Wake turbulence observed behind an upstream "extra" particle in a complex (dusty) plasma

Wake turbulence is known to form behind any object that is moving fast enough, for instance, behind an aircraft as it passes through the air [1]. It is important to study this phenomenon in many aspects, e.g., regarding the safety of the take-off of following flights or flights from crossing runways. A turbulent wake is common to observe, but its particular structure depends strongly on conditions at which the object passes through the medium. A famous example is the Karman vortex street [2] (see also, e.g., [3]). In particular, a well-studied pattern of this kind is formed behind the cylinder [4,5] or a flat plate [6] in an air or liquid stream. Decaying grid turbulence with self-similarity [7] adds more spectacular multi-wake exemplars in fluids [8,9] and soap films [10].

The wake effect involving elements of turbulent pulsations is known to be important when a sand flow interacts with an immobile aluminium cylinder [11]. A peculiarly structured Mach-cone–type wake (or "shock") is formed behind a stainless-steel rod inserted into and moving in a shallow, vibro-fluidized granular layer consisting of bronze spheres [12]. The shocks formed were interpreted as an example of Cerenkov radiation generated by an object traveling through a medium faster than the wave phase velocity. The turbulent wake formation is interesting also for explorations of the onset of turbulence [13].

The turbulent wake behind small objects is especially interesting to study as it has important implications in many physical and biophysical applications as diverse as self-propelled stochastic microdevices, turbulence around small organisms, "bacterial" quasi-turbulence, in insect flight, and many others [14–19]. Regarding the wake formed behind small-size objects there are of interest either induced turbulence [20,21] or interactions with the turbulent flow [22]. The microzones surrounding small organisms as well as the turbulent flow at small scales are also well-known challenging issues [18,19].

A great advantage of turbulence studies using complex plasmas is that the particles that transmit the interaction can be visualized directly [23,24]. For instance, vortices in complex plasmas could be an ideal test bed for studying the onset of turbulence and collective effects on the kinetic level [25]. Complex plasmas consist of micrometer-sized particles embedded in a low-temperature plasma. Under certain conditions they are highly ordered and form plasma crystals [26–28].

A Mach cone (or wake) is formed when the monolayer plasma crystal is perturbed by a supersonic charged particle-projectile [29–34]. Experimentally, the Mach cones (wakes) are often used as a robust mean to test the plasma crystal for diagnostic purposes [35,36] as well as to heat it [37]. A "repulsion-dominated" Mach cone is created by a fast "extra" particle moving below the main particle monolayer [31,32]. Such "downstream" particles are heavier than the particles in the main layer and, therefore, their tracks are only weakly perturbed by the presence of the monolayer. The wake formed behind the downstream particle is fine structured in all three directions, involving in-plane and vertical displacements of the particles in a monolayer lattice as was recently discovered [33].

An "attraction-dominated" Mach cone is initiated when the particle passes above the monolayer [29,30]. Those "upstream" particles are lighter than the particles inside the main layer, and, therefore, are easier for it to influence. It gives a unique opportunity to observe rather delicate phenomena accompanying the interaction of the plasma crystal with upstream particles, e.g., leading to constraining (channeling) of the path of the upstream particle [29]. The character of the upstream extra particle motion depends on the local structure of the lattice layer. The path of the well-channeled particle remains smooth and is followed by a regular highly structured wake [29].

In this letter we report on a turbulent wake formed behind the upstream particle-traveling above a highly ordered lattice layer. This principally new dynamical pattern is formed due to strong upstream particle-lattice layer interactions. The dynamics of the turbulent wake in complex plasmas is of great interest to study as it is perhaps the simplest paradigm of stochastic or quasi-chaotic billiards driven externally [38–40]. To the best of our knowledge, persistent turbulent wakes in a highly ordered lattice layer have not been reported so far. Implications for diagnostic purposes are also briefly discussed.

The interaction of upstream extra particles with a 2D plasma crystal was studied by using a modified Gaseous Electronics Conference (GEC) rf reference cell [29,30]. An argon plasma was sustained using a capacitively coupled rf discharge at 13.56 MHz and rf power at 20 W [29]. Monodisperse polystyrene (PS) particles were used to create 2D plasma crystals suspended above the bottom rf electrode. The PS particles have a diameter of $11.36\pm0.12\ \mu \text{m}$ and mass density of 1.05 g/cm³. The gas pressure was maintained at about 0.65 Pa; the corresponding neutral gas damping rate was 0.91 s⁻¹ [41]. After the particles were injected into the plasma, they formed a single-layer suspension with a size of 50–60 mm. In addition, some upstream extra particles that levitated above the main layer were present perturbing the layer¹ .

