Lecture 1.1 Aug. 4

Chapter 7 White: Boundary-Layer Flows

%Introduction to boundary layers

%Karman's integral method review

%Homework hint discussion

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Purpose

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The purpose of boundary layer theory is to describe the flow external to a body immersed in a fluid stream. Close to the body the flow is affected by viscous forces. Turbulence may develop. Far from the body potential flow may develop. Matching such solutions is an important part of the process of solving boundary layer problems. Often no solution is possible because the flow separates and becomes turbulent. This is a major problem to be avoided in the design of wings, but is unavoidable for blunt objects. A large number of empirical formulas and drag coefficient plots are given in the Chapter to describe lift and drag coefficients determined experimentally for a wide variety of object shapes.

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Laminar versus turbulent boundary layers

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No satisfactory boundary layer theory exists for Reynolds numbers 1 to 1000. For larger Re values, the accepted expressions for boundary layer thickness \delta (99% = u/U) are:

** \delta / x = 5.0 / Re_x ^{1/2} laminar (Blasius profile)

** \delta / x = 0.16/ Re_x ^{1/7} turbulent

where Re_x = Ux/\nu is the local Reynolds number based on the distance x from the beginning of the boundary layer.

As shown in the table of formula values, the boundary layer is thin, never over 5% :

Re_x	10^4	10^5	10^6	10^7	10^8
\delta / x _ lam	0.050	0.016	0.005
\delta / x _ turb.			0.022	0.016	0.011

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Karman momentum integral

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Recall the derivation of drag D(x) = \rho b int [0,\delta] u(U-u) dy from Chapter 3, using the Reynolds transport theorem for mass and momentum for a control volume bounded by stream surfaces h from the wall upstream and \delta from the wall downstream, b wide and L long. In terms of the momentum thickness \theta:

** D(x) = \rho b U^2 \theta, where

** \theta = int [0,\delta] u(U-u)\U^2 dy .

Increasing \theta gives increasing drag. The drag is also the integrated shear stress

** D(x) = b int[0,x] \tau_w(x) dx, or

** dD/dx = b \tau_w.

But

** dD/dx = \rho b U^2 d\theta/dx,

so

** \tau_w = \rho U^2 d\theta/dx, valid for either laminar or turbulent flow.

Recall Karman assumed a parabolic velocity profile u(x,y) = U(2y/\delta - y^2/\delta^2) which gives \theta = 2/15 \delta. The stress at the wall \tau = 2 \mu U / \delta from Karman's profile, and \rho U^2 d\theta/dx. Differentiate \theta = 2/15 \delta and integrate to give

** \delta / x = 5.5 Rex^1/2 ; which is within 10% of the Blasius profile result.

See Homework Hints.