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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 and - = 1 dy U u U u e e 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 y , and u = e U for > y : a) Find a parabola that satisfies the conditions, u = 0 at y = 0; u = e U at y = ; and = y u at y = ; b) Substitute this velocity profile into the Karman momentum integral equation, 2 =- + + - 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 as a function of Reynolds number based on x,- A relation for the displacement as a function of ,- A relation for the momentum thicknesses as a function of , 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 / * = . Note that the exact Blasius solution is .) 349 . / * = Solution: Assumptions: Incompressible flow Analysis Part a Parabola satisfying, u = 0 at y = 0; u = e U at y = ; and = y u at y = ; General parabola: c by ay u + + = 2 Enforce BCs 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 BCs at edge: Slope condition: b a + = 2 Function condition: b a U e + = 2 Two equations in two unknowns for a and b . Final solution: + - = y y U u e 2 2 Answer : + - = y y U u e 2 2 Part b Substitute velocity profile, + - = y y U u e 2 2 , into the Karman momentum integral equation, 2 =- + + - w e e e dx du u...
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- Spring '08