Lecture 1.2 Aug. 6

Chapter 7 White: Boundary-Layer Flows

Last time we began the discussion of boundary layer flows using the Karman integral boundary layer method as one example of how such flows can be treated. We found that accurate expressions for the thickness and drag coefficient could be obtained assuming the profile of velocity was parabolic.

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Blasius boundary layer

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A precise analysis of boundary layers was first obtained by Prandtl and his student Blasius. To solve the problem it was necessary to assume very thin boundary layers in steady flow. Blasius obtained the dimensionless equation:

** f''' + (1/2) f f'' = 0

by changing variables to

\eta = y (U/\nu x)^{1/2}

where u/U = f' .

Note that the distance (\nu x /U)^{1/2} is a diffusion length for vorticity or momentum during the time x/U while the fluid of the boundary layer has been in contact with the surface. The solution of Blasius' equation is close to parabolic, which explains Karman's success.

Clearly the parameter f is a dimensionless stream function. It can be obtained from the continuity equation by integration, as shown in example problem 7.22.

7.22 For the Blasius flat plate problem, does a 2D stream function \psi exist. Determine the dimensionless form, assuming \psi = 0 at the wall.

u = partial \psi / partial y ; so \psi = int u dy|x = int[0,y] (U df/d\eta) d\eta ((\nu x/U)^{1/2})

so \psi = (\nu x/U)^{1/2} int[0,\eta]df = (\nu x/U)^{1/2} f

Lift and drag coefficients for a wide variety of objects are given in Chapter 7, mostly obtained empirically.

See Homework hints for many examples of how drag coefficients are used to find forces on bodies with external flows.