Theoretical and Experimental Investigation into the flight of an X-Zylo
TTheoretical and Experimental Investigation into the flight of an X-Zylo
Nils Wagner ∗ Abstract
Flying Gyroscopes are fascinating flight objects, which, due to gyroscopicstabilization, can achieve surprisingly long flight distances when thrown withrapid spin. The most common example hereby is a traditional Frisbee disc.This paper focuses on a similar object called X-Zylo, that shows a remarkablestraight flight despite its simple geometry.The main aim of the present study is to investigate the flight behavior of theX-Zylo and to build a reliable groundwork for further quantitative parameterstudies on ring wing configurations. To achieve this goal, a six degree offreedom model to predict the flight trajectory was developed. The trajectorycomputation uses interpolated high-fidelity CFD simulation data to calculatethe acting moments and forces on the object during flight. A launch contraptionwas built to be able to validate the theory systematically and reproducible inexperiments without human factors involved in the launch.Despite the complexity of the flight, the theoretical simulations match thereal world data qualitatively, however quantitative differences still prevail. Theinvestigation shows that the deviation between theory and experiment mostlystems from uncertainties in the CFD data as well as the optical recording of theexperimental data. Despite the methods outperforming those of prior studies,advancements still have to be made in those areas in order to obtain betterquantitative accordance between theory and experiment.
Keywords: trajectory simulation, CFD, ring wing, annular airfoil, toy aerodynamics ∗ This work begun as a project for the national science fair in Germany (Jugend forscht) and was later overhauledfor publication. Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] J a n ontents heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner Besides paper airplanes several flying toys have been developed, which, thrown the right way, cantravel long distances through the air in a controlled manner. The “X-zyLo™” (from now on simplycalled
X-Zylo ) is a lesser-known example of such toys. Thrown like a football spinning along itslong axis can result in flight distances up to
100 m and above. The distance record given by themanufacturer is
655 ft or . [1].Figure 1: Picture of the X-Zylo Relevant Data: mass: m = (22 . ± .
16) g length: l = 48 . − . ( l avg = 54 . )diameter: d outer = 97 . thickness: (leading edge), .
25 mm (trailing edge)center of mass: (1 . ± .
01) mm behind leading edgespecial feature: weighted front, sinusoidal trailing edgeReynolds number ∗ : Re ≈ . · The object flies in an almost straight line and seems to lose very little height while airborne(see picture 2). With the X-Zylo being simply a thin hollow cylinder as seen in figure 1, thisflight characteristic is quite impressive. To understand the flight behavior in great detail, a sixdegree of freedom model was developed to compute the trajectory of an X-Zylo. To be able toaccurately calculate the aerodynamic forces and moments acting on the object during flight, CFDsimulations were conducted on a computing cluster. In order to test the prediction systematically,a launch device with the ability to launch the X-Zylo in a controlled manner was developedand build. The flight of the object is tracked using several cameras to obtain detailed flightinformation; this data is then corrected for camera induced errors. In the end the theoreticalpredictions are extensively compared to the observed data.Many papers already simulated the flight of a spin-stabilized disc, widely known as Frisbee™,which shows a quite similar flight behavior [2, 3]. However, a less rigorous approach on theexperimental part mostly defies a good comparison between theoretical and experimental trajectory.This work aims to resolve this issue by using a dedicated launch mechanism for better control onthe initial flight parameters. The X-Zylo itself was also subject of former investigations [4, 5],but in less depth than in the present study. Future applications could involve the optimizationof such toys as well as potential insights in annular airfoil technology experimentally used forcoleopters in the past. Furthermore the understanding of the aerodynamics of such elementaryobjects could yield insight into the flow past more complex structures.In the following work those thin hollow cylinders investigated are referred to as throwing rings or simply rings . The “X-Zylo” is hereby only a particular, commercially available ring with thespecial features stated above, which was used for all experiments conducted. observed trajectory
Figure 2: Typical trajectory for an X-Zylo launched with a small launch angle. Even thoughthe ring flies more than
40 m , the flight path is extraordinary flat. ∗ reference values: v = 17 .
35 m / s , characteristic length is the ring’s chord length l avg Page 1/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner
Two technical terms have to be differentiated, as they are of great importance for the initialflight and can be misunderstood easily. At first there is a specific launch angle , which is theangle between the velocity vector at launch and its projection onto the xy -plane (note the usedcoordinate system in figure 5). It captures how steep the X-Zylo is thrown in respect to theground. The second important angle is the Angle of Attack (AoA). The AoA is the angle betweenthe ring’s symmetry axis and the velocity vector. It has to be emphasized that the initial AoAat launch and the launch angle are independent of each other; one can throw the ring very flat,however with its axis inclined to give it a high initial AoA.As stated before, the characteristic of the X-Zylo is the stable and straight flight. This holdsfor every initial launch angle. The ring typically is thrown without an initial AoA since theaxis of the ring matches the direction, in which the ring is launched at start. This is especiallytrue for the launch system built, see section 6.4. This begs the question why the ring does notfall to the ground as any other object would, considering that the ring is virtually rotationallysymmetrical, therefore generating no lifting force. The sinusoidal tracing edge of the X-Zylobreaks this symmetry, but even without this wavy edge the ring will fly nonetheless. For simplicityonly idealized hollow cylinders with a straight trailing edge will be discussed in the following.Also the air is seen as stationary, therefore the ideal scenario is windless.
At launch it holds true that the ring has an AoA of °, meaning the symmetry axis as well as theflight direction are parallel (see figure 3a). As the ring generates no lift force, it gets acceleratedtowards the ground due to gravity. This results in a direction change of the velocity vector, whilethe gyroscopic stabilization—due to the rapid spin imparted at launch—keeps the axis directionof the ring (nearly) constant. Therefore, even after a short amount of time an increasing AoAbetween the ring’s flight direction as well as ring’s axis will form shown in figure see figure 3b.This imparts a linearly increasing lift force until the AoA is great enough to support the weight ofthe ring. This initial flight phase is further denoted as the first drop . At a specific angle this liftforce compensates the gravitational pull and the ring encounters an equilibrium phase , where theAoA is stable. If the lift force exceeds the gravitational force, the ring gets accelerated upwardsand therefore the AoA decreases, decreasing lift; and vice versa. Hence the ring flies straight foran elongated time since this equilibrium state is maintained. Drag decreases the velocity of thering, and will slowly increase the equilibrium AoA since a higher AoA is needed to generate thelift force equivalent to gravity. Ever-increasing AoA yields flow separation later during flight sothat the ring plummets quickly. ω (cid:126)F grav (cid:126)F drag (cid:126)v (cid:107) (cid:126)L xz y (a) Initial launch condition (idealized, α launch = 0 °). ω(cid:126)F grav (cid:126)F drag (cid:126)F lift (cid:126)v(cid:126)L (cid:107) (cid:126)L α (b) Ring mid-flight (equilibrium). Figure 3: Simplistic model of the flight of a throwing ring without consideration of torques acting.While the idealized initial conditions show no angle between velocity vector and ring axis, duringflight an AoA α is formed due to the gravitational acceleration.Page 2/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils WagnerIt has to be noted that when thrown by a human, the initial condition with ° AoA is notperfectly met, additionally the ring wobbles a lot and stabilizes only after a short amount oftime [4]. This makes the flight hard to predict since the launch conditions are hard to measureaccurately. The influence of an initial AoA is further investigated in section 6.4.Even though the flight can be explained well using those descriptions, which were formerlyknown (see [6]), it can be observed that the ring drifts sideways later during flight. This effectwill be explained further in the following section since it was mentioned in former publications(e.g. Tarr [5]) but not analyzed in detail. Later during flight torques on the flying ring, that impact the direction in which the ring flies,become more important. There are several key components that decide how much the directionof the ring changes in the sideways y -direction:i) Position of the Center of Pressure: As Center of Mass (COM) and
Center of Pressure (COP) do not fall together in a single point, a torque is imparted which lets the ring precess.For traditional airfoils it is known that the COP moves upstream for higher AoA [7, pp.385+386], which complicates the torque calculation. The CFD results in section 3.3.5 alsoconfirm that the COP is not fixed for the X-Zylo during flight (see figure 10). As the COP isusually found to be at approximately a quarter of the chord length for a flat plate (quarterchord point), the X-Zylo is designed to counter this by having a weighted front using athin metal band. This shifts the COM towards the quarter chord point so that the actingtorques are of small magnitude.ii)
Angular Velocity:
During launch the ring spins rapidly, but friction decreases this spinmidst flight, so that the translatoric as well as the rotational speed decreases while airborne.This results in torques becoming more prominent later on since the angular momentum ofthe ring decreases over time.iii)
Aerodynamic Forces:
Since the only forces which produce a net torque on the rotatingcylinder are aerodynamic forces, the torque is directly proportional to the magnitude of liftand drag. As those forces are very small at launch due to the small AoA, the change inangular momentum is not visible. However, the AoA increases steadily over time magnifyingaerodynamic forces. This is coupled to the decreasing translatoric velocity of the ring whichin contrast decreases lift and drag.As the aerodynamic forces act in the xz -plane at launch, the torque will purely act in the y -direction at first (see figure 4a). This torque slowly lets the ring precess, turning the ringsideways. However the velocity vector remains unchanged at first, only the ring precesses (seefigure 4b). The tilt of the symmetry axis towards the flight direction and therefore towards theoncoming air generates a sideways lift force. Only then does the velocity vector follow the ringaxis vector, letting the ring drift sideways, which can be observed. The direction in which the ringdrifts is therefore dependent on the spin direction imparted at launch along with the location ofthe COP (behind or in front of the COM). It will become visible in section 6 that as the locationof the COP changes, also the drift direction changes midst flight.A force not accounted for in this approach is the Magnus force acting on the spinning cylinderwhen swerving sideways, effectively creating a sideways incident flow component. This componenthowever is negligibly small, only when dealing with stronger sideways winds, those forces have tobe considered. In the ideal windless case, this factor can therefore be neglected.Page 3/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner (cid:126)M aero (cid:126)F aero ω y xz COMCOP (cid:126)v, (cid:126)L ∈ (cid:8) (cid:126)a ∈ R ∧ a y = 0 (cid:9) (a) Initial cruising condition with torqe acting on theCOM due to aerodynamic forces. ω(cid:126)F drag (cid:126)F lift (cid:126)v(cid:126)Lβ v y = 0 L y (cid:54) = 0 (b) Ring condition after torque changesthe direction of angular momentum. Figure 4: Simplistic model of how torques act on the ring during flight shown in birds-eyeperspective. The sideways drift angle β is exaggerated for better visibility. All vectors depictedin the picture are projections on the xy -plane, the z -component is not shown. To predict the flight behavior a program was developed which approximates the trajectorynumerically. A forward Euler method is chosen in which the solution is propagated using smalldiscrete time steps; the implementation was done in MATLAB [8]. The Euler method showed tobe sufficient for this problem, as a test using Adams-Bashforth methods of the second and thirdorder showed insignificant discrepancy between the results.
Starting the calculation, some initial parameters have to be specified, for example the velocitymagnitude at launch v launch , the launch angle α launch , the launch height h ( t ) , and the angularvelocity ω ( t ) . Furthermore, the initial time is set to t = 0 and a discrete time step ∆ t chosen forthe iterative forward Euler method. In addition, some ring parameters have to be specified, e.g.mass m , inner radius of the hollow cylinder r i , outer radius r a , and length l . Assuming a uniformmass distribution, the inertia of the hollow cylinder is then calculated to be I = m (cid:0) r i + r a (cid:1) .The position of the tip of the ring (pointon its axis in the plane of the leading edge)is named P tip . All other used ring locationsare named using P with the point specified assubscript. The normalized ring axis directionvector is called (cid:126)R axis . For the launch conditionsone gets (cid:126)v ( t ) = v launch · cos( α launch )0 v launch · sin( α launch ) , (1) (cid:126)P tip ( t ) = h ( t ) , (2) (cid:126)R axis ( t ) = (cid:126)v ( t ) | (cid:126)v ( t ) | . (3) α launch h ( t ) z x − y (cid:126)v ( t ) Figure 5: Fixed coordinate system usedthroughout the work with a schematic tra-jectory.Page 4/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils WagnerIn equation (3) the simplification that at start (cid:126)R axis is parallel to (cid:126)v ( t ) is used. Note that thisis sufficiently true for the launch construction, in a human induced launch with its perturbationsthis will not hold true (see section 6.4). With the given distance between the COM and the tip,the position of the COM can be calculated by (cid:126)P COM ( t ) = (cid:126)P tip ( t ) − (cid:12)(cid:12)(cid:12) (cid:126)P COM ( t ) − (cid:126)P tip ( t ) (cid:12)(cid:12)(cid:12) · (cid:126)R axis ( t ) . (4)A loop variable n is introduced and set to 0 initially. As long as P tip, z ( t n ) > holds, the ring isconsidered airborne and equations (5) to (12) are repeatedly solved for the next iteration untilthis condition is not satisfied any more.All current forces acting on the ring are summed up and used to calculate the velocity andpositions for the next iteration. For the acceleration one gets (cid:126)a tot ( t n ) = (cid:126)F tot ( t n ) m = (cid:126)F lift ( t n ) + (cid:126)F drag ( t n ) + (cid:126)F grav ( t n ) m (5)with (cid:126)F grav ( t n ) = − mg · (cid:126)e z using g = (9 . ± .
