Abstract
Microswimmers exhibit an intriguing, highlydynamic collective motion with largescale swirling and streaming patterns, denoted as active turbulence – reminiscent of classical highReynoldsnumber hydrodynamic turbulence. Various experimental, numerical, and theoretical approaches have been applied to elucidate similarities and differences of inertial hydrodynamic and active turbulence. We use squirmers embedded in a mesoscale fluid, modeled by the multiparticle collision dynamics (MPC) approach, to explore the collective behavior of bacteriatype microswimmers. Our model includes the active hydrodynamic stress generated by propulsion, and a rotlet dipole characteristic for flagellated bacteria. We find emergent clusters, activityinduced phase separation, and swarming behavior, depending on density, active stress, and the rotlet dipole strength. The analysis of the squirmer dynamics in the swarming phase yields KolomogorovKraichnantype hydrodynamic turbulence and energy spectra for sufficiently high concentrations and a strong rotlet dipole. This emphasizes the paramount importance of the hydrodynamic flow field for swarming motility and bacterial turbulence.
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Introduction
Active matter comprises a unique class of systems with intricate structural and dynamical features, facilitated by their elementary agents consuming internal energy, or energy from the environment, to maintain an outofequilibrium state. The interplay between the autonomous locomotion of the agents and their interactions leads to largescale selforganized swarm behavior manifested in such diverse biological systems as flocks of birds^{1,2,3,4}, school of fish^{5,6}, bacterial colonies^{7,8,9,10,11,12,13,14,15,16}, epithelial cell monolayers^{17,18,19}, and the cell cytoskeleton^{20,21,22,23}, as well as synthetic systems like robots^{24,25}, selfassembled magnetic spinners^{26}, and phoretic colloids^{27,28,29}.
Bacteria exhibit a particular mode of locomotion in dense populations denoted as swarming motility, where they exhibit rapid, coherent group migration over surfaces, with largescale swirling and streaming patterns^{7,10,11,12,30,31}. Similarly to bacterial swarming behavior^{11,13,16,32,33,34,35,36,37,38}, tissue cells^{19,39,40,41}, and filament/motorprotein mixtures^{18,22,23,42,43} exhibit collective, visually chaotic motion, nowadays often denoted as active turbulence or mesoscale turbulence, with largescale spatially and temporally random flow patterns. At first glance, the flow patterns are reminiscent of those observed in classical highReynoldsnumber hydrodynamic turbulence^{44,45,46}, despite active turbulence occurring at exceedingly small Reynolds numbers. The similarity prompted intensive studies of the collective motion of active matter systems to unravel the underlying physical mechanisms due to its prototypical character for nonlinear and nonequilibrium dynamical systems, which is considered as a major challenge for current theoretical physics^{43}.
Fundamental insight into hydrodynamic turbulence is achieved via velocity correlation functions^{47}. In particular, Kolmogorov predicted the universal powerlaw dependence for the energy spectrum E ~ k^{−κ} on the wavenumber k = ∣k∣, with κ = 5/3, obtained by Fourier transformation of the spatial velocity correlation function^{44,47}. In fact, this relation applies for two (2D) and threedimensional (3D) systems^{45}. Numerous studies on active systems reveal a wide spectrum of possible turbulent characteristics dependent on their constituents and the detailed (microscopic) interaction mechanisms, reflected in a wide range of exponents deviating from the Kolmogorov value, see Table 1. Experiments on B. subtilis and E. coli bacteria^{13,38} yield exponents significantly above and below the Kolmogorov value. Computer simulations employing various models have been performed and the energy spectrum has been calculated. Nonhydrodynamic particlebased simulations of an extension of the Vicsek model^{48}, accounting for shortrange parallel and largerange antiparallel alignment, yield the same exponent^{49} as experiments on E. coli^{13}. Simulations of selfpropelled rodlike particles give a value close to the Kolmogorov value^{13,50}. Lattice Boltzmann simulations of microswimmers represented by extended force dipoles (point particles) produce seemingly turbulent behavior for sufficiently large swimmer densities^{51} (see Table 1). For active nematics, the route to chaotic behavior has been studied experimentally and theoretically^{23,52}. Their dynamics is characterized by an intrinsic length scale l_{a}, where l_{a} is determined by the balance between the active and nematic elastic stress^{18,42}, and the creation and annihilation of topological defects. In addition, various theoretical studies have been performed with^{42} and without^{18} defects, where both yield similar energy spectra with distinct powerlaw exponents for length scales larger and smaller than l_{a} (Table 1). In contrast, we expect hydrodynamic interactions to dominate the chaotic and turbulent behavior in bacterial suspensions. Hence, it is a priori not evident that both types of chaotic dynamics exhibit the same kind of turbulent behavior, taken into account the disparity in the exponents κ and \(\hat{\kappa }\) in Table 1.
There are two particular systems of mesoscopic active particles, namely spinners — short rodlike selforganized colloidal structures rotated by an external magnetic field^{53} — and Marangoni surfers^{28}, where turbulent dynamics consistent with Kolmogorov scaling has been observed. Their Reynolds numbers \(Re \sim {{{{{{{\mathcal{O}}}}}}}}(10)\) are much smaller than that of classical inertial turbulence, but are much larger than those of microswimmer systems, where Re ≪ 1.
As a major difference to hydrodynamic turbulence, various experimental and simulation studies of active turbulence suggest the presence of a characteristic upper length scale for the vortex size, only below which the energy spectrum decreases in a powerlaw manner with increasing wavenumber k^{32,35,36}. This scale is typically on the order of ten microswimmer lengths. Theoretical studies based on a continuum approach^{13,36,43,54}, where the velocity field is described by the incompressible TonerTu equation^{55,56} combined with a SwiftHohenberg term^{57} for pattern formation, support this observation. However, in contrast to highReynoldsnumber hydrodynamic turbulence, which is governed by inertia, the internal stress due to selfpropulsion and polar alignment interactions of the active agents is important, which, combined with the fluid dynamics, determines the vortex size^{54}.
