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How fast must an oriented collection of extensile swimmers swim to escape theinstability of viscous active suspensions? We show that the answer lies in thedimensionless combination $R=\rho v_0^2/2\sigma_a$, where $\rho$ is thesuspension mass density, $v_0$ the swim speed and $\sigma_a$ the active stress.Linear stability analysis shows that for small $R$ disturbances grow at a ratelinear in their wavenumber $q$, and that the dominant instability mode involvestwist. The resulting steady state in our numerical studies is isotropichedgehog-defect turbulence. Past a first threshold $R$ of order unity we find aslower growth rate, of $O(q^2)$; the numerically observed steady state is {\itphase-turbulent}: noisy but {\it aligned} on average. We present numericalevidence in three and two dimensions that this inertia driven flockingtransition is continuous, with a correlation length that grows on approachingthe transition. For much larger $R$ we find an aligned state linearly stable toperturbations at all $q$. Our predictions should be testable in suspensions ofmesoscale swimmers [D Klotsa, Soft Matter \textbf{15}, 8946 (2019)].

Inertia drives a flocking phase transition in viscous active fluids