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  • Babe Ruth, a legend of Major League Baseball

  • He led the league in home runs during a season 12 times

  • and made a record of 60 home runs in 1927 season

  • as well as 714 home runs of his life time record

  • In the dead-ball era

  • Ruth's home run and slugging percentage dominated the league

  • He creates an unchallengeable position in the heart of baseball fans

  • and was respectfully named "Baseball Demigod"

  • Roger Maris, an outfielder of New York Yankee

  • who challenged Ruth's record after 34 years-

  • - 60 home runs in a season

  • With his home run number getting close to 60

  • however, there was no joy for fans

  • because there can be only one hero in their heart

  • Roger, up here

  • Hey it's the voice from above

  • Hey Maris up here

  • it's the Babe

  • Hey you wnat my record? you want my record?

  • Catch this you ape piece of shit

  • Stop

  • In the game 154 in 1961

  • Maris hit number 59 in the 3rd inning

  • Two outs, in the top of 9th

  • and it's the last chance for Maris

  • Orioles was behind but made an unusual substitution

  • The closer Hoyt Wilhelm was called on to stop him tying Ruth's record

  • I hate that knuckleball

  • In this weather it's gonna be dancin' all over the place

  • Even if he gets wood on it it's not goin' anywhere

  • Bushlinger

  • Wilhelm takes the sign

  • but we all know what's coming

  • Does Roger Maris have one last home rum in him

  • he fouled it back

  • Roger had a good cut on that ball, but it's dancin' all over the place

  • Wilhelm's got it

  • Out

  • The knuckleball dominated the hitter and Maris failed to tie the record on that day

  • Knuckleball is a very special pitch in baseball games

  • It may suddenly change in direction during its flight

  • and dance all over the place with erratic movement

  • so that it can fool the batter

  • In the following we will investigate knuckleball

  • base on aerodynamics

  • Furthermore, we will then introduce various pitches of baseball

  • When it comes to aerodynamics, most people are familiar with Bernoulli's principle

  • But, to establish enough background knowledge

  • we need to learn more about the fluid theories

  • besides the Bernoulli's principle

  • First let's talk about boundary layer theory

  • When a fluid flows over a obstacle, they would interact with each other

  • Let's look into a simplest case

  • When a fluid passing through a flat plate with velocity V

  • it causes friction on the surface

  • In general, fluids are viscous

  • so the flow speeds decrease gradually when it get close to the surface

  • and end up at zero

  • The region where flow speeds gradually decrease

  • is called the "Boundary layer"

  • Viscosity dominate physical phenomenon in this thin layer

  • and the Bernoulli's principle does not apply in this region

  • However, viscosity has no effect on the region outside the boundary layer

  • where the fluid behaves like a inviscid ideal fluid

  • such that the Bernoulli's principle applies

  • We will back to Bernoulli's principle soon

  • Now pay attention to the phenomenon inside the boundary layer

  • If the flow velocity V is not large

  • The airflow inside the boundary layer is stable

  • and the streamlines look like a Mille Crepe

  • This is named a "Laminar boundary layer"

  • The geometric shape of obstacles have effects on boundary layer

  • For example, when substitute the plate by a sphere

  • a new physical phenomenon may occur

  • Streamlines are drew around the sphere

  • and the green part represents a boundary layer

  • which is a important region

  • Let's study the region outside boundary layer first

  • where the Bernoulli's principle applies

  • Observe the motion of a small volume element

  • Since the speed of a baseball is much lower than sound speed

  • the compression effect can be ignore

  • and the volume of this small element keeps unchanged

  • Imagine it moves in a tube that continuously changes in diameter

  • When it moves toward A

  • its speed decreases

  • A is called the "stagnation point"

  • around which the fluid separates to both sides

  • The speed increases from A to B

  • and reach the highest value at B

  • After passing through B the speed decreases

  • and reach the minimum speed when it arrive at C

  • Bernoulli's principle states that

  • high speed corresponds to low pressure

  • and low speed corresponds to high pressure

  • Therefore, the pressure maximums occurs at A and C

  • and B has the minimum pressure

  • Interesting thing occurs in the downstream region

  • The pressure gradually increases from B to C

  • Outside the boundary layer, when the external pressure variation is large enough

  • it can force the downstream boundary layer to move in opposite direction

  • Once the inverse flow occurs

  • upstream boundary layer can be pushed away from the surface

  • This phenomenon is called "Boundary layer separation"

  • The location of "Separation point" is near B

  • and the flow is very unstable behind the separation point

  • Round and round

  • When various scales of vortices emerge, it is called "Turbulence"

  • The tail-like turbulent region behind a obstacle is also named a "Wake"

  • Since the separation points occur close to B and D

  • the wake width is comparable to the diameter of the sphere

  • As a whole, wake is a low pressure region

  • and this is the main reason for air resistance

  • However, the location of separation points

  • are sensitive to the geometric shape of obstacles

  • For a streamlined body

  • the separation occur near its tail

  • The smaller the wake width

  • the smaller the drag

  • This is the reason why airfoils and solar cars

  • are designed to be streamlined form

  • Besides the geometric shape of obstacles

  • the surface roughness also determines the location of separation points

  • Take a look at this interesting experiment

  • The left photo represents a smooth sphere in wind tunnel

  • and the separation points are close to the largest cross section of the sphere

  • The right photo shows that with a thin trip wire in the upstream

  • the separation points move backward and the wake shrinks

  • What's going on here?

  • Why a small raised surface leads to this result?

