aoe 5104 class 4 9/4/08 online presentations for today’s class: –vector algebra and calculus 2...
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AOE 5104 Class 4 9/4/08
• Online presentations for today’s class:– Vector Algebra and Calculus 2 and 3
• Vector Algebra and Calculus Crib• Homework 1 • Homework 2 due 9/11• Study group assignments have been made and
are online. • Recitations will be
– Mondays @ 5:30pm (with Nathan Alexander) in Randolph 221
– Tuesdays @ 5pm (with Chris Rock) in Whitemore 349
I have added the slides without numbers. The numbered slides are theoriginal file.
Last Class
• Changes in Unit Vectors
• Calculus w.r.t. time• Integral calculus w.r.t.
space• Today: differential
calculus in 3D
PP'
er
e
ez
dr
zrdd ee
ee dd r 0zde
dtdtdt
ttt
ttt
ttt
BABA
BAB
ABA
BAB
ABA
BABA
.
Oliver Heaviside1850-1925
Shock in a CD Nozzle
Bourgoing & Benay (2005), ONERA, France Schlieren visualizationSensitive to in-plane index of ref. gradient
In 1-D
τ 0ΔS
1grad lim dS
τ
n
In 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) x 0
1lim ( ) ( )
dff x x f x
dx x
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
Review
ΔS
0τ dSτ
1Limgrad n
Gradient
Magnitude and direction of the slope in the scalar field at a point
Gradient
• Component of gradient is the partial derivative in the direction of that component
• Fourier´s Law of Heat Conduction
ss
.e
n.Tkn
Tkq
= low
= high
ss e,
zyxdxdydz
zdxdydz
ydxdydz
x
kjikji
n
τ
1Lim
dSτ
1Limgrad
0τ
ΔS
0τ
Face 2
Differential form of the Gradient
ΔS
0τ dSτ
1Limgrad n
Cartesian 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
dydzdx
xFace
)(2
dS1
in
dydzdx
xFace
)(2
dS2
in
adding these gives dxdydzx
i
Proceeding in the same way for y and z
dxdydzy
j dxdydz
z
kandwe get , so
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
, :
rad where
Here rad maps a vector, , into another vector, , and plays the role of a derivative for the vectorfield
Gradient of a vector V V i j k
V r r V r V V r r i j k
V r V
V i j
G
G
u v w
d d d d dx dy dz
d d
d du dv
k i j k
V
u u u v v v w w wdw dx dy dz dx dy dz dx dy dz
x y y x y z x y z
u u u
x y zdu dx
v v vd dv d
x y zdw
w w w
x y z
rad
Some people prefer to use "dyadic" notation, to call rad a dyad, and to write it as follows:
rad
V
V
V i i i j i k
G
G
G
u u u
x y z
v v vy
x y zdz
w w w
x y z
u u u
x y z
where denotes the so-called dyadic product, which does not have a geometric interpretation. Often is omitted. The
first base vector indicates the co
j i j j j k k i k j k kv v v w w w
x y z x y z
mponent of being differentiated and the second indicates the direction of the derivative;
hence, Grad has nine components: three derivatives of each of its three components.
V
Vcontinued
, : ; rad where
definition:
rad ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
u v w d d d d dx dy dz
u u u v v v w wd d d d d d d d d
x y z x y z x y
Gradient of a vector V V i j k V r r V r V V r r i j k
V r i i r i j r i k r j i r j j r j k r k i r k j r
G
G ( )
rad
similarly
rad ( ) ( ) ( ) ( ) ( ) ( ) (
wd
z
u u u v v v w w wd dx dy dz dx dy dz dx dy dz d
x y z x y z x y z
u u u v v v wd d d d d d d
x y z x y z x
k k r
V r i j k V
r V r i i r i j r i k r j i r j j r j k
G
G ) ( ) ( )
rad rad
w wd d d
y z
u u u v v v w w wd dx dx dy dz d
x y z x y z x y z
u v w u v wdx dy dz dx dy dz
x x x y y y
r k i r k j r k k
r V i j k i j k i j k V r
i j
G G
Note: in general
u v wdx dy dz d
z z z
u u u u u u
x y z x y zdx
v v v v v vdy dx dy dz
x y z x y zdz
w w w w w w
x y z x y z
k V
continued
is defined so that it is consistent with the parallelogram law of addition (PLA):
and
Addition of dyads
A C B C T Sxx xy xz x xx xy xz
yx yy yz y yx yy yz
zx zy zz z zx zy z
T T T C S S S
T T T C S S S
T T T C S S S
according to the PLA , etc.
where
R A B
R A B C C C T S= T S
T S
x
y
z z
x x x
x xx x xy y xz z xx x xy y xz z
x xx xx x xy xy y xz xz z
xx xx xy xy xz xz
yx y
C
C
C
R A B
R T C T C T C S C S C S C
R T S C T S C T S C
T S T S T S
T S
To preserve the parallelogram law of addition, we need to add the corresponding
components of the dyads.
