1. 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

2. I have added the slides without numbers. The numbered slides are the original file.

3. Last Class Changes in Unit Vectors Calculus w.r.t. time Integral calculus w.r.t. space Today: differential calculus in 3D P P' erer ee ezez  dd r z

4. Oliver Heaviside 1850-1925

5. Shock in a CD Nozzle Bourgoing & Benay (2005), ONERA, France Schlieren visualization Sensitive to in-plane index of ref. gradient

6. In 1-D In 3-D Differential Calculus w.r.t. Space Definitions of div, grad and curl Elemental volume  with surface  S n dS D=D(r),  =  (r)

7.  ndS (large) Gradient 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

8. Review Gradient Magnitude and direction of the slope in the scalar field at a point

9. Gradient Component of gradient is the partial derivative in the direction of that component Fourier´s Law of Heat Conduction  = low  = high

10. Face 2 Differential form of the Gradient Cartesian system dy dx dz j i k 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 adding these gives Proceeding in the same way for y and z and we get, so

11. Differential Forms of the Gradient These differential forms define the vector operator  Cartesian Cylindrical Spherical

12. continued

16. .. A B

17. .... time t time t + Δt A A’ B’ B d fluid particle moves from here to here during Δt

19. dS n Divergence Fluid particle, coincident with  at time t, after time  t has elapsed. = proportionate rate of change of volume of a fluid particle Elemental volume  with surface  S

20. Review Divergence For velocity: proportionate rate of change of volume of a fluid particle Gradient Magnitude and direction of the slope in the scalar field at a point

21. Differential Forms of the Divergence Cartesian Cylindrical Spherical

22. Differential Forms of the Curl CartesianCylindricalSpherical Curl of the velocity vector  V = twice the circumferentially averaged angular velocity of -the flow around a point, or -a fluid particle =Vorticity Ω Pure rotationNo rotationRotation

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

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

25. Oliver Heaviside 1850-1925 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.

26. Integral Theorems and Second Order Operators

27. 1st Order Integral Theorems Gradient theorem Divergence theorem Curl theorem Stokes’ theorem Volume R with Surface S dd ndS Open Surface S with Perimeter C ndS

28. The Gradient Theorem Finite Volume R Surface S dd Begin with the definition of grad: Sum over all the d  in R: didi d  i+1 n i dS n i+1 dS We note that contributions to the RHS from internal surfaces between elements cancel, and so: Recognizing that the summations are actually infinite:

29. 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 

30. Flow over a finite wing S1S1 S2S2 S = S 1 + S 2 R is the volume of fluid enclosed between S 1 and S 2 S1S1 p is not defined inside the wing so the wing itself must be excluded from the integral

31. 1st Order Integral Theorems Gradient theorem Divergence theorem Curl theorem Stokes’ theorem Volume R with Surface S dd ndS Open Surface S with Perimeter C ndS

32. Alternative Definition of the Curl e Perimeter C e dsds Area 

33. Stokes’ Theorem Finite Surface S With Perimeter C d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: didi n dsidsi ds i+1 d  i+1

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

35. 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

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

37. Vector Operators of Vector Products

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

39. 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.

40. Class Exercise 1.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

41. 2 nd Order Integral Theorems Green’s theorem (1 st form) Green’s theorem (2 nd form) Volume R with Surface S dd ndS These are both re-expressions of the divergence theorem.