Class 4 - Vector Calculus v1

7/27/2019 Class 4 - Vector Calculus v1

1/41

AOE 5104 Class 4 9/4/08

Online presentations for todays class: Vector Algebra and Calculus 2 and 3

Vector Algebra and Calculus Crib

Homework 1

Homework 2 due 9/11

Study group assignments have been made andare online.

Recitations will be Mondays @ 5:30pm (with Nathan Alexander) in

Randolph 221

Tuesdays @ 5pm (with Chris Rock) in Whitemore 349


2/41

I have added the slides without numbers. The numbered slides are the

original file.


3/41

Last Class

Changes in Unit

Vectors

Calculus w.r.t. time

Integral calculus w.r.t.space

Today: differential

calculus in 3D

P

P'

er

e

ez

d

r

z

rdd ee ee dd r

0zde

dtdtdt

ttt

ttt

ttt

BABA

BABABA

BAB

ABA

BABA

.


4/41

Oliver Heaviside

1850-1925


5/41

Shock in a CD Nozzle

Bourgoing & Benay (2005), ONERA, France Schlieren visualizationSensitive to in-plane index of ref. gradient


6/41

In 1-D

0S

1grad lim dS

n

In 3-D

Differential Calculus w.r.t. Space

Definitions ofdiv, gradand curl

0S

1div lim . dS

D D n

0

S

1curl lim dS

D D n

Elemental volumewith surface S

n

dS

D=D(r), = (r) x 0

1lim ( ) ( )

dff x x f x

dx x


7/41

ndS

(large)

Gradient

0S

1grad lim dS

n

dS

n

Elemental volume

with surface S

ndS

(small)

ndS

(medium)

ndS

(medium)

= magnitude and direction of the slope in the scalar field at a point

ResultingndS


8/41

Review

S

0 dS

1Limgrad n

Gradient

Magnitude and direction

of the slope in the scalar

field at a point


9/41

Gradient

Component of gradient is the partial derivative

in the direction of that component

Fouriers Law of Heat Conduction

ss

.e

n.Tkn

Tkq

ss e,


10/41

zyxdxdydz

zdxdydz

ydxdydz

x

kjikji

n

1Lim

dS

1Limgrad

0

S

0

Face 2

Differential form of the Gradient

S0

dS

1Limgrad n

Cartesian system

dy

dx

dz

j

ik

P

Evaluate integral by expanding the variation in

about a point Pat the center of an elementalCartesian volume. Consider the twoxfaces:

= (x,y,z)

Face 1

dydzdx

xFace

)(2

dS1

in

dydzdxx

Face

)(2

dS2

in

adding these gives dxdydzx

i

Proceeding in the same way foryand z

dxdydzy

j dxdydzz

kandwe get , so


11/41

Differential Forms of the Gradient

These differential forms define the vector operator

sinsin r

e

+r

e

+rer

e

+r

e

+re

ze+

r

e+

re

ze+

r

e+

re

z

k+

y

j+

x

i

z

k+

y

j+

x

i

=grad

rr

zrzr

Cartesian

Cylindrical

Spherical


12/41

, :

rad where

Here rad maps a vector, , into another vector, ,and plays the role of a derivative for the vectorfield

Gradient of a vector V V i j k

V r r V r V V r r i j k

V r V

V i j

G

G

u v w

d d d d dx dy dz

d d

d du dv

k i j k

V

u u u v v v w w wdw dx dy dz dx dy dz dx dy dz

x y y x y z x y z

u u u

x y z

du dxv v vd dv d

x y zdw

w w w

x y z

rad

Some people prefer to use "dyadic" notation, to call rad a dyad, and to write it as follows:

rad

V

V

V i i i j i k

G

G

G

u u u

x y z

v v vy

x y zdz

w w w

x y z

u u u

x y z

where denotes the so-called dyadic product, which does not have a geometric interpretation. Often is omitted. The

first base vector indicates the co

j i j j j k k i k j k kv v v w w w

x y z x y z

mponent of being differentiated and the second indicates the direction of the derivative;

hence, Grad has nine components: three derivatives of each of its three components.

V

V

continued


13/41

, : ; rad where

definition:

rad ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

u v w d d d d dx dy dz

u u u v v v w wd d d d d d d d d

x y z x y z x y

Gradient of a vector V V i j k V r r V r V V r r i j k

V r i i r i j r i k r j i r j j r j k r k i r k j r

G

G ( )

rad

similarly

rad ( ) ( ) ( ) ( ) ( ) ( ) (

wd

z

u u u v v v w w wd dx dy dz dx dy dz dx dy dz d

x y z x y z x y z

u u u v v v w

d d d d d d d x y z x y z x

k k r

V r i j k V

r V r i i r i j r i k r j i r j j r j k

G

G ) ( ) ( )

rad rad

w w

d d dy z

u u u v v v w w wd dx dx dy dz d

x y z x y z x y z

u v w u v wdx dy dz dx dy dz

x x x y y y

r k i r k j r k k

r V i j k i j k i j k V r

i j

G G

Note: in general

u v wdx dy dz d

z z z

u u u u u u

x y z x y zdx

v v v v v vdy dx dy dz

x y z x y zdz

w w w w w w

x y z x y z

k V

continued


14/41

is defined so that it is consistent with the parallelogram law of addition (PLA):

and

Addition of dyads

A C B CT S

xx xy xz x xx xy xz

yx yy yz y yx yy yz

zx zy zz z zx zy z

T T T C S S S

T T T C S S S

T T T C S S S

according to the PLA , etc.

