By R. A. Wallis
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In the case of laminar flow the properties of the layer can, in the presence of a velocity gradient, be expressed in terms of local flow parameters, such as Λ and λ. , the velocity profile will to some degree be 38 BOUNDARY LAYER AND SKIN FRICTION RELATIONS dependent on the turbulence history of the flow approaching the point in question. Empirical relations, representing averaged conditions, have been formulated on the basis of experimental data. 34) For zero velocity gradient and moderate Reynolds numbers, WÄ 1/7 ; at high Reynolds numbers nx 1/9.
E. <2) For Reynolds' experiments, the flow in the pipe was "fully developed". e. at the edge of the bound ary layer, and δ is the boundary layer thickness, which in this case equals d/2. The characteristic velocity and length are therefore related to the boundary layer. This is very reason able when it is remembered t h a t the viscous forces are of no practical importance outside the region of the boundary layer. For entry flow to a pipe, this latter Reynolds number is the one with real meaning.
In cases where the velocity is variable with x, where nonuniform roughness exists, or where other complications are present, good accuracy can be obtained only by boundary layer growth calculations and by subsequent integration of the local elementary skin friction forces. Sufficient data to facilitate such calculations will be given later. The foregoing paragraphs have illustrated the manner in which non-dimensional coefficients can be used in the calcula tion of skin friction forces. Flow over various types of solid surfaces will be considered in the relevant sub-sections.
Axial Flow Fans. Design and Practice by R. A. Wallis