# Preturbulence in momentum-relaxing Navier-Stokes hydrodynamics

Vladimir A. Goy  Pacific Quantum Center, Far Eastern Federal University,
Sukhanova 8, Vladivostok, 690950, Russia  Uri Vool  Max-Planck-Institut für
Chemische Physik Fester Stoffe, Dresden, Germany  Maxim N. Chernodub  Institut
Denis Poisson UMR 7013, Université de Tours, 37200 France

###### Abstract

In a hydrodynamic regime of electronic fluids, the electron scattering on
impurities and phonons inhibits the development of oscillating preturbulent
phenomena (such as the Kármán vortex shedding) and enforces a laminar flow.
Working in a local generalization of the Navier-Stokes hydrodynamics of
incompressible two-dimensional fluids, we show that the critical relaxation
time separating these two regimes is described by a surprisingly simple
fractional power function of the Reynolds number. The critical exponent,
$\alpha\simeq 4/3$, sets particular experimental conditions for the
observation of preturbulent effects in electronic fluids.

## Introduction.

In certain solid-state systems, electronic transport exhibits collective
hydrodynamic behavior, which appears to be in sharp contrast with the expected
diffusive drift of a Fermi liquid [1]. Instead, the electron flow resembles a
fluid when the momentum-conserving electron-electron scatterings events occur
much faster than the timescale of the momentum-relaxing interactions of
electrons with phonons, impurities, and the boundaries of the system [2, 3].

The hydrodynamic behavior of electrons reveals itself via various transport
properties in conductors of finite geometries. For example, the Gurzhi effect,
which emerges in clean conducting channels, discriminates between ballistic
transport restrained by the scattering of electrons with the boundaries of the
channel and laminar flow of electronic fluid, dominated by the momentum-
conserving electron-electron scattering events [4]. The robust signatures of
the electron hydrodynamic flow have been found in various compounds via its
imprints on the thermal and electric transport [5, 6, 7, 8].

The exceptional mobility of electrons in ultraclean graphene offers another
exciting possibility to investigate their hydrodynamics [9]. Due to the
viscosity of the electronic flow [10, 11, 12, 13] a localized injection of the
electronic current can create whirlpools in the electron flow near the
contacts. The emerging electronic vortices were predicted to generate a
viscous backflow [14, 15] which leads to experimentally observable negative
resistance [16] as well as a strong violation of the Wiedemann-Franz law [17].
The viscosity of electronic flow reveals itself in hydrodynamic conductance in
narrow constrictions [18], which allows for the visual discrimination111Notice
that visually compelling hydrodynamic-like response may also emerge in the
non-hydrodynamic ballistic regime [40], via surface scattering (“para-
hydrodynamics”) [37], and in diffusive two-component systems (“pseudo-
hydrodynamics”) [41]. of the high-temperature diffusive Ohmic current from the
low-temperature laminar Poiseuille flow via spatial imagining of the
electronic current profiles [20, 21, 22].

The experimental efforts are mostly concentrated in the laminar hydrodynamical
regime characterized by low values ($\mathrm{Re}\ll 1$) of the Reynolds
number:

$\displaystyle\mathrm{Re}=\frac{UL}{\nu}\,,$ (1)

set by the characteristic linear dimension $L$ of the hydrodynamic problem,
the flow velocity $U$, and its kinematic (shear) viscosity $\nu$.

At higher Reynolds numbers, typically $\mathrm{Re}\simeq 10\dots 10^{3}$,
electronic transport may exhibit “preturbulent” phenomena [9], which may
experimentally be observed via fluctuations of the electronic current past
constrictions and localized impurities in the flow channel [23, 24]. In
ordinary fluid dynamics, the preturbulent regime is manifested via a periodic
detachment of hydrodynamic vortices, which form the so-called Kármán vortex
street [25, 26]. In condensed matter context, the formation of the Kármán
vortices has been theoretically predicted [27] and experimentally observed
[28] in the Bose-Einstein condensates of cold atoms.

