Vortex Phenomena in Planetary Atmospheres and Dusty Plasma

#### **Chapter 7**

## Vortex Dynamics in the Wake of Planetary Ionospheres

*Hector Pérez-de-Tejada and Rickard Lundin*

#### **Abstract**

Measurements conducted with spacecraft around Venus and Mars have shown the presence of vortex structures in their plasma wake. Such features extend across distances of the order of a planetary radius and travel along their wake with a few minutes rotation period. At Venus, they are oriented in the counterclockwise sense when viewed from the wake. Vortex structures have also been reported from measurements conducted by the solar wind-Mars ionospheric boundary. Their position in the Venus wake varies during the solar cycle and becomes located closer to Venus with narrower width values during minimum solar cycle conditions. As a whole there is a tendency for the thickness of the vortex structures to become smaller with the downstream distance from Venus in a configuration similar to that of a corkscrew flow in fluid dynamics and that gradually becomes smaller with increasing distance downstream from an obstacle. It is argued that such process derives from the transport of momentum from vortex structures to motion directed along the Venus wake and that it is driven by the thermal expansion of the solar wind. The implications of that momentum transport are examined to stress an enhancement in the kinetic energy of particles that move along the wake after reducing the rotational kinetic energy of particles streaming in a vortex flow. As a result, the kinetic energy of plasma articles along the Venus wake becomes enhanced by the momentum of the vortex flow, which decreases its size in that direction. Particle fluxes with such properties should be measured with increasing distance downstream from Venus. Similar conditions should also be expected in vortex flows subject to pressure forces that drive them behind an obstacle.

**Keywords:** vortex Venus plasma wake, solar wind-Venus interaction, plasma acceleration in the Venus wake

#### **1. Introduction**

Measurements conducted with the Pioneer Venus Orbiter (PVO) and the Venus Express spacecraft (VEX) around Venus have provided evidence on the existence of vortex structures in the Venus plasma wake [1–5]. Much of what has been learned led to estimate the (~ 1 RV) scale size of those features across the wake, which are shown in the left panel of **Figure 1** with a view of the velocity vectors projected on the plane transverse to the solar wind direction. The flow pattern corresponds to a vortex gyration of the velocity vectors of the solar wind H+ ions, which is also accompanied by a similar distribution of the velocity vectors of the planetary O+ ions that have been dragged by the solar wind from the Venus upper ionosphere. Comparative indications on the presence of vortex structures in the Mars plasma environment have also

**Figure 1.**

*(Left panel) Velocity vectors of H +* ≈ *1–300 eV ions measured with the VEX spacecraft in the Venus near wake projected on the YZ plane transverse to the solar wind direction. Data are averaged in 1000 × 1000 km columns at X <* −*1.5 RV (adapted from Figure 4 of [2]). (Right panel) Average direction of solar wind ion velocity vectors across the Venus near wake collected from many VEX orbits and projected in cylindrical coordinates [7].*

been inferred from the observation of ionospheric-sheath boundary oscillations from the MAVEN plasma data [6]. In this case, measurements suggest Kelvin-Helmholtz oscillations with a periodicity of ~3 min but that have not yet completed a full vortex turn. At Venus, there is evidence of a reversal in the direction of the velocity vectors of plasma particles that lead them back to the planet in the central part of the Venus wake as is shown in the right panel of **Figure 1**.

Evidence of vortex structures in the Venus wake is also available from changes in the magnetic field direction in the VEX measurements. In previous reports [4, 8], it has been pointed out that at the time when a vortex structure is identified by a local increase of the plasma density in the Venus wake, there is an accompanying

#### **Figure 2.**

*Energy spectra of the H+ and O+ ions (upper panels) with measurements in the Sept 26–2009 VEX orbit where there are strong oscillations in the magnetic field components between 02:05 UT and 02:30 UT (bottom panel). In that time span there are enhanced density and speed values of the O+ ions (third and fifth panels).*

*Vortex Dynamics in the Wake of Planetary Ionospheres DOI: http://dx.doi.org/10.5772/intechopen.101352*

decrease of the local magnetic field convected by the solar wind with distinct changes in the orientation of its components. A useful example with such properties is reproduced in **Figure 2** to show evidence of sudden changes in the orientation of the magnetic field components between 02:00 UT and 02:30 UT (bottom panel) when the plasma density and speed of planetary particles in the Venus wake show enhanced values (third and fifth panels). Despite the fact that the data has provided notable information on the general characteristics of vortex structures, there remains to address important aspects related to their origin and to the dynamics of their motion. These issues will be examined by considering the various fluid forces that are imposed by the solar wind on the planetary ions that stream in the wake. The description to be addressed here refers to the Venus observations where there is ample evidence on the geometry, distribution, and time variations of vortex structures. Despite the presence of small fossilized magnetic field areas in the Mars surface [9], much of the information for the Venus wake will apply to plasma vortices that stream along the Mars plasma wake.

#### **2. Fluid dynamic forces in the Venus wake**

A dominant feature in the motion of the solar wind particles that stream around the Venus ionosphere is that different from the pile-up of magnetic field fluxes convected by the solar wind over the dayside hemisphere, the plasma experiences local heating processes when it moves by the terminator of the Venus ionosphere. Indications of that plasma heating were first obtained from the Venera measurements through crossings of that spacecraft across the Venus wake with enhanced plasma temperature values along the flanks of the Venus ionosheath and that is reproduced in **Figure 3** [10, 11]. The heating derives from dissipation processes produced by the transport of solar wind momentum mostly over the Venus polar ionosphere where the local pile-up of the solar wind magnetic field fluxes is not strong. A suitable added information in the data of **Figure 3** is the presence of a plasma transition in the temperature and speed profiles (at ~02:00 Moscow time), which together with the vortex structure shown in **Figure 1** are unrelated to the dynamics expected from sling shot effects produced by the magnetic field fluxes draped around Venus [12]. While measurements have shown an antisolar directed motion of planetary ions in the Venus wake, a vortex flow configuration like that shown in **Figure 1** is more complex than that expected from a slingshot geometry.

As a result of the enhanced plasma temperature values, the solar wind expands by thermal pressure forces and then moves into the Venus wake from both polar regions. An implication of that displacement is that there are two separate flows of plasma particles reaching the central wake from two opposite directions along the Z-axis. They move from both polar regions where the planetary O+ ions are first subject to low values of the rotation of the Venus ionosphere, and then they are displaced to equatorial latitudes where the rotation speed of the Venus ionosphere is larger. Since both plasma flows are also streaming along the X-axis following the solar wind direction, there should be a Coriolis force that deflects them in opposite directions along the Y-axis. For both flows the deflection of the particles should not be in the same sense since in the northern hemisphere the particles will move in the –Z direction and in the southern hemisphere in the +Z-direction. In addition to such opposite deflection along the Y-axis, the streaming particles will also be influenced by the effects of the aberrated direction of the solar wind and a general Magnus force that drive all planetary ions in motion around the planet with a velocity component directed in the +Y sense ([13], see their Fig. 15; [14]). Since that effect is contrary to the direction of motion along the -Y sense imposed by the

#### **Figure 3.**

*Ion speed and temperature measured along the orbit of Venera 10 on Apr. 19, 1976. The Venera orbit in cylindrical coordinates is shown at the top. The temperature burst at position 1 was recorded during a flank crossing of the shock wave. The boundary layer is apparent by the increase in temperature and decrease in speed and is initiated by the intermediate transition at position labeled 2. The discontinuity in the boundary layer temperature profile corresponds to the boundary of the magneto-tail. (from [10]).*

Coriolis force to the O+ ions that stream in the southern hemisphere, their resulting total velocity will be smaller than that of the O+ ions in the northern hemisphere where the velocity components implied by the Coriolis and the Magnus force are oriented in the same sense along the +Y direction.

An implication of that velocity difference between both hemispheres is that the momentum flux of the O+ ions along the Y-axis in the southern hemisphere will be smaller than the momentum flux of the O+ ions in the northern hemisphere. Consequently, a fraction of the momentum flux of the O+ ion fluxes that move north along the Z-axis in the southern hemisphere from the polar region (derived from the enhanced thermal pressure force at that region) will be transferred to that in the Y-sense to compensate for the smaller values of their momentum flux with respect to the larger Y-directed momentum flux of the O+ ions that stream in the northern hemisphere. Thus, in addition that the larger momentum flux of the O+ ions in the XY plane in the northern hemisphere over that of the O+ ions in the southern hemisphere, there will also be smaller values for the momentum flux of the O+ ions that move north along the Z-axis in the southern hemisphere. Under such conditions the momentum flux of the O+ ions that are directed south in the northern hemisphere will be dominant over that directed north in the southern hemisphere. As a result, the motion of the O+ ions in the northern hemisphere will force the vortex structure to be displaced toward the –Z direction. Such an effect

*Vortex Dynamics in the Wake of Planetary Ionospheres DOI: http://dx.doi.org/10.5772/intechopen.101352*


#### **Table 1.**

*VEX coordinates along the X, Y, and Z-axis (in RV) together with the speed v of the planetary O+ ions (in km/s), measured during the inbound (left columns) and the outbound (right columns) crossings of a vortex structure in selected VEX orbits across the Venus wake (the last two columns are the extent of the vortex structure measured in the X and in the Z directions in RV).*

agrees with the data of the VEX position of the vortex structures measured in the XZ plane during the 2006 and 2009 orbits listed in **Table 1** and that is also represented in their profiles in **Figure 4** showing how they become directed to lower –Z values with increasing distance downstream from Venus ([8], see their **Figure 3**). As a result, the outcome of the different momentum flux of the planetary O+ ions in the northern and in the southern hemispheres in the Venus wake is related to the

#### **Figure 4.**

*Position of the VEX spacecraft projected on the XZ plane during its entry (inbound) and exit (outbound) through a corkscrew plasma structure in several orbits. The two traces correspond to four orbits in 2006 and four orbits in 2009 listed in Table 1 [8].*

effects produced by the Coriolis and the Magnus force and that lead to the southbound displacement of the vortex structures shown in **Figure 4**.

#### **3. Cross section of the vortex structures along the Venus wake**

Data corresponding to the 2006–2009 orbits discussed in **Figure 4** have been further addressed to examine the shape of the vortex structures along the Venus wake with results that are presented in **Table 1**. We have separately collected information obtained for the inbound (left side) and for the outbound (right side) columns in each set of orbits. Values for the extent between both crossings along the X and Z-axis are indicated in the two last columns.

A notable aspect of the data in **Table 1** is that the values of the X-coordinate for the inbound and the outbound crossings of the four orbits during 2006 are larger than those for 2009. The implication is that the vortex structure is located closer to Venus during conditions approaching the solar cycle minimum by 2009. The same conclusion has also been inferred from the relative position of the 2006 and 2009 profiles in **Figure 4**. At the same time, the values of the X-coordinate during the inbound crossings in all eight orbits of **Table 1** (left side **c**olumns) are larger than those of the outbound crossings. Such difference derives from the tilted orientation of the trajectory of the VEX spacecraft on the XZ plane, which is directed toward Venus from the wake as it moves from the southern to the northern hemispheres (the inbound crossing of VEX is encountered at a larger distance downstream from Venus in the southern hemisphere as it then moves to a closer distance to Venus during its outbound crossing in the northern hemisphere).

A detail calculation has now been conducted to the data in the orbits of **Table 1** to estimate the width and the location of the vortex structures measured in both the 2006 and the 2009 orbits. The results are shown in **Figure 5** and indicate that there is a tendency for the vortex structures in the 2006 orbits to occur farther away from Venus than those in the 2009 orbits. At the same time there is an indication that the time width ΔT between the inbound and the outbound encounters marked by the vertical coordinate in **Figure 5** occurs at smaller values in the 2006 orbits, which trace the wake farther away from Venus. Such would be the case in a corkscrew flow configuration in fluid dynamics where its width becomes smaller with downstream distance from an obstacle as is represented in **Figure 6**.

