Boundary-layer theory pressure distribution

Re < 40, and the predicted pressure distribution from potential-flow theory. The ability of the boundary-layer theory to predict separation is probably its most important characteristic. 

This quantity was plotted in Fig. 10-3.21 Also shown is the measured pressure distribution for Re in the range 30 to 40. It is evident that the measured and predicted distributions are in close agreement over the front portion of the cylinder. Thus it is not surprising that boundary-layer theory, based on the potential-flow pressure distribution, should be quite accurate up to the vicinity of the separation point. It is this fact that explains the ability of boundary-layer theory to provide a reasonable estimate of the onset point for separation, as we shall demonstrate shortly. 

The parameter k is called the von Karman constant, and the value that fits most of the data is 0.41. The corresponding value of B is 5.0. Intermediate between these layers is the buffer layer, where both shear mechanisms are important. The essential feature of this data correlation is that the wall shear completely controls the turbulent boundary layer velocity distribution in the vicinity of the wall. So dominant is the effect of the wall shear that even when pressure gradients along the surface are present, the velocity distributions near the surface are essentially coincident with the data obtained on plates with uniform surface pressure [82]. Within this region for a flat plate, the local shear stress remains within about 10 percent of the surface shear stress. It is noted that this shear variation is often ignored in turbulent boundary layer theory. 

Although the solution (5 74) seems to be complete, the key fact is that the pressure gradient V.s//0) in the thin gap, and thus p(0 xs, 0, is unknown. In this sense, the solution (5-74) is fundamentally different from the unidirectional flows considered in Chap. 3, where p varied linearly with position along the flow direction and was thus known completely ifp was specified at the ends of the flow domain. The problem considered here is an example of the class of thin-film problems known as lubrication theory in which either h(xs) and us, or h(xs, 0) and uz are prescribed on the boundaries, and it is the pressure distribution in the thin-fluid layer that is the primary theoretical objective. The fact that the pressure remains unknown is, of course, not surprising as we have not yet made any use of the continuity equation (5-69) or of the boundary conditions at z = 0 and h for the normal velocity component ui° ... 

An example of the pressure distribution predicted by potential-flow theory and the experimentally measured distribution for a case in which the boundary-layer separates was shown in Fig. 10-3. 

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