Peculiarities of Streamline Flowing of Bodies with Droplets

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1		Sergei V. Ryzhkov	--	2908	2023-02-24 08:42:40	\|
2	format	Jason Zhu	-3 word(s)	2905	2023-02-27 04:47:25	\|

This entry is adapted from the peer-reviewed paper 10.3390/sym15020388

The gas dynamics of heterogeneous flows with dispersed admixtures in the form of solid particles or liquid droplets has been one of the most rapidly developing areas of the mechanics of multiphase (two-phase) media. This is due to the numerous engineering applications of such flows (steam generators, facilities for the thermal pretreatment of coal, heat exchangers with two-phase working fluid, sand and bead blasting facilities, dust collectors of different types) and their wide distribution in nature (tornados, dust storms, volcanic eruptions, snow, rain). One of the most important classes of two-phase flows is the flow of gas with an admixture of solid particles or droplets near the limiting surfaces and when flowing around bodies and obstacles.

flow past body gas-dynamic spraying Filtration

1. Introduction

In computational theoretical studies of two-phase flows without walls, often, no distinction is made in calculations of flows with solid particles and droplets. This is justified only 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 trace) or mechanical (collision) interaction between particles (droplets). Real two-phase flows (in technical devices and in experiments) usually contain particles (droplets) of different sizes, i.e., they are polydisperse. In addition, real flows are accompanied by phase and chemical transformations, which change the size of the dispersed phase. The mentioned circumstances greatly complicate the matter, because the difference in size leads to a difference in velocity (and temperature in the case of non-isothermal flow), which leads to enormous growth in the collision cross-section.

Below, the results of research that sheds light on the qualitative differences in the process of interaction of particles and drops with the surface of a streamlined body are presented and analyzed.

There are recently published papers ^[1]^[2]^[3]^[4]^[5] in which various aspects of the collision of single drops with curvilinear surfaces and small bodies of various shapes were studied.

In contrast to a colossal number of studies (e.g., ^[6]^[7]^[8]) where the impact of a droplet against a flat surface was studied, in ^[1], the features of a normal impact of droplets against curved solid surfaces were considered. The physics of droplet impact with curved surfaces was studied for different Weber numbers (

We < 15

), radii of curvature, and surface wettability. Using the original theoretical approach and the axisymmetric lattice Boltzmann method (LBM), it was concluded that all the varying parameters had a great influence on the processes of spreading and bouncing of the droplet. The parametric studies revealed the presence of five interaction modes, from complete deposition to complete rebound. The results of ^[1] provide important information about the structure of the curved surface to control the behavior of the droplet and the time of its contact with the surface.

In research ^[2], the characteristics of the axisymmetric impact of a water drop against a thin vertical dry solid cylinder were studied numerically using the volume of fluid (VOF) method. It is obtained that the surface of the droplet undergoes continuous deformation during impact on a thin cylindrical target, which leads to various stages: free fall, impact, cap formation, encapsulation, opening, and detachment. The ratio of cylinder diameter to droplet diameter (

D_{c} / d_{d}

) was varied from 0.13 to 0.4 in order to observe different droplet deformation patterns. The effect of the boundary angle, ratio

D_{c} / d_{d}

, and dimensionless criteria (

We

Oh

, and

Bo

) on the maximum strain coefficient was studied. The data obtained showed, in particular, that the maximum strain factor increases with an increasing

We

and decreasing marginal angle. The process of the maximum spreading of a droplet due to its collision with a dry stationary spherical particle was studied numerically using the level contour reconstruction method (LCRM) in ^[3]. The Weber number

We = 30 - 90

, Ohnesorge number

Oh = 0.0013 - 0.7869

, drop size to particle size ratio

d_{d} / d_{p} = 0.1 - 0.5

, as well as fluid viscosity and surface curvature were varied in the calculations. Calculations showed that the maximum flowout increases at smaller particles for both capillary and viscous modes. The increase in maximum spreading is mainly determined by surface rim deformation for the capillary mode and viscous dissipation for the viscous mode. An empirical correlation is also presented, which can be applied to the impact of a droplet on both a particle and a flat surface.

Experimental and numerical studies of drop impact on a cone were performed in ^[4]. The Weber number and cone angle were varied. In particular, it was found that in the phase at which the droplet leaves the surface in the form of a ring, its contact time is reduced by 54% compared to a flat surface. The influence of the Weber number and cone angle on the contact time of a droplet with the cone surface was studied.

In Ref. ^[5], numerical and theoretical studies of the behavior of a water droplet upon impact with small cylindrical superhydrophobic targets were performed. The effect of the Weber number and the ratio of the target diameter to the droplet diameter (less than unity) on the droplet impact behavior, including the droplet profile and the deformation coefficient, was investigated. The results show that a larger Weber number accelerates the spreading and droplet fall and promotes droplet decay. Increasing the diameter ratio delays the spreading and droplet fall from the target side, thereby increasing the deformation and rebound of the droplet. Increasing both the Weber number and the diameter ratio contributes to increasing the maximum strain coefficient.