#### **Figure 5.**

*Segments measured between the inbound and the outbound crossings of vortex structures by VEX in the eight orbits included in Table 1 (they are identified by changes in the particle flux intensity measured in the energy spectra of the O+ ions). Their position along the X-axis shows that the 2009 orbits (marked in blue) are located closer to Venus and that the width of the 2006 orbits (marked in red) is smaller since they have lower* Δ*T values and are encountered further downstream along the wake.*

*Vortex Dynamics in the Wake of Planetary Ionospheres DOI: http://dx.doi.org/10.5772/intechopen.101352*

#### **Figure 6.**

*View of a corkscrew vortex flow in fluid dynamics. Its geometry is equivalent to that of a vortex flow in the Venus wake with its width and position varying during the solar cycle. Near minimum solar cycle conditions, the vortex is located closer to Venus (right side) and there are indications that its width becomes smaller with increasing distance downstream from the planet. Such is the case for the 2006 orbits (marked in red) traced in Figure 5 and that were conducted before the solar cycle minimum at 2009–2010, thus implying that the vortex flow becomes thinner when it is detected further downstream along the wake.*

A view that can be proposed from such difference is that the width of the vortex structures is larger during the 2009 orbits (when they are located closer to Venus) rather than in 2006 (when they occurred further downstream). A general description of vortex structures measured in the Venus wake is that as shown in **Figure 4,** they are formed closer to Venus during minimum solar cycle conditions by 2009 and at the same time they are wide features. Different properties are encountered as the solar cycle progresses since they will now be formed further downstream from Venus and as shown in **Table 1** and in **Figure 5,** they will now extend across a smaller cross section within the wake. More extended calculations are still required to examine the geometric properties of the vortex structures listed in the 20 VEX orbits reported by Pérez-de-Tejada and Lundin [8]. In particular, it will be necessary to address whether the width of the vortex structures becomes narrower when they are measured further downstream from Venus. It was pointed out in that report that the width of the vortex structures becomes narrower when they are measured further downstream from Venus.

Thus, the thickness of the vortex structures gradually decreases with distance downstream from Venus and eventually fade away and diffuse with the solar wind plasma. Further studies of more extended data are required to examine the evolution of the vortex structures far downstream along the Venus tail. A view of the distribution of vortex structures along the Venus wake as they follow the motion of plasma particles can be inferred by comparing their cross section formed between the inbound and the outbound crossings on the XZ plane for the different orbits in **Table 1**. The result of that comparison is presented in **Figure 7** where the width ¨Δ¨ of the vortex structures along the Z-axis is compared with that of their extent ¨δ¨

#### **Figure 7.**

*Values of the extent ¨*δ*¨ of vortex structures along the wake (X-axis) and their width ¨*Δ*¨ along the Z-axis as derived from the data in Table 1. Thin features (small ¨*Δ*¨ values) are obtained in the 2009 orbits while wider vortices correspond to the 2006 orbits.*

along the X-axis. It is notable that this latter quantity is correlated with their width along the Z-axis implying that thin vortices have a shorter extent along the wake and that wide vortices have a larger extent in that direction. A peculiar characteristic of the ¨δ¨ and the ¨Δ¨ values in the trace shown in **Figure 7** is that **Δ >** δ (by comparison, a linear relation between them is shown by the straight line for the case in which they have the same value). Thus, as is indicated in the last two columns of **Table 1,** there is a tendency that along the VEX trajectory the vortex structures have a larger width along the Z-axis (that difference may be due to the tilted orientation of the VEX trajectory on the XZ plane or to enhanced Δ values produced by an asymmetric shape of the vortex structure in that plane).

#### **4. Velocity values of plasma particles along the Venus wake**

In **Table 1,** there is evidence that the speed values of the planetary O+ ions vary between 15 and 30 km/s by the inbound and the outbound crossings of vortex structures in the 2006 and 2009 orbits. Such values correspond to measurements conducted along the sun-Venus direction (X-axis) and thus are not related to changes produced by the vortex motion whose speed values vary across the wake. Vortex motion involves speed values of the order of ~200 km/s, which are derived from the transit time of particles around structures with a 1 RV planetary radius during a ~ 3 min rotation period T [6]. With such high-speed values the plasma particles contain a large fraction of the momentum flux brought in by the solar wind and that has been employed to produce the vortex motion that they follow within the wake. As noted above, the vortex features are displaced in a consistent unified manner along the wake with much smaller speed values.

In addition to the different speed of the vortex structures and that of the particles that move within them, it should be noted that vortex motion marks a response different from that expected from motion produced by the convective electric field of the solar wind. Rather than following the directional motion of the solar wind with a gyrotropic trajectory at nearly the same speed as is predicted in that view the available momentum flux is employed to produce, instead, an alternate vortex flow configuration that is displaced coherently with more moderate speeds.

Despite the fact that there is no indication in the data of **Table 1** on the manner in which the speed values of the vortex structure change with distance along the Venus wake, it is possible to obtain that information from the varying values of the width ¨Δ¨ of the vortex structures that is implied from those obtained

#### *Vortex Dynamics in the Wake of Planetary Ionospheres DOI: http://dx.doi.org/10.5772/intechopen.101352*

during 2006 and 2009. From the data in **Table 1** we can assume Δ = 0.765 RV as the average value for the width of the vortex structures during the 2006 orbits and Δ = 0.532 RV for that obtained in the 2009 orbits. Since such change implies a scale size decrease by a factor of 0.50% in the area ¨A¨ of the vortex structures, we can apply conservation of mass flux ρv 2 A = cst to suggest that their speed will increase to nearly double values assuming that the plasma density remains the same. By applying this procedure, it is possible to argue that as the vortex structures become narrower with downstream distance from Venus, the kinetic energy of planetary particles that stream along the wake will be increased as a result of the decreased values of the cross section of the vortex structures. Thus, it should not be unexpected to measure higher-speed planetary ions moving in the far reaches of the Venus wake.

While most measurements of the vortex structures reported in **Table 1** are applicable to conditions that occur near the midnight plane of the Venus wake (small values of the Y-coordinate in the data of **Figure 2**), there are a few cases (orbits Sept 25–26 in 2009 and Aug 28 in 2006 in **Table 1**) where such structures are encountered at large distances away from that plane; that is, with larger Y-values. Those features also involve enhanced values of the O+ ion density values separated from the Venus ionosphere and that are included in the corresponding panels of figures like those shown in **Figure 2**. In particular, the location where the enhanced density

#### **Figure 8.**

*Projected orientation of VEX orbits on the ecliptic plane XY with data of vortex structures in Table 1 during Sept 252,009 (labeled A) and Sept 262,009 (labeled B). The inbound and the outbound VEX crossings have the subscript 1 and 2 for each orbit. In both cases the traces are not directed along the midnight plane but have been shifted in the +Y direction following the effects of the Magnus force on the rotating Venus ionosphere (from [14, 15]).*

values of the O+ ions are observed in those orbits has now been placed in the XY (ecliptic) plane in **Figure 8** to show that they are also shifted to the +Y direction. That region coincides with that where other features produced by the interaction of the solar wind with the Venus ionosphere are shifted in that direction (transterminator plasma flow, [16]; ionospheric polar channels, [15]).

Since polar plasma channels are mostly distributed near the midnight plane of the Venus wake rather than by its flanks [15], vortex structures should follow them and as shown in the data of **Table 1** maintain small Y values. However, vortex structures should also be subject to the effects of the aberrated direction of the solar wind and also from the fluid dynamic Magnus force both being directed in the +Y direction. **Figure 8** shows the position of the inbound and outbound VEX encounters of vortex structures along the Sept 25, 2009 and also Sept 26, 2009 orbits that trace the wake with a ~ 10° angle from the midnight plane. It should be noted that both crossings of each orbit occur nearly at the same position in the XY plane despite the fact that they are located at different values along the Z-axis. Thus, vortex structures develop as a near planar structure. Different conditions are encountered in the August 24, 2006 orbit where a wide vortex structure extends along all three coordinates. In fact, similar distances are measured between the inbound and the outbound crossings in the X and in the Y axis. As a result, the structure should not be viewed as a feature that mostly extends in the X direction but that equally applies along the Y direction. Thus, vortex structures may turn out to be complicated features distributed along the wake.

#### **5. Summary of results**

A basic issue that is ultimately responsible for the fluid dynamic interpretation employed to account for the motion of the plasma particles within vortex structures is related to their acceleration. Rather than solely applying the convective electric field E = V × B of the solar wind along their trajectories (V and B being its velocity and magnetic field intensity), it is remarkable that other sources are required to account for the complicated features that are measured. In particular, slingshot trajectories applied to the ionospheric plasma by the magnetic polar regions of Venus and Mars are not sufficient to explain the intricate configuration that is produced from the particle motion and that gives place to the complex vortex flow configuration of their velocity vectors indicated in the left panel of **Figure 1**. A more internal contact between the solar wind and the planetary ions is necessary to deflect their streamlines in a manner that the projection of the velocity vectors on the YZ plane accounts for the peculiar aspect of a vortex flow.

At the same time, while the convective electric field of the solar wind is useful to describe differences in the density and speed of the accelerated planetary ions between the hemispheres where it has a different direction away or toward the planet [12], it is not sufficient to justify the generation of a sharp plasma boundary as that reproduced in **Figure 3** from the Venera measurements. Charge exchange activity between the solar wind hydrogen ions and heavy planetary particles is not satisfactory because it is unrelated to the notable temperature increases reported in those measurements. Instead, concepts based on a fluid dynamic approach rely on arguments that seem to be more accessible to a different acceleration process by invoking wave-particle interactions as the origin of the manner in which both particle populations share their properties. Such condition is expected from the oscillations and fluctuations in the direction and magnitude of the magnetic field measured around the wake [17, 18] and serves to produce the transfer of statistical properties among both plasma populations.

*Vortex Dynamics in the Wake of Planetary Ionospheres DOI: http://dx.doi.org/10.5772/intechopen.101352*

In terms of that description it is of interest to note that it has been useful to account for the following main aspects discussed in this report and that are related to: (i) There is a correlation indicated in **Figure 7** between the width Δ and the extent δ of the vortex structures along the Z axis and the X axis, with Δ > δ values implied from the data of **Table 1**; (ii) as noted in **Figure 4,** vortex structures are measured closer to Venus near solar minimum conditions by 2009; (iii) a notable property in the distribution of vortex structures in the Venus wake is the tendency of their width to become smaller with increasing downstream distance from Venus as can be inferred from the position of their segments in **Figure 5**. That difference implies that the thickness of the vortex region decreases along the wake and thus is reminiscent of a corkscrew flow in fluid dynamics represented in **Figure 6**; (iv) an important consequence in the shape of that region is that mass flow conservation across the vortex structure implies larger speed values of particles that move along the wake (particles with larger speeds should be detected far downstream from Venus); (v) as noted above in **Figure 8,** planetary O+ ion fluxes can also be significantly shifted along the Y-axis in response to effects produced by the aberrated direction of the solar wind and the Magnus force on the motion of planetary O+ ions that are dragged by the solar wind.

#### **Acknowledgements**

We wish to thank Gilberto A. Casillas for technical work provided. Financial support was available from the INAM-IN108814-3 Project.

#### **Author details**

Hector Pérez-de-Tejada1 \* and Rickard Lundin2


\*Address all correspondence to: hectorperezdetejada@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

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[3] Pérez-de-Tejada H et al. Plasma vortex in the Venus wake. Eos. 1982;**63**(18):368

[4] Pérez-de-Tejada H, Lundin R, Intriligator D. Plasma vortices in planetary wakes, Chapter 13. In: Olmo G, editor. Open Questions in Cosmology. Croatia: INTECH Pub; 2012

[5] Pérez-de-Tejada H et al. Solar wind driven plasma fluxes from the Venus ionosphere. Journal of Geophysical Research. 2013;**118**:1-10. DOI: 10.1002/ 2013JA019029

[6] Ruhunusiri S et al. MAVEN observations of partially developed Kelvin-Helmholtz vortices at Mars. Geophysical Research Letters. 2016;**43**:4763-4773. DOI: 10.1002/ 2016GL068926

[7] Pérez-de-Tejada H et al. Vortex structure in the plasma flow channels of the Venus wake, Chapter 1. In: Pérezde-Tejada H, editor. Vortex Structures in Fluid Dynamic Problems. Croatia: INTECH Pub; 2017. DOI: 10.5772/66762

[8] Pérez-de-Tejada H, Lundin R. Solar cycle variations in the position of vortex structures in the Venus wake, Chapter 3. In: Bevelacqua J, editor. Solar System Planets and Exoplanets. Croatia: INTECH-OPEN Pub; 2021. DOI: 10.5772/96710

[9] Acuña M et al. Mars observer magnetic field investigation. Journal of Geophysical Research. 1992;**97**(E5):7799 [10] Romanov SA et al. Interaction of the solar wind with Venus. Cosmic Research. 1979;**16**:603 (Fig 5)

[11] Verigin M et al. Plasma near Venus from the Venera 9 and 10 wide angle analyzer data. Journal of Geophysical Research. 1978;**83**:3721

[12] Dubinin E et al. Plasma in the near Venus tail: VEX observations. Journal of Geophysical Research. 2013;**118**(12): 7624

[13] Miller K, Whitten R. Ion dynamics in the Venus ionosphere. Space Science Reviewes. 1991;**55**:165

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#### **Chapter 8**

## Vortex Dynamics in Dusty Plasma Flow Past a Dust Void

*Yoshiko Bailung and Heremba Bailung*

#### **Abstract**

The beauty in the formation of vortices during flow around obstacles in fluid mechanics has fascinated mankind since ages. To beat the curiosity behind such an interesting phenomenon, researchers have been constantly investigating the underlying physics and its application in various areas of science. Examining the behavior of the flow and pattern formations behind an obstacle renders a suitable platform to realize the transition from laminar to turbulence. A dusty plasma system comprising of micron-sized particles acts as a unique and versatile medium to investigate such flow behavior at the most kinetic level. In this perspective, this chapter provides a brief discussion on the fundamentals of dusty plasma and its characteristics. Adding to this, a discussion on the generation of a dusty plasma medium is provided. Then, a unique model of inducing a dusty plasma flow past an obstacle at different velocities, producing counter-rotating symmetric vortices, is discussed. The obstacle in the experiment is a dust void, which is a static structure in a dusty plasma medium. Its generation mechanism is also discussed in the chapter.