The diversity of obtained energy spectra and characteristic power laws (Table 1) indicates a strong dependence of the collective behavior on the detailed microswimmer interactions. Yet, it is not clear to which extent and under what circumstances hydrodynamic interactions are important.
In this article, we perform extensive coarsegrained mesoscale hydrodynamic simulations by employing the multiparticle collision dynamics (MPC) approach for fluids^{58,59,60} to elucidate the collective, turbulent motion of microswimmers in monolayer films. The microswimmers are described in a coarsegrained manner through the squirmer model^{60,61,62,63,64}. Particular attention is paid to the influence of the microswimmers’ hydrodynamic flow field on their collective behavior, i.e, the active stress and the rotlet dipole resulting from the rotating flagella (bundles) and the counterrotating cell body in flagellated bacteria^{65,66,67,68}. In general, hydrodynamics plays a decisive role in the collective behavior of microswimmers^{60,63,69}. While dry spherical active Brownian particles (ABPs) exhibit motilityinduced phases separation (MIPS)^{15,70,71,72,73,74,75}, spherical microswimmers in the presence of hydrodynamics show cluster formation^{60}, but no phase separation^{60,76}. However, anisotropic, spheroidal squirmers exhibit enhanced clustering compared to similar ABP systems due to hydrodynamic attraction and steric interactions^{60}. Hence, it is important to unravel the effect of shape, active stress, and of a rotlet dipole in dense microswimmer systems on their emergent collective properties, since bacteria in films exhibit swarming — a rapid, coherent group migration over surfaces in dense populations, with largescale swirling and streaming patterns^{10,11,16,31} — rather than clustering and phase separation^{11,13,16,32,33,34,35,36,38}.
By systematically varying the squirmer density, the active stress, and the rotlet dipole strength, our simulations provide insight into their influence on the collective dynamics of microswimmers. In particular, the combination of active stresses and a nonzero rotlet dipole suppresses phase separation and promotes swarming motility.
The analysis of the swarming phases reveals turbulentlike motion, where the energy spectrum displays powerlaw decays below the characteristic length scale discussed above, however, with an exponent depending on the squirmer concentration. We find the value κ = 5/3 for our largest density, strong active stress, and a nonzero rotlet dipole, consistent with the Kolmogorov prediction. Based on our analysis, in the Discussion and Conclusions section, we propose criteria which a dense, visually chaotic systems should satisfy to be possibly classified as turbulent.
Results
In our simulations, N_{sq} prolate spheroidal squirmers with the semimajor, b_{z}, and minor, b_{x}, axis are confined in a threedimensional narrow slit between two parallel walls and periodic boundary conditions along the x and z direction (Fig. 1). As described in the Methods section, the prescribed squirmer surface velocity yields swimming with the velocity v_{0}, an active stress of strength β, and a rotlet dipole of strength λ. The embedding fluid is explicitly modeled via the multiparticle collision dynamics (MPC) method^{58,59}, applying the stochasticrotation variant with angular momentum conservation (MPCSRD+a)^{77,78}. Further details of the model and implementation are presented in the Methods section.
Structural properties
The simulation snapshots of Fig. 2 illustrate emergent structures for the various considered packing fractions, active stresses, and rotlet dipole strengths. Distinct motility patterns can be identified:

(i)
Gas of small clusters for ϕ ≲ 0.3.

(ii)
Motilityinduced phase separation (AMIPS) for ∣β∣≥1, λ = 0, ϕ ≳ 0.3. Since here the shape of the spheroids implies squirmer alignment and the formation of polar motile clusters, we use the notation AMIPS to distinguish it from the case of isotropic, nonaligning particles, which form immobile clusters (MIPS)^{15,71,72}.

(iii)
Swarming motility for ∣β∣ > 1, λ = 4, ϕ ≳ 0.6.
In a general sense, the clusters formed by AMIPS can exhibit swarming behavior, because they are rather dynamic and exhibit translational and rotational motion. Their size increases with increasing packing fraction and are systemspanning for ϕ ≳ 0.5, consistent with our previous studies^{60}.
In the dense swarming phase, clusters of squirmers migrate collectively, thereby forming dynamic swirling and streaming patterns^{10,12,16,31}. A quantitative criterion for the classification into AMIPS and swarming motility will be provided in terms of the clustersize distribution function (cf. Sec. Clustersize distribution). Some of the small clusters for ϕ ≲ 0.3 exhibit cooperative motion, where a few squirmers move together for some time. In general, the rotlet dipole enhances cluster formation, and squirmers align side by side, which is clearly visible for ϕ ≲ 0.4. The precise mechanism for this cooperative motion is unexplored, but could depend on squirmer wall interactions. In contrast, for larger packing fractions the rotlet dipole suppresses AMIPS and enhances swarming motility.
Local packing fraction
Clustering and AMIPS of the squirmers are analyzed quantitatively by a Voronoi tessellation of the accessible volume^{60,74,79,80}. Figure 3 provides examples of density distributions for the average packing fractions ϕ = 0.4 and 0.6. The pronounced peak at the local packing fraction ϕ_{loc} ≈ 0.75 for ϕ = 0.4, β = −1, and λ = 0 indicates AMIPS (Fig. 3(a)), with a dense phase in contact with a dilute phase, consistent with the snapshots of Fig. 2. Results for large ∣β∣ imply a disintegration of the large aggregate and ultimately, for β < −3, P_{ϕ} displays a maximum at the average packing fraction, which indicates the absence of phase separation. Similarly, at ϕ = 0.6, the peaks in Fig. 3(b) for λ = 0 indicate phase separation, even for β as negative as β = −5. The rotlet dipole prevents formation of large clusters, but even for β = −5 and λ = 4 a broad range of cluster sizes exists.
