This tutorial on describing mean flow development begins with an overview of free and wall-bounded turbulent shear flows of technological interest, including a brief discussion of the relevant applications. While the major themes of the presentation will generally apply to all turbulent shear flows, the focus is on wall-bounded flows, as they present the greatest challenges and opportunities. Because of the strong empirical component to turbulent wall-flow research, the difficulties associated with obtaining data at high Reynolds number are described, as are the current ways that these difficulties are being addressed. The closure problem is introduced, and basic elements of the 1st and 2nd order closure schemes commonly used in practice are described. The modeling notion of “distance from the wall scaling” is shown to be consistent with dimensional arguments. Asymptotic methods for estimating mean flow behaviors are discussed. These are extensively employed in wall-turbulence research owing to the importance of engineering prediction in high Reynolds number regimes. The deficiencies inherent to constructing models based upon additional modeling assumptions or phenomenological or similarity hypotheses are described relative to the loss of a first-principles basis. This is used to motivate opportunities for applied mathematical analyses that retain connection to the governing mean equations. The value of such analyses is exemplified through some simple examples. The value of direct numerical simulation data in such efforts is made apparent.
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