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Sample Exam 2 Questions with Answers.
Typology: Exams
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Allotted Time: 3 Hours
The exam is closed book and closed notes. Students are allowed one (double-sided) formula sheet.
There are five questions on this exam. Answer any four (each for 25 points). If you answer all five, the best four will be considered.
State all your assumptions, and explain your reasoning clearly.
a) Show that for an irrotational flow that is also incompressible, the velocity potential also satisfies the Laplace equation. b) Furthermore, in two-dimensional, incompressible flows, a stream-function (x,y) exists such that 𝜕 𝜕𝑥 = −𝑣^
𝜕 𝜕𝑦 = 𝑢 so that and satisfy the Cauchy-Riemann equations:
where w(z) is a complex analytic function of z = x+iy. Interpret any four of the following flows, write down expressions for the velocity components, and draw streamlines where possible: I. w(z) =z II. w(z) = z^2 III. w(z) = m log(z) IV. w(z) = i log(z) V. w(z) = U a (z/a + a/z), with real positive constants U and a.
Hint: Feel free to use polar coordinates z=r exp(i) for some of the above cases to simplify analyses.
b. the local isentropic stagnation pressure c. the friction force on the duct wall between sections 1 and 2
What additional information and assumptions would you need if you are asked to determine the distance between the sections 1 and 2? Where would you find these additional quantities?
Show the steps you would have followed to find the distance between the sections 1 and 2. Use either a symbol or a logical value for any missing quantity (for this question only).
