# Relativistic Wind Farm Effect: Possibly Turbulent Flow of a Charged,
Massless Relativistic Fluid in Graphene

Mark Watson  Department of Physics, University of Colorado, Colorado Springs,
Colorado 80904, USA

###### Abstract

At low Reynolds numbers, the wind flow in the wake of a single wind turbine is
generally not turbulent. However, turbines in wind farms affect each other’s
wakes so that a turbulent flow can arise. In the present work, an analogue of
this effect for the massless charge carrier flow around obstacles in graphene
is outlined. We use a relativistic hydrodynamic simulation to analyze the flow
in a sample containing impurities. Depending on the density of impurities in
the sample, we indeed find evidence for potentially turbulent flow and discuss
experimental consequences.

##  I Introduction

Graphene is a sheet of graphite, one carbon atom thick, with interesting
physical and electrical properties. Its provides an excellent experimental
platform to observe electric transport properties and test theories on
conductivity at moderately high temperatures. The two-dimensional solid is
made of carbon atoms arranged in a honeycomb lattice, showing a high
conductivity through a simple band structure. Of focused interest is the
charge neutral state where each carbon atom contains exactly one electron so
that half of the energy levels are occupied. In this state the band gap
disappears at points on the Fermi surface called Dirac points, and the
dispersion relation is nearly linear surrounding the points and form a conical
Fermi surface at low energies. Deviations from the linear dispersion begin to
appear at a very high temperature, $\approx 10^{5}K$, validated experimentally
in [1][2]. The electric charge is transported through the conical band
structure by highly mobile, massless chiral quasi-particles [3] referred to as
Dirac particles. The number density of the quasi-particles is determined using
electrical properties such as the Hall resistance, which normally assumes an
integer value in a two-dimensional solid, but in graphene is found in
multiples of $1/2$, known as the fractional Quantum Hall Effect [4]. In the
presence of a strong magnetic field the wave functions of the quasi-particles
behave like simple quantum harmonic oscillators that take on discrete energy
levels based on the strength of the magnetic field. This effect, called the
Shubnikov-De Haas effect, is used to determine the effective mass of the
charge carrying quasi-particles.

The dynamics of the quasi-particles constituting the electric current can be
described collectively with hydrodynamics, but since the quasi-particles have
no mass and are subject to Dirac’s equation their collective velocity
distribution cannot be given by a Maxwell-Boltzmann distribution. Hence,
despite having typical velocities much less than the speed of light, the flow
of these particles is better described with relativistic hydrodynamics.
Because of this the study of the electric current in graphene can also be
applicable to some astrophysical systems such as gamma ray bursts, supernovae
or the flow of particles around the event horizon of a black hole. These
systems share the property of a relativistic hydrodynamic flow, but are of
course, much less accessible to experiment. See [5] [6] for recent textbooks
on relativistic hydrodynamics.

The transport of the charge carrying quasi-particles within graphene
demonstrates a high conductivity that stays above a minimum value, even when
carrier concentrations approach zero [3]. Additionally, the viscosity to
entropy ratio is determined to be very small; smaller than that of superfluid
helium [7]. A sample of graphene will contain impurities embedded within the
honeycomb lattice structure or within the supporting substrate that will
affect the current based on their number density, size, placement and electric
properties. In a hydrodynamic flow, if the average fluid speed around these
obstructions is large enough compared to the viscous damping effect, a
turbulent flow is possible.

In the present study we note a dependency of turbulence in the Dirac fluid in
graphene on the position of an obstruction relative to other impurities
present in the sample. The effect is similar to the susceptibility of a wind
farm to turbulence based on the placement of the turbines. An engineer must
take into account the effect the wake a turbine has on downwind turbines when
determining their placement. At high wind velocities a single turbine can
create a vortex street, or regularly sized and spaced vortices, within the
flow of its wake (fig. 1 left). If another turbine is placed relatively close
to its upstream neighbor near its wake, a turbulent flow is shown to be
possible (fig. 1 right) which will affect the efficiency of the farm [8].

