AOE 5104 Class 3 9/2/08

1
AOE 5104 Class 3 9/2/08

• Online presentations for todays 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 _at_ 530pm (with Nathan Alexander)
• Tuesdays _at_ 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

2
Unnumbered slides contain comments that I
inserted and are not part of Professors
Devenports original presentation.
3
Last Class

• Vectors, inherent property of direction
• Algebra
• Volumetric flow rate through an area
• Taking components, eqn. of a streamline
• Triple products, A.BxC, Ax(BxC)
• Coordinate systems

4
Cylindrical Coordinates

• Coordinates r, ? , z
• Unit vectors er, e?, ez (in directions of
  increasing coordinates)
• Position vector
• R r er z ez
• Vector components
• F Fr erF? e?Fz ez
• Components not constant, even if vector is
  constant

z
F
ez
e?
er
R
z
r
?
y
x
5
Spherical Coordinates
Errors on this slide in online presentation

• Coordinates r, ? , ?
• Unit vectors er, e?, e? (in directions of
  increasing coordinates)
• Position vector
• r r er
• Vector components
• F Fr erF? e?F? e?

z
er
e?
F
r
e?
?
r
?
y
x
6
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LOW REYNOLDS NUMBER AXISYMMETRIC JET
J. KURIMA, N. KASAGI and M. HIRATA
(1983) Turbulence and Heat Transfer Laboratory,
University of Tokyo
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
• (1,0,1) b) (1,?,1) c) (1,?/2,0) d)
  (0,?/2,0)

z

1. (9.8,0,0)
2. (-9.8,0,0)
3. (0,-9.8,0)
4. (9.8,0,0)

ez
e?
er
9.8m/s2
R
z
r
?
y
x
9
Vector Algebra in Components
B
n
A
10
3. Vector Calculus

• Fluid particle Differentially Small Piece of the
  Fluid Material

11
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Concept of Differential Change In a Vector. The
Vector Field.
???(r,t)
Scalar field
VV(r,t)
Vector field
V
dV
VdV

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

13
Change in Unit Vectors Cylindrical System
de?
erder
e?
ez
e?de?
der
er
P'
e?
P
z
er
r
d?
?
14
Change in Unit Vectors Spherical System
er
e?
r
e?
?
r
See Formulae for Vector Algebra and Calculus
?
15
Example
Fluid particle Differentially small piece of the
fluid material
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.
VV(t)
RR(t)
Cartesian System
O
Cylindrical System
... This is an example of the calculus of vectors
with respect to time.
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

17
High Speed Flow Past an Axisymmetric Object
Finned body of revolution fired from a gun into a
supersonic wind tunnel flow for a net Mach number
of 2. The plastic shell casing is seen
separating. Vincenti, NASA
Shadowgraph picture is from An Album of Fluid
Motion by Van Dyke
18
Line integrals
19
Integral Calculus With Respect to Space
D(r)
D(r)
O
n
B
r
DD(r), ? ?(r)
ds
dS
d?
Surface S Volume R
A
Line Integrals
For closed loops, e.g. Circulation
20
For closed loops, e.g. Circulation
Mach approximately 2.0
Picture is from An Album of Fluid Motion by Van
Dyke
21
Integral Calculus With Respect to Space
D(r)
D(r)
O
n
B
r
DD(r), ? ?(r)
ds
dS
d?
Surface S Volume R
A
Surface Integrals
For closed surfaces
e.g. Volumetric Flow Rate through surface S
Volume Integrals
22
n
dS
Mach approximately 2.0
Picture is from An Album of Fluid Motion by Van
Dyke
23
Differential Calculus w.r.t. Space Definitions of
div, grad and curl
In 1-D
In 3-D
24
Alternative to the Integral Definition of
Grad We want the generalization of
continued
25
Alternative to the Integral Definition of
Grad Cylindrical coordinates
continued
26
Alternative to the Integral Definition of
Grad Spherical coordinates
27
Gradient
? high
?ndS (medium)
?ndS (large)
n
Resulting ?ndS
? low
dS
?ndS (small)
Elemental volume ?? with surface ?S
?ndS (medium)
magnitude and direction of the slope in the
scalar field at a point
28
Gradient

• Component of gradient is the partial derivative
  in the direction of that component
• Fouriers Law of Heat Conduction

29
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
Differential form of the Gradient
Cartesian system
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)
dz
k
i
j
P
adding these gives
Face 2
Proceeding in the same way for y and z
and
we get
, so
dx
Face 1
dy
31
An element of volume with a local Cartesian
coordinate system having its origin at the
centroid of the corners, O
Point M is at the centroid of the face
perpendicular to the y-axis with coordinates (0,
?y/2, 0) Other points in this face have the
coordinates (x, ?y/2, z)
.
O
Gradient of a Differentiable Function, F
continued
32
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Differential Forms of the Gradient
Cartesian
Cylindrical
Spherical
These differential forms define the vector
operator ?
35
Divergence
Fluid particle, coincident with ?? at time t,
after time ?t has elapsed.
n
dS
Elemental volume ?? with surface ?S
proportionate rate of change of volume of a
fluid particle
36
Differential Forms of the Divergence
Cartesian
Cylindrical
Spherical
37
Differential Forms of the Curl
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 ?

Pure rotation
No rotation
Rotation
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Curl
Elemental volume ?? with surface ?S
e
n
dS
Perimeter Ce
Area ??
ds
?h
n
dSds?h
radius a
v? avg. tangential velocity
twice the avg. angular velocity about e