Answered: IBL, Flat Plate. Apply the integral…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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IBL, Flat Plate. Apply the integral boundary layer analysis to a flat plate turbulent flow as follows. Assume the turbulent profile u/U = (y/δ)1/6 to compute the momentum flux term on the RHS of IBL, but on the LHS of IBL, use the empirical wall shear stress, adapted from pipe flow: ?w = 0.0233 ⍴U2 (v/Uδ)1/4 where the kinematic viscosity ν = μ/⍴. It is necessary to use this empirical wall shear relation because the turbulent power law velocity profile blows up at the wall and cannot be used to evaluate the wall shear stress. Compute (a) (δ/x) as a function of Rex; (b) total drag coefficient, CD, L as a function of ReL; (c) If ReL = 6 x 107 compare values for this IBL CD,L and those empirical ones given in Table 9.1 for both smooth plate and transitional at Rex = 5 x 105 cases. Note: You must show all the algebra in evaluating the IBL to get full credit. Ans OM: (a) (δ/x) ~ 10-1/(Rex)1/5; (b) CD,L ~ 10-2/(ReL)1/5; (c) CD,IBL ~ 10-3; CD,Smooth ~ 10-3; CD,Trans ~ 10-3

$**IBL, Flat Plate** Apply the integral boundary layer analysis to a flat plate turbulent flow as follows. Assume the turbulent profile $ u/U = (y/\delta)^{1/6} $ to compute the momentum flux term on the RHS of IBL, but on the LHS of IBL, use the empirical wall shear stress, adapted from pipe flow: $ \tau_w = 0.0233 \, \rho U^2 \, (\nu/U\delta)^{1/4} $ where the kinematic viscosity $ \nu = \mu/\rho $. It is necessary to use this empirical wall shear relation because the turbulent power law velocity profile blows up at the wall and cannot be used to evaluate the wall shear stress. Compute (a) $ (\delta/x) $ as a function of $ Re_x $; (b) total drag coefficient, $ C_{D,L} $ as a function of $ Re_L $; (c) If $ Re_L = 6 \times 10^7 $ compare values for this IBL $ C_{D,L} $ and those empirical ones given in Table 9.1 for both smooth plate and transitional at $ Re_x = 5 \times 10^5 $ cases. *Note: You must show all the algebra* in evaluating the IBL to get full credit. **Ans OM:** - (a) $ (\delta/x) \sim 10^{-1}/(Re_x)^{1/5} $ - (b) $ C_{D,L} \sim 10^{-2}/(Re_L)^{1/5} $ - (c) $ C_{D,IBL} \sim 10^{-3}; \, C_{D,Smooth} \sim 10^{-3}; \, C_{D,Trans} \sim 10^{-3} $$

Transcribed Image Text:**IBL, Flat Plate** Apply the integral boundary layer analysis to a flat plate turbulent flow as follows. Assume the turbulent profile $ u/U = (y/\delta)^{1/6} $ to compute the momentum flux term on the RHS of IBL, but on the LHS of IBL, use the empirical wall shear stress, adapted from pipe flow: $ \tau_w = 0.0233 \, \rho U^2 \, (\nu/U\delta)^{1/4} $ where the kinematic viscosity $ \nu = \mu/\rho $. It is necessary to use this empirical wall shear relation because the turbulent power law velocity profile blows up at the wall and cannot be used to evaluate the wall shear stress. Compute (a) $ (\delta/x) $ as a function of $ Re_x $; (b) total drag coefficient, $ C_{D,L} $ as a function of $ Re_L $; (c) If $ Re_L = 6 \times 10^7 $ compare values for this IBL $ C_{D,L} $ and those empirical ones given in Table 9.1 for both smooth plate and transitional at $ Re_x = 5 \times 10^5 $ cases. *Note: You must show all the algebra* in evaluating the IBL to get full credit. **Ans OM:** - (a) $ (\delta/x) \sim 10^{-1}/(Re_x)^{1/5} $ - (b) $ C_{D,L} \sim 10^{-2}/(Re_L)^{1/5} $ - (c) $ C_{D,IBL} \sim 10^{-3}; \, C_{D,Smooth} \sim 10^{-3}; \, C_{D,Trans} \sim 10^{-3} $

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