# Investigation of the Inverse Magnus Effect on a Rotating Sphere in Hypersonic Rarefied Flow
## 1 Introduction  
Hypersonic technology has been attracting wide attention for decades. As the scientific foundation of hypersonic flight, hypersonic aerodynamics has become a hotspot in the research on fluid mechanics and aerospace engineering. Many advances have been made to fully understand the various phenomena involved in hypersonic flows, such as thermochemical nonequilibrium processes under high-enthalpy conditions [1], hypersonic boundary-layer transition [2; 3; 4], and hypersonic propulsion [5]. Among these research areas, hypersonic rarefied flow [6; 7] is academically challenging and also significant for engineering applications. Hypersonic rarefied flow has features that are shared by all hypersonic flows, and additionally, it is characterized by gas rarefaction due to the high altitude of flight. In rarefied gas flows, the translational motion of gas molecules can be highly nonequilibrium, leading to the breakdown of macroscopic governing equations for continuum gas flows. Knowledge based on continuum gas dynamics can be problematic for rarefied flows; thus, some phenomena found in rarefied flows are thought to be counter-intuitive [8]. In order to predict the trajectories of asteroids [9] and space debris [7; 10], it is important to simulate hypersonic flow fields and compute the aerodynamic characteristics at high altitudes. For the highly rarefied flows over asteroids or space debris at the early stage of their atmospheric entries, gas-kinetic-based numerical methods [11] and experiments in low-density wind tunnels [10] are very useful research tools. In these investigations [7; 9; 10], the geometries of asteroids, space debris, or their fragments are usually treated as spheres. Along with their hypersonic velocities, these unmanned atmospheric-entry objects can also have remarkable spinning rates during their flights. In both continuum and rarefied flows, the rotation of a solid body can produce lift, thus changing its trajectory. It is worth noting that the aerodynamic laws governing rarefied flows over a rotating cylinder/sphereare qualitatively different from the Magnus effect [12; 13] observed in the continuum flow regime. This difference leads to theoretical and numerical investigations on the so called \"inverse Magnus effect\" [14; 15] within the context of hypersonic rarefied flows. Another motivation to study hypersonic rarefied flows and the inverse Magnus effect comes from the utilization of extremely low earth orbits and the upper atmosphere [16]. The altitude of the upper atmosphere ranges from 100 km to 200 km. Neither traditional orbiting spacecrafts nor emerging hypersonic vehicles can fulfil the requirements for long-endurance and maneuvering flight between 100 km and 200 km. To help fill this gap, it is crucial to further understand and make use of aerodynamic forces in the low-density atmosphere. Unfortunately, the lift-to-drag ratios (\(L/D\)) of known aerodynamic configurations (e.g., waveriders) cannot meet the requirements for achieving both long endurance and maneuverability within this altitude interval (100 km-200 km). [PERSON] et al. [16] pointed out that reducing the accommodation coefficient of the solid surface can elevate the \(L/D\) of the vehicle in the upper atmosphere. However, the accommodation coefficient needs to be as low as about 0.2 to achieve \(L/D>1\). Such a value of the accommodation coefficient is much lower than the existing data for surfaces made of common materials. In this paper, another potential approach to increase \(L/D\) in hypersonic rarefied flows will be explored, which relies on the inverse Magnus effect for rotating surfaces. In the context of continuum aerodynamics, the Magnus effect in the flow over a rotating cylinder/sphere has been well studied [17]. Generally speaking, the rotation of the solid surface drives the ambient fluid to circulate because of the no-slip condition. As a consequence, the velocity distribution, and thus the pressure distribution, around the body become asymmetric.