The particle dynamics was recorded in high resolution using a high-speed Photron camera at a frame rate of 250 fps. The recorded frame sequences were then used to track the particle positions from one frame to another as explained in detail in ref. [29]. Before calculating the particle velocities, the obtained tracks were filtered by applying a standard filtering procedure to reduce the influence of industrial noise (∼ 50 Hz). Only the high-quality filtered data were used further to compute the particle velocity field and its differential characteristics discussed below. The experimental parameters are listed in table 1.

Table 1:.  Compressional and transverse sound speed C_L,T, projectile mean velocity $\langle V\rangle$ and velocity V^inst at time of fig. 1, the spatial damping increment κ, and the thermal diffusivity χ for the "normal" channeling (case 1, fig. 1 right panels) and "abnormal" strong scattering event (case 2, fig. 1, left panels).

case	$C_L/C_T$	$\langle V\rangle$	Vinst	κ	χ
	mm/s	mm/s	mm/s	mm−1	mm2/s
1	27/6α	29	28	0.3β	16β
2	18/5γ	24	22	3.7	3.0

^αMeasured in ref. [29]; ^βmeasured in ref. [30]; ^γwith accuracy ±15%; with accuracy ±24%.

Examples of laminar and turbulent wakes, emerging behind a fast moving particle, are shown in fig. 1. This figure exhibits the vorticity and divergence maps of the velocity fields formed behind the either strongly scattered (fig. 1, left panels) or channeled (fig. 1, right panels) upstream particle. Strikingly different patterns are apparent there.

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A peculiar wake with a collision-dominated quasi-chaotic pattern inside its core is activated behind the strongly scattered particle (fig. 1, left panels). Curiously, the divergence and the vorticity maps are difficult to distinguish. This indicates an approximate equipartition of energy and momentum between the shear and compression components of the turbulent velocity field. Typically, the size of the wake core does not exceed about 2–3 mm; a relatively weak and more extended "halo" is also present; see fig. 2.

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In contrast, an extended fine structure, patterning the wake, is noticeable behind the channeled upstream particle (fig. 1, right panels). The structure of the compressional wide-angle Mach cone represented by the divergence map corresponds very well to theoretical expectations [34]. The vorticity map helps to discover also a narrow-angle shear Mach cone [42] located mostly along the particle-projectile path. The appearance of the shear Mach cone originates in a peculiar character of deformation of the channel walls caused by the upstream particle passing through the channel [29]. The half-open angle θ of the cones in fig. 1 follows the well-known Mach cone rule:

$$\begin{equation} \sin\theta_{L,T}=C_{L,T}/v_d, \end{equation} \tag{ 1 }$$

where C_L,T is the speed of sound, v_d the projectile velocity, and L,T stands for longitudinal/transverse waves. Note that simultaneous compressional and shear wakes excited by a moving spot of laser light have been reported in [42]. The neat "coexistence" of the naturally excited shear and compressional Mach cones is reported for the first time.

Turbulent wakes are as frequent as the laminar wakes discovered in [30], and likewise, their appearance cannot be controlled experimentally² .

To quantify the existence of turbulence in the wake, we calculated the isotropic part of the phonon energy of the crystal (see fig. 1, bottom panels). In the case of the channeled upstream particle the main part of the energy corresponds to the thermal phonon spectrum ($\propto\exp\left(-k/k_T\right)$ [43]), while in the case of the strongly scattered upstream particle almost all phonons exhibit the direct cascade power law $(\propto k^{-5/3})$ Kolmogorov [44] spectrum. It is worth noting also that the reduced kinetic temperature of the turbulent wake is about two orders of magnitude higher than that of the laminar Mach cone (see fig. 3), and that its thermal diffusivity is anomalously low (see table 1 and fig. 4).

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Dynamically, the highly collisional wake observed in the given experiments is equivalent to a time-dependent billiard system, and should be treated correspondingly [45]. The description of the wake in our case could be significantly simplified, though. The energy of the wake particles changes essentially at the time scale of a single collision, the particles are "forgetting" their initial velocities, and, hence, the wake energy distribution evolves diffusively. In other words, the energy distribution can be described by a Fokker-Planck equation, or, under certain simplifying conditions, by a diffusion model [46].

