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Chapter 7: Problem 25
The speed of deep ocean waves depends on the wave length and gravitational acceleration. What are the appropriate dimensionless parameters?
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The following dimensionless groups are often used to present data on centrifugal pumps: flow coefficient \(\varphi=\frac{Q}{\omega D^{3}},\) head coefficient \(\psi=\frac{g H}{\omega^{2} D^{2}},\) power coefficient \(\xi=\frac{\dot{W}}{\omega^{3} D^{5}},\) efficiency \(r_{i}=\) \frac{\rho g Q H}{\dot{W}}, \text { specific speed } N_{s}=\frac{\omega \sqrt{Q}}{(g h)^{3 / 4}}, \text { specific diameter } D_{s}=\frac{\omega \sqrt{Q}}{(g h)^{1 / 4}} Show that the last three groups can be formed from combinations of the first three groups.
A stream of atmospheric air is used to keep a ping-pong ball aloft by blowing the air upward over the ball. The ping-pong ball has a mass of \(2.5 \mathrm{g}\) and a diameter \(D_{1}=3.8 \mathrm{cm},\) and the air stream has an upward velocity of \(V_{1}=0.942 \mathrm{m} / \mathrm{s}\). This system is to be modeled by pumping water upward with a velocity \(V_{2}\) over a solid ball of diameter \(D_{2}\) and density \(\rho_{b_{2}}=2710 \mathrm{kg} / \mathrm{m}^{3} .\) In both cases, the net weight of the ball \(W_{b}\) is equal to the air drag, $$\mathrm{W}_{b}=\frac{\mathrm{C}_{\mathrm{D}} \rho A V^{2}}{2}$$where \(\mathrm{C}_{\mathrm{D}}=0.60, \rho\) is the fluid density, \(A\) the ball's projected area, and \(V\) the velocity of the fluid upstream from the ball. Determine all possible combinations of \(V_{2}\) and \(D_{2}\). [Hint: A force balance involving the drag on the ball, the buoyant force on the ball, and the weight of the ball is needed.]
An incompressible fluid oscillates harmonically \(\left(V=V_{0}\right.\) \(\sin \omega t, \text { where } V \text { is the velocity })\) with a frequency of 10 rad/s in a 4-in.- -diameter pipe. A \(\frac{1}{4}\) scale model is to be used to determine the pressure difference per unit length, \(\Delta p_{\ell}\) (at any instant) along the pipe. Assume that $$\Delta p_{\ell}=f\left(D, V_{0}, \omega, t, \mu, \rho\right)$$ where \(D\) is the pipe diameter, \(\omega\) the frequency, \(t\) the time, \(\mu\) the fluid viscosity, and \(p\) the fluid density. (a) Determine the similarity requirements for the model and the prediction equation for \(\Delta p_{\ell}\) (b) If the same fluid is used in the model and the prototype at what frequency should the model operate?
The drag characteristics of an airplane are to be determined by model tests in a wind tunnel operated at an absolute pressure of \(1300 \mathrm{kPa}\). If the prototype is to cruise in standard air at 385 \(\mathrm{km} / \mathrm{hr},\) and the corresponding speed of the model is not o differ by more than \(20 \%\) from this (so that compressibility effects may be ignored), what range of length scales may be used if Reynolds number similarity is to be maintained? Assume the viscosity of air is unaffected by pressure, and the temperature of air in the tunnel is equal to the temperature of the air in which the airplane will fly.
A screw propeller has the following relevant dimensional parameters: axial thrust, \(F\), propeller diameter, \(D\), fluid kinematic viscosity, \(v,\) fluid density, \(\rho,\) gravitational acceleration, \(g,\) advance velocity, \(V,\) and rotational speed, \(N .\) Find appropriate dimensionless parameters to present the test data.
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