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Unformatted text preview: ME 309 Fall 2008 Section 2 Homework # 30 Due Friday 14 November 2008 Problems 30-1 and 30-2 . The displacement thickness and momentum thickness of a boundary layer are given by: ∫ ∞ - = * 1 dy U u e δ and ∫ ∞ - = 1 dy U u U u e e θ where e U is the velocity at the edge of the boundary layer. For a parabolic profile of the form: c by ay u + + = 2 for δ ≤ ≤ y , and u = e U for δ > y : a) Find a parabola that satisfies the conditions, u = 0 at y = 0; u = e U at y = δ ; and = ∂ ∂ y u at y = δ ; b) Substitute this velocity profile into the Karman momentum integral equation, 2 =- + + - ∫ ∫ ρ τ δ δ δ w e e e dx du u udy dx d u dy u dx d to find a differential equation for the boundary layer thickness. c) Simplify your result in Part b to a boundary layer with constant edge velocity and find : - A relation for δ as a function of Reynolds number based on x,- A relation for the displacement as a function of δ,- A relation for the momentum thicknesses as a function of δ , and - A relation for the skin friction coefficient as a function of Reynolds number Compare your results with the coefficients from Blasius exact solution in the text (Table 9.2, p 402) and calculate the percent error. (Answer for displacement thickness: 3 / 1 / * = δ δ . Note that the exact Blasius solution is .) 349 . / * = δ δ Solution: Assumptions: Incompressible flow Analysis Part a Parabola satisfying, u = 0 at y = 0; u = e U at y = δ ; and = ∂ ∂ y u at y = δ ; General parabola: c by ay u + + = 2 Enforce BC’s at wall: c b a + + = or c = 0 Parabola now of form: by ay u + = 2 and derivative is: b ay dy du + = 2 Enforce BC’s at edge: Slope condition: b a + = δ 2 Function condition: δ δ b a U e + = 2 Two equations in two unknowns for a and b . Final solution: + - = δ δ y y U u e 2 2 Answer : + - = δ δ y y U u e 2 2 Part b Substitute velocity profile, + - = δ δ y y U u e 2 2 , into the Karman momentum integral equation, 2 =- + + - ∫ ∫ ρ τ δ δ δ w e e e dx du u...
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- Spring '08