Figure 1: The left shows a vortex street effect a single wind turbine can have
on the air flow in its wake when the wind velocity is large enough. The right
sketches a turbulent wind flow caused by additional wind turbines positioned
in the wake of another. The vortex street can turn into a turbulent flow.

Turbulence is a non-linear, chaotic flow disruption that results in the
creation of vortices that can cause fluctuations in the current. It is
difficult to identify the presence of turbulence qualitatively. The Reynolds
number is the unitless ratio of the characteristic length $L$ and average
(macroscopic) fluid speed of the system $U$ to the viscous damping term $\nu$;
$Re=\frac{LU}{\nu}$. It is commonly used to identify the boundary between a
turbulent flow and a simple chaotic or Laminar flow. A large ratio indicates
an over-matched damping term, and turbulence is thought to be present when it
is larger than $5000$. If the ratio is smaller than that threshold, but larger
than $10$ or $100$, the flow is thought to be in the preturbulent regime. Any
flow with a Reynolds number below that is not believed to be turbulent or
preturbulent. Though charge neutral graphene is an exceptionally conductive
material and the charged quasi-particles are able to obtain a large velocity
compared to the viscosity, it is believed that it is unlikely to be capable of
producing a turbulent flow [9].

However, a recent study by Mendoza, Herrmann, and Succi [10] modeled a $5\mu
m$ impurity obstructing a quasi-particle flow of velocity $10^{5}$ $m/s$ using
a two-dimensional relativistic hydrodynamic numerical model. They noted a
pattern of vortex shedding, a phenomenon of preturbulence, by direct
observation and through fluctuations in the current density. They calculate a
Reynolds number on the order of $100$ and proposed that, under these
circumstances, graphene could potentially produce a flow in the preturbulent
range.

The present report employs a similar hydrodynamic modeler based on the lattice
Boltzmann method to reproduce the flow of a charged Dirac fluid in an
experimentally realistic sample of graphene, incorporating impurities embedded
within the honeycomb lattice (or in the substrate). Using experimentally
realistic parameters, though the estimate for the Reynolds number in this work
is smaller, we find a signal of possible preturbulent flow, corroborating the
results of [10]. What is new in this work is that we also consider _multiple_
impurities in a single sample and investigate if a relativistic flow around
these can give rise to turbulent (as opposed to preturbulent) flow signals.

##  II Model and Methods

A turbulent flow is characterized by chaotic changes in pressure and velocity
throughout the fluid as a result of large kinetic energy in parts of the fluid
that overcome the viscous damping. The Reynolds number provides the most
commonly accepted metric to predict the presence of turbulence. Classically
the Reynolds number is determined, as stated, using the kinematic viscosity
$\nu$ defined as the ratio of the dynamic viscosity $\eta$ to the fluid’s mass
density $\rho$; $\nu=\frac{\eta}{\rho}$. For fluid systems consisting of
massless particles, formulations of the Reynolds number commonly replace the
mass density with the entropy density $s$. Therefore, the expression for the
dimensionless viscosity is $\frac{\eta}{s}$, and the Reynolds number is
expressed as $Re=\frac{UL}{(c^{2}/T)(\eta/s)}$, where $\frac{c^{2}}{T}$
balances the units. The Reynolds number found in [10] is determined with this
formulation. Alternatively, one is able to retain the classical expression of
kinematic viscosity for a fluid of massless particles by defining the mass
density in terms of the number density of the quasi-particles and their
“effective” mass; $\rho=nm_{e}$. The number density is determined
experimentally using techniques such as the measurement of the Hall
resistance, while the effective electron mass is determined through methods
such as the Shubnikov-De Haas effect. The resulting Reynolds number is a ratio
incorporating the velocity of a volume of massless particles with respect to
inertial mass to the kinematic viscosity defined in terms of the effective
mass with respect to the particles’ electric properties. This more classical
form of the Reynolds number is more readily compared to that of traditional
fluids.

