New class of turbulence in active fluids

Abstract

Turbulence is a fundamental and ubiquitous phenomenon in nature, occurring from astrophysical to biophysical scales. At the same time, it is widely recognized as one of the key unsolved problems in modern physics, representing a paradigmatic example of nonlinear dynamics far from thermodynamic equilibrium. Whereas in the past, most theoretical work in this area has been devoted to NavierStokes flows, there is now a growing awareness of the need to extend the research focus to systems with more general patterns of energy injection and dissipation. These include various types of complex fluids and plasmas, as well as active systems consisting of self-propelled particles, like dense bacterial suspensions. Recently, a continuum model has been proposed for such "living fluids" that is based on the Navier-Stokes equations, but extends them to include some of the most general terms admitted by the symmetry of the problem [Wensink HH, et al. (2012) Proc Natl Acad Sci USA 109:14308-14313]. This introduces a cubic nonlinearity, related to the Toner-Tu theory of flocking, which can interact with the quadratic Navier-Stokes nonlinearity. We show that as a result of the subtle interaction between these two terms, the energy spectra at large spatial scales exhibit power laws that are not universal, but depend on both finite-size effects and physical parameters. Our combined numerical and analytical analysis reveals the origin of this effect and even provides a way to understand it quantitatively. Turbulence in active fluids, characterized by this kind of nonlinear self-organization, defines a new class of turbulent flows.D espite several decades of intensive research, turbulence-the irregular motion of a fluid or plasma-still defies a thorough understanding. It is a paradigmatic example of nonlinear dynamics and self-organization far from thermodynamic equilibrium also closely linked to fundamental questions about irreversibility (1) and mixing (2). The classical example of a turbulent system is a Navier-Stokes flow, with a single quadratic nonlinearity, wellseparated drive and dissipation ranges, and an extended intermediate range of purely conservative scale-to-scale energy transfer (3). However, many turbulent systems of scientific interest involve more general patterns of energy injection, transfer, and dissipation. A fascinating example of these kinds of generalized turbulent dynamics can be observed in dense bacterial suspensions (4). Although the motion of the individual swimmers in the background fluid takes place at Reynolds numbers well below unity, the coarse-grained dynamics of these self-propelled particles display spatiotemporal chaos, i.e., turbulence (5-7). Nevertheless, the correlation functions of the velocity and vorticity fields display some essential differences compared with their counterparts in classical fluid turbulence (8, 9). Moreover, the collective motion of bacteria in such suspensions exhibits long-range correlations (10), appears to be driven by internal instabilities (11), and depends strongly...

Vortex dynamics and Lagrangian statistics in a model for active turbulence

“…[17] and then normalize our numerical results based on dynamically emerging length and time scales. In summary, the procedure is as follows.…”

“…Consistent with ref. [17] we select a length scale l = 5π/k c . A velocity scale can be defined dimensionally as v 0 = Γ 3 0 /Γ 2 , which also selects a time scale l/v 0 .…”

“…(1) using this length scale and time scale reduces the two parameters Γ 0 and Γ 2 to constant numbers 0.045 and 9 × 10 −5 , respectively, thus decreasing the number of free parameters to three. If not noted otherwise, we choose the set of parameters λ 0 = 3.5, α = −1.0 and β = 0.5, which already has been investigated in [17]. We normalize our numerical results with respect to the dominant length scale in the system.…”

“…Supplementary material in the form of two .mp4 files is available from the Journal web page at http://dx.doi.org/10.1140/epje/i2018-11625-8 a e-mail: michael.wilczek@ds.mpg.de its name, active turbulence displays an intrinsic lengthscale selection absent in hydrodynamic turbulence, which is characterized by the formation of stable vortices of approximately constant sizes [15,17]. This typical scale is larger than the scale on which the driving occurs as the result of an upscale energy transport.…”

Gaussian theory for spatially distributed self-propelled particles

“…The result of the model is surprising because it introduces a new class of transitions with continuous symmetry breaking in two dimensions [28,29]. Later, many different studies were done to capture other interesting behavior of active matter [27,[30][31][32][33][34][35][36][37][38][39][40][41][42].…”