1. 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)

2. Review Gradient Divergence Curl
For velocity: twice the circumferentially averaged angular velocity of a fluid particle = Vorticity Ω
Magnitude and direction of the slope in the scalar field at a point
For velocity: proportionate rate of change of volume of a fluid particle
Gradient
Divergence
Curl

3. Differential Forms of the Divergence
Cartesian
Cylindrical
Spherical

4. Curl Elemental volume  with surface S e n dS Perimeter Ce Area 
dS=dsh
radius a
v avg. tangential velocity
= twice the avg. angular velocity
about e

5. Review Gradient Divergence Curl
For velocity: twice the circumferentially averaged angular velocity of a fluid particle = Vorticity Ω
Magnitude and direction of the slope in the scalar field at a point
For velocity: proportionate rate of change of volume of a fluid particle
Gradient
Divergence
Curl

6. Integral Theorems and Second Order Operators
AOE 5104 Class 5 9/5/06
Assignments:
Collect HW1
Homework 2, due 9/12/06
Online presentation “Equations of Motion 1”

7. George Gabriel Stokes 1819-1903

8. 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

9. 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i+1
Recognizing that the summations are actually infinite:
ni+1dS
di

10. Assumptions in Gradient Theorem
A pure math result, applies to all flows
However, S must be chosen so that  is defined throughout R S
Submarine
surface 

11. 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

12. 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

13. Alternative Definition of the Curl
Perimeter Ce
Area  ds

14. 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:
dsi+1
di+1
Since the summations are actually infinite, and replacing  with the more normal area symbol S:
dsi
di

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

16. 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
2D flow over airfoil with =0 C
The vorticity doesn’t imply anything about the circulation around C

17. Flow over a finite wing C S
Wing with circulation must trail vorticity. Always.

18. Vector Operators of Vector Products

19. Convective Operator
= change in density in direction of V, multiplied by magnitude of V

20. Second Order Operators
The Laplacian, may also be applied to a vector field.
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.

21. Class Exercise
Make up the most complex irrotational 3D velocity field you can. ?
We can generate an irrotational field by taking the gradient of any scalar field, since
I got this one by randomly choosing
And computing

22. 2nd Order Integral Theorems
Green’s theorem (1st form)
Green’s theorem (2nd form)
Volume R
with Surface S
ndS d
These are both re-expressions of the divergence theorem.