Physics of knuckleballs

Zigzag paths in sports ball trajectories are exceptional events. They have been reported in baseball (from where the word knuckleball comes from), in volleyball and in soccer. Such trajectories are associated with intermittent breaking of the lateral symmetry in the surrounding flow. The different scenarios proposed in the literature (such as the effect of seams in baseball) are first discussed and compared to existing data. We then perform experiments on zigzag trajectories and propose a new explanation based on unsteady lift forces. In a second step, we exploit wind tunnel measurements of these unsteady lift forces to solve the equations of motion for various sports and deduce the characteristics of the zigzags, pointing out the role of the drag crisis. Finally, the conditions for the observation of such trajectories in sports are discussed.


Introduction
In many sports, players use aerodynamic effects to lure opponents and score. For example in soccer, a free kick may get around the defensive wall thanks to a side spin [1]. Among the different paths [2], one of the most intriguing paths is probably the zigzagging knuckleball. This rare event was first introduced in baseball by the Major League Baseball pitcher E Cicotte, in 1908 [3]. By mastering this pitch, Cicotte deserved the nickname ' Knuckles ' and finished his career with more than 200 wins [4]. After Cicotte, several pitchers mastered knuckleballs and the pitch became a nightmare for batters. Despite this old history, the scientific observation of zigzag paths on a baseball pitch has only been done recently by Nathan et al [5]. In this article, the authors measured the angular direction of pitches as observed by the batter as a function of the ball releasing speed for a 'normal' pitcher and a pitcher mastering knuckleballs. Nathan et al observe that the 'normal' pitcher Jon Lester is able to send the baseball with different techniques characterized by a precise releasing speed and a precise angular direction. This feature is different from the one observed with Tim Wakefield, famous for his knuckleballs. In his case, the ball is launched at a precise speed of 66 mph ( -29.5 m s 1 ) and follows a random angular direction characteristic of the knuckle effect. The releasing speed of knuckleballs is far below the maximal launching speed of baseballs observed in the field (105 mph ; 47 m s −1 ). Nathan also determined that lateral deviations of knuckleballs are smaller than 1.2 cm which is 0.17 ball diameter [5]. Despite the slenderness of zigzag motions, the randomness of this pitch explains the difficulty for batters to hit them. Apart from baseball, zigzagging paths have been reported in volleyball and observed when the ball is launched with no spin [6]. Depra and co-workers measured these trajectories on a volleyball field and observed lateral deviations of the order of the ball diameter for a velocity of about 16 m s −1 [7]. Again, this velocity is far below the maximal velocity observed in volleyball, which is 37 m s −1 . Similar trajectories have finally been observed in soccer by Barber et al [8]. In this sport, zigzag paths occur with particular balls (such as the official match ball of the 2010 FIFA World Cup, Jabulani) when kicked with no spin [9]. Finally, knuckleballs have also been reported in cricket for a delivering speed of about 30 m s −1 [10].
We list in table 1 sports which involve balls of different diameters (   D 4cm 24cm) and different velocities (   --U 10 m s 91 m s possible impact of aerodynamic forces in many of the sports listed in table 1 [2], knuckleballs are only reported for baseball, cricket, volleyball and soccer. Our goal is to understand why. The typical conditions and characteristics of zigzagging paths are gathered in the last columns of table 1. We notice that knuckleballs mostly occur in a narrow range of velocities (column 7), significantly below U max , the maximum speed of each game (column 5). The typical length λ and amplitude δ of a zigzag are reported in the last two columns of table 1. Even if the lateral deviations δ are small, the unpredictability of this effect hinders receivers and goalkeepers. This phenomenon led one of the greatest baseball players, Willie Stargell, to describe knuckleballs as ' a butterfly with hiccups '. The purpose of this paper is to identify a scenario able to account for these characteristics including the rareness of knuckleballs in sports. We first discuss the possible origins of the zigzags (section 2) including seams (section 2.1) and other sources of asymmetry (section 2.2). The experimental set-up able to produce knuckleballs is presented in section 3 together with the experimental results on zigzags. The equations of motion for sport balls are solved in section 4 taking into account unsteady lift forces. The different asymptotic regimes are discussed together with predictions of the mean lateral deviation and the typical wavelength of the zigzag path for various initial conditions (angles and velocities). We then propose a phase diagram in which sports where a knuckle effect may impact the game are identified (section 5).
2. Several possible origins 2.1. Effect of seams A zigzag path is obtained provided a lateral unsteady asymmetry of the flow surrounding the ball exists. The first source of asymmetry considered to account for knuckleballs in baseball was the seams [12], a scenario recently revisited by Morissey et al [13].
Watts and Sayers measured the steady lift force in a wind tunnel for a baseball, as a function of its orientation θ with the flow (figure 1). They showed that lift oscillates with the ball orientation ( figure 1(b)). Thus, when the ball slowly rotates in air, it undergoes an alternative lift, which leads to a zigzag path. The lateral deviations expected from this scenario can be estimated. We consider the ball motion along a direction z perpendicular to the baseball velocity U and to the gravitational acceleration g (figure 1(a)) and we assume along this direction a lift force withẇ q = 4 (for a single rotation of the ball, the lift experiences about four oscillations due to the presence of four seams, as shown in figure 1(b)). The equation of motion of the ball projected along the z-direction is is the mass of a ball of diameter D and density r b . Equation (1) yields the lateral deviation of the ball At = -U 20.7 m s 1 , the measurements by Watts and Sayers provide figure 1(c)). In order to produce a complete zigzag on a distance λ corresponding to the distance between the pitcher and the batter Table 1. Characteristics of different ball games. Sports in which knuckleballs are observed are indicated in red (column 1). Ball diameter D (column 2) and mass M (column 3), ball to air density ratio r r b with r = -1.2 kg m 3 (column 4), maximal speed of the ball U max recorded on sport fields (column 5), corresponding Reynolds number n = Re DU max with ν the kinematic viscosity of air (n =´--1.5 10 m s 5 2 )(column 6). For sports where zigzag paths are reported (in red), the range of ball velocity U for which knuckleballs are obtained is reported in column 7 together with the typical length λ of a zigzag (column 8) and the range of normalized lateral deviation d D (column 9). Data are extracted from [2,[5][6][7][8]11]. , on the order of the ball diameter. This scenario may explain the zigzag paths reported on the pitch. However, it does not predict why knuckleballs are observed in a narrow window of ball velocity significantly below the maximal velocity achievable in game (table 1). Indeed, a shot at = U U 0 m a x will maximize F L,0 and thus δ as determined by Watts and Sayers ( figure 1(c)). In addition, knuckleballs were also reported for balls without seams, which suggests that another scenario could be at play to explain this phenomenon.

