137
ANSWERS TO PROBLEMS REFERRED TO IN THE TEXT OF
CHAPTER 14
Please note: There are two problems that are described within the text of Chapter 14; these are
not specifically itemized at the end of the chapter. In this Solutions Manual, they will be referred
to by page numbers.
14.1 (See page 509 in the text).
Statement of the problem: Derive Eq. (14.15)
SOLUTION
2 3/ =
14.2 (See page 521 in the text.)
Statement of the problem: Drive the transformed full velocity potential equation. I.e., transform
Eq. (8.17) to (, ) space.
SOLUTION: Writing Eq. (8.17) for a two-dimensional flow, we have
138
2
2
2
2
Similarly,
x
x
In Eq. (4), the terms labeled as B and C are mixed derivatives, which can be evaluated by the
chain rule as follows:
2
2
2
2
2
2
y
Then
2
D
140
2
y
2
y
However,
y
y
2
Substituting Eqs. (9) and (10) into (8)
2
2
This is the transformed second derivative with respect to y. Let us now proceed to find the
transformed second derivative with respect to x and y.
2
x
2
x
2
2
D
141
Substituting Eqs. (2), (3), (7), (11), and (13) into (1), we have
2
This is the full velocity potential equation in terms of and as the new independent variables,
where the metrics x, y, xy, x, y, xy are given by the transformation