Physics > QUESTIONS & ANSWERS > AAE 333 Fluid Mechanics (All)

AAE 333 Fluid Mechanics

Document Content and Description Below

[40 pts] A large water tank empties slowly through a small hole under the force of gravity. The flow is steady, and the volumetric flow rate ??̇ (in m3/s) depends on the exit velocity V, the gravit ... ational acceleration g, the depth of the water h, and the diameter of the hole D. the viscosity µ, and the surface tension σ (force/unit length). (a) Express the nondimensional volumetric flow rate ??̇ in terms of its dependence on the other nondimensional groups. (b) A test is to be made on a 1/4th scale model. What is the ratio of the model volumetric flow rate to the prototype volumetric flow rate that would be needed to obtain dynamical similarity? (c) (i) If you now decide that, instead of the volumetric flow rate, you are interested in the mass flow rate ??̇ , and the density ?? must therefore be included in the dimensional analysis, will the number of nondimensional groups change? (ii) Without going through the full derivation again, use the results of part (a) to express the nondimensional mass flow rate ??̇ in terms of its dependence on the other nondimensional groups. (d) If you later discover that the mass flow rate ??̇ depends on the viscosity ?? and the surface tension ?? (dimensions of force per unit length), in addition to ??, ??, ??, ℎ, and ??, find the mass flow rate ??̇ in terms of its dependence on the other nondimensional groups. Hint: you may avoid the full Buckingham Pi derivation by using the nondimensional groups from part (c) and compiling additional groups with “new” variables and “old” repeating variables simply by inspection. Solution: (a) ??̇ = ??(??, ??, ℎ, ??). N=5 variables. Denote L=length, T=time, we have [??̇] = L3 T [ℎ] = L; [??] = L; [??] = L T ; [??] = L 2T . Of the three fundamental units (mass, length, and time), mass is not present here, so K=2. Therefore, we should have N-K=3 nondimensional groups. Select ??, ?? as repeating variables. Show that we can obtain each fundamental unit with these two variables: [??] = L; �????� = T. Form the nondimensional Pi products: Π1 = [????1????1ℎ] = ????1(????−1)??1?? = 1 => ??1 + ??1 + 1 = 0; ??1 = 0 => ??1 = −1 => Π1 = ℎ ?? . Π2 = [????2????2??] = ????2(????−1)??2????−2 = 1 => ??2 + ??2 + 1 = 0; −??2 − 2 = 0 => ??2 = 1; ??2 = −2 => Π2 = ????−2?? = ???? ??2 . [Show More]

Last updated: 3 years ago

Preview 1 out of 9 pages

Buy Now

Instant download

We Accept: