Diabolo is a popular game in which the object can be spun at up to speeds of 5000 rpm. This high spin velocity gives the diabolo the necessary angular momentum to remain stable. The shape of the diabolo generates an interesting air ow pattern. The viscous air applies a resistive torque on the fast spinning diabolo. Through computational uid dynamics (CFD) simulations it's shown that the resistive torque has an interesting dependence on the angular speed of the diabolo. Further, the geometric shape of the diabolo a ects the dependence of torque on angular speed. 1 1 Introduction The aerodynamics around spinning objects has long been studied using CFD and experimental techniques due to its complexity and a lack of an- alytical solution for the turbulent nature of ows involved. Objects of particular interest include cylinders [1] and spheres [2] [3] [4]. Improved understanding of the aerodynamics of spinning and moving spherical objects lends to better de- sign of sports objects such as cricket balls [5], golf balls [6], [7], [8], and soccer balls [9], [10], [11], and [12]. For example, it's long been known that the dimpled surface of a golf ball [7] [8] will disrupt laminar ow around the golf ball and cause the onset of turbulent ow to happen near the leading edge of the golf ball. Such early 1darrenjia@bernardsboe.com, Ridge High School, NJ USAonset of turbulent air ow signi cantly reduces the air resistance so the golf ball can y further. A traditional East Asian past time involves spinning a diabolo on a thin thread while the person performs many challenging movements. The aerodynamics aspect of a spinning diabolo has not been paid much tension in the past. In this work, we present a set of preliminary results concerning the aerodynamics of a spinning dia- bolo and the relationship between the spin angu- lar velocity of the diabolo and the torque applied on the diabolo due to air resistance. 2 Simulation Setup The aerodynamics of a spinning diabolo is simu- lated using the open source CFD software Open- FOAM [13] [14] [15]. We have previously imple- mented a solver [16] based on nite di erence methods which su ered from the problem that the grid resolution is insucient to resolve the boundary layer behavior near the solid walls of 2 2.2 Meshing 2 SIMULATION SETUP the diabolo. OpenFOAM is a popular open source CFD package that supports a nite volume [13] ap- proach to aerodynamics simulation. In addition, the nite volume of the mesh [17] is adaptive and can be re ned near the walls of the diabolo to capture the details of the ow eld near the wall, where large gradient of the ow eld vari- ables is present. To set up the simulation, we rst create a ge- ometry of the diabolo using blender (open source CAD software) [18]. Then a nite volume mesh with re nement zone near the diabolo is created to be used in the OpenFOAM solver. 2.1 Geometry of a Diabolo We used the measurements of a Mister Babache [19] diabolo along with a reference image of a Flight Lander diabolo to create a diabolo geom- etry in blender as shown in gure 1. Figure 1: Geometry of the diabolo used in simu- lation: length 14.07cm, inner diameter 10.58 cm, outer diameter: 11.53 cm, narrow section diam- eter: 0.95cm2.2 Meshing As shown in gure 2, a re ned boundary layer is formed around the wall of the diabolo. Then a re ned mesh in the shape of a cylinder surrounds the diabolo. The radius of this re ned mesh is about 10 cm. The resolution of the mesh beyond the cylindrical re nement zone is lower. On the sectional view of the mesh, the largest cells have a resolution of 6 :451031:85102 in meters. Each re nement level increases the resolution by a scale of 2. Near the diabolo, the resolution becomes 3 :231039:23103me- ter. Finally the boundary layer mesh has a high- est resolution of 1 :621034:61103meter. Overall the mesh used in the simulation for base diabolo geometry has 127313 cells. Figure 2: Meshing of the diabolo: the resolu- tion of the mesh increases near the wall of the diabolo, including the re ned boundary layer on the diabolo. 2.3 Solver OpenFOAM SIMPLE solver uses a nite volume [13] approach to solve the steady-state incom- pressible viscous uid ow [14] [15]. This is a suitable algorithm for the diabolo simulation as we are looking for a ow eld solution when 3 2.4 Flow pattern 2 SIMULATION SETUP the angular speed of the diabolo becomes rela- tively steady. The governing equations of the incompressible viscous uids are the typical in- compressible steady Navier-Stokes equations: r~ v= 0 (1) r(~ v ~ v) =rp+r(r~ v) (2) In the SIMPLE algorithm, the pressure- velocity coupled equation is solved by decou- pling the pressure and momentum elds through predictor-corrector steps. During the momen- tum predictor step, Hmatrix is solved from the momentum equation which has been matrixized on the nite volume mesh. r(~ v ~ v)r(r~ v) =rp (3) MV =rP (4) A=diag(M) (5) H=AVMV (6) AVH=rP (7) Now we can start the iterative process solving for pressure Pand velocity V. Start with the momentum