In ^[9], using high-velocity video imaging, the effect of the appearance of droplets with near-zero velocities in the flowing of bodies by gas droplet flows was discovered for the first time. The formation of levitating droplets was due to the emergence of falling and reflected by the model droplets. It has been suggested that the main mechanism of the appearance of droplets with near-zero velocities is an exchange of momentum as a result of the collision of droplets having an opposite direction and velocity values that are close in magnitude. The effect of increasing the size of large levitating droplets due to the merging of falling droplets with them as a result of multiple collisions was found.

Geometrical, kinematic, and temporal characteristics of collision processes accompanying gravitational droplet sedimentation on a model with a hemispherical face were studied in ^[10]. Data on the velocities and sizes of both small (secondary droplets) and large (fragments) droplets in the case of high (close to dynamic Leidenfrost temperature) model temperatures have been obtained. Data were obtained on the effect of droplet size on the velocity recovery coefficient during their interaction with a curved surface. The experiments revealed a decrease in the velocity recovery coefficient with the increasing inertia of the droplets due to a greater loss of momentum, attributed to a longer interaction with the surface. The effect of mismatching the touch and rebound points of the droplets during their interaction with the curvilinear surface of the model was revealed. It was found that with increasing drop size, the effect of gravity increases, which leads to a change in the drop’s rebound to its flow. This effect contributes to an increase in the interaction time of the droplet with the model and the distance between the points of contact and detachment from the surface.

2. Filtration

Traditional air filtration systems often use fibrous filters and are based on the interaction of particles with individual fibers. Several filtration mechanisms are usually distinguished: inertial, blocking, diffusion, and electrostatic. For a droplet of the size of the order of a few micrometers, the inertial mechanism is considered to be the dominant filtration mechanism ^[11]. The collision of a droplet with a single fiber is determined by two criteria—the Stokes number

S t k_{f D}

S t k_{f R}

, which is responsible for the droplet’s ability to follow the carrier air current lines, and the droplet’s Reynolds number

R e_{d}

in the oncoming flow, on which its aerodynamic drag depends. Numerous studies have shown (e.g., ^[12]) that the droplet trapping efficiency increases with increasing

S t k_{f D}

, since, in this case, the particles cannot move around the cylinder. It is also known that at a fixed Stokes number

S t k_{f D} = c o n s t

, the droplet entrapment efficiency decreases with increasing Reynolds number due to an increase in the drag force acting on the droplets. The described classical theory of the single-fiber model is widely used to calculate droplet trapping by fiber filters (e.g., ^[13]^[14]). A limitation of the single-fiber model is that it does not account for changes in the flow field by neighboring fibers and the real distribution of droplets behind the fiber downstream.

In ^[15], the effect of the upstream cylinder on the efficiency of droplet capture by the test cylinder was investigated. It is shown that the efficiency of droplet capture by the test cylinder depends on its relative displacement in the transverse direction (normal to the flow) relative to the front cylinder. At relatively small displacements, there is complete shielding, i.e., the droplets do not collide with the test cylinder. As the displacement increases, the droplet trapping efficiency increases and exceeds the corresponding value for an isolated cylinder at medium displacements and a Stokes number close to unity. At larger displacements and larger Stokes numbers, the effect of the front cylinder on aerosol capture by the test cylinder decreases and eventually disappears. The main achievement is the detection of extremely high trapping ratios (over 100%) and the determination of the ranges of Stokes numbers and displacements at which this effect is observed.

3. Icing

One of the most well-known and serious safety problems in modern aviation is aircraft icing in flight ^[16]^[17]^[18]^[19]. Supercooled water droplets contained in clouds under certain conditions can freeze, hitting the nose part of the fuselage, wings, elements of the fins, and parts of aircraft engines. The formation of the ice crust leads to a number of negative consequences—changes in the streamline regime, reduced wing lift, loss of thrust, reduced controllability, weight increase, etc. Note that ice formation on the surface of the compressor inlet guide apparatus and the nacelle shell of an aircraft engine can occur during adiabatic air expansion, even at positive ambient air temperatures. Subsequently, the ice crust can collapse and enter the engine, causing possible damage to the compressor blades and even engine failure.

There are three basic types of ice—loose ice, transparent (glassy) ice, and mixed ice ^[17].

Loose ice has a brittle, porous, milky white structure, which is a mixture of tiny ice particles; it is formed by the crystallization of small supercooled droplets on the leading edges, when the amount of water after the beginning of the solidification process is not sufficient to form a continuous water layer. The use of de-icing systems prevents the formation of such ice or easily removes it if it occurs.

Transparent (glassy) ice has a smooth surface; it forms when large (diameter greater than 20 μm) supercooled droplets freeze on the contour of the streamlined profile, when the solidification of an individual droplet on the surface occurs gradually and some of the surface droplets have time to spread on the surface before freezing. When heat is applied, such ice forms new droplets and streams that migrate downstream and form new ice in the form of ridged growths (“barrier ice”).

Mixed ice is a combination of loose and glassy ice; it is formed by the presence of droplets of different sizes in clouds. In its pure form, loose ice is formed when FV moves in high cumulonimbus clouds, and glassy ice is formed in low layers in rain clouds.

