Numerical Investigation of the Magnus Effect on Dimpled Spheres
Total Page：16
File Type：pdf, Size：1020Kb
NUMERICAL INVESTIGATION OF THE MAGNUS EFFECT ON DIMPLED SPHERES Nikolaos Beratlis Elias Balaras Mechanical and Aerospace Engineering Department of Mechanical Arizona State University and Aerospace Enginering Tempe, AZ, USA George Washington University [email protected] Washington, DC, USA [email protected] Kyle Squires Mechanical and Aerospace Enginering Arizona State University Tempe, AZ, USA [email protected] ABSTRACT 0.5 An efficient finite-difference, immersed-boundary, k 0.4 3 k2 Navier-Stokes solver is used to carry out a series of sim- k1 ulations of spinning dimpled spheres at three distinct flow 0.3 D regimes: subcritical, critical and supercritical. Results exhibit C 0.2 all the qualitative flow features that are unique in each regime, golf ball smooth namely the drag crisis and the alternation of the Magnus 0.1 effect. 0 5 6 10 Re 10 Figure 1. Variation of CD vs Re for a smooth sphere Achen- INTRODUCTION bach (1972), –; spheres with sand-grained roughness Achen- Golf ball aerodynamics are of particular importance not −5 −5 bach (1972), −− (k1 = k=d = 1250×10 , k2 = 500×10 , only due to the quest for improved performance of golf balls − k = 150 × 10 5); golf ball Bearman & Harvey (1976), ·−. but also because of the fundamental phenomena associated 3 with the drag reduction due to dimples. Dimples are known to lower the critical Reynolds number at which a sudden drop in the drag is observed. Figure 1, for example, shows the tion caused by dimples. Using hot-wire anemometry they drag coefficient, CD, as a function of the Reynolds number, measured the streamwise velocity within individual dimples Re = UD=n, (where D is the diameter of the golf ball, U the and showed that the boundary layer separates locally within velocity and n the kinematic viscosity of the air) for a golf ball the dimples. They concluded that turbulence is generated by and for spheres with various degrees of roughness [Achen- a shear layer instability which causes momentum transfer of bach (1972); Bearman & Harvey (1976)]. It can be seen that high speed fluid towards the surface of the sphere. Further dimples are more effective in lowering the critical Reynolds support for this mechanism was provided by recent direct nu- number compared to sand-roughened spheres with grain size merical simulations (DNS) by Smith et al. (2010). In their k equal to the dimple depth d. In addition, dimples have the work, flow visualization identified the formation of thin shear benefit of not increasing drag considerably in the supercriti- layers at the leading edge of the dimples, which become un- cal region (the region after the minimum in CD is reached), stable and are effective in energizing the near-wall flow. although that minimum value of CD is not as low as that of a When a sphere or golf ball is spinning around an axis per- smooth sphere. pendicular to the flow, a significant lift force is also generated. It is generally believed that dimples cause the laminar Robins (1805) showed in the 18th century that the spherical boundary layers to transition to turbulence, energizing the shots fired from bent gun barrels experience a lift force. A near wall flow and adding momentum, which helps over- century later Magnus (1853) employed a spinning cylinder to come the adverse pressure gradient and delay separation. Re- demonstrate the existence of lift and attributed this phenom- cently Choi et al. (2006) carried out a series of experiments ena on the Bernoulli’s principle. When a cylinder spins it im- on dimpled spheres to clarify the mechanism of drag reduc- parts velocity on the side moving with the flow and removes 1 velocity from the side moving against the flow. The higher (c) velocity on one side means lower pressure which creates a lift force in the direction of the side spinning against to the side spinning with the flow. This phenomena is commonly termed the Magnus or Robbins effect. It wasn’t until the introduc- tion of boundary-layer theory by Prandlt in 1904 that a better explanation was given for the Magnus effect. It is now under- stood that lift is generated by the asymmetric separation of the boundary layer on opposite sides of a cylinder or a sphere. Davies (1949) carried out experiments in which spinning golf balls and smooth spheres were dropped inside a wind tun- nel. He estimated the lift on the balls by measuring the drift of the golf ball on the landing spot. The Reynolds number was 88 × 103 and the dimensionless spin rate a = wD=2U Y varied up to 0.55, where w is the rotational speed in rpm. The interesting point in his observations was the existence of a X negative Magnus effect for the case of a smooth sphere and a < 0:35, that is