We consider the problem of computing the soccer ball trajectory, that is, given the initial velocity, the initial angular velocity, the wind velocity, and the soccer ball geometry, we compute the soccer ball positions at different times during the ball flight. For the solution of this problem, the classical model of projectile motion considering only the gravity force has been equipped with fluid dynamics forces including eventual wind effects. Understanding these forces is essential for improving both gameplay and equipment design. The starting point of the proposed method is a deep revision of fluid dynamics results to provide an effective computational procedure for the evaluation of the drag force and the lift force on the soccer ball. The resulting computational framework is a quite flexible tool that allows the easy management of the main conditions affecting the ball trajectory. So, it can support the training activity and the ball design. Some numerical experiments show the efficacy of this computational tool.

1. Introduction

Fluid dynamics offers a concrete scientific framework for understanding the behavior of fluids in motion and their interaction with solid objects; the corresponding results directly influence the design, optimization, and innovation processes within numerous fields, demonstrating the profound importance of fluid dynamics from an application-oriented perspective. For example, in aerospace and aeronautical engineering, fluid dynamics is pivotal to reduce drag in aircraft and to improve fuel efficiency and lower emissions. Maritime engineering also relies heavily on fluid dynamics to design ships and underwater vehicles that navigate the oceans more efficiently, reducing fuel consumption and minimizing their environmental impact [1]. In the energy sector, fluid dynamics is integral to the design and operation of wind turbines generators [2] or to design geothermal exchangers [3, 4]. Similarly, in hydroelectric power generation, it is used to optimize the flow through turbines [5]. The principles of fluid dynamics are also fundamental elements in environmental sciences [6, 7], where they are used to model groundwater flow, weather patterns, climate change, and the spread of pollutants in air and water bodies. Research on sailboats and fluid dynamics has led to fascinating insights, such as outfitting sailboats with sensors to capture wind measurements, and advanced techniques to design sail shapes [8], to mention just a few of them.

In the context of sports science, fluid dynamics enables the analysis of how air and water resistance affect the performance of athletes and the equipment they use [9]. In particular, for sports balls [10] and for soccer balls [11–13], as in our case study, the interaction of the ball with the fluid is a fundamental study that helps to show how variations in ball design influence its trajectory, speed, and stability, contributing to the development of sports equipment that enhances performance and fairness in competition.

The analysis of the motion of objects through fluids is a leitmotif in several interesting fluid dynamics applications, where two principal aerodynamic forces come into play: drag, usually denoted with D, and lift, denoted with L. So, the modeling of the fluid-structure interaction plays a key role in these applications and allows the characterization of different numerical methods [14–16] to be employed for these problems.

More specifically, drag force D or fluid resistance acts in opposition to the direction of the velocity of the object. The magnitude of the drag force D depends mainly on the speed and shape of the object, as well as the density and viscosity of the surrounding fluid. Conversely, the lift force L operates perpendicular to the direction of motion, potentially causing deviations from the object trajectory. It arises from the differential pressure created on opposing sides of the object boundary when it traverses the fluid. The object shape and also its rotation play a crucial role in determining the lift force magnitude and direction [17].

The concrete evaluation of the drag force D and of the lift force L cannot be done by explicit analytical procedures, so they are usually estimated by two alternative approaches: laboratory experiments with a wind tunnel or numerical simulations of a wind tunnel by a computational fluid dynamic tool. The results obtained for D and L have to be considered in the dynamical system for the motion of the object. This general approach has been already used in several models for the trajectory of sports balls, with particular attention devoted to the evaluation of the drag coefficient C_D and the lift coefficient C_L [18–21], and a special attention to the effect of wind [22]. An interesting review on different ball sports is also provided in [23]. Our contribution is in line with such studies and is characterized by a simple approach to compute the values of the drag coefficient C_D and the lift coefficient C_L through fluid dynamic simulations and then to dynamically use them in the numerical solution of the dynamical model for the soccer ball trajectory.

