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

曾經奪得12次全壘打王的頭銜

and made a record of 60 home runs in 1927 season

並且在1927年創下單季60支全壘打的紀錄

as well as 714 home runs of his life time record

在其職業生涯總共累積了714支全壘打

In the dead-ball era

在那個被稱為「死球時代」的時期

Ruth's home run and slugging percentage dominated the league

貝比．魯斯的長打能力遠遠的超越同時代的打擊者

He creates an unchallengeable position in the heart of baseball fans

於是 他在球迷的心中建立了不可挑戰的神格地位

and was respectfully named "Baseball Demigod"

被尊稱為「棒球之神」

Roger Maris, an outfielder of New York Yankee

羅傑．馬里斯 洋基隊的外野手

who challenged Ruth's record after 34 years-

在30多年後挑戰貝比．魯斯的紀錄

- 60 home runs in a season

單季60支全壘打

With his home run number getting close to 60

然而 眼看著馬里斯的全壘打數字步步逼近

however, there was no joy for fans

球迷的心中並沒有喜悅

because there can be only one hero in their heart

因為英雄只能有一個

Roger, up here

嗨 羅傑 看上面

Hey it's the voice from above

這是來自天堂的聲音 哈哈哈...

Hey Maris up here

嘿 馬里斯 看上面

it's the Babe

貝比．魯斯在此

Hey you wnat my record? you want my record?

你想破我的紀錄嗎? 想破嗎?

Catch this you ape piece of shit

垃圾來了給我接好 猿猴

Stop

暫停

In the game 154 in 1961

1961年 洋基第154場比賽

Maris hit number 59 in the 3rd inning

馬里斯在第三局擊出了第59號全壘打

Two outs, in the top of 9th

九局上半 兩出局

and it's the last chance for Maris

馬里斯最後一次打擊機會

Orioles was behind but made an unusual substitution

為了防止他追平紀錄

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

金鶯隊在落後的情況下做出不尋常的調度

I hate that knuckleball

王牌投手 霍依．威漢登板救援

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

我痛恨蝴蝶球

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

今天的風會讓蝴蝶球亂飄

Bushlinger

就算能碰到球 球也飛不到哪去

Wilhelm takes the sign

小人

but we all know what's coming

威漢收到暗號

Does Roger Maris have one last home rum in him

但大家都知道會配什麼球

he fouled it back

羅傑能從他手中擊出最後一支全壘打嗎

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

打成後方界外

Wilhelm's got it

那一球的揮擊角度很好 問題是球四處亂竄

Out

威漢接到球

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

出局

Knuckleball is a very special pitch in baseball games

受到蝴蝶球的壓制 馬里斯只能鎩羽而歸

It may suddenly change in direction during its flight

在棒球的世界中 蝴蝶球是一種極為特殊的球路

and dance all over the place with erratic movement

它能夠忽左忽右 或者突然的下墜

so that it can fool the batter

以幾乎不太規則的方式運動

In the following we will investigate knuckleball

也就是這樣能夠愚弄打擊者

base on aerodynamics

接下來 我們將從空氣動力學的角度出發

Furthermore, we will then introduce various pitches of baseball

了解蝴蝶球的原理

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

並且進一步的 介紹並分析其他各種變化球球路

But, to establish enough background knowledge

講到流體力學 一般人最熟知的就是柏努力定律

we need to learn more about the fluid theories

但是在柏努力定律之外

besides the Bernoulli's principle

還有其他重要的流體理論需要了解與認識

First let's talk about boundary layer theory

以便具備足夠的基礎知識

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

我們先來談邊界層理論

Let's look into a simplest case

當流體流過障礙物時 會與障礙物發生交互作用

When a fluid passing through a flat plate with velocity V

我們先來研究最簡單的情況

it causes friction on the surface

當流體以速度V流過一個平板

In general, fluids are viscous

流體會跟物體表面發生摩擦

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

因為流體一般都是有黏滯性的

and end up at zero

所以越往平板表面靠近 流速會越來越慢

The region where flow speeds gradually decrease

直到趨近於零

is called the "Boundary layer"

這個流速逐步遞減的範圍

Viscosity dominate physical phenomenon in this thin layer

稱之為「邊界層」

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

在這個薄層內 黏滯力主導物理現象

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

有黏滯力的區域就不適用柏努力定律

where the fluid behaves like a inviscid ideal fluid

然而 在邊界層外面

such that the Bernoulli's principle applies

黏滯力沒有發揮什麼作用

We will back to Bernoulli's principle soon

流體的行為接近於無黏滯性的理想流體

Now pay attention to the phenomenon inside the boundary layer

所以可以使用柏努力定律加以描述

If the flow velocity V is not large

我們稍後會再回來談柏努力定律

The airflow inside the boundary layer is stable

現在先讓我們來關注邊界層內的現象

and the streamlines look like a Mille Crepe

當流速V不大的情況下

This is named a "Laminar boundary layer"

