Answered: 3.4 Specification of a laminar boundary…

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:2) 3.4 Specification of a laminar boundary layer profile as an inflow condition is often used in incompressible flow simulations. Let us consider the Blasius profile, which is a similarity solution for the steady laminar boundary layer on a flat plate. u(x, y) Fig. 3.25 Flat-plate laminar boundary layer 1. Assume two-dimensional, steady flow with no imposed pressure gradient as shown in Fig. 3.25. Show that the Navier-Stokes equations inside the bound- ary layer can be reduced to ди ди 8²u U +v =V Əx Əy მy2 U ap მ = 0, and ди Əx Əv dy

Transcribed Image Text:მ 2. Consider the streamfunction (with u = transform from (x, y) to (x, n), where n = √ux/Ux Using the above and a streamfunction in the form = √xUf(n), show that the governing equations can be reduced to ff" +2f"" = 0 with boundary conditions of f(0) = f'(0) = 0 and lim, f(n) = 1. This equation is referred to as the Blasius equation and its solution is known as the Blasius profile. and v= 3. Solve the Blasius equation numerically. Note that this is a boundary-value prob- lem. y == Rex. u Plot as a function of 1,√e a as a function of n, C, as a function of Re, C as a function U₂ U₂ of Re, as a function of Re,, as a function of Re,, where, Re X du is the skin friction coefficient, T = μ| dy y=0 coefficient of the plate, is the momentum thickness. -L(₁-7) y=01 U₂ = and a coordinate UL. number, Re, =is Reynolds number based on plate length, L is the plate length, C, = V Ux is local wall shear stress, Cp = dy is the displacement thickness, 0 = is local Reynolds Tdx PUL -L-(1-7) U₂ Jx=0 Tw ŽPUZ is the drag dy

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