Three-dimensional flow structures of turbulence in precessing spheroids

We conduct direct numerical simulations of turbulence sustained in slowly precessing spheroids with the absolute value of the ellipticity eta being between zero (i.e., a sphere) and 0.2. Using a flexible grid generation algorithm, we can effectively simulate flows in an arbitrarily shaped container. This enables us to investigate the ellipticity dependence of the precession-driven flow in spheroids with the spin and precession axes being at a right angle. The numerical results are in excellent agreement with experimental data under the same flow conditions. In particular, we numerically realize hysteresis loops, which have been well known since the seminal experiments by Malkus [Science 160, 259 (1968)], connecting two qualitatively different states in a precessing spheroid with non-negligible ellipticity vertical bar eta vertical bar larger than about 10/root Re(where Re denotes the Reynolds number defined by using the spin angular velocity). Our numerical simulations reveal the three-dimensional turbulent flow structures in these states. One is a high-energy state where the mean flow is approximated by a uniform-vorticity flow. The other is a low-energy state with twisted mean-flow streamlines, which lead to fully developed turbulence when the Reynolds number is high enough. The mean-flow structures in the low-energy state are common irrespective of the ellipticity; namely, the main component of the mean flow is a circulation about the axis perpendicular both to the spin and precession axes, but the torsion of the mean-flow streamlines is larger for smaller eta. For sufficiently high Reynolds numbers, the low-energy state and therefore developed turbulence are sustained for the Poincare number (the precession rate) larger than about vertical bar eta vertical bar/2. On the other hand, stronger precession leads to the significant reduction of turbulence in a central region of the container. Hence, a container with smaller vertical bar eta vertical bar is adequate to sustain developed turbulence with weak precession.