x yy yy yz yz
zx zx zy zy zz zz
T S T S
T S T S T S
continued
of rad where denotes velocity
1 1
2 2
1 1rad
2 2
1
2
u u v u wu u ux y x z xx y z
v v v u v v v w
x y z y x y z y
w w w u wx y z z x
Decomposition V V
V
G
G
1 10
2 2
1 10
2 2
1 1 10
2 2 2
u v u w
y x z x
v u v w
x y z y
v w w w u w v
z y z x z y z
rad train ot read strain of and rotation of
train is symmetric, . ot is antisymmetriij jiS S
V V V V V
V V
G St R
St R c, .
train has six distinct components. ot has only three distinct components.
Note:
Curl
ij jiR R
w v u w v u
x y z y z z x x
u v w
V V
i j k
V i j
St R
01
and ot 02
0
Let represent any vector; then Curl
01
ot 02
0
z y
x y z z x
y x
x y z x y z z y y z x z z x y x x y
x y z
z y
z x
y x
y
A A A A A A A A A
A A A
k i j k V
i j k
A i j k V A i j k
V A
R
R1 1
Curl2 2
x z y y z
y x z z x
z y x x y
A A A
A A A
A A A
V A
continued
.
.Bv
AvA
B
dr
and are two points in the same fluid separated
by the infinitesimal position vector
rad
ot train
1 Curl train
2
B A A
A
A
A B
d
d d
d d
d d
r
v v v v v r
v v r v r
v v r v r
G
R S
S
:
1) translation at the velocity
12) rigid-body rotation at the angular velocity Curl
2: the vector Curl is simultaneously
we can identify the three components of the motion of a continuum
v
ω v
note v r
A
d to both Curl and ;
the fact that Curl is to means that this term does not cause
to change length
3) deformation given by train
train
v r
v r r
r
v r
v v ω r v r
S
SB A
d
d d
d
d
d d
using Taylor series and the PLA, we can write
A B
B A
t d d t
d
d v r r v
v v v
.
.
.
.
time t
time t + ΔtA
A’
B’
Bd
fluid particlemoves fromhere
to hereduring Δt
dr
d r
B tv
A tv
τ 0ΔS
1div lim dS
τ
V V n
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
Review
τ 0ΔS
1div lim . dS
τ
V V n
Divergence
For velocity: proportionate rate of change of volume of a fluid particle
τ 0ΔS
1grad lim dS
τ
n
Gradient
Magnitude and direction of the slope in the scalar field at a point
Differential Forms of the Divergence
sin
sin sin sin
yx z
r zr z
2r
r2
divA = A
AA A+ + i + j + k .Ax y z x y z
1 1 erA A A+ + + + .Ae er r r z r r z
1 1 1 eA er A A+ + + + .Aer r r r r rr
Cartesian
Cylindrical
Spherical
Differential Forms of the Curl
sin
sin
sin
r z r
2
r z rx y z
r r ri j e e e e e ek1 1
curlA = A = = =x y z r r z rr
A rA A rA rA AA A A
τ 0ΔS
1curl lim dS
τA A n
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
Curlτ 0
ΔS
1curl lim dS
τ
V V n
dS
n
enPerimeter Ce
dsh
dS=dsh
Area
radius a
v avg. tangential velocity= twice the avg. angular velocity
about e
Elemental volume with surface S
τ 0ΔS
1.curl lim . dS
τ e V e V n
τ 0ΔS
1.curl lim . dS
h
e V V e n
τ 0ΔS
1.curl lim . ds h
h
e V V e n
e
Ce
τ 0 τ 0C
1.curl lim .d lim
e V V s
20 0
1.curl lim 2 2lim
a a
vv a
a a
e V
Review
0ΔS
1div lim . dS
τ
V V n
Divergence
For velocity: proportionate rate of change of volume of a fluid particle
0ΔS
1grad lim dS
τ
n
Gradient
Magnitude and direction of the slope in the scalar field at a point
0ΔS
1curl lim dS
τ
V V n
Curl
For velocity: twice the circumferentially averaged angular velocity of a fluid particle = Vorticity Ω
Oliver Heaviside1850-1925
Writes Electromagnetic induction and its propagation over the course of two years, re-expressing Maxwell's results in 3 (complex) vector form, giving it much of its modern form and collecting together the basic set of equations from which electromagnetic theory may be derived (often called "Maxwell's equations"). In the process, He invents the modern vector calculus notation, including the gradient, divergence and curl of a vector.
Integral Theorems and Second Order Operators
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
C
SSd 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
2 2 - dn nR
S
d S
These are both re-expressions of the divergence theorem.
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