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HW_32_Soln

# HW_32_Soln - ME 309 Fall 2008 Section 2 Homework 32 Due...

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ME 309 Fall 2008 Section 2 Homework # 32 Due Wednesday 19 November 2008 Problem 32-1 a) Show that the log law of the boundary layer: 5 . 5 ln 5 . 2 + = + + y u can be expressed in terms of skin friction and the Reynolds number based on the boundary layer thickness as: 5 . 5 2 Re ln 5 . 2 2 + = f f C C δ . To do this, evaluate the log law at y = δ where the velocity equals the free stream value. b. This equation between the Reynolds number and the skin friction coefficient can be approximately fit to the more accessible relation, 6 / 1 Re 02 . 0 - = δ f C . Using this skin friction relation in the momentum integral relation for a flat plate (zero pressure gradient) along with a one-seventh power law velocity profile, find relations for δ , θ, and C f as function of Re x . Also find a relation between the local skin friction coefficient at the end of the plate, C fL and the drag coefficient for the entire plate length, C D . Solution: Given Log-law profile: 5 . 5 ln 5 . 2 + = + + y u Evaluate log-law at boundary layer edge, δ where the velocity equals its free stream value 5 . 5 ln 5 . 2 + = ν δ τ τ u u u but: 5 . 5 ln 5 . 2 2 2 + = = u u u u u w w ρ τ ν δ τ ρ τ Thus, log law immediately give: 5 . 5 2 Re ln 5 . 2 2 + = f f C C δ Curve fit of C_f vs Re_delta: 6 / 1 Re 02 . 0 - = δ f C Combine with Karman momentum (for zero pressure gradient) dx d C f θ = 2 : Use power law to describe turbulent velocity profile: 7 / 1 = δ y u

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HW_32_Soln - ME 309 Fall 2008 Section 2 Homework 32 Due...

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