lift in the direction opposite to that predicted by the Magnus effect. He attributed the negative Magnus ef- Figure 2. Typical grid resolution in a dimple. fect on the possibility that the effective Reynolds number on each side of the sphere is different creating a distinct pressure distribution on each side that results in negative lift. time advancement. All terms are treated explicitly using the Bearman & Harvey (1976) used large scale models of Runge-Kutta 3rd order scheme, except for the viscous and golf balls with spherical and hexagonal dimples attached to convective terms in the azimuthal direction which are treated two thin wires. A motor was placed inside the hollow shells implicitly using the 2nd order Crank-Nicholson scheme [see of the ball to enable spinning at the desired rotational speeds. Aksevoll & Moin (1996)]. All spatial derivatives are dis- The Reynolds number varied from 3:8 × 104-2:4 × 105 and a cretized using second-order central-differences on a staggered from 0-2. They generally observed that as the spin rate in- grid. creases so do the lift and drag, although for the lowest Re and The code is parallelized using a classical domain decom- for spin rates less than a ∼ 0:23 negative lift was observed. position approach. The domain is evenly divided along the ax- They speculated that the generation of negative lift was due ial direction and communication between processors is done to the fact that the boundary layer on the side spinning with using MPI library calls. Most tasks, like the computation of the flow remained laminar while the one on the side spinning the right hand side and the advancement of the velocity in the against the flow was tripped into a turbulent state. However, predictor and corrector steps, can be performed independently no visual or quantitative information was provided to support on each processor. To facilitate that, each subdomain contains this conjecture. one layer of ghost cells to its left and right. The value of More recently several other experimental studies have any variable at the ghost cells is obtained by simple ’ejection’ been carried out on spinning golf balls in order to quantify from the neighboring subdomain, which is part of the cost of the effect of dimples and rotation on the drag and lift [i.e. communication between processors. Stanczak et al. (1998); Aoyama (1998)]. The main prob- We considered four different Reynolds numbers, Re = lem with the experiments, however, is the inherent limitation DU=n, and one dimensionless rotation rate, a. Their val- to measure quantities near and within the dimples when the ues were chosen in a way that all three distinct flow regimes, golf ball is spinning. Computational studies can bridge this namely the subcritical (Re = 1:7×104 and a = 0:12568), crit- gap and provide detailed spatial and temporal information for ical (Re = 4:5 × 104 and a = 0:12568 and Re = 6:5 × 104 spinning golf balls. The recent large-eddy simulations (LES) and a = 0:12568) and supercritical (Re = 1:7 × 105 and a = by Aoki et al. (2010) and DNS by Smith et al. (2010) are 0:12568) are covered (see Table 1 for a summary of the com- characteristic examples. The present study extends the work putations). The dimensional parameters for the supercritical by Smith et al. (2010) to rotating golf balls. We will present case correspond to the velocity and rotation rate of a golf ball results at four Reynolds numbers covering the subcritical, crit- in flight after a typical stroke. The dimple diameter is approx- ical and supercritical flow regimes at constant rotation rate, a. imately d ∼ D=10, and the dimple depth, h = D=160. The de- tails of the geometry can be found in Smith et al. (2010). The Eulerian grid is stretched in the axial and radial directions to METHODOLOGIES AND SETUP resolve the key flow features. Figure 2 shows an example of The flow around a spinning golf ball is governed by the computational grid near a dimple. the Navier-Stokes equations for viscous incompressible flow. For all cases the domain extends 10D upstream and 30D In the present formulation the governing equations are dis- downstream of the golf ball (the center of the golf ball is lo- cretized on a structured grid in cylindrical coordinates, and cated at r=D = 0, z=D = 0). The number of grid points used the boundary conditions on the surface of the golf ball, which in each case is shown in Table 1. As the Reynolds number in- is not aligned with the grid lines, are introduced using an creases the grid resolution around the golf ball and total grid immersed-boundary method [see Yang & Balaras (2006)]. size increases too. For the supercritical case the resolution in An exact, semi-implicit, projection method is used for the the radial and axial directions is the highest of all.