class 5 - integral theorems

7/27/2019 Class 5 - Integral Theorems

1/22

AOE 5104 Class 5 9/9/08

Online presentations for next class:

Equations of Motion 1

Homework 2 due 9/11 (w. recitations, thisevenings is in Whitemore 349 at 5)

Office hours tomorrow will start at 4:30-

4:45pm (and I will stay late)


2/22

Review

S

0 dS.

1Lim nVV

div

Divergence

For velocity: proportionate

rate of change of volume

of a fluid particle

S

0 dS

1Limgrad n

Gradient

Magnitude and direction

of the slope in the scalar

field at a point

S

0 dS

1Lim nVV

curl

Curl

For velocity: twice the

circumferentially averaged

angular velocity of a fluid

particle= Vorticity


3/22

Differential Forms of the

Divergence

A.r

e+

r

e+

re

A.z

e+r

e+

re

A.

z

k+

y

j+

x

i

A

r

1+

A

r

1+

r

Ar

r

1

z

A+

A

r

1+

r

rA

r

1

z

A+

y

A+

x

A

A.=Adiv

r

zr

r2

2

zr

zyx

sinsin

sin

sin

Cartesian

Cylindrical

Spherical


4/22

a

vav

acurl

curl

hsh

curl

h

curl

curl

curl

aa

020

Ce0

C

0

S

0

S

0

S

0

S

0

2Lim21

Lim.

Limd.1

Lim.

d.1

Lim.

dS.1

Lim.

dS.

1Lim.

dS1Lim

e

Ve

sVVe

neVVe

neVVe

nVeVe

nVV dS

n

Curle

n

Perimeter Ce

ds

h

dS=dsh

Area

radius a

v avg. tangential

velocity= twice the avg. angular velocity

about e

Elemental volume

with surface S


5/22

Review

S

0 dS.

1Lim nVV

div

Divergence

For velocity: proportionate

rate of change of volume

of a fluid particle

S

0 dS

1Limgrad n

Gradient

Magnitude and direction

of the slope in the scalar

field at a point

S

0 dS

1Lim nVV

curl

Curl

For velocity: twice the

circumferentially averaged

angular velocity of a fluid

particle= Vorticity


6/22

Integral Theorems and

Second Order Operators


7/22

George Gabriel Stokes1819-1903


8/22

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


9/22

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+1dS

We note that contributions to the

RHS from internal surfaces between

elements cancel, and so:

SRdSgrad nd

Recognizing that the summations

are actually infinite:

SR

d dSgrad n


10/22

Assumptions in Gradient Theorem

A pure math result, applies to all flows

However, S must be chosen so that isdefined throughout R

SR d dSgrad n

S

Submarine

surface

SR

d dSgrad n


11/22

Flow over a

finite wing

S1

S2

SR

pdp dSn

S = S1 + S2

R is the volume of fluid

enclosed between S1and S2

S1

p is not defined inside the

wing so the wing itself

must be excluded from the

integral


12/22

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


13/22

Ce0

C

0 Limd.1

Lim.

e

sAAe curl

Alternative Definition of the Curl

e

Perimeter Ce

ds

Area


14/22

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

eC

d.. sAAn

n

dsi

dsi+1

di+1 CS

d sAAn d..

CS Sd sAnA d..


15/22

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..

CC

S

Sd

sVn d..

0.

dS

S

n

CS Sd sAnA d..


16/22

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

CS

Sd sAnA d..

2D flow over airfoil with =0

C

?d..

CS

Sd sVn

The vorticity doesnt

imply anything about the

circulation around C


17/22

Flow over a

finite wing

CS

Sd sVnV d..

C

S

Wing with circulation must trail vorticity.Always.


18/22

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.(

+=)(


19/22

Convective Operator

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

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

B

z

A

y

A

x

A=B).A(

).(A

zA

yA

xA=

).A(

zyx

zyx

21

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


20/22

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 i r rotat ional. We refer to as the scalar po tent 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 po tent ial.


21/22

Class Exercise

1. Make up the most complex irrotational 3D velocityfield 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


22/22

2nd Order Integral Theorems

Greens theorem (1st form)

Greens theorem (2nd form)

Volume R

with Surface S

d

ndS

S

RSd d

n

2

S

RSd d

n-

n

22

These are both re-expressions of the divergence theorem.

class 5 - integral theorems

Documents