Consider the boundary layer over a flat plate at 45° angle as shown. The exact flow field in this configuration is described by the Falkner-Skan similarity solution with n = 1/3 (see Figure 10.8 of the textbook, the Falkner-Skan profile chart). The objective is to find the approximate solution to this problem using the Thwaites method and calculate its error. 11/4 Assume that for this approximate solution the free stream velocity is U₂(x) = ar" where a is an unknown constants and n = 1/3. Use the Thwaites method to find the momentum 0/x and 8/x displacement thicknesses as well as the friction coefficient cf = 0.5, as functions of Re₂ = U₂x/v, where is the shear stress at the wall. (No need to interpolate the Thwaites method table values; you can pick the nearest numbers.) Using the Falkner-Skan profile chart approximate the friction coefficient c; (by estimating the slope of the corresponding velocity profile at the wall). How does this value compare with your prediction in part (a)? Find the error involved in the Thwaites method for this flow field. This can be done by substituting 0, 8 and Tw from part (a) into the the von Karman integral equation and calculating the residual R (R is the difference between the the left and right hand sides of the von Karman integral equation). Express the error as a function of U₂ and Re..

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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Transcribed Image Text:Consider the boundary layer over a flat plate at 45° angle as shown. The exact flow field in this configuration is described by the Falkner-Skan similarity solution with n = 1/3 (see Figure 10.8 of the textbook, the Falkner-Skan profile chart). The objective is to find the approximate solution to this problem using the Thwaites method and calculate its error. Ve 11/4 Assume that for this approximate solution the free stream velocity is U₂(x) = ax" where a is an unknown constants and n = 1/3. Use the Thwaites method to find the momentum 0/x and 8*/x displacement thicknesses as well as the friction coefficient cf = 0.5, as functions of Re₂ = Uer/v, where is the shear stress at the wall. (No need to interpolate the Thwaites method table values; you can pick the nearest numbers.) Using the Falkner-Skan profile chart approximate the friction coefficient c; (by estimating the slope of the corresponding velocity profile at the wall). How does this value compare with your prediction in part (a)? Find the error involved in the Thwaites method for this flow field. This can be done by substituting 0, 8* and T from part (a) into the the von Karman integral equation and calculating the residual R (R is the difference between the the left and right hand sides of the von Karman integral equation). Express the error as a function of U₂ and Re..

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