Turbulence can also be identified through its effect on fluctuations in the
current density. Through non-linear dynamics a turbulent flow creates self-
organizing vortices within the fluid, and its effects on the current density
are characterized by the emergence of multiple traveling modes in Fourier
space. The non-linear effects in the flow can produce a breaking of the
longitudinal symmetry in a Laminar flow causing vortices to spontaneously
emerge. The vortex structures cohere to each other, clockwise rotating
structures to other clockwise structures, and counterclockwise to
counterclockwise. The structures eventually detach at larger scales, shedding
away from the vortex at periodic intervals. This is referred to as vortex
shedding. When the vortices begin this pattern of coherence and shedding, the
flow creates fluctuations in the current density with a defining spectral
signal in frequency space and in wave number space. When the flow reaches a
state of turbulence, the modes migrate from a few initial prominent modes and
the spectrum becomes broadband. If the broadband signal is present in both
frequency space and in k-space the flow is considered turbulent as well as
chaotic [11]. However, if the signal is narrow banded in either space, that
is, the modes do not migrate temporally or spatially, then the flow (though
still potentially chaotic) is not considered turbulent. Note that while the
k-space spectrum is easily determined in a modeled system where the data is
directly sampled, it is more difficult to assess the wavenumber of the current
density on a real sample.

To explore the possibility of a turbulent flow of the charged quasi-particles
in a sample of graphene under realistic conditions we simulate the
hydrodynamic equations of motion using an adaptation of the relativistic
lattice Boltzmann method described by Romatschke, Mendoza and Succi [12]. The
Relativistic Lattice Boltzmann Model (RLBM) is a hydrodynamic numerical
modeler based on kinetic theory [13] [14]. It is a variation of the popular
Lattice Boltzmann Method (LBM) which is derived from a discretized form of the
Boltzmann equation that describes the time evolution of the number density of
a group of particles as a probability distribution function $f(x,v,t)$. The
function represents the probability that a given particle is in a particular
state in phase space, and the equation is an expression of the conservation of
particle number, momentum and energy. The collision term employed in a lattice
Boltzmann model is a greatly simplified version of the collision term defined
in Boltzmann’s equation using the probability distribution function’s
relaxation to equilibrium. In the RLBM, the probability distribution function
for a fluid in local equilibrium follows a Jüttner distribution instead of a
Maxwell-Boltzmann distribution.

In this work the RLBM model reproduces a two-dimensional quasi-particle flow
in a $40\mu m$ by $10\mu m$ sample of graphene with one or two rigid
impurities obstructing the flow. The sample is simulated with a $1024\times
256$ node lattice with a spacing of $0.038\mu m$. Initial tests incorporate a
single circular impurity with a diameter $D$ of $0.5\mu m$ placed within the
sample in a region near the inflow boundary, referred to as the “obstruction
region”. Subsequent tests are conducted with a second impurity placed at
controlled distances from the first within the same region. At normal
temperatures the impurities within a graphene sample are believed to be
charged, creating charge puddles in the surrounding region that affect the
electric flow. The impurities are largely sourced from the substrate [15]
[16], but can also be embedded within the sample itself. The size of the
impurities can vary greatly depending on the foreign material, but they
typically stay below approximately $0.5\mu m$ [9]. The size and placement of
the impurities are difficult to control in an experimental setting, but the
model seeks to simulate the effects of one or two quasi-isolated impurities on
the current density in ideal but realistic conditions in order to determine if
the detection of a turbulent signal is possible. Therefore the diameter of the
obstacle is chosen to maximize its turbulence producing potential while
maintaining a realistic size. The velocity of the charged flow in the Dirac
liquid can be relatively high owing to a large effective electric coupling
constant. Flow speeds on the order of $10^{2}$ $m/s$ are common [17] [18], but
the flow can approach velocities as large as 10% of the Fermi velocity,
$v_{F}\approx 1.1\times 10^{6}m/s$ [9]. In order to maximize the possibility
of a turbulent signal, the model introduces the largest realizable fluid
velocity of $10^{5}$ $m/s$ ($0.1$ $v_{F}$) into the sample at the same
magnitude along the inflow border. The borders that are perpendicular to the
inflow border use periodic boundary conditions, effectively simulating an
infinitely wide sample with multiple, regularly placed obstacles, but at a
distance where the wakes created by the obstacles cannot affect each other.
Each lattice node in a region occupied by an impurity implements bounce-back
boundary conditions.

