AOE 5104 Class 5 9/9/08

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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 430-445pm
  (and I will stay late)

2
Review
3
Differential Forms of the Divergence
Cartesian
Cylindrical
Spherical
4
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
5
Review
6
Integral Theorems and Second Order Operators
7
George Gabriel Stokes1819-1903
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1st Order Integral Theorems
Volume R with Surface S

• Gradient theorem
• Divergence theorem
• Curl theorem
• Stokes theorem

ndS
d?
Open Surface S with Perimeter C
ndS
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The Gradient Theorem
Finite Volume R Surface S
Begin with the definition of grad
Sum over all the d? in R
d?
We note that contributions to the RHS from
internal surfaces between elements cancel, and so
nidS
d?i1
Recognizing that the summations are actually
infinite
ni1dS
d?i
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Assumptions in Gradient Theorem

• A pure math result, applies to all flows
• However, S must be chosen so that ? is defined
  throughout R

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Flow over a finite wing
S1
S1
S2
S S1 S2
R is the volume of fluid enclosed between S1 and
S2
p is not defined inside the wing so the wing
itself must be excluded from the integral
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1st Order Integral Theorems
Volume R with Surface S

• Gradient theorem
• Divergence theorem
• Curl theorem
• Stokes theorem

ndS
d?
Open Surface S with Perimeter C
ndS
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Alternative Definition of the Curl
e
Perimeter Ce
Area ??
ds
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Stokes Theorem
Finite Surface S With Perimeter C
Begin with the alternative definition of curl,
choosing the direction e to be the outward normal
to the surface n
n
Sum over all the d? in S
d?
Note that contributions to the RHS from internal
boundaries between elements cancel, and so
dsi1
d?i1
dsi
Since the summations are actually infinite, and
replacing ? with the more normal area symbol S
d?i
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Stokes Theorem and Velocity

• Apply Stokes Theorem to a velocity field
• Or, in terms of vorticity and circulation
• What about a closed surface?

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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

The vorticity doesnt imply anything about the
circulation around C
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Flow over a finite wing
C
S
Wing with circulation must trail vorticity.
Always.
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Vector Operators of Vector Products
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Convective Operator
change in density in direction of V, multiplied
by magnitude of V
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Second Order Operators
The Laplacian, may also be applied to a vector
field.

• So, any vector differential equation of the form
  ??B0 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
  ?.B0 can be solved identically by writing B??A.
  
• We say B is solenoidal or incompressible.
• We refer to A as the vector potential.

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Class Exercise

1. Make up the most complex irrotational 3D velocity
   field you can.

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2nd Order Integral Theorems

• Greens theorem (1st form)
• Greens theorem (2nd form)

Volume R with Surface S
ndS
d?
These are both re-expressions of the divergence
theorem.