Clustersize distribution
The clustersize distribution function
represents the fraction of squirmers belonging to a cluster of size n, where p(n) is the number of clusters of size n. The distribution is normalized such that \(\mathop{\sum }\nolimits_{n = 1}^{{N}_{sq}}{{{{{{{\mathcal{N}}}}}}}}(n)=1\). We use a distance and an orientation criterion to define a cluster: a squirmer belongs to a cluster, when its closest distance to another squirmer of the cluster is d_{s} < 1.8(2^{1/6} − 1)σ_{s} and the angle between the orientations of the two squirmers is <π/6 (see Methods section for the definition). The latter allows us to identify different clusters even at high packing fractions.
The clustersize distribution function is a useful quantity to characterize the motility pattern of a microswimmer system^{16,81}. In the homogeneous phase, the distribution function decays exponentially, whereas a second peak (bimodal distribution) indicates the formation of giant clusters (AMIPS). At the percolation transition, \({{{{{{{\mathcal{N}}}}}}}}\) becomes scale free and decays by a power law, \({{{{{{{\mathcal{N}}}}}}}} \sim {x}^{\gamma }\)^{81}. The swarming phase is characterized by a powerlaw decay with an exponential cutoff and a characteristic scale determined by an average vortex size^{16}. The distribution functions presented in Fig. 4 confirm our above conclusions on the emergent phases and motility patterns.
For ϕ = 0.4 and (β, λ) = (−1, 0), (−1, 4), ϕ = 0.6, λ = 0, and all considered β, as well as (β, λ) = (−1, 4), (−3, 4), we obtain bi and multimodal distributions with a powerlaw decay (cf. Table 2) at small cluster sizes and a high probability for giant clusters (N_{sq} = 270, ϕ = 0.4 and N_{sq} = 833, ϕ = 0.6). This indicates AMIPS^{16,60}. The large polar clusters are mobile, but the systems lack the characteristic largescale swirling patterns of swarming (cf. Supplementary Movie 1 and Supplementary Movie 2). The distribution functions for ϕ = 0.4, (β, λ) = (−5, 0), (−5, 4) decay in a qualitative different manner. They are well fitted by the function^{82}
This functional form is observed in various clusterforming processes^{81}. The function interpolates between the powerlaw decay found for percolating clusters and an exponential suppression of larger clusters. Table 2 presents the fit parameters for the various curves of Fig. 4. The exponential largen decay for ϕ = 0.4, (β, λ) = (−5, 0), (−5, 4) with a small value of x_{1} reflects the predominance of very small clusters — such systems are considered as a gas of clusters. In contrast, the clustersize distribution for ϕ = 0.6, (β, λ) = (−5, 4) decreases over a broad range of n in a powerlaw fashion reflecting the presence of a wide distribution of cluster sizes (x_{1} ≈ 80), and only larger clusters are exponentially suppressed — this system is in the swarming phase. The major difference to systems with (β, λ) = (−1, 4), (−3, 4) at this concentration is the more pronounced suppression of large clusters, which renders the overall system more dynamic.
The probability distribution functions of the local packing fraction (Fig. 3) and clustersize distribution functions (Fig. 4) clearly reveal a marked effect of the rotlet dipole on the collective behavior of the squirmers. In particular, AMIPS is suppressed, but formation of highly dynamic clusters prevails, with a rather broad distribution of cluster sizes for high squirmer densities.
Dynamical properties
Rotational diffusion
An individual squirmer in the slit exhibits rotational diffusion around a minor body axis. Interactions between squirmers, either steric or by their flow fields, change their diffusive behavior substantially^{60,63}. Figure 5(a) displays the time dependence of the autocorrelation function 〈e(t) ⋅ e(0)〉 of the propulsion direction of the squirmers. The various curves reflect a marked dependence of the rotational dynamics on the active stress and the rotlet dipole strength. The correlation function of the systems for (β, λ) = (−1, 0), (−3, 0), (−1, 4) exhibit a nonsingleexponential decay. Steric interactions between squirmers with a preference to cluster formation as well as between finitesize clusters lead to a rotation of whole clusters, which implies a faster decay of the rotational correlation compared to thermal fluctuations alone (cf. Supplementary Movie 4)^{83}.
We characterize the rotational motion by fitting the initial decay of the correlation function with the exponential
as displayed in Fig. 5(a). The factor \({C}_{R}^{0}\approx 1.03\) is included to account for a nonexponential decay for very short times. Squirmers with large active stresses and a rotlet dipole ((β, λ) = (−5, 0), (−3, 4), (−5, 4)) exhibit an exponentially decaying correlation function C_{R} over more than an order of magnitude. The extracted rotational diffusion coefficients D_{R} obey \({D}_{R}/{D}_{R}^{0} > 1\) (Fig. 5(b)), which reveals an accelerated rotational motion by shapeinduced steric interactions and hydrodynamic flow fields. Note that \({D}_{R}^{0}\) in a dilute system is independent of β. The diffusion coefficient D_{R} increases with increasing squirmer concentration, reaches a packing fractiondependent maximum and decreases again for larger ϕ. The snapshots of Fig. 2 suggest that the maxima in Fig. 5(b) are related to the threshold of cluster formation. An increasing number of squirmer contacts with increasing ϕ (ϕ ≲ 0.5) implies a faster reorientation. However, at larger ϕ, the emerging clusters, which move collectively and more persistently, lead to a reduction of D_{R}. The larger D_{R} values for larger ∣β∣ demonstrate the substantial contribution of active stress to the reorientation of the squirmers. At smaller ϕ and β < −1, the presence of a rotlet dipole with λ = 4 evidently reduces D_{R} compared to that for λ = 0, which is associated with the appearance of small clusters of sidebyside swimming squirmers (cf. Fig. 2 and Supplementary Movie 4). In contrast, at high packing fractions, a rotlet dipole implies a larger D_{R} as a consequence of an enhanced orientational motion of smaller clusters, specifically at large ∣β∣ = 5.