Other possible origins
Other sources of asymmetry can be considered to account for knuckleballs. The flow profile around a sphere depends on the Reynolds number. Two characteristic features are generally stressed in the flow past a sphere: first, the stagnation points where the flow stops; secondly, the separation points where the boundary layer separates from the sphere. These characteristic points are represented in figure 2 with blue and red dots respectively. The up-down asymmetry of stagnation or separation points relative to the flow direction indicates the existence of a lift. For a smooth sphere and Reynolds numbers lower than unity, the flow profile is symmetric (figure 2(a)) and the particle experiences no lift force. In this case, the stagnation points are located at the poles of the sphere and there is no separation of the flow from the solid. When < Re 300, a symmetric recirculation zone appears behind the sphere (figure 2(b)). In this case, the flow separates from the sphere close to the equator as symbolized by red dots. For > Re 300, the alternate vortex shedding breaks the up-down symmetry and generates unsteady lift forces (figure 2(c)) [14]. If the sphere spins, its rotation also breaks the flow lateral symmetry, displaces the stagnation points and induces a steady lift force referred as a Magnus effect (figure 2(d)) [15]. As considered previously, the presence of surface roughnesses (seams for example) delays the separation of the boundary layer, moves the separation points out from the equator and provokes a lateral force (figure 2(e)) [16]. Finally, at Ŕe 3 10 5 , an intermediate value between 10 4 and 10 6 (table 1) the boundary layer transits from laminar to turbulent state and the separation point moves downstream [17]. In the drag crisis, intermittent reattachments of the boundary layer generate temporarily asymmetric situations as shown in figure 2(f). Among these different sources of asymmetry, we try to identify the one which is likely to produce knuckleballs with the characteristics displayed in table 1.

