Characteristics of flow over a rotating sphere with smooth or grooved surface

Abstract

Flow over a rotating object such as a sphere or a cylinder is of a significant interest in many engineering areas including external ballistics, flying machines, ship stabilization and the saltation of particles. The very phenomenon that has attracted most attention in this flow is a lateral deflection of the object due to the lift force generated by the rotation. For a clockwise-rotating sphere moving from right to left, the pressure difference between the upper and lower sides exerts a upwards lift on the sphere, which is called the Magnus effect. However, at some specific Reynolds numbers and spin ratios (the ratio of the rotational velocity to the translational one), a flying rotating sphere deflects in a direction opposite to that predicted by the Magnus effect, which is known as the inverse Magnus effect. To elucidate when and why the inverse Magnus effect occurs, in Part I, the drag and lift forces on a rotating sphere with smooth surface and the corresponding flow field are measured by varying the spin ratio. This counterintuitive phenomenon occurs because the boundary-layer flow moving against the surface of a rotating sphere undergoes a transition to turbulence, whereas that moving with the rotating surface remains laminar. The turbulence energizes the flow and thus the main separation occurs farther downstream, inducing faster flow velocity there and generating negative lift force. Empirical formulae are derived to predict the location where the flow separates as a function of the Reynolds number and the spin ratio. Using the formulae derived, the condition for the onset of the inverse Magnus effect is suggested based on the negative lift generation mechanism. The characteristics of flow over a rotating sphere is also of great importance in many sports such as golf, baseball, soccer and volleyball since the drag and lift forces determine the trajectory of the ball. In particular, the dimples on a golf ball dramatically reduce the drag and thus the flight distance of the dimpled sphere increases by about two times, compared with that of a smooth sphere. Although they reduce the drag on the ball and increase the flight distance, dimples are known to deteriorate the putting accuracy owing to their edges. Therefore, it is very desirable to design a new golf ball that has both high putting accuracy and good aerodynamic performance. In Part II, I investigate the aerodynamics of a newly designed golf ball that does not have dimples but grooves on its surface. The smooth part of its surface is approximately 1.7 times that of a golf ball with dimples and thus accurate putts are naturally expected for the golf ball with grooves. The drag and lift forces on two versions of this golf ball are directly measured in the ranges of real golf-ball velocity and rotational speed, and are compared with those of smooth and dimpled balls. At zero spin, the drag coefficient of grooved balls shows a rapid fall-off at a Reynolds number similar to that of a dimpled ball and maintains nearly a constant value (lower by 50% than that of smooth ball) at higher Reynolds numbers. At non-zero spin, the lift-to-drag ratio of one version of grooved ball is higher by 5 – 20% in the supercritical Reynolds number regime than that of a dimpled ball, but it is lower otherwise. With the measured drag and lift forces, the trajectories of grooved balls are predicted and compared to those of smooth and dimpled balls for the same initial conditions. The flight distances of grooved balls are larger by 148 – 202% than that of smooth ball and shorter by 6 – 10% than that of a dimpled ball.