Mean square displacement
The meansquare displacement of the squirmers at high packing fractions (ϕ ≥ 0.6, Fig. 6) exhibit the typical ballistic motion for short times and a crossover to a diffusive motion for long times \(t{D}_{R}^{0}\,\gtrsim\, 0.1\)^{15,71}, at least for systems with λ = 4. (The resolution of the longtime behavior of the phase separated systems for λ = 0 requires longer simulations.) There is only a slight difference in the swimming speed of the various squirmers at short times. The presence of a rotlet dipole causes an earlier deviation from a strict ballistic motion toward a ballisticlike motion with an exponent somewhat smaller than 2 as time increases compared to squirmers without such a dipole. The systems with (β, λ) = (−5, 0), (−1, 4), (−3, 4), (−5, 4) exhibit a crossover from a ballistic or near ballistic to a diffusive motion at a displacement roughly corresponding to 12b_{z}, i.e., 6 squirmer lengths. We may consider this as a characteristic length scale in the system, separating the scale of persistent motion from that of diffusive motion. The crossover for (β, λ) = (−1, 0), (−3, 0) occurs at longer times.
The attempt to fit the meansquare displacement of Fig. 6 by the expression of an ABP^{15,71} failed for β > 1, in particular for λ = 4. As reflected by the density dependence of D_{R}, the longtime dynamics is strongly affected by the formation of cluster and their collective dynamics. However, the restriction of the fit to the crossover regime from ballistic to diffusive motion yields rotational diffusion coefficients in agreement with those extracted from the shorttime behavior of the correlation function C_{R} (Fig. 5).
Velocity distribution function
Thermal fluctuations and squirmer interactions imply strongly varying instantaneous velocities, both in direction as well as in magnitude. Hence, for the calculation of the velocity distribution function, we determine a swimming velocity by the finitedifference quotient of displacements
where r_{i} is the centerofmass position of squirmer i. During the selected time interval \({{\Delta }}t=1{0}^{3}\sqrt{m{a}^{2}/({k}_{B}T)}\), a squirmer moves at most the distance 2b_{z}/3.
For lower packing fractions ϕ ≤ 0.5, Fig. 7(a) displays the distribution function P(v) of the Cartesian inplane velocities \({{\Delta }}v=({v}_{x/z}{\bar{v}}_{x/z})\), where \({\bar{v}}_{x/z}\) are the ensemble and timeaveraged velocities along the Cartesian directions x and z. The averages \({\bar{v}}_{x/z}\) are very small for all considered parameter sets. Since the two spatial dimensions are equivalent, P(v) is averaged over the x and z direction. For ϕ = 0.1, we find pronounced nonGaussian, bimodal distributions. It reflects the swimming of the squirmers with nearly constant velocity magnitude v_{0} along their major semiaxis. This is emphasized by the distribution function of the velocity modulus v = ∣v∣ (inset Fig. 7(a)). This behavior is not unique for squirmers, but generic and also displayed by ABPs^{84}. Thermal fluctuations, and hydrodynamic and steric interactions between squirmers modify the swimming velocity, hence, P(v) is broadened and asymmetric with respect to the average of the modulus of the swimming velocity. With increasing density, the modulus decreases and the two peaks of the bimodal distribution gradually merge, exhibiting a flat central regime for certain parameters. Our data show that the rotlet dipole enhances the variations in v.
At the highest packing fraction ϕ = 0.68, compare Fig. 7(b), in particular for (β, λ) = (−5, 4), the squirmers strongly interact with each other and the distribution function P(v) becomes Maxwellian and P(v) Gaussian. The latter not only requires pronounced changes of the swimming direction, but more importantly, of the modulus v. The crossover from a bimodal to a Gaussian distribution is gradual and depends on the microswimmer parameters β and λ. The broad tails of the distribution functions for λ = 0 and the small deviations from a Gaussian for β = −3 and −5 reflect a persistent motion of squirmers in large clusters with a preference toward larger velocities v. The system with (β, λ) = (−1, 0) provides an example for a strongly correlated dynamics of squirmers in huge clusters (Fig. 2), with a very slow sampling of the velocity distribution. As a consequence, we obtain broad tails and large variations in the vicinity of Δv = 0 in the distribution function. Here, many more realizations and longer simulation times have to be considered to converge to the final stationary state. It is worth mentioning that experimental and theoretical studies of persistent random walks with a broad distribution of relaxation times predict nonGaussian distribution functions with a broad tail^{85,86}. Indeed, the orientational correlation function of the squirmers decays in a nonsingle exponential manner for various pairs of β and λ, specifically for λ = 0, corresponding to a wide distribution of rotational diffusion coefficients, as shown in Fig. 5(a). This reflects a more intricate dynamics of the squirmers in these systems, which could suffice to result in nonGaussian velocity distribution functions. Here, further studies are required to resolve the influence of clusters on the dynamics of the squirmers and the velocity distribution function.
We like to emphasize that the velocity distribution functions in the regime of swarming motility, especially for systems with ϕ = 0.68 and (β, λ) = (−3, 4), (−5, 4), are very well described by a Gaussian, despite pronounced collective swimming. Evidently, steric and flowfield interactions induce sufficient randomness, which correspondingly leads to large variations in the swimming velocity, specifically in the modulus. This aspect is particularly relevant, because velocities in both bacterial^{13,36} and highReynoldsnumber turbulence are Gaussian distributed.
