AOE 5104 Class 3 9/2/08

• Online presentations for today’s class:– Vector Algebra and Calculus 1 and 2

• Vector Algebra and Calculus Crib• Homework 1 due 9/4• Study group assignments have been made and are

online. • Recitations will be

– Mondays @ 5:30pm (with Nathan Alexander)– Tuesdays @ 5pm (with Chris Rock)– Locations TBA– Which recitation you attend depends on which study group

you belong to and is listed with the study group assignments

Unnumbered slides contain comments that I inserted and are not part of Professor’s Devenport’s original presentation.

4

Cylindrical Coordinates

R

er

eez

• Coordinates r, , z

• Unit vectors er, e, ez (in directions of increasing coordinates)

• Position vector R = r er + z ez

• Vector components F = Fr er+F e+Fz ez

Components not constant, even if vector is constant

r

z

F

x

y

z

5

Spherical Coordinates

r

er

e

e

rF

• Coordinates r, ,

• Unit vectors er, e, e (in directions of increasing coordinates)

• Position vector r = r er

• Vector components F = Fr er+F e+F e

Errors on this slide in online presentation

yx

z

11

1

r r x y z

x r r

r rr

r

F F F F F F

F F F F

x r y r r r

r r r

hh r r

hh r

sin cos , sin sin , cos

sin cos sin sin cos

sin cos sin sin cos

cos cos cos sin sin

F e e e i j k

F i e i e i e i

r i j k

r re i j k

re i j k

1

x r

r

h rh

F F F F

sin cos sin

sin cos cos cos sin

r

r re i j

7

J. KURIMA, N. KASAGI and M. HIRATA (1983)Turbulence and Heat Transfer Laboratory, University of Tokyo

LOW REYNOLDS NUMBER AXISYMMETRIC JET

8

Class ExerciseUsing cylindrical coordinates (r, , z)

Gravity exerts a force per unit mass of 9.8m/s2 on the flow which at (1,0,1) is in the radial direction. Write down the component representation of this force at

a) (1,0,1) b) (1,,1) c) (1,/2,0) d) (0,/2,0)

R

er

eez

r

z

9.8m/s2

a) (9.8,0,0)

b) (-9.8,0,0)

c) (0,-9.8,0)

d) (9.8,0,0)

xy

z

Vector Algebra in Components

1 1 2 2 3 3

1 2 3

1 2 3

1 2 3

2 3 3 2 1 3 1 1 3 2 1 2 2 1 3

cos

where is the smaller of the two angles

between and

sin where is determined by the

right-han

A B A B

A B

e e e

A B

e e e

A B A B n n

A B A B A B

A A A

B B B

A B A B A B A B A B A B

d rule

A

B

n

3. Vector Calculus

Fluid particle: Differentially Small Piece of the Fluid Material

Concept of Differential Change In a Vector. The Vector Field.

V

-2

-1

0

1

2y

/ L

-2

0

2-T / U L0

1

2

z / L

V+dV

dV

V=V(r,t)

=(r,t)Scalar field

Vector field

Differential change in vector• Change in direction• Change in magnitude

13

PP'

er

e

ez

d

r

z

Change in Unit Vectors – Cylindrical System

rdd ee

ee dd r

0zde

e+de

er+der

er

e

de

der

14

Change in Unit Vectors – Spherical System

sin

cos

sin cos

e e e

e e e

e e e

r

r

r

d d d

d d d

d d d

r

er

e

e

r

See “Formulae for Vector Algebra and Calculus”

15

Example

kjir zyx

dt

drV

zr zr eer

R=R(t)

Fluid particleDifferentially small piece of the fluid material

V=V(t) The position of fluid particle moving in a flow varies with time. Working in different coordinate systems write down expressions for the position and, by differentiation, the velocity vectors.

