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.
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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
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