The Reynolds number is determined for each test to predict the presence of
turbulence using the kinematic viscosity based on the number density as
described. The kinematic viscosity in graphene is found to be $0.000132$
$\frac{\hbar c^{2}}{eV}$ in natural units ($c=k_{B}=\hbar=1$) [7], readily
obtained from the value of the diffusion constant for momentum
$\eta=2.633\times 10^{3}\frac{eV^{2}}{c^{2}\hbar}$ found in [7], the number
density $n=38.93\frac{eV^{2}}{\hbar^{2}c^{2}}$ determined in [19], and the
effective mass of a charged Dirac quasi-particle participating in the electric
flow, usually given in terms of the mass of an electron $m_{e}$, where
$m_{e}=5.11\times 10^{5}\frac{eV}{c^{2}}$. The effective electron mass is
taken here to be $1.0$ $m_{e}$. The diameter of the embedded impurity is the
most appropriate choice for the system’s characteristic length for the
Reynolds number formulation; $U=D$.

The state data for each node in the lattice is collected throughout the
simulation and the local macroscopic moments are determined and recorded. The
current density $\bm{j}$ is calculated along the lattice nodes at the outflow
border, and the frequency of the fluctuations is determined in Fourier space
against time. The spatial fluctuation of the current density is determined
along the lattice nodes in the region of the sample down stream from the
current inflow referred to as the “current density sampling region”. The
spectrum of the current density fluctuations in frequency space and in k-space
are recorded and plotted for qualitative inspection to look for evidence of
mode generation and migration, indicative of a turbulent flow.

##  III Results

###  III.1 Single Impurity

A simulation of a Dirac fluid flow in a sample of graphene containing a single
impurity of diameter $0.5\mu m$ shows a breaking of longitudinal symmetry that
develops into vortices in the wake, and forms a flow pattern known as a von
Kármán vortex street. The vortices form on both sides of the obstacle’s wake
at a roughly consistent size and placement, alternating on either side of the
wake, and shedding at regular intervals (fig. 2). The vortex creation,
coherence, and shedding produces temporal fluctuations in the current density
that are detected at the outflow border and spatial fluctuations found
throughout the sampling region (fig. 3). The fluctuations produce one or
perhaps two prominent modes in the frequency spectrum, but they appear to
broaden to create a slope of about $\omega^{-5/3}$ in small portions of the
spectrum. There is, however, a single prominent mode in the wave number
spectrum indicating the flow around the obstacle does not produce a turbulent
signal in the current density. The Reynolds number for this system is readily
calculated in terms of modified natural units based on the Fermi velocity (see
Appendix A) using the diameter of the impurity as the characteristic length
($L=D=0.5\mu m=3562.5\frac{\hbar v_{F}}{eV}$), the average flow speed
($U=0.1v_{F}$), and the number-density-dependent kinematic viscosity
($\nu=11.8\frac{\hbar v_{F}^{2}}{eV}$). It is found to be

$Re=\frac{LU}{\frac{\eta}{\rho}}=\frac{\left(3562.5\frac{\hbar
v_{F}}{eV}\right)\left(0.1v_{F}\right)}{\left(11.8\frac{\hbar
v_{F}^{2}}{eV}\right)}=6.37.$ (1)

Figure 2: A velocity magnitude heatmap depicting the flow of charge carrying
quasi-particles on a simulated sample of graphene at $0.1v_{F}$ around an
impurity of diameter $0.5\mu m$ in the obstruction region. A large velocity
around the obstacle creates a regular pattern of vortices in the wake
affecting current density fluctuations detected in the sampling region. The
von Kármán vortex street flow pattern is too regular to be considered
turbulent, producing only a single mode in wave number space. Figure 3: The
effects of a single impurity of diameter $0.5\mu m$ on the current density
with respect to time $\bm{j}(t)$ (top left) and space $\bm{j}(x)$ (bottom
left). The frequency spectrum of $\bm{j}(\omega)$ (top right in log scale)
shows one or two prominent modes that are beginning to migrate. One prominent
mode is seen in the wave number spectrum of $\bm{j}(k_{x})$ (bottom right in
log scale) indicating the chaotic flow does not show signs of turbulence. The
current density $\bm{j}$ is in units of the Fermi velocity $v_{F}$, and
$t,x,\omega,$ and $k_{x}$ are in modified natural units (see Appendix A).