equation, V=A1HA1rP (8) Substituting the Vequation into continuity equation leads to a Poisson equation of pressure that can be solved by under relaxation method. rV=r(A1HA1rP) = 0 (9) r(A1rP) =r(A1H) (10) In the predictor step, the momentum equation is solved using initial pressure Pand velocity boundary condition to nd velocity V MV =rP (11)The velocity eld Vis then used in the Poisson equation to solve pressure P r(A1rP) =r(A1H) (12) The pressure eld is then used in equation ( 8) to correctVat the boundary. This is the correc- tor step. This process is iterated until a solution ofVandPconverges for the computational do- main. In this particular simulation, the k[20] tur- bulence mode is enabled. The Reynolds number Re=vL for a spinning diabolo at hundreds of rad/s teogether with the geometry we used far exceeds the laminar regime of air ow. 2.4 Flow pattern The contour plots in gure 3 show pressure, speed, and vorticity when the diabolo spins at != 600 rad/s. We clearly observe the presence of vortex shedding with its signature wavy pat- tern. As the angular velocity increases, the mag- nitude of pressure change from reference pres- sure, air speed, vorticity and wall shear stress all increase. The high pressure regions at the top and bottom of the computational domain are artifacts of the simulation where the mesh cells between the rotating re nement mesh and outer boundary of the computational domain rotate relative to the re nement geometry. This arti- fact can be more readily seen by examining the z-component of velocity which are opposite in sign above and below the diabolo. Further, the distance between origin (center of the diabolo) and the inner boundary of the high speed region above or below is exactly the radius of the cylin- drical re nement mesh. The ow eld beyond the re nement mesh is not analyzed due to the arti cial nature of the result from OpenFOAM in this region. 4 2.4 Flow pattern 2 SIMULATION SETUP Figure 3: Air pressure (top left), speed (top right), close up speed near the wall of the diabolo (bottom left), and vorticity (bottom right) at != 600 rad/s. 5 3 EFFECT OF GEOMETRY ON AERODYNAMICS 2.5 Torque Figure 4: Torque on the base length diabolo model over varying angular velocities. As the angular velocity increases from 0 to 600 rad/s, torque increases from 0 to 0.007 Nm. Ln OmegaLn Torque -10-8-6-4-20 3.5 4.0 4.5 5.0 5.5 6.0Ln torque 1.6*x + -15.2 R² = 1Ln Torque vs. Ln Omega Figure 5: A log scale graph of the Torque vs !data is shown here. Using the relationship found by the trendline, an equation can be cre- ated relating Angular Velocity and Torque. = 2:5107!1:6. As the diabolo is spinning, air friction applies a resistive torque on the diabolo due to wall shear stress. As time passes, the torque exerted on the diabolo stabilizes and converges on a value fromthe elliptic solver in OpenFOAM, allowing us to record the torque for any given angular veloc- ity. We simulated the diabolo rotating along its lengthwise axis between 30 rad/s and 600 rad/s with an interval of 30 rad/s between data points. Figure 4 shows how the torque calculated along this axis varies as a function of angular veloc- ity!based on the measured values shown in table 1. The smooth pattern of the curve sug- gests a power law relationship. By applying log- arithmic functions of the torque and angular ve- locity values, we can produce the relationship ln( ) =bln(!) +a. This further leads to the following relationship = ! (13) As shown gure 6, this equation correctly models the torque found through OpenFOAM simulation with high correlation. For the base diabolo geometry, the two coecients through regression are = 2:5107and = 1:6 We rst use a base diablo geometry con guration to examine the torque-angular velocity relation- ship. In the next section, the geometry of the di- abolo is varied and the e ect of geometry on the torque-angular velocity is further investigated. 3 E ect of geometry on aerody- namics After establishing the power law relationship be- tween torque and angular velocity, we now mod- ify the base geometry of the diabolo by scal- ing the diabolo model. Diabolos with varying lengths are created by scaling the diabolo along its lengthwise axis, while keeping the radius of the diabolo the same. The length scales are shown in table 2. We can then use these ge- 6 3.1 E ect of length 3 EFFECT OF GEOMETRY ON AERODYNAMICS !