Contrary to the flow of ordinary fluids, which experience momentum-relaxing
interactions only with the channel boundaries and macroscopic obstacles in the
flow, the electronic fluids can also lose their momentum at a microscopic
level via inelastic scattering of electrons on thermal phonons and impurities
of a crystal lattice. These momentum-relaxing events can substantially inhibit
the development of the preturbulence regime.

The aim of our paper is to investigate the preturbulent regime in two-
dimensional incompressible fluids in the presence of a uniform background of
momentum-relaxing events. We consider Navier-Stokes (NS) fluids extended with
a simple momentum-relaxing term and study its effect in the realm of
electronic flows in graphene. Due to the minimal and model-independent nature
of the extension, this simple model can serve as a universal benchmark case of
NS fluids with dissipation to be compared with more elaborate models [23, 24].

## Navier-Stokes hydrodynamics with relaxation.

A one-component electronic fluid is characterized by the electronic density
$n$ of the charge carriers with the mass $m$, the particle current density
$\boldsymbol{j}=n{\boldsymbol{u}}$ and the local velocity
${\boldsymbol{u}}={\boldsymbol{u}}({\boldsymbol{r}})$. We consider a fluid
propagating slower than plasmonic velocities which possesses incompressible
flow, ${\boldsymbol{\nabla}}\cdot{\boldsymbol{u}}=0$ [13, 14]. Its
hydrodynamic regime is described by the extended Navier-Stokes equation:

$\biggl{(}\frac{\partial}{\partial
t}+{\boldsymbol{u}}\cdot{\boldsymbol{\nabla}}\biggr{)}{\boldsymbol{u}}-\nu\Delta{\boldsymbol{u}}=-\frac{1}{mn}{\boldsymbol{\nabla}}P-\frac{e}{m}{\boldsymbol{E}}-\frac{1}{\tau_{{\mathrm{mr}}}}{\boldsymbol{u}}\,,$
(2)

which incorporates forces experienced by an element of the fluid including (i)
internal shear stress due to velocity gradients with the magnitude set by the
kinematic shear viscosity $\nu$, (ii) the pressure-gradient
${\boldsymbol{\nabla}}P$, (iii) the background electrostatic field
${\boldsymbol{E}}$, and (iv) friction due to electron scattering on phonons
and crystal defects.

The parameterization of the friction term by the single quantity, the
scattering time $\tau_{{\mathrm{mr}}}$, turns out to be remarkably successful
in explaining experimental results in the linear-response regime [16, 18, 13,
29].

A steady laminar flow can be described by the simplified time-independent NS
equation (2), with the material derivative and pressure-gradient terms
neglected [16]:
$\sigma_{0}{\boldsymbol{E}}({\boldsymbol{r}})-eL_{\mathrm{Gu}}^{2}\Delta{\boldsymbol{j}}({\boldsymbol{r}})+e{\boldsymbol{j}}({\boldsymbol{r}})=0$.
The spatial hydrodynamic length scale is set by the (diffusive) Gurzhi length

$\displaystyle L_{\mathrm{Gu}}=\sqrt{{\nu}\tau_{\mathrm{mr}}}\,.$ (3)

which, for example, determines the spatial size of the electronic whirlpools
in graphene [14]. At vanishing viscosity, $\nu\to 0$, or at high momentum-
relaxing collision rate, $\tau_{\mathrm{mr}}\to 0$, the hydrodynamic length
vanishes, $L_{\mathrm{Gu}}\to 0$, and the NS equation reduces to the usual
Ohmic drift of electric current,
${\boldsymbol{J}}\equiv-e{\boldsymbol{j}}=\sigma_{0}{\boldsymbol{E}}$, set by
the Drude-like conductivity $\sigma_{0}=n_{0}e^{2}\tau_{\mathrm{mr}}/m$.