02) m / s . The velocity and position of the COMfor the next time step are then given by (cid:126)v ( t n +1 ) = (cid:126)v ( t n ) + ∆ t · (cid:126)a tot ( t n ) , (6) (cid:126)P COM ( t n +1 ) = (cid:126)P COM ( t n ) + ∆ t · (cid:126)v ( t n ) . (7)Section 3.2 will focus on the magnitude of lift and drag as well as the COP; those quantities areinterpolated from the CFD results. The direction of the lift and drag vector can be calculatedusing the Gram-Schmidt process resulting in (cid:126)F lift ( t n ) = (cid:126)R axis ( t n ) − (cid:126)v ( t n ) | (cid:126)v ( t n ) | · (cid:68) (cid:126)R axis ( t n ) , (cid:126)v ( t n ) (cid:69)(cid:12)(cid:12)(cid:12) (cid:126)R axis ( t n ) − (cid:126)v ( t n ) | (cid:126)v ( t n ) | · (cid:68) (cid:126)R axis ( t n ) , (cid:126)v ( t n ) (cid:69)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) (cid:126)F lift (cid:0) α ( t n ) , | (cid:126)v ( t n ) | (cid:1)(cid:12)(cid:12)(cid:12) , (8a) (cid:126)F drag ( t n ) = − (cid:126)v ( t n ) | (cid:126)v ( t n ) | · (cid:12)(cid:12)(cid:12) (cid:126)F drag (cid:0) α ( t n ) , | (cid:126)v ( t ) | (cid:1)(cid:12)(cid:12)(cid:12) , (8b)where (cid:104) (cid:126)a,(cid:126)b (cid:105) denotes the standard scalar product in euclidean space of vectors (cid:126)a and (cid:126)b . The liftand drag forces are dependent on the angle α ( t n ) between the ring axis direction vector and thevelocity vector. One gets α ( t n ) = arccos (cid:68) (cid:126)R axis ( t n ) , (cid:126)v ( t n ) (cid:69)(cid:12)(cid:12)(cid:12) (cid:126)R axis ( t n ) (cid:12)(cid:12)(cid:12) · | (cid:126)v ( t n ) | . (9)Note that as stated in section 2.2, the lift force does not always act in the xz -plane. After thering tilts sideways this will be accounted for in the direction of the aerodynamic forces (8a+8b)as well as the AoA (9) used in the calculation of lift and drag.Due to torques acting on the ring the direction of the ring axis (cid:126)R axis changes over time. Withthe angular momentum (cid:126)L ( t n ) being parallel to the axis, one deduces (cid:126)L ( t n ) = (cid:12)(cid:12)(cid:12) (cid:126)L ( t n ) (cid:12)(cid:12)(cid:12) · (cid:126)R axis ( t n ) = I · ω ( t n ) · (cid:126)R axis ( t n ) . (10)Of significant relevance is the position of the COP, which is evaluated using the interpolatedCFD results. A stated before, COM and COP do not fall together, therefore creating a torque (cid:126)M ( t n ) acting on the ring: (cid:126)M ( t n ) = (cid:16) (cid:126)P COM ( t n ) − (cid:126)P COP ( t n ) (cid:17) × (cid:16) (cid:126)F lift ( t n ) + (cid:126)F drag ( t n ) (cid:17) . (11)Page 5/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils WagnerThis torque changes the angular momentum so that one gets (cid:126)L ( t n +1 ) = (cid:126)L ( t n ) + ∆ t · (cid:126)M ( t n ) . Thenew direction of the ring axis (cid:126)R axis ( t n +1 ) is then the normalized angular momentum vector.For the sake of simplicity the magnitude of the angular frequency is seen as decoupled fromthe complicated motion of the X-Zylo. Using Sliding Mesh simulations (see section 3.3.6), themean Wall Shear Stress τ w is calculated for different rotational frequencies and interpolated forthe trajectory simulation. Using only the angular velocity one can then calculate the additionaltorque acting on the ring, which in turn reduces its angular momentum using the simplifiedrelation (cid:12)(cid:12)(cid:12) (cid:126)L ( t n+1 ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:126)L ( t n ) (cid:12)(cid:12)(cid:12) − τ w (cid:0) ω ( t n ) (cid:1) · A ring · ω ( t n ) · r a (cid:113)(cid:0) ω ( t n ) · r a (cid:1) + | (cid:126)v ( t n ) | (12)with the rings surface area A ring and the ratio of the rotational velocity ω ( t n ) · r a to the totalvelocity on the rings surface. Mind that equation (12) only holds for small AoA as the ratio ofrotational velocity and total velocity only holds for zero AoA flight. Therefore it is only a roughapproximation for larger AoA scenarios later in flight.At last the loop variable n gets incremented by one, the time incremented by ∆ t , and afterthat all steps will repeat. As can be seen in the theoretical calculation in section 3.1, the magnitude of the lift and dragforce as well as the COP is needed for any arbitrary angle α ( t n ) and any flow velocity | (cid:126)v ( t n ) | .As all magnitudes change gradually, the approach will be to calculate lift and drag coefficientas well as the COP for discrete AoA using CFD ( C omputational F luid M echanics) and then tointerpolate the results. The change in Reynolds number and therefore the change in drag andlift coefficient for different flow velocities is neglected. As the velocity magnitude of the ring liesbetween / s and
18 m / s for the whole flight duration in most standard cases, this assumptionshould yield fairly accurate results. Nonetheless, this is a potential source of error one has tokeep in mind, see section 6.5. Several sources have derived analytical approximations for the lift and drag forces of hollowcylinder configurations which will be shortly mentioned and later compared to the CFD results insection 3.3.5. For clarity, the lift and drag coefficients ( C D and C L ) are written in square bracketsas only the force equations are shown.One of the first analytical descriptions was done by Ribner [9], who used concepts of Prandtllifting-line theory in order to derive the lift force for a thin ring airfoil of diameter d and (chord)length l . This resulted in the expression F lift = (cid:34) dπl dπl · π α (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) C L [9] · ρv · dl = (cid:34) π α πλ (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) C L [10] · ρv · dl , (13)where λ = l/d is the aspect ratio between chord length and diameter of the ring, S = dl thereference surface area for the coefficient calculation, v the velocity magnitude of the oncomingair, and ρ the air density. Mind differences in the definition of the reference wing surface area S in different publications, sometimes S (cid:48) = πdl as well as S (cid:48)(cid:48) = 2 dl are used. Moreover, in otherliterature the definition for the aspect ratio λ is sometimes defined as the reciprocal fraction.Formula (13) was also found by Pivko [10], who additionally calculated the induced drag force tobe F drag, induced = λ · (cid:32) π α πλ (cid:33) · ρv · dl (14)Page 6/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerusing the found correlation C D = λ/ · C L .Weissinger [11] developed a refined theory on general ring wing configurations and found theformulae (13) and (14) to be a special case for λ −→ . A more accurate approximation for the liftforce for small aspect ratios ( λ < ) was found to be F lift = (cid:34) π α πλ + λ · arctan(1 . λ ) (cid:35) · ρv · dl . (15)For rotationally symmetric rings the drag can be calculated analogical as in formula (14) using C D = λ/ · C L [11].Tarr derived another analytic expressions for lift and drag forces in his book “What MakesThe Amazing X-Zylo Fly” [5], specifically to examine the X-Zylo, yielding F lift = [4 πα ] · ρv · dl , (16a) F drag = 4 π · ρ · v · l · α (cid:124) (cid:123)(cid:122) (cid:125) induced drag + π · µ · v / · d · l / ν / · ( S upper + S lower ) (cid:124) (cid:123)(cid:122) (cid:125) viscous drag (16b)where µ is the dynamic viscosity and ν the kinematic viscosity of air, S upper and S lower are thevelocity gradients on the upper and lower surface. Those gradients were numerically computed byTarr for an AoA of . ° and a flow velocity of
15 m / s , yielding S upper = 0 . and S lower = 0 . .The same values were used for the comparison seen in figure 10 while neglecting changes in thevelocity gradients based on the angle of attack α and the flow velocity v .For later comparison with the CFD results, the coefficients were calculated for a temperatureof ° C , yielding ρ = 1 .
204 kg m − , µ = 1 . · − kg m − s − and ν = 1 . · − kg m − . To get reliable values for the aerodynamic forces as well as the COP, CFD simulations areused. It is undoubted that todays CFD solvers (when used correctly) can outperform even thebest analytical models due to complex turbulence modeling and consideration of viscous effects.Therefore even for higher AoA beyond flow separation, approximate results can be obtained.
The geometry of an X-Zylo, which is used for all simulations other than the validation cases, ispresented in detail in figure 6. All meshes are created in the same process. SALOME [12] is usedto create a structured quad surface mesh using quadrangle mapping, which can be seen in figure7a for the X-Zylo. From there ANSYS Fluent [13] is used to generate the volume mesh (see figure7c) as well as the CFD calculations itself. The boundary prism layer consists out of 20 layersusing a geometric layer height increase of 1.2. Hereby the initial height is set to satisfy y + ≈ . ,which for the X-Zylo results in an initial layer height of .
01 mm resulting in a total boundarylayer height of approximately .
87 mm (see figure 7b). A sphere of radius . is used as far-field;the volume mesh is an unstructured tetrahedral mesh. The simulations were conducted on theLinux-Cluster of the LRZ ( L eibnitz- R echen Z entrum Garching, DE), utilizing the HPC resourcesto cut computing times. Page 7/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner b) side viewa) front view d f r o n t ,i nn e r = . mm d r e a r ,i nn e r = . mm d o u t e r = . mm . − . . c) profile .
75 mm0 .
25 mm l max = 61 . l avg = 54 . l min = 48 . sinusoidaltrailing edgeleading edgeinner side faceouter side face inner side faceouter side face Figure 6: Details of the geometry of an X-Zylo. The thickness is scaled by a factor of 4 in thedrawings for better visibility. The sinusoidal trailing edge consists of 5 full sine waves withampitude
13 mm . The profile drawing shows the lower part of the cross-section of the ring. (a) Surface mesh (b) Boundary prism layers (c) Volume mesh
Figure 7: Mesh for the X-Zylo ( M2 ), which showed small error from finer meshes as well as ajustifiable computing time. The simulation settings were kept constant throughout the whole work, only specific setting thatchange from case to case (e.g. the validation cases) are mentioned separately in the respectiveparagraphs. For the simulation itself the transient pressure-based RANS ( R eynolds- A veraged N avier S tokes) solver is used. The Pressure-Velocity-Coupling was set to use the SIMPLE( S emi- I mplicit M ethod for P ressure- L inked E quations) solver, the relevant discretisation schemes(pressure, density, momentum and energy) were set to second order. The ring itself was modeledPage 8/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerto be a no slip wall while the sphere served as a pressure-far-field as inlet—therefore the air ismodeled as an ideal gas. To initialize the flow field, ANSYS Fluent’s standard initialization fromthe pressure-far-field was used. As all simulations were conducted transient, the time-step was setfor the maximum CFL (Courant) number to be between 1 and 10, depending on the convergenceof the problem. All obtained values are for a temperature of ° C . As turbulence modeling is akey component using the RANS solver, and many different models are implemented in ANSYSFluent, two validation cases were simulated using a variety of turbulence models. There exist several sources that experimentally evaluate different forms of annular airfoils. Earlyexperiments done by Fletcher [14] were performed using annular airfoils with Clark-Y cross-sectionand aspect ratios of 1/3, 2/3, 1, 3/2, and 3. Chord length and diameter varied from .
235 m to .
704 m , the flow velocity used was . / s . Therefore the dimensions of the used modelsas well as the flow velocity greatly exceed the operating conditions of an X-Zylo with Reynoldsnumbers 11 to 33 times greater than in the present case. A more recent paper by Traub [15]examines annular wings with an Eppler-68 section and aspect ratios 1/2 and 1. Those wings weremerged into a NACA 0012 cross-section on the vertical sides allowing the Eppler-68 profile to benormal at the bottom and the top section. Also a revolution of a NACA 0012 profile ( λ = 1 / )was tested by Traub [15, 16]. The chord length used for every model was . , therefore thedimensions resemble the studied case in this work better. The free-stream velocity used in thewind tunnel was
40 m / s , which is still well above the flow velocities for the X-Zylo, yielding aReynolds number still 4.2 times greater than needed. Another experimental investigation wasdone by Latoine [17] on small circular hollow cylinders with flat-plate cross-section. The largestcylinder had a diameter of in ( ≈ .