In particular, we present a complete computational framework to evaluate the trajectory of the soccer ball from the knowledge of its initial velocity, its geometrical features, and the wind velocity. This work diverges from conventional approaches that usually rely on the analyses of high-speed cameras, offering a comprehensive computational alternative for investigating fluid dynamic effects. The proposed procedure dynamically evaluates drag and lift forces on the ball, trying to move beyond the limitations of observational techniques. The capacity of this method to account for a wide range of velocities and spin rates not only broadens its theoretical appeal but also extends its applicability to practical applications in sports training and equipment design.

The article is organized as follows. In the next section, the mathematical model for the evaluation of the soccer ball trajectory is presented together with the corresponding approximation procedures at the base of the presented numerical simulations. In Section 3, the results for the drag and lift coefficients are reported and some examples of soccer ball trajectories are shown. Then, the results obtained are discussed along with the description of some aspects of the model that can be improved. Finally, in Section 4, some conclusions and future perspectives of this study are provided.

2. Methods

We start by introducing the main notations used in the paper. We consider a soccer ball moving in the air. The velocity of the soccer ball is denoted with , its angular velocity is denoted with ; the soccer ball has mass m and radius R, so its transverse area is A = πR². The expressions of the forces exerted on the soccer ball are based on the gravity constant g, the air density ρ, and the air viscosity μ. Moreover, we also need two further fluid dynamic concepts: the boundary layer δ and the Reynolds number Re. The boundary layer is a thin region of fluid that surrounds a solid object as the fluid flows past it. This concept was firstly introduced by Prandtl, but then, additional efforts have been made from different points of view such as experimental observation and large-scale computation. A review on these different approaches can be found in [24]. In our case study, the approach followed is the one provided in [9, 19] The Reynolds number is a dimensionless parameter that characterizes the flow regime of a fluid; in [17], a discussion on turbulence for a smooth sphere is provided, with transition to turbulent regime typically around Re ≈ 2 · 10⁵. In our case study, as Re ∈ [7.5 · 10⁴, 7.5 · 10⁵], the flow is predominantly turbulent in the region of interest. These parameters are reported in Table 1 for the reader convenience.

Table 1. Physical parameters used in the proposed method and the relative range of values.

Short description	Symbol	Reference value
Acceleration of gravity	g	9.81 m/s²
Mass of the soccer ball	m	0.43 kg
Radius of the soccer ball	R	0.11 m
Air density	ρ	1.225kg/m³
Air viscosity	μ	1.79 · 10⁻⁵kg/ms
Width of the boundary layer	δ	10⁻⁵ m
Reynolds number	Re	In the order of 10⁵

We initially consider the soccer ball trajectory obtained by the usual model for the projectile motion, and then, we refine this model by taking into account the influence of fluid dynamic effects including an eventual wind velocity field. A projectile is an object launched into the air and subject only to the force of gravity. From standard arguments in dynamics, we have that the projectile velocity u satisfies the following equation:

()

where the resulting force

is the gravity force having direction

, that is, the negative z-axis which is the vertical Cartesian axis, m is soccer ball mass, and g is the gravity constant. We can easily show that, from Equation (1), a parabolic trajectory is obtained. We want to generalize this simple model by introducing the fluid dynamics effects in the soccer ball including the influence of wind. In particular, the drag force D is obtained by using the drag force equation [25], that is

()

where C_D is the drag coefficient, u is the soccer ball velocity, ρ is the air density, A = πR² is the area of the transverse section of the ball, and R is the radius of the soccer ball.

The lift force is expressed by using the lift force equation proposed [23], that is,

()

where C_L is the lift coefficient and ω is the angular velocity of the soccer ball. Thus, the right-hand side F_res of Equation (1) is now given by the sum of 3 contributions:

•
Gravitational force: .
•
Drag force, expressed by (2).
•
Lift force, expressed by (3).

In addition, we have to take into account the presence of wind, whose velocity vector u_w is assumed to be known and constant during the ball flight. The presence of wind has the effect to modify both the drag and lift forces as proposed in [22]; so, in Formulas (2) and (3), the vector u is replaced with the vector u + u_w. Therefore, we obtain the following model:

()

where T > 0 is the flight time of the soccer ball and the vector u₀ is the initial condition that depends on the ball shoot.