Experimental results
We recorded the motion of the ball with two high-speed cameras Photron SA3 looking from the front and from the side of the trajectory. The two cameras were synchronized with an external trigger and recorded the sequence at 1000 Hz in full resolution (1280 × 1024 pixels). Figure 4(a) shows the recorded trajectory from the side. One observes that the trajectory is slightly asymmetric with respect of the maximum, consistently with predictions of Cohen et al [18] when the initial velocity of the ball U 0 is comparable to its free fall velocity . From the side trajectory, we deduce the Reynolds number as a function of the horizontal distance x as shown in figure 4(b). The decrease of the Reynolds number along the trajectory is due to the drag which acts on the characteristic length r rD 6 m b [18]. A front chronophotography of the ball till it reaches the top of the trajectory is shown in figure 5(a), revealing the non-straight path followed by the ball. Using the ball as a scale evolving over time, we deduced its lateral deviation along the z-direction as a function of the horizontal distance x and we report the data in figure 5(b). Initial conditions are similar, yet trajectories are different. This erratic behavior is the signature of knuckleballs. Lateral deviations of the ball are typically 0.1 to 0.4 ball diameter, that is, a few centimeters. Zigzags are observed on a traveled distance along the x-direction of about 7 m. Even if lateral deviations are small, they can strongly disturb volleyball receivers or soccer goalkeepers who are used to Magnus effect, but not to erratic phenomena.

Equation of motion
The goal of this section is to solve the trajectory of a ball experiencing a drag F D and an unsteady lift force F L in addition to its weight g M . At large Reynolds numbers the drag force is expressed as ) and C D the drag coefficient. In this regime, the lift force is defined as where are the characteristic lengths associated to drag and lift forces. Without lift, equations (3), (4) and (5) reduce to a ' Tartaglia ' path [18]. In the general case, solving these equations requires to know the time evolution of the lift coefficients ( ) Lift coefficients typically reported in the literature are time-averaged quantities. However, owning For the flow, the lateral motion of the ball is thus quasi-steady. We thus use the spectrum of lift force fluctuations measured using wind tunnel tests in order to solve the equations of motion. Norman and McKeon reported the spectral density Φ of the lift as a function of the dimensionless frequency St=fD/U (also known as Strouhal number) and the Reynolds number for a smooth sphere in the sub-critical, critical and super-critical regimes [19]. Examples of these measurements are shown in figures 7(a) and (b). These data allow us to express the instantaneous lift coefficients by the way of a Fourier decomposition    However, the numerical reproduction of zigzag trajectories does not allow us to make any prediction on the moment they occur, or why we only observe them in a few sports. In order to understand more deeply what provokes knuckleballs, we consider below the different contributions of the lift force spectrum separately. Indeed, we observe in figure 7(a) that the spectral density Φ of the lift force for <Ŕe 3 10 5 exhibits three distinct features: a white noise behavior in the range < St 0.1, a peak at  St 0.2 which corresponds to vortex shedding [14] and a power law decrease at large Strouhal numbers ( > St 0.3). For >Ŕe 3 10 5 , the peak corresponding to vortex shedding disappears but both the low frequency white noise and the large frequency decrease remain ( figure 7(b)).

The vortex shedding limit
For the sake of clarity, the impact of vortex shedding on the trajectory is considered in the naive case where the ball moves at a constant velocity = U e U x with only a lift force along the z-direction. In such a situation, the equations of motion (3), (4) and (5) reduce to is the ball density. Assuming that the spectrum of lift is a single peak for . In this context, the equation of motion (7) yields The normalized lateral deviation d D produced by vortex shedding at  St 0.2 depends on the inverse of the ball-air density ratio r r b . Horowitz and Williamson proved that for light spheres (r r < 1 b ), this phenomenon provokes alternate deviations of about a ball diameter [20]. A review of rising and falling nonrectilinear trajectories of a sphere in a fluid was proposed by Ern et al [21]. In sports, the ratio r r b is much larger than unity ( r r < < 46 570 b , see table 1), so that we expect negligible lateral deviations (  d D 1). In conclusion, the classical vortex shedding is not responsible for knuckleballs.

The white noise limit
In this section, we consider the low frequency limit of the spectrum of lift force, which is close to a white noise (figures 7(a) and (b)). As in section 4.3, each frequency f in the lift force is associated to a normalized maximal deviation˜( For a traveled distance x, the minimal frequency corresponding to a complete zigzag is = f U x; combining it with equation (11), we get˜( The lateral deviation δ still depends on the inverse of the density ratio r r b and on the lift coefficientC L but it is also a function of the distance x over which balls are observed: the larger x, the larger the lateral deviations, in agreement with what can be observed in figure 5(b). This behavior is verified more quantitatively in figure 8 where the mean lateral deviation d m is presented as a function of the ball position along the x-axis. The quantity d m is calculated over N=1000 trajectories obtained with numerical methods presented in shows a maximum close to Ŕe 3 10 5 which corresponds to the drag crisis regime. The intermittent reattachment of the boundary layer is responsible for this effect [19]. Thereby, if a ball is launched in the range of velocity corresponding to the drag crisis ( Ŕe 3 10 5 for a smooth sphere), we expect lateral deviations to be increased.