O

... This is an example of the calculus of vectors with respect to time.

dt

drV

zr dt

dz

dt

dr

dt

dreee

Cartesian System

Cylindrical System

zr

r dt

dz

dt

dr

dt

dre

ee

kjidt

dz

dt

dy

dt

dx

ee dd r

16

Vector Calculus w.r.t. Time

• Since any vector may be decomposed into scalar components, calculus w.r.t. time, only involves scalar calculus of the components

dtdtdt

ttt

ttt

ttt

BABA

BAB

ABA

BAB

ABA

BABA

.

17

High Speed Flow Past an Axisymmetric Object

Shadowgraph picture is from “An Album of Fluid Motion” by Van Dyke

1

1

lim where is evaluated at any

point on and the largest 0 as

similarly for other integrals

lim etc.

I s s

s s

V s V s

B i n

i i in

iA

i i

B i n

i in

iA

d

n

I d

Line integrals

divide the path into

small segments:

is a typical onesi

n

si

Vi

A

B

19

Integral Calculus With Respect to Space

D=D(r), = (r)

n

dS

Surface SVolume R

D(r)

ds

O

r

D(r)

d

Line Integrals

s D s D sB B B

A A Ad d d

For closed loops, e.g. Circulation V sd

A

B

20

Picture is from “An Album of Fluid Motion” by Van Dyke

Mach approximately 2.0

For closed loops, e.g. Circulation sV d.

21

Integral Calculus With Respect to Space

D=D(r), = (r)

n

dS

Surface SVolume R

D(r)

ds

O

r

D(r)

d

Surface Integrals

n D n D nS S S

dS dS dS Se.g. Volumetric Flow Rate through surface S S

dSnV.

For closed surfaces

Volume Integrals

DR Rd d

A

B

22

Picture is from “An Album of Fluid Motion” by Van Dyke

Mach approximately 2.0

n

dS

. n Dn D nS S S

dS dS dS

23

In 1-D

τ 0ΔS

1grad lim dS

τ

nIn 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) 0

1lim ( ) ( )

x

dff x x f x

dx x

Alternative to the Integral Definition of Grad

We want the generalization of

0

0

1lim ( ) ( )

lim ( ) ( ) Grad

Grad

Grad

rr r r r

i j k

i j k

x

df dff x x f x df dx

dx x dx

df f f f d

f f fdf dx dy dz f dx dy dz

x y z

f f ff

x y z

continued

Alternative to the Integral Definition of Grad

Cylindrical coordinates

0

cos sin

cos sin sin cos

lim ( ) ( ) Grad

Grad

1Grad

r

r i j k e k

r i j i j k

e e k

r r r r

e e k

e e k

r

r

r

r

r r z r z

d dr rd z

dr rd dz

df f f f d

f f fdf dr d dz f dr rd dz

r z

f f ff

r r zcontinued

Alternative to the Integral Definition of GradSpherical coordinates

sin cos sin sin cos

1sin cos sin sin cos 1

1cos cos cos sin sin

1sin cos sin

sin cos sin

r i j k

r re i j k

r re i j k

r re i j

r i

r rr

r

r r r

hh r r

h rh r

h rh

d dr

sin cos cos cos cos sin sin

sin sin cos sin

, , Grad sin

1 1 Grad

sin

j k i j k

i j e e e

e e e

e e

r

r

r

rd

r d dr rd r d

F F FdF r dr d d F dr rd r d

r

F F FF

r r r e

27

ndS(large)

Gradient

τ 0ΔS

1grad lim dS

τ

n

dS

n = low

= high

Elemental volume with surface S

ndS(small)

ndS(medium)

ndS(medium)

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

Resulting ndS

28

Gradient

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

• Fourier´s Law of Heat Conduction

e s s

n

Tq k k T

n

The integral definition given on a previous slide can also be used to obtain the formulas for the gradient.

On the next four slides, the form of GradF in Cartesian coordinates is worked out directly from the integral definition.

