AOE 5104 Class 5 9/9/08

• Online presentations for next class:– Equations of Motion 1

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

• Office hours tomorrow will start at 4:30-4:45pm (and I will stay late)

Review

ΔS

0τ dS.τ

1Lim nVV

div

Divergence

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

ΔS

0τ dSτ

1Limgrad n

Gradient

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

ΔS

0τ dSτ

1Lim nVV

curl

Curl

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

Differential Forms of the Divergence

A.r

e +

re +

re

A.z

e + re+

re

A.z

k + y

j + x

i

Ar

1+A

r

1+

rAr

r

1

zA+A

r

1+

rrA

r

1

zA+

yA+

xA

A. = Adiv

r

zr

r2

2

zr

zyx

sinsin

sin

sin

Cartesian

Cylindrical

Spherical

a

vav

acurl

curl

hsh

curl

hcurl

curl

curl

aa

020

Ce0

C

0

ΔS

0τ

ΔS

0τ

ΔS

0τ

ΔS

0τ

2Lim21

Lim.

Limd.1

Lim.

d.1

Lim.

dS.1

Lim.

dS.τ

1Lim.

dSτ

1Lim

e

Ve

sVVe

neVVe

neVVe

nVeVe

nVV dS

n

Curle

nPerimeter Ce

dsh

dS=dsh

Area

radius a

v avg. tangential velocity

= twice the avg. angular velocity about e

Elemental volume with surface S

Review

ΔS

0τ dS.τ

1Lim nVV

div

Divergence

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

ΔS

0τ dSτ

1Limgrad n

Gradient

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

ΔS

0τ dSτ

1Lim nVV

curl

Curl

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

Integral Theorems and Second Order Operators

George Gabriel Stokes1819-1903

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

Volume Rwith Surface S

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

CS

Sd 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)

Volume Rwith Surface S

d

ndS

S

RSd d

n2

S

RSd d

n-

n22

These are both re-expressions of the divergence theorem.