The spatially developing compressible plane mixing layer with a convective Mach number of 0.7 is investigated by direct numerical simulation. A pair of equal and opposite oblique instability waves is introduced to perturb the mixing layer at the inlet. The full evolution process of instability, including formation of $ \mrm{\Lambda} $ -vortices and hairpin vortices, breakdown of large structures and establishment of self-similar turbulence, is presented clearly in the simulation. In the transition process, the flow fields are populated sequentially by $ \mrm{\Lambda} $ -vortices, hairpin vortices and ‘flower’ structures. This is the first direct evidence showing the dominance of these structures in the spatially developing mixing layer. Hairpin vortices are found to play an important role in the breakdown of the flow. The legs of hairpin vortices first evolve into sheaths with intense vorticity then break up into small slender vortices. The later flower structures are produced by the instability of the heads of the hairpin vortices. They prevail for a long distance in the mixing layer until the flow starts to settle down into its self-similar state. The preponderance of slender inclined streamwise vortices is observed in the transversal middle zone of the transition region after the breakup of the hairpin legs. This predominance of streamwise vortices also persists in the self-similar turbulent region, though the vortices there are found to be relatively very weak. The evolution of both the mean streamwise velocity profile and the Reynolds stresses is found to have close connection to the behaviour of the large vortex structures. High growth rates of the momentum and vorticity thicknesses are observed in the transition region of the flow. The growth rates in the self-similar turbulence region decay to a value that agrees well with previous experimental and numerical studies. Shocklets occur in the simulation, and their formation mechanisms are elaborated and categorized. This is the first three-dimensional simulation that captures shocklets at this low convective Mach number.

Immersed boundary-lattice Boltzmann simulations are used to examine the effects of particle rotation, at low particle Reynolds numbers, on flows in ordered and random arrays of mono-disperse spheres. The drag force, the Magnus lift force and the torque on the spheres, are determined at solid volume fractions up to the close-packed limits of the arrays. The rotational Reynolds number based on the angular velocity and the diameter of the spheres is used to characterize the rotational movement of spheres. The results show that the normalized Magnus lift force produced by particle rotation is approximately in direct proportion to the rotational Reynolds number, while the normalized drag force and torque acting on spheres are barely affected by this number. The Magnus lift force is negligible relative to the magnitude of the drag force when the rotational Reynolds number is low. However, it can be very significant, and even larger than the drag force, as the rotational Reynolds number increases up to $O(10^{2})$ , especially for low solid volume fractions. Based on the simulation results, relations for the Magnus lift force and the torque for both ordered arrays and random arrays of rotating spheres at solid volume fractions from zero to close-packed limits are formulated. Further, the drag force relations in the literature are revised based on existing theories and the present simulation results for both arrays of spheres.

Direct numerical simulations with an immersed boundary-lattice Boltzmann method are used to investigate the effects of particle rotation on flows past random arrays of mono-disperse spheres at moderate particle Reynolds numbers. This study is an extension of a previous study of the authors [Q. Zhou and L.-S. Fan, “Direct numerical simulation of low-Reynolds-number flow past arrays of rotating spheres,” J. Fluid Mech. 765, 396–423 (2015)] that explored the effects of particle rotation at low particle Reynolds numbers. The results of this study indicate that as the particle Reynolds number increases, the normalized Magnus lift force decreases rapidly when the particle Reynolds number is in the range lower than 50. For the particle Reynolds number greater than 50, the normalized Magnus lift force approaches a constant value that is invariant with solid volume fractions. The proportional dependence of the Magnus lift force on the rotational Reynolds number (based on the angular velocity and the diameter of the spheres) observed at low particle Reynolds numbers does not change in the present study, making the Magnus lift force another possible factor that can significantly affect the overall dynamics of fluid-particle flows other than the drag force. Moreover, it is found that both the normalized drag force and the normalized torque increase with the increase of the particle Reynolds number and the solid volume fraction. Finally, correlations for the drag force, the Magnus lift force, and the torque in random arrays of rotating spheres at arbitrary solids volume fractions, rotational Reynolds numbers, and particle Reynolds numbers are formulated.