From Formula (4), we can see that also the time-evolution of the angular velocity ω is required, so the balance equation for the angular momentum has to be considered:

()

where I = 2/3mR² is the moment of inertia of the soccer ball and T_res is the resulting torque. In particular, T_res can be described by the following relation:

()

where δ is the boundary layer; see [19] for a detailed discussion. So, we obtain the following equation for ω:

()

where similarly to the model for the linear momentum, T > 0is the flight time of the soccer ball and the vector ω₀ is the initial angular velocity that depends on the ball shoot.

The system of Equations (4), (5), (8), and (9) gives an initial value problem for the velocity u and the angular velocity ω. We note that, supposing m, R, μ, and δ constants with respect to time, Equations (8) for ω, with the corresponding initial Condition (9), can be solved explicitly, obtaining in this way the following time evolution of the angular velocity:

()

So, the numerical solution of the Equation (4) provides an approximation of the velocity field u. From the knowledge of the initial position x(0), a further time-integration can be computed for the expected trajectory of the soccer ball.

The time-integration of (4) can be easily done by numerical methods for initial value problems [26] when the drag coefficient C_D and the lift coefficient C_L are provided together with the value of the wind velocity vector u_w. To this aim, we used the predictor–corrector method of Dormand and Prince, which is based on explicit Runge–Kutta formulas of fourth and fifth order; see [27] for details.

The coefficients C_D and C_L are actually functions of u = ‖u‖ and C_L also of ω = ‖ω‖, and they are computed by fluid dynamics simulations, as described in the remaining part of this section. The soccer ball is defined by the 3D model obtained from [28], where we have prescribed the parameters roughness height, K_s, and roughness constant C_s, following the ideas presented in [29]. Additionally, since the Reynolds number Re > 10⁴, we consider the fluid flow in the turbulent regime. So, the standard k-ε model [30] is considered as turbulence model. These fluid dynamics simulations have considered different values of the velocity u = ‖u‖ for the computation of the drag coefficient C_D and different values of u = ‖u‖ and ω = ‖ω‖ for the computation of the lift coefficient C_L. The computation of such coefficients is based on the drag force Equation (2) and the lift force Equation (3), through a direct numerical evaluation of the drag force D and the lift force L, by using the air flow pressure computed on the surface of the ball. The numerical simulations are done with fixed direction of both u and ω since it does not affect the value of the drag coefficient C_D and the lift coefficient C_L. A more detailed discussion on the computation of the drag and lift coefficients is provided in the appendix.

Let u_n, n = 0, 1, ⋯, N, ω_m, m = 0, 1, ⋯, M be the discrete values for the velocity and the angular velocity, respectively, used in the computation of C_D and C_L; let C_D,n, C_L,n,m, n = 0, 1, ⋯, N, m = 0, 1, ⋯, M be the corresponding values obtained by fluid dynamic simulations. From these grid values, a piecewise-linear interpolation is used to obtain functions C_D(u), u ∈ [u₀, u_N], C_L(u, ω), u ∈ [u₀, u_N], and ω ∈ [ω₀, ω_M], where u₀ and u_N and ω₀ and ω_M are the minimum and the maximum expected values for the velocity and the angular velocity, respectively. These functions allow a simple and fast evaluation of C_D and C_L in the solution procedure of Problems (4) and (5).

Thus, once the look-up tables for C_D and C_L are done and the corresponding interpolation functions have been set, the integration of (4) and (5) can be used to obtain an efficient computational tool, where given the input vectors:

•
The initial position x₀;
•
The initial velocity u₀;
•
The initial spin ω₀;
•
The vector for the constant wind u_w.

The time evolution of the soccer ball’s position, velocity, and spin is computed.

3. Results and Discussion

We present the results obtained with the method proposed in the previous section, focusing on the evaluation of drag and lift coefficients and the analysis of ball trajectory under various conditions.

The implementation of the k-ε model for the fluid dynamic simulations is the one provided by Ansys Fluent [31], which is a widely used computational fluid dynamics software specialized in fluid flow and related phenomena.