**Keywords:** vortices, vorticity, fluid flow, Reynolds number, plasma, dusty plasma, obstacle, dust void, viscosity

#### **1. Introduction**

Vortices are common in fluid motion that originates due to the rotation of fluid elements. They occur widely and extensively in a broad range of physical systems from the earth's surface to interstellar space. A few examples include spiral galaxies in the universe, red spots of Jupiter, tornadoes, hurricanes, airplane trailing vortices, swirling flows in turbines and in different industrial facilities, vortex rings formed by the firing of certain artillery or in the mushroom cloud resulting from a nuclear explosion. The physical quantity that characterizes the rotation of fluid elements is the vorticity ω = ∇ u where u is the fluid velocity. Qualitatively, it can be said that in the region of vortex formation, the vorticity concentration is high compared with its surrounding fluid elements. Vortices formed behind obstacles to a fluid flow are also an interesting observation in various aspects of daily life. Study on the fluid flow around obstacles dates back to the fifteenth century when Leonardo da Vinci drew some sketches of vortex formation behind obstacles in flowing fluids. It has been an interesting and challenging problem in fluid mechanics and is of basic importance in several areas such as the study of aircraft designing, oceanography, atmospheric dynamics, engineering, human blood circulation [1–4]. Analyzing the behavior of flow around such obstacles also provides a medium to

study the physical mechanism of transition from laminar to turbulent flow. If a stationary solid boundary lies in the path of a fluid flow, the fluid stops moving on that boundary. Thereby, a boundary layer is formed and its separation from the solid boundary generates various free shear layers that curl into concentrated vortices. These vortices then evolve, interact, become unstable and detach to turbulence. The dynamics of fluids is very diverse and the detail characteristics of transition to turbulence are quite complicated, which also differ from flow to flow. Such understandings can only be realized by experiments and computational models. However, there are a few unifying themes in the theory and a few routes to turbulence that are shared by many flows. One such theme is that when the Reynolds number (the parameter measuring the speed of a class of similar flows with steady configuration) increases, the temporal and spatial complexity of the flows increases eventually leading to turbulence. At a low Reynolds number, a pair of counter-rotating vortices forms behind the cylinder. As the Reynolds number increases, the vortices become unstable and gradually evolve into a von Karman vortex street [5–9]. The topic of flow past an obstacle is of utmost importance from the experimental point of view also. Its understanding is applicable in the stability of submerged structures, vortex-induced vibrations, etc. [10].

Vortices have been extensively studied and explored in the liquid state of matter. However, scientists have also extended their research to study the formation and behavior of vortices in the fourth state of matter, the plasma. Measurements done in space have shown that plasma vortices appear in the earth's magnetosphere as well as along the Venus wake. On both planets, the solar wind encounters different obstacles. For earth, it is the earth's magnetic field and for Venus, the interaction takes place with the ionized components in the upper layer of the planet's atmosphere. Plasma vortices in earth-based laboratories have also been studied theoretically and experimentally [11–15]. Plasma is said to cover more than 99% of the matter found in the universe and dust particles are the unavoidable, omnipresent ingredients in it. Hence, in most cases, plasma and dust particles exist together, and these particles are massive (billion times heavier than the protons). Their size ranges from tens of nanometer to as large as hundreds of microns. Foreign particles in the plasma environment get charged up by the inflow of electrons and ions present in the plasma. The presence of these charged and massive particles increases the complexity of the plasma environment, and hence, this class of plasma has been named as 'complex plasmas' or 'dusty plasmas'. They involve in a rich variety of physical and chemical processes and are thus investigated as a model system for various dynamical processes [16, 17]. Phase transition is an important and characteristic feature in dusty plasmas, due to which it is considered as a versatile medium to study all the three different phases (solid, liquid and gas) in just a single phase. They also behave as many particle interacting systems and provide a unique platform to study various organized collective effects prevalent in fluids, clusters, crystals, etc., in greater spatial and temporal resolution. With the help of laser light scattering, it is possible to visualize the micrometer or nanometer-sized dust particles through proper illumination. This allows to study the various phenomena in dusty plasma in greater spatial and temporal resolution since they appear in a slower time scale owing to their heavier mass [18, 19]. Along with a variety of dynamic phenomena that includes waves, shocks, solitons, etc., dusty plasma medium also supports the formation of vortices. Self-generated vortices have been observed in many dusty plasma experiments, which have been dealt with significant attention. The main cause of such vortex formation is the nonzero curl of the various forces acting on the electrically charged dust particles that are commonly found in radiofrequency (RF) discharges, microgravity conditions and subsonic dusty plasma flow with low Reynolds numbers [20, 21]. The nonzero component of the curl

*Vortex Dynamics in Dusty Plasma Flow Past a Dust Void DOI: http://dx.doi.org/10.5772/intechopen.101551*

induces a rotational motion to the charged dust particles, which leads to the formation of the vortices. Depending on the different plasma production mechanisms and dust levitation (floating of dust particles in the plasma medium), the causes of the rotation of dust particles vary accordingly. Most importantly, the problem of fluid flow around obstacles can be investigated at the most elementary individual particle level in dusty plasmas. The existence of a liquid phase of dusty plasmas provides us the suitable conditions for the study. However, the obstacles used for such study in dusty plasmas are different from the solid obstacles in the hydrodynamic fluid medium.

In this chapter, we will concentrate mainly on dusty plasmas, their characteristics and a model system to study fluid flow around an obstacle at the particle level. After the introductory portion in Section 1, the fundamentals of dusty plasma are discussed in Section 2. In Section 3, the production of a dusty plasma medium by RF discharge will be discussed. Then in Section 4, we will discuss about the type and behavior of the obstacle which is used in dusty plasma flows. In Section 5, we will discuss the dusty plasma flow and the pattern formation behind the obstacle. The final section then summarizes the chapter as a whole.

#### **2. Fundamentals of dusty plasma**

#### **2.1 Dusty plasma**

First, let us start with a very brief explanation of plasma! Basically, plasma is an assembly of a nearly equal number of electrons and ions, and the charge neutrality is sustained on a macroscopic scale. In the absence of any external disturbance, that is, under equilibrium conditions, the resulting total electric charge is zero. The microscopic space-charge fields cancel out inside the plasma and the net charge over a macroscopic region vanishes totally. The quasi-neutrality condition at equilibrium is given by,

$$n\_{\boldsymbol{\epsilon}} \approx n\_{\boldsymbol{i}} \tag{1}$$

where *ne* and *ni* are the electron and ion densities, respectively.

The 'plasma'state of matter differs from ordinary fluids and solids by its natural property of exhibiting collective behavior. These collective effects result in the occurrence of various physical phenomena in the plasma, resulting in the longrange of electromagnetic forces among the charged particles. The very first example of plasma that is obvious to refer is the Sun, the source of existence of life. The protective layer to the earth's atmosphere, known as the ionosphere, also remains in the form of ionized particles, that is plasma. Moreover, natural plasmas exist in interstellar space, stars, intergalactic space, galaxies, etc. On earth, the common form of natural plasma is lightning, fire and the amazing Aurora Borealis. Artificial plasmas are generated by applying electric or magnetic fields through a gas at low pressures. These are commonly found in street lights, neon lights, etc. Neon light is a gas discharge light, which is actually a sealed glass tube with metal electrodes at both the ends of the tube and filled with one or several gases at low pressure.

As already mentioned before, dust particles in space as well as in earth's atmosphere, are unavoidable. These particles in plasma form a new field, that is dusty plasma or complex plasma. Dusty plasma is defined as a normal electron-ion plasma with charged dust components added to it. Naturally, dust grains are metallic, conducting, or made of ice particulates. Until and unless these are manufactured in laboratories, their shape and size vary. Depending on the surrounding plasma environments (due to the inflow of electrons and ions), dust particulates are either negatively or positively charged. These charged particles as a whole affect the plasma and result in collective and unusual behavior. When observed from afar, dust particles can be considered as point charges. As they are charged by the plasma species (electrons and ions), the charge neutrality condition is now modified, which is given by,

$$\mathbf{Q}\_d \mathbf{n}\_{d0} + \mathbf{e} \mathbf{n}\_{eo} = \mathbf{e} \mathbf{n}\_{io} \tag{2}$$

where *ne***0**, *ni***<sup>0</sup>** and *nd***<sup>0</sup>** are the equilibrium densities of electrons, ions and dust, respectively, 'e' is the magnitude of electron charge, *Qd* ¼ *eZd* is the charge on the dust's surface and *Zd* is the dust charge number. It is important to highlight that the charge of the particles depends significantly on the plasma parameters. And the basic physics of the dusty plasma medium entirely rests on the *Qdnd***<sup>0</sup>** term of the charge neutrality condition.

Plasma possesses the fundamental property of shielding any external potential by forming a space charge around it. This particular property provides a measure of the distance over which the influence of the electric field of a charged particle (dust particle in our case) is experienced by other particles (electrons and ions) inside the plasma. Typically, this length is known as the dust Debye length *λD*, within which the dust particles can rearrange themselves to shield all the existing electrostatic fields. The negatively charged heavier dust particles are assumed to form a uniform background and the electrons and ions, which are assumed to be in thermal equilibrium, simply obey the Boltzmann distribution. The dust Debye length is given by,

$$
\lambda\_{\rm D} = \frac{\lambda\_{\rm De} \lambda\_{\rm Di}}{\sqrt{\left(\lambda\_{\rm De}^2 + \lambda\_{\rm Di}^2\right)}} \tag{3}
$$

where *λDe* and *λDi* are electron and ion Debye lengths, respectively. These are expressed as,

$$
\lambda\_{Dt} = \sqrt{\frac{k\_B T\_e}{4\pi n\_{eo}e^2}}\tag{4}
$$

$$
\lambda\_{Di} = \sqrt{\frac{k\_B T\_i}{4\pi n\_{io}c^2}}\tag{5}
$$

*Te*,*<sup>i</sup>* represents the electron and ion temperatures, respectively, *neo*,*io* are the electron and ion densities, respectively, and *kB* is the Botlzmann constant.

A pictorial representation of a dusty plasma medium is shown in **Figure 1**.

In a dusty plasma medium, the charged particles interact with each other *via* the electrostatic Coulomb force. However, due to the inherent shielding property of the plasma electrons and ions, the charged particles are shielded and hence, the interaction energy among them is known as Screened Coulomb or Yukawa potential energy. Consider two dust particles having the same charge *Qd* and separated by a distance '*a*'. The screened Coulomb potential energy is given by,

$$P.E = \frac{Q\_d^2}{4\pi\varepsilon\_0 a}e^{-\kappa} \tag{6}$$

#### **Figure 1.**

*Schematic of a dusty plasma medium. The pink-shaded portion is the plasma medium. The green ball is the dust particle that is negatively charged. λ<sup>D</sup> is the dust Debye length.*

where *<sup>κ</sup>* <sup>¼</sup> *<sup>a</sup> <sup>λ</sup><sup>D</sup>* is the screening strength. The dust thermal energy is given by,

$$K.E = k\_B T\_D \tag{7}$$

where *TD* is the dust temperature. The ratio of the P.E to the K.E is termed as the Coulomb Coupling parameter, given by

$$
\Gamma = \frac{Q\_d^2}{4\pi\varepsilon\_0 a k\_B T\_D} e^{-\kappa} \tag{8}
$$

Depending on the coupling parameter, a dusty plasma system remains in a weakly coupled state or a strongly coupled one. When Γ< 1, the thermal energy of the dust particles is greater than the potential energy of the system and the system is said to be weakly coupled. On the other hand, when the potential energy exceeds the thermal energy, that is Γ> 1 the system becomes strongly coupled. So, from Eq. (8), we can see that dust charge, screening parameter and the dust temperature play an important role in determining the system's coupling state. As Γ exceeds a critical value Γ*c*, called the critical coupling parameter, a dusty plasma system attains a crystalline state. However, this critical value for crystallization is dependent on the screening parameter [22]. For 1<Γ< Γ*c*, the system remains in a fluid (liquid or gas) state.

Thus, we see that by adjusting the dusty plasma parameters, we can obtain a fluid state of the dusty plasma medium experimentally. This provides us a unique model to study vortex formation behind an obstacle in the particle most level.

#### **3. Production of a dusty plasma medium**

Laboratory dusty plasmas differ from space and astrophysical dusty plasmas in a significant manner. The discharges done in the laboratory have geometrical boundaries. The composition, structure, conductivity, temperature, etc., of these geometries affect the formation and transport of the dust grains. Also, the external circuit, which produces and sustains the dusty plasma, imposes boundary conditions on the dusty discharge, which vary spatially as well as temporally. Dusty plasmas in the laboratory are generally produced by two main discharge techniques—direct current (DC) discharge and RF discharge. In this chapter, we will mainly focus on the production technique by RF discharge method in a DUPLEX device.

As the name suggests, DUPLEX is an abbreviation for Dusty Plasma Experimental Chamber. It comprises of a cylindrical glass chamber, 100 cm in length and 15 cm in diameter. The glass chamber configuration of the DUPLEX device provides a suitable and great access for optical diagnostics. One end of the cylindrical chamber is connected to the vacuum pump systems and the other end is closed by a stainless steel (SS) flange with Teflon O-ring between the glass chamber and the SS flange. On this closed end, there are ports for pressure gauge fitting, probe insertion and electrical connections. A radio frequency power generator (frequency: 13.56 MHz, power: 0–300 W) and an RF matching network are used for the plasma discharge. The RF antennas used in this setup are aluminum strips of 2.5 cm width and 20 cm length typically placed on the outer surface of the glass chamber. A schematic of the setup is shown in **Figure 2**. This strip acts as the live electrodes.

Initially, the chamber pressure is reduced to a value of about <sup>10</sup><sup>3</sup> mbar with the help of a rotary pump. Argon is used as the discharge gas, by injecting which the desired chamber pressure can be maintained. A grounded base plate is also inserted into the chamber (about 30 cm length, 14.5 cm width and 0.2 cm thickness), which acts as the grounded electrode and the region above it is selected as the experimental region. Applying a radiofrequency power (13.56 MHz and 5 W) between the aluminum strips (working as live electrodes) and the grounded base plate, a capacitively coupled RF discharge plasma is produced. Due to the application of the RF field, initially, the stray electrons inside the chamber get energized and in turn ionize the gas molecules present in the chamber. The aluminum strips used as live electrode outside give the flexibility to change the electrode position whenever required. Also, it facilitates in forming a uniform plasma over an extensive area of the grounded plate, that is the experimental region. The plasma parameters can be varied manually by tuning the discharge conditions, *viz*. RF power and neutral pressure.

Dust particles used in the experiment are gold-coated silica dust particles ( 5 micron diameter). These are initially put inside a buzzer that is fitted to the grounded base plate. After the production of the plasma, a direct current (DC) voltage of (6–12)V is applied to the buzzer, which ejects the dust particles from it

**Figure 2.** *Schematic of a DUPLEX setup. The pink-shaded portion is the argon plasma.*

*Vortex Dynamics in Dusty Plasma Flow Past a Dust Void DOI: http://dx.doi.org/10.5772/intechopen.101551*

through a hole. When these dust particles enter into the plasma environment, electrons and ions flow towards it and charge up the particles. In the laboratory, the dust particles are usually negatively charged as the electrons are lighter and highly mobile than the ions. These negatively charged dust particles are acted upon by two forces mainly, the upward electric field force (*QdE*) due to the sheath electric field (*E*) of the grounded base plate and the downward gravitational force (*mdg*). Dust particles levitate at the position where these two forces exactly balance. The dust particles are illuminated by laser light scattering, and the dust dynamics are recorded in high-speed cameras. **Figure 3** shows the levitation of dust particles in a plasma medium. Above the dust layer, the purple color signifies argon discharge plasma. The dark region below the dust layer and above the plate is the sheath (where ionization does not take place) where a strong electric field (*E*) is present. The charged dust particles levitate at the interface region ( 0.8 cm above the plate) between plasma and the sheath where the force balance occurs. This is shown by a dashed line.

#### **4. Obstacle in dusty plasma flows**

The obstacle used in dusty plasma flow experiments is actually a dust void. A void is a dust-free region, which is encountered spontaneously in certain experimental conditions or can be produced externally also [23–28]. In the past couple of decades, there have been a few studies on the interaction of a dusty plasma medium with dust voids. In 2004, Morfill et al. studied a laminar flow of liquid dusty plasma with a velocity 0.8 cms<sup>1</sup> around a spontaneously generated lentil-shaped void [29]. They observed the formation of a wake behind the void that is separated from the laminar flow region by a mixing layer. The flow also exhibited stable vortex flows adjacent to the boundary of the mixing layer. Another study was made in 2012

**Figure 3.** *Photograph of a dust layer levitation in plasma.*

by Saitou et al. where they externally placed a thin conducting wire of 0.2 mm diameter and 2.5 cm length. They made the dust particles flow with velocity in the range (5–15) cms<sup>1</sup> but did not observe any vortex formation behind the obstacle. What they observed was a bow shock in front of it [30]. In the very next year itself (2013), Meyer et al. also did a similar experiment with a different configuration and dust flow mechanism (velocity <sup>10</sup>–25 cms<sup>1</sup> ) than Saitou's [21]. They produced a dust void by placing a 0.5-mm-diameter cylindrical wire transverse to the flow. They too observed a bow shock and a tear-shaped wake in front and behind the obstacle, respectively. Moreover, Charan et al. in 2016 did a molecular dynamics simulation study where they used a square obstacle and observed von Karman vortex street at low Reynolds number (i.e. low velocity) compared with normal hydrodynamic fluids [31]. Then in 2018, Jaiswal et al. investigated dust flow towards a spherical obstacle over a range of flow velocities (4–15)cms<sup>1</sup> and different obstacle biases [32]. The spherical obstacle also generated a dust-free area in its vicinity. They too observed bow shock formation in front of the obstacle but no vortex formation behind it. In 2020, Bailung et al. also investigated the study of dust flow around a dust void with a unique flow mechanism (dust flow velocity <sup>3</sup>–10 cms<sup>1</sup> ) in a DUPLEX setup [33]. Dust particles are allowed to flow towards an already existing stationary dust layer. They could observe the formation of a counter-rotating pair of vortices behind the obstacle in a particularly narrow range of velocity (4–7) cms<sup>1</sup> . Above and below this range, their vortices are not observed. Due to the interplay between these two forces, a circular void is generated around the pin. At the void boundary, these two forces equate with each other.

In the next section, we will study the results of Bailung et al. in detail, but before that let us understand the mechanism of the formation of dust void due to the insertion of an external cylindrical wire. A cylindrical pin inside the plasma attains a negative potential for the plasma and a sheath is formed in its vicinity. Due to the negative potential of the pin, ions try to drift towards it giving rise to a force on the dust particles named as ion drag force. This force is directed radially inward with the pin as the centre. Also, the negatively charged dust particles experience a repulsive electrostatic force from the pin which is directed radially outward. The interplay between these two forces generates a circular void around the pin. At the void boundary, these two forces equate with each other. A typical configuration of pin insertion through the grounded plate of a DUPLEX chamber is shown in **Figure 4**. The pin is externally connected to a DC bias voltage. By varying the bias voltage, the size of the void can be altered according to experimental requirements. Typically, at a RF power of 5 W and chamber pressure 0.02 mbar, the diameter of the dust void in floating condition (i.e. no external bias) is 1.7 cm. A typical example of a dust void is shown in **Figure 5**. However, unlike the solid obstacles in the case of hydrodynamic fluids, the dust void is not a rigid kind of obstacle. As already seen, the boundary of the void is maintained by a delicate force balance

*Vortex Dynamics in Dusty Plasma Flow Past a Dust Void DOI: http://dx.doi.org/10.5772/intechopen.101551*

**Figure 5.**

*Snapshot of a dust void formed in DUPLEX chamber. The bright spot in the Centre is the reflection of laser light from the cylindrical pin. The photograph is taken from the top of the chamber.*

between the outward electrostatic force and inward ion drag force. An incoming dust flow, depending on the velocity of the flow, would cross the void boundary and penetrate into the void.

#### **5. Vortices in the wake of a dust void**

Due to the non-rigidity of the dust void boundary, the behavior of the flow near the obstacle is somewhat different than conditions of hydrodynamic fluid with a rigid obstacle. Despite this difference, the transition from laminar to turbulence is observed in the wake of the obstacle in the case of dusty plasma flow also. As the flow approaches the void boundary, the middle section of the flow slightly penetrates into the void region and slips through the void boundary layer on both sides. The trajectory of the flow (in the mid-section) is deflected in front of the void due to the repulsive force exerted by the sheath electric field of the void and then flows downstream surrounding the void. The curved dust flow again meets behind the void and continues with the flow. As observed by Bailung et al. at a very narrow range of velocity (4–7) cms<sup>1</sup> , a counter-rotating vortex pair is seen to appear. Below and above this range, the dust particles do not form any vortices. A typical example of three different conditions is shown in **Figure 6**.

In each of the images, dust particles flow from right to left shown by dashed arrows. The top image (a) depicts a flow with dust flow velocity 3.5 cms<sup>1</sup> and the snapshot is at time t = 1370 ms from a reference time (t = 0, when dust flow reaches the right edge of the images). The middle image (b) shows dust fluid flow velocity 4.5 cms<sup>1</sup> at t = 1033 ms showing a vortex pair formation behind the void. The vortices are shown by the two arrow marks. It is observed that vortices are not formed for larger flow velocity 8 cms<sup>1</sup> (image (c)). For such high

*Typical snapshots showing structures formed behind the void at (a) 3.5cms<sup>1</sup> , (b) 4.5cms<sup>1</sup> , (c) 8 cm<sup>1</sup> .*

velocities, flow trajectories behind the void are elongated and dynamics in the wake is rather complex due to cross-flow at high speed. The bright illuminated point at the centre of each image is the reflection of laser light from the pin. Two horizontal lines that appear in all the images are due to laser reflection from the wall of the glass chamber. It is noted that dust flow with unsteady laminar velocity, which is (4–7)cms<sup>1</sup> , and optimum dust density in the experimental region above the grounded plate is required to generate the vortex behind the void.

For a better understanding, a pictorial representation showing the dusty plasma streamlines around dust void at three different velocities are shown in **Figure 7(a)**– **(c)** of **Figure 7** corresponds to the observation shown in (a), (b) and (c) of **Figure 6**.

#### **Figure 7.**

*A pictorial illustration of the dusty plasma flow interaction with the dust void at different flow velocities. (a) Laminar flow, (b) unsteady laminar flow with filamentary vortex-type structure in the upstream and vortex pair in the downstream and (c) turbulent flow.*

At a lower dust flow velocity, the void in the upstream is slightly compressed and trajectories of the streamlines flowing close to the void (boundary layer) curl behind the void. However, no structure formation in the wake appears here. Dust particles, after meeting behind the void, just continue with the flow smoothly. For critical flow speed (b), flow dynamics in the upstream void boundary is quite different. Streamlines that hit perpendicularly at the void flow some distance into the void region. They reconstruct the boundary during the flow and get ejected backward making the streamline bifurcation to occur much ahead of the void boundary. The curved streamlines, which are ejected backward, again flow along with the incoming dust flow close to the boundary layer. This critical reorientation in the front of the void generates a suitable condition for the formation of the vortex pair behind the void. Particles get slowed down in this region and these slower

particles flow close to the boundary layer around the void and contribute in the formation of the vortex pair. At higher velocities (c), that is above the critical range for vortex formation, all the particles that hit the upstream void boundary are flushed away by the flow along with it. The streamlines intersect and crossover at a distance far behind the void and there is no formation of any stable structure. It is well known that in hydrodynamic fluids, at much higher velocities, vortex streets are observed. However, here such streets are not observed to form. This may be due to the restriction of the experimental geometry. The transition from laminar to turbulence is well known in fluid dynamics. But studying it in dusty plasma provides the chance to observe the individual particle-level trajectory. In turn, the dynamics can be studied in greater detail.

To see the dust dynamics in greater detail, let us look at the vortex formation behind the dust void step by step. At the outset of the formation, the slower particles moving along the curved boundary layer interact with the stationary particles behind the void and start to swirl on each side. The flow front then meets in the wake region behind the void (**Figure 8(a)**) and gradually traverses a swirling circular path. This is evident in the dotted arrow marks in (Figure b). After duration of 966 ms from the start of the flow, two counter-rotating vortices complete their formation (Figure c). Only the slower particles flowing close to the boundary layer participate in this swirling motion due to the nonzero curl of the forces. Those particles away from the boundary layer move faster and do not contribute to the swirling. With increasing time and inflow of more particles, the swirling finally grows into a distinct pair of the vortex with an eye in the middle (Figure (d)). As the flow progresses by maintaining a constant inflow of particle flux, the vortex pair sustains till 1167 ms. The one shot of dust flow in the experiment done by Bailung et al. lasted for about 2 sec.

Hence, gradually when the particle flux started decreasing, the vortex pair starts to die out. It is faintly visible till 1233 ms (Figure g). The time for the growth of the vortex pair is 200 ms (from the time the particles meet behind the void) and survives for duration of 200 ms (depending on the duration of accelerated dusty plasma fluid flow). Finally, they disappear after 1300 ms. The rotational frequency measured for the vortices is about 3 Hz.

It is already mentioned that the advantage of studying vortex dynamics in dusty plasma lies in the fact that particles can be individually tracked. Different particle tracking software and computational models are available, which can

#### **Figure 8.**

*The parallel arrows depict the direction of the dust flow. (a) when both the oppositely curling flow front meet behind the void. Dotted curve traces in (b) indicate flow trajectories. The arrows in (c) - (g) show the vortex pair. the vortices vanish with time when flow is nearly over (h).*

*Vortex Dynamics in Dusty Plasma Flow Past a Dust Void DOI: http://dx.doi.org/10.5772/intechopen.101551*

generate the velocity vectors of the trajectory of the particles and hence can give a quantitative interpretation of the experimentally observed results. One such particle tracking platform is OpenPIV (Open Particle Image Velocimetry) in MATLAB [34]. This helps to study the evolution of the vortex pair along with its vorticity. But to perform successful PIV from images, the recorded videos of the dust flow dynamics should have a high-quality resolution and must be in high speed. A PIV analysis performed on a video recorded at 100 frames per second is shown in **Figure 9**.

Each image in the figure is an average PIV result of 10 consecutive image frames. The position of the void and the pin position are drawn by a red-dashed circle and a red dot, respectively. The velocity vectors show the trajectory of the dust particles and the color code gives the value of the vorticity at different times in units of s�<sup>1</sup> . The slowing down of the particles in front of the void is clearly seen by comparing the velocity vectors' lengths in **Figure 9(a)** and **(b)**. The backflow of the incoming dust particles mentioned earlier (due to repulsive sheath electric field force of the pin) is also observed in (b). The curling of dust particles leading to vortex formation is evident from (c) and (d). The vorticity of the fully formed vortex pair is about � (20–25)s�<sup>1</sup> , which is shown by the color bar in (e) and (f)). This is nearly equal to twice the measured angular frequency. With the decrease of the dust flow influx, the vortex structure deforms (vorticity �15 s�<sup>1</sup> ) and breaks away into smaller vortices (vorticity �10s�<sup>1</sup> ) as seen from (g) to (i). Vortices finally disappear in (j), evident from the vorticity value which almost tends to 0.

Reynolds number is the characteristic parameter that helps to predict flow patterns. It is the ratio of the inertial forces to the viscous forces and is given by,

$$Re = \frac{\rho v\_d L}{\eta} \tag{9}$$

where *ρ* is mass density, *vd* is dust velocity, *L* is the obstacle dimension, that is the void diameter and *η* is the viscosity of the dust fluid.

In case of dusty plasma fluids, the viscosity is estimated from the formula,

$$
\eta = \sqrt{3} \hat{\eta} m\_d n\_d \alpha\_E a^2 \tag{10}
$$

where ^*η* is the normalized shear viscosity, *md* is the dust mass, *nd* is the dust density, *<sup>ω</sup><sup>E</sup>* <sup>¼</sup> *<sup>ω</sup>pd<sup>=</sup>* ffiffiffi <sup>3</sup> � � <sup>p</sup> is the Einstein frequency, *<sup>ω</sup>pd* is the dust plasma frequency and *a* is the interparticle distance. The normalized shear viscosity in dusty plasma fluid is a function of the Coulomb coupling parameter Γ, which has been estimated for a range of coupling parameters in different conditions *via* simulation [35, 36]. For typical plasma parameters of DUPLEX chamber, that is,

$$m\_d = 1.7 \times 10^{-13} \text{ kg}$$

$$n\_d = 9 \times 10^9 m^{-3}$$

$$o\_{pd} = 247.5 \, s^{-1}$$

$$a = 3 \times 10^{-4} \, m$$

The viscosity is calculated to be 9 � <sup>10</sup>�<sup>9</sup> Pas.

Thus, the Reynolds number for dust flow velocity � (3–10) cms�<sup>1</sup> is estimated to be lying in the range 50–190. The vortex pair formation appears in a critical range of 60–90.

#### **Figure 9.**

*PIV analysis showing the time evolution of the vortices for a duration of 1 sec. The velocity vectors and vorticity profile drawn from (a) (0.53-0.62) sec (b) (0.63-0.72) sec (c) (0.73-0.82) sec (d) (0.83-0.92) sec (e) (0.93-1.02) sec (f) (1.03-1.12) sec (g) (1.13-1.22) sec (h) (1.23-1.32) sec (i) (1.33-1.42) sec (j) (1.43- 1.52) sec are shown. The color bar shows the value of vorticity in 1/s. The dotted circle in (a) shows the original position of the void boundary before the flow and the dot at the center of the circle depicts the pin position.*

In the case of hydrodynamic fluid, the range of Reynolds number for vortex formation is 5–40, which is much lower compared with that in dusty plasma fluid. This is because the ratio *ρ=η* (which is the kinematic viscosity) is one order larger in the case of dusty plasma fluids than that in hydrodynamic fluids. The estimated kinematic viscosity for dusty plasma fluids is 0.088 cm2 s 1 , whereas the kinematic viscosity for water is 0.008 cm<sup>2</sup> s 1 .

### **6. Conclusion**

The study of vortices in the problem of flow past an obstacle is significant as it provides a platform to investigate the transition from laminar to turbulence. Formation of vortices in the wake region behind an obstacle appears in the unsteady laminar regime of flows and has been widely studied in hydrodynamic fluids. However, dusty plasma medium, which is a component of the fourth state of matter, provides a unique stage to study such phenomena at the particle level. A special property of this medium is that it can remain in both fluids (liquid- or gas-like) as well as the crystalline state. By mere adjustment of plasma conditions, the desired state can be obtained. The individual tracking of micron-sized dust particles by methods such as PIV (Particle Image Velocimetry) yields the particle trajectory in form of velocity vector fields. This gives a very clear picture of the behavior of flow near obstacle boundaries. However, the obstacles used in dusty plasma flow experiments differ from those in hydrodynamic fluid experiments in the sense that unlike those in hydrodynamics, the obstacle boundaries in dusty plasma are non-rigid. Any foreign pin or wire inserted into the plasma would possess a negative potential with respect to the plasma. Dust particles in its vicinity are repelled due to electrostatic force and form a dust-free region around it, called the dust void. This dust void, whose boundary is delicately maintained by dusty plasma forces, acts as a non-rigid type of obstacle. Dusty plasma flows also generate counter-rotating vortices in the wake region behind a dust void at a particular range of velocities. Below and above this range, no structure formation is seen to appear. The particle behavior causing the formation of the vortices is better understood by tracking particles in consecutive frames. The estimated Reynolds number value for vortices to appear in the wake of a void in a dusty plasma medium is estimated to lie in the range 60–90. This is quite larger than the Reynolds number range for hydrodynamic fluids which is roughly about 5–40. This higher range in dusty plasma medium is attributed to the higher kinematic viscosity of dusty plasma fluids. However, in dusty plasma experiments, Von Karman vortex streets (observed in the turbulent regime of hydrodynamic fluids) are not yet explored. If such experiments could be successfully performed, then there will be immense scope of understanding turbulence at the particle-most level and with a better perspective. Although to study turbulent dynamics, high-speed cameras with high-quality resolution would be necessary.

### **Author details**

Yoshiko Bailung<sup>1</sup> and Heremba Bailung<sup>2</sup> \*

1 Department of Physics, Goalpara College, Goalpara, India

2 Dusty Plasma Laboratory, Physical Sciences Division Institute of Advanced Study in Science and Technology, Guwahati, Assam, India

\*Address all correspondence to: hbailung@yahoo.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Vortex Dynamics in Dusty Plasma Flow Past a Dust Void DOI: http://dx.doi.org/10.5772/intechopen.101551*

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Section 4

## Vortices in Aerospace Engineering

#### **Chapter 9**

## Wingtip Vortices of a Biomimetic Micro Air Vehicle

*Rafael Bardera, Estela Barroso and Juan Carlos Matías*

#### **Abstract**

Wingtip vortices are generated behind a wing that produces lift. They exhibit a circular pattern of spinning air that generates an additional drag force, the induced drag, reducing the aerodynamic performance of an aircraft. Moreover, the wingtip vortices can pose a hazard to airplane maneuvers, mainly in take-off and landing operations. This chapter describes a review of the lifting-line vortex theory applied to a biomimetic Micro Aerial Vehicle (MAV) with Zimmerman planform. Therefore, the horseshoe vortex model is deeply explained and the estimations of vortex velocity distribution, lift, and induced drag are obtained with this simple model. These results are compared with experimental data obtained from wind tunnel testing by using Particle Image Velocimetry (PIV). Finally, the vorticity maps in the wake of this MAV are obtained from PIV measurements.

**Keywords:** tip vortices, biomimetic, micro aerial vehicle, induced drag, vorticity

#### **1. Introduction**

The aeronautic industry has developed a growing interest in Unmanned Aerial Vehicles (UAVs). These vehicles have been designed for multiple missions where the human factor is not required. Therefore, in dangerous missions, unhealthy environments, or inaccessible areas, human accidents can be avoided. The UAVs can be distinguished into different categories according to their performance characteristics. In this context, the relevant design parameters are weight, manufacturing costs, and size. Mainly, the manufacturing costs have been the key point for that engineers and researchers could be focused on developing smaller vehicles in order to perform unmanned activities. This group of smaller vehicles is known as Micro Aerial Vehicles (MAVs) [1–3]. Their main features are low aspect ratio (AR) and low range operation. Research centers and universities have been able to investigate new designs of MAVs due to their low manufacturing costs and small size. This is the case of aerodynamic challenges posed in the work of Moschetta [4]. The MAV designs have taken into account the fixed-wing, coaxial, biplane, and tilt-body concept. Marek [5] performed experimental tests to determine the aerodynamic coefficients in six different types of platforms. The Zimmerman and elliptical planforms resulted in having the highest lift coefficient. Hence, Hassannalian and Abdelkefi [6] designed and manufactured a fixed-wing MAV based on the Zimmerman planform. Also, other authors designed the Dragonfly MAV using Zimmerman planform [7, 8].

The chapter will begin with a description of the biomimetic Micro Air Vehicle (MAV) [9, 10], then the horseshoe theory will be explained and applied to the

studied vehicle. Consequently, the experimental facility, the Particle Image Velocimetry technique, and the description of the experimental tests will be defined. Then, the vorticity and several vortex models will be defined and applied to the experimental data obtained from the Wind Tunnel. At the end of the chapter, the formulation which relates the axial vorticity and the circulation will be presented and finally, the lift coefficient will be obtained.

#### **2. Micro air vehicle geometry**

The geometry of the studied Micro Air Vehicle (MAV) is based on Zimmerman planform and Eppler 61 airfoils for the wing configuration and Whitcomb II airfoils for the fuselage (see **Figure 1**).

**Figure 1.** *Biomimetic MAV model (dimensions in mm).*

**Figure 2.** *Zimmerman planform.*


#### **Table 1.**

*MAV features.*

The Zimmerman wing consists of two halves of ellipses connected at one point of reference which corresponds to 1/4 of the maximum wing root chord (*cr* = 200 mm) for one half of the ellipse and 3/4 of the *cr* for the other half of the ellipse. **Figure 2** shows the planform of the micro air vehicle and their dimensions. The rest of the geometrical features are shown in **Table 1**.

#### **3. The horseshoe vortex: Biot-Savart law**

In this section, a previous formulation of the wingtip vortex will be presented. The 3D wings can be modeled by vortex filaments. The horseshoe is the simplest mathematical model of potential flow to represent the aerodynamics of a wing aircraft. That consists of the bound vortex (vortex filament of the wing) and the trailing vortices formed by the semi-infinite filament vortex that represents the wingtips.

The horseshoe is a 3-D vortex that can be represented with an arbitrary shape according to the Helmholtz vortex theorems:


In this context, the velocity field of a 3-D vortex by applying the Biot-Savart Law is defined by the following expression Eq. (1), [11].

$$\overrightarrow{V}(x,y,z) = \frac{\Gamma}{4\pi} \int\_{-\infty}^{+\infty} \frac{d\overrightarrow{l}\times\overrightarrow{r}}{\left|\overrightarrow{r}\right|^{3}}\tag{1}$$

where *r* ! is extended from the integration point to the field point and the arc length element *d l* ! points follow the direction of positive circulation.

Taking into account the straight vortex of **Figure 3**, *h* is defined as the nearest perpendicular distance from the vortex line and *θ* is the angle between the radius vector *r* ! and the vortex line (are defined in Eq. (4) and (5)).

$$r \equiv \left| \overrightarrow{r} \right| = \frac{h}{\sin \theta} \tag{2}$$

$$l = -\frac{h}{\tan \theta} \tag{3}$$

$$dl = \frac{h}{\sin^2 \theta} \tag{4}$$

$$\begin{aligned} \left(d\stackrel{\rightarrow}{l} \times \stackrel{\rightarrow}{r} = \left(dl\ r\sin\theta\right)\hat{\theta}\right. \end{aligned} \tag{5}$$

Now, the velocity field can be recalculated as Eq. (6):

$$\overrightarrow{V} = \frac{\Gamma}{4\pi h} \hat{\theta} \Big|\_{0}^{\pi} \sin \theta \, d\theta = \frac{\Gamma}{2\pi h} \hat{\theta} \tag{6}$$

To reproduce the wingtip vortices of the studied MAV, a simple model based on the superposition of the freestream flow (*U*∞) and a horseshoe vortex is described. The horseshoe vortex can be defined as the sum of three segments that can be seen in **Figure 4**: two free-trailing vortices at each tip of the wing (segment AB and segment CD) that are connected by a bound vortex spanning the wing (segment BC). As explained previously, the circulation *Γ* along the entire vortex line is constant, and the vortex lines have to extend downstream to infinity (see **Figure 3**). This potential solution is not very effective since the local lift to span is constant over the wingspan and in the real MAV model, the local lift is zero at the tip of the wings. A scheme of the horseshoe vortex model is defined in **Figure 4**.

The velocity field downstream of the wing in *x* = constant planes is similar to the potential solution generated by a horseshoe vortex except near the vortex axes. Now, to obtain the vertical velocity distribution of the potential vortex in our MAV, it is necessary to know the wing chord (*c* = 0.2 m), wingspan (*b* = 1.6*c*), and chord distance downstream of the trailing edge of the wing (*x* = 3*c*). Therefore, the following two non-dimensional variables (*η* and *ζ*) need to be defined (Eq. (7)):

$$
\eta = \frac{\varkappa}{a} = \textbf{3.75}; \zeta = \frac{\nu}{a} \tag{7}
$$

where *<sup>a</sup>* is the semi-wingspan, *<sup>a</sup>* <sup>¼</sup> *<sup>b</sup>* <sup>2</sup> (see **Figure 2**), *η* and *ζ* are the nondimensional coordinates according to the x-axis and y-axis, respectively.

*Wingtip Vortices of a Biomimetic Micro Air Vehicle DOI: http://dx.doi.org/10.5772/intechopen.102748*

**Figure 4.** *Scheme of the horseshoe vortex model.*

Then, the non-dimensional vertical velocity *ψ ζ*ð Þ can be defined as Eq. (8):

$$
\omega(\eta) = \frac{w(\zeta)}{\frac{\Gamma}{4\pi t}}\tag{8}
$$

which presents a different formulation depending on the vortices defined in each of the segments (see **Figure 3**):

$$\mathbb{V}\_{AB}(\eta) = \frac{-\mathbf{1}}{\left(\zeta + \mathbf{1}\right)} \left[\mathbf{1} + \frac{\eta}{\sqrt{\left(\zeta + \mathbf{1}\right)^2 + \left(\eta\right)^2}}\right] \tag{9}$$

$$\boldsymbol{\Psi}\_{CD}(\boldsymbol{\eta}) = \frac{\mathbf{1}}{\left(\boldsymbol{\zeta} - \mathbf{1}\right)} \left[ \mathbf{1} + \frac{\boldsymbol{\eta}}{\sqrt{\left(\boldsymbol{\zeta} - \mathbf{1}\right)^2 + \left(\boldsymbol{\eta}\right)^2}} \right] \tag{10}$$

$$\mathcal{W}\_{\rm BC}(\eta) = \frac{-1}{\eta} \left[ \left[ \frac{\eta + 1}{\sqrt{\left(\zeta + 1\right)^2 + \left(\eta\right)^2}} \right] - \left[ \frac{\eta - 1}{\sqrt{\left(\zeta - 1\right)^2 + \left(\eta\right)^2}} \right] \right] \tag{11}$$

Finally, the total non-dimensional vertical velocity is defined as the sum of the three velocities of the vortices (Eq. (12)):

$$
\Psi(\zeta) = \Psi\_{AB}(\zeta) + \Psi\_{BC}(\zeta) + \Psi\_{CD}(\zeta) \tag{12}
$$

In the following **Figure 5**, the total non-dimensional vertical velocity distribution of this MAV is presented only for the section located at 3c downstream of the trailing edge of the wing and for the angle of attack of 10°.

To obtain a better understanding of the flow behavior of these vortices and how they interact between them, in **Figure 6** the non-dimensional vertical velocity only of the AB free-trailing vortex region is presented. The blue line shows the velocity distribution of the AB free-trailing vortex while the dashed red and black lines correspond to the velocity of the bound vortex (BC in **Figure 4**) and the CD freetrailing vortex, respectively. It is clearly noted that both vortices, bound vortex, and CD free-trailing vortex are not affecting the AB free trailing vortex since their

#### **Figure 5.**

*The non-dimensional vertical velocity at 3c downstream of the trailing edge of the MAV wing.*

#### **Figure 6.**

*The non-dimensional vertical velocity at 3c downstream of the trailing edge of the MAV wing.*

velocity values are very small. As a consequence, in that region only the flow presence from the AB free-trailing vortex itself.

#### **4. The experimental set-up**

In this section, the experimental setup will be presented. All experimental tests were carried out in a Low-Speed Wind Tunnel at the Instituto Nacional de Técnica Aeroespacial (INTA) in Madrid (Spain). This wind tunnel has a closed circuit with an elliptical open test section of 6 m2. The DC engine, which is situated at the opposite side of the test section, works at 420 V, allowing a maximum airflow speed of 60 m/s with a turbulence intensity lower than 0.5%. **Figure 7** shows the Low-Speed Wind Tunnel of INTA.

*Wingtip Vortices of a Biomimetic Micro Air Vehicle DOI: http://dx.doi.org/10.5772/intechopen.102748*

**Figure 7.**

*Components of the low-speed wind tunnel of INTA.*

The MAV model was tested with a freestream velocity of the wind tunnel of 10 m/s (*U*<sup>∞</sup> <sup>¼</sup> 10 m*=*s), which results in a Reynolds number of 1.3 � <sup>10</sup><sup>5</sup> based on the wing root chord ð Þ *cr* ¼ 0*:*20 *m* . This analysis was performed for the cruise configuration (with an angle of attack equal to 0°). The experimental tests consisted in obtaining various transversal planes of the flow field at different sections downstream of the trailing edge of the wing.

The test experiments were carried out by using a full-scale model made in plastic material (PLA) by means of additive manufacturing and attached to a wood board by means of a streamlined support strut (see **Figure 8**). Only half of the model was studied due to its symmetry. Moreover, the MAV model and the wood board had to be painted in black in order to avoid reflections of the laser plane. The CCD camera was located behind the model (**Figure 8**), parallel to the flow stream of the wind tunnel.

The flow field velocity was determined by Particle Image Velocimetry (PIV). PIV is an advanced experimental technique that has the advantage of measuring the velocity field in a non-intrusive manner. This technique measures the velocity of the flow by analyzing flow images pairs [12]. For achieving this, PIV requires tracer particles that have to be seeded in the airflow. Olive oil was chosen for the

**Figure 8.** *Experimental setup.*

generation of the tracer particles. A laser sheet has to be generated in order to go through the tracer particles and illuminate them. Two Neodymium-Yttrium Aluminum Garnet (Nd:YAG) lasers and an optical system were used for achieving this. The two lasers Nd:YAG has a pulse energy of 190 mJ within a time gap of 22 μs. The location of the tracer particles has to be recorded by a high-resolution camera with 2048 � 2048 pixels equipped with a lens Nikon Nikkor 50 mm 1:1.4D. A crosscorrelation implemented via Fast Fourier Transform (FFT) is carried out over small image regions in order to obtain the averaged displacement of the tracer particles. The field of view (FOV) of the camera was 190 � 190 mm<sup>2</sup> . The flow images are divided into interrogation window of 32 � 32 pixels. By using the Nyquist Sampling Criteria, these interrogation windows are overlapped by 50%. Moreover, the peak of correlation is adjusted to the subpixel accuracy by Gaussian approximation. A final post-processing task to remove spurious data and fill the empty vectors is needed. Therefore, a local mean filter based on a neighbor kernel window of 3 � 3 vectors was applied.

#### **5. The vorticity in the wingtip wake**

The vorticity is defined as the curl of the flow velocity, by the following expressions (Eq. (13) and Eq. (14)),

$$
\overrightarrow{\phi} = \nabla \times \overrightarrow{\mathbf{V}}\tag{13}
$$

$$\overrightarrow{\boldsymbol{\alpha}} = \left(\frac{\partial \boldsymbol{w}}{\partial \boldsymbol{\eta}} - \frac{\partial \boldsymbol{v}}{\partial \boldsymbol{x}}\right) \overrightarrow{\boldsymbol{i}} + \left(\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}} - \frac{\partial \boldsymbol{w}}{\partial \boldsymbol{x}}\right) \overrightarrow{\boldsymbol{j}} + \left(\frac{\partial \boldsymbol{v}}{\partial \boldsymbol{x}} - \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{y}}\right) \overrightarrow{\boldsymbol{k}}\tag{14}$$

The two-dimensional (2D *y-z* plane) streamwise vorticity *ω<sup>x</sup>* ¼ *ξ* can be determined from measured velocities by solving Eq. (15), which depends on the velocity spatial derivatives, as follows,

$$\vec{\xi} = \frac{\partial w}{\partial \mathbf{y}} - \frac{\partial v}{\partial \mathbf{z}} = \left(\nabla \times \vec{V}\right) \bullet \stackrel{\rightarrow}{i} \tag{15}$$

#### **Figure 9.**

*Non-dimensional axial vorticity measured by PIV at 1.4 c downstream of the trailing edge of the model* ð Þ *U*<sup>∞</sup> ¼ 10*m=s*,*cruise* : *α* ¼ *β* ¼ 0° *.*

The axial vorticity had to be obtained with central differencing in crossflow velocities. The non-dimensional form of axial vorticity component ~*ξ* is given by the following expression (Eq. (16)):

$$\tilde{\xi} = \frac{\left(\frac{b}{2}\right) \bullet \xi}{U\_{\infty}} \tag{16}$$

where *b* is the whole spanwise of the model and *U*<sup>∞</sup> is the frestream velocity.

**Figure 9** shows the non-dimensional axial vorticity after taking PIV measurements in the wake downstream when the vehicle was flying in a cruise condition. It can be seen that the peak of maximum axial vorticity (red region) takes place at the wingtip, and from there it starts to decrease.

#### **6. Circulation and vorticity**

By analyzing the flow downstream of the aircraft model, this flow can be studied as the 2D wingtip wake and the vorticity is related to the velocity circulation from Stokes theorem by the following expression (Eq. (17)), [11].

$$
\Gamma = \oint\_C \vec{V} \cdot d\vec{l} = \iint (\nabla \times \vec{V}) \cdot \vec{n} \cdot dA \tag{17}
$$

where *C* is a closed curve, *V* ! is the flow velocity on a small element defined on the closed curve, and *dl* is the differential length of that small element. As the plane streamwise is the 2D-*yz* plane, we have *ω* ! ¼ *ξ i* ! , and the unit normal vector *n* ! ¼ *i* ! , then (Eq. (18)),

$$
\Gamma = \oint\_C \vec{V} \cdot d\vec{l} = \iint \vec{w} \cdot \vec{n} \cdot dA = \iint \xi \cdot dA \tag{18}
$$

#### **7. Evolution of the vorticity**

The Navier-Stokes equations in vector form for an incompressible flow are given by,

$$\nabla \bullet \overrightarrow{V} = \mathbf{0} \tag{19}$$

$$\frac{\partial \vec{V}}{\partial t} + \vec{V} \cdot \nabla \vec{V} = -\nabla \left(\frac{p}{\rho} + \text{gz}\right) + \nu \nabla^2 \vec{V} \tag{20}$$

The vorticity equation (Eq. (13)) is obtained by taking the curl of the Navier-Stokes equation, as follows,

$$\nabla \times \left(\nabla \bullet \vec{V}\right) = \mathbf{0} \tag{21}$$

$$\nabla \times \left(\frac{\partial \vec{V}}{\partial t} + \left(\vec{V} \cdot \nabla\right) \vec{V} = -\nabla \left(\frac{p}{\rho} + \text{gz}\right) + \nu \nabla^2 \vec{V}\right) \tag{22}$$

By calculating each term, where the conservation of vorticity is Eq. (23),

$$\nabla \times \left(\frac{\partial \vec{V}}{\partial t}\right) = \frac{\partial \vec{\phi}}{\partial t} \tag{23}$$

$$\nabla \times \left( \left( \overrightarrow{\mathbf{V}} \cdot \nabla \right) \overrightarrow{\mathbf{V}} \right) = \left( \overrightarrow{\mathbf{V}} \cdot \nabla \right) \overrightarrow{\boldsymbol{\phi}} - \left( \overrightarrow{\boldsymbol{\phi}} \cdot \nabla \right) \overrightarrow{\mathbf{V}} \tag{24}$$

$$\nabla \times \left( -\nabla \left( \frac{p}{\rho} + \text{gz} \right) \right) = \mathbf{0} \tag{25}$$

$$\nabla \times \left(\nu \nabla^2 \overrightarrow{V}\right) = \nu \nabla^2 \overrightarrow{\phi} \tag{26}$$

and finally, the vorticity equation is,

$$\frac{\partial \overrightarrow{\boldsymbol{\phi} \boldsymbol{\phi}}}{\partial t} + \left(\overrightarrow{\boldsymbol{V}} \cdot \nabla\right) \overrightarrow{\boldsymbol{\phi}} = \left(\overrightarrow{\boldsymbol{\phi}} \cdot \nabla\right) \overrightarrow{\boldsymbol{V}} + \nu \nabla^2 \overrightarrow{\boldsymbol{\phi}} \tag{27}$$

The law of vorticity evolution is a convective vector diffusion equation given by the following expression,

$$\frac{D\overrightarrow{\boldsymbol{\phi}}}{Dt} = \left(\overrightarrow{\boldsymbol{\phi}} \cdot \nabla\right) \overrightarrow{\boldsymbol{V}} + \nu \nabla^2 \overrightarrow{\boldsymbol{\phi}}\tag{28}$$

The viscous term (*ν*∇<sup>2</sup>*ω* !) causes the vortex to diffuse through the fluid flow. By using index notation, the vorticity equation for 3D flow is given by,

$$\frac{\partial \alpha\_i}{\partial t} + \mu\_j \frac{\partial \alpha\_i}{\partial \mathbf{x}\_j} = \alpha\_j \frac{\partial \mu\_i}{\partial \mathbf{x}\_j} + \nu \frac{\partial^2 \alpha\_i}{\partial \mathbf{x}\_k \partial \mathbf{x}\_k} \tag{29}$$

For a 2D flow, the stretching term is absent, and the corresponding equation is,

$$\frac{\partial \alpha\_i}{\partial t} + \mu\_j \frac{\partial \alpha\_i}{\partial \mathbf{x}\_j} = \nu \frac{\partial^2 \alpha\_i}{\partial \mathbf{x}\_k \partial \mathbf{x}\_k} \tag{30}$$

Equivalently, in vector form, for a 2D flow we have the velocity is perpendicular to the vorticity, so *V* ! � *ω* ! ¼ 0 .The velocity is *V* ! ¼ ð Þ 0,*V*, *W* and vorticity *ω* ! ¼ ð Þ *ωx*, 0, 0 , so that,

$$
\overrightarrow{a} \cdot \nabla \overrightarrow{V} = \mathbf{0} \tag{31}
$$

$$\frac{D\overrightarrow{a\nu}}{Dt} = \frac{\partial\overrightarrow{a\nu}}{\partial t} + \left(\overrightarrow{V} \cdot \nabla\right)\overrightarrow{a\nu} = \nu\nabla^2\overrightarrow{a\nu} \tag{32}$$

where the operator *<sup>D</sup> Dt* <sup>¼</sup> *<sup>∂</sup> <sup>∂</sup><sup>t</sup>* þ *V* ! � ∇ � � is the material derivative and it describes the evolution along the flow lines.

#### **8. Decay of wingtip vortices**

The study of the decay of wingtip vortices under the assumption of 2D flow with *<sup>ω</sup><sup>y</sup>* <sup>¼</sup> *<sup>ω</sup><sup>z</sup>* <sup>¼</sup> 0, velocity *Vx* <sup>¼</sup> 0 and *<sup>∂</sup>=∂<sup>x</sup>* <sup>¼</sup> 0, can be performed by the 2D viscous diffusion of vorticity equation, given by,

*Wingtip Vortices of a Biomimetic Micro Air Vehicle DOI: http://dx.doi.org/10.5772/intechopen.102748*

$$
\frac{\partial \overrightarrow{ab}}{\partial t} = \nu \nabla^2 \overrightarrow{ab} \tag{33}
$$

$$\frac{\partial \mathbf{u}}{\partial t} = \nu \cdot \Delta \mathbf{u} \tag{34}$$

Where *ω* ¼ *ω<sup>x</sup>* and Δ is the Laplacian operator.

Assuming axisymmetric flow, in cylindrical coordinates,

$$\frac{\partial \alpha \boldsymbol{\alpha}}{\partial t} = \frac{\nu}{r} \cdot \frac{\partial}{\partial \mathbf{r}} \left( r \frac{\partial \alpha}{\partial r} \right) \tag{35}$$

The initial vorticity for the study of decay point vortex in an unbounded domain is given by a 2D delta function in the plane *yz*,

$$
\rho \left( \vec{\mathbf{x}}, t = \mathbf{0} \right) = \Gamma\_0 \delta(\mathbf{y}) \delta(\mathbf{z}) \tag{36}
$$

Introducing the dimensionless similarity variable [13],

$$
\varepsilon = \frac{r}{\sqrt{\nu t}} \tag{37}
$$

and the nondimensional combination *ωνt=Γ*<sup>0</sup> can be expressed as an unknown function *g* of the variable *ϵ*, defined as

$$
u \mathfrak{u} / \Gamma\_0 = \mathfrak{g}(\mathfrak{e})\tag{38}$$

So that,

$$
\rho = \frac{\Gamma\_0}{\nu t} \mathbf{g}(\epsilon) = f(t) \,\mathbf{g}(\epsilon) \tag{39}
$$

Calculating the derivatives quantities from the earlier equation,

$$\frac{\partial \boldsymbol{\alpha}}{\partial t} = \frac{\partial \boldsymbol{f}(t)}{\partial t} \mathbf{g}(\boldsymbol{\varepsilon}) + \boldsymbol{f}(t) \frac{\partial \mathbf{g}(\boldsymbol{\varepsilon})}{\partial t} = -\frac{\Gamma\_0}{\nu t} \frac{\mathbf{1}}{\mathbf{t}} \mathbf{g}(\boldsymbol{\varepsilon}) + \boldsymbol{f}(t) \frac{d \mathbf{g}(\boldsymbol{\varepsilon})}{d \boldsymbol{\varepsilon}} \frac{\partial \boldsymbol{\varepsilon}}{\partial t} \tag{40}$$

$$\frac{\partial \boldsymbol{\alpha}}{\partial t} = -f(t)\frac{1}{t}\mathbf{g}(\boldsymbol{\epsilon}) - f(t)\frac{\boldsymbol{\epsilon}}{2t}\frac{d\mathbf{g}(\boldsymbol{\epsilon})}{d\boldsymbol{\epsilon}} = -f(t)\frac{1}{t}(\mathbf{g} + \boldsymbol{\epsilon}\mathbf{g}'/2) \tag{41}$$

On the other hand,

$$\frac{\partial \boldsymbol{\alpha}}{\partial r} = f(t) \frac{\partial \mathbf{g}(\boldsymbol{\varepsilon})}{\partial r} = f(t) \left( \frac{\partial \boldsymbol{\varepsilon}}{\partial r} \frac{d \mathbf{g}(\boldsymbol{\varepsilon})}{d \boldsymbol{\varepsilon}} \right) = f(t) \left( \frac{\boldsymbol{\varepsilon}}{r} \frac{d \mathbf{g}(\boldsymbol{\varepsilon})}{d \boldsymbol{\varepsilon}} \right) \tag{42}$$

$$\frac{\partial \boldsymbol{\alpha}}{\partial r} = f(t) \frac{\partial \mathbf{g}(\boldsymbol{\varepsilon})}{\partial r} = f(t) \left( \frac{\partial \boldsymbol{\varepsilon}}{\partial r} \frac{d \mathbf{g}(\boldsymbol{\varepsilon})}{d \boldsymbol{\varepsilon}} \right) = f(t) \left( \frac{\boldsymbol{\varepsilon}}{r} \frac{d \mathbf{g}(\boldsymbol{\varepsilon})}{d \boldsymbol{\varepsilon}} \right) \tag{43}$$

$$\frac{\partial}{\partial r}\left(r\frac{\partial \boldsymbol{\omega}}{\partial r}\right) = \frac{\partial \boldsymbol{\varepsilon}}{\partial r}\frac{d}{d\boldsymbol{\varepsilon}}\left(\boldsymbol{f}(t)\,\boldsymbol{\varepsilon}\,\,\frac{d\mathbf{g}(\boldsymbol{\varepsilon})}{d\boldsymbol{\varepsilon}}\right) = \frac{\boldsymbol{\varepsilon}}{r}\boldsymbol{f}(t)\frac{d}{d\boldsymbol{\varepsilon}}\left(\boldsymbol{\varepsilon}\,\,\frac{d\mathbf{g}(\boldsymbol{\varepsilon})}{d\boldsymbol{\varepsilon}}\right) \tag{44}$$

And substituting in (35), the following expression is derived,

$$2(\epsilon \mathbf{g'})' + \epsilon^2 \mathbf{g'} + 2\mathbf{g}\epsilon = \mathbf{0} \tag{45}$$

