Almost Voodoo--The Science of a Bending Soccer Ball (Part 2)

In my quest to comprehend the secrets of a bending ball, I stumbled upon SoccerNASA, an online soccer kick simulation program developed, oddly enough, by the folks at NASA. Apparently the intent of this program is to encourage student interest in the study of aerodynamics. Anyway it’s a wonderful thing, and further proof, if proof was wanting, that this is a great country.

As you may recall from my last post, I used Luis Suarez’s brilliant goal in the Uruguay v South Korea World Cup match as the paradigm for how to bend a soccer ball. To review...Suarez received the ball at the left side of the box, pushed the ball to his right, and struck it at the edge of the penalty box about midway between the penalty arc and the left side of the box. Professor John Eric Goff, who analyzed World Cup goals for the Wall Street Journal, estimated that Suarez’s shot left his foot at 54 mph and bent 8.7 feet from a straight trajectory before it hit the far post and rebounded into the goal.

It seems a reasonable extension of my original (nerdy) inquiry into Suarez’s goal to model it using the SoccerNASA program. The program allows you to simulate a direct free kick by varying conditions related to a ball’s flight such as starting position relative to goal, velocity, spin, spin axis, vertical angle of kick, and altitude. Technically the ball was moving when Suarez kicked it, but not so fast that it differs appreciably from a free kick (at least for my purposes here).


SoccerNASA simulation of Suarez kick--top view, no spin.
Note the ball misses the far post by about 9 ft.

 Using SoccerNASA I set it up so that without any spin Suarez’s ball would have missed the far post by about 9 feet (relatively easy to do as this is half the depth of the goalie box). The program allowed me to select the actual Uruguay v S. Korea match location of Port Elizabeth, South Africa (197 feet above sea level). I set the spin axis at -10° from vertical, assuming that the natural motion of a kick imparts a small amount of top spin (0° from vertical is pure side spin). Then I ran the simulator with different vertical angles and spin (rpm) until I found a combination where the ball just curled into the goal at the far post, i.e., it “looked” like Suarez’s goal.

SoccerNASA simulation of Suarez kick--top view, 310 rpm spin



SoccerNASA simulation of Suarez kick--field view, 310 rpm spin

 The combination of spin and vertical angle that best recreates Suarez’s goal is 18.5° and 310 rpm. The ball never exceeds the height of the goal (8 ft) and strikes the far post about 4 feet above the ground. From the moment the ball was kicked the goalkeeper had 1.2 seconds to react.

Almost all models based on theory involve some simplifying assumptions. SoccerNASA assumes that a kicked soccer ball typically travels fast enough that the boundary layer of air around it falls in the turbulent flow regime, i.e., faster than about 30 mph. Previously I suggested (as have many others) that the transition from a turbulent to a laminar (smooth) boundary layer accounts for some of the eccentric movement of balls in flight. For Suarez’s goal the original assumption appears reasonable as drag only slows the ball from 54 to 39 mph over the 24 yard flight.

Two empirical constants, the drag coefficient, Cd, and the lift coefficient, Cl, account for the real world behavior of a soccer ball as opposed to an ideal smooth sphere in flight. Wind tunnel tests demonstrate that the drag coefficient for a rough sphere (a sports ball) in laminar boundary layer flow is about 0.5. As air speed changes the boundary layer flow to turbulent, the drag coefficient drops to 0.25. The lift coefficient for a soccer ball is also about 0.25 and to my knowledge doesn’t change with flow regime.

Data table--SoccerNASA simulation of Suarez kick

In the SoccerNASA program the change in a ball’s flight behavior could be accomplished by changing the drag coefficient, Cd, from 0.25 to 0.5 at the appropriate point in a ball’s flight, but this potentially gets complicated as different makes of soccer balls may transition between laminar and turbulent flow at different speeds. For instance, the new Jabulani ball used at the 2010 World Cup likely flies differently than the traditional 32-panel stitched ball.

The drag and lift forces are also dependent on the ball’s velocity—lift is a function of velocity, drag is a function of velocity squared. As a consequence the lift and drag forces on the spinning ball are changing throughout its flight at different rates (see table above). No wonder a goalkeeper may have trouble following a well-struck bending ball.

Suarez scored his goal in Port Elizabeth, almost at sea level. World Cup games were also played in Johannesburg at 5,558 ft above sea level (higher than Denver). If Suarez strikes the ball identically in Johannesburg, it generates less lift and less drag. His shot goes wide. The match goes into overtime.

Almost Voodoo--the Science of Bending a Soccer Ball (Part 1)

In the 80th minute of the Round of 16 match between South Korea and Uruguay, Uruguayan striker Luis Suarez received a headed ball at the left side of the box, moved quickly to his right, and fired a curling right-footed shot that deflected off the far post into the goal. It proved the game winner.

Suarez's goal: the ball hits far post, Suarez ends up outside box at right 

Suarez’s shot was a remarkable demonstration of soccer technique, made the more impressive by the rainy conditions. As analyzed by Professor John Eric Goff of Lynchburg College, the ball left Suarez’s foot traveling 54 mph and bent 8.7 feet on its 24 yard flight to the far post.