Figure 1 shows the values computed for the drag coefficient C_D with varying Reynolds numbers corresponding to the values of u ∈ (5, 50) m/s. This result is similar to the one proposed in [32], where the drag coefficient is also quite stable even if slightly higher than the one obtained in our simulations. Figure 2 shows the lift coefficient C_L for the fixed Reynolds number values of Re = 1.66 · 10⁵, corresponding to u = 10m/s; Re = 3.33 · 10⁵ corresponds to u = 20m/s, and Re = 6.65 · 10⁵ corresponds to u = 45m/s and ω = 10^°, 20^°, ⋯, 100^°/s. The results obtained are slightly lower but still similar to the ones reported in [18].

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

The values of the drag coefficients C_D computed by fluid dynamic simulations corresponding to u = 5, 10, ⋯, 50, values expressed in meters per second.

As already mentioned in the previous section, these results of fluid dynamic simulations are used in the solution of Problems (4) and (5). For the integration method of this problem, we have used the Runge–Kutta–Fehlberg method [33], implemented in the ode45 function of the MatLab software [34] with absolute error tolerance equal to 10⁻⁶ and relative error tolerance equal to 10⁻³. Finally, the trajectory of the soccer ball is obtained through a numerical integration of the velocity. The results show different cases, and each one is presented with the following plots:

•
A three-dimensional plot to show the overall progress of the trajectory of the soccer ball;
•
A side-view plot that highlights the behavior of topspin and backspin shots as well as the effects of tailwind and headwind.
•
A view from above plot to highlight the effects of side-spin and lateral wind.

An overview of the experiment considered for the soccer ball trajectory is given in Table 2. Figure 3 shows the effects of frontal wind for a long throw with no spin and initial velocity vector ‖u₀‖ = 26m/s. The three curves describe the cases of no wind, tailwind, and headwind; more precisely, the three corresponding wind velocity vectors are as follows.

Table 2. Summary of fluid dynamic effects considered in the experiment.

Case	Wind	Lift force	Drag force
1	No/tailwind/headwind	No	Yes
2	No	Yes/no	Yes
3	No	No	Yes/no
4	No	Yes/no	Yes
5	No/lateral wind	No	Yes

We can notice that this wind of moderate intensity, that is, 5 m/s, affects distinctively the distance traveled by the soccer ball of around 5–6 m. The three trajectories coincide in the view from above plot since no side spin and no lateral wind effects are applied. Figure 4 describes a similar type of shot; the initial velocity vector u₀ is the same of the previous example; however, in this case, we suppose the absence of wind; therefore, u_w = (0, 0, 0), and the ball has either no spin, backspin, that is, ω = (0, −80, 0), or topspin, that is, ω = (0,80,0). We can notice the impact of the lift force in the trajectory with a difference in the distance traveled by the ball of around 1.5 m with respect to the case of no spin. Also, in this case, the three trajectories in the view from above plot coincide since we are working with an idealized situation of perfect spin. Figure 5 shows the same type of shot of Figure 3. The initial velocity is unchanged; however, in this case, we deactivate the drag force on the ball’s trajectory to highlight the effect of this force. Additionally, we suppose

Figure 6 shows a different situation. In this case, we describe a direct shot towards the goal, and we want to highlight how the spin applied to the soccer ball is of fundamental importance in determining whether one succeeds in hitting the goal or not. The shot starts from a distance of 20 m to the center of the goal, and the three cases presented are the shots with the following side spin applied:

In all cases, the initial velocity vector is ‖u₀‖ ≈ 19m/s, and we suppose the absence of wind.

Finally, Figure 7 shows how even a mild lateral wind can affect the trajectory of the ball in a decisive way if we aim a relatively small target such as the goal. In this scenario, the shot is aimed from the same position as the one in the previous case with the same initial velocity but with no spin. More precisely, in this case, the following wind velocities are considered:

We can see that the shot with no lateral wind was wrongfully aimed outside of the top corner of the goal. The lateral wind deflects the trajectory of the ball bringing it in one case even more outside of the goal and in the other case inside the goal, making the shot dangerous.