where

R A B

R A B C C CT S= T S

T S

x

y

z z

x x x

x xx x xy y xz z xx x xy y xz z

x xx xx x xy xy y xz xz z

xx xx xy xy xz xz

yx y

C

C

C

R A B

R T C T C T C S C S C S C

R T S C T S C T S C

T S T S T S

T S

To preserve the parallelogram law of addition, we need to add the corresponding

components of the dyads.

x yy yy yz yz

zx zx zy zy zz zz

T S T S

T S T S T S

continued


15/41

of rad where denotes velocity

1 1

2 2

1 1

rad 2 2

1

2

u u v u wu u u

x y x z xx y z

v v v u v v v w

x y z y x y z y

w w w u w

x y z z x

Decomposition V V

V

G

G

1 10

2 2

1 1

02 2

1 1 10

2 2 2

u v u w

y x z x

v u v w

x y z y

v w w w u w v

z y z x z y z

rad train ot read strain of and rotation of

train is symmetric, . ot is antisymmetriij jiS S

V V V V V

V V

G St R

St R c, .

train has six distinct components. ot has only three distinct components.

Note:

Curl

ij jiR R

w v u w v u

x y z y z z x x

u v w

V V

i j k

V i j

St R

01

and ot 02

0

Let represent any vector; then Curl

01

ot 02

0

z y

x y z z x

y x

x y z x y z z y y z x z z x y x x y

x y z

z y

z x

y x

y

A A A A A A A A A

A A A

k i j k V

i j k

A i j k V A i j k

V A

R

R1 1

Curl2 2

x z y y z

y x z z x

z y x x y

A A A

A A A

A A A

V A

continued


16/41

Bv

AvA

B

dr

and are two points in the same fluid separated

by the infinitesimal position vector

rad

ot train

1Curl train

2

B A A

A

A

A B

d

d d

d d

d d

r

v v v v v r

v v r v r

v v r v r

G

R S

S

:

1) translation at the velocity

12) rigid-body rotation at the angular velocity Curl

2

: the vector Curl is simultaneously

we can identify the three components of the motion of a continuum

v

v

note v r

A

d to both Curl and ;

the fact that Curl is to means that this term does not cause

to change length

3) deformation given by train

train

v r

v r r

r

v r

v v r v r

S

SB A

d

d d

d

d

d d


17/41

using Taylor series and the PLA, we can write

A B

B A

t d d t

d

d v r r v

v v v

.

.

time t

time t+tA

A

B

Bd

fluid particle

moves from

here

to here

duringt

dr

d r

B tv

A tv


18/41


19/41

0S

1div lim dS

V V n

dS

n

Divergence

Fluid particle, coincident

with at time t, after time

thas elapsed.

= proportionate rate of change of volume of a fluid particle

Elemental volume

with surface S


20/41

Review

0

S

1div lim . dS

V V n

Divergence

For velocity: proportionate

rate of change of volume

of a fluid particle

0

S

1grad lim dS

n

Gradient



field at a point


21/41

Differential Forms of the

Divergence

sinsin sin sin

yx z

r zr z

2

rr2

divA = A

AA A+ + i + j + k .Ax y z x y z

1 1 erA A A+ + + + .Ae e

r r r z r r z

1 1 1 eA er A A+ + + + .Aer r r r r r r

Cartesian

Cylindrical

Spherical


22/41

Differential Forms of the Curl

sin

sin

sin

r z r

2

r z rx y z

r r ri j e e e e e ek

1 1curlA = A = = =x y z r r z rr

A rA A rA rA AA A A

0S

1curl lim dS

A A n

Cartesian Cylindrical Spherical

Curl of the veloci ty vectorV =twice the circumferentially averaged angular

velocity of

-the flow around a point, or

-a fluid particle

=Vort ici tyPure rotation No rotation Rotation


23/41

Curl

0S

1curl lim dS

V V n

dS

n

e

nPerimeter Ce

ds

h

dS=dsh

Area

radius a

v avg. tangential

velocity= twice the avg. angular velocity

about e

Elemental volume

with surface S

0

S

1.curl lim . dS

e V e V n

0S

1.curl lim . dS

h

e V V e n

0S

1.curl lim . ds h

h

e V V e n

e

Ce

0 0

C

1.curl lim .d lim

e V V s

20 0

1.curl lim 2 2lim

a a

vv a

a a

e V


24/41

Review

0S

1div lim . dS

V V n

Divergence

For velocity: proportionate

rate of change of volume

of a fluid particle

0S

1grad lim dS

n

Gradient



field at a point

0S

1curl lim dS

V V n

Curl

For velocity: twice the

circumferentially averaged

angular velocity of a fluid

particle= Vorticity


25/41

Oliver Heaviside

1850-1925

Writes Electromagnetic induction and its propagation over the

course of two years, re-expressing Maxwell's results in 3(complex) vector form, giving it much of its modern form and

collecting together the basic set of equations from which

electromagnetic theory may be derived (often called "Maxwell's

equations"). In the process, He invents the modern vector

calculus notation, including the gradient, divergence and curl of a

vector.