(rad/s) (N m) 30 5.42E-05 60 1.66E-04 90 3.22E-04 120 5.17E-04 150 7.41E-04 180 9.89E-04 210 1.26E-03 240 1.55E-03 270 1.87E-03 300 2.20E-03 360 2.91E-03 390 3.29E-03 420 3.69E-03 450 4.12E-03 480 4.60E-03 510 5.10E-03 540 5.63E-03 570 6.20E-03 600 6.79E-03 Table 1: Torque calculated from wall shear stress in OpenFOAM as a function of the diabolo's spin angular velocity. ometries to examine the e ect of length on the torque-angular velocity relationship. Similarly, diabolos of varying radii are created by scaling the diabolo radially, while keeping the length of the diabolo constant. The outer radius, inner radius, and narrow section of the diabolo are updated. The radius scales are shown in ta- ble 3. These geometries are used to examine the e ect of radius on the torque-angular veloc- ity relationship. Angular VelocityTorque 0.0000.0020.0040.0060.008 100 200 300 400 500 600Torque Predicted TorqueTorque PredictionFigure 6: The simulated torque found in Open- FOAM and the predicted torque created by our equation match well. Angular VelocityTorque 0.00 E+002.50E-035.00E-037.50E-031.00E-021.25E-02 100 200 300 400 500 6000.25 Length 0.5 Length 0.75 Length 1 Length 1.25 Length 1.5 Length 1.75 Length 2 LengthTorque vs Angular Velocity Figure 7: Torque as a function of angular ve- locity for varying diabolo lengths. 3.1 E ect of length As shown in gure 7, increasing the length of the diabolo increases the torque exerted on it. The relation between torque and angular velocity re- mains the same ( = ! ) however the values of and vary with di erent diabolo lengths. The value of alpha increases linearly with dia- bolo length as shown in table 2, giving us the following relationship on 7 3.1 E ect of length 3 EFFECT OF GEOMETRY ON AERODYNAMICS Length Scale (dimensionless) (N m) (dimensionless) 0.25 1.38E-07 1.613 0.5 1.95E-07 1.580 0.75 2.11E-07 1.592 1 2.41E-07 1.600 1.25 2.47E-07 1.606 1.5 2.62E-07 1.623 1.75 2.80E-07 1.633 2 3.14E-07 1.628 Table 2: and as a function of varying length scale from the base diabolo geometry. increases due to increased contact surface area. stays around 1.6 for all lengths suggesting length has no e ect on . Length scale is the ratio between varied lengths against base diabolo length. Radius Scale (dimensionless) (N m) (dimensionless) 0.25 2.82E-09 1.529 0.5 2.17E-08 1.584 0.75 8.17E-08 1.596 1 2.41E-07 1.599 1.25 4.67E-07 1.618 1.5 8.04E-07 1.658 1.75 1.45E-06 1.665 2 2.21E-06 1.703 Table 3: and as a function varying radius from the base diabolo geometry. Both variables increase with radius scale. Radius scale is the ratio between varied radii against base diabolo radius. = 8:64108L+ 1:39107(14) The value of stays around 1.6 as shown in table 2. There is no strong correlation between and length which suggests that remains con- stant at about 1.6 for all lengths. Furthermore, a relationship between torque and length can be derived when !is held con- stant. The torque exerted on the diabolo in- creases linearly with diabolo length when an- gular velocity is 600 rad/s with relation =3:97103L+ 3103. A similar linear relationship for other angular velocities exists between length Land torque . The torque on the diabolo can be expressed more generally as =a(!)L+b(!) (15) wherea(!) andb(!) are regressed against ! to produce the following equations a= 9:21108!1:66(16) 8 4 CONCLUSION AND FUTURE WORK b= 1:37107!1:55(17) 3.2 E ect of radius Angular VelocityTorque 0.002.50E-025.00E-027.50E-021.00E-011.25E-01 100 200 300 400 500 6000.25 Radius 0.5 Radius 0.75 Radius 1 Radius 1.25 Radius 1.5 Radius 1.75 Radius 2 RadiusTorque vs Angular Velocity Figure 8: Torque as a function of angular veloc- ity for varying radii. As shown in gure 8, increasing the radius of the diabolo increases the torque exerted as ex- pected. The general relation between torque and angular velocity remains the same = ! . The value for increases as radius increases as shown in table 3 yielding the equation after regression = 2:27107r3:23(18) The value for increases linearly with radius as shown in table 3, giving = 0:0869r+ 1:52 (19) A similar relation can be derived between torque and radius when !is held constant. The torque exerted on the diabolo when the angu- lar velocity is held at 600 rad/s increases with radius, creating a smooth curve with relation = 7:60103r3:72We see this same smooth curve for every other angular velocity. The torque dependence on ra- diusrcan then be expressed more generally as =a(!)rb(!)(20) wherea(!) is regressed against !to produce a= 3:38107!1:56(21) b(!) increases from 3.46 to 3.72 as !increases from 30 rad/s to 600 rad/s with a correlation of 0.793 from linear regression. 4 Conclusion and future work By varying the angular speed of the base dia- bolo geometry, we have observed a power law re- lationship between torque and angular velocity = ! . By further examining the dependence of torque on the shape of the diabolo, we can conclude that both coecients are functions of the geometry of the diabolo. The diabolo used in our simulation has a smooth wall which could perform better if we used a dimpled surface to reduce air resistance. We are not aware of any diabolos made of a dim- pled surface. It'll be interesting to further inves- tigate the e ect of dimpled wall on the aerody- namics of diabolo. One critical element missing from this work is experimental validation of the dependence found from CFD simulation. We suggest setting up experiments to verify if the predicted torque- angular velocity relationship agrees with obser- vation. One approach is to measure the mechan- ical power required to sustain a constant angular speed described by equation 22. P= != ! != ! +1(22) 9