3.8 - Method of Images

• Source near wall:

  
  

  $$\phi = \frac{m}{2\pi }\left( {\ln \sqrt {x^2 + \left( {y - b}... ...^2} } { + \ln \sqrt {x^2 + \left( {y + b} \right)^2} } \right)$$
  
  

  
  

  
  
  

  1.

  Question: verify that the boundary condition of no flux on the wall is satisfied.

  (a)

  Hint: no flux trough the wall $\rightarrow \phi_{y} = 0$ on $y = 0$.

  (b)

  Hint: evaluate $\frac{\partial \phi}{\partial y}$.

• Vortex near a wall: (ground effect)

  
  

  $$\phi = U_x + \frac{\Gamma }{2\pi }\left( {\tan ^{ - 1}\frac{y - b}{x} - \tan ^{ - 1}\frac{y + b}{x}} \right)$$
  
  
  
  
  
  
  

  1.

  Question: verify that the boundary condition of no flux on the wall is satisfied.

  (a)

  Hint: no flux trough the wall $\rightarrow \phi_{y} = 0$ on $y = 0$.

  (b)

  Hint: evaluate $\frac{\partial \phi}{\partial y}$.

• Circle near a wall: (radius a)

  
  

  $$\phi \cong Ux\left( {1 + \frac{a^2}{x^2 + \left( {y - b} \rig... ...} + \frac{a^2}{x^2 + \left( {y + b} \right)^2}} \right) \notag$$
  
  

  
  

  
  
  

  This solution satisfies the boundary condition on the wall ( $\frac{\partial \phi}{\partial n} = 0$), and the degree it 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, $\phi \sim \frac{1}{d},\nabla \phi \sim \frac{1}{d^2}$

  1.

  Question: verify the boundary condition of no flux at the wall.

  (a)

  Hint: evaluate the component of the velocity vector in the $y$ direction.

  (b)

  Hint: set $y = 0$ in the expression for the velocity vector in the $y$ direction.

  Homework 1: At the circle boundary the no flux boundary condition should be satisfied. Check to which degree is the boundary condition of no flux at the circle boundary satisfied.

  More than one wall:

Homework 2: Use the method of images to derive the potential function for a circle in a free-stream of velocity $U$ in the $+x$ direction, but between two walls parrallel to each other and to the free stream (first example in the figure above).

Keyword Search

 

• Solution: method of image for a 2D source near a wall
• Solution: method of image for a 2D vortex near a wall
• Solution: method of image for a circle in a free stream near a wall - boundary condition at the wall.

3.9 Forces on a body undergoing steady translation - D'Alembert's ``paradox''

• 3.9.1 Fixed bodies & translating bodies - Galilean transformation
• 3.9.2 Forces

3.10 Lift due to circulation