The simulations required roughly 14 min for the evaluation of each study case for the drag coefficient C_D and the lift coefficient C_L, whereas the evaluation of the soccer ball trajectory required roughly 45 s. Regarding the computational time, the numerical simulations were performed on a computer equipped with operating system Windows 10, processor, Intel Core i7-7500U 2.7 GHz, with 12 GB DDR4 RAM.

The results described so far have yielded significant insights into the aerodynamics of the soccer ball, specifically focusing on the drag and lift coefficients, as well as the ball trajectory under a variety of conditions. In particular, the results for the drag and lift coefficients are in good agreement with previous research such as [18, 32]. Moreover, the results for the ball trajectory seem to realistically capture the effects of wind and fluid dynamic forces on the ball trajectory. However, our approach needs further investigations, and these promising results need other validation steps.

From a methodological standpoint, the robustness of the proposed method has to be analyzed with respect to different computational options, such as turbulence models and wall conditions, in the evaluation of the computed drag and lift coefficients. Also, different integration methods should be compared to analyze their effects on the simulated ball trajectories.

From the validation standpoint, the main limitation is the experimental range’s breadth; the conditions tested, though varied, do not encompass the full spectrum of real-world scenarios that could impact the drag and lift forces acting on the soccer ball trajectory. This gap could lead to underestimate environmental factors such as wind variability, altitude, and humidity with the consequent limitation in the model applicability. We note that the results obtained in this paper are completely based on a numerical simulation approach. This allows an accessible evaluation tool for the soccer ball trajectory so this is the strength of the proposed method; however, the final validation of such results needs the comparison with experimental measurements using high-speed video footage in wind tunnel and in soccer fields.

4. Conclusion

This paper presented a model that accurately approximates drag and lift coefficients for a soccer ball by fluid dynamics simulations to a better understanding of the soccer ball dynamics. More precisely, by incorporating these coefficients into the differential equations governing ball trajectory, we have obtained accurate predictions of its flight path that provide insights for enhancing athletic performance. This dual contribution of refining theoretical models and offering practical applications has potential implications for training, equipment design, and the broader field of sports science.

An interesting application of the proposed methodology is the possibility to provide a detailed examination of how the ball’s surface properties affect its trajectory. It is worth noting that for every major tournament, a new official soccer ball is introduced with varied surface properties to improve the fluid dynamics of the ball. However, this is not a trivial evaluation, and, in the past, there were cases where the new surface had terrible fluid dynamics performances, like the ball known as Jabulani used in the 2010 World Cup [35]. Therefore, the evaluation of the dynamic interplay between ball design and trajectory outcomes could contribute valuable insights.

The proposed method after the aforementioned validation phases can provide a valuable tool in the design of soccer balls and for athletes training. The improvement by the integration of artificial intelligence and machine learning models is another interesting approach to enhance the predicting capability of the method under a vast array of conditions not feasible for direct experimentation, such as high variable wind conditions and rain effects. These predictive models could facilitate the identification of optimal conditions for performance. Through these efforts, the research could tap into innovative methodologies, such as virtual reality for simulating environments and interactive models for athlete training.

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

All the authors contributed equally for this paper, and listing order is alphabetical.

Funding

No funding was received for conducting this study.

Acknowledgments

We would like to thank Ansys to make freely available its products during the thesis work of Riccardo Piombin for an easy evaluation of drag and lift coefficients through fluid dynamics simulations. We would like to thank Creaate to make freely available the 3D model [28] for the soccer ball. Pierluigi Maponi is a member of Gruppo Nazionale Calcolo Scientifico-Istituto Nazionale di Alta Matematica (GNCS-INdAM).

Appendix A: The Computation of Drag and Lift Coefficients

The computation of the drag coefficient C_D and the lift coefficient C_L is based on the incompressible Navier–Stokes equations.