## Scales.

The vortex shedding occurs in the non-linear regime of the NS equation (2)
which can be conveniently rewritten in the following dimensionless form:

$\displaystyle\mathrm{Re}\left(\frac{\partial}{\partial
t_{0}}+{\boldsymbol{u}}_{0}\cdot{\boldsymbol{\nabla}}_{0}\right){\boldsymbol{u}}_{0}-\Delta_{0}{\boldsymbol{u}}_{0}=-\mathrm{Rl}\,{\boldsymbol{u}}_{0}+{\boldsymbol{f}}_{0}\,,\quad$
(4)

where dimensionless quantities carry the subscript “0”:
${\boldsymbol{x}}=L{\boldsymbol{x}}_{0}$, $t=T_{L}t_{0}$,
${\boldsymbol{u}}=U{\boldsymbol{u}}_{0}$, including the dimensionless force
${\boldsymbol{f}}_{0}=-T_{L}({\boldsymbol{\nabla}}_{0}P)/(mn\nu)-eLT_{L}{\boldsymbol{E}}/(m\nu)$.
The hydrodynamics of fluid (4) is characterized by two dimensionful time
scales: the kinematic time $T_{L}=L/U$, which sets, for example, the scale for
the shedding frequency of the Kármán vortices [see Eq. (11)], and the
kinematic relaxation time scale $\tau_{L}=L^{2}/\nu$, which determines their
attenuation. In Eq. (4), we also introduced the dimensionless momentum-
relaxation number,

$\displaystyle\mathrm{Rl}=\frac{\tau_{L}}{\tau_{{\mathrm{mr}}}}=\frac{L^{2}}{\nu\tau_{\mathrm{mr}}}=\frac{L^{2}}{L_{\mathrm{Gu}}^{2}}\,.$
(5)

given by the ratio of kinematic, $\tau_{L}$, and actual, $\tau_{\mathrm{mr}}$,
momentum-relaxation times. This parameter separates regimes of fast
($\tau_{L}\gg\tau_{\mathrm{mr}}$ or $\mathrm{Rl}\gg 1$) and slow
($\tau_{L}\ll\tau_{L}$ or $\mathrm{Rl}\ll 1$) momentum relaxation. Thus, the
hydrodynamics can be fully described by only two dimensionless numbers: The
Reynolds number $\mathrm{Re}$ (1) and the momentum-relaxation number
$\mathrm{Rl}$ (5).

## The critical line of preturbulence.

We extensively simulated the NS fluid (4) to elucidate the effects of the
momentum relaxation term on the onset of preturbulence. We consider a circular
impurity of the diameter $L$ placed in the center of a wide straight channel
of the large width $W=10L$ and use no-slip boundary conditions with vanishing
velocity at the boundaries.

In the absence of momentum relaxation, $\tau_{{\mathrm{mr}}}\to\infty$, the
flow becomes unstable at high Reynolds numbers (1),
$\mathrm{Re}>\mathrm{Re}_{c}$, where the critical value $\mathrm{Re}_{c}$
depends on the geometry of the hydrodynamic system. For a disk-shaped two-
dimensional obstacle in a channel of infinite width, the Kármán vortices start
to form if the Reynolds number exceeds the critical value
$\mathrm{Re}^{(\infty)}_{c}\simeq 47$ [30]. In our geometry with a finite-size
channel, the critical Reynolds number in the relaxation-free
($\tau_{\mathrm{mr}}=\infty$) regime is:

$\displaystyle\mathrm{Re}_{c}=52.4(5)\,,\qquad({\textrm{at}\
}\tau_{{\mathrm{mr}}}\to\infty)\,.$ (6)

We found numerically that a finite momentum relaxation time
$\tau_{{\mathrm{mr}}}$ drastically affects the value of the critical Reynolds
number by inhibiting the formation of the Kármán vortices in agreement with
Ref. [24]. The phase diagram of the preturbulence in the
$(\mathrm{Re},\mathrm{Rl})$ plane is shown in Fig. 1 (the amplitude of the
flow velocity is explored in more detail in Supplementary Figure S1).

Figure 1: Phase diagram of the vortex formation in the plane of the Reynolds
number $\mathrm{Re}$, Eq. (1), and the momentum-relaxation number
$\mathrm{Rl}$, Eq. (5), past a circular impurity of a diameter $L$ in the
channel of the width $W=10L$. The critical line, described by the
phenomenological fractional power law (7) with parameters (8), separates the
laminar flow (the white region above the transition line at low
$\tau_{{\mathrm{mr}}}$, high $\mathrm{Rl}$) and the preturbulent region with
higher $\tau_{{\mathrm{mr}}}$ (lower $\mathrm{Rl}$), where the flow produces
the Kármán vortices. The examples of the laminar flow and the oscillatory
Kármán flow are shown in the insets (a) and (b), respectively. The color
displays the magnitude of velocity oscillations at the point $r=L$ in the
middle of the channel past the obstacle.