###  III.2 Multiple Impurities

A second obstacle is introduced in a subsequent model with the same impurity
size and lattice spacing as in the initial test, $0.5\mu m$ and $0.038\mu m$
respectively. The new obstacle is placed next to, but slightly offset behind
the first with respect to the flow in the obstruction region at a separation
distance of about $3.38\mu m$ (fig. 4). The Reynolds number for this system,
found to be $12.74$, is similar to the ratio determined for the initial,
single-obstacle test, falling to the edge of the lower boundary of what can be
considered preturbulent. However, we see a less regular vortex pattern and the
von Kármán vortex street flow pattern is no longer present. The fluctuations
in the current density show emerging modes in both frequency space and in wave
number space which appear to broaden, forming a spectral slope conforming to
the power of $-5/3$ as they migrate (fig. 5). There is one or two slightly
prominent outlier modes in frequency space, but the higher wave number modes
are broadband. The spectrum in k-space conforms well to a $|k|^{-5/3}$ slope
that one might expect for mode creation in a nonlinear system. Comparing the
current density fluctuations in fig. 2 with fig. 4, we conclude that the
resulting flow in this arrangement of obstacles, despite a borderline
disqualifying Reynolds number, may be considered turbulent.

Figure 4: A velocity magnitude heatmap of a quasi-particle flow around two
impurities of size $0.5\mu m$, showing an irregular pattern of vortices in the
wake of the two obstacles suggesting turbulence may be present in the flow.
Vortex creation and shedding is evident as in the same system modeling a
single impurity, but the irregularity of the coherence of the vortex
structures cause a much less regular pattern in the fluctuations in the
current density. Figure 5: The current density of a quasi-particle flow around
two impurities of size $0.5\mu m$ with respect to time $\bm{j}(t)$ (upper
left), with respect to space along the axis parallel to the flow $\bm{j}(x)$
(lower left), with respect to frequency $\bm{j}(\omega)$ (upper right in log
scale), and with respect to wave number $\bm{j}(k_{x})$ (lower right in log
scale). The current density in frequency space, $\bm{j}(\omega)$, shows the
formation of multiple modes propagating from a single mode creating a rough
conformity to a $-5/3$ slope in log scales. The current density in k-space,
$\bm{j}(k_{x})$, is broadband with a close adherence to a $|k|^{5/3}$ slope
expected for migrating modes in a non-linear system. The broadband spectra in
frequency space and in k-space hint at a potential turbulent signal. The
current density $\bm{j}$ is in units of the Fermi velocity $v_{F}$, and
$t,x,\omega,$ and $k_{x}$ are in modified natural units (see Appendix A).

We find the presence of turbulence in a two-dimensional solid such as graphene
to be sensitive to obstacle placement. A series of subsequent tests place the
second obstacle at various other distances from the first in directions both
parallel and perpendicular with respect to the inflow. The second obstacle is
positioned such that the flow on the front side of the obstacle is impacted to
some degree by the wake created by the first. When the trailing obstacle is
positioned within approximately $2\mu m$ of the leading obstacle no turbulent
or preturbulent signal is detected in $\bm{j}$. At this range the current
density’s wave number spectrum shows a single dominant mode, and the frequency
spectrum shows a few distinct prominent modes with initial signs of
broadening, but is not broadband (fig. 6). The spectra are very similar to
that created by a single obstacle (see fig. 3), implying the close proximity
of the obstacles has the same effect on the current density as a single,
larger obstacle. Additionally, there is no evidence of a turbulent signal when
the obstacles are situated at a large distance with respect to the
characteristic length of the system. For models examined in this work, a
turbulent signal is not detected when the obstacles are separated at distances
greater than $5\mu m$. The current density fluctuations recorded in a
simulation of a model with obstacles $6.02\mu m$ apart show a single prominent
mode in wave number space but a somewhat broadband spectrum in the high
frequency range in frequency space, with the exception of a single outlier
mode; also similar to the current density fluctuations caused by a single
obstacle (fig. 7). The contribution of the non-linear effects caused by the
wake of first obstacle evidently dissipates at larger distances.