  • The reason is that the state of boundary layer has been changed

  • It is originally a laminar flow in the upstream boundary layer

  • however, since the raised surface creats small scale vortices

  • the boundary layer becomes turbulent

  • This is called a "Turbulent boundary layer"

  • Fluid flows faster outside the boundary layer because no viscous effect there

  • A turbulent boundary layer is stirred by vortices

  • which makes it mixed with the outer and faster fluid

  • Therefore, the average velocities of boundary layer increases

  • and get more momentum toward downstream

  • So that the separation points retreat and the wake shrinks

  • The purpose of dimple design on a golf ball

  • is to create a turbulent boundary layer and reduce the drag

  • Baseballs are not smooth spheres, too

  • The seam lines on the surface of a balseball can disturb boundary layer

  • In general, baseball seams can cause the retreat of separation points

  • Baseball seams are not uniformly distributed on the surface

  • and this could result in the wake deflection

  • Moreover, the rotations of a baseball are classified into two types

  • One is four-seamer: There are 4 seams passing by in one full rotation

  • The other one is two-seamer: There are 2 seams passing by in one full rotation

  • The changes in attack angle results in non-symmetric force

  • which can be measured by wind tunnel experiments

  • This figure shows the force versus attack angle

  • for a four-seamer at zero rotation speed

  • As you can see, the force has four periods of variation in one full rotation

  • The detail explanations for that figure are given in the following computer graphics

  • The air flows in steadily from the left

  • In the beginning the attack angle is zero

  • and the seams are symmetrically distributed

  • therefore, the wake is right behind the ball without being deflected

  • But the wake deflects dramatically when the attack angle changes

  • The largest upward deflection occurs when the attack angle is about 22 degree

  • What is the reason for wake deflection ?

  • The air in upstream boundary layer becomes turbulent after passing by the seams

  • Notice the seam in the lower half part is more close to the stagnation point

  • and the lower half turbulent boundary leads the upper one

  • With longer path and more chance for mixing

  • the boundary layer gets more velocity and momentum moving downward

  • so the lower half part separation point retreats more

  • On the contrary

  • the upper half part separation point is in advance and the average velocity is lower

  • After mixing of these two sides of boundary layer flow in the wake

  • the average velocity deviates upward

  • and this is the explanation for wake deflection

  • To realize how the force acting on the ball

  • one can regard the complex interaction processes as a black box

  • The horizontal air flow deflects upward after passing through the black box

  • implies that a upward force acting on the air flow

  • Simultaneously, there must be a reaction force in the black box exert on the ball

  • which is equal in magnitude and opposite in direction

  • as specified by Newton's third law

  • This is the result of non-symmetric seam distribution

  • Keep on varying the attack angle

  • the force vanishes at about 45 degree

  • beyond which the wake deflects to another side

  • and the force reachs its maximum at about 68 degree

  • Then the force vanishs again at 90 degree

  • that it has goes through a period

  • As a result, there are 4 periods in a full rotation of 360 degree

  • In the case of two-seam rotation for a knuckleball

  • there are two main periods of force variation

  • The maximum force is equivalent to 2/3 of baseball weight

  • and the period is two times larger than the case of four-seam rotation

  • Define the coordinate system first before follow-up discussions

  • Use the rectangular coordinate system

  • and let X axis pointing toward home plate

  • Y axis pointing toward first base

  • Let XY plane be parallel to the ground

  • so Z axis stands vertically

  • For this coordinate system

  • The lift force is in the positive Z direction

  • the gravity is in the negative Z direction

  • the lateral force is in the positive/negative Y direction

  • and the drag force is in the negative X direction

  • Now consider the rotation axis of a knuckleball is perpendicular to the ground

  • Observe the trajectory from top view

  • and assume the ball spins half a rotation during the flight

  • The wind tunnel experiment implies that

  • there are two periods of left and right motion for a four-seam knuckleball

  • In the case of two-seam rotation

  • the knuckleball trajectory has one period of variation

  • In practice

  • a pitcher would try to throw the ball as no spin as possible

  • The way is to push the ball with fingertip

  • but it makes the ball hard to be controlled

  • Since the spin axis may not be fixed in some specific direction

  • and since the force is sensitive to attack angle

  • the knuckleball trajectory becomes hard to be predicted

  • This is a demonstration of knuckleball's movement

  • by Tim Wakefield in a Japan TV show

  • From the research of knuckleball we learn that at extremely low spin rate

  • the seam locations or say attack angle

  • determines the force on the baseball

  • In baseball games, however

  • other kinds of pitches have spin rate over 10 times higher than knuckleball

  • and a new physical effect of force enters in-

  • - the "Magnus force"

  • To get rid of the disturbance by seams

  • let's begin with a smooth sphere

  • When it start to spin

  • The B-side surface and the ambient air move in the same direction

  • while C-side moves reversely

  • Since the fluid inside boundary layer is governed by viscosity

  • the B-side boundary layer flows faster than that of C-side toward downstream

  • and carries more momentum

  • Therefore, the B-side separation point becomes relative backward

  • so the wake deflects downward after mixing of these two sides of air

  • and the transverse force arises

  • This is called the "Magnus effect"

  • that resulted from the rotation of moving object in the fluid

  • The magnitude of Magnus force is probably proportional to

  • the air flow speed V and the angular frequency ω of the ball

  • So the empirical formula for Magnus force is given by

  • 1/2 times "Magnus coefficient CM"

  • "air density ρ"

  • "baseball cross-sectional area A"

  • "radius R"

  • "angular frequency ω" and the "flow speed V"

  • The Magnus coefficient is a dimensionless quantity

  • and is the only one parameter to be determined from experiments

  • Rotation is the motion characterize with direction

  • and can be well described by a "Vector"

  • For example, a spin vector S