30

τ 0ΔS

τ 0

1grad lim dS

τ

1lim

τ

n

i j k i j k

dxdydz dxdydz dxdydzx y z x y z

Face 2

Differential form of the Gradient

τ 0ΔS

1grad lim dS

τ

nCartesian 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

e1

dS ( )2

n iFac

dxdydz

x

e 2

dS ( )2

n iFac

dxdydz

x

adding these givesi dxdydz

xProceeding in the same way for y and z

j dxdydz

y

k dxdydz

zandwe get , so

An element of volume with a local Cartesian coordinate system having its origin at the centroid of the corners, O

O .

x

y

z

Δx

Δz

Δy

M

Point M is at the centroid of the face perpendicular to the y-axis with coordinates (0, Δy/2, 0)

Other points in this face have the coordinates (x, Δy/2, z)

F y axis

F x y z

We introduce a local Cartesian coordinate system and then use

a Taylor series to express at a point in the face

in terms of its value and the values of its derivatives at the origin:

( , , ) 2

2

0 0 0 0 0 0 0 0 00 0 0

2

F F y FF x z O l

x y z

O l x y z

( , , ) ( , , ) ( , , )( , , ) ( )

where ( ) denotes the remainder which contains the factors , ,

in at least quadratic formcontinued

Gradient of a Differentiable Function, F

2

4

0 0 0 0 0 0 0 0 00 0 0

2

0 0 0

2

y

y axis

F x y z dS

F F y FF x z O l dxdz

x y z

F y x zF x z O l

y

z x

2 2- z - x

2 2

the integral over the face ( ) :

( , , )

( , , ) ( , , ) ( , , ) ( , , ) ( )

( , , ) ( )

the integ

n j

n

j

j

40 0 00 0 0

2

0 0 0

y

y y

y axis

F y x zF x y z dS F x z O l

y

FF x y z dS F x y z dS y

y

ral over the face ( ) :

( , , ) ( )( , , ) ( , , ) ( )

the sum of the integrals over these two faces is given by

( , , )( , , ) ( , , ) (

n j

n j

n n 4x z O l ) ( )

4

4

4

0 0 0

0 0 0

0 0 0

y y

x x

z z

FF x y z dS F x y z dS y x z O l

y

FF x y z dS F x y z dS y x z O l

x

FF x y z dS F x y z dS y x z O l

z

y

( , , )( , , ) ( , , ) ( ) ( )

similarly

( , , )( , , ) ( , , ) ( ) ( )

( , , )( , , ) ( , , ) ( ) ( )

n n j

n n i

n n k

0 0

1l

allfac s

x z

F F FF F x y z dS O l

x y z

F F FF

x y z

e

volume of the element

Grad lim ( , , ) lim ( )

Grad

n i j k

i j k

34

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

35

τ 0ΔS

1lim dS

τ

V V ndiv

dS

n

Divergence

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

= proportionate rate of change of volume of a fluid particle

Elemental volume with surface S

36

Differential Forms of the Divergence

A.r

e + re +

re

Ar

1+A

r

1+

rAr

r

1

A.z

e + re+

re

zA+A

r

1+

rrA

r

1

A.z

k + y

j + x

izA+

yA+

xA

A. = Adiv

rr

2

2

zrzr

zyx

sinsin

sin

sin

Cartesian

Cylindrical

Spherical

37

Differential Forms of the Curl

ArrAA

r

erere

r

1=

ArAA

zr

eere

r

1=

AAA

zyx

kji

= A = Acurl

r

r

2

zr

zr

zyx

sin

sin

sin

ΔS

0τ dSτ

1Lim nAAcurl

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

38

e

τ 0ΔS

τ 0ΔS

τ 0ΔS

τ 0ΔS

Ce

τ 0 τ 0C

1lim dS

τ

1lim . dS

τ

1lim . dS

1lim . d

1lim .d lim

V V n

e V eV n

e V V e n

e V V e n

e V V s

e

curl

curl

curlh

curl s hh

curl

c2τ 0 τ 0

1lim 2 2 lim

V

vurl v a

a a

dS

n

Curl

enPerimeter Ce

dsh

dS=dsh

Area

radius a

v avg. tangential velocity

= twice the avg. angular velocity about e

Elemental volume with surface S