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3.21 From Eq. (3.92), the stream function for the flow over a circular cylinder is:
At θ = /2,
Streamline 2 goes through point (1.2R, /2).
Hence, the volume flow rate between 2 and 1 is
The volume flow rate between 3 and 1 is twice this amount:
Thus,
Since the general expression for the streamfunction is given by Eq. (3.92), the
coordinate directly above the top of the cylinder through which 3 flows is
obtained from
Multiply both sides by r/R:
or:
From the quadratic formula:
Only the plus sign makes physical sense here. Thus,
The point directly above the top of the cylinder through which 3 flows is
Note that the spacing between 3 and 2 over the top of the cylinder is
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3.22 From Section 3.13, the maximum velocity in the flowfield over a circular
cylinder occurs at the top and bottom of the cylinder, and is equal to 2 V∞.
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3.23 In Section 3.6, the condition for incompressible flow is V = 0. If this
Indeed, in the solution provided in Equation 2.1, the expression for V is
obtained as