The phase diagram possesses a laminar phase (at
$\tau_{\mathrm{mr}}<\tau_{{\mathrm{mr}},c}$) and the pre-turbulent Kármán
phase (at $\tau_{\mathrm{mr}}>\tau_{{\mathrm{mr}},c}$). We established that
the critical line that separates these phases is described by the
fascinatingly simple fractional power law:

$\displaystyle\frac{\mathrm{Rl}\ }{\mathrm{Rl}_{0}}=\left(\frac{\mathrm{Re}\
}{\mathrm{Re}_{c}}\right)^{\alpha}-1,\qquad\mathrm{Re}\geqslant\mathrm{Re}_{c}\,,$
(7)

where the best-fit parameters are as follows:

$\displaystyle\mathrm{Rl}_{0}=3.63(7)\,,\qquad\alpha=1.348(2)\simeq 4/3\,,$
(8)

and the critical Reynolds number is given in Eq. (6). The excellence of the
fit of the numerical data by the phenomenological scaling (7) is demonstrated
in Fig. S2.

The relative amplitude of the velocity oscillations, $\delta U/U$, created by
the vortices produced past the impurity, takes its weakest values near the
critical line (7). The amplitude gets substantially enhanced as either the
Reynolds number Re or the relaxation time $\tau_{\mathrm{mr}}$ increases.

In a nondissipative limit, when the relaxation time is long,
$\tau_{{\mathrm{mr}}}\gg\tau_{L}$ ($\mathrm{Rl}\ll 1$), the vortices start to
form closer to the lowest critical Reynolds number (6). At higher Reynolds
values, the preturbulent vortex regime sets in at a shorter momentum-
relaxation time (for example, according to Fig. 1, the critical relaxation
time is $\tau_{{\mathrm{mr}},c}\simeq\tau_{L}/60$ or $\mathrm{Rl}_{c}\simeq
60$ at $\mathrm{Re}=400$).

Equations (1), (5) and (7) imply that the Kármán vortex phase is realized at
the initial flow velocities higher than the critical value $u>u_{c}$:

$\displaystyle
u_{c}(\tau_{{\mathrm{mr}}})=\mathrm{Re}_{c}\,\frac{\nu}{L}\left(\frac{1}{\mathrm{Rl}_{0}}\,\frac{L^{2}}{\nu\tau_{{\mathrm{mr}}}}+1\right)^{1/\alpha}\,.$
(9)

In the absence of momentum relaxation, $\tau_{{\mathrm{mr}}}\to\infty$, the
critical line (9) is reduced to
$u_{c}(\tau_{{\mathrm{mr}}})=\mathrm{Re}_{c}\,\frac{\nu}{L}$ which defines the
critical point (6) $\mathrm{Re}=\mathrm{Re}_{c}$, as expected. In the
opposite, fast-relaxation limit, $\tau_{{\mathrm{mr}}}\ll\tau_{L}$ (at
$\mathrm{Rl}\gg 1$), one gets the following asymptotic scaling regime:

$\displaystyle
u_{c}(\tau_{{\mathrm{mr}}})=u_{0}\,\nu^{\frac{\alpha-1}{\alpha}}L^{\frac{2-\alpha}{\alpha}}\tau^{-\frac{1}{\alpha}}_{{{\mathrm{mr}}}}\quad\
\mbox{for}\quad\tau_{{\mathrm{mr}}}\ll L^{2}/\nu,\quad$ (10)

with the dimensionless
$u_{0}=\mathrm{Re}_{c}\,\mathrm{Rl}_{0}^{-1/\alpha}=20.1(2)$.

Figure 2:  The Strouhal number St, which determines the frequency of the
vortex oscillations (11), is shown in the (Re, Rl) plane of the phase diagram
of Fig. 1 along with the examples of the Kármán vortex streets and Kármán
tadpoles at the points $A,B,\dots,K$ (the color marks the absolute value of
the velocity flow). The inset shows a low-Re corner of the diagram.

