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Chapter 7: Problem 47

You are to conduct wind tunnel testing of a new football design that has a smaller lace height than previous designs (see Videos \(V 6.1\) and \(V 6.2\) ). It is known that you will need to maintain Re and St similarity for the testing. Based on standard college quarterbacks, the prototype parameters are set at \(V=40 \mathrm{mph}\) and \(\alpha=300 \mathrm{rpm},\) where \(V\) and \(\omega\) are the speed and angular velocity of the football. The prototype football has a 7 -in. diameter. Due to instrumentation required to measure pressure and shear stress on the surface of the football, the model will require a length scale of 2: 1 (the model will be larger than the prototype). Determine the required model free stream velocity and model angular velocity.

Short Answer

Expert verified
To maintain Reynolds and Strouhal similarity, the required model free stream velocity (\(V_m\)) and model angular velocity (\(\omega_m\)) should be equal to \(V_p*D_m/D_p\) and \(\omega_p\) respectively.

Step by step solution

01

Calculate Reynolds Number

The Reynolds number for the prototype can be calculated with the formula \(Re_p = \frac {V_pD_p}{\nu}\), where \(\nu\) is the kinematic viscosity, \(V_p\) is the speed of the prototype and \(D_p\) is the characteristic length, here the football diameter.
02

Calculate Reynolds Number for Model

The Reynolds number for the model should be the same as the prototype for flow similarity. Thus, simultaneous equations can be set up to solve for \(V_m\), the model speed: \(Re_m = Re_p\) and \(Re_m = \frac {V_mD_m}{\nu}\). Since model is larger by a factor of scale 2, \(D_m = 2*D_p\). Substitute \(Re_p\) and \(D_m\) into the equation and solve for \(V_m\).
03

Calculate Strouhal Number

The Strouhal number, which accounts for vortex shedding frequency, for the model must also equal that of the prototype \(St_m = St_p\). The formula for Strouhal number is \(St = \frac {fD}{V}\), where \(f\) is the frequency of vortex shedding, \(D\) is the characteristic length and \(V\) is the speed.
04

Calculate Model Angular Velocity

The frequency of vortex shedding is related to the football's spin, or angular velocity (\(\omega\)), as \(f = \frac {\omega}{2\pi}\). Therefore, the angular velocities of model and prototype should be the same for Strouhal similarity (\(\omega_m=\omega_p\)).

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Most popular questions from this chapter

(See The Wide World of Fluids article titled "Ice Engineering." Section \(7.9 .3 .)\) A model study is to be developed to determine the force exerted on bridge piers due to floating chuaks of ice in a river. The piers of interest have square cross sections. Assume that the force, \(R\), is a function of the pier width, \(b\), the thickness of the ice, \(d\), the velocity of the ice, \(V\), the acceleration of gravity, \(g,\) the density of the ice, \(\rho_{i},\) and a measure of the strength of the ice, \(E_{i},\) where \(E_{i}\) has the dimensions \(F L^{-2}\) (a) Based on these variables determine a suitable set of dimensionless variables for this problem. (b) The prototype conditions of interest include an ice thickness of 12 in. and an ice velocity of \(6 \mathrm{ft} / \mathrm{s}\). What model ice thickness and velocity would be required if the length scale is to be \(1 / 10 ?(\mathrm{c})\) If the model and prototype ice have the same density, can the model ice have the same strength properties as that of the prototype ice? Explain.

The excess pressure inside a bubble (discussed in Chapter 1 ) is known to be dependent on bubble radius and surface tension. After finding the pi terms, determine the variation in excess pressure if we (a) double the radius and (b) double the surface tension.

At a sudden contraction in a pipe the diameter changes from \(D_{1}\) to \(D_{2} .\) The pressure drop, \(\Delta p,\) which develops across the contraction is a function of \(D_{1}\) and \(D_{2},\) as well as the velocity, \(V\), in the larger pipe, and the fluid density, \(\rho,\) and viscosity, \(\mu .\) Use \(D_{1}, V,\) and \(\mu\) as repeating variables to determine a suitable set of dimensionless parameters. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable?

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.]

At a large fish hatchery the fish are reared in open, water-filled tanks. Each tank is approximately square in shape with curved corners, and the walls are smooth. To create motion in the tanks, water is supplied through a pipe at the edge of the tank. The water is drained from the tank through an opening at the center. (See Video \(\vee 7.9 .)\) A model with a length scale of 1: 13 is to be used to determine the velocity, \(V\), at various locations within the tank. Assume that \(V=f\left(\ell, \ell_{i}, \rho, \mu, g, Q\right)\) where \(\ell\) is some characteristic length such as the tank width, \(\ell\), represents a series of other pertinent lengths, such as inlet pipe diameter, fluid depth, etc.. \(\rho\) is the fluid density, \(\mu\) is the fluid viscosity, \(g\) is the acceleration of gravity, and \(Q\) is the discharge through the tank. (a) Determine a suitable set of dimensionless parameters for this problem and the prediction equation for the velocity. If water is to be used for the model, can all of the similarity requirements be satisfied? Explain and support your answer with the necessary calculations. (b) If the flowrate into the full-sized tank is 250 gpm, determine the required value for the model discharge assuming Froude number similarity. What model depth will correspond to a depth of 32 in. in the full sized tank?
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