Figure 6: Current density fluctuations recorded from a modeled flow obstructed
by two impurities positioned relatively close. The proximity produces a single
dominant mode in wave number space (shown in log scale) similar to what is
seen in $\bm{j}(k_{x})$ from a single obstacle (fig. 3), implying the effect
of closely placed obstacles on the current density is similar to that caused
by a single obstacle. The multi-modal spectrum in frequency space (shown in
log scale) is also similar the $\bm{j}(\omega)$ spectrum for a single
obstacle. The units of the plot are indicated in Appendix A. Figure 7: Current
density fluctuations from a model simulating a flow around two impurities
situated at a large distance relative to their diameter. In this arrangement
two discrete modes appear prominent in the wave number spectrum of $\bm{j}$
shown in log scale, and multiple modes are visible in the frequency spectrum
of $\bm{j}$, also in log scale. A turbulent signal is not detected for
obstacles placed too far apart. The units of the plot are indicated in
Appendix A.

Fig. 8 depicts the current density in frequency space and in k-space of five
runs with increasing obstacle placement. At the smallest distance, $1.68\mu
m$, the frequency space spectrum is multi-modal while the k-space spectrum has
a single prominent mode, resembling the spectral effects of a single obstacle.
At a slightly larger distance, $2.34\mu m$, the k-space spectrum becomes
broadband and conforms to the slope of $|k|^{-5/3}$. More modes are present in
the frequency spectrum, but it is not definitively broadband. A broadband
signal is present in both frequency space and wave number space for objects
placed at a distance of $3.38\mu m$ (also shown in fig. 4), indicating
turbulence. As the distance increases from $3.38\mu m$ to $6.02\mu m$, the
frequency spectrum reverts back to multiple modes, and the broadband spectrum
in k-space dissipates into a single mode. The turbulent signal is gone when
the objects are separated at this distance. At the largest separation tested,
$8.39\mu m$, a small number of prominent modes are present in both spectra of
$\bm{j}$ so that it also resembles the current density spectra created by a
single obstacle. Further tests investigating different placement
configurations (not shown) indicate that detection of turbulence is sensitive
to other positional features such as alignment. Because one obstacle must be
within the influence of the other’s wake, if the objects are adjacent and too
far apart, a turbulent signal is not detected.

Figure 8: The effect of object separation of two impurities on the current
density spectra. A turbulent signal is detected when the spectrum in both
frequency space ($\bm{j}(\omega)$ in log scale in the second column) and wave
number space ($\bm{j}(k_{x})$ in the third column; log scale) are broadband.
There is no turbulent signal when the impurities are less than approximately
$2\mu m$ apart as evidenced by the lack of a broadband signal in frequency
space. A broadband signal is similarly absent when the obstacles are separated
by greater than approximately $5\mu m$.

Note that the identification of a broadband spectrum contains an element of
judgment, and a determination for the presence of turbulence using the
spectrum is not rigorous. Indeed, [10] indicates the presence of vortex
shedding is evidence of preturbulence since it is characteristic of
preturbulent phenomena. Because each model configuration shows multiple modes
in frequency space, and because the most prominent mode for each in k-space
could be considered an outlier, it is possible that each depicted obstacle
configuration can be considered capable of producing a turbulent signal.
Increasing the number of lattice nodes in the model will produce a higher
resolution spectrum and subsequently more confidence in the interpretation,
but it would also be more computationally expensive.