## Shedding frequency and momentum relaxation.

The vortex production is characterized by a periodic pattern of the unsteady
flow of the vortex shedding. In ordinary fluids, the main vortex shedding
frequency,

$\displaystyle f=\mathrm{St}\frac{U}{L}\equiv\frac{\mathrm{St}}{T_{L}}\,,$
(11)

is expressed via the dimensionless Strouhal number “St” [26]. The Strouhal
number, which can be computed numerically, depends on the geometry of the
obstacle.

The typical frequency of the vortex shedding (11) is set by the kinematic
timescale $T_{L}$ which depends on the velocity of the fluid and the size of
the obstacle. For a disk-shaped impurity in the absence of the momentum
relaxation, $\tau_{{\mathrm{mr}}}=\infty$, the Strouhal number takes values
around $\mathrm{St}\simeq 0.2$ in a wide range of values of the Reynolds
number [31]. In a momentum-nonconserving fluid, the finite relaxation time
scale $\tau_{\mathrm{mr}}$ implicitly affects the vortex shedding frequency
(11). Contrary to the soft equation of state [24], the fluid oscillations in
our model (4) feature a single production frequency (11).

Figure 2 shows the Strouhal number calculated numerically across the whole
phase diagram. In the low-Re corner of the diagram, corresponding to slow
relaxation with high $\tau_{\mathrm{mr}}$, the Strouhal number takes the
lowest value around $\mathrm{St}\simeq 0.15$. In contrast, in the opposite
high-Re and low-$\tau_{{\mathrm{mr}}}$ corner, the oscillations of the vortex
flow are characterized by a twice higher Strouhal number, $\mathrm{St}\simeq
0.3$. Finally, in the rest of the phase diagram, the oscillations are close to
the standard value $\mathrm{St}\simeq 0.2$.

## Kármán vortices and Kármán tadpoles.

Figure 2 also illustrates qualitative features of the fluid flow in various
points $A$-$K$ of the phase diagram. The momentum-relaxing laminar flow above
the critical line (at $\mathrm{Re}>\mathrm{Re}_{c}$ and
$\tau_{\mathrm{mr}}\neq 0$ with $\tau_{\mathrm{mr}}<\tau_{{\mathrm{mr}},c}$,
point $A$ in Fig. 2), appears to be qualitatively similar to the ordinary
laminar phase below the critical Reynolds number
($\mathrm{Re}<\mathrm{Re}_{c}$ at $\tau_{\mathrm{mr}}=\infty$, point $B$). The
only difference between these phases appears, expectedly, in a faster
dissolution of the long-distance laminar tail at $\tau_{\mathrm{mr}}\neq 0$
(well pronounced at point $F$ at a smaller $\tau_{\mathrm{mr}}$).

In the preturbulent phase, the momentum relaxation produces a visible
qualitative effect on the structure of the vortex shedding. For example, at
$\mathrm{Re}=100$, the vortices become individually non-distinguishable as
they form a single oscillating trail which we dub the “Kármán tadpole”, which
is well-visible in point $C$. As the relaxation time $\tau_{\mathrm{mr}}$
increases, the tadpole breaks into individual vortices (point $D$), which
become more pronounced as $\tau_{\mathrm{mr}}$ reaches the non-dissipative
limit (point $E$). Thus, at $\mathrm{Re}>\mathrm{Re}_{c}$, an increasing
relaxation time $\tau_{\mathrm{mr}}$ turns a laminar flow ($B$) into a Kármán
tadpole ($C$), which then gets gradually transformed into the usual Kármán
vortex street ($D$). The effect of the momentum relaxation on the vortex
attenuation is also seen in the series of points $F$-$H$ and $I$-$K$ at higher
Reynolds numbers.

## Large holes.

A non-zero momentum relaxation also produces an attenuating effect for large
holes, which may have important consequences relevant to the experimental
observation of preturbulence. In Fig. 3, we show the velocity flow around a
large disk-shaped impurity for several momentum-relaxation times
$\tau_{\mathrm{mr}}$. An impurity of diameter $L$ in a channel of slightly
larger width $W=4/3L$ separates the fluid flow into two narrow constrictions.
The constrictions inject the fluid with an enhanced velocity past the obstacle
close to the walls of the channel, leading to the formation of the Kármán
vortex street.

