AOE 5104 Class 4 9/4/08 Online presentations for today’s class: –Vector Algebra and Calculus 2...

AOE 5104 Class 4 9/4/08

• Online presentations for today’s 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 and

are online. • Recitations will be

– Mondays @ 5:30pm (with Nathan Alexander) in Randolph 221

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

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

Last Class

• Changes in Unit Vectors

• Calculus w.r.t. time• Integral calculus w.r.t.

space• Today: differential

calculus in 3D

PP'

er

e

ez

dr

zrdd ee

ee dd r 0zde

dtdtdt

ttt

ttt

ttt

BABA

BAB

ABA

BAB

ABA

BABA

.

Oliver Heaviside1850-1925

Shock in a CD Nozzle

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

In 1-D

τ 0ΔS

1grad lim dS

τ

n

In 3-D

Differential Calculus w.r.t. Space Definitions of div, grad and curl

τ 0ΔS

1div lim . dS

τ

D D n

τ 0ΔS

1curl lim dS

τ

D D n

Elemental volume with surface S

n

dS

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

1lim ( ) ( )

dff x x f x

dx x

ndS(large)

Gradient

τ 0ΔS

1grad lim dS

τ

n

dS

n = low

= high


ndS(small)

ndS(medium)

ndS(medium)

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

Resulting ndS

Review

ΔS

0τ dSτ

1Limgrad n

Gradient

Magnitude and direction of the slope in the scalar field at a point

Gradient

• Component of gradient is the partial derivative in the direction of that component

• Fourier´s Law of Heat Conduction

ss

.e

n.Tkn

Tkq

= low

= high

ss e,

zyxdxdydz

zdxdydz

ydxdydz

x

kjikji

n

τ

1Lim

dSτ

1Limgrad

0τ

ΔS

0τ

Face 2

Differential form of the Gradient

ΔS

0τ dSτ

1Limgrad n

Cartesian system

dy

dx

dz

j

ik

P

Evaluate integral by expanding the variation in about a point P at the center of an elemental Cartesian volume. Consider the two x faces:

= (x,y,z)

Face 1

dydzdx

xFace

)(2

dS1

in

dydzdx

xFace

)(2

dS2

in

adding these gives dxdydzx

i

Proceeding in the same way for y and z

dxdydzy

j dxdydz

z

kandwe get , so

Differential Forms of the Gradient

These differential forms define the vector operator

sinsin re +

re +

re

re +

re +

re

ze +

re+

re

ze +

re+

re

zk +

yj +

xi

zk +

yj +

xi

= grad

rr

zrzr

Cartesian

Cylindrical

Spherical

, :

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 zdu dx

v 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

Vcontinued

, : ; 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 wd 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 wd d d

y 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

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

and

Addition of dyads

A C B C T Sxx 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 C T 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

of rad where denotes velocity

1 1

2 2

1 1rad

2 2

1

2

u u v u wu u ux 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 wx y z z x

Decomposition V V

V

G

G

1 10

2 2

1 10

2 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

.

.Bv

AvA

B

dr

and are two points in the same fluid separated

by the infinitesimal position vector

rad

ot train

1 Curl 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

ω 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 ω r v r

S

SB A

d

d d

d

d

d d

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 + ΔtA

A’

B’

Bd

fluid particlemoves fromhere

to hereduring Δt

dr

d r

B tv

A tv

τ 0ΔS

1div lim dS

τ

V V n

dS

n

Divergence

Fluid particle, coincidentwith at time t, after timet has elapsed.