Active turbulence
The characteristic features of the swimmer flow fields at higher densities are illustrated in Fig. 8. The clusters depicted in Fig. 8(a) exhibit a visually chaotic collective motion with regions of low and high velocity (Fig. 8(b)) and vorticity (Fig. 8(c)) (see Supplementary Movie 3, Supplementary Movie 5, and Supplementary Movie 6 for the packing fraction ϕ = 0.68). The patterns are similar to those observed in experiments on bacteria^{13,16,32,35,36}, previous simulations^{13,49}, and continuum theory^{13,33,36}.
Spatial velocity correlation function
Quantitative insight into the turbulent dynamics of the squirmers is obtained by their spatial velocity correlation function, a concept well established in classic hydrodynamic turbulence^{13,44,46,47}. For the discrete particle system, we define the spatial velocity correlation function as^{50,74,87}
Moreover, we introduce a normalized velocity correlation function as \({C}_{v}^{0}(R)={C}_{v}(R)/{c}_{0}\), with \({c}_{0}={\sum }_{i}\langle {{{{{{{{\boldsymbol{v}}}}}}}}}_{i}^{2}\rangle /Nsq\). (For a homogeneous and isotropic system, C_{v}(R) is a function of R = ∣R∣ only.) Results of \({C}_{v}^{0}\) for the packing fractions ϕ = 0.4 and 0.6 are presented in Fig. 9. Three distinct decay patterns can be identified: (i) a very slow decay over roughly the whole system (ϕ = 0.4, (β, λ) = (−1, 0), Fig. 9(a); ϕ = 0.6, (β, λ) = (−1, 0), (−3, 0), Fig. 9(b)), (ii) a decay, where correlations functions are negative for R ≲ L/2 (ϕ = 0.4, (β, λ) = (−1, 4); ϕ = 0.6, (β, λ) = (−1, 4), (−3, 4)), and (iii) correlations functions, which assume negative values over a certain interval, but are positive for R ≈ L/2 (ϕ = 0.4, (β, λ) = (−5, 0), (−3, 4), (−5, 4); ϕ = 0.6, (β, λ) = (−5, 4)). The case (i) corresponds to longrange correlations over the entire simulation box, consistent with AMIPS and the appearance of a large cluster (Fig. 4). As shown in Fig. 9(b), such \({C}_{v}^{0}(R)\) can be fitted by the function
Specifically for ϕ = 0.6, we obtain the parameters of Table 3. The respective velocity correlation functions decay approximately exponentially, with characteristic lengths scales between 2.3 and 5.6 swimmer lengths. The smaller value ξ/(2b_{z}) = 2.3 for λ = 4 indicates that a nonzero rotlet dipole implies weaker spatial correlation and, hence, smaller clusters. The distinct decay patterns support our conclusion on the motility mode as discussed in relation with the clustersize distribution functions (Fig. 4). However, a clearcut separation of swarming and cluster dynamics is difficult to establish based on C_{v}(R).
An important feature of bacterial turbulence is a finite vortex size, which marks a characteristic length scale in the system and is reflected in a minimum of the velocity correlation function^{13,34,36,87}. Our simulations yield such a minimum, e.g., for ϕ = 0.6, 0.68, (β, λ) = (−5, 4). Hence, we expect such squirmer system to exhibit active turbulence. A characteristic length scale can also exist for lower densities, e.g., for ϕ = 0.4, (β, λ) = (−5, 0), (−5, 4), where only small clusters are present. We do not denote the dynamics of such systems as turbulent according to the criteria provided in the Discussion and Conclusions section.
Energy spectrum
Insight into the turbulent behavior is gained by the energy spectrum
which is obtained as Fourier transform of the spatial velocity correlation function (5)^{47}, and manifests the distribution of kinetic energy over different length scales. In the calculation of E(k), we apply a leftshift of the correlation function C_{v}(R) (Fig. 9) such that the decay starts at R = 0 in order to avoid artifacts in the Fourier transformation by a truncated correlation function. As for bacterial suspensions, the energy injection scale is the length scale of a microswimmer (2b_{z}), which yields the characteristic (maximum) wavenumber k_{c} = π/b_{z} for our squirmers.
Figure 10 displays the energy spectrum for (β, λ) = (−5, 4) and the two packing fractions ϕ = 0.6, 0.68, and various system sizes. The simulations show two powerlaw regimes for a given density, namely \(E(k) \sim {k}^{\hat{\kappa }}\) for k < k_{m} and E(k) ~ k^{−κ} for k_{m} < k < k_{c}, with k_{m} corresponding to the position of the maximum of E(k). Such a maximum in E(k) is a feature of microswimmer active turbulence, and reflects a characteristic vortex size^{13,35,36}. Our simulations yield approximate vortex sizes of 5 (10b_{z}) and 10 squirmer lengths (20b_{z}) for ϕ = 0.6 and 0.68, respectively. They are roughly consistent with the patterns of Fig. 8, the crossover from ballistic to diffusive motion in the meansquare displacement of Fig. 6, and the minimum of the correlation function of Fig. 9(b). Vortex sizes on the order of 5–10 microswimmer lengths are also found in experiments^{13,35,38}.
For k_{m} < k < k_{c}, corresponding to R > 2b_{x}, our simulations yield turbulent flow patterns (Fig. 8). The exponent of the scaling regime depends on the squirmer density, with the values κ = − 2 for ϕ = 0.6 and κ = − 5/3 for ϕ = 0.68. The latter is consistent with the KolmogorovKraichnan prediction for classical 2D turbulence^{45}. This is in contrast to the wide range of exponents found in simulations and experiments (cf. Table 1). Density seems to play an important role for the observed turbulent behavior. The squirmers of both densities exhibit swarming motility, namely, collective motion with largescale swirling and streaming patterns. However, only the dynamics in the higher density system exhibits the exponent κ = 5/3.
