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Figure 3. Spectra of the four contributions to the energy balance Eq. (9) (Eq. (S32)). These results are for the largest system size with 2048 x 2048 grid points at activity number A = 3.2 - 10°. Fig. S2 shows results for smaller systems. The active energy injection is en- tirely balanced by dissipation at every scale; no energy is transferred between scales. The energy injection rate is maximal at the selected wavelength \,;.

Figure 3 Spectra of the four contributions to the energy balance Eq. (9) (Eq. (S32)). These results are for the largest system size with 2048 x 2048 grid points at activity number A = 3.2 - 10°. Fig. S2 shows results for smaller systems. The active energy injection is en- tirely balanced by dissipation at every scale; no energy is transferred between scales. The energy injection rate is maximal at the selected wavelength \,;.

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Figure 1. Stationary patterns upon the spontaneous flow instability. (A) Growth rate Eq. (8) of perturbations transverse to the director (¢ = 7/2) for a contractile system (S = —1) with R = 1. The crit- ical wavelength ~ @, decreases with the activity number A. Points indicate numerical results (Methods), with discrete wave numbers for a system of size L. (B) Stripe pattern and the corresponding spon- taneous shear flow for A = 200. (C) Stable vortex pattern at higher activity number A = 500. Small bars indicate the director (top). Lines with arrows indicate streamlines (bottom). Vortex patterns).
Figure 1. Stationary patterns upon the spontaneous flow instability. (A) Growth rate Eq. (8) of perturbations transverse to the director (¢ = 7/2) for a contractile system (S = —1) with R = 1. The crit- ical wavelength ~ @, decreases with the activity number A. Points indicate numerical results (Methods), with discrete wave numbers for a system of size L. (B) Stripe pattern and the corresponding spon- taneous shear flow for A = 200. (C) Stable vortex pattern at higher activity number A = 500. Small bars indicate the director (top). Lines with arrows indicate streamlines (bottom). Vortex patterns).
Figure 2. Disordered pattern of director domains with a characteristic wavelength. (A) Snapshot of the angle field for a system size of 2048 x 2048 grid points at activity number Amax = 3.2 - 10°. (B) The spectrum of the Frank elastic energy Eq. (4) (Eq. (S29), units of K/Lmax) is peaked at an intrinsic wavelength \; ~ £- independent of system size. Here, £. is held fixed (Cc = Lmax/WAmax © Dmax /566), such that the activity number A = L* /£2 increases with system size L = n&. The wave number is rescaled by the largest system size Lomax, Such that the axis shows the mode number in the largest system (n = 2048).
Figure 2. Disordered pattern of director domains with a characteristic wavelength. (A) Snapshot of the angle field for a system size of 2048 x 2048 grid points at activity number Amax = 3.2 - 10°. (B) The spectrum of the Frank elastic energy Eq. (4) (Eq. (S29), units of K/Lmax) is peaked at an intrinsic wavelength \; ~ £- independent of system size. Here, £. is held fixed (Cc = Lmax/WAmax © Dmax /566), such that the activity number A = L* /£2 increases with system size L = n&. The wave number is rescaled by the largest system size Lomax, Such that the axis shows the mode number in the largest system (n = 2048).
Figure 4. Universal scaling of the flow spectra at large scales. (A) Snapshot of the stream function field for a system size of 2048 x 2048 grid points at activity number Amax = 3.2 - 10°. The small background ripples correspond to coherent vortices of typical size given by the scale of maximal energy injection 4;. However, large-scale correlated flow patches also form due to the non-local character of viscous flow. Lines with arrows indicate a few streamlines that highlight these large-scale circulations. (B-C) The spectra of kinetic energy and enstrophy Eq. (10) (Eqs. (S35) and (S38)) exhibit a peak at the maximal injection scale A;. At smaller and, importantly, also at larger scales, the spectra feature distinctive universal scaling regimes. As in Fig. 2B, @. is held fixed (€¢ = Dmax/VAmax © Dmax/566), such that the activity number A = L*/€2 increases with system size L = nA. The wave number is rescaled by the largest system size Lmax, such that the axis shows the mode number in the largest system (n = 2048).
Figure 4. Universal scaling of the flow spectra at large scales. (A) Snapshot of the stream function field for a system size of 2048 x 2048 grid points at activity number Amax = 3.2 - 10°. The small background ripples correspond to coherent vortices of typical size given by the scale of maximal energy injection 4;. However, large-scale correlated flow patches also form due to the non-local character of viscous flow. Lines with arrows indicate a few streamlines that highlight these large-scale circulations. (B-C) The spectra of kinetic energy and enstrophy Eq. (10) (Eqs. (S35) and (S38)) exhibit a peak at the maximal injection scale A;. At smaller and, importantly, also at larger scales, the spectra feature distinctive universal scaling regimes. As in Fig. 2B, @. is held fixed (€¢ = Dmax/VAmax © Dmax/566), such that the activity number A = L*/€2 increases with system size L = nA. The wave number is rescaled by the largest system size Lmax, such that the axis shows the mode number in the largest system (n = 2048).
Figure S1. Probability distribution of the active power J for different system sizes L = nA. Here, ¢. is held fixed (C2 = Dmax/VAmax © Imax /566), such that the activity number A = L? / 2 increases with system size L. Negative and positive values of I respectively cor- respond to active energy absorption and injection into the system. Therefore, on average, active stresses inject energy to drive the flows.
Figure S1. Probability distribution of the active power J for different system sizes L = nA. Here, ¢. is held fixed (C2 = Dmax/VAmax © Imax /566), such that the activity number A = L? / 2 increases with system size L. Negative and positive values of I respectively cor- respond to active energy absorption and injection into the system. Therefore, on average, active stresses inject energy to drive the flows.
Figure $2. Spectra of the four contributions to the energy balance Eq. (9) (Eq. (S32)) for different system sizes L = nA. Here, @ is held fixed (€. = Lmax/WVAmax © Dmax/566), such that the activity number A = L?/€? increases with system size L. The wave number is rescaled by the largest system size max, such that the axis shows the mode number in the largest system (n = 2048).
Figure $2. Spectra of the four contributions to the energy balance Eq. (9) (Eq. (S32)) for different system sizes L = nA. Here, @ is held fixed (€. = Lmax/WVAmax © Dmax/566), such that the activity number A = L?/€? increases with system size L. The wave number is rescaled by the largest system size max, such that the axis shows the mode number in the largest system (n = 2048).
Figure S3. Power spectrum of the vorticity source Eq. (S43) for different system sizes L = nA. Here, @. is held fixed (@. = Lmax/V Amax © LDmax/566), such that the activity number A = L? /€2 increases with system size L. The wave number is rescaled by the largest system size Lmax, such that the axis shows the mode number in the largest system (n = 2048).
Figure S3. Power spectrum of the vorticity source Eq. (S43) for different system sizes L = nA. Here, @. is held fixed (@. = Lmax/V Amax © LDmax/566), such that the activity number A = L? /€2 increases with system size L. The wave number is rescaled by the largest system size Lmax, such that the axis shows the mode number in the largest system (n = 2048).
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