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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). Gradient. Divergence. Curl.
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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)
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
Differential Forms of the Divergence Cartesian Cylindrical Spherical
Curl Elemental volume with surface S e n dS Perimeter Ce Area ds h n dS=dsh radius a v avg. tangential velocity = twice the avg. angular velocity about e
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
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
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 di+1 Recognizing that the summations are actually infinite: ni+1dS di
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
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
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
Alternative Definition of the Curl e Perimeter Ce Area ds
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 di+1 dsi Since the summations are actually infinite, and replacing with the more normal area symbol S: di
Stokes´ Theorem and Velocity • Apply Stokes´ Theorem to a velocity field • Or, in terms of vorticity and circulation • What about a closed surface?
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
Flow over a finite wing C S Wing with circulation must trail vorticity. Always.
Convective Operator = change in density in direction of V, multiplied by magnitude of V
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.
? 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.
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.