邊界層內的氣流是穩定的

The geometric shape of obstacles have effects on boundary layer

流體的流線就像千層派一樣

For example, when substitute the plate by a sphere

這樣的邊界層稱之為「層流邊界層」

a new physical phenomenon may occur

障礙物的幾何形狀會影響邊界層

Streamlines are drew around the sphere

例如說從平板變成球體的時候

and the green part represents a boundary layer

會有新的物理現象發生

which is a important region

我們把球體周圍的流線畫出來

Let's study the region outside boundary layer first

綠色的部位代表邊界層

where the Bernoulli's principle applies

它是重要的區域

Observe the motion of a small volume element

我們先來探討邊界層外面的區域

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

那是可以運用柏努力定律的地方

the compression effect can be ignore

觀察一小塊體積單元的運動

and the volume of this small element keeps unchanged

因為棒球運動遠低於音速

Imagine it moves in a tube that continuously changes in diameter

不必考慮壓縮效應

When it moves toward A

這一塊小體積在運動過程中保持體積不變

its speed decreases

而且可以把它想像成是在口徑不斷變化的管子中移動

A is called the "stagnation point"

當它靠近A點時

around which the fluid separates to both sides

速度逐漸變慢

The speed increases from A to B

A點又叫做「停滯點」

and reach the highest value at B

流體會從停滯點開始往兩側分離

After passing through B the speed decreases

當它離開A點後速度逐漸增加

and reach the minimum speed when it arrive at C

到達B點時速度最高

Bernoulli's principle states that

然後通過B點之後又逐漸減速

high speed corresponds to low pressure

到達C點時流速又降到最低

and low speed corresponds to high pressure

柏努力定律是說

Therefore, the pressure maximums occurs at A and C

高流速的地方壓力小

and B has the minimum pressure

低流速的地方壓力大

Interesting thing occurs in the downstream region

所以

The pressure gradually increases from B to C

A跟C點的壓力最大

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

B點壓力最小

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

有趣的地方是發生在下游的區域

Once the inverse flow occurs

從B到C壓力是逐漸變大的

upstream boundary layer can be pushed away from the surface

邊界層在這樣的外部壓力變化下

This phenomenon is called "Boundary layer separation"

如果這個壓力差足夠大的話

The location of "Separation point" is near B

會發生邊界層往上游逆流的現象

and the flow is very unstable behind the separation point

一旦發生逆流

Round and round

上游的邊界層就會被推擠離開球體表面

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

這種現象叫做「邊界層分離」

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

「分離點」大約會發生在B點附近

Since the separation points occur close to B and D

在分離點之後的流體非常的不穩定

the wake width is comparable to the diameter of the sphere

會不停打轉

As a whole, wake is a low pressure region

然後生出各種大大小小的渦流

and this is the main reason for air resistance

這種情況稱之為「亂流」

However, the location of separation points

因為它是出現在障礙物的後方

are sensitive to the geometric shape of obstacles

所以這整個亂流區域還有個特別的名子

For a streamlined body

叫做「尾流」

the separation occur near its tail

因為分離點就發生在B跟D點附近

The smaller the wake width

所以尾流的寬度大致就和球體的寬度差不多

the smaller the drag

整體來說

This is the reason why airfoils and solar cars

尾流是一個低壓區域

are designed to be streamlined form

這是造成空氣阻力的主要原因

Besides the geometric shape of obstacles

然而分離點發生的位置

the surface roughness also determines the location of separation points

很容易因為障礙物的外形而變化

Take a look at this interesting experiment

像是流線形的物體

The left photo represents a smooth sphere in wind tunnel

分離點會發生在尾端附近

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

尾流變小了

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

阻力也就大為縮減

the separation points move backward and the wake shrinks

這就是為什麼機翼

What's going on here?

以及太陽能車要設計成流線型的原因

Why a small raised surface leads to this result?

除了障礙物的外形

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

另一個決定分離點的因素是障礙物表面的粗糙程度

It is originally a laminar flow in the upstream boundary layer

先來看一組有趣的實驗

however, since the raised surface creats small scale vortices

左邊的圖是光滑球體的實驗

the boundary layer becomes turbulent

分離點大約發生在球體最大截面積的地方

This is called a "Turbulent boundary layer"

