=9.32$ m/s, ${\mathrm{A}}_v=9.96$) sand particles, the above characteristics took the following values: ${\tau }_D^{}=38$ s, $m_{\Sigma D}^{}=2.41\cdot {10}^6$ kg, $H_D^{}=162$ m, $H_{DC}^{}=53.9$ m, $R_{DC}^{}=185$ m, $u_{\phi max}^{}=103$ m/s, and $u_{\phi c}^{}=83.2$ m/s.

Some new mathematical models and numerical methods have been realized and described in Refs. [104,105,106]. Experiments on the effect of vertical and horizontal grid obstacles on the stability of model free unsteady vortices were performed in [107]. It is known that the presence of large particles in a turbulent flow [82,83,84] can lead to the effect of additional turbulence energy generation.

In [107], in order to mitigate the three-dimensional unsteady atmospheric vortex, it was proposed that obstacles in the form of vertical grids consisting of vertical and horizontal elementary turbulence transformers (ETT) be used as a certain analogue of large particles in the two-phase flow. Thus, in the process of interaction of the atmospheric vortex formation with the grid obstacle, the conversion of the large-scale turbulent energy of the vortex into the energy of the secondary small-scale currents takes place.

5. Conclusions

A review of the currently available research on the behavior of particles and droplets in concentrated vortex structures, as well as their inverse effect on the parameters of the carrier gas flow, was performed.

In all of the described works, the calculation of the motion characteristics of the dispersed and carrier phases of the two-phase vortex flow was based on both the Euler–Lagrange and Euler–Euler consideration.

Only a few studies have calculations which take into account the inverse effect of the dispersed phase on the parameters of the carrier flow. In even fewer studies, the role of interparticle (interdroplet) collisions was taken into account. In all of these studies, it was clearly shown that the presence of particles (droplets) can lead to a radical restructuring of the continuous medium flow, and that taking into account the influence of particles (when concentration reaches certain values) is extremely important.

Almost all of the described studies revealed the effect of ejection of particles from vortex centers, as well as their accumulation in regions close to the braking points. To date, the main dimensionless criteria responsible for the behavior of particles and their inverse effect on various vortex structures have been found. The importance of taking into account the three-dimensionality of two-phase vortex flows is shown.

One of the most important goals of two-phase flow research is to develop simple but reliable models of the interaction of small droplets and solid particles with the carrier gas turbulence field. Apparently, successful progress in further improving the models of two-phase turbulent flows will be based on a deep understanding of the relationship between centrifugal forces, aerodynamic drag forces, gravitational forces, and other forces acting on particles during their interaction with the vortex structures of the turbulent field of the carrier gas, characterized by different spatial and temporal scales.

This review makes it possible to formulate the directions for further studies of two-phase concentrated vortex structures:

(1)

study of two-phase vortices in a wide range of changes in the inertia of the dispersed phase (determined primarily by the Stokes number Stk_f) and the intensity of vortex structures (determined by the vortex Reynolds number Re_Γ);

(2)

study of two-phase vortices at moderate and high values of volume concentration, i.e., in the case of the presence of the back influence of the dispersed phase on the carrier gas characteristics (two-way coupling) and the presence of interparticle (or interdroplet) collisions (four-way coupling);

(3)

study of the characteristics of monodisperse and polydisperse particles (droplets) in multi-vortex structures as an approximation to a real two-phase turbulent flow, which is of great practical importance.

In conclusion, we note the obvious need for experimental studies to study the features of the movement of particles of different inertia in concentrated vortices of different types, as well as their reverse effect on their characteristics using modern diagnostic methods (PIV and its numerous modifications).