152 m) and an aspect ratio of / . Since the used flowvelocity was only . / s , the Reynolds number based on the chord length is in this case 4 timessmaller than that of an X-Zylo in flight. Additionally, in this work the aspect ratio of λ = 1 / isfar from the ratio of an X-Zylo with λ = 0 . .Due to the lack of experimental data in the exact Reynolds number regime of the X-Zylo,both data sets by Traub (closed NACA0012 revolution) and Latoine were chosen to test the setup.The Reynolds number of the studied case then lies between the numbers of both validation cases. . . . . . . . . angle of attack [ ◦ ] . . . . . li f t c o e ffi c i e n t C L Experiment [17]laminar (0 eq.)Spalart-Allmaras (1 eq.)k − ω − SST (2 eq.)Transition SST (4 eq.)RSM (7 eq.) (a) Comparison of the lift coefficient − . − . . . . . . . angle of attack [ ◦ ] . . . . . d r a g c o e ffi c i e n t C D Experiment [17]Experiment (adjusted)laminar (0 eq.)Spalart-Allmaras (1 eq.)k − ω − SST (2 eq.)Transition SST (4 eq.)RSM (7 eq.) (b) Comparison of the drag coefficient
Figure 8: Comparison of the CFD results to the experimental results obtained by Latoine [17].In figure 8 the CFD results compared to the experimental data on the small hollow cylinderby Latoine [17] are presented. The tested cylinder has a diameter of .
152 m and a thickness of .
51 mm ; the mesh consisted of 3.85 million cells. Wind tunnel experiments were performed with awind speed of . / s and a turbulence intensity of . ; those parameters were also set in thesimulation. From there 2000 time steps with a step size of · − s (CFL ≈ ) were calculated.Page 9/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils WagnerSeveral turbulence models, which have great influence on the CFD solution, are compared. Asthe Reynolds number is very low, also a laminar solution is computed which fits the experimentaldata well for very small AoA up to °. However, for larger AoA the solution diverges as smallflow separations form, especially when comparing the lift coefficient. All simulations involvingturbulence modeling can predict the large flow separation occurring at approximately ° onward,but overestimate the decrease in lift. The Transition SST ( S hear S tress T ransport) model herebyis the best model while the R eynolds- S tress- M odel (RSM) greatly underestimates the lift forceafter separation. For smaller AoA this trend is reversed, RSM and k- ω -SST can predict the liftslope accurately while Transition SST and S palart- A llmaras (SA) compute a lift slope far greaterthan captured in the experiment. In this low Reynolds regime the flow is not fully turbulent,which is an explanation for the deviation between the models. All models can predict the dragsufficiently well; except the RSM, which heavily over-predicts drag. The Transition SST model isespecially good at capturing the drag at small AoA. It has to be noted that the drag should befully axis-symmetric to an AoA of ° since the hollow cylinder investigated is symmetric, howeverthe measured drag coefficient in the experiment showed a shift of approximately . ° and is notsymmetric. When adjusting this shift, the drag values fit the simulation data almost perfectly(see figure 8b). From this validation case a laminar calculation as well as a simulation using theRSM turbulence model can be excluded. Therefore, the second validation case was only simulatedusing the remaining three turbulence models SA, k- ω -SST and Transition SST.In the second validation case a closed NACA0012 revolution ( λ = 1 / , chord length . )including its mount was simulated at wind tunnel conditions of
40 m / s wind speed and a turbulenceintensity of . [16]. 3000 time steps with a step size of · − s (CFL ≈ ) were calculated.A finer mesh consisting of 8.21 million cells was used due to a more sophisticated geometry. Theresults are displayed in figure 9. One can deduce that the setup works very well for low AoA wherelittle flow separation is present. The lift slope deviates from the experimentally measured oneby . (SA), . (k- ω -SST) and . (Transition SST). Calculating the average deviationof the drag coefficient up to an angle of ° one gets errors of . (SA), . (k- ω -SST) and . (Transition SST). For higher AoA all turbulence models struggle again due to large flowseparations. − angle of attack [ ◦ ] − . . . . . . li f t c o e ffi c i e n t C L Experiment [16]Spalart-Allmaras (1 eq.)k − ω − SST (2 eq.)Transition SST (4 eq.) (a) Comparison of the lift coefficient − angle of attack [ ◦ ] . . . . . . d r a g c o e ffi c i e n t C D Experiment [16]Spalart-Allmaras (1 eq.)k − ω − SST (2 eq.)Transition SST (4 eq.) (b) Comparison of the drag coefficient
Figure 9: Comparison of the CFD results to the experimental results obtained by Traub [16].Both validations show, that the simulation can qualitatively predict the flow even with largeflow separations occurring at higher AoA. From a quantitative standpoint the simulations are onlyvalid for small AoA which is the expected behavior. Nonetheless, both SA and the k − ω − SSTturbulence model produce good results in both validation cases. Transition SST showed goodperformance in specific situations (low AoA drag for low Reynolds number case), but was not asconsistent. As the lift in the first validation case was best approximated using the k- ω -SST model,it will be the choice for the simulations of the X-Zylo. It was later observed that for the casePage 10/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerof the X-Zylo at small AoA, the Transition SST model predicts a drag coefficient 25% smallerthan that calculated with the k- ω -SST model. As the drag coefficient in the small Reynoldsnumber case was better approximated with Transition SST and the difference is significant, thecalculations were repeated using the Transition SST model to compare both results with thefound experimental behavior (see section 6).As further validation a paper by Werle [18] summarized most of the available experimentaland CFD data to test the predictions by Weissinger [11]. It was shown that the lift slope wasalmost exactly predictable for all cases using Weissingers approximation (15). Therefore also thiscan be used as further validation for the obtained CFD results for the X-Zylo. A small mesh independence study for the two SST turbulence models was done on four mesheswith varying sizes, ranging from 2 to 17 million mesh cells, roughly doubling every step. The flowvelocity in all further computations involving the X-Zylo was set to .
35 m / s (Mach 0.05 at thepressure far-field) and a turbulence level of was used. Those values were estimated as thestandard environment of an X-Zylo flying in a sports hall. For every mesh a CFL number of 1was used in the calculation and . with an AoA of ° were simulated. The results obtainedusing the k- ω -SST model are visible in table 1, the results for the Transition SST calculations areonly summarized. It can be observed that especially drag and lift force can be well approximatedusing even the coarsest mesh. In contrast the error for the COP— calculated from the finestmesh ( M4 )—is still high using meshes M1 or M2 . Since the COP was seen less important thanthe aerodynamic forces, the coarse mesh ( M2 ) was used in all further simulations as it was agood balance between simulation time and the mesh induced error. Later it was discovered thatthe COP is an extraordinary sensitive parameter and one should have opted for the fine mesh( M3 ), see section 6.3. The error for forces calculated using the medium mesh ( M2 ) lie well below with both turbulence models ( . for k- ω -SST, . for Transition SST), therefore theinfluence of the turbulence model exaggerates the mesh induced error.Mesh refinement number of cells C D C L COP location % error(% error) (% error) (% error)coarse ( M1 ) 2,268,018 0.08742 0.3714 22.99 % 3.69 %(3.51 %) (0.57 %) (6.98 %)medium ( M2 ) 4,237,776 0.08474 0.3702 22.24 % 1.35 %(0.33 %) (0.24 %) (3.49 %)fine ( M3 ) 8,389,523 0.08404 0.3694 21.72 % 0.53 %(0.50 %) (0.03 %) (1.07 %)very fine ( M4 ) 16,960,108 0.08446 0.3693 21.49 % − Table 1: Results of the mesh independence study for the k- ω -SST turbulence model. The erroris calculated from the finest mesh ( M4 ), the total error is the arithmetic average of the singledeviations. The COP is given by the percentual location on chord. The same calculations wereperformed using the Transition SST Turbulence Model, obtaining slightly higher total errormargins of .
90 % ( M1 ), .
23 % ( M2 ), and .
16 % ( M3 ). However, independent of the turbulencemodel, the largest error is always seen for the calculation of the COP. The CFD simulations for the X-Zylo were carried out using 5500 time steps (each . · − s ,CFL 1.5) totaling a simulated flight time of about .
24 s . The mean over the last 500 time stepswas calculated to average small numerical fluctuations as well as periodic effects. As noted beforein section 3.3.3, both SST models show a significant discrepancy in their drag behavior for smallPage 11/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils WagnerAoA. Since neither solution can be disregarded using the validation cases, all computations weremade for both turbulence models. While in both validation cases the drag estimates for small AoAusing the Transition SST model were conceivably smaller than k- ω -SST, in the small Reynoldsnumber regime of the case by Latoine, this estimate fits the data better. As the Transition SSTmodel is designed to operate well in the turbulence transition range where the X-Zylo mostlyoperates, it is not unlikely that it performs well in this case. The final conclusion can be drawnwhen the calculated trajectories using the CFD data are compared to the experimental results insection 6. angle of attack α [ ◦ ] . . . . . . . li f t c o e ffi c i e n t C L angle of attack α [ ◦ ] . . . . . . . d r a g c o e ffi c i e n t C D angle of attack α [ ◦ ] C O P l o c a t i o n o n c h o r d [ % ] angle of attack α [ ◦ ] . . . . . . . . , d r a g c o e ff . C D li f t c o e ff . C L . . . . . . . d i ff e r e n c e C F D s o l u t i o n s [ % ] d i ff e r e n c e C F D s o l u t i o n s [ % ] . . . . . d i ff e r e n c e C F D s o l u t i o n s [ % ] C O P l o c a t i o n o n c h o r d [ % ] C D (CFD, k − ω − SST) C L (CFD, k − ω − SST)COP (CFD, k − ω − SST) C D (CFD, Trans − SST) C L (CFD, Trans − SST)COP (CFD, Trans − SST) C D Weissinger [11] C L Weissinger [11] C L Ribner [9] C D Tarr [5] C L Tarr [5]∆ CFD
Figure 10: Obtained CFD results using ANSYS Fluent [13] in comparison to several analyticapproximations. The dotted solid line graphs show the deviation of the Transition SST model tothe k- ω -SST model. All CFD data is interpolated using cubic splines to be used in the trajectorycalculation in section 6.The results of the simulations as well as the analytic approximations from several sources canbe seen in figure 10. While the CFD results use a non-rotating X-Zylo with wavy trailing edge,the other sources are calculated for a thin ring wing with length . and diameter
97 mm .As Hirata, et al. [4] calculated the drag and lift coefficient of a simplified model with a length of
60 mm and a diameter of
100 mm using motion analysis, the values are scaled accordingly to fitthe real dimensions of an X-Zylo. It also has to be mentioned that source [4] only lists a singlevalue for C D and C L at an AoA of °. Therefore this data is not listed in figure 10.At first, it can be seen that while both SST turbulence models have a different drag behaviorat small AoA, they coincide well for higher AoA. The error for the lift force only exceeds 3% forAoA smaller than °; the error for drag also plummets to under 10% for AoA greater than °.Page 12/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils WagnerFor the COP location also a small error of under 4% for AoA greater than ° is seen. Altogether,the error gets smaller the higher the AoA gets with the exception being the lift coefficient. Hereboth turbulence models differ in the prediction of the flow separation, however not significantly.It can also be noted that the X-Zylo seems to behave like a traditional biplane wing with its liftcurve showing two different slopes before and after large flow separation occurs.As can be seen the approach by Tarr [5]—equations (16a) and (16b)—greatly overestimatesthe lift and induced drag force generated by the X-Zylo, while underestimating viscous drag. Forthe lift slope the deviation to the CFD results is more than , also the viscous drag is onlyhalf of what was computed (42% for k- ω -SST, 56% for Transition SST). The approach by Hirata,et al. [4] also underestimates drag and lift forces. When comparing the sole data point given tothe Fluent simulation a difference of for the lift force (both models) and (k- ω -SST) or (Transition SST) difference for the drag force is calculated. The analytical approximations ofthe lift force found by Ribner [9], Pivko [10] and especially Weissinger [11] coincide well with thesimulation data obtained. The difference in lift slope of the refined formula (15) by Weissingerand the CFD simulation is less than 1% for both SST models which is remarkable. This alsoserves as additional validation since formula (15) was found to be very precise before by Werle [18].Nonetheless, only Tarr approximates viscous drag of the X-Zylo, therefore the drag curve for theWeissinger approximation starts in the origin. The induced drag force calculated using formula(14) with the correction term for C L by Weissinger also matches the induced drag calculatedusing CFD for an AoA smaller than °, but diverges from the obtained simulation results forangles greater than °.Figure 11: Pathlines around a non-rotating X-Zylo showing the turbulent intensity along theirpath for different AoA. The re-circulation zones are clearly visible with their size increasing forrising AoA.Figure 11 shows pathlines colored by turbulence intensity around the X-Zylo for high AoA toqualitatively capture the flow separation. It can be seen that as the flow slowly separates in thecenter of the upper and lower wing segment, the side flow swirls into this area due to the lowerpressure region created. For AoA smaller than ° this effect is small and the separation bubbleis only concentrated to the center portion of the wing while growing outwards for higher AoA. Asthe study of the flow separation is not a focus in this paper, this will not be investigated further. To account for the rotation of the X-Zylo during flight, a Sliding Mesh approach was used to simu-late the ring with different rotational frequencies. The simulations were conducted for an AoA of °using the k- ω -SST model and a mesh containing 5.6 million cells, the results can be seen in figure 12.The percentual difference between the non-rotating solution and the simulation with non-vanishingPage 13/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner rotational frequency [ rads ] d e v i a t i o n f r o m r a d s s o l u t i o n [ % ] revolutions per second [Hz] C D deviationC L deviationCOP deviationC D interpolationC L interpolationCOP interpolation Figure 12: Results for a rotating X-Zylo at anAoA of ° using a Sliding Mesh approach.rotational frequency is shown. Especiallythe influence of the rotation towards thedrag is not negligible as the rotational fre-quency of an X-Zylo shot by the launchmechanism can exceed 50 revolutions persecond (see table 2). It is seen that theincrease in drag stems from a longer dis-tance traveled by the air over the chord asthe air is deflected near the surface by therotating ring. This also increases lift andshifts the COP, however this influence isrelatively small. The interpolation for C D and C L was done using a cubic polynomial,the COP fitting only uses a quadratic poly-nomial. Those interpolations are then usedin the program to simulate the trajectory.Moreover, the wall shear stress was captured for the rotating X-Zylo to calculate the decrease inangular velocity over time for different rotational frequencies (see equation (12)). launch slide X-Zylo mount + slide rubber bands stopperpulleyspringV-beltrubber rollerdrilling machinetriggerpivotablecounterweightcentralguiding rod base plate side arm