In the small kvalue regime, we obtain the exponents \(\hat{\kappa }=1\) for ϕ = 0.68 and \(\hat{\kappa }=5/3\) for ϕ = 0.6, which reflect an increase of the energy with increasing k. The dependence k^{5/3} is consistent with that observed theoretically and experimentally in Ref. ^{13}, as well as in simulations^{49}. However, other studies yield rather different dependencies (Table 1). Theoretical models suggest that the smallk slope is governed by finitesystemsize effects, i.e., depends in the boundary condition and physical parameters^{43}. The curves in Fig. 10 reflect a weak dependence on the system size.
The presences of a smalldistance cutoff, where energy input by the squirmers occurs, and the peak in E(k), corresponding to a characteristic vortex size, limits the krange over which the energy spectrum decays in a powerlaw manner. This is in stark contrast to classical highReynoldsnumber turbulence, where the energy cascade extents over many orders of magnitude.
Discussion
We have performed largescale mesoscale hydrodynamics simulations of spheroidal squirmers in a narrow slit in order to analyze the emerging structures, motility patterns, and turbulent behavior for various packing fractions, active stresses, and rotletdipole strengths.
Our studies reveal a strong dependence of the motility pattern on the microswimmer concentration and their propulsioninduced flow field. The classification of the distinct motion pattern into the various categories — swimming and collective motion of very small clusters (cluster gas), phase separation by activity and anisotropic swimmer shape (AMIPS), and swarming — is accomplished by visual inspection of snapshots (Fig. 2) and the characteristic features of the clustersize distribution function (Fig. 4). AMIPS appears for small active stresses, ∣β∣ ≲ 3, and all packing fractions ϕ > 0.2. Here, we expect enhanced cluster formation for larger system sizes rather than active turbulence. Squirmers with stronger forces dipoles, ∣β∣ ≳ 3, at concentrations ϕ < 0.4 exhibit small clusters and strong cooperative effects for λ = 4. At higher packing fractions, ϕ > 0.6, swarming motility appears for the rotletdipole strength λ = 4, where clusters of squirmers move collectively, and even exhibit active turbulence for ϕ = 0.68 and β = − 5 (Fig. 2). Importantly, the rotlet dipole suppresses AMIPS. As shown in Fig. 10, this behavior is independent of system size.
Our simulations clearly reveal the difficulty to characterize turbulence in active systems. Even more fundamental is the question, which criteria should be applied to classify a mesoscale system as turbulent. Considering microswimmer systems, visually chaotic flow patterns are evidently not sufficient. Inspired by experimental systems displaying bacterial turbulence, our simulation results for squirmer, and systems exhibiting classical hydrodynamic turbulence, we propose the following “minimal” criteria for bacterial turbulence:

Reynolds numbers Re ≪ 1^{10,13}

high microswimmer density: closely packed swimmers with average distances smaller than their size^{16}

presence of visually chaotic flow patterns^{18,43,52} with largescale collective behavior

characteristic vortex size^{35,36} and a velocity correlation function which becomes negative on intermediate distances

Gaussian velocitydistribution function of the microswimmer’s Cartesian velocity components^{13,36}

energy spectrum with powerlaw decay E(k) ~ k^{−κ}, κ > 0, on length scales below the characteristic vortex size.
The presence of small and large lengthscale cutoffs by the microswimmer and vortex size implies a universal, scalefree behavior only over a limited range of length scales.
Analyzing the swarming motion of the squirmers, we find nonGaussian distribution functions for the velocities parallel to the confining walls for ϕ ≤ 0.6. According to our criteria, we classify such systems as nonturbulent. However, we obtain a Gaussian velocity distribution for ϕ = 0.68 and (β, λ) = (−5, 4) (Fig. 7). The energy spectrum of that system exhibits a powerlaw decay with the exponent κ = 5/3, characteristic for KolmogorovKraichnantype turbulence in the inertial range. Hence, this systems fulfills all the above criteria, and we consider it as fully turbulent.
The slope of the powerlaw regime depends on the squirmer density. At the smaller packing fraction ϕ = 0.6 and (β, λ) = (−5, 4), the energy spectrum decreases faster, with the exponent κ = 2. At the same time, the velocity distribution function is nonGaussian. Thus, the system is not showing active turbulence in the above sense, yet, is exhibiting swarming behavior. This suggests a tight link between the energy spectrum and the velocity distribution function, a relation which needs further considerations.
As typically observed in turbulent bacterial suspensions^{13,35,36}, we also obtain a maximum in the energy spectrum at 5–10 squirmer lengths, as well as a negative spatial velocity correlation function, in agreement with the presence of a characteristic vortex size.
The effective inertia due to the collective active motion could play an important role, since the crossover from the active ballistic motion — equivalent to inertia of a passive system — to active diffusion appears on the length scale of approximately 6 squirmers lengths, which is comparable to the characteristic vortex size. Yet, the Reynolds number on the scale of a vortex (approximately 10 microswimmer lengths) is still smaller than unity. Here, more detailed theoretical studies of a suitable model are required to assess the relevance of the various interactions on active turbulence.
Despite the similarities of our squirmer systems with bacterial suspensions, there is one major difference, namely, the swimming speed of bacteria increasing in the swarming phase, whereas it decreases in our case^{88}. This may point toward a particular role of bacterial flagella in the propulsion of the dense bacterial system.