右圖是在上游的地方加上一圈細線

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

結果分離點向後退

A turbulent boundary layer is stirred by vortices

尾流變小了

which makes it mixed with the outer and faster fluid

這是怎麼回事呢？

Therefore, the average velocities of boundary layer increases

為什麼球面上的微小凸起會產生這樣的變化？

and get more momentum toward downstream

原因就在於邊界層的狀態改變了

So that the separation points retreat and the wake shrinks

原本應該是穩定的「層流邊界層」

The purpose of dimple design on a golf ball

因為球面上的凸起部分製造出小尺度的渦流

is to create a turbulent boundary layer and reduce the drag

使得邊界層演變成亂流的狀態

Baseballs are not smooth spheres, too

這稱之為「亂流邊界層」

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

我們知道邊界層外面的流體不受黏滯力的影響

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

所以有較大的流速

Baseball seams are not uniformly distributed on the surface

亂流邊界層因為受到渦流的攪動

and this could result in the wake deflection

使得它與周圍較快的流體發生混合

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

這樣一來

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

邊界層的平均速度變快了

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

有更大的動量往下游衝擊過去

The changes in attack angle results in non-symmetric force

如此便造成分離點向後退

which can be measured by wind tunnel experiments

尾流也跟著縮小了

This figure shows the force versus attack angle

高爾夫球在表面上設計了很多的小凹洞

for a four-seamer at zero rotation speed

目的也是為了製造出亂流邊界層

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

以達到減少阻力的目的

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

然而

The air flows in steadily from the left

棒球也不是光滑球體

In the beginning the attack angle is zero

表面有約一公釐高的縫線會干擾邊界層

and the seams are symmetrically distributed

通常的情況

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

棒球的縫線也會導致分離點的位置延後

But the wake deflects dramatically when the attack angle changes

縫線在棒球表面的分布是不均勻的

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

這會使得尾流發生偏折的現象

What is the reason for wake deflection ?

然而

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

棒球的旋轉又可以分為兩種方式

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

一種是旋轉一圈的時候

and the lower half turbulent boundary leads the upper one

有四條縫線劃過

With longer path and more chance for mixing

稱之為「四縫線球」

the boundary layer gets more velocity and momentum moving downward

另一種是旋轉一圈

so the lower half part separation point retreats more

有兩條縫線劃過

On the contrary

稱之為「二縫線球」

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

攻角的變化會使棒球受到不對稱的作用力

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

這可以藉由風洞實驗加以測量

the average velocity deviates upward

這張圖是代表四縫線球

and this is the explanation for wake deflection

在轉速等於零時

To realize how the force acting on the ball

各種角度所產生的作用力

one can regard the complex interaction processes as a black box

可以看到棒球自轉一周

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

作用力重覆變化四個週期

implies that a upward force acting on the air flow

我們現在運用電腦動畫

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

更詳細的來說明這個實驗結果

which is equal in magnitude and opposite in direction

空氣固定的由左邊流進來

as specified by Newton's third law

一開始角度等於零的時候

This is the result of non-symmetric seam distribution

縫線在兩側的分布是對稱的

Keep on varying the attack angle

所以尾流沒有偏折

the force vanishes at about 45 degree

位於球的正後方

beyond which the wake deflects to another side

一旦改變球的攻角

and the force reachs its maximum at about 68 degree

尾流很快的就發生偏折

Then the force vanishs again at 90 degree

在大約22度附近向上偏折最大

that it has goes through a period

要如何解釋偏折的原因呢？

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

上游的邊界層氣流在經過縫線之後

In the case of two-seam rotation for a knuckleball

會變成亂流邊界層

there are two main periods of force variation

請注意下半邊的縫線比上半邊更接近停滯點

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

也就是下半邊的亂流邊界層提早發生

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

走過更長的路徑

Define the coordinate system first before follow-up discussions

有更多的機會發生混合

Use the rectangular coordinate system

因此邊界層得到更大的速度與動量向下游流去

and let X axis pointing toward home plate

下半邊的分離點就延後了

Y axis pointing toward first base

相反的

Let XY plane be parallel to the ground

上半邊的分離點較早

so Z axis stands vertically

流速也較慢

For this coordinate system

兩側的邊界層氣流在尾流中混合後

The lift force is in the positive Z direction

平均速度偏向上方

the gravity is in the negative Z direction

這就是尾流發生偏折的原因

the lateral force is in the positive/negative Y direction

要理解棒球所受到的作用力

and the drag force is in the negative X direction

我們可以把中間複雜的交互作用過程當成黑盒子看待

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

原本水平流進來的空氣

Observe the trajectory from top view

經過黑盒子後向上偏折

and assume the ball spins half a rotation during the flight

這表示空氣受到向上的作用力

The wind tunnel experiment implies that

從牛頓第三運動定律可以知道

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

在這個黑盒子中

In the case of two-seam rotation

一定有大小相等

the knuckleball trajectory has one period of variation

方向相反的反作用力

In practice

作用在棒球上

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

這就是縫線的不對稱分布所造成的結果

The way is to push the ball with fingertip

繼續改變攻角

but it makes the ball hard to be controlled

到達45度時作用力回復為零

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

再來尾流會向另一側偏折

and since the force is sensitive to attack angle

到68度時作用力最大

the knuckleball trajectory becomes hard to be predicted

90度時又回復為零

This is a demonstration of knuckleball's movement

這樣子就走完一個週期

by Tim Wakefield in a Japan TV show

所以轉一圈360度作用力就有四個週期的變化

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

若是改成以二縫線的方式旋轉

the seam locations or say attack angle

實驗結果發現自轉一周有兩個大的週期變化

determines the force on the baseball

作用力的最大值相當於棒球重量的2/3

In baseball games, however

週期比四縫線大一倍

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

我們先把座標系定義好以方便之後的討論

and a new physical effect of force enters in-

使用直角座標系

- the "Magnus force"

X方向指向本壘板

To get rid of the disturbance by seams

Y方向指向一壘側

let's begin with a smooth sphere

並且令XY平面與地面平行

When it start to spin

Z軸垂直於地表

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

這樣定義的座標系

while C-side moves reversely

升力剛好在正Z方向

Since the fluid inside boundary layer is governed by viscosity