Numerical trajectories
The numerical integration of the equations of motion (3), (4) and (5) is done for the ball properties listed in table 1. Figure 9 shows the numerical results in the (O, x, z) plane (top view) for soccer, volleyball, baseball and table tennis. In each situation, the numerical method has been driven four times with a different choice in the spectral phase Ψ, which allows us to appreciate the mean lateral deviation produced by the lift forces and the typical zigzagging wavelength. For a soccer ball ( figure 9(a)), lateral deviations are about a ball diameter after 30 m. Between one and two changes in transverse direction are observed. Compared to soccer, lateral deviations are increased for a volleyball while wavelengths are reduced ( figure 9(b)). On the distance which separates the batter and the pitcher in baseball, balls have about one change in lateral direction and typical excursions are about 0.5 ball diameter (about 3 cm) ( figure 9(c)). The predictions show qualitative agreement with observations of knuckleballs in the field (see table 1), which confirms that the unsteady lift experienced by a sphere can be responsible for knuckleballs. Numerical solutions for a table tennis ball exhibit transverse deviations lower than 0.5 ball

Conditions for observation
As knuckleballs are not observed in all sports, we finally discuss the conditions to observe zigzags. Assuming that this phenomenon comes from the white noise present in the spectrum of lift forces described in section 4.4, we use equation (12) for the prediction of the lateral deviation δ of a ball which has followed a zigzag motion by a distance x. The equation can be written in the following waỹ is a typical distance fixed by the ball diameter D and the solid-fluid density ratio r r b and reported for the different sports in table 2. One deduces from equation (13) that zigzag with a significant lateral deviation (d > D) requires trajectories longer than λ. However, the distance traveled by the ball is not the only criterion needed to observe a tortuous path. Indeed, equation (13) shows that the spectral intensityC L must be large enough in order to produce a significant deviation. More precisely, it is the spectral intensity of lift components of frequencies lower than l = f U that is important. The two conditions are discussed in what follows.
The criteria for long trajectories can be approached by comparing the minimal zigzag wavelength λ to the typical distance L shot from which the different sport balls are shot to score. The distances λ and L shot and their ratio are reported in table 2. For sports where l > L 1 shot , it will be possible to observe a complete zigzag and most sports (except bocce, handball and basketball) satisfy this criterion.
As for the spectral intensityC L of lift forces at low frequency ( l < f U ), Norman and McKeon showed that C L reaches a maximum for Ŕe 3 10 5 ( figure 7(c)). Thereby, the ball velocity will impact non-trivially the zigzag properties and we try here to estimate how. We study numerically the effect of the initial velocity of the ball U 0 on the mean lateral deviation d m respectively to the trajectory expected without lift. The quantity d m is calculated over N=1000 trajectories obtained from numerical solutions of equations (3), (4) and (5) with the procedure described in section 4.4. Figure 10 As seen in figure 10(a), d m increases with the traveled distance x as expected from equation (12). The evolution of d m with Re 0 is non-monotonic and it reaches a maximum for Ŕe 3.0 10 5 . This observation directly results from the fact that fluctuations of the lift coefficients reach a maximum at the drag crisis ( figure 7(c)). We evidenced the asymmetry of the flow past a sphere in the drag crisis with an experiment achieved in the wind tunnel of the LME laboratory in Valenciennes. We measure the angle q w of the wake past a Table 2. Characteristics of knuckle trajectories deduced from the numerical method presented in section 4 (column 1): ball diameter D (column 2), solid-air density ratio r r b (column 3), typical length L shot from which a shot is performed to score (column 4), ball velocity U shot for which the zigzag effect is larger (d d = max ) (column 5), zigzag wavelength l p r r = D 2 4 3 b (column 6), ratio between the typical shot distance and the wavelength l L shot (column 7) and maximal normalized mean deviation d D max in the range of Reynolds number encountered in the game and for a traveled distance corresponding to λ (column 8), ratio between the characteristics time of a zigzag and the reaction time t t l t = U knuckle shot (column 9). Sports highlighted in red are those where knuckleballs are expected to be observed according to our model. smooth sphere relatively with the direction of the flow U (defined in figure 10(b)) as a function of the Reynolds number. One observes in figure 10(c) that the asymmetry in the wake dramatically increases wheñŔ e 3 10 5 . Thereby, the knuckle effect of a non-spinning ball is enhanced if the range of initial velocity corresponds to the drag crisis. One knows that the critical Reynolds number of the drag crisis strongly depends on the texture of the ball [24]. This effect occurs in a range of velocity between 15m s −1 and 25 m s −1 for soccer [25], between 16 m s −1 and 18 m s −1 for volleyball [6] and between 30 and 35m s −1 for baseball [16]. All these ranges of velocity are in good agreement with the ones where knuckleballs are observed [table 1]: increase of lateral deviations in the drag crisis regime appears to be essential to observe zigzag paths in sports.

