AOE 5104 Class 5 9/9/08

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. 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 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 dS=dsh radius a v avg. tangential velocity = twice the avg. angular velocity about e

5. 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 Review

6. Integral Theorems and Second Order Operators

7. George Gabriel Stokes1819-1903

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

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 e 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 dsi Since the summations are actually infinite, and replacing  with the more normal area symbol S: 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. C 2D flow over airfoil with =0 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 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. ? We can generate an irrotational field by taking the gradient of any scalar field, since I got this one by randomly choosing And computing Class Exercise • Make up the most complex irrotational 3D velocity field you can.

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.