Mechanical Engineering Department
Mechanical Engineering 390
Fluid Mechanics
Spring 2008 Number: 11971 Instructor: Larry Caretto
Jacaranda (Engineering) 3333Mail CodePhone: 818.677.6448
E-mail: 8348Fax: 818.677.7062
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Solution to Quiz Eight –Dimensional Analysis and Similitude
Recall the rule of thumb that gas flows may be treated as incompressible flows if the Mach number is less than 0.3. If the maximum velocity in a wind tunnel is 160 mph, what is the maximum prototype velocity than can be simulated under the conditions shown in the table below? In all cases the fluid for both the model and prototype is air and the temperature is 70oF.
Case / Model pressure (psia) / Prototype pressure (psia) / Length scale, ℓm/ℓp1 / 14.696 / 14.696 / 1/3
2 / 14.696 / 14.696 / 1/8
3 / 14.696 / 2.9392 / 1/3
4 / 14.696 / 2.9392 / 1/8
Wind tunnel flows require Reynolds number similarity. For Reynolds number similarity between the model (subscript m) and the prototype (no subscript) we must have
Although viscosity depends on temperature it is only a weak function of pressure and the pressure dependence can be ignored for this problem; thus = m. Using the ideal gas equation for density, = P/RT for both the model and the prototype and cancelling R,T, and that are the same for both the model and the prototype gives
Applying this equation to all four cases shown above with the maximum model velocity of 160 mph gives the following prototype velocities for the four cases: 53.3 mph, 20 mph, 267 mph, and 100 mph. Checking for compressibility effects we first determine the model Mach number. At 70oF we find the sound speed from Table B.3 on p. 762 of the text: c = 1128 ft/s = 769 mph. (The conversion factor that 30 mph = 44 ft/s was used to get the value of c in mph. For this sound speed a Mach number of 0.3 corresponds to a speed of (0.3)(769 mph) = 231 mph. Thus compressibility effects will not be present for the model. Furthermore, there is only one computed prototype speed, V = 267 mph in case 3 that exceeds this 231 mph limit. For this case, the model speed will be limited to 138 mph.
The relationship between the drag force on the prototype, FD,p, and the drag force on the model, FD,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. Using this result and the ideal gas law for density as used previously gives the following result after cancellation of the RT product that are the same for the prototype and the model.
In the first two cases where the model and the prototype pressure are the same (Pm.P = 1), the drag force on the prototype is the same as the drag force on the model. For the second two cases where Pm/P = 5, the drag force on the prototype is five times the drag force on the model.