##  IV Conclusions

To summarize, we were able to detect a potential turbulent flow in the
relativistic hydrodynamics of massless charged quasi-particles in an ultra
pure, idealized sample of graphene. Though the Reynolds number of the modeled
systems were on the lower edge of the preturbulent range, evidence of
turbulence emerges as a result of interactions in the wake of multiple
obstacles placed in the sample. For two similarly sized impurities a signal of
turbulence is dependent upon their separation and position, while apparently
independent of size or shape. This dependence creates a more complex
formulation of turbulence that cannot be represented by the Reynolds number
metric alone, and has consequences for the conductivity of graphene. The
sensitivity to turbulence based on impurity placement is similar to the
sensitivity of the placement of wind turbines in a wind farm. A notable
difference, though, is the importance of higher altitudes above the turbines
on the air flow through the farm. The electric flow within two-dimensional
graphene is not affected by out of plane effects.

A preturbulent signal was detected in a test modeled after [10], in agreement
with the findings of that work. Further investigation is needed to study the
effect of semi-rigid obstacles and momentum relaxation on the production of a
turbulent signal. The resonance of a semi-rigid obstacle caused by the flow
would amplify the current density fluctuations, and the effect would likely be
multiplied by the presence of additional semi-rigid obstacles. However, the
effect would be balanced or mitigated by momentum relaxation effects caused by
the obstacle and the lattice itself. The lattice structure of another two-
dimensional solid such as a Kagome metal should also be investigated with the
employed RLBM numerical modeler. The two-dimensional molecular lattice of a
Kagome metal creates a linear dispersion relation at low energies, similar to
that of graphene, but it has a much stronger electric coupling constant that
enables a faster Dirac fluid flow [20]. With a similar viscosity, this implies
a higher likelihood of detecting a preturbulent or perhaps a turbulent flow.

We thank Paul Romatschke for his instruction, helpful discussions and
expertise on RLBM modeling. We would also like to thank Andrew Lucas and Mark
Hoefer for their helpful comments and insight. This material is based upon
work supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics program under Award Number DE-SC-0017905. All simulations were
run on the Eridanus cluster at the University of Colorado Boulder.

##  V Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Model Scaling

As a technical note, the selected scaling of the RLBM numerical scheme
employed in this study is with respect to the Fermi velocity, so the numerical
parameters are given in a slightly modified form of natural units where
$v_{F}=k_{B}=\hbar=1$. The factor $\frac{c_{F}}{c}=3.333\times 10^{3}$
converts all values in natural units of $c$ (where $c$ is the speed of light)
to the modified form of natural units. Length parameters are given in terms of
$\frac{\hbar v_{F}}{eV}$ and the current density $\bm{j}$ is in terms of Fermi
velocity $v_{F}$. Units of time, frequency and wave number are
$\frac{\hbar}{eV},\frac{eV}{\hbar},$ and $\frac{eV}{\hbar v_{F}}$
respectively.

##  Appendix B Viscidity

Though the more classical form of the kinematic viscosity using mass density
is used to determine the Reynolds number, the viscous term in the employed
RLBM model is specified by the ratio of the viscosity over entropy density
$\frac{\eta}{s}$, as described, and as is more common for the description of
the dissipation of massless particles. Therefore, the number-density-dependent
kinematic viscosity used in the calculation of the Reynolds number is
converted to an “effective” viscosity-entropy ratio
$\frac{\eta}{s}=\frac{T_{0}}{v_{F}^{2}}\frac{\eta}{\rho}$ (where $T_{0}=300K$)
for the RLBM model. This “effective” viscosity-entropy ratio is referred to in
the numerical modeling parameters as “viscidity”, denoted by $g$, to identify
it as the model’s viscous parameter, but differentiating it from a true
viscosity-entropy or viscosity-mass density ratio. The viscidity of the
modeled system is determined in 2.