The wake immediately behind the projectile blows up forming a compact core which gradually transforms into an asymmetric halo. The latter extends backwards slowly decaying along the path of travel; see fig. 1 and fig. 2. The "core+halo" structuring persists while the core slowly collapses when the projectile path approaches the monolayer center. This can be interpreted as a weak large-scale variation of the crystal transport coefficients. The heat transport can then be modeled as follows:

$$\begin{equation} \partial_tT=-2\gamma T+\nabla\cdot\left[\chi\nabla T\right]+S(t,\text{r}), \end{equation} \tag{ 2 }$$

where T is the kinetic temperature of the monolayer particles, t the time, γ the friction drag coefficient [41], χ the turbulent (anomalous) thermal diffusivity, S the source of perturbation, and r symbolizes the radius vector. Let us suppose, for example, a point-like perturbing source: $S(t,\mathbf{r})\propto \delta\left(x+Vt\right)\delta\left(y\right)$, where x and y are the coordinates in the monolayer plane, $V\equiv \mathbf{V}_x$ is the velocity of the projectile, and δ is the Dirac δ-function. Assuming further a locally constant thermal diffusivity χ, one readily finds an analytical solution:

$$\begin{equation} T(t,\mathbf{r})\propto \exp\left(-\frac{V\eta}{2\chi}\right)K_0\left(\kappa R\right), \quad R=\sqrt{\eta^2+y^2}, \end{equation} \tag{ 3 }$$

where $\eta=x+Vt$, K₀ is the modified Bessel function, and

$$\begin{equation} \kappa =\sqrt{\frac{2\gamma}{\chi}+\frac{V^2}{4\chi^2}}. \end{equation} \tag{ 4 }$$

The solution (3), (4) is comparatively simple, approximates well the waveform of the wake (see fig. 2) and, therefore, serves as a convenient mean to measure the heat transport coefficient.

It is important to mention that the projectile velocity V and the damping rate γ are independent of the heat transport model. Therefore, the only free parameter in the model (3), (4) is the thermal diffusivity coefficient χ. This fact can be conveniently used for diagnostic purposes.

The interparticle separation, which is approximately the mean free path of the wake particles, is $a\approx 0.65\ \text{mm}$, the wake core size is $w=2\text{--}4\ \text{mm}$, and the typical length of the projectile travel path $L=20\text{--}30\ \text{mm}$ in the turbulent wake experiment. Therefore, the following inequalities hold: a < w < L. Based on this we conclude that i) the diffusion wake model is applicable, at least qualitatively; ii) since the wake size is much less than the length of the projectile path of travel, the thermal diffusivity coefficient can be treated as locally constant.

Note that the energy distribution, being highly asymmetric along the path of travel (3), is fairly symmetric transversally (cf. [30]). Moreover, the transverse distribution decays approximately exponentially; see fig. 2(a). Therefore, it is advantageous to explore the transverse energy distribution, which significantly simplifies the fitting procedure. The thermal diffusivity coefficient can be obtained by merging the measured spatial damping increment of the experimental energy distributions to the theoretical spatial damping increment (4). The results of these calculations are shown in fig. 4(b). The spatial resolution in this method is limited by the wake core size.

The high collision rate of the particles inside the wake core makes it possible to suggest that turbulence has enough time to become quasi-isotropic. This means, in particular, that the core size must be of the order of the Kolmogorov turbulence scale length $L\simeq L_K\approx (\nu^3/\epsilon)^{1/4}$ [44], where ν is the wake viscosity, $\epsilon \approx \nu\Omega^2$ is the (reduced) energy damping rate, and Ω is the vorticity. The latter, in turn, could be roughly estimated as $|\Omega|\approx V/L$. From relationship (4) it follows then that the wake Prandtl number is

$$\begin{equation} P=\frac{\nu}{\chi}\simeq\frac{VL}{\chi}=\frac{2}{\sqrt{1+\xi}}, \quad \xi=\frac{8\gamma\chi}{V^2}. \end{equation} \tag{ 5 }$$

As friction is weak $\xi\ll1$, the wake Prandtl number $P\simeq 2$. To compare, it is certainly higher than the Prandtl number of the argon gas in which the discharge was triggered [47] and is less than, e.g., that of water [48].

As expected, the anomalous wake appears to be more viscous than a liquid complex plasma [49].

To conclude, the interaction of an extra upstream particle with a plasma crystal results in the appearance of an unusual dynamic pattern which we termed the turbulent wake. The complex plasma of the turbulent wake exhibits the anomalously low thermal diffusivity which makes it overheated compared to the normal laminar wake (Mach cone). Experiments with turbulent wakes are a promising mean to observe the anomalous transport in complex plasmas.

We acknowledge support from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 267499 and from the National Natural Science Foundation of China (NNSFC), Grant No. 11405030. GEM wishes to acknowledge the support from the Russian Science Foundation under grant 14-43-00053.