We like to emphasize that hydrodynamic interactions are paramount for microswimmer swarming and active turbulence, specifically the active stress and the rotlet dipole determine their swarming motility. However, for KolmogorovKraichnantype characteristics to emerge, in addition, density plays a major role, and ensures an isotropic and homogeneous dynamics on lengths scales larger than approximately a squirmer length. Our simulations provide a benchmark for further theoretical and simulation studies on bacterial turbulence to elucidate the interplay between hydrodynamic stress — specifically a rotlet dipole —, alignment interactions by anisotropic swimmer shapes, and volume exclusion.
Methods
Microswimmer model: prolate squirmer
The prescribed surface velocity of the prolate spheroidal squirmer, a homogeneous colloidal particle of mass M, is given by^{61,62,63,64}
in terms of spheroidal coordinates τ, ζ, φ (1 ≤ τ < ∞, −1 ≤ ζ ≤ 1, 0 ≤ φ < 2π) (Fig. 1(a))^{63,89}. For a squirmer with propulsion direction e = (0, 0, 1), the Cartesian coordinates of a point on the spheroid surface \({{{{{{{{\boldsymbol{r}}}}}}}}}_{s}={({x}_{s},{y}_{s},{z}_{s})}^{T}\) are
with \({\bar{r}}_{s}=\sqrt{{x}_{s}^{2}+{y}_{s}^{2}}\), r_{s} = ∣r_{s}∣, \(\tau ={\tau }_{0}={b}_{z}/\sqrt{{b}_{z}^{2}{b}_{x}^{2}}\), and the lengths b_{z} and b_{x} along the semimajor and minor axis (Fig. 1(a)). The terms with the coefficients B_{1} and β (β < 0, pusher) account for swimming in the direction e and an active stress, respectively^{63,77,89}. The rotletdipole term (second term on the righthand side of Eq. (8)) accounts for the torquefree nature of swimming bacteria with a cell body counterrotating with respect to the flagellar bundle^{66}. Equation (8) is a straightforward generalization of the expression derived for spheres with the independent parameter λ^{90}. It is a solution of Stokes’ equations and, with the boundary condition on the spheroid’s surface (8), the velocity field of such a squirmer in an infinite fluid is given by \({{{{{{{{\boldsymbol{v}}}}}}}}}_{R}({{{{{{{\boldsymbol{r}}}}}}}})=3\lambda z\bar{r}{{{{{{{{\boldsymbol{e}}}}}}}}}_{\varphi }/{r}^{5}\) in a reference frame, where e is aligned with the z axis of the bodyfixed reference frame (see Fig. 1), \(\bar{r}=\sqrt{{x}^{2}+{y}^{2}}\), and r = ∣r∣. The swimming velocity of a squirmer is related to B_{1} as
To insure quasitwodimensional motion between the walls (Fig. 1(b)), a strong repulsive interaction between squirmers and walls is implemented by the truncated and shifted LennardJones potential
for y < 2^{1/6}σ_{w} and zero else, where y is the closest distance between a wall and the surface of a squirmer. Here, σ_{w} and ϵ_{w} determine to the length and energy scale, respectively. Hence, squirmers never touch a wall.
Squirmer volumeexclusion interactions are described by a separationshifted LennardJones potential with parameters σ_{s} and ϵ_{s}, where y → d_{s} + σ_{s} in Eq. (10), and d_{s} is the distance between the two closest points on the surfaces of two interacting spheroids^{63,89}.
The solidbody equations of motion of the squirmers — the centerofmass translational motion and the rotational motion described by quaternions — are solved by the velocityVerlet algorithm^{63,89}.
Fluid model: multiparticle collision dynamics
The fluid is modeled via the multiparticle collision dynamics (MPC) method, a particlebased mesoscale simulation approach accounting for thermal fluctuations^{58,59}, which has been shown to correctly capture hydrodynamic interactions^{91}, specifically for active agents and systems^{66,90,92,93,94,95,96,97,98,99,100,101}.
We apply the MPC approach with angular momentum conservation (MPCSRD+a)^{77,78}. The algorithm proceeds in two steps — streaming and collision. In the streaming step, the MPC point particles of mass m propagate ballistically over a time interval h, denoted as collision time. In the collision step, fluid particles are sorted into the cells of a cubic lattice of lattice constant a defining the collision environment, and their relative velocities, with respect to the centerofmass velocity of the collision cell, are rotated around a randomly oriented axes by a fixed angle α. The algorithm conserves mass, linear, and angular momentum on the collisioncell level, which implies hydrodynamics on large length and long time scales^{58,91}. A random shift of the collision cell lattice is applied at every collision step to ensure Galilean invariance^{102}. Thermal fluctuations are intrinsic to the MPC method. A celllevel canonical thermostat (MaxwellBoltzmann scaling (MBS) thermostat) is applied after every collision step, which maintains the temperature at the desired value^{103}. The MPC method is highly parallel and is efficiently implemented on a graphics processing unit (GPU) for a highperformance gain^{104}.
The fluid and the squirmers are confined in a narrow slit with noslip boundary conditions of the fluid at the walls. Squirmerfluid interactions appear during streaming and collision. While streaming squirmers and fluid particles, fluid particles are reflected at a squirmer’s surface by application of the bounceback rule and addition of the surface velocity u_{s}(8). To minimize slip, phantom particles are added inside of the squirmers, which contribute when collision cells penetrate squirmers. In all cases, the total linear and angular momenta are included in the squirmer dynamics. More details are described in Ref. ^{63} and the supplementary material of Ref. ^{89}.