$g=\frac{\eta}{s}=\frac{T_{0}}{c_{F}^{2}}\frac{\eta}{\rho}=\left(2.5862\times
10^{-2}\frac{eV}{k_{B}}\right)\left(11.88\frac{\hbar
c_{F}^{2}}{eV}\right)=0.30724\frac{\hbar}{k_{B}}$ (2)

## References

  * [1] Yuanbo Zhang, Yan-Wen Tan, Horst L. Stormer, and Philip Kim.  Experimental observation of the quantum hall effect and berry’s phase in graphene.  Nature, 438:201–204, January 2005.
  * [2] R. Krishna Kumar, D. A. Bandurin, F. M. D. Pellegrino, Y. Cao, A. Principi, H. Guo, G. H. Auton, M. Ben Shalom, L. A. Ponomarenko, G. Falkovich, K. Watanabe, T. Taniguchi, I. V. Grigorieva, L. S. Levitov, M. Polini, and A. K. Geim.  Superballistic flow of viscous electron fluid through graphene constrictions.  Nature Physics, 13:1182–1185, January 2017.
  * [3] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov.  Two-dimensional gas of massless dirac fermions in graphene.  Nature, 438:197–200, Nov 2005.
  * [4] R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard, and J. H. English.  Observation of an even-denominator quantum number in the fractional quantum hall effect.  Phys. Rev. Lett., 59:1776–1779, Oct 1987.
  * [5] Luciano Rezzolla and Olindo Zanotti.  Relativistic hydrodynamics.  Oxford University Press, 2013.
  * [6] Paul Romatschke and Ulrike Romatschke.  Relativistic fluid dynamics in and out of equilibrium: and applications to relativistic nuclear collisions.  Cambridge University Press, 2019.
  * [7] Markus Müller, Jörg Schmalian, and Lars Fritz.  Graphene: A nearly perfect fluid.  Phys. Rev. Lett., 103:025301, Jul 2009.
  * [8] Richard JAM Stevens and Charles Meneveau.  Flow structure and turbulence in wind farms.  Annual review of fluid mechanics, 49:311–339, 2017.
  * [9] Andrew Lucas and Kin Chung Fong.  Hydrodynamics of electrons in graphene.  Journal of Physics: Condensed Matter, 30(5):053001, Jan 2018.
  * [10] M. Mendoza, H. J. Herrmann, and S. Succi.  Preturbulent regimes in graphene flow.  Phys. Rev. Lett., 106:156601, Apr 2011.
  * [11] Mark Hoefer.  personal communication.
  * [12] P. Romatschke, M. Mendoza, and S. Succi.  Fully relativistic lattice Boltzmann algorithm.  Physical Review C, 84(3):034903, Sep 2011, 1106.1093.
  * [13] Sauro Succi.  The lattice Boltzmann equation: for fluid dynamics and beyond.  Oxford university press, 2001.
  * [14] Roberto Benzi, Sauro Succi, and Massimo Vergassola.  The lattice boltzmann equation: theory and applications.  Physics Reports, 222(3):145–197, 1992.
  * [15] Kentaro Nomura and A. H. MacDonald.  Quantum transport of massless dirac fermions.  Phys. Rev. Lett., 98:076602, Feb 2007.
  * [16] Shaffique Adam, E. H. Hwang, V. M. Galitski, and S. Das Sarma.  A self-consistent theory for graphene transport.  Proceedings of the National Academy of Sciences, 104(47):18392–18397, 2007, https://www.pnas.org/content/104/47/18392.full.pdf.
  * [17] Inanc Meric, Melinda Y. Han, Andrea F. Young, Barbaros Ozyilmaz, Philip Kim, and Kenneth L. Shepard.  Current saturation in zero-bandgap, top-gated graphene field-effect transistors.  Nature Nanotechnology, 3(11):654–659, 2008.
  * [18] Vincent E. Dorgan, Myung-Ho Bae, and Eric Pop.  Mobility and saturation velocity in graphene on sio2.  Applied Physics Letters, 97(8):082112, 2010, https://doi.org/10.1063/1.3483130.
  * [19] Daniel E. Sheehy and Jörg Schmalian.  Quantum critical scaling in graphene.  Phys. Rev. Lett., 99:226803, Nov 2007.
  * [20] Domenico Di Sante, Johanna Erdmenger, Martin Greiter, Ioannis Matthaiakakis, Rene Meyer, David Rodriguez Fernand ez, Ronny Thomale, Erik van Loon, and Tim Wehling.  Turbulent hydrodynamics in strongly correlated Kagome metals.  arXiv e-prints, page arXiv:1911.06810, Nov 2019, 1911.06810.