()

where Ω ∈ ℝ³ is the domain of study, W ⊂ Ω is the soccer ball, t is the time variable, ρ is the density of the fluid, u is the velocity vector, p is the pressure field, μ is the fluid viscosity, and f is the body force vector per unit mass acting on an infinitesimal fluid element; in the following, f = 0 is considered. For the boundary conditions, we prescribe the unperturbed flow conditions at ∂Ω, that is,

()

with u^∗ being a given constant. Additionally, we impose a specific behavior of the fluid on the soccer ball surface ∂W called no-slip condition, that is,

()

In the numerical simulations, Problems (A.1)–(A.4) are equipped with an initial condition u(x, 0) = u^∗, where u^∗ ranges from 5 to 50m/s (see Tables A1 and A2), and this is the same vector appearing in (A.3). So, the Reynolds number

()

is such that Re > 10⁴ since, as reported in Table 1, ρ = 1.225kg/m³, μ = 1.79 · 10⁻⁵kg/ms and R = 0.11 m, indicating that the fluid flow is in the turbulent regime. To account for turbulence, the RANS (Reynolds-Averaged Navier–Stokes) model and the standard k-ε model have been used; see [30, 36, 37] for a detailed discussion of the RANS model and the standard k-ε model. Moreover, the boundary Condition (A.4) gives steep velocity gradients nearby the soccer ball surface, and in order to reduce the numerical instabilities of this boundary layer, we use the wall function approach [38, 39], which allows the prescription of the two parameters, that is, the roughness height K_S and the roughness constant C_S to take into account the physical properties of real obstacles.

The fluid dynamic simulation gives an approximation , of the velocity field and the pressure, respectively; in particular, the approximated pressure is integrated over the soccer ball surface to compute the module of both the drag force D and the lift force L. Therefore, from the drag force Equation (2), it is possible to evaluate the drag coefficient C_D, and from the lift force Equation (3), the lift coefficient C_L is obtained.

In the following, the complete tables for the drag and lift coefficients computed as explained above. We notice that the results obtained are in a good agreement with the ones reported in [18, 32].

Table A1. Drag coefficient values C_D computed for the soccer ball with fluid dynamic simulation.

u (m/s)	5	10	15	20	25	30	35	40	45	50
C_D	0.1992	0.1969	0.1949	0.1938	0.1930	0.1924	0.1921	0.1917	0.1915	0.1913

Table A2. Lift coefficient values C_L computed for the soccer ball with fluid dynamic simulation.

ω\u	5	10	15	20	25	30	35	40	45	50
10	0.100	0.045	0.028	0.025	0.018	0.012	0.010	0.008	0.006	0.004
20	0.148	0.092	0.070	0.053	0.042	0.034	0.028	0.023	0.020	0.017
30	0.204	0.127	0.098	0.078	0.063	0.053	0.045	0.040	0.034	0.030
40	0.258	0.153	0.120	0.095	0.081	0.069	0.060	0.051	0.047	0.042
50	0.310	0.181	0.136	0.110	0.092	0.088	0.071	0.064	0.060	0.053
60	0.340	0.209	0.153	0.124	0.106	0.095	0.084	0.074	0.068	0.063
70	0.370	0.235	0.170	0.140	0.120	0.105	0.092	0.084	0.078	0.073
80	0.395	0.260	0.190	0.178	0.132	0.115	0.102	0.094	0.087	0.081
90	0.400	0.283	0.216	0.170	0.143	0.126	0.113	0.103	0.095	0.089
100	0.410	0.300	0.234	0.186	0.153	0.135	0.122	0.111	0.102	0.095

Open Research

Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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A Computational Framework to Predict the Soccer Ball Trajectory

Abstract

1. Introduction

2. Methods

3. Results and Discussion

4. Conclusion

Conflicts of Interest

Author Contributions

Funding

Acknowledgments

Appendix A: The Computation of Drag and Lift Coefficients

Open Research

Data Availability Statement

References

Figures

References

Information

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A Computational Framework to Predict the Soccer Ball Trajectory

Abstract

1. Introduction

2. Methods

3. Results and Discussion

4. Conclusion

Conflicts of Interest

Author Contributions

Funding

Acknowledgments

Appendix A: The Computation of Drag and Lift Coefficients

Open Research

Data Availability Statement

References

Figures

References

Related

Information