Fluid Dynamics Sol ch6

Fluid Dynamics Sol ch6 - Chapter 6 Viscous Flow in Ducts...

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Chapter 6 Viscous Flow in Ducts 6.1 In flow past a sphere, the boundary layer becomes turbulent at about Re D 2.5E5. To what air speed in mi/h does this correspond to a golf ball whose diameter is 1.6 in? Do the pressure, temperature, and humidity of the air make any difference in your calculation? Solution: For air at 20 ° C, take ρ = 1.2 kg/m 3 and µ = 1.8E 5 kg/m s. Convert D = 1.6 inches to 0.0406 m. The critical Reynolds number is D VD 1.2V(0.0406) m Re 2.5E5 , or V 92 1.8E 5 s Ans. === =≈ mi 206 h Since air density and viscosity change with pressure, temperature, and humidity, the calculation does indeed depend upon the thermodynamic state of the air. 6.2 Air at approximately 1 atm flows through a horizontal 4-cm-diameter pipe. (a) Find a formula for Q max , the maximum volume flow for which the flow remains laminar, and plot Q versus temperature in the range 0 ° C T 500 ° C. (b) Is your plot linear? If not, explain. (a) First convert the Reynolds number from a velocity form to a volume flow form: 2 4 , therefore Re 2300 for laminar flow (/ 4 ) d QV d Q V d d ρρ µπ π == = Maximum laminar volume flow is given by . (a) Ans max 2300 Q 4 = πµ d With d = 0.04 m = constant, get for air from Table A-2 and plot Q versus T ° C: Fig. P6.2
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364 Solutions Manual Fluid Mechanics, Fifth Edition The curve is not quite linear because ν = µ / ρ is not quite linear with T for air in this range. Ans. (b) 6.3 For a thin wing moving parallel to its chord line, transition to a turbulent boundary layer occurs at a “local” Reynolds number Re x , where x is the distance from the leading edge of the wing. The critical Reynolds number depends upon the intensity of turbulent fluctuations in the stream and equals 2.8E6 if the stream is very quiet. A semiempirical correlation for this case [Ref. 3 of Ch. 6] is crit 21 / 2 1/2 2 1 (1 13.25 ) Re 0.00392 x ζ −+ + where is the tunnel-turbulence intensity in percent. If V = 20 m/s in air at 20 ° C, use this formula to plot the transition position on the wing versus stream turbulence for between 0 and 2 percent. At what value of is x crit decreased 50 percent from its value at = 0? Solution: This problem is merely to illustrate the strong effect of stream turbulence on the transition point. For air at 20 ° C, take = 1.2 kg/m 3 and = 1.8E 5 kg/m s. Compute Re x,crit from the correlation and plot x tr = Re x /[ (20 m/s)] versus percent turbulence: Fig. P6.3 The value of x crit decreases by half (to 1.07 meters) at 0.42% . Ans .
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Chapter 6 Viscous Flow in Ducts 365 6.4 For flow of SAE 30 oil through a 5-cm-diameter pipe, from Fig. A.1, for what flow rate in m 3 /h would we expect transition to turbulence at (a) 20 ° C and (b) 100 ° C? Solution: For SAE 30 oil take 3 891 kg/m ρ = and take µ = 0.29 kg/m s at 20 ° C (Table A.3) and 0.01 kg/m-s at 100 ° C (Fig A.1). Write the critical Reynolds number in terms of flow rate Q : 3 33 4 4(891 / ) (a) Re 2300 , (0.29 / )(0.05 ) solve 0.0293 . (a) crit VD Q kg m Q Dk g m s m m QA n s s ρρ µπ ==== == m 106 h 3 3 4 4(891 / ) (b) Re 2300 , (0.010 / )(0.05 ) solve 0.00101 . (b) crit VD Q kg m Q g m s m m n s s π 3 m 3.6 h 6.5 In flow past a body or wall, early transition to turbulence can be induced by
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This note was uploaded on 05/07/2008 for the course MAE 101a taught by Professor Sakar during the Spring '08 term at UCSD.

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Fluid Dynamics Sol ch6 - Chapter 6 Viscous Flow in Ducts...

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