Figure 13: Schematic of the launch construction. The different parts are labeled as well as theactual object for launch marked in red. The functionality of the parts is covered in the text.To be able to throw the ring in a reproducible and controlled fashion, a launch device wasconstructed, which can be seen in picture 14. Especially the swerving motion of the ring at launchwhen thrown by a human as well as a non-vanishing initial AoA (see section 6.4) complicate theearly flight behavior. Page 14/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner
Figure 14: Picture of the cocked launch device which was developed and constructed for thisproject. The X-Zylo colored in red sits on a mount which is then accelerated forward by rub-ber bands. The rotation is achieved using a drill and a V-belt.To explain the mechanism in a nutshell, schematic 13 shows the most important parts. The ringsits loosely on a mount which can be altered for different ring geometries. This mount can slideon a central guiding rod which is oriented in the direction the ring should be launched. Theinitial launch angle can be chosen between ° and °. The whole mount will be spun by a V-beltconnected to a drilling machine, a counterweight maintains tension in the belt. When the mountspins at the desired speed, the weight is lifted, releasing the tension in the belt. The trigger canbe pulled to release the cocked launch slide, catapulting the launch slide as well as the mountforwards using rubber bands (Thera-Band Gold). A stopper at the end of the guiding rod stopsthe mount spontaneously and releases the X-Zylo; a spring dampens the hit on the stopper. Toevaluate the velocity and angular frequency of the ring at launch, a slow motion video of the ringis captured (see section 5.2). A scale in the xz -plane is set up to calculate the initial velocity fromthe video footage. When fully cocked, the launch mechanism has a draw force of approximately − , depending on the number of rubber bands used on each side arm. The reproducibility of the launch construction was tested by launching the X-Zylo five timesusing the same settings and comparing the initial launch values (table 2) as well as the observedtrajectories (figure 15). Full reproducibility would be achieved if the standard deviation of thelaunch parameters would be smaller than the estimated uncertainty on each individual parameterPage 15/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner − − − − − − − − − − − − − − − − launch 1launch 2launch 3launch 4launch 5 z c oo r d i n a t e [ m ] y c oo r d i n a t e [ m ] x c o o r d i n a t e [ m ] x coordinate [m] x coordinate [m] z c oo r d i n a t e [ m ] y c oo r d i n a t e [ m ] Figure 15: Comparison of the trajectory of five launches with the launch contraption. To testthe reproducibility of the device all launches were done using the same settings. The positionsshown are the COM locations.evaluated using the camera setup (see section 5.2). This would then cut the uncertainty on initialparameters due to the reliability of the construction.From the trajectories it is visible that the goal of achieving reproducibility with the launchmechanism was not met. While the flight distance varies only slightly, especially the sidewaysdrift is severely different for the launches. When comparing the data shown in table 2, it becomesobvious that the disagreement in sideways drift stems from the huge difference in the initialrotational frequency. This problem can be traced back to the belted motor drive mechanism. Thecounterweight has to be lifted manually in order to lower the drill, releasing the tension of theV-belt. In the same instance the trigger has to be pulled. However, this is hardly possible asthe trigger mechanism is poorly operable. Therefore, the tension in the V-belt is often releasedbefore the trigger can be pulled, decreasing the rotation of the mount rapidly. Table 2 shows thatgood reproducibility is almost achieved for the launch velocity, the launch angle still shows a highstandard deviation due to the outlier launch 3. However, the difference in the launch angle forinitial velocity launch angle init. rot. frequency launch time v launch [m/s] α launch [ ◦ ] ω ( t ) / (2 π ) [Hz] distance [m] aloft [s]launch 1 . ± . . ± . . ± . . ± . . ± . launch 2 . ± . . ± . ± . ± . . ± . launch 3 . ± . . ± . ± . ± . . ± . launch 4 . ± . . ± . ± . ± . . ± . launch 5 . ± . . ± . ± . ± . . ± . median . ± . . ± . ±
22 38 . ± . . ± . Table 2: Comparison of five launches using the launch contraption with the same settings. Theinitial values for the velocity, the launch angle and the rotational frequency were determinedwith uncertainty. The median of all five launches as well as the standard deviation is given forcomparison with the uncertainty estimated for each observation. Note that as launch 1 wasstudied in detail (see section 6.1), the initial parameters were determined with greater precision.the other four launches is small enough to be satisfactory. All in all the reproducibility fails dueto the rotational velocity (standard deviation more than ten times the individual uncertainty),where the V-belt mechanism has to be improved. Nevertheless, the contraption is still morereliable compared to a human induced launch and moreover satisfies the constraint of an initialAoA close to °. This will prove necessary later, see section 6.4.Page 16/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner Before conducting the experiments, the mass of the X-Zylo was measured using a high-resolutionbalance. Measuring two traditional X-Zylo’s, an average mass of (22 . ± .
16) g was found.Another X-Zylo was carefully disassembled and the plastic and metal part measured independently.With the mass of the metal ring being (16 . ± .
01) g and the plastic weighing (6 . ± .
01) g one can deduce the COM being (12 . ± .
1) mm behind the leading edge. The glue mass betweenthe metal ring and the plastic hull was negligible. Comparing the results to the measurementsdone by Tarr [5] shows good agreement in the X-Zylo’s mass. A larger discrepancy for the COMlocation is seen, which was found to be “at an axial distance of . from the leading edge” [5].For the experiments the X-Zylo was colored red for better visibility and a black line was addedto calculate the rotational frequency from the slow motion footage. It was however found inearlier tests that the additional mass from the red paint was not negligible and shifted the COM.By weighing the painted X-Zylo it was found that the mass increased by .
24 g ( . ) and theCOM shifted from . to . behind the leading edge which significantly changed theobserved drifting behavior (see section 6.3). The trajectory of the ring in the xz -plane was captured using a GoPro HERO 8 Black in linearmode (4k, 60fps). Linear mode uses the dedicated GoPro-intern software to eliminate the barreldistortion typically encountered with their cameras. Therefore only insignificant warp effectsare still visible which interfere with a qualitative analysis of the flight. Additionally, a SamsungGalaxy S10 and S9 are both used as close-up high-speed cameras capturing 960fps at a resolutionof 720p. Another Samsung Galaxy S7 (4k, 30fps) is used to capture the sideways drift of theX-Zylo during flight in the yz -plane. Finally, a GoPro HERO 4 Silver (linear mode, 1080p, 30fps)was used as a backup camera to probe different locations during flight. The camera setup withtheir positions can be found in figure 16. From now on almost exclusively the abbreviations S10,S9, S7, GoPro8 and GoPro4 are used instead of the full camera names mentioned above. .
96 0 .
98 1 .
00 1 .
02 1 . . . . . . Positions in the gym calibration pointslaunch constructionSamsung Galaxy S9 (SloMo)Samsung Galaxy S10 (SloMo)GoPro Hero 8 BlackGoPro Hero 4 Silver (Day 1)GoPro Hero 4 Silver (Day 2)Samsung Galaxy S7 . . . . . . . . . . . . . . Figure 16: Sketch (to scale) of the camera and calibration setup in the school gym for the finalexperiments. All camera positions shown were fixed, only the Samsung Galaxy S9 was movedthroughout the experiments.Using the open-source software Tracker [19], the trajectories could be manually evaluatedframe by frame. The experiments were conducted in a school gym to reduce external factors,Page 17/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerformer outside tests revealed bad flight behavior due to even slightly windy conditions. Toquantitatively capture the trajectory, both the xz - and yz -plane were equipped with calibrationpoints marking different distances. Additionally, a two-meter-long colored calibration bar wasused for the slow motion footage to accurately compute the launch velocity (see figure 19).However, just observing the trajectory is not enough. Several effects have to be accounted forto correct different imaging errors. As camera distortions are hard to deal with and are thoughtto have negligible impact since they are already internally corrected for the GoPro footage, theyare ignored. Only purely geometrical corrections independent of the camera are discussed in thefollowing. Ring outside the Plane of Measurement
The plane of measurement is here defined to be the plane perpendicular to the camera’s view,in which the calibration points are located. Therefore the camera only captures the projectionof the ring onto this plane of measurement. If the ring is not located in this plane, one has toaccount for this via the intercept theorem, which is demonstrated in figure 17. .
96 0 .
98 1 .
00 1 .
02 1 . . . . . . real X-Zylo positonsobserved x positionscorrect x positions real trajectorymeasurement planecamera d camera ∆ dx camera ∆ xx correct x observed launch origin xy Figure 17: Using the intercept theorem and equation (17) one can correct the observed X-Zylopositions. This is necessary as the X-Zylo leaves the measurement plane and therefore only aprojection is measured, which is illustrated in this figure.As the ring swerves during flight, it leaves the xz -plane captured by the GoPro8. This resultsin a correction term (see equation (17)) which has to be applied to the data. Mind that both ∆ x and ∆ d are signed quantities. In figure 17 ∆ d is negative and ∆ x positive. x correct = x observed − ∆ x = x observed + ( x observed − x camera ) · ∆ dd camera . (17)As the S7 films the sideways drift while the GoPro8 is capturing the xz -projection, both camerastogether can recreate the trajectory corrected for the swerving motion. The S7 however is alsosubject to the intercept theorem since the calibration points are set up at the opposite end of thegym. Therefore the image corrections of both cameras are coupled. The rectifications are appliedsequentially; first the S7 footage is adjusted with the x -coordinate of the GoPro8 video, then theGoPro’s footage is corrected with the swerving motion of the S7. One could apply the correctionsagain, however those higher order terms are of negligible magnitude.Page 18/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner .
96 0 .
98 1 .
00 1 .
02 1 . . . . . . real X-Zylo positonsobserved x positionscorrect x positions camera flight pathmeasurement plane d camera ∆ dr a (cid:126)v correct (cid:126)v correct = (cid:126)v observed · d camera − ∆ d − r a d camera (cid:126)v correct = (cid:126)v observed · d camera − ∆ d + r a d camera xy Figure 18: The intercept theorem also applies if the plane of measurement is not exactly located inthe flight path of the X-Zylo. If the camera is very close to the flight path, one has an additionaleffect of observing different parts of the ring during flight.
Ring in close Proximity to the Camera
From the S7 footage it was seen that the X-Zylo trajectory was offset from the measurementstick set up (see figure 20). This results again in an application of the intercept theorem for theclose-up cameras, illustrated in figure 18. However, this was not the only effect observed. Whenlooking closely at picture 19, which shows a time series of the tracked X-Zylo seen from the S10’sslow motion footage, one spots that the X-Zylo is tracked at different locations throughout theflight. As the object is probed almost precisely at its foremost point every frame, the spot trackedwith this method changes. This is also illustrated in figure 18. Before passing the camera, theside facing the measurement plane is tracked while after that the opposite side facing the camerais tracked. As only the velocities in the slow motion footage were important, a simple applicationof the intercept theorem results in the corrected velocities. This correction is important even ifthe difference between the tracked points is only the diameter of the ring, therefore circa .This comes due to the close proximity of the camera to the flight path. As for example the S9 islocated only about . from the measurement plane, one gets an offset of approximately . which—for standard launch conditions with a velocity magnitude of
16 m / s —results in an errorof .
24 m / s , that can not be neglected. To compare the experimental results to the predictions made by the theoretical model, one launchis studied in great detail. Furthermore, the influence of the launch angle is investigated andcompared to the model. At last, the sideways drifting behavior as well as the impact a humaninduced launch causes are studied.
Launch 1 seen in section 4.2 is used for the detailed study of a single launch. This is useful asthe initial rotational frequency is sufficiently low to observe the drifting behavior in detail. Inaddition, the launch angle is optimal to achieve a long distance shot while also gaining enoughheight for greater insight into the vertical components. Figure 19 shows a time series of thislaunch observed by the S10 camera, while figures 20 and 21 present the trajectory of the X-Zylo.Page 19/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner tip position every 10 th frametip position every frame (5 . ± .
