lecture11

# lecture11 - 2.20 Marine Hydrodynamics Spring 2005 Lecture...

This preview shows pages 1–5. Sign up to view the full content.

Lecture 11 - Marine Hydrodynamics Lecture 11 3.11 - Method of Images m Potential for single source: φ = ln x 2 + y 2 2 π m Potential for source near a wall: φ = m ln x 2 + ( y b ) 2 + ln x 2 + ( y + b ) 2 2 π b b Added source for 0 = φ dy d x y m m symmetry Note: Be sure to verify that the boundary conditions are satisfied by symmetry or by calculus for φ ( y ) = φ ( y ). 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Vortex near a wall (ground effect): φ = U x + Γ tan 1 ( y b ) tan 1 ( y + b ) 2 π x x b b Γ U x y - Γ Added vortex for symmetry Verify that = 0 on the wall y = 0. dy 2 2 φ a a Circle of radius a near a wall: = Ux 1 + + x 2 + ( y b ) 2 x 2 + ( y + b ) 2 b b U y x y This solution satisfies the boundary condition on the wall ( ∂φ = 0), and the degree it ∂n satisfies the boundary condition of no ﬂux through the circle boundary increases as the ratio b/a >> 1, i.e., the velocity due to the image dipole small on the real circle for b >> a . For a 2D dipole, φ d 1 , φ d 1 2 . 2
More than one wall: b' b U b' b U b' Example 1: Example 2: Example 3: b b b b - Γ Γ b' b' b' b' b' b' b' b' Γ - Γ b b b b 3.12 Forces on a body undergoing steady translation “D’Alembert’s paradox” 3.12.1 Fixed bodies & translating bodies - Galilean transformation . y y x U o x o z z Fixed in space Fixed in translating body x = x` + Ut 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Reference system O: v, φ, p Reference system O’: v , φ , p U S O X U S O’ X 2 φ = 0 v · ˆ n = ∂φ ∂n = U · ˆ n = ( U, 0 , 0) · ( n x , n y , n z = Un x on Body v 0 as | x | → ∞ φ 0 as | x | → ∞ ) 2 φ = 0 v · ˆ n = ∂φ ∂n = 0 v ( U, 0 , 0) as | x | → ∞ φ → − Ux as | x
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern