**April 8 Homework Solutions**
**7.1 The Reynolds number, VD/, is a very important parameter in fluid mechanics. Verify that the Reynolds number is dimensionless using both the FLT and MLT system for basic dimensions and determine its value for ethyl alcohol flowing at a velocity of 3 m/s through a 2-in diameter pipe.**
In the FLT system, the dimensions of density, velocity, diameter and viscosity are FL^{-4}T^{2}, LT^{-1}, L, and FL^{-2}T. Substituting these dimensions into the definition of the Reynolds number gives:

In the MLT system, the dimensions of density, velocity, diameter and viscosity are ML^{-3}, LT^{-1}, L, and ML^{-1}T^{-1}. Substituting these dimensions into the definition of the Reynolds number gives:

Thus, the Reynolds number is dimensionless regardless of the unit system used.

For ethyl alcohol the density and viscosity are 789 kg/m^{3} and 0.00119 N·s/m^{2} = 0.00119 kg/m·s, and the reynolds number is

**1.01x10**^{5}.

**7.35 The flowrate over the spillway a dam is 27,000 ft**^{3}/min. Determine the required flow rate for a 1/25 scale model that is operated in accordance with Froude number similarity.

For Froude number similarity between the model (subscript m) and the actual dam (no subscript) we must have

For a 1/25 scale model, all dimensions will have this scale. The flow rate in the model can be written as Q_{m} = V_{m}A_{m}, where A_{m} is the area of the water flow over the spillway in the model. The water flow in the actual dam can be written as Q = VA, where A is the actual water area over the spillway. The ratio of model area to actual area will be equal to the length scale squared.

Substituting the given value of 1/25 for the length scale gives the following result for the flowrate ratio between the model and the actual spillway.

.

Thus the model flow rate will be 0.00032(27,000 ft^{3}/min) or **Q**_{m}** = 8.64 ft**^{3}**/min**.

**7.38 If an airplane travels at a speed of 1120 km/h at an altitude of 15 km, what is the required speed at an altitude of 8 km to satisfy Mach number similarity? Assume the air properties correspond to those for the U.S. standard atmosphere.**
For Mach number similarity, the ratio of the speed, V, to the sound speed, c, must be the same. If we assume that air behaves as an ideal gas we can compute the sound speed from the equation c^{2} = kRT, where k is the heat capacity ratio. Thus, for Mach number similarity we can write

From Table C.2 for the U.S. standard atmosphere we find the temperatures at 15 km and 8 km, respectively, are -56.50^{o}C = 216.65 K and -36.94^{o}C = 236.21 K. The values of k are given as a function of temperature in Table B.4. At the two lowest temperatures, -40^{o}C and -20^{o}C the value of k is 1.401. We can assume that the two temperatures here, -56.50^{o}C and -36.94^{o}C will have the same value of k. With this assumption, our unknown velocity, V_{2}, can be found as follows.

=

**7.46 For a certain model study involving a 1:5 scale model it is known that Froude number similarity must be maintained. The possibility of cavitation is also to be investigated, and it is assumed that the cavitation number must be the same for the model and prototype. The prototype fluid is water at 30**^{o}C, and the model fluid is water at 70^{o}C. If the prototype operates at an ambient pressure of 101 kPa(abs), what is the required ambient pressure for the model system.
For Froude number similarity between the model (subscript m) and the prototype (subscript p) we must have

For cavitation number similarity we must have the same cavitation number in model and prototype.

Manipulating this equation gives

From the Froude number similarity we can replace the ratio of velocity squatred by the length scale. This gives the following equation for the ambient pressure in the model.

We can find the water properties needed for this equation from Table B.2. For water at 30^{o}C the density is 995.7 kg/m^{3} and vapor pressure is 4243 Pa(abs). For water at 70^{o}C the density is 977.8 kg/m^{3} and vapor pressure is3.116x10^{4} Pa(abs).

Substituting these water properties and the length sacle of 1:5 into the equation for the ambient pressure in the model gives

**p**_{amb,m}** = 5.02x10**^{4}** Pa(abs)**

**7.55 The drag on a small completely submerged solid body having a characteristic length of 2.5 mm and moving with a velocity of 10 m/s through water is to be determined with the aid of a model. The length scale is to be 50, which indicates that the model is to be ***larger* than the prototype. Investigate the possibility of using either an unpressurized wind tunnel or a water tunnel for this study. Determine the required velocity in both the wind and water tunnels, and the relationship between the model drag and the prototype drag for both systems. Would either type of test facility be suitable for this study?

For completely submerged flow we must have Reynolds number similarity between the model (m) and prototype (p). With the stated scale factor, the Reynolds number scaling gives the following relationship between the velocity of the model and that of the prototype.

If we use the same temperature for both the prototype and the model in the water tunnel, the kinematic viscosity will be the same in both cases and the velocity scale will be the same as the length scale: 1/50. This means that the model velocity will be 1/50^{th} of the prototype velocity of 10 m/s. So, for the water tunnel,** V**_{m}** = 0.200 m/s**.

If we use a wind tunnel, the velocity scale will depend on the viscosity ratio. Using the standard data for kinematic viscosity we have for the prototype in water, _{p} = 1.12x10^{-6} m^{2}/s and _{m} = 1.46x10^{-5} m^{2}/s for the model in air. This gives the following velocity scale.

Multiplying the prototype velocity of 10 m/s by this scale factor gives **V**_{m}** = 2.61 m/s** as the model velocity in the wind tunnel.

Both of these velocities seem reasonable and we could use either as the test medium. The computation of the drag force for the prototype from drag force measurements on the model requires that the loss coefficient be the same for both the model and the prototype.

The relationship between the drag force on the prototype, F_{D,p}, and the drag force on the model, F_{D,m}, is based on having similarity in the drag coefficient.

The area ratio between the model and the prototype is just the square of the length scale. For the water tunnel, the density ratio will be one and the velocity ratio was found above to be 1/50. For the water tunnel then

For the **water tunnel, F**_{D,p}** = F**_{D,m}.

For the wind tunnel, the air density is 1.23 kg/m^{3} and for the prototype in water the density is 999 kg/m^{3}. In this case the drag force relationship between the model and the prototype is

For the **wind tunnel, the prototype drag force is 4.77 times the model drag force**.

**7.57 If the unpressurized wind tunnel of Problem 7.56 were replaced with a tunnel in which the air can be pressurized isothermally to 8 atm(abs), what range of air velocities would be required to maintain Reynolds number similarity for the same prototype velocities given in Problem 7.56 – 20 mph and 90 mph. For the pressurized tunnel the maximum characteristic model length that can be accommodated is 2 ft, whereas the maximum characteristic prototype length remains at 20 ft.**
This is another completely submerged flow for which we must have Reynolds number similarity between the model (m) and prototype (p). With the stated maximum lengths of 2 ft for the model and 20 ft for the prototype, the Reynolds number scaling gives the following relationship between the velocity of the model and that of the prototype.

We can find the density from the ideal gas law, = p/RT. We can substitute this equation for both densities in the velocity-scale equation and cancel terms since the both use the same fluid (hence R_{m} = R_{p}) and the temperatures are the same for both the model and the prototype (hence _{m} = _{p}.) Finally we can substitute the values for p_{p} = 1 atm(abs) and p_{m} = 8 atm(abs to get a numerical value for the velocity scale.

Thus, the model velocities corresponding to the desired prototype velocities of 20 mph and 90 mph are simply 1.25 times these velocities or **25 mph and 112.5 mph**.