Where 0 denotes a derivative respect to, and finally, the equation is integrated

$$\mathbf{g}(\epsilon) = \mathbf{A} \cdot \exp\left(\frac{-\epsilon^2}{4}\right) \tag{46}$$

The condition of the flow circulation is equal to *Γ*<sup>0</sup> at any time, gives,

$$
\Gamma\_0 = \int\_0^\infty a 2\pi r \, dr = 4\pi A \Gamma\_0 \tag{47}
$$

so that *A* ¼ 1*=*4*π,* and the solution of the *g*ð Þ*ϵ* function is,

$$g(\epsilon) = \frac{1}{4\pi} \exp\left(\frac{-r^2}{4\nu t}\right) \tag{48}$$

Finally, the solution of vorticity is given by the axisymmetric Lamb-Osteen vortex by,

$$
\rho = \frac{\Gamma\_0}{4\pi\nu t} \,\, \exp\left(\frac{-r^2}{4\nu t}\right) \tag{49}
$$

The swirl velocity is,

$$V\_{\theta} = \frac{\Gamma\_0}{2\pi r} \left( 1 - \exp\frac{-r^2}{4\nu t} \right) \tag{50}$$

and the circulation is,

$$
\Gamma = \Gamma\_0 \left( 1 - \exp \frac{-r^2}{4\nu t} \right) \tag{51}
$$

The swirl velocity can be rewritten as,

$$V\_{\theta} = \frac{\Gamma\_0}{2\pi r} \left( 1 - \exp\left(-1.2526 \left(\gamma\_{r\_\circ}\right)^2\right) \right) \tag{52}$$

where *rc* is the core radius, defined as the distance from the vortex center to the point with the higher swirl velocity, and given by,

$$r\_{\varepsilon} = 2.24\sqrt{\nu t} \tag{53}$$

#### **9. Analysis of vortex models and experimental data**

The velocity components which define a 2-D vortex are typically the swirl velocity *Vθ*, the axial velocity *Vz* and the radial *Vr* velocity. The last two components usually are neglected in many applications as they are very small compared to swirl velocity, and are defined as follows,

$$V\_{\theta} = \frac{\Gamma}{2\pi r} \tag{54}$$

$$V\_r = \mathbf{0} \tag{55}$$

*Wingtip Vortices of a Biomimetic Micro Air Vehicle DOI: http://dx.doi.org/10.5772/intechopen.102748*

$$V\_x = \mathbf{0} \tag{56}$$

Several tip vortex models are usually studied, but this chapter is only focused on some of them, displayed in **Figure 10**. The first method is the Rankine vortex model, being the simplest formulation with a finite core. Therefore, the vortex is divided into two parts: the viscous core and the potential vortex. The viscous core is rotating as a solid body near the vortex center while the potential vortex remains away from the vortex center. The velocity in the potential vortex is decreasing hyperbolically with the radial coordinate [14, 15]. Therefore, the following expressions represent the swirl velocity distribution *V<sup>θ</sup>* in the tip vortex,

$$V\_{\theta}(\tilde{r}\,) = \left(\frac{\Gamma}{2\pi r\_c}\right) \bullet \tilde{r}\,\, 0 \le \tilde{r} \le 1\tag{57}$$

$$V\_{\theta} = \frac{\Gamma}{2\pi r} \,\,\tilde{r} > 1\tag{58}$$

Where *rc* is the viscous core radius and <sup>~</sup>*<sup>r</sup>* <sup>¼</sup> *<sup>r</sup> rc* is the non-dimensional radial coordinate.

The second vortex model is the Lamb-Oseen vortex which is a simplified solution of one-dimensional Navier-Stokes equations for laminar flow which is defined by the following expression,

$$V\_{\theta}(\tilde{r}\,) = \left(\frac{\Gamma}{2\pi r}\right) \bullet \left[1 - e^{-a(\tilde{r})^2}\right] \tag{59}$$

where *α* ¼ 1*:*2526 is the Oseen parameter.

An alternative tip vortex formulation is given by Vatistas in Ref. [15]. This method is based on a group of desingularized algebraic swirl velocity profiles for vortices which present continuous distributions of flow quantities. The swirl velocity is defined by,

$$V\_{\theta}(\tilde{r}) = \frac{\Gamma}{2\pi r\_{\circ}} \bullet \frac{\tilde{r}}{\left(1 + \tilde{r}^{2n}\right)^{1/n}} \tag{60}$$

where *n* is an integer.

**Figure 10.** *Swirl velocity distribution inside a tip vortex was obtained by several tip vortex models.*

The Scully vortex model is the previous formulation when the integer is *n* = 1, and it is defined as,

$$V\_{\theta}(\tilde{r}) = \left(\frac{\Gamma}{2\pi r\_c}\right) \bullet \frac{\tilde{r}}{\left(1 + \tilde{r}^2\right)}\tag{61}$$

when the integer is *n* = 2, the swirl velocity of the vortex formulation is,

$$V\_{\theta}(\tilde{r}) = \left(\frac{\Gamma}{2\pi r\_c}\right) \bullet \frac{\tilde{r}}{\sqrt{1+\tilde{r}^4}}\tag{62}$$

It is important to notice that when the integer *n* ! ∞, the swirl velocity distribution corresponds to the Rankine method.

**Figure 11** shows the flow field velocity in a normal section to the flow located at 3 chords downstream of the MAV model. The 2d vortex can be observed clearly and the color scale indicates that the velocity is increasing near the center of the vortex.

It is possible to obtain a better visualization of the flow field distribution by looking at **Figure 12**. This PIV map is obtained for the angle of attack of 10°. The plotted streamlines reveal the location of the vortex center (places at Y = Z = 0 mm), the region of the vortex core (yellow region), and the external region (green area).

Extracting the data value of the swirl velocity as measured by the PIV technique we can obtain **Figure 13** when the experimental data are plotted with curves of theoretical vortex models. The blue scatter dots which its trend is approached by a 6th-degree polynomial (red continue line).

Also, the distributions of the swirl velocity obtained by the theoretical vortex models as Rankine, Lamb-Oseen, and Scully are represented in **Figure 11**.

The analysis of this graph shows the wingtip vortex method which presents the most accurate fit to the MAV is obtained with the tip vortex model of Scully. Subsequently, there is a deviation between the two approaches (experimental data and Scully) which depends on the distance from the vortex center. The ratio between both of them is assessed by the parameter *k r*ð Þ defined as

**Figure 11.** *Wingtip vortex in the MAV wake at 3c.*

*Wingtip Vortices of a Biomimetic Micro Air Vehicle DOI: http://dx.doi.org/10.5772/intechopen.102748*

**Figure 12.** *Velocity distribution at 3c.*

**Figure 13.** *Experimental data and theoretical vortex models.*

$$k(r) = \frac{(V\_{\theta})\_{polynomial}}{(V\_{\theta})\_{\text{Sually}}} \tag{63}$$

where ð Þ *V<sup>θ</sup> polynomial* and ð Þ *V<sup>θ</sup> Scully* are the distributions of swirl velocity obtained in the test experiments and by the theoretical model proposed by Scully, respectively.

Finally, the distribution of experimental swirl velocity is fitted to the Scully model by the function called ð Þ *<sup>V</sup><sup>θ</sup> experimental*�*Scully* defined as,

$$(\left(V\_{\theta}\right)\_{\text{experimental}-\text{Sually}} = k(r) \bullet \left(V\_{\theta}\right)\_{\text{Sually}}\tag{64}$$

#### **10. Lift coefficient**

The lift of an airfoil can be determined by the Kutta-Joukowski theorem [11] relating the velocity and the circulation, as follows,


**Table 2.**

*Results of the tip vortex analysis in the wake of MAV.*

$$L' = \rho U\_{\text{\textquotedblleft}} \Gamma \tag{65}$$

By applying the earlier formulation, the total lift of the wing *L* can be obtained from the following expression

$$L = (\rho U\_{\Leftrightarrow} \Gamma) \bullet b \tag{66}$$

where b is the wingspan.

The lift coefficient *CL* is obtained by dividing the lift by *q*∞*Sref* ,

$$\mathbf{C}\_{L} = \frac{(\rho U\_{\infty} \Gamma) \bullet b}{q\_{\infty} \mathbf{S}\_{\text{ref}}} \tag{67}$$

where *<sup>q</sup>*<sup>∞</sup> is the dynamic pressure (*q*<sup>∞</sup> <sup>¼</sup> <sup>1</sup>*=*2*ρU*<sup>2</sup> <sup>∞</sup>) and *Sref* is the reference wing surface.

**Table 2** shows the values of the main parameters obtained from the tip vortex analysis, including the lift coefficient, *CL:*

#### **11. Conclusions**

Wingtip vortices generated behind an aircraft wing affect the aerodynamic performance of the aircraft while endangering take-off and landing maneuvers of the subsequent aircraft.

In this chapter, it is reviewed the theoretical background of the horseshoe vortex and several vortex models applied to a Biomimetic Micro Air Vehicle (MAV) with Zimmerman planform. The formulation of the vorticity in the wingtip wake of the MAV model has been presented as well as the expression which relates the axial vorticity and the circulation.

All experimental tests have been carried out in the Low-Speed Wind Tunnel of the Instituto Nacional de Técnica Aeroespacial (INTA) with a full-scale MAV model. Particle Image Velocimetry has been used to obtain the transversal flow field at 3 chords downstream of the trailing edge of the MAV model. The swirl velocity distribution according to the horseshoe vortex model and several vortex models (Rankine, Lamb-Oseen, Scully, and Vatistas) is plotted. The experimental results have shown that the Scully vortex has the most similar behavior to the MAV wing

*Wingtip Vortices of a Biomimetic Micro Air Vehicle DOI: http://dx.doi.org/10.5772/intechopen.102748*

vortex. The distribution of the transversal velocity as well as the axial vorticity for the section at 3 chords are presented by PIV maps. Finally, the lift coefficient by using the Kutta-Joukowski theorem is obtained.

### **Acknowledgements**

This investigation was funded by the Spanish Ministry of Defense under the program "464A 64 1999 14205005 Termofluidodinámica" with internal code IGB 99001 of Instituto Nacional de Técnica Aeroespacial "Esteban Terradas" (INTA-National Institute for Aerospace Technology of Spain).

### **Conflict of interest**

The authors declare that they have not known existing or potential Conflicts of Interest, including financial or personal factors, as well as any relationship which could influence their scientific work.

#### **Author details**

Rafael Bardera\*, Estela Barroso and Juan Carlos Matías National Institute for Aerospace Technology, Madrid, Spain

\*Address all correspondence to: barderar@inta.es

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[8] Flake J, Frischknecht B, Hansen S, Knoebel N, Ostler J, Tuley B. "Development of the Stableyes Unmanned Air Vehicle", 8th International Micro Air Vehicle Competition. Tucson, AZ: The University of Arizona; 2004. pp. 1-10 [9] Barcala-Montejano MA, Rodríguez-Sevillano A, Crespo-Moreno J, Bardera Mora R, Silva-González AJ. Optimized performance of a morphing micro air vehicle. Unmanned Aircraft Systems (ICUAS), 2015 International Conference. IEEE Journal of Intelligent Material Systems and Structures. Denver Marriott Tech Center, Denver, Colorado, USA: IEEE; June 9-12, 2015. pp. 794-800. DOI: 10.1109/ ICUAS.2015.7152363

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[13] Pozrikidis C. Introduction to Theoretical and Computational Fluid Dynamics. second ed. USA: Oxford University Press (OUP); 1997. p. 268

[14] Djojodihardjo H. Analysis and visualization studies of near field aircraft trailing vortices for passive wake alleviation. Novosibirsk, Russia: The 13th Asian symposium on visualization, June 22–26; 2015

[15] Mahendra JJ, Gordon L. Generalized viscous vortex model for application to free-vortex wake and aeroacoustic calculations. University of Maryland, College Park, Maryland: Alfred Gessow Rotorcraft Center, Department of Aerospace Engineering, Glenn L. Martin Institute of Technology; 2002

### *Edited by İlkay Bakırtaş and Nalan Antar*

This book discusses vortex dynamics theory from physics, mathematics, and engineering perspectives. It includes nine chapters that cover a variety of research results related to vortex dynamics including nonlinear optics, fluid dynamics, and plasma physics.

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Vortex Dynamics - From Physical to Mathematical Aspects

Vortex Dynamics

From Physical to Mathematical Aspects

*Edited by İlkay Bakırtaş and Nalan Antar*