10) m(2 . ± .
02) m α launch = 11 . ◦ ± . ◦ Figure 19: Time Series of an X-Zylo immediately after launch. Every frame the position ismanually evaluated using Tracker [19], which are the small black points seen in the figure. Thelarger red points show the tracked X-Zylo position every th frame. The uncertainty on thecalibration scales stems from being in slightly different measurement planes (see figure 20) as wellas an estimated scaling error due to camera distortions (see section 9.2).The calibration points—as already seen in figure 16—are also marked in the pictures. Usingthe scale uncertainty due to camera distortions, the lengths are afflicted with an error using thevalues from table 4. Mind that the trajectories seen show only the raw footage and are thereforenot adjusted to the errors mentioned in section 5.2.To get the initial launch parameters one can view the footage of the different cameras. As theparameters change rapidly after launch, only the first five frames of each video are used. From thoseframes the median as well as the standard deviation for the different values are computed using the (5 . ± .
08) m(3 . ± .
05) m observed trajectorycalibration points yz Figure 20: Examined launch inthe xy -plane (raw footage of S7). Student’s t-distribution assuming statistical scatteringof the values. Additionally, a systematic uncertainty isestimated for each camera. This yields v launch = (16 . ± . ± . m/s (S9) , = (15 . ± . ± . m/s (S10) , = (15 . ± . (cid:124) (cid:123)(cid:122) (cid:125) stat ± . (cid:124) (cid:123)(cid:122) (cid:125) sys ) m/s (GoPro8) . Using those values the weighted mean can be calculated,resulting in a launch velocity of (15 . ± .
21) m / s . Nodifference between systematic and statistical error ismade after applying the weighted mean. Repeating theprocedure for the launch angle one gets α launch = (11 . ± . ± . ◦ (S9) , = (11 . ± . ± . ◦ (S10) , = (11 . ± . ± . ◦ (GoPro8) , which then results in α launch = (11 . ± . ◦ . Theinitial rotational frequency after applying the weightedaverage is ω ( t ) / (2 π ) = (18 . ± .
4) Hz . Additionally, more insignificant uncertainties are setPage 20/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner (40 . ± .
4) m(5 . ± .
05) m (3 . ± .
03) m
S9S10 S7 observed trajectorycalibration points xz Figure 21: Examined launch in the xz -plane (raw footage of GoPro8). The close up cameras aremarked as well as the calibration distances shown with uncertainty.for the ring mass, the location of the COM, the launch height, the gravitational constant andthe ring thickness. Using those uncertainties one can calculate the theoretical trajectory withuncertainty as described in section 9.1. However, the error margins for the CFD simulations alsohave to be specified, e.g. using the mesh study (see table 1). The uncertainties for the TransitionSST model were set to be ± for C D and C L , and ± for the location of the COP; the valuesfor k- ω -SST are set to ± and ± respectively. For both turbulence models the uncertaintyfor the mean wall shear stress τ w was set to ± as equation (12) is only approximately correct. experimentally measured values velocity: ---angle: --- frequency: ---X-Zylo more settingsinitial values launch velocity [m/s] 15.94 +/- 0.21launch angle [°] 11.77 +/- 0.34ring mass [g] 23.97 +/- 0.01center of mass [cm] 1.30 +/- 0.01spin frequency [Hz] 18.5 +/- 0.4ring length [cm] 5.45 +/- -ring inner radius [cm] 4.75 +/- -ring thickness [mm] 1.00 +/- 0.05launch heigth [m] 1.40 +/- 0.01 aerodynamic forces torque calculation compare parabola uncertainty calculation equal axis show legend X-Zylo Trans-SST center of mass (x-coordinate)x axis : center of mass (y-coordinate)y axis : center of mass (z-coordinate)z axis :time interval 0.001[s] calculate experimental data: 03.09.2020 launch 1 show exp. data velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylo experimentally measured values velocity: ---angle: --- frequency: ---X-Zylo more settingsinitial values launch velocity [m/s] 15.94 +/- 0.21launch angle [°] 11.77 +/- 0.34ring mass [g] 23.97 +/- 0.01center of mass [cm] 1.30 +/- 0.01spin frequency [Hz] 18.5 +/- 0.4ring length [cm] 5.45 +/- -ring inner radius [cm] 4.75 +/- -ring thickness [mm] 1.00 +/- 0.05launch heigth [m] 1.40 +/- 0.01 aerodynamic forces torque calculation compare parabola uncertainty calculation equal axis show legend X-Zylo Trans-SST center of mass (x-coordinate)x axis : center of mass (y-coordinate)y axis : center of mass (z-coordinate)z axis :time interval 0.001[s] calculate experimental data: 03.09.2020 launch 1 show exp. data velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylovelocity: 15.94 m/sangle: 11.77° frequency: 18.5 Hzring used: X-Zylo experimentally measured values velocity: ---angle: --- frequency: ---X-Zylo more settingsinitial values launch velocity [m/s] 17.5 +/- 0.3launch angle [°] 9 +/- 0.4ring mass [g] 22.73 +/- 0.01center of mass [cm] 1.3 +/- 0.01spin frequency [Hz] 60 +/- 3ring length [cm] 5.45 +/- 0ring inner radius [cm] 4.75 +/- 0ring thickness [mm] 1 +/- 0.05launch heigth [m] 1.4 +/- 0.01 aerodynamic forces torque calculation compare parabola uncertainty calculation equal axis show legend X-Zylo Comparison front of ring (x-coordinate)x axis : front of ring (x-coordinate)y axis : - - - z axis :time interval 0.001[s] calculate experimental data: no comparison no comparison show exp. data velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: ---velocity: ---angle: --- frequency: ---ring used: --- Figure 22: 3D Trajectory of the X-Zylo in comparison with the theoretical predictions. Theinterface seen here is an interactive MATLAB GUI written to easily probe the rings flight indetail. Using the drop-down menus enables the user to plot almost all combinations of variablescalculated. For reasons of space from now on only the graph—then plotted in Python—is shownwithout the GUI.The theoretical trajectories for both turbulence models with their confidence interval are seenin figure 22. The observed trajectory is plotted without uncertainty for better visibility. Theinteractive GUI developed for the program shows the different initial values with uncertainty aswell as the different calculation settings. As this 3D view of the trajectory is only lightly helpful,figure 23 shows the different quantities as 2D plots for better comparison.Figure 23 presents the observed trajectory with derived quantities as well as the predictionsby both turbulence models. Additionally, for comparison a conventional flight parabola withoutfluid forces is shown in several subplots. The uncertainty on the simulated quantities is calculatedusing the uncertainty on the initial launch values and the procedure described in section 9.1. Theerror margins of the observed values are calculated using the procedure in section 9.2, in whichthe camera induced error is estimated. In the following every subplot of figure 23 is individuallyevaluated. Page 21/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner − − − − − − − − − − − − − − − − simulation (Transition SST)simulation (k − ω − SST) observed values (S10, SloMo)observed values (S9, SloMo) observed values (GoPro 8)observed values (GoPro 4) observed values (S7 + GoPro8)parabola comparison x -coordinate (distance) [m] z - c oo r d i n a t e ( h e i g h t) [ m ] x -coordinate (distance) [m] − − y - c oo r d i n a t e ( s i d e w a y s d r i f t) [ m ] − − y -coordinate (sideways drift) [m] z - c oo r d i n a t e ( h e i g h t) [ m ] time [s] x - v e l o c i t y v x [ m / s ] time [s] − . − . − . . . . z - v e l o c i t y v z [ m / s ] time [s] v e l o c i t y m a g n i t ud e | ~ v | [ m / s ] .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 . time [s] . . . . . . . . . x - v e l o c i t y v x [ m / s ] .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 . time [s] . . . . . . . . . z - v e l o c i t y v z [ m / s ] .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 . time [s] . . . . . . . . . . v e l o c i t y m a g n i t ud e | ~ v | [ m / s ] time [s] − − − − − − v e l o c i t y a n g l e α ~ v [ ◦ ] .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 . time [s] v e l o c i t y a n g l e α ~ v [ ◦ ] time [s] − y - v e l o c i t y v y [ m / s ] time [s] a n g u l a r f r e q u e n c y ω [ r a d / s ] time [s] a cc e l e r a t i o n m a g n i t ud e | ~ a t o t | [ m / s ] time [s] a n g l e o f a tt a c k α [ ◦ ] t r a j e c t o r yv e l o c i t y p r o fi l e s a u x ili a u r y i n f o a b cd e fg h ij k lm n o Figure 23: Comparison of the theoretical quantities with the observed trajectory data for a singlelaunch. Plots a-c show the trajectory in the xz -, xy - and yz -plane while plots d-l show differentvelocity profiles. In plots m-o additional information not probed for the real trajectory is visible.Page 22/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner23a) The projection of the observed trajectory into the xz -plane is shown. As can be seen, afterall corrections are applied to the raw footage as described in section 5.2, both cameraperspectives (GoPro8 and GoPro4) coincide really well in the given confidence interval.When comparing the experimental data with the theoretical predictions, it can be seenthat during early flight, a good match could be achieved. However, later during flight thepaths diverge outside the joint error margin. The decrease in height (second drop) is alsomore rapid in the observation compared to both predictions. This is only lightly visiblewith the given aspect ratio of the plot, when having equal axis this becomes more obvious.Nonetheless, the overall shape of the trajectory is in good agreement with the theory. Mindthat the z -axis is heavily scaled, therefore the trajectory seems not as flat as it actually is.Lastly when comparing both turbulence models, it is seen that k- ω -SST gives betterpredictions since it predicts a higher drag value for low AoA. Therefore the distance coveredby the ring is smaller which better resembles reality. Also it is seen that even though bothmodels differ quite significantly for low AoA drag prediction, the influence is rather slim.This comes due to the first drop, as small AoA are only present for a short amount of time,see figure 23o.23b) When looking at the projection of the trajectory in the xy -plane, it is visible that thequalitative behavior of the X-Zylo can be modeled well. The X-Zylo first swerves to theright, then turns mid-flight and rapidly swerves left at the end of flight, as seen in figure20. This can be explained using the model described in section 2. As the COP movesupstream, it passes the COM of the ring at an AoA of approximately . °. Consequently,the torque acting on the ring changes direction in mid-flight when the COM is passed bythe COP which results in a direction change of the swerving motion. This torque signchange is further denoted as torque flip . It can be seen that the sideways drift is strongerthan predicted, however the ring hits the ground earlier in the experiment. Therefore, theX-Zylo is not able to drift as far at the end of flight as in the simulation. The sidewaysdrift also influences the observation in the xz -plane, as all spatial directions are coupled.An increase in y -velocity reduces the other velocity directions accordingly and is thereforevisible in the xz -projection.The uncertainty on the theoretical prediction is huge in this case due to the high errormargin imposed on the COP location ( ± for k- ω -SST, ± for Transition SST). Inaddition to the uncertainty for the mean wall shear stress—which describes the decrease inrotational velocity—one gets this large uncertainty. It is also visible that the additional error on the Transition SST’s COP data make a big difference as the uncertainty almostdoubles due to nonlinear behavior. It is later seen that the location of COP and COM arevery sensitive flight parameters, see section 6.3.23c) The yz -projection is less interesting on its own, but it completes the whole picture. Forbetter visibility only the uncertainty bounds are shown. One can see the behavior alsoobserved in the real world, seen in figure 20. Mind that the values shown in the plot arecorrected for perspective, therefore both figures do not look perfectly alike. The qualitativebehavior is predicted correctly, however another time it is seen that the swerving to theright is more pronounced than expected. Using this representation it is also better visiblethat the ring looses height faster compared to the simulation at the end of flight. Eventhough the observed sideways drift is more rapid at the end of flight, the slope late flightshows a steeper descend.23d) As can be expected the x -velocity decreases over time due to drag acting on the ring.Again the behavior seen in the observed trajectory is also seen in the theoretical prediction.The disparity seen seems to stem from a difference in the launch velocity. The theoreticalprediction shows an almost constant offset from the observed velocity, indicating that theused launch velocity was overestimated using the slow motion footage. Even though thePage 23/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnersmaller launch velocity captured by the GoPro8 was accounted for in the calculation ofthe initial parameters, due to a really large uncertainty the influence of that measurementmade a negligible influence in the weighted average. This shows that the uncertainty of thelaunch velocity for the other cameras was likely underestimated.Another detail that can be observed in the plot is that the x -velocity decreases morerapidly shortly before hitting the ground as the sideways drift of the X-Zylo accelerates.This can also be seen in the theoretical prediction where this behavior is starting to formjust as the ring hits the ground, therefore a bit later as in the observed trajectory. Theparabolic trajectory shows a constant x -velocity as no drag forces are applied.23e) The z -velocity shows an interesting behavior. At launch the vertical velocity is quite large,but as expected by the theoretic prediction in section 2, the ring at first loses height ratherquickly (first drop). The asymptotic behavior for t → is seen to fit the parabolic trajectory,which was predicted. Only after about . this initial drop is absorbed by the increasinglift force due to the rising AoA. The second drop is visible later during flight starting fromabout . onward. There the ring begins to lose height faster than seen before, which canbe explained by the flow separation forming at higher AoA. Later in plot 23o this becomesmore obvious when looking at the AoA. The mentioned equilibrium phase of the flight inbetween both drop phases can be seen quite well using this representation.Comparing theory and experiment, both curves match remarkably well until the seconddrop phase sets in. During the second drop the decrease in z -velocity is underestimated bythe simulation, showing that the impact of the flow separation occurring was underrated.23f) The velocity magnitude follows a similar pattern as the x -velocity. However, due to theincreasing negative z -velocity at the end of flight the velocity magnitude plateaus witha tendency to rise again. A similar behavior is seen in the observed trajectory. Due toadditional effects stemming from the x - and y -velocity, the observed velocity magnitudedrops again shortly before hitting the ground. This effect is not visible in the theoreticalpredictions and is not fully understood.23g) This plot shows the observed x -velocity obtained by the slow motion footage (see figure 19).As the velocity range is small and the uncertainties are