Parameters
Multiple squirmers with the semimajor axis b_{z} = 6a and semiminor axis b_{x} = 2a are distributed in a narrow slit of width L_{y} = 8a, where a is the length of the MPC fluid collision cell. Parallel to the walls, periodic boundary conditions are applied. We set σ_{w} = 1.8a and ϵ_{w} = 18k_{B}T. Squirmer propulsion requires fluid particles adjacent to its surface. To avoid MPC particle depletion when two squirmers approach each other, we introduce a safety layer of thickness d_{v} = 0.25a around every squirmer, corresponding to the effective squirmer semiaxes b_{z} + d_{v} and b_{x} + d_{v}, respectively. The squirmersquirmer LennardJones parameters are set to σ_{s} = 0.5a, ϵ_{s} = 5k_{B}T. d_{s} (see microswimmer model) is now the distance between two closest points on the surfaces of the two interacting squirmers with effective (larger) semiaxes^{60,63,89}.
We employ a high average particle number 〈N_{c}〉 = 60 in a collision cell^{60}. Furthermore, we choose a small collisiontime step \(h=0.02\sqrt{m{a}^{2}/({k}_{B}T)}\) and the large rotation angle α = 130^{∘}. This results in the fluid viscosity \(\eta =127.8\sqrt{m{k}_{B}T/{a}^{4}}\) and the 2D rotational diffusion coefficient around a minor axis \({D}_{R}^{0}=5.2\times 1{0}^{6}\sqrt{{k}_{B}T/(m{a}^{2})}\). This is in close agreement with the theoretical value of a spheroid \({D}_{R}^{0}=5.5\times 1{0}^{6}\sqrt{{k}_{B}T/(m{a}^{2})}\).
For a squirmer, we choose \({B}_{1}=4.5\times 1{0}^{3}\sqrt{{k}_{B}T/m}\), corresponding to the swimming speed \({v}_{0}=4\times 1{0}^{3}\sqrt{{k}_{B}T/m}\), which yields the Péclet number \(Pe={v}_{0}/(2{b}_{z}{D}_{R}^{0})=64\) and the Reynolds number Re = 2b_{z}v_{0}〈N_{c}〉/(a^{3}η) = 0.023. The active stress values β = −1, −3, −5, covering approximately the estimated values from experiments and simulations (see below), and the rotlet dipole strengths λ = 0, 4 are considered. Typically, simulations with the box size L = 160a are performed for the 2D packing fractions ϕ = N_{sq}πb_{x}b_{z}/L^{2} = 0.1, 0.2, 0.3, 0.4, and 0.5, corresponding to the squirmer numbers N_{sq} = 66, 140, 200, 270, and 341. In order to reduce/avoid finitesize effects and to confirm our conclusions, we considered other system sizes, specifically, for higher densities significantly larger systems are simulated with L = 230a for N_{sq} = 833, 954, L = 460a and N_{sq} = 3332, 3816 (both ϕ = 0.6, 0.68), as well as L = 920a for N_{sq} = 15264 (ϕ = 0.68). (Note that the largest system contains 4 × 10^{8} MPC fluid particles.) A passive spheroid is neutrally bouyant with M = 6031m, and the MPC time step h is used in the integration of the squirmers’ equations of motion. Presented data are collected over the total time interval \(1{0}^{6}\sqrt{m{a}^{2}/({k}_{B}T)}\), after an extended equilibration period where the systems reached a stationary state. During this time, in dilute solution a squirmer actively diffuses about 150 body lengths, which corresponds to twice the size of the largest system, while for the largest packing fraction a squirmer travels about 1/3 of the system size.
Estimation of squirmer parameters for E. coli from simulations and experiments
In the farfield, the microswimmer flow field is dominated by the forcedipole term of strength^{6,15,65,105}
where P = f_{D}l_{D} is the magnitude of the force dipole of force f_{D} and length l_{D}. The latter parameters can be determined from experiments^{65} and simulations^{66}. The farfield expansion of the flow field of a spheroidal squirmer provides the relation between χ and the active stress parameter β^{63}:
With the approximation of the bacteria cell body by a spheroid, Eq. (12) provides an estimation of β for a given χ.

From simulations — An E. colitype cell model with the body length l_{b} = 2.4μm, cell body diameter d_{b} = 0.9μm, the swimming speed v_{0} = 40μm/s, forcedipole strength f_{D} = 0.57pN, and forcedipole length l_{D} = 3.84μm^{66}, yields β ≈ − 6. We use the cell body length rather the length of body plus flagellar bundle, guided by the discussion of E. coli rotation in Ref. ^{65}.

From experiments — E. coli bacteria are characterized by l_{b} = 3 μm, d_{b} = 1 μm, v_{0} = 22 μm/s, f_{D} = 0.42 pN, and l_{D} = 1.9 μm^{65}, which gives β ≈ − 3.
In both cases, the viscosity of water is used. These β values approximately fall into the range of active stresses considered in our simulations.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The custom code for the simulations on GPUs is available from the corresponding author upon reasonable request.
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Acknowledgements
This work has been supported by the DFG priority program SPP 1726 “Microswimmers – from Single Particle Motion to Collective Behaviour”. The authors gratefully acknowledge the computing time granted through JARAHPC on the supercomputer JURECA at Forschungszentrum Jülich.
Funding
Open Access funding enabled and organized by Projekt DEAL.
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R.G.W. and G.G. designed the study. K.Q. and E.W. wrote the simulation code and K.Q. performed the simulations. K.Q., R.G.W, and G.G. analyzed and discussed the results. R.G.W, G.G., and K.Q. wrote the paper.
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Qi, K., Westphal, E., Gompper, G. et al. Emergence of active turbulence in microswimmer suspensions due to active hydrodynamic stress and volume exclusion. Commun Phys 5, 49 (2022). https://doi.org/10.1038/s42005022008207
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DOI: https://doi.org/10.1038/s42005022008207
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