So how did Luis Suarez manage this feat with his foot (sorry, couldn’t resist)? A spinning spherical object, that is a ball, moving through a compressible fluid, that is air, makes a fascinating subject for scientific inquiry. To this end many engineering types have studied baseballs, golf balls, tennis balls, and soccer balls in flight.

Presumably Suarez struck the ball off center with the instep of his right foot causing the ball to rise with a counterclockwise rotation. How hard to strike the ball and how far off center to obtain the desired swerve from a straight-line flight we can reasonably assume was the result of years of practice.

As a soccer ball moves through air it is subject to two forces, drag due to air viscosity and lift created by the ball’s spin (also called the Magnus Effect). Viscosity is a measure of a fluid’s resistance to flow, what we might characterize as a fluid’s “stickiness.”

Let’s look at drag first and momentarily ignore the rotation of the ball. Drag results from the friction between the ball and the air flowing around it. The nature of the drag force changes with the speed of the soccer ball through air as shown below.

Flow regimes at varying air speeds over a sphere

At very low speed the layer of air next to the ball’s surface, called the boundary layer, flows smoothly around the ball. This is laminar flow (see A above). In this flow regime the only drag force is the pull of the air stream on the ball’s surface.

As speed increases to something more typical of a ball in flight, say 20 mph, the boundary layer separates from the ball at about 90° from the direction of flight and produces a wide turbulent wake (see D above). Turbulence is chaotic fluid flow characterized by the formation of eddies. The wake is an area of low pressure behind the ball. It creates a large pressure differential between the upstream and downstream sides of the ball which opposes the flight of the ball. This is often called pressure drag.

At even higher speed through the air, say 35 mph, the air within the boundary layer becomes turbulent (see E above). Without going into the details here, a turbulent boundary layer doesn’t separate as early from the ball’s surface, thus creating a smaller wake and correspondingly smaller area of low pressure behind the ball. In other words at the moment the boundary layer changes from laminar to turbulent, the pressure drag drops dramatically.

This phenomenon is counterintuitive as we normally associate laminar (or smooth) flow around an object with low drag. In fact one of the factors that promotes the transition to turbulent flow is surface roughness. This is why the modern golf ball is manufactured with dimples on its surface, i.e., a dimpled golf ball will travel further than a smooth golf ball. The trip wire in the wind tunnel test shown below acts like dimples on a golf ball. In the case of a soccer ball, the stitches of the standard 32-panel soccer ball create a moderately rough surface.

Wind tunnel experiments: a) large wake from laminar boundary layer,
b) smaller wake from a turbulent boundary layer induced by trip wire

Now let’s consider the rotation of the ball. As the ball moves through air, its spinning surface tugs at the boundary layer. This creates an asymmetric condition around the ball as the spin force impedes air flow along one side of the ball and assists air flow on the other side of the ball.

This simple model of air flow around a spinning ball does
not account for boundary layer separation and wake formation

The often-cited explanation for why a soccer ball deflects in flight references the Bernoulli Principle, according to which the faster moving air on one side of the ball creates lower pressure than the slower moving air on the opposite side, the consequence of which is a net force, called lift, in the direction of the faster moving air. Essentially it is the same lift force produced when an airfoil moves through air.

A more likely explanation of Magnus Effect as a consequence
of asymmetric boundary layer separation

At best this is a partial explanation for the Magnus Effect, as it doesn’t account for boundary layer separation. At the point where the boundary layer separates from the ball the Bernoulli lift disappears. An explanation that better conforms to wind tunnel experiment is that the spin changes the points of boundary layer separation on the ball’s surface, moving it downstream on the side that assists the air flow and upstream on the side that opposes air flow. This has the effect of turning the airstream towards the side of the spinning ball that opposes the air flow. The momentum change of the airstream must be balanced by an equivalent momentum change in the ball (per Newton’s 3rd law) which is accomplished by the ball moving sideways in the direction of the side assisting the airflow. This action is analogous to turning a ship’s rudder. The rudder redirects the flow of water behind the ship and pushes the boat’s stern in the opposite direction.

Back to Suarez’s shot…as the ball left his foot at the 54 mph it was likely spinning counterclockwise at about 600 rpm (at this point a SWAG). This caused the ball to curve to Suarez’s left due to Magnus Effect forces described above. We presume at some point in flight the boundary layer transitioned from turbulent to laminar flow causing the pressure drag to increase about 150% (like someone putting on brakes). The ball’s spin likely doesn’t dissipate as rapidly as its forward speed with the result that the bend in the ball’s flight appears most pronounced as it nears the goal face. It is also possible that the Magnus Effect is stronger in the laminar boundary layer flow regime (although I didn’t read anything suggesting this is the case).

To complicate matters it is possible that the change in boundary layer flow from turbulent to laminar may not happen simultaneously on both sides of the ball due to the spin force. This creates a situation where the relative position of flow separation and the resultant airstream behind the ball may suddenly flip-flop, causing the ball to momentarily move in the opposite direction, i.e., appear to wobble in flight.

Spin--it’s all in the game.

What you do when you score a game winning World Cup goal