fairly big, no error-bars were puton the experimental data. When looking at the S10’s data, the theoretical behavior wasapproximately met, however a dip is seen when the X-Zylo is in the center of the camera’sfield of view. This effect most probably occurs as some additional camera effects are notcorrected or are underestimated. As most probably the velocities observed in the centerof the field of view are more accurate, this would also backup the claim that the launchvelocity could be overestimated in the slow motion footage.The data obtained by the S9 is even more questionable as a far faster drop in velocitythan expected is observed. Here the error stems from the position of the camera, as itwas seen in the footage that the camera was slightly tilted sideways, therefore producinga skewed image. As both cameras are very close to the trajectory, the camera distortionproduces a relatively big error.Another problem can be seen when looking at the plot itself. The experimental datashows a quantized manner as only discrete velocities are seen, which is non-physical. Theresolution of the camera produces this pattern, as a pixelated image can only entail discretelength differences. Therefore, one has to track the X-Zylo almost pixel-perfect, whichinduces even more error.It is quite obvious that the used method—capturing the launch of the X-Zylo withcameras—to calculate the initial values accurately is not sufficient. Even though correctionterms are applied, the camera induced errors still dominate the obtained results. The closerthe camera is to the trajectory, the stronger those effects become.Page 24/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner23h) The z -velocity matches the theory better as camera errors like barrel distortion and thesideways tilt of the S9 mostly influence the x -coordinate. While the X-Zylo flight spansalmost the whole field of view horizontally, it only covers a small distance vertically. Thevertical components are only inflicted with serious errors for high launch angles.The mentioned asymptotic behavior for t → is best visible using this representation.While the parabolic trajectory shows a linear decrease in z -velocity, the X-Zylo’s increasingAoA dampens this decrease until a force balance is formed. This is the transition betweenthe first drop into the equilibrium phase.23i) The dip in x -velocity observed by the S10 as well as the fast drop for the S9 are still visiblein the velocity magnitude plot. As the x -component of the velocity dominates during earlyflight, the plot does not significantly alter from the observed early-time behavior of the x -velocity itself seen in figure 23g.23j) One can also look at the xz -projection of the velocity vector. The angle between the x -axisand this projection is defined by α (cid:126)v = arctan( v z /v x ) and will be denoted as velocity angle .Using this quantity, the different flight phases are nicely visible, especially the transitionfrom the equilibrium phase towards the second drop. The first drop is less pronounced asin plot 23e showing the z -velocity, however still visible. In this representation it can also bespotted that the observed data only really diverges after flow separation occurs and thesecond drop is initiated. The difference between theory and experiment in the plots 23d+facross the whole flight duration can be traced back to the velocity magnitude, which waslikely overestimated from the slow motion footage. Using this representation the velocitymagnitude has no influence, only the ratio of x - to z -velocity is important. Therefore thisrepresentation is most likely more accurate.23k) When looking at the velocity angle in the slow motion footage, the first drop is observedreally well with both cameras. The theory and the experiment coincide almost perfectlyand again the asymptotic behavior for t → matches free fall.23l) The y -velocity also follows the expected contour, however there the first qualitative differencebetween theory and experiment is seen. Instead of the y -velocity increasing linearly duringthe second drop phase, the acceleration decreases and tends to zero, so that the velocityplateaus. It is unknown if the velocity remains stable or reverses a second time, whichwould then need a new explanation. With the given data no conclusion can be found forthis issue as especially the sideways drift is hard to model. The Transition-SST model isseen to fit better to the real drift behavior than the k- ω -SST turbulence model.23m) The angular frequency was only measured at launch and not during the rest of the flight.Therefore, it is unknown whether the simplified model yields accurate results. Mind thateven though the behavior seems linear, this is not the case, especially for a longer flightduration.23n) The experimental values for the acceleration magnitude are not shown since they scattersignificantly as even small imperfections in the tracking amplify when deriving the quantities.In the theoretical prediction the first drop is perfectly seen. As the X-Zylo does not generatelift at start, the acceleration magnitude is .
81 m / s (plus additional drag force). In the first . this magnitude drops rapidly as the AoA of the ring increases. The behavior late inflight is more complex and the different components of the acceleration would need to bereviewed independently for a deeper analysis. Those are however not included here.23o) At last the theoretical AoA of the ring is shown. Here also the different phases are illustratedquite well. The ring starts off with an AoA of °; this condition was preimposed and anassumption made for the launch with the launch device. In section 6.4 this assumption isPage 25/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerdiscussed further. It is seen that as expected the AoA increases rapidly shortly after launch,then increases only slightly during the equilibrium phase. After that the second drop setsin, apparent from the rapid rise in AoA. This rise sets in after an AoA of approximately ° is exceeded due to the flow separation forming, which was seen in figure 23. It isperfectly visible that the effects seen in the CFD data necessarily also show in the simulatedtrajectory.All in all the behavior of the X-Zylo was qualitatively predicted very well. The differentphases mentioned in section 2 are clearly distinguishable in both the theory and the experiment.Moreover, the duration of the different phases could also be predicted well. This in turn showsthat the CFD simulation was able to predict the flow separation qualitatively as already seen inthe validation cases. However, it is seen that even though the qualitative flight was predictedcorrectly, especially the velocity of the ring at launch seems to be overestimated from the slowmotion footage. This can be seen when looking at the trajectory in the xz -plane (23a) and thevelocity magnitude plot for the whole trajectory (23f), which shows an offset for the whole flightduration. It is theorized that amplified camera errors due to the proximity of the slow motioncameras to the trajectory interfere with an accurate measurement of the initial parameters. Hintsfor such behavior are seen in figure 23g with the observed dip in the center of the field of view.An important detail is the asymptotic behavior seen for t → . Both the simulation and theobserved data asymptotically match the free fall simulation at launch, confirming a first drop. Notonly does this confirm the qualitative theory (see section 2.1), but also becomes very importantwhen studying a human induced launch. Resolving the additional effects one has to accountfor in a human induced launch (see section 6.4), it will become clear that without the launchcontraption, a useful analysis of the flight would likely be impossible as exactly this asymptoticbehavior is violated. As the launch angle proves to be an essential parameter in the flight of an X-Zylo and the launchmechanism is build especially to control that factor, several launches with different initial launchangles are compared. The initial launch values taken from the slow motion footage can be readfrom table 3. As seen the launch angles varied from ° to . ° with almost equal step size. Thelaunch angle initial velocity init. rot. freq. launch time α launch [ ◦ ] v launch [m/s] ω ( t ) / (2 π ) [Hz] distance [m] aloft [s]flat ( L1 ) . ± . . ± . ± . ± . . ± . medium ( L2 ) . ± . . ± . ± . ± . . ± . steep ( L3 ) . ± . . ± . ± . ± . . ± . steepest ( L4 ) . ± . . ± . ± . ± . . ± . Table 3: Initial launch values with uncertainty for the compared launches taken from the slowmotion footage. The launch distances and the flight time are included for comparison.initial velocity however shows a variance depending on the launch. Also the rotational frequencyis very different for the launches which was already seen in section 4.2. The predicted trajectoriesas well as the experimental data can be seen in figure 24 and 25. At first, it is seen that thepredicted trajectory shows a farther flight distance than observed in all cases. As seen in section6.1, several effects play a major role for the underestimated flight distance. In particularly anunderrated flow separation for high AoA, a miscalculation of the launch velocity and a differencein drift behavior are the most prone errors. The steepest launch L4 represents the underratedflow separation really well, as the trajectory only deviates from the theory shortly before thesecond drop sets in. The sideways drift behavior observed in the medium flat launch L2 seenPage 26/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerin figure 25 shows, that the drift has non-negligible influence on the drop. As the ring driftsless significantly into the positive y -direction compared to the simulated trajectory, the observeddrop is less rapid than simulated. Modeling the sideways drift more accurately could in fact alsoimprove the overall performance of the prediction as all flight directions are strongly coupled. x coordinate (distance) [m] z c oo r d i n a t e ( h e i g h t) [ m ] observed trajectorysimulation (Transition SST) simulation (k − ω − SST) parabola comparison α launch . ◦ α launch . ◦ α launch . ◦ α launch . ◦ Figure 24: Comparison of the trajectories for different launch angles using equal axis for a betterview on the flight. Due to this scale the error-bars in z direction are scaled by a factor of 3 forbetter visibility.For the sideways drift of the trajectory (see figure 25), the observed values are still in theconfidence interval of the prediction made by the Transition-SST turbulence model, however onlydue to the huge uncertainties given by the model. As additional information it is shown where the y -acceleration vanishes and the torque on the ring flips (vertical lines). This happens when theCOP passes through the COM of the ring. For the observed trajectory an additional confidenceinterval is shown as the torque flip is not sharply visible in the data. One can deduce that thetorque flip is mostly theorized too early during flight. The ring generally drifts stronger intothe negative y -direction before turning, showing that either the COM or COP data is erroneous.Even minimal shifts in either parameter results in visible differences of the sideways drift (seesection 6.4).Figure 26 represents the experimental observation of the velocity angle for all four launches.The theoretical prediction again shows good agreement, however the deviation seen is larger thanformerly observed in figure 23e. This is especially the case for the equilibrium phase of the lowlaunch angle cases ( L1 , L2 ). Here the first drop lasts slightly longer than expected, thereforecreating a faster decline in the velocity angle than predicted. A good explanation was not foundas it would be expected that for lower AoA the prediction would be better as the CFD simulationsshould yield more accurate results. In the case L4 the faster drop during late flight can be seenagain in the steeper slope of the velocity angle during the second drop phase.Concerning the time duration of the different phases one clearly observes that as expectedthe equilibrium phase gets shorter for higher launch angles. Here it comes into play that theX-Zylo generates less lift in the z -direction for a steeper climb, therefore the AoA to support thePage 27/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner x coordinate (distance) [m] − . − . − . . . . . y c oo r d i n a t e ( s i d e w a y s d r i f t) [ m ] x coordinate (distance) [m] − . − . . . . . . y c oo r d i n a t e ( s i d e w a y s d r i f t) [ m ] x coordinate (distance) [m] − . − . . . . . y c oo r d i n a t e ( s i d e w a y s d r i f t) [ m ] x coordinate (distance) [m] − y c oo r d i n a t e ( s i d e w a y s d r i f t) [ m ] observed trajectorysimulation (Transition SST) simulation (k − ω − SST) torque flip
COM > COP COM < COP COM > COP COM < COPCOM > COP COM < COP COM > COP COM < COP α launch . ◦ α launch . ◦ α launch . ◦ α launch . ◦ Figure 25: Sideways drift behavior of the X-Zylo in flight for different launch angles. It is markedwhere the COP moves past the COM on the chord and therefore the drift behavior changes. It isshortly noted with COM > COP when the COM is in front of the COP, and vice versa. Notethat it is implicitly assumed that the COP always lies on the symmetry axis of the X-Zylo. Thisassumption was used for all computations and is thought to sufficiently depict the real situation.weight has to be higher. This increases the equilibrium AoA and therefore the drag on the ringduring the whole equilibrium phase. This can be seen as the velocity angle in the equilibriumphase is higher for higher launch angles, which directly translates to the AoA as the ring axisvector is virtually constant during the first two flight phases and only changes in the third phase.The transition phase leading to the second drop is seen to be longer for small launch angles,with the second drop phase being longer for higher launch angles as the ring gained more heightduring flight. The equilibrium phase independently of the launch angle shows an almost uniformduration. A notable detail in figure 26 is that in all launches, the asymptotic behavior of thevelocity angle for t → again matches that of a parabolic trajectory. The deviation seen for thefirst frames of the high launch angle cases is also small enough to still be in the given uncertaintybounds. Therefore, the preimposed condition of an ° AoA is approximately met.Figure 27 shows a parameter analysis of the flight distance of an X-Zylo with set initialparameters and variable launch angle. The flight distance with respect to the launch angle isplotted for launch angles in the range [ °, °]. For comparison the curve for a parabolic trajectoryis shown, as well as the error margins for the flight distances. It is visible that instead of theflight distance rising with higher launch angle, a small launch angle is favorable.Page 28/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner time [s] − − − − v e l o c i t y a n g l e α ~ v [ ◦ ] time [s] − − − − v e l o c i t y a n g l e α ~ v [ ◦ ] time [s] − − − − − v e l o c i t y a n g l e α ~ v [ ◦ ] time [s] − − − − − − v e l o c i t y a n g l e α ~ v [ ◦ ] observed valuessimulation (Transition SST) simulation (k − ω − SST) parabola comparison st dropequilibrium phase nd drop st dropequilibrium phase nd drop st dropequilibrium phase nd drop st dropequilibrium phase nd drop α launch . ◦ α launch . ◦ α launch . ◦ α launch . ◦ Figure 26: Comparison of the velocity angle of the ring during flight for different launch angles.The first .