At a distance $r\simeq(5-6)L$ from the obstacle, the flow becomes laminar. A
decreasing relaxation time inhibits the preturbulence effect, leading to a
pure laminar flow of two joining sub-flows above the critical value
$\mathrm{Rl}\gtrsim 15$. Our results, illustrated in Fig. 3, imply that the
most favorable point for the observation of vortices is $r\simeq(2-3)L$ past
the impurity. The relevant frequencies/amplitudes are of the same order as for
small holes with important geometry-dependent corrections that are explored in
more detail in Supplementary Fig. S4.

Figure 3: Absolute value of the flow velocity around a large circular impurity
of the diameter $L$ in a channel of the width $W=4/3L$ for various momentum-
relaxation numbers $\mathrm{Rl}$. $\mathrm{Re}=100$, determined formally via
the hole size (1), is fixed for all maps, and we observe preturbulence with
the critical value $\mathrm{Rl}_{c}\approx 15$. The velocity color map follows
Fig. 2.

## Realistic experimental parameters and measurement techniques.

With this quantitative understanding of the threshold for Kármán vortices, let
us now examine if realistic parameters can be chosen that cross this threshold
in an experimental system, focusing on high quality low carrier-density
conductors such as graphene sheets and GaAs quantum wells. Initially, a large
flow velocity $U$ is required, and velocities of $U\approx(1-5)\times
10^{5}~{}\mathrm{m/s}$ have been previously achieved in GaAs [32, 33] and
graphene [34, 35, 36]. The Reynolds number can be increased by constrictions
of larger diameter $L$, but this also increases the relaxation timescale
$\tau_{L}$, thus placing a larger constraint on the relaxation time
$\tau_{\mathrm{mr}}$. This trade-off is explored in detail in Supplementary
Figure S3, which shows the required velocity $U$ and viscosity $\nu$ to cross
the vortex-shedding threshold for various constriction diameters $L$ and
momentum relaxation times $\tau_{{\mathrm{mr}}}$.

As $\tau_{{\mathrm{mr}}}$ is currently experimentally limited to
$1{-}10\,\mathrm{ps}$ [5, 16], a diameter length scale of $L\approx
1\,\mu\mathrm{m}$ seems optimal. This experimentally favorable condition is
achieved in the asymptotic scaling regime (10) which gives
$u_{c}\propto\nu^{1/4}L^{1/2}\tau^{-4/3}_{{{\mathrm{mr}}}}$ due to the
relatively large (8) scaling exponent $\alpha>1$. The observation of
preturbulence requires then viscosity $\nu\approx 10^{-4}\mathrm{m^{2}/s}$.
Such a value could be possible [16], but is currently the most significant
deviation of needed parameters from existing measured values, consistent with
a previous analysis of preturbulence [24]. Improvements in relaxation time can
allow for larger constriction sizes, thus reducing the viscosity constraint.

As the effect of Kármán vortices is local to the area past the impurity, the
strong effects will be difficult to measure in regular transport measurements
but will require local probes such as electric potential [20], scanning SQUID
magnetometry [37], or magnetometry with nitrogen-vacancy (NV) centers [21, 22,
8]. The latter is particularly advantageous due to the high sensitivity of the
NV to microwave radiation at frequencies corresponding to its spin transitions
for a given applied magnetic field. Thus, in addition to the measurement of
the DC current shape, the oscillations themselves can be locally observed,
with the ability to tune their frequency in-situ by varying the applied
current and thus the velocity $U$. The oscillation frequency (11) of the
vortices for this typical geometry is $f\approx 10\,\mathrm{GHz}$, which is
conveniently obtainable for NV centers with the applied field. The observation
of a local microwave signal (when only a DC signal is applied) that is tunable
with the applied current will be a smoking-gun signature of preturbulent
electron hydrodynamics.

