The previous flow example has limited application since the body isn

1
Rankine Ovals

• The previous flow example has limited application
  since the body isnt closed but extends
  downstream to x??.
• A simple way of closing the body is to add a sink
  downstream of the source and of equal strength.
• For symmetry purposes, we will also move the
  origin to be halfway between the source and sink.
  Thus
• Since the center of the source and sink are no
  longer on the origin, the definitions of r and ?
  in terms of x and y also change.

2
Rankine Ovals 2

• For any point (x,y) in space, the following
  relations apply
• And the stream function solution is

3
Rankine Ovals 3

• The stream (and potential) function solution to
  this case looks lie
• This body type is known as a Rankine Oval.

4
Rankine Ovals 4

• First lets find the length of this body.
• Note that for this flow, there are two
  stagnations points one at both the leading edge
  and trailing edges.
• To determine the location of these points on the
  y axis, once again find where u0.
• At the stagnation point (xs,0)

5
Rankine Ovals 5

• With reduction, this expression becomes
• Or
• The streamline which forms the body shape is
  found from the value of psi at the right
  stagnation point
• Since psi is zero on the surface, the following
  relationship exists

6
Rankine Ovals 6

• Or, the difference in the two angles is
• Where a nondimensional source strength, q, is
  defined by
• Taking the tangent of the first equation,
  substituting for the tangent of the angles, and
  then solving for x, the following equation is
  found for the surface
• Which is really x(Y) rather than Y(x) but is
  the best we can do.

7
Rankine Ovals 6

• Just for completeness, the u and v components of
  this flow velocity are
• And, once again, the pressure coefficients can be
  found from
• But I really doubt it this can be simplified to a
  short simple equation.

8
Doublets and Cylinders

• While Rankine ovals are not a common aerodynamic
  shape, they do lead to one that is cylinders.
• First, consider a source/sink pair and consider
  the limit as they closer and closer (xo ? 0)
  while their strengths increase such that Q 2xo
  ? constant, we get
• This limit can be evaluated by looking at the
  details of the triangle formed

9
Doublets and Cylinders 2

• By right triangle rules
• In the limit, d? is very small, r1 r2 r, and
  ?1 ?2 ?.
• Thus
• Or similarly for the potential function

10
Doublets and Cylinders 3

• The streamlines defined by this Doublet look
  like
• One nice feature of a doublet is that it has a
  zero net mass flux what flows out flows back
  in!
• The other nice feature is when a doubled is
  combined with a uniform flow.

11
Doublets and Cylinders 4

• When a uniform flow is superimposed on a doublet,
  the flow looks like that around a circular
  cylinder.

12
Doublets and Cylinders 5

• The stream and potential functions for the flow
  just shown are usually written as
• As part of your homework, you will show that the
  velocity on the surface of the cylinder is given
  by
• And thus the surface pressure coefficient is