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**Unformatted text preview: **Chapter 5 Dimensional Analysis and Similarity 5.1 For axial flow through a circular tube, the Reynolds number for transition to turbulence is approximately 2300 [see Eq. (6.2)], based upon the diameter and average velocity. If d = 5 cm and the fluid is kerosene at 20 C, find the volume flow rate in m 3 /h which causes transition. Solution: For kerosene at 20 C, take = 804 kg/m 3 and = 0.00192 kg/m s. The only unknown in the transition Reynolds number is the fluid velocity: tr Vd (804)V(0.05) Re 2300 , solve for V 0.11m/s 0.00192 = = = 3 2 m Then Q VA (0.11) (0.05) 2.16E 4 3600 4 s Ans. = = = m 0.78 hr 3 P5.2 A prototype automobile is designed for cold weather in Denver, CO (-10 C, 83 kPa). Its drag force is to be tested in on a one-seventh-scale model in a wind tunnel at 20 C and 1 atm. If model and prototype satisfy dynamic similarity, what prototype velocity, in mi/h, is matched? Comment on your result. Solution : First assemble the necessary air density and viscosity data: s m kg E m kg RT p K T s m kg E m kg RT p K T m m p p = = = = = = = = = = 5 80 . 1 ; 205 . 1 ) 293 ( 287 101350 ; 293 : tunnel Wind 5 75 . 1 ; 10 . 1 ) 263 ( 287 83000 ; 263 : Denver 3 3 Convert 150 mi/h = 67.1 m/s. For dynamic similarity, equate the Reynolds numbers: . / 8 . 22 / 2 . 10 for Solve 5 80 . 1 ) )( 1 . 67 )( 205 . 1 ( | Re 5 75 . 1 ) 7 ( ) 10 . 1 ( | Re Ans h mi s m V E L VL E L V VL prototype m m m m p p p = = = = = = = Chapter 5 Dimensional Analysis and Similarity 371 This is too slow , hardly fast enough to turn into a driveway. Since the tunnel can go no faster, the model drag must be corrected for Reynolds number effects. Note that we did not need to know the actual length of the prototype auto, only that it is 7 times larger than the model length. 5.3 An airplane has a chord length L = 1.2 m and flies at a Mach number of 0.7 in the standard atmosphere. If its Reynolds number, based on chord length, is 7E6, how high is it flying? Solution: This is harder than Prob. 5.2 above, for we have to search in the U.S. Stan-dard Atmosphere (Table A-6) to find the altitude with the right density and viscosity and speed of sound. We can make a first guess of T 230 K, a (kRT) 304 m/s, U = 0.7a 213 m/s, and 1.51E 5 kg/m s. Then our first estimate for density is 3 C UC (213)(1.2) Re 7E6 , or 0.44 kg/m and Z 9500 m (Table A-6) 1.51E 5 = = Repeat and the process converges to 0.41 kg/m 3 or Z 10100 m Ans . 5.4 When tested in water at 20 C flowing at 2 m/s, an 8-cm-diameter sphere has a measured drag of 5 N. What will be the velocity and drag force on a 1.5-m-diameter weather balloon moored in sea-level standard air under dynamically similar conditions?...

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