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. The drag coefficient C_D is defined as a stress divided by \rho U^2/2, by tradition. The factor of 1/2 is misleading, since it implies that energy has something to do with drag. The logical definition would make the stress, or force per unit area, dimensionless using the inertial stress tensor \rho v v. No factor of 1/2 appears in the inertial stress tensor.

The peculiar shape of the drag coefficient for a sphere in Figure 7.16 has to do with the behavior of the boundary layer and its separation from the sphere. Starting from the analytical result C_D = (drag force/projected area)/\rho V^2/2 = 24/Re_D corresponding to Stokes law for laminar forces on spheres we see drag coefficient flattens out to a constant value of about 0.8 until the Reynolds number reaches about 2x10^5, where it suddenly decreases to about 0.2. This is because the boundary layer has become turbulent upstream of the separation point. The strong diffusivity of the turbulent boundary layer delays separation of the boundary layer until downstream of the 90 degree angle, compared to upstream for the laminar boundary layer separation (see the photographs of Fig. 7.14, p415 of the text for a bowling ball with and without nose sand roughness). This phenomenon is called the "drag crisis", and is exploited in sports activities involving high speed balls. The dimples on golf balls trip the boundary layer to reduce drag, just as the seams on baseballs and cricket balls are used by pitchers and bowlers to make the balls difficult to hit.

Illustrations of lift and drag can be found in the homework hints.