Taking into account the existence of different types of ice and using the concept of the frozen ice fraction, three icing modes are distinguished ^[18]—wet, liquid, and dry.

Wet mode is realized when the surface temperature is equal to the water crystallization temperature. In this case, the fraction of frozen ice varies from zero to one. The supercooled droplets that have fallen to the surface combine to form large droplets. Large droplets partially solidify and are transformed by air flow into streams or films, under which a thin ice layer is formed.

Liquid mode is characterized by the fact that the temperature of the streamlined surface is higher than the solidification temperature of water. In this case, the proportion of frozen ice is zero. Water is present on the surface in the form of drops, streams, or a film. Ice on the surface melts in the water film.

Dry mode is realized when the temperature of the outer ice layer is lower than the crystallization temperature of water. In this case, the fraction of frozen ice is equal to one. The supercooled droplets precipitated on the surface solidify, being transformed into loose ice and not reaching a sufficiently large size to be carried away by the external flow.

In study ^[19], the results of calculating the icing of the cylinder and the NACA 0012 profile when flowing with a viscous compressible air flow carrying droplets in a two-dimensional formulation are presented. The developed model takes into account the interaction of the carrier phase and droplets, the form of existence of moisture on the profile, the mode of ice build-up, and changes in the geometry of the streamlined body. As a result of the calculations, the characteristic shapes of ice films (growths) for various modes of icing—dry, wet, and mixed—were obtained.

In paper ^[20], the main physical processes accompanying the flow of bodies in a supercooled cloud are studied. The interaction of water nanodrops consisting of a different number of molecules (54, 159, 349, 647, and 1080) with the surface, which can be considered as the beginning of the icing process, has been studied by the method of molecular dynamics. It is shown that with a decrease in the size of the nanocapsule and an increase in the potential energy of the interaction of droplet molecules with surface wall molecules, the wetting edge angle and the size of the contact area increase. It is concluded that an increase in the wetting angle leads to an increase in the velocity of the droplets and prevents them from freezing.

One of the most unpleasant forms of icing is the so-called barrier ice, which is formed, as a rule, as a result of the controlled melting of ice formed on the leading edge of the wing. Efforts to address it are complicated by the fact that the areas of its formation are unknown in advance and poorly predictable compared to the locations of formation of ordinary ice (leading edges of the wing, stabilizer, etc.).

At the very beginning of the study of the problem of icing, concerns were expressed about finding a material to which ice would not stick ^[21]. There was an attempt to use fluorocarbon materials (for example, Teflon), well known for their water-repellent properties, in this capacity. However, studies have shown that the adhesion of Teflon with respect to drip ice (formed by droplets moving on the surface) does not differ from the adhesion of other materials. A further goal of the researchers was to search for a material with low adhesion to ice formed when a two-phase flow flows around the FV, i.e., under other physical conditions ^[22]^[23]^[24]^[25]^[26]^[27]^[28].

In ^[21], the results of an experimental study of the effectiveness of using nanomodified surfaces to combat icing are presented. The conditions of barrier ice formation have been studied experimentally. The flow rate in the experiments was 80 m/s; the temperature was minus 20 °C; the water content (mass water content) was 0.57 g/m³. The model was a wing profile: the nanomodified sample under study was installed on top of the profile, and an ordinary plate of untreated duralumin was placed below for comparison. The leading edge of the wing was heated by an ohmic heater. The characteristic appearance of the barrier ice formed is schematically. The effect of the periodic self-cleaning of superhydrophobic surfaces from ice was found under the same conditions in which an ice barrier grows on the surfaces of ordinary materials. The marginal wetting angle of superhydrophobic surfaces was, at the same time, more than 160°. This effect and direct measurements of the adhesion strength confirmed its significant decrease in the case of the nanomodified superhydrophobic material.

In Ref. ^[29], a physical and mathematical model of a cooling and solidifying liquid film entrained by air along the heated surface of a streamlined body with a given distribution of heat flux density on the surface is developed. In ^[30], the motion of nonspherical particles in a two-phase flow was studied. The coefficients of recovery of the velocity components of ice crystals colliding with the surface of a solid body in a wide range of values of the control parameters of the process have been calculated. In ^[31], a physical and mathematical model of the spatial and temporal evolution of an ice layer growing during the collision of individual droplets with the surface of a solid body, sliding along the surface and solidifying on it, which leads to two-dimensional roughness (bumpy ice), was developed. In ^[32], the effect of ice crystals in the air flow on the evolution of barrier ice on the surface of a wing model in an aero-cooling tube was investigated numerically. The results confirm that the change in the mass of ice deposits detected in the experiment when crystals are introduced into the flow is associated with the absorption of a film of water formed on the surface of the solid, part of the crystals’ mass at low flow rates (increase in barrier ice), and the spilling of the film at high speeds (decrease in barrier ice).

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Aleksey Yu. Varaksin

Sergei V. Ryzhkov

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1. Introduction

2. Filtration

3. Icing

References