aoe 5104 class 5 9/9/08 online presentations for next class: –equations of motion 1 homework 2 due...
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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.