The two-layer concept is a framework for interpreting events and constructing mathematical models of turbulent wall layers. In this paper an asymptotic theory is constructed employing the idea that the interaction between the layers is the most important aspect. It is shown that the matching process for the layers can be used to define a characteristic scale, u*, and to produce an equation that relates u* to the known parameters; U, v, h, e, and dp/dx. At infinite Reynolds number the scale u* is equal to uτ, the friction velocity, but they are distinct at moderate Reynolds numbers. The theory produces very simple results. For instance, the overlap velocity laws are logarithmic with an invariant von Ka´rma´n constant; at low Reynolds numbers the additive constant changes while the slope remains the same. The effect of low Reynolds numbers on the Reynolds stress in the overlap layer is also analyzed. A composite expansion explains the strong Reynolds number effect on the stress profiles. This occurs because the mixing of outer and inner layer phenomena take place at different locations as the size of the overlap region changes. The location of the maximum Reynolds stress is given by y+max = (Re/k)1/2. The overlap region was not found to be a region of constant stress, as put forth in many heuristic arguments.

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