= proportionate rate of change of volume of a fluid particle


Review

τ 0ΔS

1div lim . dS

τ

V V n

Divergence

For velocity: proportionate rate of change of volume of a fluid particle

τ 0ΔS

1grad lim dS

τ

n

Gradient


Differential Forms of the Divergence

sin

sin sin sin

yx z

r zr z

2r

r2

divA = A

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

1 1 erA A A+ + + + .Ae er r r z r r z

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

Cartesian

Cylindrical

Spherical

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 ek1 1

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

A rA A rA rA AA A A

τ 0ΔS

1curl lim dS

τA A n

Cartesian Cylindrical Spherical

Curl of the velocity vector V = twice the circumferentially averaged angular velocity of

-the flow around a point, or-a fluid particle

=Vorticity ΩPure rotation No rotation Rotation

Curlτ 0

ΔS

1curl lim dS

τ

V V n

dS

n

enPerimeter Ce

dsh

dS=dsh

Area

radius a

v avg. tangential velocity= twice the avg. angular velocity

about e


τ 0ΔS

1.curl lim . dS

τ e V e V n

τ 0ΔS

1.curl lim . dS

h

e V V e n

τ 0ΔS

1.curl lim . ds h

h

e V V e n

e

Ce

τ 0 τ 0C

1.curl lim .d lim

e V V s

20 0

1.curl lim 2 2lim

a a

vv a

a a

e V

Review

0ΔS

1div lim . dS

τ

V V n

Divergence

For velocity: proportionate rate of change of volume of a fluid particle

0ΔS

1grad lim dS

τ

n

Gradient


0ΔS

1curl lim dS

τ

V V n

Curl

For velocity: twice the circumferentially averaged angular velocity of a fluid particle = Vorticity Ω

Oliver Heaviside1850-1925

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.

Integral Theorems and Second Order Operators

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 Rwith Surface S

d

ndS

C

SSd sAnA d..

Open Surface Swith Perimeter C

ndS

The Gradient Theorem

ΔS

0τ dSτ

1Limgrad n

Finite Volume RSurface S

d

Begin with the definition of grad:

Sum over all the d in R:

R ΔSR

i

dSgrad n id

di

di+1

nidS

ni+1dS

We note that contributions to the RHS from internal surfaces between elements cancel, and so:

SR

dSgrad n d

Recognizing that the summations are actually infinite:

SR

d dSgrad n

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

Submarinesurface

SR

d dSgrad n

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

1st Order Integral Theorems

• Gradient theorem

• Divergence theorem

• Curl theorem

• Stokes’ theorem

R S

dSd n

R S

dSd nAA ..

R S

dSd nAA


d

ndS

C

SSd sAnA d..

Open Surface Swith Perimeter C

ndS

Ce

0

C

0 Limd.1

Lim.e

sAAe curl

Alternative Definition of the Curl

e

Perimeter Ce

ds

Area

Stokes’ TheoremFinite Surface SWith 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

SS

ideC

d.. sAAn

n

dsi

dsi+1di+1

CS

d sAAn d..

C

SSd sAnA d..

Stokes´ Theorem and Velocity

• Apply Stokes´ Theorem to a velocity field

• Or, in terms of vorticity and circulation

• What about a closed surface?

C

SSd sVnV d..

C

CS

Sd sVnΩ d..

0. dSS

nΩ

C

SSd sAnA d..

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

C

SSd sAnA d..

2D flow over airfoil with =0

C?d..

CS

Sd sVnΩ

The vorticity doesn’t imply anything about the circulation around C

Flow over a finite wing

C

SSd sVnV d..

C

S

Wing with circulation must trail vorticity. Always.

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.(

+ = )(

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 of V, multiplied by magnitude of V

Second Order Operators

2

2

2

2

2

22.

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 irrotational. • We refer to as the scalar potential.

• So, any vector differential equation of the form .B=0 can be solved identically by writing B=A. • We say B is solenoidal or incompressible. • We refer to A as the vector potential.

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 of any scalar field, since 0

I got this one by randomly choosing

zyxe x /132sin And computing

kjiVzyx

2nd Order Integral Theorems

• Green’s theorem (1st form)

• Green’s theorem (2nd form)


d

ndS

S

RSd d

n2

2 2 - dn nR

S

d S

These are both re-expressions of the divergence theorem.

AOE 5104 Class 4 9/4/08 Online presentations for today’s class: –Vector Algebra and Calculus 2...

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Transcript of AOE 5104 Class 4 9/4/08 Online presentations for today’s class: –Vector Algebra and Calculus 2...