35 s after launch are magnified and the different flight phases marked. launch angle α launch [ ° ] fl i g h t d i s t a n c e [ m ] maximaparabola comparisonsimulation (Transition SST) simulation (k − ω − SST) . ◦ ± . ◦ . ◦ ± . ◦ Figure 27: Parameter analysis of the flight distancein respect to the launch angle for both turbulencemodels and a traditional parabolic trajectory. The initial values used for the parameteranalysis were set to be v launch = (16 . ± .
3) m / s ,ω ( t ) / (2 π ) = (50 ±
3) rad / s ,h ( t ) = (1 . ± .
01) m . The uncertainty for the launch angle wasset to ± . °. All other parameters wereset as if the colored X-Zylo was used. Asseen in figure 24, the flight distance de-creases for larger launch angles, howeverthe maximum is predicted to be at a stillsmaller launch angle of . ° ± . °. Unfor-tunately no launch angles smaller than °were experimentally recorded during thefinal test run, therefore no comparison canbe made for those cases. The parabolictrajectory shows a deviation from the ex-pected optimal launch angle of ° due to the height offset at launch. It can be seen that theX-Zylo operates well only in a small launch angle interval of about [ °, °]. It is also seen thatthe flight distance is much more sensitive to launch angles lower than the optimal value than toPage 29/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerhigher angles. This is due to two effects. On one hand a higher launch angle yields a smallerequilibrium AoA, decreasing drag over an elongated flight period. However, for small launchangles the ring cannot cover a long flight distance as it looses height during the first drop. Thisleads to a trajectory that curves to the ground early in flight and therefore cannot fully utilizethe equilibrium phase. The optimal trade-off strongly varies depending on the ring mass, thelaunch height and the initial velocity magnitude. As an in depth parameter analysis would gobeyond the scope of this paper, this is not covered in more detail. The sideways drift is heavily dependent on the location of the COM. This is shown in figure 28,where two different X-Zylo launches are shown. The initial conditions for both launches only differslightly ( ∆ v launch = 2 . , ∆ α launch = 1 . , ∆ ω ( t ) = 13 . ), however one X-Zylo is coloreduniformly while the second one is unaltered. As stated in section 5.1, coloring the X-Zylo shifted the − − . − . . . . . . . observed trajectorytorque flipsimulation (Transition SST) simulation (k − ω − SST) x coordinate (distance) [m] y c oo r d i n a t e ( s i d e w a y s d r i f t) [ m ] coloreduncolored comparison Figure 28: Sideways drift behavior for the flight of acolored X-Zylo compared with a launch of an unalteredX-Zylo. COM by downstream, which hasa big influence on the observed flightpattern. The unaltered ring turns sig-nificantly earlier, which is an effectof the COM being further upstream.As predicted by the CFD data shownin figure 10, the COP moves up-stream with increasing AoA. The po-sitions in flight where a change in y -acceleration is seen are marked withthe respective uncertainty. These arethe positions where the COP movespast the COM and the torque on thering flips. An earlier torque flip forthe uncolored X-Zylo is seen as wouldbe expected. As can be seen in figure15, the flight is not very sensitive forthe rotational velocity of the ring, only the drift amplitude shows an almost linear dependence onthe initial rotational frequency. It is seen that the location of the COM is a much more sensitiveparameter. A core assumption made in all theoretical calculations is that the initial AoA of the ring is °.However, it is seen in the work by Hirata, et. al. [4] that this assumption does not hold for alaunch by a human. To evaluate how the trajectory and the AoA behave for different initialparameters, figure 29 shows trajectories for an initial launch angle of ° and an initial velocitymagnitude of
16 m / s . The initial AoA is varied from − ° to ° in discrete steps and the resulting xz -projection of the theoretical trajectory shown. Additionally, the AoA is shown as it covers theearly flight in more detail.It is seen that the trajectory of the ring bends upwards early in flight when the initial AoA isvery high. This can be explained using the model developed in section 2.1. As the AoA at launchis higher than the equilibrium AoA, this produces a larger lift force than the opposite gravitationalpull and therefore the ring is accelerated upward. This then results in a smaller AoA and returnsthe ring to the equilibrium AoA over time. The first flight phase is therefore seen to be a first riseof the ring instead of a drop. It is visible that the ring rapidly moves towards an equilibrium AoAno matter the initial AoA. From there on the flight is normal again, only the first flight phaseis altered. However, it can be seen that even a change in initial AoA of ° gives tremendouslyPage 30/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerdifferent results in the trajectory. Moreover, one can observe that the equilibrium AoA is differentfor the different initial AoA values. This effect is due to the different velocity magnitudes thering has when encountering the equilibrium. As the drag is approximately rising quadratically forhigher AoA, the trajectory with an initial AoA of ° shows a higher equilibrium AoA as thering lost more speed early on in the first rise. Additionally, with the trajectory bending upward,again the lift does contribute less towards maintaining a straight flight as the z -component isdiminished. As already stated in section 6.2, it is again seen that the first flight phase before anequilibrium is achieved lasts approximately .
25 s no matter the initial AoA or the launch angle.Similar to a pendulum a stronger deviation from the equilibrium position results in a greaterrestoring force. x coordinate (distance) [m] z c oo r d i n a t e ( h e i g h t) [ m ] .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 . time [s] − a n g l e o f a tt a c k α [ ° ] -6-303691215 i n i t i a l A o A a t l a un c h α ( t ) [ ° ] st rise st dropequilibrium phase ** Figure 29: Comparison of trajectories with different initial AoA. A ° initial AoA is the coreassumption of all former calculations. However, when the ring is launched by a human, thisassumption must not hold, causing different phenomena to occur.A real human-induced launch is visible in figure 30, carried out by Hirata, et. al. [4]. It isvisible that the flight behavior is remarkably different from the observed behavior with the launch x -coordinate (distance) [m] − − z - c oo r d i n a t e ( h e i g h t) [ m ] observed trajectory Hirata, et al [5]simulation (k − ω − SST)simulation (Transition SST)
Figure 30: Launch by a human recorded by Hirata, et.al. [4]. The launch can be modeled with a high initialAoA of approximately ° and a negative launch angleof about − °. device. The X-Zylo rises after launchbefore encountering the equilibriumphase, just as seen in figure 29. Whentrying to reconstruct the found tra-jectory one finds good agreement fora negative launch angle of − ◦ aswell as for a high initial AoA ofabout ◦ . Those values—even ifonly approximate—show the signifi-cance of a launch device, with whichthe initial AoA can be controlled. Asseen in figure 29, an initial AoA ofmore than ° would lead to an ex-tremely fast rise of the X-Zylo. Be-cause of that, experimentally eval-uating the initial parameters wouldbecome impossible, as the velocityvector used to find the launch anglechanges drastically in the first . of flight. Additionally, the initial AoA of the ring itself is hard to determine. As significant initialparameters cannot be determined accurately, a quantitative comparison between theory andexperiment is unlikely to be of good quality. One could only use a maximum likelihood analysisto determine the most suitable initial parameters, however without knowing the experimentalparameters, a comparison is still not possible.It has to be emphasized that as in all launches conducted using the dedicated launch device,the asymptotic behavior for t → was approximately that of a parabolic trajectory. However,Page 31/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerthe implications of a non-zero initial AoA as seen in figure 29 would be strongly visible. Thisshows that the condition of an initial ° AoA is suitably met with the launch apparatus, as evensmall deviations would result in a different asymptotic behavior. Several problems still prevail, which have to be considered for further analysis of the flight. Anon-comprehensive list with some open issues is described in the following:i)
CFD Data:
The CFD Data is seen to be the weak point of the theoretical model. Eventhough the model was validated and even rotational effects were taken into consideration,still a deviation can be seen especially for the high AoA flight. Here more sophisticatedmethods as for example LES simulations would be needed for better coverage of the effects.Also in contrary to what was thought before, the COP is a very sensible parameter and hasto be known with high precision. Therefore, the medium M2 mesh with its . (k- ω -SST)or . (Transition SST) error from the finest mesh is not accurate enough. A betterresolution here would be needed for this parameter as well as an additional validation withexperimental COP data. Moreover, the assumption that C D and C L do not depend onthe flow velocity has to be investigated. As was seen in figure 23f, the velocity magnitudeduring flight ranges from / s to
16 m / s . In contrast, the CFD analysis was performed fora flow velocity of .
15 m / s , which could yield more slight deviations that add up over time.Also, the influence of the model’s rotation was only calculated for an AoA of °, a differentbehavior could be present for other AoA.ii) Launch Device:
To alter the launch device for higher reproducibility would be farfetched, only the belted motor drive mechanism could be improved easily to retain a morestable rotational frequency. However, as the initial values are calculated for every launchindividually, reproducibility is not seen as a key factor needed for better experiments. It ismore important to satisfy the initial ° AoA condition needed to make good predictions. Informer iterations of the launch mechanism, this condition was not met and strong effects asseen in figure 29 were visible. Those issues were resolved, however it is not known what thedeviation from the condition at this final point really is. As it is not feasible to observe aninitial AoA of less than °, the only possibility would be to further improve the device toensure the condition is met accordingly all the time.iii) Tracking the Trajectory:
As five different cameras were used with non-identical lensesand intern software it is impractical to correct the trajectory reliable. The used correctionsas stated in 5.2 only concern geometrical corrections, camera induced errors were neglected.At best one would use two synchronized high resolution cameras that observe the trajectory.With additional markers on the X-Zylo one would then be able to use more advancedtracking software with computer vision algorithms to probe the flight accurately. Thiswould then also ensure reproducibility as a set algorithm performs the tracking and correctionprocedure rather than a human tracking frame by frame manually. However, this wouldneed a dedicated camera setup currently not available to the author. Future improvementsin LiDAR ( Li ght D etection A nd R anging) technology could also be a suitable trackingalternative, bypassing some imaging errors one encounters with cameras.iv) Initial flight parameters:
The initial flight parameters as well as their uncertainty arekey factors for the theoretical prediction of the flight behavior and are seen as the biggesterror source involved. Using close-up cameras, only the rotational frequency is reliablyaccessible; the initial launch velocity and the launch angle can only be estimated roughly.Several factors as for example the close proximity of the cameras to the trajectory andtracking the object manually hinder this process. If one tracks the X-Zylo frame by framePage 32/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagnerin the slow motion footage, the time resolution is very high. However, as the distancetraveled is only some pixels wide, the velocity deviates much and shows a discrete spectrum.If however only every tenth frame is probed, the time resolution is to small to observethe first drop in its full extent since the initial quantities change rapidly after launch. Forexample in the experiments the initial launch angle changed about . ° in the first . aloft (see figure 26). Additionally, camera distortions and other geometric considerationsmentioned in section 5.2 due to the proximity of the camera further increase the uncertaintyon the gained values. Therefore, it is necessary to either use a dedicated high-speed camerasetup for the launch or to use alternate means of probing the initial values. One couldthink of several light barriers in close distance to the launch origin to probe the velocityreliably. The launch angle could theoretically be measured using the central guiding rod ofthe launch mechanism. For now this was not done as the stability of the device does notallow for accurate measurements using the apparatus itself.v) Incomplete Theoretical Model:
The theoretical model is known to be incomplete asminor effects were not accounted for, for example a small Magnus force when driftingsideways, therefore having a small velocity component perpendicular to the ring’s axis. Itwas also preimposed that the COP is always located on the ring’s symmetry axis, whichdoes not have to be true. As those additional effects should have insignificant influence onthe flight, they were neglected. However, it is not certain if for different flight scenariosthose effects become important, e.g. for crosswinds outside. There could also be additionalforces and torques at play that were not considered.