Phase diagram for knuckleballs
The minimal flight length and the speed required to observe a knuckle effect in sport can be summarized in a phase diagram. We reported in table 2 the maximal normalized deviation d D max experienced when the projectile has traveled by a distance λ from its releasing point. The quantity d max is the maximal value of the mean deviation of the ball d m in the range of Reynolds number encountered in the considered sport (see table 1). One notices that d D max is lower than 0.5 in bocce, handball, basketball, squash, tennis and table tennis. For such balls, a zigzag path never disturbs players. The two criteria to obtain a knuckle effect are gathered in figure 11, where the coordinates are the ratios d D max and l L shot for each sport. In this plot, sports where knuckleballs may perturb the game are located in the domain where l > L 1 shot and d > D 1 max . In the others domains of the diagram, knuckleballs cannot be observed, either because of small transverse deviations or because wavelengths are longer than the field. For sports producing knuckleballs, it is required to launch the ball at very low spin and in the good range of initial velocities. These conditions may be difficult to obtain, especially in sports where the time of execution of a technical movement has to be minimized. Also, one may guess that it will be easier to launch a ball without spin by kicking it with a flat surface such as a racket rather than with a curved surface such as a bat. Despite that the link between the shape of the impactor and the easiness to produce knuckleballs is out of the scoop of this study, one may think that zigzag trajectories are simpler to produce in volleyball than in baseball for such a reason.
Surprisingly, golf balls belong to sports where knuckleballs may occur while no study reports such a behavior. Several arguments can be advanced to explain this enigma: first golf balls always experience a large spin, which produces a Magnus lift force much larger than the lift force unsteadiness discussed in this paper; secondly the resolution of the human vision does not allow to distinguish the lateral motion of the ball at a large distance; and of course there is no catcher in golf to be disturbed by a zigzag flight.

Characteristic time
So far, the characteristics time over which knuckleballs occur have not been discussed. Such a time can be estimated as t l = U knuckle shot and compared with the typical reaction time of a player  t 1 s as introduced earlier by Darbois Texier et al [11]. One expects that if t t > knuckle , players can react to ball motions and catch zigzagging paths. On the contrary, if the reaction time τ is longer than the characteristics zigzagging time t knuckle , knuckleballs can lure players. The ratio of the two characteristics times t t knuckle is reported in the last column of table 2. One notices that in all sports where knuckle effect plays a role, the ratio t t knuckle is smaller than one. Such an observation implies that in sports where zigzag paths can be observed, they systematically disturb the game. More particularly, one remarks that the ratio of the zigzagging time over the reaction time is smaller in baseball than in volleyball and soccer. Catching a knuckleball in baseball requires shorter reflexes than in volleyball and soccer and may explain why such phenomenon were first reported in baseball.

Conclusion
We have considered different possible origins for knuckleballs. These zigzag trajectories are associated to asymmetric and unsteady flows of air around the ball. We first quantified zigzags with an experiment, and then modeled them theoretically. Using unsteady lift forces, we obtain a quasi-quantitative agreement with experimental trajectories of a non-spinning ball traveling through the air. Using the numerical integration, we estimated the lateral deviations and the wavelengths expected for each sport ball, and predict how the unsteadiness of lift forces can produce a change in lateral directions within a field and with a sufficient magnitude to disturb players. In these sports, we showed that the obtention of a large knuckle effect requires the ball to be launched in a particular range of initial velocities corresponding to the drag crisis of the ball. This criterion, plus the absence of initial spin, explains the rareness of knuckleballs in sport. It also explain why the effect is only observed in a narrow range of ball velocities.
In baseball, the proposed scenario and the one involving the seams are both compatible. Knuckleballs in this sport may thus have two different origins, keeping their complexity veiled and remaining a nightmare for batters.
we thank Francois Gallaire, Lloyd Smith and Alan Nathan for many relevant discussions. Finally, the authors acknowledge Fawaz Massouh and Ivan Dobrev for providing us the access to the wind tunnel of the ENSAM school and Laurent Keirsbulck for the access to the wind tunnel of the LME laboratory which both give us experimental confirmations of the previous scenario.
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