## Conclusions.

Our paper explored the turbulent-dissipative regime in two-dimensional
incompressible electronic fluids by taking the most straightforward local
extension of the NS equation with the local momentum-relaxation term (4)
arising from electron scattering on impurities and phonons. We suggested
characterizing the relaxation rate in terms of the new momentum relaxation
number Rl (5). We found the critical relaxation number Rl that discriminates
the laminar flow from the pre-turbulent regime, is described by a surprisingly
simple fractional power law (7) of the Reynolds number Re with the exponent
$\alpha\simeq 4/3$ which intriguingly matches the Yaglom turbulence exponent
for dissipation rates of conserved quantities [38, 39].

Close to the critical transition line, the street of shredded Kármán vortices
collapses to a new dynamical structure, a single vibrating Kármán tadpole. In
the local relaxation model (4), the oscillations are characterized by a single
global frequency that does not depend on the distance to the impurity.

The novel preturbulent-dissipative regime accessible with electron
hydrodynamics shows rich behavior that can be effectively modeled by two
dimensionless parameters. This simple model will serve as a benchmark for the
exploration of this regime and a useful tool to motivate, guide, and interpret
future experimental studies.

###### Acknowledgements.

## Acknowledgments.

The work of VAG was supported by Grant No. 0657-2020-0015 of the Ministry of
Science and Higher Education of Russia.

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Supplementary Information

Preturbulence in momentum-relaxing Navier-Stokes hydrodynamics

by Vladimir A. Goy, Uri Vool, and Maxim N. Chernodub

##  Appendix A A. Amplitude of velocity fluctuations

Figure S1: The relative amplitude of the velocity fluctuations of the flow,
$\delta u/U$ (with the initial flow velocity $U$), calculated at the distances
$r=L,3L,5L,10L$ past the hole of the diameter $L$ in the middle of the channel
as the function of the Reynolds number Re and the momentum-relaxation number
$\mathrm{Rl}$. The right column represents a zoom in on the small-Re corner of
the corresponding plot at the left column.

##  Appendix B B. The best fit of the transition line.

Figure S2: The best fit of the transition line which separates, in the plane
of the Reynolds parameter $\mathrm{Re}$ and the relaxation parameter
$\mathrm{Rl}$, the regions of the preturbulent and laminar phases in Fig. 1
and Fig. 2. The fit is performed by the fractional power function (7) with the
best-fit parameters given in Eq. (8).

##  Appendix C C. The transition line with physical parameters for electronic
fluids

|  |
---|---|---
|  |
Figure S3: The critical velocity $u_{c}$ of the initial electronic flow vs.
viscosity $\nu$ at various relaxation times $\tau_{{\mathrm{mr}}}$ for the
circular hole of various diameters $L=0.1,\dots,100\,\mu$m as derived from the
fractional power law (7) which is assumed to be valid in the whole range of
parameters used. The straight segments mark the validity of the asymptotic
fractional-power-low regime (10) which is realized at short relaxation times
$\tau_{{\mathrm{mr}}}$ and characterized by the scaling
$u_{c}(\tau_{{\mathrm{mr}}})\propto\nu^{\frac{\alpha-1}{\alpha}}=\nu^{\gamma}$
with the fixed exponent $\gamma\equiv 0.258(1)\simeq 1/4$. The electronic
fluid enters the preturbulence regime at $u>u_{c}$.

##  Appendix D D. Frequencies and magnitude of velocity oscillations for
various sizes of holes.

Figure S4: The Strouhal number (11) of flow oscillations at $\mathrm{Re}=100$
(a) in the dissipationless limit, $\tau_{{\mathrm{mr}}}=\infty$, as a function
of the ratio $L/W$ of the hole diameter $L$ and the width of the channel $W$;
(b) as a function of the relaxation number $\mathrm{Rl}$ for various ratios
$L/W$. (c) and (d): The same as in (a) and (b), but for the magnitude of the
velocity oscillations at the point at a distance $3L$ past the hole (cf. Fig.
3 in the main text). The frequency does depend on the geometry (a) but is
largely independent of the relaxation time $\tau_{{\mathrm{mr}}}$ (b). The
oscillations of the flow velocity exhibit substantial dependence on both
parameters. The insets show examples of the fluid flow at particular
geometries and also the limits at a vanishing diameter of the hole.