THE MAGNUS EFFECT EXPLAINED
Regardless of the sport, the interaction between the spinning ball and the fluid creates a pressure difference on either side of the ball. This pressure difference creates a lift force perpendicular to the direction of its motion.
Bernoulli’s equation can be used to describe this behaviour.
Bernoulli’s equation is a principle in fluid dynamics relating pressure, velocity, and height of a moving fluid. Bernoulli stated that the total energy per unit volume of a fluid remains constant along a streamline.
In simpler terms, its total energy remains constant. This results in an inverse relationship between pressure and velocity.
When a spinning object moves through a fluid domain, it creates a flow pattern surrounding it.
On one side of the ball, where the fluid travels in the direction of the spin, its velocity increases meaning its pressure decreases according to Bernoulli’s equation.
On the opposite side of the ball, the velocity is working against the direction of the spin so reduces, meaning its pressure increases according to Bernoulli.
This pressure difference creates a lift force perpendicular to the direction of the fluid flow, and thus we have the Magnus effect.
This lift force is responsible for curving the trajectory of the balls flight as it passes through the fluid.
CFD ANALYSIS OF THE MAGNUS EFFECT IN SOLIDWORKS
Now topspin and backspin are key applications of the Magnus effect in tennis, but the simplest way to show this effect is probably with a free-kick in football.
Let’s illustrate the lift force induced by the Magnus effect with SOLIDWORKS Flow Simulation.
CFD is a numerical simulation technique used to solve the Navier-Stokes equations which describe the conservation of mass, momentum, and energy for a dynamic fluid situation, thus satisfying Bernoulli’s principle.
By hand, these equations are incredibly complex to solve, but SOLIDWORKS Flow Simulation is able to closely approximate solutions to them in relatively short timeframes.
We’ve used a representation of a size 5 football – simply a spherical object with a diameter of 220 mm.
A typical football is made up from an array of patches, sewn together. To simplify the problem, we used a uniform sphere with a wall roughness of 0.5 mm.
Let’s say a football is kicked at 70 mph (roughly 31 m/s) in the X direction.
The ball travels through the medium of air, so this was selected in flow simulation setup wizard.
The outer face of the ball is modelled as a real wall that is rotating with an angular velocity of 30 rad/s around the Z axis (4.77 rev/s).
The characteristic length (CL) of the ball (220mm) was used to determine a suitable size for the fluid domain.
Upstream and to the side of the ball, the domain should be a minimum of 1.5xCL. Downstream from the ball, the domain should be a minimum of 4xCL to properly capture the flow following the ball.
The study goal was to find the force in the Y direction, which would illustrate the lift induced by the Magnus effect and therefore indicating the ball should move in the direction of its spin.
ANALYSING THE RESULTS
After running the analysis, the result plots for pressure and velocity clearly demonstrate Bernoulli’s principle.
When the fluid is travelling with the direction of spin, the pressure decreases to its lowest, meaning velocity increases to its maximum.
At a speed of 31 m/s, an angular velocity of 30 rad/s, the lift force from the Magnus effect converged at a result of 3.075N.
EFFECT OF SPIN RATE AND BALL SPEED
Using the parametric study functionality in SOLIDWORKS Flow Simulation, the study was repeated varying the angular velocities of the ball. The force results are tabulated below.
The results trend linearly, indicating that angular velocity has a direct correlation with the pressure difference experienced on each side of the ball.
The pressure difference is then responsible for the lift force acting upon the ball; the same reason as to why aeroplanes are able to fly.
Further investigation into effects of larger angular velocities and greater ball speeds could be carried out, to determine at what Reynolds number this correlation begins to deviate.
It is anticipated, that for high ball speeds, vortex shredding could provide further problems with forces acting on the ball, and potentially would increase forces in certain directions.
IMPACT OF THE MAGNUS EFFECT ON PROFESSIONAL SPORTS
With the knowledge that spin can affect a ball’s flight, skilled professionals can use it to their advantage.
For example, in football, a free-kick taker can apply spin to bend or dip the ball around the wall, allowing the ball to be hit harder, giving it a better chance of passing the keeper.
Cricket bowlers and baseballer pitchers can use this phenomenon to make it harder for a batsman.
Tennis and ping pong players can apply top spin to dip the ball over the net.
Some famous examples of this phenomenon range from bend it like Beckham, that Roberto Carlos free-kick, Tiger Woods’ fairway bunker shot at the 2019 WGC
, and the late Shane Warnes bowling clinics.
The real-world applications of this phenomenon are endless, but ultimately manipulating a ball’s flight can give athletes a competitive advantage.
This also means sporting goods manufacturers must be aware of the phenomenon to create consistent and predictable balls.
ADVANCED SIMULATION ALTERNATIVES TO SOLIDWORKS
CFD simulations are a powerful tool that can be used to understand a range of problems.
By understanding the airflow engineers can design products with specific aerodynamic properties that will give them desired characteristics.
SOLIDWORKS Flow Simulation isn’t the only CFD software application.
If you need an even more advanced and powerful application, then SIMULIA xFlow on the 3DEXPERIENCE platform is the perfect choice.
We just had to see how beautiful our tennis ball looks experiencing the Magnus effect within xFlow.
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