26/41

Integral Theorems and

Second Order Operators


27/41

1st Order Integral Theorems

Gradient theorem

Divergence theorem

Curl theorem

Stokes theorem

R S

dSd n

R S

dSd nAA ..

R S

dSd nAA

Volume R

with Surface S

d

ndS

C

SSd sAnA d..

Open Surface S

with Perimeter CndS


28/41

The Gradient Theorem

S

0 dS

1Limgrad n

Finite Volume R

Surface S

d

Begin with the definition of grad:

Sum over all the d in R:

R SR i dSgrad nid

di

di+1nidS

ni+1

dS

We note that contributions to the

RHS from internal surfaces between

elements cancel, and so:

SR dSgrad ndRecognizing that the summations

are actually infinite:

SR

d dSgrad n


29/41

Assumptions in Gradient Theorem

A pure math result, applies to all flows

However, S must be chosen so that is defined

throughout R

SR d dSgrad n

S

Submarine

surface

SR

d dSgrad n


30/41

Flow over a

finite wing

S1

S2

SR

pdp dSn

S = S1 + S2

R is the volume of fluid

enclosed between S1

and S2

S1

p is not defined inside the

wing so the wing itself

must be excluded from the

integral


31/41

1st Order Integral Theorems

Gradient theorem

Divergence theorem

Curl theorem

Stokes theorem

R S

dSd n

R S

dSd nAA ..

R S

dSd nAA

Volume R

with Surface S

d

ndS

C

SSd sAnA d..

Open Surface S

with Perimeter CndS


32/41

Ce0

C

0 Limd.1

Lim.

e

sAAe curl

Alternative Definition of the Curl

e

Perimeter Ce

ds

Area


33/41

Stokes TheoremFinite Surface S

With Perimeter C

d

Begin with the alternative definition of curl, choosing the

direction e to be the outward normal to the surface n:

Sum over all the d in S:

Note that contributions to the RHS

from internal boundaries between

elements cancel, and so:

Since the summations are actually infinite, and

replacing with the more normal area symbol S:

eC

0 d.1

Lim. sAAn

di

SSid

eCd.. sAAn

n

dsi

dsi+1

di+1 CS d sAAn d..

CS

Sd sAnA d..


34/41

Stokes Theorem and Velocity

Apply Stokes Theorem to a velocity field

Or, in terms of vorticity and circulation

What about a closed surface?

CS

Sd sVnV d..

C

CS

Sd

sVn d..

0.

dS

S

n

CS Sd sAnA d..


35/41

Assumptions of Stokes Theorem

A pure math result, applies to all flows

However, C must be chosen so that A is defined over all S

CSSd sAnA d..

2D flow over airfoil with =0

C?d..

C

SSd sVn

The vorticity doesnt

imply anything about the

circulation around C


36/41

Flow over a finite wing

C

SSd sVnV d..

C

S

Wing with circulation must trail vorticity.Always.


37/41

Vector Operators of Vector

Products

B).A(-)A.(B-A).B(+)B.(A=)BA(

B.A-A.B=)BA.(

)A(B+)B(A+)A.B(+)B.A(=)B.A(

A+A=)A(

A.+A.=)A.(

+=)(


38/41

Convective Operator

)]A.(B-.B)(A+)BA(-

)A(B-)B(A-)B.A([=

Bz

Ay

Ax

A=B).A(

).(A

zA

yA

xA=

).A(

zyx

zyx

21

.V= change in density in direction ofV, multiplied by magnitude ofV


39/41

Second Order Operators

2

2

2

2

2

2

2.zyx

The Laplacian, may also be

applied to a vector field.

0.

0

).(

).(

2

A

AAA

A

So, any vector differential equation of the form B=0

can be solved identically by writing B=.

We say B is i r rotat ional.

We refer to as the scalar potent ial.

So, any vector differential equation of the form .B=0

can be solved identically by writing B=A. We say B is solenoidalorincompress ib le.

We refer to A as the vector potent ial.


40/41

Class Exercise

1. Make up the most complex irrotational 3D velocity field you can.

2223sin /3)2cos( zyxxyxe x kjiV ?

We can generate an irrotational field by taking the gradient ofany

scalar field, since 0

I got this one by randomly choosing

zyxe x /132sin

And computing

kjiVzyx


41/41

2nd Order Integral Theorems

Greens theorem (1st form)

Greens theorem (2nd form)

Volume R

with Surface S

d

ndS

S

RSd d

n

2

2 2 - dn nR

S

d S

These are both re-expressions of the divergence theorem.

Class 4 - Vector Calculus v1

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Transcript of Class 4 - Vector Calculus v1