It was found that the conceptual explanation of the flight as well as the theoretical simulationcan predict and explain the observed flight behavior in great detail. Only minor details like theplateauing y -velocity during late flight are not yet fully understood. However, it is not unlikelythat those remaining anomalies are an effect of the imperfect camera setup used. The flight of anX-Zylo was seen to be very sensitive to several initial parameters, especially the initial launchangle, the initial AoA, and the location of the COM. Particularly the initial AoA, which can bekept close to ° with a launch device, makes a quantitative analysis of the whole flight almostimpossible when launching the X-Zylo by hand. Therefore, even without good reproducibility,the launch mechanism still is a key part of the experiment as it allows to neglect the impact of acritical parameter.From a quantitative point of view the flight was found to be modeled sufficiently precise,however several areas have to be improved upon. At first the second drop cannot be capturedin its full extent as the flow separation strongly impacts the second half of the flight. It is seenthat the CFD data can predict the moment of separation correctly, however the impact of theseparation is underestimated. Therefore, most simulations show a longer flight distance and aless significant drop at the end of flight compared to the observed data. Another observation isthat the sideways drift is stronger than predicted, therefore the COM and COP locations have tobe calculated more accurately. Another weak point is the used camera setup. A more dedicatedsetup with additional calibration points and an automated tracking procedure would be neededto advance the experimental measurements. Especially during early flight it is hard to capturethe velocity and the initial launch angle with great detail, as both values change rapidly in thefirst . of flight.All in all it is expected that advancements in the CFD data and the camera setup wouldentail a remarkable accordance between theory and experiment. While a qualitative parameteranalysis can indeed be made with the given setup (see figure 27), a detailed quantitative analysiscan not yet be achieved. Therefore, the goal of this work was only partially achieved with furtherwork being necessary. Page 33/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner Acknowledgments
I want to thank the whole SciComp research group at the TU Kaiserslautern, especially OleBurghardt and Tim Albring, for their correspondence and indispensable help to start this project.Additionally, I would like to thank CADFEM as well as Dr. Thomas Frank and the LRZfor providing the license for the ANSYS software and access to High Performance Computingresources. Furthermore, advice given by Prof. Dr. Nicolas Gauger and Prof. Dr. ChristianBreitsamter regarding the publication of this work was greatly appreciated. I wish to extend myspecial thanks to Hermann Steffen and Christopher Reinbold for their insightful comments onearlier versions of this manuscript. [1] William Mark Corporation (online). Accessed on March 20, 2020. url : .[2] William Crowther and Jonny Potts. “Simulation of a spinstabilised sports disc”. In: SportsEngineering
10 (Jan. 2007), pp. 3–21. doi : .[3] Mont Hubbard and S. Hummel. Simulation of Frisbee Flight . Jan. 2000. url : .[4] Katsuya Hirata et al. “Field observation and numerical analysis of a rotating pipe in flight”.In: Journal of Fluid Science and Technology doi : .[5] David London Tarr. What makes the Amazing X-Zylo fly . 1.0. Columbia-Capstone, 2016. isbn : 978-0-9821148-3-4.[6] Peter Kämpf. Accessed on March 20, 2020. 2016. url : https://aviation.stackexchange.com/questions/24782/how-does-a-ring-paper-airplane-fly-for-a-long-distance .[7] H. Schlichting and E.A. Truckenbrodt. Aerodynamik des Flugzeuges: Erster Band: Grundla-gen aus der Strömungstechnik Aerodynamik des Tragflügels . 3rd ed. Klassiker der TechnikTeil 1. ISBN: 978-3-540-67374-3. Springer Berlin Heidelberg, 2001. isbn : 9783540673743. url : .[8] The MathWorks, Inc. MatLab . Version R2018b. Sept. 13, 2018. url : https://mathworks.com/products/matlab.html .[9] Herbert S. Ribner. “The Ring Airfoil in Nonaxial Flow”. In: Journal of the AeronauticalSciences doi : .[10] Svetopolk Pivko. “Zur Abschätzung der aerodynamischen Eigenschaften dünner kreiszylin-drischer, schrägangeströmter Ringflügel”. In: ZAMM - Journal of Applied Mathematicsand Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik doi : .[11] J. Weissinger. Zur Aerodynamik des Ringflügels. Die Druckverteilung dünner, fast drehsym-metrischer Flügel in Unterschallströmung . Vol. 198. Forschungsberichte des Wirtschafts-und Verkehrsministeriums Nordrhein-Westfalen. ISBN: 978-3-663-04040-8. VS Verlag fürSozialwissenschaften, 1955. isbn : 9783663040408. doi : .[12] Open Cascade SAS. SALOME . Version 8.3.0. Sept. 6, 2017. url : .[13] ANSYS, Inc. Fluent . Version 2019.R3. Sept. 10, 2019. url : . Page 34/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagner[14] Herman S. Fletcher. “Experimental Investigation of lift, drag and pitching moment of fiveannular airfoils”. In: NACA TN 4117 (Oct. 1957). url : https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930084906.pdf .[15] Lance W. Traub. “Experimental Investigation of Annular Wing Aerodynamics”. In: Journalof Aircraft doi : .[16] Lance W. Traub. “Experimental Study of a Morphing Annular Wing”. In: Journal of Aircraft
56 (Sept. 2019), pp. 1–7. doi : .[17] Edmund V. Latoine. “Wind Tunnel Tests of Wings and Rings at Low Reynolds Numbers”. In: Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications . 2001, pp. 83–90. doi : .[18] Michael J. Werle. “Aerodynamic Loads and Moments on Axisymmetric Ring-Wing Ducts”.In: AIAA Journal doi : .[19] Douglas Brown and Wolfgang Christian. Tracker . Version 5.0.6. Aug. 15, 2018. url : https://physlets.org/tracker/ . An important step in the trajectory simulation is calculating an uncertainty corridor in whichthe X-Zylo is predicted, using the given uncertainties measured in the experiment. Therefore, arough error approximation is necessary, as all the initial parameters for the launched X-Zylo aswell as the drag and lift data are afflicted with error.The trajectory is calculated using a function f , which inputs are all the initial parameters { a, b, c, ... } and which output is the trajectory (and auxiliary information): (cid:126)P com ( x ( t ) , y ( t ) , z ( t )) = f ( a, b, c, ... ) . Let us consider an initial uncertainty ∆ a for the parameter a . The used approach is to call f three times, once with the parameter a and then twice with parameters a ± ∆ a . This yields threedifferent trajectories: (cid:104) (cid:126)P com (cid:105) = f ( a, b, c, ... ) , (cid:104) (cid:126)P com (cid:105) ± ∆ a = f ( a ± ∆ a, b, c, ... ) . This is done for every parameter with an inscribed uncertainty while the time step ∆ t iskept constant so that one ends up with many trajectories (cid:2) (cid:126)P com (cid:3) , (cid:2) (cid:126)P com (cid:3) ± ∆ a , (cid:2) (cid:126)P com (cid:3) ± ∆ b , . . . .Therefore, it is now possible to calculate a vectorial difference between the baseline trajectory (cid:2) (cid:126)P com (cid:3) and the ones induced with error for each time step t n (cid:104) ∆ (cid:126)P com ( t n ) (cid:105) ± ∆ a = (cid:104) (cid:126)P com ( t n ) (cid:105) ± ∆ a − (cid:104) (cid:126)P com ( t n ) (cid:105) = ∆ x ( t n )∆ y ( t n )∆ z ( t n ) ± ∆ a , which is also calculated for every other trajectory, labeled with corresponding subscripts. Nowthe approximate maximum uncertainty at time t n —assuming independent uncertainties for theinitial parameters—for e.g. the x -coordinate is calculated to be: (cid:104) ∆ x ( t n ) (cid:105) ± tot = (cid:114)(cid:16)(cid:2) ∆ x ( t n ) (cid:3) ± ∆ a (cid:17) + (cid:16)(cid:2) ∆ x ( t n ) (cid:3) ± ∆ b (cid:17) + ... = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) i ∈{ a,b,c,... } (cid:16)(cid:2) ∆ x ( t n ) (cid:3) ± ∆ i (cid:17) > . Page 35/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils WagnerTo not mix between trajectories over- or undershooting the baseline trajectory (cid:2) (cid:126)P com (cid:3) , it ischecked that all differences contributing to the errors ∆ ± tot are of the same sign. Consequently (cid:2) ∆ x ( t n ) (cid:3) − tot (cid:2) ∆ x ( t n ) (cid:3) + tot (cid:2) ∆ z ( t n ) (cid:3) + tot (cid:2) ∆ z ( t n ) (cid:3) − tot (cid:2) ∆ y ( t n ) (cid:3) + tot (cid:2) ∆ y ( t n ) (cid:3) − tot (cid:2) (cid:126)P com ( t n ) (cid:3) Figure 31: Visualization of the uncertaintyapproximation at time t n . x ( t n ) ± (cid:2) ∆ x ( t n ) (cid:3) ± tot are upper and lowerbounds for the x -coordinate of the X-Zyloat time t n . Figure 31 shows all six limitingpoints that get calculated and in which boundsthe X-Zylo is most likely to be found duringflight. One could now lay a generalized ellip-soid through those points, however for the sakeof simplicity the bounding box around thosepoints is used as the approximation. In the2D case of looking at the trajectory in the xz -plane, the upper and lower trajectory limitswill be the points (cid:104) (cid:126)P com ( t n ) (cid:105) upper = (cid:104) (cid:126)P com ( t n ) (cid:105) + (cid:104) ∆ (cid:126)P com ( t n ) (cid:105) + tot , (cid:104) (cid:126)P com ( t n ) (cid:105) lower = (cid:104) (cid:126)P com ( t n ) (cid:105) − (cid:104) ∆ (cid:126)P com ( t n ) (cid:105) − tot as they are the corners of the bounding box.This approach can be generalized beyond the trajectory to other parameters that are outputtedby the simulation, however those cases are easier due to them being plotted against time ratherthen other parameters also inflicted with error. Not only the simulation is inflicted with error, also the usage of cameras to observe the flight’strajectory gives erroneous information. In section 5.2, it is touched upon how the observedtrajectory is corrected and which imaging errors are considered. However, as some camera inducederrors were left out of the calculation and the applied corrections still contain small uncertainties,the influence of those effects has to be approximated. The calculation is only shown for theX-Zylo’s position measurements, the velocity errors were calculated similarly. Several error sourcesfor the location of the X-Zylo are seen as independent, each afflicted with a certain error:i)
Uncertainty while tracking:
This error is purely human-induced. As the X-Zylo isonly seen as a small black blob in the tracking software, it is hard to accurately trackthe ring. Especially when the background changes color—e.g. when the ring flies in frontof a basketball net— the position is only vaguely visible. In addition, manual trackingis not perfect and not the same spot is tracked every frame, which further increases theuncertainty. Still this error is relatively small compared to other uncertainties. All in all a uncertainty is added due to this tracking procedure for all coordinates.ii)
Barrel distortion:
Even though both GoPro’s shoot in linear mode, a very small barreldistortion is still visible which affects the calibration scale as well as the observed positionof the X-Zylo. The error on the scale is discussed further in point iii), only the error on thetracked position is approximated here.As the barrel distortion effects increase towards the edges of the camera’s field of view, theadded uncertainty depends on the difference between the position of the camera x camera and the tracked object x correct . A linear error term is then added δx distortion = | x correct − x camera | · R distortion (18)with the error rate R distortion . This rate has to be approximated for every camera usedas well as every coordinate due to the barrel distortion acting differently for differentcoordinates. The approximated rates can be read from table 4.Page 36/37heoretical and Experimental Investigation into the flight of an X-Zylo Nils Wagneriii) Scale error:
As already mentioned in point ii), the small barrel distortion adds anuncertainty to the calibration scale used. From measuring each calibration segment it wasfound that the uncertainty is under for the GoPro8 and about for the GoPro4.However, it was also seen that the larger error of the GoPro4 does not stem only from thescale error. Only about again is from the scale error, the rest is due to error v). Usingthis scale uncertainty an additional linear error term δx scale = x correct · R scale (19)is added to the whole uncertainty. The scale error R scale is dependent on the camera itselfand on the rotation of the camera.iv) Error due to X-Zylo’s drift:
As can be seen in equation (17) the x -correction involves—additionally to the observed X-Zylo position—the different camera locations seen in figure16. As those were measured beforehand, even a large measurement error of about . would add a negligible error, however the value ∆ d is inflicted with greater error as it hasto be taken from the measurement of another camera (S7). As no dedicated equipment wasused and both cameras were not synchronized, an error of for ∆ d is used.v) Camera rotation:
It has a big influence if the camera is rotated slightly and is not setperfectly perpendicular to the flight path. This also influences the scale, again due to theintercept theorem. Therefore, if it is visible in the footage that the camera is turned slightly,the scale error factor R rotation has to be added to the value R scale as seen in iii).Summarized we get a total uncertainty on the position of the X-Zylo, e.g. the x -coordinate isafflicted with an approximate error of δx total = δx tracking + δx distortion + δx drift + δx scale + δx rotation (20) = 1 cm + | x correct − x camera | · R distortion + | x correct − x camera | · | ∆ d | d camera · (0 . R rotation ) + x correct · R scale . Note that all errors and free parameters are only estimated based on the footage to make aneducated approximation of the real uncertainty. A better consideration of the uncertainties isneeded. R dist,x R dist,y R dist,z R scale,x R scale,y R scale,z R rot,x R rot,y R rot,z GoPro8 0.005 0.005 − − − GoPro4 0.008 0.008 − − − S7 − − − − − − R dist = R distortion and R rot = R rotationrotation